Error analysis in college algebra in the higher education institutions in la union
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This doctoral study looked into the error categories of the students in College ALgebra ...

This doctoral study looked into the error categories of the students in College ALgebra

It provided an Instructional Intervention Plan as the output of the study

It also provided a model framework on how specific error categories in students' solutions can be addressed, the Ragma's Error Interventions Model

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Error analysis in college algebra in the higher education institutions in la union Document Transcript

  • 1. ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF LA UNION A Dissertation Presented to the Faculty of the Graduate School Saint Louis College City of San Fernando, La Union In Partial Fulfillment of the Requirements for the Degree Doctor of Education Major in Educational Management by FELJONE GALIMA RAGMA January 11, 2014
  • 2. ii INDORSEMENT This dissertation entitled, ―ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF LA UNION,‖ prepared and submitted by FELJONE GALIMA RAGMA, in partial fulfillment of the requirements for the degree DOCTOR OF EDUCATION major in EDUCATIONAL MANAGEMENT, has been examined and is recommended for acceptance and approval for ORAL EXAMINATION. NORA ARELLANO-OREDINA, Ed.D. Adviser This is to certify that the dissertation entitled, ―ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF LA UNION,” prepared and submitted by FELJONE GALIMA RAGMA, is recommended for ORAL EXAMINATION. MARIA LOURDES R. ALMOJUELA, Ed.D. Chairperson JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D. Member Member AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member Noted by: ROSARIO C. GARCIA, DBA Dean, Graduate School Saint Louis College
  • 3. iii APPROVAL SHEET Approved by the Committee on Oral Examination as PASSED with a grade of 96% on January 11, 2014. MARIA LOURDES R. ALMOJUELA, Ed.D. Chairperson JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D. Member Member AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member Accepted and approved in partial fulfillment of the requirements for the degree DOCTOR OF EDUCATION MAJOR IN EDUCATIONAL MANAGEMENT. ROSARIO C. GARCIA, DBA Dean, Graduate Studies Saint Louis College This is to certify that FELJONE GALIMA RAGMA has completed all academic requirements and PASSED the Comprehensive Examination with a grade of 96% on June 15, 2013 for the degree DOCTOR OF EDUCATION major in EDUCATIONAL MANAGEMENT. ROSARIO C. GARCIA, DBA Dean, Graduate Studies Saint Louis College
  • 4. iv ACKNOWLEDGMENT The researcher wishes to express his sincerest gratitude to the following persons who contributed much in helping him structure the research. Dr. Nora A. Oredina, dissertation adviser, for always affirming and supporting; and for giving necessary suggestions to better this study. Dr. Maria Lourdes R. Almojuela, chairperson of the dissertation panel, for her valuable critique, and most especially, for directing the researcher to the correct structure of the research. Dr. Aurora R. Carbonell, Dr. Augustina C. Dumaguin, Dr. Daniel B. Paguia, Dr. Rosario C. Garcia and Dr. Jovencio T. Balino, the panelists, for their brilliant thoughts. The validators of the questionnaire and the research output for giving suggestions that improved the study. Presidents, registrars, academic deans, department chairpersons, instructors and students of the Private Higher Education Institutions in La Union, for lending some of their precious time in dealing with the pre-survey and the questionnaires. Mrs. Edwina M. Manalang and Mrs. Marilyn Torcedo, for sparing some time for brainstorming for the built-in theory of the study.
  • 5. v Mesdames Grace, Lea, Melody, Graziel, Jay Ann, Abegail, Sister Grace, Mafe, and Sir Roghene, the researcher’s friends, who gave him inspiration. Mr. & Mrs. Felipe and Norma Ragma, the researcher’s parents, for always being there when the researcher needed some push. Kuya Darwin, Ate Felinor and Ate Nailyn, the researcher’s siblings, for always following up the researcher’s progress. And lastly, to GOD Almighty, for giving the needed strengths in the pursuit of this endeavor. F. G. R.
  • 6. vi D E D I C A T O N To my Parents, Mr. & Mrs Felipe and Norma Ragma and To my siblings, Darwin, Felinor and Nailyn This humble work is dedicated to all of you! F.G.R.
  • 7. vii ABSTRACT TITLE : ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF LA UNION Total Number of Pages: 374 Text Number of Pages : 358 AUTHOR : FELJONE G. RAGMA ADVISER : NORA ARELLANO-OREDINA, Ed.D. TYPE OF DOCUMENT : DISSERTATION TYPE OF PUBLICATION: Unpublished ACCREDITING INSTITUTION: SAINT LOUIS COLLEGE City of San Fernando, La Union CHED, Region I KEY WORDS : Error Analysis, Math Performance, Error Categori- zation, Educational Management, Instructional Intervention Plan, Mathematics Teaching Interven- tions, etc. Synopsis The descriptive study identified and analyzed the error categories of students in College Algebra in the Higher Education Institutions of La Union as basis for formulating a validated Instructional Intervention Plan. Specifically, it determined the a) level of performance of the students in College Algebra along elementary topics in sets and Venn diagrams, real numbers, algebraic expressions, and polynomials; special product patterns; factoring patterns; rational expression; linear
  • 8. viii equations in one unknown; systems of linear equations in two unknowns; and exponents and radicals; b) the capabilities and constraints of the students in College Algebra; and, c) the error categories of the students along reading, comprehension, mathematising, processing and encoding. Data were collected using a researcher-made, all-word-problem test. The participants were 374 first year students enrolled in College Algebra for first semester, school year 2013-2014. The data gathered were treated statistically using frequency count, mean, percentage and the Newmann’s tool for error analysis. It found out that the students had fair performance in elementary topics, special products and factoring while poor performance in rational expressions, linear equations and systems of linear equations and very poor performance in exponents and radicals; thus, the students, in general, had poor performance. The performances of the student in the specified topics were all considered as constraints. Mathematising and comprehension were the major error categories of the students in elementary topics, processing and reading errors in special products, reading and Mathematising in factoring, reading and Mathematising in rational expressions, reading and comprehension in linear equations; and reading and Mathematising in systems of linear equations and exponents and radicals. In general, their major error categories in College Algebra were along reading and Mathematising. Moreover, the instructional plan is
  • 9. ix found to have very high validity. Based on the findings, it was concluded that the students cannot competently deal with elementary topics, special product and factoring patterns rational expressions, linear equations, systems of linear equations and radicals and exponents. Additionally, the instructional intervention plan is a very good material that addresses problems on performance and errors. Based on the conclusions, it is recommended that the schools should adopt the Instructional Intervention Plan and let their mathematics instructors attend the two-day seminar-workshop. The students should exert more effort in understanding the different concepts in their College Algebra course. They should spend more time dealing with drills and exercises. The mathematics teachers should suit their instructional strategies to the needs of the students. The English teachers must also intensify in their classes the basic skill of reading with comprehension. A study should be conducted to determine the effectiveness of the instructional intervention plan. And, a similar study should be conducted in other branches of Mathematics, applied sciences and English.
  • 10. x TABLE OF CONTENTS Page TITLE PAGE………………………………………………………………… i INDORSEMENT…………………………………………………………… ii APPROVAL SHEET…………....................................................... iii ACKNOWLEDGMENT…………………………………………………… iv DEDICATION……………………………………………………………… vi ABSTRACT………………………………………………………………… vii TABLE OF CONTENTS………………………………………………….. x LIST OF TABLES…………………………………………………………. xiv LIST OF FIGURES……………………………………………………….. xvi CHAPTER I INTRODUCTION……………………………………………… 1 Background of the Study.……......………….......... 1 Theoretical Framework……………………………..... 8 Conceptual Framework……………………………….. 15 Statement of the Problem…………........................ 19 Assumptions……………………………………........... 21 Importance of the Study……………...................... 21 Definition of Terms…………………………………..... 23 II METHOD AND PROCEDURES…………………………… 27 Research Design……………………………………… 27
  • 11. xi Page Sources of Data………………………………………. 28 Locale and Population of the Study……………... 28 Instrumentation and Data Collection ..……….... 29 Validity and Reliability of the Questionnaire. Administration and Retrieval of the Questionnaire ……………………………… 30 31 Data Analysis …………………………………………. Data Categorization………………………………..... 32 33 Parts of the Instructional Intervention Plan….………………………………………………. 36 Ethical Considerations…………………………...... 37 III RESULTS AND DISCUSSION…………………………….. 39 Level of Performance of Students in College Algebra…………………………………………….. 39 Elementary Topics……………………………… 39 Special Product Patterns……………………… 41 Factoring Patterns ……………………………… 44 Rational Expressions…………………………… 46 Linear Equations in One Variable…………… 48 Systems of Linear Equations in Two Unknowns………………..………………….. 50 Exponents and Radicals………………………. 51
  • 12. xii Page Summary on the Level of Performance of Students in College Algebra …………. 52 Capabilities and Constraints of Students in College Algebra………………………………….. 54 Error Categories in College Algebra……………… 56 Elementary Topics……………………………… 56 Special Product Patterns……………………… 63 Factoring…………………………………………. 67 Rational Expressions…………………………… 74 Linear Equations in One Variable Systems 80 Systems of Linear Equations in Two Unknowns…………………………………… 85 Exponents and Radicals………………………. 91 Summary on the Error Categories in College Algebra ……………………………. 93 Validated Instructional Intervention Plan ……… 96 Instructional Intervention Plan …………………… Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan……… Sample Flyer of the Two-Day Seminar/ Workshop ……………………………………….. Level of Validity of the Instructional Inter- vention Plan ……………………………………… 99 296 299 300 IV SUMMARY, CONCLUSIONS AND RECOMMEN- DATIONS……………………………………………….. 301
  • 13. xiii Page Summary………………………………………………. 301 Findings………………………………………………… 302 Conclusions…………………………………………… 302 Recommendations…………………………………… 303 BIBLIOGRAPHY……………………………………………… 305 APPENDICES………………………………………………… 313 A Sample Computations on the: Reliability of the College Algebra Test … 313 Validity of College Algebra Test ……….. List of Suggestions Made by the Validators and the Correspond- ing Action/s by the Researcher ……. B Letter to Students-Respondents to Administer College Algebra Test ……….. The College Algebra Test ……………………… 314 315 317 317 Math I – College Algebra Test (Table of Specifications) ………………….. C Letter to the Presidents/School Heads of the HEIs understudy to Gather Data/Information …………………………. 324 326 D Sample of Corrected College Algebra Test… 336 CURRICULUM VITAE…………………………………….. 354
  • 14. xiv LIST OF TABLES Table Page 1 Distribution of Respondents ………………………… 29 2 Level of Performance of Students in Elementary Topics ……………………………………………….. 40 3 4 Level of Performance of Students in Special Product Patterns ………………………………….. Level of Performance of Students in Factoring Patterns …………………………………………….. 42 45 5 Level of Performance of Students in Rational Expressions ……………………………………….. 47 6 7 Level of Performance of Students in Linear Equations in One Variable …………………….. Level of Performance of Students in Systems of 49 Linear Equations …………………………………. 51 8 Level of Performance of Students in Exponents and Radicals ……………………………………….. 52 9 Summary Table on the Level of Performance of Students in College Algebra ……………………. 53 10 Capabilities and Constraints of Students in College Algebra …………………………………… 55 11 Error Categories in Elementary Topics………..….. 57 12 Error Categories in Special Product Patterns……. 64 13 Error Categories in Factoring Patterns…………..... 68 14 Error Categories in Rational Expressions ……………………………………….. 75
  • 15. xv 15 Error Categories in Linear Equations in One Variable……………………………………………. Page 81 16 Error Categories in Systems of Linear Equations in Two Variables ........................ 86 17 Error Categories in Exponents and Radicals…….. 92 18 Summary Table on the Error Categories in College Algebra………………………………….. 94 19 Level of Validity of the Instructional Intervention Plan………………………………………………… 300
  • 16. xvi LIST OF FIGURES Figure Page 1 Ragma’s Error Intervention Model…………………………… 13 2 The Research Paradigm ……………………………………….. 18
  • 17. 1 CHAPTER I INTRODUCTION Background of the Study Education, in its general sense, is a form of learning in which knowledge, skills, and values are imparted to a person or group of persons through teaching, training, or research. Many countries adhere to the principle that education is the key to a nation’s success. Some experts even correlate the number of literate people to the nation’s economic growth since national advancements are most commonly achieved by people who have trainings and intellectual advancements (www.educationworld.com). Furthermore, the central goal of education is to help a person develop critical thinking, reasoning and problem-solving skills. Hence, education prepares a person for life. One subject that helps people prepare for life is Mathematics. Mathematics is the science that deals with the logic of shape, quantity, reasoning and arrangement. It is concerned chiefly on how ideas, processes and analyses are applied to create useful and meaningful knowledge that man can use throughout his life (Prakash, 2010). It has also become one of the powerful tools of man in cultural adaptation and survival. Recorded history narrates that mathematical
  • 18. 2 discoveries have been at the forefront of every civilized society and in use even in the most primitive of cultures. The needs of mathematics arose based on the wants of society. The more complex a society is, the more complex is the mathematical need. Primitive tribes needed little more than the ability to count, but also relied on mathematics to calculate the position of the sun and the physics of hunting (Hom, 2013). Mathematics has played a very important role in building up modern civilization by perfecting the sciences. In this modern age of Science and Technology, emphasis is given on sciences such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, is also an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, "Mathematics is the science of all sciences and the art of all arts." (Wells, 2006). Furthermore, Mathematics is the language and the queen of the Sciences. According to the famous Philosopher Kant, "A Science is exact only in so far as it employs Mathematics." So, all scientific education and studies which do not commence with Mathematics is said to be defective at its foundation (Wells, 2006). Thus, neglect of mathematics causes injury to all knowledge.
  • 19. 3 It is undeniable that Mathematics expresses itself everywhere, in almost every facet of life - in nature and in the technologies in our hands. It is the building block of everything in our daily lives, including mobile devices, architecture, art, money, engineering, sports and many others. Without mathematics, man can go astray (Petti, 2009). Mathematical literacy is a must element in providing the students with the basic skills to live their life. It is one of the basic pillars for the student on which his life is, and would be standing. So the base of this pillar needs to be really strong and clear. Mathematics helps the student in developing conceptual, computational, logical-analytical, reasoning and problem-solving skills. One Mathematics subject that trains such skills is College Algebra. College Algebra is a pre-requisite subject in higher education institutions. The National Center for Academic Transformation (2009) labels it as the gateway course for freshmen in the tertiary level. This means that a student who aspires to be a degree holder must pass successfully through the course. This is the main reason why most countries, through their ministry or department of education, have mandated the inclusion of College Algebra in the course curriculum. No one can negate the importance of College Algebra. Cool (2011), enumerates some of the uses of algebra in today’s world. Algebra is used in companies to figure out their annual budget which involves their
  • 20. 4 income and expenditure. Various stores use algebra to predict the demand of a particular product and subsequently place their orders. It also has individual applications in the form of calculation of annual taxable income and bank interest on loans. Algebraic expressions and equations serve as models for interpreting and making inferences about data (Okello, 2010). Further, algebraic reasoning and symbolic notations also serve as the basis for the design and use of computer spreadsheet models. Therefore, mathematical reasoning developed through algebra is necessary through life, affecting decisions people make in many areas such as personal finance, travel, cooking and real estate, to name a few. Thus, it can be argued that a better understanding of algebra improves decision-making capabilities in society (The Journal of Language, Technology & Entrepreneurship in Africa, Vol. 2, No.1, 2010). In addition, Algebra is one of the most abstract strands in mathematics. This very nature of the subject makes it difficult for students to appreciate and love Algebra. With this, Prakash (2010) remarked that the place of mathematics in education is in grave danger. The teaching and learning of College Algebra, with insufficient skills and high anxiety levels, degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. A testament to this worsening scenario is the global move for educational reforms.
  • 21. 5 Countries around the world are alarmed by the lowering performance of their students, especially in College Algebra. In America alone, educational experts are tasked to improve performance in Mathematics (Arithmetic, Algebra, Geometry and the like) so they can bring back the glory days of the United States in topping Surveys of Countries along students’ academic performance (Serna, 2011). Bressoud (2012) added that even though there are interventions, College Algebra failure rates are disappointing. Further, in a University in Africa of Fall 2007, College Algebra examination results showed that only 23% of the students performed well. This poor performance calls for the establishment of the reason why College Algebra is challenging to many students (Kuiyan, 2007). In addition, Shepherd (2005) revealed that most students do not excel in their Algebra course. Most of them cannot perform indicated operations, especially when fronted with word problems. Students find it hard to solve problems in Algebra. Some just do not answer at all. These situations reflect poor understanding of and performance in the course (The Journal of Language, Technology & Entrepreneurship in Africa, Vol. 2, No.1, 2010). Although there are many causes of student difficulties in mathematics, the lack of support from research fields for teaching and learning is noticeable (The Journal of Science and Mathematics Education, 2010). Egodawatte (2009) emphasized that getting the level
  • 22. 6 of performance among students would not help much in Mathematics Education; researches need to dig deeper into the reasons by characterizing students’ errors and misconceptions. With this situation, error analysis is very essential. Egodawatte (2009) added that using error analysis, it would be possible for teachers to design effective instruction or instructional intervention to avoid this dismal performance. Thus, it can be construed that research on student errors is a way to clearly plot out a more valid action plan that could address issues on students’ mathematics performance. Mathematical errors are a common phenomenon in students’ learning of mathematics. Students of any age irrespective of their performance in mathematics have experienced getting mathematics wrong. It is natural that analyzing students’ mathematical errors is a fundamental aspect of teaching for mathematics teachers (Hall, 2007). The Philippines is also not exempted from this global predicament on the dismal performance in College Algebra. Garcia (2012) mentioned that Filipino students enrolled in College Algebra regarded the subject as challenging and a difficult subject which contributed to their low performance. In addition, the national survey conducted by Drs. Lambitco, Laz and Malab (2009) on the readiness of Filipino students in College Algebra revealed that the students are not ready to take up College Algebra course. Further, according to Professor Ramos
  • 23. 7 (2012), 40-50% of the students enrolled in College Algebra failed. According to him, this performance is caused by poor instruction and cognitive unpreparedness. This low performance was also highlighted when Leongson (2003) revealed that Filipino students excelled in knowledge acquisition but fared considerably low in lessons requiring higher-order-thinking skills. On the provincial scene, Picar (2009) strongly presented in his study that students’ anxiety in College Algebra is high but their performance is low. Pamani (2006) also mentioned that more than 60% of the college freshmen in La Union have low to fair competence. Pamani (2006) stressed that these results point out to a problematic situation in education. These facts are also strengthened by Bucsit (2009) when she revealed that out of 195 college freshmen in the Private Schools in La Union, 113 or 58% of the students have fair performance. In addition, Oredina (2011) revealed that the performance of SLC students in College Algebra was at the moderate level only. Furthermore, the researcher, being a College Algebra instructor, observes that many students still have many misconceptions along certain topics in College Algebra, even if most of the course contents are just a recap of high school mathematics. To note, some students omitted the signs when performing operations. Others did not know what to do
  • 24. 8 when presented with a word problem while many were not able to craft their own procedures in solving the given problems The aforementioned situationers on College Algebra performance prompted the researcher to conduct an error analysis in College Algebra in the Higher Education Institutions (HEIs) of La Union as basis for formulating an instructional intervention plan. Theoretical Framework M. Anne Newman’s (1977) theory of errors and error categories maintains that when a person attempts to answer a standard, written, mathematics question, he has to be able to pass through a number of successive hurdles, namely Reading (or Decoding), Comprehension, Transformation or ―Mathematising,‖ Processing, and Encoding. From these successive stages, students commit varied errors. According to the theory, the reading errors are committed when someone could not read a key word or symbol in the written problem to the extent that this prevented him/her from writing anything on his/her solution sheet or from proceeding further along an appropriate problem-solving path; the comprehension errors are committed when someone had been able to read all the words in the question, but had not grasped the overall meaning of the words; thus, he can only indicate partially what are the given and what are unknown in the problem; the transformation or
  • 25. 9 mathematising errors are committed when someone had understood what the questions wanted him/her to find out but was unable to identify the operation, or sequence of operations or the working equation needed to solve the problem; the processing errors are committed when someone identified an appropriate operation, or sequence of operations or the working equation, but did not know the procedures necessary to carry out these operations or equation accurately; and, the encoding errors are committed when someone correctly worked out the solution to a problem, but could not express this solution in an acceptable written form. In some case, if the answer is not in its accepted simplified form and does not indicate the unit. Researchers which made use of the abovementioned theory were Clement (2002), Ashlock (2006), Hall (2007) and Egodawatte (2011). All of their studies were able to find out the specific error categories of their student-respondents. Furthermore, Vygotsky (1915) and Kolb’s (1939) constructivist theory proposes that a person can construct and conditionalize knowledge, especially after learning or experiencing something. As applied to this study, the students are believed to be capable of showing the desired competence after learning the contents of College Algebra from their instructors.
  • 26. 10 Dewey (1899) and Roger’s (1967) active learning and experiential learning theories propose that students are able to learn something and apply what they have learned if they are engaged with their experiences. As applied in the study, the problems in the researcher-made test were anchored to the real-life encounters of the college students. Also, Bruner’s (1968) intellectual development theory discusses that intellect is innately sequential, moving from inactive through iconic to symbolic representation. He felt that it is highly probable that this is also the best sequence for any subject to take. The extent to which an individual finds it difficult to master a given subject depends largely on the sequence in which the material is presented. Further, Bruner also asserted that learning needs reinforcement. He explained that in order for an individual to achieve mastery of a problem, feedback must be reviewed as to how they are doing. The results must be learned at the very time an individual is evaluating his/her performance. This theory supports the idea that solving written problems are successive in nature. This also gave the idea to the researcher on how to check the all-word problem test. Further, Bandura’s (1963) social learning theory holds that knowledge acquisition is a cognitive process that takes place in social context and can purely occur through observation or direct instruction.
  • 27. 11 As applied in the study, the instructional interventions are student- centered so that learning becomes more active. In addition, when one attempts to address concerns on student’s errors, instructional intervention can be a good scheme. Egodawatte (2009) stresses that error analysis can pave away to clearly conceptualize an action plan such as designing effective instruction or plotting out instructional intervention. This idea by Egodawatte (2009) structures the foundation of the output of the study. Howell (2009) describes instructional intervention as a planned set of procedures that are aimed at teaching specific set of academic skills to a student or group of students. An instructional intervention must have the following components: it is planned – planning implies a decision- making process. Decisions require information (data); therefore, an instructional intervention is data-based or research-based set of teaching procedures; it is sustained – this means that an intervention is likely implemented in a series of lessons over time; it is focused– this means that an intervention is intended to meet specific set of needs for students; it is goal-oriented – this means that the intervention is intended to produce a change in knowledge from some beginning or baseline state toward some more desirable goal state; and, it is typically a set of procedures rather than a single instructional component/
  • 28. 12 strategy. Moreover, according to Manitoba Education Website (2010), an instructional intervention plan contains the purpose or the background, intervention objectives, specific topics, the error categories, the sample of error, the proposed instructional strategy and or activities, and the procedures of implementing the strategy. (http://www. edu. gov. mb.ca/k12/specedu/bip/sample.html.) The aforecited theories find their essence in the teaching and the learning of mathematics and in the specific categories in the research’s aim of identifying and analyzing errors. These also gave the researcher the main reasons of formulating the research tool composed of all word problems. Generally, they serve as the building blocks in structuring this research. Further, the concept of instructional intervention plan serves as the core idea in designing the output of this study. Furthermore, these theories served as foundations in formulating the proposed model of the researcher, the Ragma’s Error Intervention Model. Figure 1 illustrates the model. The model, a corollary of Newmann’s (1977), highlights that when someone answers a written mathematical problem, he has to undergo different but successive stages such as reading, comprehension, mathematising, processing and encoding stages. In simple words, someone has to read the problem, understand what the problem says,
  • 29. 13 Figure 1. Ragma’s Error Intervention Model INSTRUCTIONAL INTERVENTION (Game-based, visual/spatial-based, motivational instruction, technology-based, cooperative learning, tutorials, differentiated teaching, understanding-centered, processing-centered, reading strategies, experiments, dyads, observations, and scaffolding) CAUSES OF ERRORS (low Interest, attitude, high anxiety, Insufficient recall, misconception, deficient mastery, carelessness) Encoding Processing Stage Comprehension Stage Reading Mathematising Stage Error CategoriesStages in Problem Solving Encoding Errors Processing Errors Mathematising Errors Comprehension Errors Reading Errors Better Performance in College Algebra Mathematics Word Problems
  • 30. 14 14 structure the working equation, solve and then finalize the answer/s. In each of these successive stages, errors can be committed. These errors are caused by low interest, high anxiety, negative attitude, insufficient recall, misconception, poor mastery, and carelessness. To exemplify, when someone does not bother to answer the problem, he is not interested in mathematics or has high anxiety towards math. If he fails to completely analyze what the problem is all about, he cannot completely recall the essential mathematical details. If he cannot create a working equation, he has poor mastery and deficient mathematical skills. If he cannot proceed to the starting point of the mathematical solution, he cannot recall the formulas or is unable to formulate the working equation. If he cannot correctly and completely solve the problem, he has deficient mastery and is careless in handling mathematical algorithms. And, if he is unable to write a valid or unaccepted final answer, he is careless or lacks the necessary mathematical skills. Moreover, the different error categories and their causes can be addressed through the varied instructional interventions. To illustrate, reading errors caused by high anxiety and disinterest can be addressed by providing motivational instructional activities and games; differentiated instruction can also be a good instructional scheme. Comprehension errors caused by misconception can be addressed by
  • 31. 15 concept attainment and processing. Mathematising errors caused by poor mastery and insufficient recall can be addressed by direct instruction, memory-bank game and the think-pair-share activities, to name a few. Processing errors caused by poor mastery and insufficient recall can be addressed by error targeting and correcting, explicit instruction, etc. And lastly, encoding errors caused by carelessness can be solved by solve-and-compare, cooperative learning groups, etc. When all the error categories in each problem-solving stage together with their respective causes are addressed through the instructional interventions, better performance of the students in College Algebra will be achieved. Conceptual Framework Answering a standard, written, mathematics question requires a person to undergo a number of successive stages: reading, comprehension, mathematising, processing, and encoding. From these successive stages, students commit varied errors. The reading errors are committed when someone could not read a key word or symbol in the written problem to the extent that this prevented him/her from writing anything on his/her solution sheet or from proceeding further along an appropriate problem-solving path. The comprehension errors are committed when someone had been able to read all the words in the question, but had not grasped the
  • 32. 16 overall meaning of the words; thus, he can only indicate partially what are the given and what are the unknown in the problem. The transformation or mathematising errors are committed when someone had understood what the questions wanted him/her to find out but was unable to identify the operation, or sequence of operations or the working equation needed to solve the problem. The processing errors are committed when someone identified an appropriate operation, or sequence of operations or the working equation, but did not know the procedures necessary to carry out these operations or equation accurately. The encoding errors are committed when someone correctly worked out the solution to a problem, but could not express this solution in an acceptable written form. In some case, if the answer is not in its accepted simplified form and does not indicate the unit. This makes mathematics teaching challenging. Thus, for learning to take place, all the stages and aspects of problem analysis and problem solving must be well understood by the students. Moreover, when someone aspires to help students to improve on their performance, one needs to dig deeper into the reasons behind the dismal performance. According to Newmann (1977), the type of errors
  • 33. 17 committed by the students when solving word problems can give baseline data to teachers to help them improve on their mathematical skills. Egodawatte (2009) and Hall (2007) stressed that mathematical errors are a common phenomenon in mathematics learning. Students of any age have experienced getting mathematics wrong (Hall, 2007). It is natural that analyzing students’ mathematical errors is a fundamental aspect of teaching for mathematics teachers. Error Analysis is then an effective assessment approach that allows one, especially teachers, to determine whether students are making consistent mistakes when performing computations. By pinpointing the error category or pattern of an individual student’s errors, one can then directly teach the correct procedure for solving the problem or can even formulate an effectively designed instructional intervention scheme (Egodawatte, 2009). It is in this light that the study is thought of, formulated and set up. This conceptualization is logically designed in the Research Paradigm in Figure 2. The paradigm made use of the Input-Process-Output (IPO) model. The input is composed of the performance of the students along elementary topics, special product patterns, factoring, rational expressions, linear equations, systems of linear equations in two unknowns and exponents and radicals. It also incorporates the error
  • 34. 18 Patterns PROCESS OUTPUTINPUT Validated Instructional Intervention Plan for College Algebra in the Higher Education Institutions of La Union 1. Interpretation and Analysis of the Performance of the students along the specified topics 2. Identification and Analysis of the capabilities and constraints based on the level of performance 3. Identification and Analysis of error categories of the students 4. Preparation and Validation of Instructional Intervention Plan 1. Performance of the students along: a. Elementary topics a.1. sets and Venn diagrams a.2. Real numbers a.3. Algebraic expressions a.4. Polynomials b. Special Product c. Factoring Patterns d. Rational Expressions e. Linear Equations in One Unknown f. Systems of Linear Equations in Two Unknowns g. Exponents and Radicals 2. Error Categories along the specified topics in College Algebra along a. reading b. comprehension c. transformation d. process e. encoding Figure 2. The Research Paradigm
  • 35. 19 categories of the students along the specified topics in Math 1 or College Algebra along reading, comprehension, mathematising, processing and encoding. These variables are indeed necessary to determine the performance and error categories of the students in College Algebra. The process incorporated the interpretation and analysis of the performance of the students in College Algebra, the identification and analysis of the capabilities and constraints and the identification, categorization and analysis of errors in College Algebra. It also holds the process of conceptualizing and validating the output of the study. The output of the study, therefore, is a validated instructional intervention plan for the Higher Education Institutions of La Union. Statement of the Problem This study identified and analyzed the error categories of students in College Algebra in the Higher Education Institutions of La Union as basis for formulating a Validated Instructional Intervention Plan. Specifically, it sought answers to the following questions: 1. What is the level of performance of the students in College Algebra along: a. Elementary Topics; a.1. Sets and Venn Diagrams a.2. Real Numbers
  • 36. 20 a.3. Algebraic Expressions a.4. Polynomials b. Special Products; c. Factoring Patterns; d. Rational Expressions; e. Linear Equations in One Unknown; f. Systems of Linear Equations in Two Uknowns; and g. Exponents and Radicals? 2. What are the capabilities and constraints of the students in College Algebra? 3. What are the error categories of the students along the topics in College Algebra along: a. Reading; b. Comprehension; c. Mathematising or Transformation; d. Processing; and e. Encoding? 4. Based on the findings, what validated instructional intervention plan can be proposed? a. What is the level of validity of the instructional intervention plan along face and content?
  • 37. 21 Assumptions The researcher was guided with the following assumptions: 1. The level of performance of the students in College Algebra is satisfactory. 2. The capabilities are along elementary topics while the constraints are along factoring, special products, and systems of linear equations in two unknowns. 3. The major error categories of the students are mathematising and processing errors. 4. A validated instructional intervention plan addresses the errors of the students in College Algebra. Importance of the Study This piece of work will greatly benefit the CHED, administrators, heads, teachers, students, the researcher and future researchers. The Commission on Higher Education (CHED). This study will give the commission an idea of the reasons or causes of low performance in College Algebra, which will help in developing improvements along curriculum and human resource. The school administrators of the HEIs in La Union. This study will provide them with data that can be used as input to the curricular programs.
  • 38. 22 The Mathematics department heads. This study will give them insights about the performance and errors in College Algebra, which will help them in designing mathematics instruction that suits the identified errors of the students. The Mathematics instructors. This study will give them baseline data of the performance and errors of their students in College Algebra. The output of the study, on the other hand, will make them more prepared in addressing the errors since instructional interventions are proposed for their utilization. The students of the HEIs in La Union. This study will lead them to a thoughtful understanding of mathematics since their errors will be known. They will also be helped in improving their performance since the instructional interventions will address their identified errors. The researcher, a Mathematics instructor of Saint Louis College (SLC). This study will make him more knowledgeable of his students’ performance and errors. This will also give him the opportunity to structure an error intervention model that addresses students’ errors which contributes to the improvement of the fields of mathematics teaching and learning. The future researchers. This study will motivate them to pursue their research since this study can be used as basis for their future
  • 39. 23 study. This can also give them an idea on how to structure their own instructional plan based on their students’ needs and interests. Definition of Terms To better understand this research, the following items are operationally defined: Capabilities. These refer to a performance with a descriptive equivalent of satisfactory performance and above. College Algebra. This is a 3-unit requisite subject in college which includes elementary topics, special product and factoring patterns, rational expressions, linear equations in one unknown, systems of linear equations in two unknowns and exponents and radicals. Elementary topics. These topics include concepts on sets, real number system and operations, and polynomials. Algebraic expressions. These are expressions containing constants, variables or combinations of constants and variables. Polynomials. These are algebraic expressions with integer exponents. Real numbers. These are the numbers composing of rational and irrational numbers. Sets. These are collection of distinct objects.
  • 40. 24 Venn diagrams. These are diagrams proposed by the mathematician A. Venn, which are used to show relationships among sets. Factoring patterns. These include the topics in factoring given a polynomial. These include common monomial factor, perfect square trinomial, general trinomial, factoring by grouping and factoring completely. Linear equations in one unkown. This includes topics on equations with one variable such as 2x- 4 = 10 and 5x - 2x = 36. The main thrust of this topic is for an unkown variable to be solved in an equation. Rational expressions. These are expressions involving two (2) algebraic expressions, whose denominator must not be equal to zero. The topics included are simplifying and operating on rational expressions. Special product patterns. These topics include the patterns in multiplying polynomials easily. These patterns include the sum and difference of two identical terms, square of a binomial, product of two binomials, cube of a binomial and square of a trinomial. Systems of linear equations in two unknowns. This topic discusses how the solution set of a given system is solved. The methods
  • 41. 25 that are used in this certain topics include graphical, substitution and elimination methods. Constraints. These refer to a performance with a descriptive equivalent of fair performance and below. Error analysis. It is a diagnostic procedure aimed at determining specific inaccuracies of the students in College Algebra. The analysis is made using the Newmann Error Analysis tool (1977). Error categories. These are the classes of inaccuracies according to Newmann (1977). These error categories are reading, comprehension, transformation or ―mathematising‖, process and encoding. Encoding errors. These are committed when someone correctly worked out the solution to a problem, but could not express this solution in an acceptable written form. In some case, if the answer is not in its accepted simplified form and does not indicate the unit of measurement. Comprehension errors. These are committed when someone had been able to read all the words in the question, but had not grasped the overall meaning of the words; thus, can only indicate partially what are the given, what are unknown in the problem Processing errors. These are committed when someone identified an appropriate operation, or sequence of operations or the
  • 42. 26 working equation, but did not know the procedures necessary to carry out these operations or equation accurately Transformation errors. These are committed when someone had understood what the questions wanted him/her to find out but was unable to identify the operation, or sequence of operations or the working equation needed to solve the problem Reading errors. These are committed when someone could not read a key word or symbol in the written problem to the extent that this prevented him/her from writing anything on his solution sheet or from proceeding further along an appropriate problem- solving path. Higher Education Institutions (HEIs). This refers to the twelve (12) respondent academic colleges and universities, public or private, in La Union offering College Algebra for the school year 2013-2014. Instructional intervention plan. This plan contains the teaching approaches that address dismal performance. It is composed of the background, the general objectives, the specific topics, the error categories and causes, the sample error, the intervention and the assessment strategy. This serves as the output of the study.
  • 43. 27 CHAPTER II METHOD AND PROCEDURES This chapter presents the research design, sources of data, data analysis, the parts of the instructional intervention plan and ethical considerations. Research Design The descriptive method of investigation was used in the study. This design aims at gathering data about the existing conditions. Leary (2010) defines such design as one that includes all studies that purport to present facts concerning the nature and status of anything. This design is appropriate for the study since it is aimed at gathering pertinent data to describe the performance and errors of students in College Algebra. Further, the quantitative research approach was also used. Hohmann (2006) defines quantitative research approach as a component of descriptive design making use of numerical analysis. It is aimed at analyzing input variables using quantitative techniques such as averages, percentages, etc. This approach is apt for this study since it makes use of quantitative techniques to show the performance and errors of the students in College Algebra.
  • 44. 28 Sources of Data Locale and Population of the Study. The population of this study was composed of College Algebra students enrolled in the Higher Education Institutions (HEIs) of La Union for the first semester, school year 2013-2014. The total population of 5,849 students was pre-surveyed in this study; however, since the population reached 500, random sampling was employed. To generate the sample population, the Slovin’s formula (Leary 2010) was used. n = 𝑁 1+𝑁(𝑒2) where: n = the sample population N = the population 1 = constant e = level of significance @ .05 Using the Slovin’s formula, a total of 374 students distributed among the 12 respondent Higher Education Institutions of La Union constituted the respondents of this study. Table 1 reveals the distribution of the sample population.
  • 45. 29 Table 1. Distribution of Respondents Respondent HEIs N n Institution A 78 5 Institution B 482 31 Institution C 230 15 Institution D 900 58 Institution E 609 39 Institution F 1349 86 Institution G 65 4 Institution H 196 13 Institution I 51 3 Institution J 1536 98 Institution K 170 11 Institution L 183 12 Total 5849 374 Instrumentation and Data Collection A pre-survey was conducted to gather the contents of the syllabus in College Algebra in each of the HEIs. The researcher was able to meet the math instructors, department heads/chairs and academic deans who gave data pertinent to the scope of College Algebra. The conglomerated topics indicated in all the syllabi served as basis in the topics specified in the research tool. (Please see appended table of specifications) To gather the data pertinent to the level of performance and the error categories, a researcher-made test was made. The researcher-made test is an all-word-problem 20-item test, 5 points per item, covering all the topics in College Algebra. Most of the questions were based on the word problems from College Algebra books. All problem questions were
  • 46. 30 aligned along the synthesis-evaluation/evaluating-creating level under the Bloom’s Taxonomy. As such, the questions dug into the overall conceptualization and utilization of algebraic concepts and principles to be able to carry out such problem. Hence, an item combined several related subtopics to ensure that the scope of the course was still covered. The whole test was administered by the math instructors handling the classes through the permission of the presidents or concerned authority in the HEI. The test was good only for one hour and did not allow the use of calculators. Validity and Reliability of the Questionnaire. To ensure the validity of the research tool, it was presented to the members of the panel and to experts in the field of mathematics. The experts are professors of mathematics. Further, the suggestions made by the validators were incorporated in the test (see suggestions in the appendix). The computed validity rating was 4.32, interpreted as high validity (please see appended computation). This means that the research tool was able to measure what it intended to measure. Moreover, to establish its reliability, it was pilot-tested to thirty (30) students of Saint Louis College. The thirty (30) students were not included as respondents of the study. The internal consistency or reliability was determined using the Kuder-Richardson 21 formula. The formula is (Monzon-Ybanez 2002):
  • 47. 31 𝐾𝑅21 = 𝑘 𝑘−1 1 − 𝑥 𝑘−𝑥 𝑘𝜎2 where: k = number of items 𝑥 = mean of the distribution 𝜎2 = the variance of the distribution Thus, the computed reliability coefficient was 0.72 (please see appended computation). This means that the test was highly reliable, which pinpoints that the test was internally consistent and stable. Administration and Retrieval of the Questionnaire. With the necessary endorsement from the Dean of the Graduate School (Dr. Rosario C. Garcia) of Saint Louis College, City of San Fernando, La Union, the researcher sought permission from the president or head of the different twelve (12) respondents-institutions to float the questionnaire. The copies of the questionnaire was handed to the deans/program heads of the various college institutions who were also requested to administer the said questionnaire to the respondents of which the answered questionnaires were retrieved on a specified date as it was scheduled by the deans/program heads of the various respondents-institutions.
  • 48. 32 Tools for Data Analysis The data gathered, collated and tabulated were subjected for analysis and interpretation using the appropriate statistical tools. The raw data were tallied and presented in tables for easier understanding. For problem 1, frequency count, mean and rate were utilized to determine the level of performance in College Algebra. The formula for mean is as follows (Ybanez, 2002): M = ∑x N Where: M – mean x – sum of all the score of the students N – number of students For problem 2, the capabilities and constraints were deduced based on the findings, particularly on the level of performance in College Algebra. An area was considered a capability when it received a descriptive rating of satisfactory and above; otherwise, the area was considered a constraint. For problem 3, the Newmann Error Analysis Tool (1977) was used to identify the errors and error categories of the students. (Please see the error categories in the definition of terms.) Moreover, frequency count, average and rate were used to determine the error categories of the students.
  • 49. 33 The MS Excel Worksheet and StaText were employed in treating the data. Data Categorization For the scoring/checking of the test, the scheme below was used: Point Assignment Error Category 0 Reading Error 1 Comprehension Error 2 Mathematising Error 3 Processing Error 4 Encoding Error 5 No Error For the level of performance in each topic in College Algebra, the following scale systems were utilized. Elementary Topics/ Factoring Score Range Level of Performance Descriptive Equiva- lent Rating 16.00-20.00 Outstanding Performance (OP) Capability 12.00-15.99 Satisfactory Performance (SP) Capability 8.00 -11.99 Fair Performance (FP) Constraint 4.00-7.99 Poor Performance (PP) Constraint 0-3.99 Very Poor Performance (VPP) Constraint
  • 50. 34 Special Products and Patterns/Rational Expressions/Linear Equations in One Variable Score Range Level of Performance Descriptive Equiva- lent Rating 12.00-15.00 Outstanding Performance (OP) Capability 9.00-11.99 Satisfactory Performance (SP) Capability 6.00-8.99 Fair Performance (FP) Constraint 3.00-5.99 Poor Performance (PP) Constraint 0.00-2.99 Very Poor Performance (VPP) Constraint Systems of Linear Equations in Two Variables Score Range Level of Performance Descriptive Equiva- lent Rating 8.00-10.00 Outstanding Performance (OP) Capability 6.00-7.99 Satisfactory Performance (SP) Capability 4.00-5.99 Fair Performance (FP) Constraint 2.00-3.99 Poor Performance (PP) Constraint 0-1.99 Very Poor Performance (VPP) Constraint Exponents and Radicals Score Range Level of Performance Descriptive Equiva- lent Rating 4.00-5.00 Outstanding Performance (OP) Capability 3.00-3.99 Satisfactory Performance (SP) Capability 2.00-2.99 Fair Performance (FP) Constraint
  • 51. 35 Score Range Level of Performance Descriptive Equiva- lent Rating 1.00-1.99 Poor Performance (PP) Constraint 0-0.99 Very Poor Performance (VPP) Constraint For the general performance in College Algebra, the scales below were used: Score Range Level of Performance 80.00-100.00% Outstanding Performance (OP) 60.00-79.99% Satisfactory Performance (SP) 40.00-59.99% Fair Performance (FP) 20.00-39.99% Poor Performance (PP) 0-19.99% Very Poor Performance (VPP) The scale for interpretation on the reliability of the College Algebra test was: 1.00 - Perfect Reliability (PR) 0.91-0.99 - Very High Reliability (VHP) 0.71-0.90 - High Reliability (HR) 0.41-0.70 - Marked Reliability (MR) 0.21-0.40 - Low Reliability (LR) 0.01-0.21 - Negligible Reliability (NR) 0.00 - No Reliability (NoR)
  • 52. 36 For the validity of the College Algebra test and the Instructional Intervention Plan, the scale below was used: Points Ranges Descriptive Equiva- lent Rating 5 4.51-5.00 Very High Validity (VHV) 4 3.51-4.50 High Validity (HV) 3 2.51-3.50 Moderate Validity (MV) 2 1.51-2.50 Poor Validity (PV) 1 1.00-1.50 Very Poor Validity (VPV) Parts of the Instructional Intervention Plan The instructional intervention plan contains the purpose or the background, intervention objectives, specific topics, the error categories, the sample error, the proposed instructional strategy and or activities, the procedures of implementing the strategy and the assessment strategy. The instructional intervention plan is based on the level of performance of the students in College Algebra, the culled-out capabilities and constraints and the different error categories in each topic of College Algebra. The foremost constraints and the two primary error categories in each topic are given more emphasis on the instructional intervention plan as seen on the number of indicated
  • 53. 37 interventions. There are still interventions for those considered as capabilities for sustainability. Ethical Considerations To establish and safeguard ethics in conducting this research, the researcher strictly observed the following: The students’ names were not mentioned in any part of this research. The students were not emotionally or physically harmed just to be a respondent of the study. There were HEIs which decided not be included in the study due to some concerns and other priorities. This decision of opting not to join in the study was respected by the researcher. Coding scheme was used in reflecting the respondent HEI in the table for distribution of respondents. Proper document sourcing or referencing of materials was done to ensure and promote copyright laws. A communication letter was presented to the Registrar’s Office or President’s Office to ask authority to gather the needed data on the contents of the syllabi and number of students enrolled in College Algebra. A communication letter was presented to the President’s Office asking permission to float the questionnaire.
  • 54. 38 The research instrument was subjected to validity and reliability. Their suggestions were incorporated in the instrument. A list of summary and the corresponding actions of the researcher is appended. The instructional intervention plan was subjected for acceptability. All the suggestions were incorporated.
  • 55. 39 CHAPTER III RESULTS AND DISCUSSION This chapter presents the statistical analysis and interpretation of gathered data on the level of performance in College Algebra and the error categories in each specified topic. Level of Performance of Students in College Algebra The first problem considered in this study dealt on the level of performance of students in College Algebra along elementary topics - sets and Venn diagrams, real numbers, algebraic expressions, and polynomials; special product patterns, factoring patterns; rational expressions; linear equations in one unknown; systems of linear equations in two unknowns; and, exponents and radicals. Elementary Topics Table 2 shows the performance of the students in College Algebra along elementary topics. It shows that the students had a mean score of 8.69 or 43.45%, a fair performance in elementary topics. This implies that the students had not achieved to the optimum the needed skills in elementary topics. It also reflects that the students had poor performance in sets and Venn diagrams. This means that the students were not capable of representing data relationships and solving problems
  • 56. 40 Table 2. Level of Performance of Students in Elementary Topics Subtopic Mean Score Rate Descriptive Equivalent Sets and Venn Diagrams (5) 1.78 35.60% Poor Real Number System (5) 2.87 57.40% Fair Algebraic Expressions (5) 1.64 32.80% Poor Polynomials (5) 2.4 48.00% Fair Overall 8.69 43.45% Fair involving sets and Venn diagrams. Moreover, they had fair performance in real number system. This means that the students could visualize, to a moderate extent, the number line and perform operations on real numbers. Further, they had poor performance in algebraic expressions. This implies that the students could not perform well translations and operations involving algebraic expressions. On the other hand, they had fair performance in polynomials. This suggests that the students could moderately recognize quantities represented by polynomials and perform mathematical processes involving polynomials. The findings of the study corroborate with the study of Oredina (2011) revealing that the students had moderate level of competence in Elementary topics. She mentioned that the students needed to achieve to
  • 57. 41 the fullest the needed competence in elementary topics in College Algebra. Further, the findings of the study conform to the study of Elis (2013) revealing that the students had moderate performance in Algebraic expressions. He stressed that this was caused by negative attitude towards Mathematics. On the other hand, the study of Pamani (2006) does not run parallel to the findings of the study stating that the students had high competence in pre-algebra, which included sets, real numbers, algebraic expressions, etc. She explained that such level of performance reflected that the students were highly capable of determining concepts and performing mathematical procedures along these specified topics. The findings of the study do not also harmonize with the study of Okello (2010) revealing that 73% of the students failed in almost all topics in College Algebra such as prerequisites, factoring and systems of equations. Special Product Patterns Table 3 shows the performance of the students in College Algebra along special product patterns. It reveals that the students had a mean score of 7.41 or 49.40%, a fair performance in special product patterns. This means that the students could not correctly perform special
  • 58. 42 Table 3. Level of Performance of Students in Special Product Patterns Subtopic Mean Score Rate Descriptive Equivalent Product of Two binomials (5) 2.69 53.50% Fair Square of a trinomial (5) 2.13 42.60% Fair Cube of a Binomial (5) 2.59 51.80% Fair Overall 7.41 49.40% Fair product patterns implying that the students failed to master the skills along special products. Further, it reveals that the students had fair performance along product of two binomials. This implies that the students could not productively use the FOIL method in getting the product of binomials, implying that they cannot multiply and simplify two alike or different binomials. Also, they had fair performance along the square of a trinomial. This entails that the students cannot use the (F + M +L)2= (F2 + M2 + L2 + 2FM + 2FL + 2ML) pattern reasonably. Moreover, they also had fair performance along the cube of a binomial. This indicates that the students cannot use the (F ± L)3= (F3 ± 3F2L ± 3FL2 ± L3) pattern correctly. Since the performance was within the fair level only, it can be construed that the students had not attained to the fullest the skills along the utilization of such patterns.
  • 59. 43 The findings of the study adhere to the study of Wood (2003) emphasizing that the students performed fairly in College Algebra, especially in special product and factoring patterns. He mentioned that the students’ level of performance dug into a level of 39% and below. The findings of the study also corroborate with the study of Pamani (2006) stressing that the students had moderate competence in special products. She mentioned that the students failed to master to the fullest the needed skills in all the special product patterns. Further, the study jibes with Oredina (2011) stating that the students had moderate competence in special products. This means that the students can handle special product patterns but had not fully mastered the desired competencies. The students had very low competence in squaring a binomial, low competence in monomial multiplier, low competence in sum and difference of 2 binomials, high competence in product of 2 different binomials but very high competence on cube of a binomial and square of a trinomial. Further, the study also agrees with the study of Bucsit (2009) stating that the students had poor performance in special products. She stated that this very dismal performance pointed out to the fact the students could not really perform multiplication using polynomials. She further explained that the students had not very well understood the concepts and processes involved in special products.
  • 60. 44 Factoring Patterns Table 4 illustrates the performance of the students in College Algebra along factoring patterns. It shows that the students had a mean score of 8.03 or 40. 15%, interpreted as a fair performance. This means that the students could perform, to a restrained extent, factoring patterns, pinpointing that the students failed to master, to the fullest, all the skills along factoring. It also shows that the students had poor performance in difference of two perfect squares. It can be inferred that the students could not distinguish and factor correctly polynomials of the form (x2-y2). Further, the students had fair performance in perfect square trinomial. This stresses that the students could not optimally recognize and factor patterns of the form (F2 ± √2FL + L2). It also reveals that the students had fair performance in factoring general trinomials. This means that they were deficient along the required skills. It also reveals that the students had poor performance in factoring by grouping. This implies that the students failed to distinguish expressions within a polynomial that can be grouped together for the purposes of simplification through factoring. The study harmonizes with Gordon (2008) emphasizing that the students had dismal performance in concepts involving algebraic expressions, factoring and special product patterns.
  • 61. 45 Table 4. Level of Performance of Students in Factoring Patterns Subtopic Mean Score Rate Descriptive Equivalent Difference of 2 Perfect Squares (5) 1.05 21.00% Poor Perfect Square Trinomial (5) 2.64 52.80% Fair General Trinomial (5) 2.67 53.40% Fair Factoring by Grouping (5) 1.67 33.40% Poor Overall 8.03 40.15% Fair These findings also agree with the study of Pamani (2006) revealing that students had moderate performance in factoring. It was stressed that the students could perform factoring but needed to do more in order for the students to attain the desired level of competency. The findings of the study are in contrast with the study of Oredina (2011) stating that the students had high competence in factoring patterns. This means that the students could do well and perform very satisfactorily factoring exercises. It also does not jibe with the finding of the study of Bucsit (2009) stating that the students had poor performance in factoring. She stated that the students could not very well recognize and perform factoring patterns.
  • 62. 46 Rational Expressions Table 5 shows the performance of the students in College Algebra along rational expressions. It shows that the students had a mean score of 4.73 or 31. 53%, interpreted as a poor performance in rational expressions. This pinpoints that the students failed to correctly simplify and perform operations involving rational expressions or expressions involving fractions. Further, it reflects that the students had fair performance in simplification of RAEs. This means that the students could not simplify competently rational expressions to their simplest form by performing cancellation and reduction. It also mirrors that the students had poor performance in operations of RAEs. The students could not proficiently add, subtract, multiply and divide rational algebraic terms or expressions. It also shows that the students had very poor performance in simplification of complex RAEs. This means that the students failed to perform procedures and algorithms pertinent to the simplification of complex fractions. The findings of the study run parallel to the study of Laura (2005) stressing that students’ performance in College Algebra was in crisis. He explained that the cohort of students passing College Algebra was only about 33.33%. He pinpointed that factoring and rational expressions
  • 63. 47 Table 5. Level of Performance of Students in Rational Expressions (RAEs) Subtopic Mean Score Rate Descriptive Equivalent Simplification of RAEs (5) 2.43 48.6% Fair Operations of RAEs (5) 1.52 31.40% Poor Simplification of Complex RAEs (5) 0.78 15.60% Very Poor Overall 4.73 31.53% Poor were the most difficult for the students. The findings jibe with the study of Bucsit (2009) revealing that her respondents had poor performance along rational or fractional expressions. She stressed that the students had deficient skills as regards performing operations and simplifying involving rational expressions. The students were not able to deal with finding the correct LCDs to simplify correctly the expressions. Contrary, the findings do not relate to the study of Oredina (2011) showing that the students had moderate competence in rational expressions. This means that the students had not fully acquired the needed competence along the indicated areas. It was stressed that the students could not correctly manipulate rational expressions, simplify such and operate using the fundamental operations.
  • 64. 48 Linear Equations in One Variable Table 6 shows the performance of the students in College Algebra along linear equations in one variable. It shows that the students had a mean score of 3.29 or 21. 93%, interpreted as a poor performance in linear equations. This implies that the students had not mastered the mathematical ways of representing data and forming linear equations to be able to interpret and solve worded problems. It also unveils that the students had poor performance in distance, mixture, and age problems. This pinpointed to the fact the students were deficient in analyzing, representing, crafting working equations and solving problems related to linear equations in one variable. They could not see how variables were related to each other; they failed to see meaning among the algebraic verbal and numerical expressions that could serve as their basis for structuring the solution of certain problems. The study agrees with Bucsit’s (2009) since it revealed that the students were poor along word problems in linear equations in one variable. She underlined that the students lacked the necessary skills in understanding and translating expressions into useful data relevant to the solution of a certain problem. It also corroborates with the study of Pamani (2006) revealing that the students had fair competence along linear equations. She stressed
  • 65. 49 Table 6. Level of Performance of Students in Linear Equations in One Variable Subtopic Mean Score Rate Descriptive Equivalent Distance Problem (5) 1.06 21.20% Poor Mixture Problem (5) 1.09 21.80% Poor Age Problem (5) 1.14 22.80% Poor Overall 3.29 21.93% Poor that this performance points to the failure of students to understand the complexities of word problems. The findings of the study do not relate to the study of Oredina (2011) revealing that the students had moderate competence in linear equations in one variable. It was emphasized that students’ performances were fair-to-good only along this area. They had moderate competence in solution of linear equations in one variable including coin, distance and age problems, low competence in problems on involving work, mixture, geometric relations and solid mensuration but had high competence in number relation. She remarked that the students could deal correctly with formulating, manipulating and finalizing formulas and the linear equations in one unknown that best fit the main thrusts of the word problems
  • 66. 50 Systems of Linear Equations in Two Variables Table 7 shows the performance of the students in College Algebra along systems of linear equations in two variables. It shows that the students had a mean score of 3.55 or 35.50%, interpreted as a poor performance in systems of linear equations in two variables. This implies that the students failed to represent and solve problems using systems of linear equations. It can also be understood that the students failed to perform elimination, substitution and other pertinent methods used in solving systems of linear equations. The findings of the study relate to the study of Denly (2009) stating that the students performed unsatisfactorily in number system, equations and inequalities. He noted that students did not consider correctly the properties needed in solving equations. This finding also harmonizes with Pamani’s study (2006) revealing that the students had fair performance in systems of linear equations. She stressed that the students were not able to apply the correct mathematical methods to be able to get the correct solution sets to the systems. This study does not run parallel to the study of Oredina (2011) disclosing that the students had moderate competence in Systems of Linear Equations in Two Variables. This means that the students did
  • 67. 51 Table 7. Level of Performance of Students in Systems of Linear Equations in Two Variables Subtopic Mean Score Rate Descriptive Equivalent Applied Problems on fare (5) 1.28 25.60% Poor Applied Problems on numbers (5) 2.27 45.40% Fair Overall 3.55 35.50% Poor not achieve to the maximum the needed competencies in College Algebra. They had moderate competence in graphing systems of linear equations and solving worded problems; they also had low competence in slope and systems in two (2) variables. Exponents and Radicals Table 8 unveils the performance of the students in College Algebra along exponents and radicals. It discloses that the students had a mean score of 0.39 or 7.80%, a very poor performance. This means that the students had not mastered the needed skills for them to deal with exponential and radical expressions competently. They were deficient in manipulating expressions and equations involving exponents and radicals. They were not able to correctly treat data inside the radical symbols and express correctly the square of certain expressions.
  • 68. 52 Table 8. Level of Performance of Students in Exponents and Radicals Subtopic Mean Score Rate Descriptive Equivalent Exponential and Radicals (5) 0.39 7.80% Very Poor Overall 0.39 7.80% Very Poor The findings corroborate with the study of Li (2007) stating that students had difficulty in dealing with exponents and radicals. He explained that the students did not master the mathematical principles behind simplification of such concepts. This dismal performance points out to the fact that mastery was not attained. In addition, the findings also jibe with the study of Pamani (2009) showing that the students had fair performance in exponential and radical expressions and equations. It was stressed that students failed to understand the rudiments of these algebraic concepts. Summary on the Level of Performance of Students in College Algebra in the HEIs in La Union Table 9 shows the summary of the level of performance of students in College Algebra. It can be clearly gleaned from the table that generally, the students had a mean score of 36.08 or 36.08%, interpreted as poor performance. This implies that students did not really achieve to the
  • 69. 53 Table 9. Summary Table on the Level of Performance of Students in College Algebra TOPIC Mean Score Rate Descriptive Equivalent Elementary Concepts (20) 8.69 43.45% Fair Special Product Patterns (15) 7.41 49.40% Fair Factoring (20) 8.03 40.15% Fair Rational Expressions (15) 4.73 31.53% Poor Linear Equation in One Variable (15) 3.28 21.93% Poor Systems of Linear Equations (10) 3.55 35.50% Poor Exponents and Radicals (5) 0.39 7.80% Very Poor Overall 36.08 36.08% Poor maximum the needed or the desired competencies of the subject, especially that such score did not even reach the mean score of 50 or 50%. This can be attributed to the fact that all the items were word problems that require higher-order thinking and mathematical skills. Wood (2003) stressed that when students are prompted with knowledge or computation questions, students’ success rate is 86% or even higher; but, when students are prompted with word problems, their success rate dips down to a low of 39%. This is easy to understand since word problems synthesize all the necessary skills, from knowledge to evaluation, to be able to carry out the solution to a given problem. It is in
  • 70. 54 word problems where students are able to apply all the necessary competencies learned to a situation that requires higher-order-thinking skills. Further, the students scored highest along special product patterns; but, still within the fair level. It can be understood that the students’ foremost moderate skill is along this subject matter. On the contrary, they scored lowest along exponents and radicals. This means that they had not gained competence in this area. This can be attributed to insufficient time. Capabilities and Constraints of Students in College Algebra The second problem in this study covered the capabilities and constraints of students in College Algebra. Table 10 discloses the capabilities and constraints in College Algebra as culled out from the level of students’ performance. It can be clearly read from the table that all content areas were regarded as constraints since the performance was within the fair-to-very-poor levels only. Their foremost constraint was along exponents and radicals. This means that they were weak along treating exponential and radical expressions. Although still treated as a constraint, they performed a little better along special product patterns. The findings of the study corroborate with the study of Bucsit (2009) stating that the students performed moderately in number
  • 71. 55 Table 10. Capabilities and Constraints of Students in College Algebra TOPIC Mean Score Rate Classification Elementary Concepts 8.69 43.45% Constraint Special Product Patterns 7.41 49.40% Constraint Factoring 8.03 40.15% Constraint Rational Expressions 4.73 31.53% Constraint Linear Equation in One Variable 3.28 21.93% Constraint Systems of Linear Equations in Two Variables 3.55 35.50% Constraint Exponents and Radicals 0.39 7.80% Constraint system, poor in special product and factors, poor in linear equations and systems, and fair in rationals, radicals and exponents. It can be deduced that the constraints of the students in this study were along all the topics in College Algebra. Also, the study agrees with Denly (2009) when he revealed that all students had difficulty in all the content areas in College Algebra. She mentioned that College Algebra is indeed in crisis since most of the students could not hurdle the demands of algebraic manipulations, logic, and analysis of the different variables, especially in written word problems.
  • 72. 56 Error Categories in College Algebra The third problem considered in this study is on the error categories of the students along elementary topics in College Algebra. Elementary Topics Table 11 shows the error categories of students along elementary topics. It reveals that 85 or 22.72% of the errors in elementary topics were along mathematising, 69.50 or 18.58% were along comprehension, 68 or 18.18% were along reading, 64 or 17.11% were along encoding, and 61 or 16.31% were along processing. It also shows that 26.50 or 7.09% were not considered errors. This means that most of the students committed Mathematising errors along elementary topics, implying that they were able to understand what the questions wanted them to find out; but failed to identify the series of operations or formulate the working equation needed to solve the problem. Specifically, 149 errors in sets and Venn diagrams were along Mathematising errors. This means that the students were not able to draw the relationships of the given data using the correct Venn diagrams. Some made use of tables instead of Venn Diagrams. Others had not written any equation, solution or diagram after identifying the given data of the problem. Others also wrote an incorrect working equation such as ―250 - 160 - 150 - 180 = x‖, ―250-20 = 30‖ and
  • 73. 57 Table 11. Error Categories in Elementary Topics Subtopic Error Categories R C M P E N Sets and Venn Diagram 94 51 149 35 18 27 Real Number System 45 15 62 91 142 19 Algebraic Expressions 62 185 41 35 20 31 Polynomials 71 27 89 83 75 29 Average 68 69.50 85 61 64 26.50 Rate 18.18% 18.58% 22.72% 16.31% 17.11% 7.09% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error ―160+150+180+75+90+20=775‖. Others did not write any equation after presenting the data. This was caused by poor recall and mastery of the course content. It is also good to note that 94 errors were along reading. This means that the students had poor understanding regarding the problem given, which led them not write any data from the given. It also implies that the students really did not know what to do, leaving the item unanswered. This highlights deficient mastery of the subject matter. Moreover, 51 errors were committed along comprehension errors. This implies that the students were able to read the problem but had not completely understood the problem. This means that they were unable to
  • 74. 58 completely write the needed data. They missed out writing data such as ―20 customers chose all the brands‖. This was caused by deficient mastery and carelessness. Also, 35 errors were committed along processing errors. They were able to write the correct working equation; however, failed to correctly write the solution. Students wrote on their diagrams incorrect difference such as ―10‖ instead of ―5‖ for the remaining number of people who chose Samsung brands. This was caused by carelessness and deficient mastery of operations on sets. Lastly, 18 errors were committed along encoding errors. The students were not able to write the final answer in an acceptable form. The students just left the answer 5 inside the Venn Diagram. Others just indicated ―5‖ instead of indicating ―5 people chose other brands or love other brands‖ as the final answer. This was caused by carelessness and lack of critical thinking. It also shows that 142 errors in real number system were along encoding errors. This implies that the students failed to write the final answer in an acceptable form. Most students only indicated ―11‖ as their final answer instead of writing ―11 units‖. This was due to lack of critical thinking among the students. It is also good to note that 91 errors in this course content were along processing. It means that they were unable to correctly perform the needed operations to be able to solve the problem. The students committed errors on getting the distance of 9 from -2 and
  • 75. 59 10 from 8. Instead of writing ―9- (-2) = 11‖ and ―10 -8 = 2‖, students wrote ―9- (-2) = 7‖ and ―10 + 8 = 18‖. Others also performed counting but failed to consider the principle of counting from a number line, implying an incorrect distance of 10 and 3 units. Some also left the answers ―9 units‖ and ―2 units‖ unadded even if the question was asking them to get the sum of the distances. Also, 62 errors were along Mathematising errors. The students did not write anything as a working equation. Others wrote an incorrect one such as ―7 + (-2) =d1 and10 + 8 = d2‖. Such error was caused by poor recall of concepts and deficient mastery. Moreover, 45 errors were committed along reading. This means that the students left the item unanswered. This means that the students did not know what to do. Lastly, 15 errors were committed along comprehension. They were able to indicate only 7 and -2, but not 10 and 8. Others indicated the distance to be from -2 being the least coordinate and 10, being the highest coordinate. This was caused by deficient skill in mathematical understanding. Further, it also reveals that 185 errors in algebraic expressions were along comprehension. This means that the students were able to read all the words in the question, but had not grasped the Overall meaning of the words; they only indicated partially what were the given, what were unknown in the problem. Most of the students had written an
  • 76. 60 incomplete representation of the phrase ―the height is (x+9) cm more than the base‖. Instead of writing ―(x+9) + (2x-5)‖, most of them wrote ―(x+9) cm‖ only. This was due to insufficient understanding of mathematical expressions or poor skills along mathematical translations. It is revealing that 62 errors were along reading. Students left this item unanswered. This means that the students did not know what to do. This error was caused by poor mastery or deficient recall. Moreover, 41 errors were committed along Mathematising. Students were not able to correctly indicate the formula for the area of a right triangle. Others wrote ―A = bh, c2= a2+ b2 and A= 3s‖ instead of ―A = ½ bh‖. Others did not write any formula after indentifying the given from the problem. This was due to poor recall. Further, 35 errors fall along processing errors. Students committed errors in multiplying (2x-5) and (3x +4). Instead of writing ―2x2 -7x -20‖, they wrote ―2x2 -23x -20, 2x2 +7x -20 and 2x2 -7x +20‖. Others also committed errors in adding (2x-5) and (x+9). Instead of writing ―3x + 4‖, they wrote ―3x-4‖. Others overdid their analysis by applying the concept of the relationship and the measurement of the 3 sides; so they wrote 2x-5< x+9. This was due to deficient mastery and carelessness. Lastly, 20 errors were along encoding errors. Students failed to indicate the correct unit of measurement. The students wrote the answer in ―cm‖ instead of ―cm2‖. They also forgot to
  • 77. 61 write the unit of measurement. This was due to lack of critical thinking and carelessness. Moreover, 89 errors along polynomials were along Mathematising errors. Most of the students failed to write the working equation. Others wrote an incorrect equation such as ―(x4-1)-(x+1)‖ instead of ―(x4- 1)/(x+1)‖. This was caused by poor mastery and deficient recall. It is also seen that 83 errors were along processing errors. Students performed incorrect synthetic division while others performed incorrect factoring for ―(x4-1)‖ such as ―(x3)(x-1)‖ and ―(x + 1)(x -1)(x+ 1)(x + 1)‖. Others performed incorrect cancellation in (x4-1)/(x+1). They immediately cancelled x4 and x and subtracted 1 and -1; thereby, generating answers x3 and x3-1. Others had written the correct working equation but had not proceeded to the correct solution path. This was due to carelessness and deficient mastery. In addition, 75 errors were along encoding. Students just wrote ―x3- x2+x-1 or (x2+1)(x-1)‖ without the word ―ice cream‖. Others had correctly performed division but had not copied the correct sign, so instead of writing ―(x3-x2 + x-1) ice cream‖, they wrote ―x3-x2-x-1) ice cream‖. Lastly, 27 errors were along comprehension. Students failed to completely write the data from the given problem. This was due to laziness and carelessness.
  • 78. 62 These results agree with the study of White (2007) revealing that most misconceptions of his respondents along College Algebra were along reading/ comprehension, transformation and carelessness in writing the final answers. He revealed that most problems involving situations were misunderstood by the students. He explained that these errors appeared because the students did not have the critical ability to deduce major concepts from a given problem. He also explained that the students’ insufficient exposure to this kind of problem and poor mastery caused the errors. Further, the findings of the study corroborate with Peng (2007) revealing that students left items on Venn Diagrams, Polynomials and Algebraic Expression integrating other concepts on Geometry, Measurement and Basic Numerical Analysis unanswered. The unanswered items pointed out to insufficient or even no knowledge of the concepts. He explained that the items were unanswered because students were new to this type of problem presentation or may not had exposed well to diagram analysis. This type of error, according to Peng (2007), is termed as ―beginning error for interpretation and logic‖. This also relates to the study of Hall (2007) stressing that one of the foremost problems of his students was their inability to understand the language of mathematics. For some students, mathematical disability was as a result of problems with the language of mathematics. Such
  • 79. 63 students had difficulty with reading, writing and speaking mathematical terminologies which normally were not used outside the mathematics lesson. They were unable to understand written or verbal mathematical explanations or questions and therefore cannot translate these to useful data. Special Product Patterns Table 12 unveils the error categories of the students in special product patterns. It can be seen from the table that 151.33 or 40.46% errors were committed along processing, 78.33 or 20.94% were along reading, 47.67 or 12.75% were along Mathematising, 36 or 9.63% were along encoding and 16.67 or 4.46% were along comprehension. It is also good to note that 44 or 11.76% were not considered as errors. This means that majority of the students committed processing errors in special product patterns. They were able to read, understand and set up the working equation but failed in proceeding to the correct solution path, leaving incorrect answers. Specifically, the table shows that 201 errors in product of 2 binomials were committed along processing errors. Students incorrectly multiplied (3x2-5) to (3y+4) and (2x2+45) to (5y+2). Others committed errors in evaluating (3x2-5); instead of writing ―(3(10)2-5 = 295)‖, they wrote ―900-5 = 895‖. They also failed to multiply the measure of the lot
  • 80. 64 Table 12. Error Categories in Special Product Patterns Subtopic Error Categories R C M P E N Product of Two Binomials 64 16 15 201 26 52 Square of a Trinomial 86 18 82 146 30 12 Cube of a Binomial 85 16 46 107 52 68 Average 78.33 16.67 47.67 151.33 36 44 Rate 20.94% 4.46% 12.75% 40.46% 9.63% 11.76% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error by its respective price, leaving the solution process incomplete. This was due to lack of critical thinking and deficient skill. Moreover, 64 errors were along reading. The students left the item unanswered. This implies that the students did not know what to do. This was caused by poor mastery of content. It can also be gleaned that 16 errors were along comprehension and 15 errors were along Mathematising. The students failed to get the gist of the problem. The students, due to their misunderstanding of the focus of the problem, failed to craft the working equation or remember the formula suited to the problem. Further, 146 of the committed errors in square of a trinomial were along processing errors. The students failed to correctly square a
  • 81. 65 trinomial. Most of them answered (2x-4y+6z)2 as (4x2+16y2+36z2), worse (4x2-8xy2+12y2) instead of 4x2+16y2+36z2+16xy+24xz-48yz. Others also wrote 4x2+16y2+36z2–8xy +12xz -24yz. Others performed correctly the pattern but failed to employ the rules of signs. This was caused by deficient mastery of the subject matter. It is also noted that 86 errors were along reading. This means that some students left the item unanswered. The students had not understood fully the problem or did not really know how to deal with the problem. This was caused by poor competence. Also, 82 errors were along Mathematising. The students failed to write the correct formula. Instead of writing A= ∏r2, most of them wrote A= 2∏r, and A= 2∏r2. This was misalignment of formulas. Others also were not able to write any formula or working equation. This was caused by deficient recall. 30 errors were also committed along encoding errors. Most of them failed to write the unit of measurement of the final answer. Others also committed parenthetical error, a kind of encoding error. Instead of writing (4x2+16y2+36z2–16xy +24xz -48yz)∏ cm2, they wrote 4x2+16y2+36z2–16xy +24xz -48yz∏ cm2 . This was due to carelessness and lack of critical thinking. Lastly, 18 errors were along comprehension. The students failed to completely identify all the given from the data. They just listed (2x-4y + 6z). Others even wrote (2x+4y+6z). This was due to carelessness among students.
  • 82. 66 The table also shows that 107 errors in cube of a binomial were along processing errors. The students failed to correctly cube the binomial (2x+4). Most of them just wrote (8x3+63) or worse (8x3+12) and (6x3+12) and (8x+64). The students failed to apply the pattern of (F+L)3 = (F3+3F2L+3FL2+L3). This was caused by poor competence. In addition, 85 errors were along reading. The students left the items unanswered. They did not know what to do to be able to arrive at the correct answer. This was caused by poor mastery. It can also be noted that 46 errors were along Mathematising errors. The students failed to write the correct formula, V = s3. The students wrote s2 or (s)(s). Some also wrote V= 3s3 and V= 4s. This was due to poor retention of formulas taught to them even in the elementary. Also, 52 errors were along encoding errors. Students failed to write the final answer with the correct unit of measurement. Others wrote cm, cm2 or none at all. This was due to lack of criticality and carelessness among students. Lastly, the 16 errors were committed along comprehension. The students failed to write completely the given data. Instead of writing (2x +4), some wrote (2x-4), (2+4), (x+4). This was due to carelessness. The findings of the study corroborate with the study of Egodawatte (2011) divulging that most students committed transformation and processing errors along word problems involving algebraic expressions, factoring and special products. He explained that the students failed to
  • 83. 67 remember and apply perfectly the special product and factoring patterns. He further stressed that the students committed these kinds of errors because the students had difficulty in carrying out several steps involved in the mathematical process. He specifically itemized that the students were poor in simplification, performing operations, exponential laws as applied in factoring and product patterns, incorrect distribution and invalid cancellation. Also, the study of Allen (2007) harmonizes with the finding of the study revealing that most students committed processing errors when dealing with special products and factoring. He stressed that students did not apply the correct rules in simplification of polynomials, algebraic expressions, special products and factoring. He showed that many students expanded (x+3)2 as x2+9 or worse x+6. Many of the errors were caused by poor mastery of the mathematical principles in the said topics. Factoring Patterns Table 13 exposes the error categories of students in factoring patterns. It shows that the students committed 128.25 or 34.29% reading errors, 78 or 20.85% Mathematising errors, 60 or 16.17% encoding errors, 39 or 10.42% processing errors and 25.58 or 6.75% comprehension errors. It also shows that 43 or 11. 50% were not considered errors. This implies that majority of the students failed to
  • 84. 68 Table 13. Error Categories in Factoring Patterns Subtopic Error Categories R C M P E N Difference of two Perfect Squares 182 46 111 17 15 3 Perfect Square Trinomial 88 29 53 37 92 75 General Trinomial 95 12 57 34 100 76 Factoring by Grouping 148 14 91 68 35 18 Average 128.25 25.25 78 39 60.5 43 Rate 34.29% 6.75% 20.85% 10.42% 16.18% 11.50% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error understand the applied problems along factoring. Majority left the items unanswered since they did not know what to do. This is caused by poor competence. This is even attested by the fact that only 43 students got the item correctly. It can also be read from the table that 182 errors in factoring difference of two perfect squares were along reading errors. This means that the students left the items unanswered. They did not understand what the problem wants them to do or they did not know what to do. This is due to the lack of competence of students. Moreover, 111 errors were along Mathematising errors. This means that the students failed to
  • 85. 69 correctly write the correct formula or working equation demanded by the problem. They failed to write the formula for the area of the rhombus, A= ½ d1d2. Others wrote the formula for the area of the square, A = s2. This is clear sign of misalignment of formulas. This was due to insufficient recall. This was due to poor exposure to this kind of geometric figure. Also, 46 errors are along comprehension. This means that the students did not fully understand the focus of the problem. This is attested by the incomplete data or incorrect data written on their answer sheets. Someonly wrote (2x2-162), forgetting (x-9). Others wrote (2x2-162) and (x+9). This is due to carelessness. Further, 17 errors were along processing. Most of the students after substituting the values to the formula, committed factoring errors. Instead of writing 2(x2-81), they wrote 2 (x2-162). They were able to factor out 2 from the first expression but not in the 2nd expression. Others also left the items as (2(x2-162))/(x- 9). This means that the students failed to recognize the common factors in the numerator which later on leads to the cancellation of the expressions both for the numerator and denominator. This was due to insufficient mastery in factoring. Lastly, 15 errors were along encoding errors. This means that the students were able to correctly carry out the solution process but failed to write the final answer in an unacceptable form. Students forgot to indicate the unit of measurement, units2. This was due to carelessness and lack of criticality,
  • 86. 70 Moreover, it can also be gleaned from the table that majority of the errors along perfect square trinomial were along encoding. The students failed to indicate the correct unit of measurement of the answer. Instead of writing (2x-5) m, the students wrote simply (2x-5). Others wrote the incorrect unit such as ―m2‖ and ―cm‖. This was due to carelessness and lack of critical thinking. In addition, 88 errors were along reading. This means that the students left the items unanswered. The students had not understood the meaning of the problem which led them to leave the item unanswered. They did not know how to hurdle such applied problem. This was due to poor performance. Additionally, 53 errors were along Mathematising. The students did not write the formula or the working equation of the problem. Some incorrectly wrote the formula. Instead of writing A = s2, they wrote A = 4s. Others had incorrect derivation of the formula for ―s‖. Instead of writing s =√A, they wrote s = A/2. This was caused by poor recall and poor competence. Also, 37 errors were along processing errors. They failed to get the factored form of the PST (4x2-20x+25). They divided the expression by 2 instead of performing factoring. Lastly, 29 errors were along comprehension. Students failed to completely understand what the problem is asking them. They also incorrectly copied the given data. So instead of writing (4x2-20x+25), some wrote (4x2+20x+25) and (4x- 20x+25).
  • 87. 71 Likewise, it is also reflected in the table that 100 errors in factoring general quadratic trinomial were along encoding. They were able to correctly get the answer (x+5) but failed to write the correct unit of measurement, cm. This was due to lack of reflection among the students. In addition, 95 errors were along reading. Students never wrote something that leads to the solution of the problem. This implies that the students did not know how to deal with the problem. Further, 57 errors were along Mathematising. Students failed to correctly write the working equation. Some did not write any formula while the others wrote an incorrect one. The students wrote (x2+3x-40) - (x-8) instead of (x2+3x-40)/ (x-8). This was in spite of the presence of the word ―divide‖ in the problem. This was due to poor competence and analytical thinking. Also, 34 errors were along processing. Students failed to correctly factor (x2+3x-40) leaving it unfactored and unsimplified with the denominator. Students also incorrectly cancelled x2 with x and 40 with 8 in their equation, (x2+3x-40)/ (x-8). This was invalid cancellation. This implies that the students really did not know how to factor trinomials of this form. This was due to poor competence and mastery. Lastly, 12 errors were along comprehension. Students failed to correctly write the two given data correctly. Instead of writing (x2+3x-40) and (x-8), students wrote (x2+3x-40) and (x+8) or (x2-3x-40) and (x-8). This was due to carelessness. Others wrote the number ―2‖ as an
  • 88. 72 important detail in the problem solution besides the fact that it only details the equal measurements of the string when divided into two; writing a given as (x2+3x-40)/2. Additionally, the table also shows that 148 errors in factoring by grouping were along reading errors. The students did not understand what the problem is asking them to do. Others really did not know the answer. Students even ignored a problem when prompted with series of algebraic expressions such as x2+2xy+y2+x+y. Pamani (2006) stressed that students with high anxiety and poor mathematical performance often ignore expressions which were lengthy and contain complex expressions and exponents. The errors were caused by high anxiety and poor exposure to such kind of problem. Also, 91 errors were along Mathematising. Students were not able to write any working equation to solve the problem. Others performed subtraction instead of division despite the implication of ―2 equal shares‖ in the problem; the working equations used were ―x2+2xy+y2+x+y – x+ y‖ and‖ x2+2xy+y2+x+y-xy‖. This was caused by poor understanding and mastery. Likewise, 68 errors were along processing. Students failed to factor correctly and completely x2+2xy+y2+x+y. Others invalidly cancelled ―x+y‖ in (x2+2xy+y2+x+y) with (x+y), resulting in an incorrect answer x2+2xy+y2. This was a clear reflection of misuse of cancellation rules. Others also
  • 89. 73 wrote the correct common factor (x+y) but failed to correctly factor the remaining expressions. This was caused by poor mastery of factoring by grouping. Further, 35 errors were along encoding. Students were able to correctly factor the given expressions but failed to write the correct unit of measurement. Lastly, 14 errors were along comprehension. Students did not completely and accurately analyze what the problem wanted them to do. Students incompletely wrote the given while the others wrote additional unnecessary data such as ―2‖ resulting in a data (x2+2xy+y2+x+y)/2. The findings of the study corroborate with the study of Egodawatte (2011) divulging that most students committed transformation and processing errors along word problems involving algebraic expressions, factoring and special products. He explained that the students failed to remember and apply perfectly the special product and factoring patterns. He mentioned that students generated incorrect factored forms of x2+x, which were x(x+x) and worse, x(1). He stated that the students ―oversimplified‖ the answer. They lacked critical analysis as to when and how to end the factoring process correctly. He also explained that ―overdoing‖ existed as he pointed out to incorrect cancellation of expressions.
  • 90. 74 This also agrees with the study of McIntyre (2005) revealing that his respondents had misconceptions in writing final answers in algebraic expressions and factoring patterns. The answer ―x+y‖ was still reduced to xy. He explained that in factoring patterns and algebraic expressions, students never leave an answer with an addition symbol present; the two variables must be physically conjoined. According to him, students felt that x+y can still be combined through the indicated operation. This error, according to him, was caused by misassociation of arithmetic principles; ―7+3= 10‖ is misassociated to ―x+y = xy‖. Rational Expressions Table 14 shows the error categories of students in rational expressions. It can be gleaned from the table that 165.33 or 44.21% of the errors were along reading, 90 or 24.06% were along Mathematising, 41 or 10.96% were along processing, 22 or 5.88% were along encoding and 19.67 or 5.26% were along comprehension. It is also worthy to note that 36 or 9.63% were not considered errors. This means that majority of the students committed reading errors in simplifying rational algebraic expressions. This implies that the students left the item unanswered. They had not understood clearly and comprehensively the problem that hindered them to write even a single data from the problem. This was caused by the lack of exposure to such kinds of problems. According to
  • 91. 75 Table 14. Error Categories in Rational Expressions Subtopic Error Categories R C M P E N Simplification of RAEs 105 18 51 84 46 70 Operations of RAEs 138 17 161 20 17 21 Simplification of Complex RAEs 253 24 58 19 3 17 Average 165.33 19.67 90 41 22 36 Rate 44.21% 5.26% 24.06% 10.96% 5.88% 9.63% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error some reactions of professors after retrieving the questionnaires, the students failed to recognize the operations or the mechanical procedures when expressions were converted to word problems. Blakelock (2013) agrees with this observation of the professors when she mentioned that when students just learned direct operation, direct cancellation or simplification in the class, students would be hard up dealing with such kind of expressions when written in word problems. It can also be seen from the table that 105 errors in simplification of rational algebraic expressions (RAEs) were along reading. This means that most of the students left the item unanswered. This implies that the
  • 92. 76 students failed to write any data given by the problem. This means that they had not understood what the problem is all about. It also means that the students were not interested to solve problems involving fractions or fractional expressions. Hall (2007) emphasized that most students had difficulty dealing with exponents, fractions and radicals. Most students, who find difficulty with these, often abandon solving such problems. Further, 84 errors were along processing errors. This implies that the students failed to correctly solve the given problems. Most of them performed incorrect cancellation in (12x4y6/7xy) and (21/6x3y5). Others placed the incorrect exponents in the denominator instead of in the numerator such as (2/3xy). Others incorrectly placed the cancelled form of 21/7 as 1/3 instead of 3/1 or 3. Additionally, 51 errors were along Mathematising errors. Students failed to write down the correct working equation of the problem. Others wrote the incorrect working equation such as (12x4y6/7xy) ÷ (21/6x3y5) or (12x4y6/7xy) - (21/6x3y5) or (12x4y6/7xy) = (21/6x3y5). This is due to poor analytical skills. Further, 46 errors were along encoding. This means that the answers were not written in a correct form. Others did not write the unit of measurement. Others did not simplify 6/1 pesos. Lastly, 18 errors were along comprehension errors. Students failed to fully understand the given in the problems. Others wrote only partial given such as (12x4y6/7xy) alone or (21/6x3y5) alone. Others wrote
  • 93. 77 (12x4y6/7xy) and (21/6x3y5) but without their corresponding units. This was due to carelessness. The table also reflects that 161 errors in operation of RAEs were along Mathematising. The students failed to correctly write the working equation of the problem. It is surprising that even if the students came from different schools with different instructors, the students commonly wrote the equation (1/2x)(8x/2) instead of (5/2x)(80x/2). This means that the students failed to transform verbal expressions to numerical expressions correctly. This was due to poor mathematical skills. Also, 138 errors were along reading. The students failed to write any data from the given problem. This means that the students failed to understand the given problem which impeded them to deal with the problem. Further, the 20 errors were committed along processing. The students failed to correctly perform the mechanical procedures in solving the given problem. Others placed (5/2x) ÷ (80/2x) instead of (5/2x) x (80x/2). Others evaluated the value of 5 in (1/2x) and 10 in (8x/2). This was due to carelessness and poor performance. Additionally, 17 errors were committed along comprehension and encoding errors. This means that they incompletely wrote the data, excluding 5 and 10 pesos as vital in the solution of the problem. This also implied that the students left the final answer without the correct unit. This was due to carelessness.
  • 94. 78 The table also exposes that 235 errors in simplification of complex RAEs were along reading. The students left the items unanswered. This means that they were not interested in solving the problems especially so that the problem involves fractional expressions. They also forgot how to deal with interest problems involving fractional items. This is due to their low performance. Also, 58 errors were along Mathematising. This means that the students failed to write the formula for interest, I = PRT. Others did not write the formula and just multiplied the given. Others wrote the formula I = 1 + PRT and I = PR. This agrees with the number of errors along reading. Additionally, 24 errors were along comprehension. The students failed to correctly indicate all the data in the problem. They had not written correctly (1- 1/3) and wrote only 1/3 instead. Most of them did not indicate a representation for time, which should had been ―x years‖. This is due to insufficient critical analysis. Also, 19 errors were along processing. The students failed to correctly compute the answer to the given problem. Others incorrectly substituted the given to the formula such as P = I/RT as (1/6 x 12,000) =P/[(1- 1t/3)]. This was due to deficient mastery. Lastly, only 3 errors were along encoding. The 3 errors failed to write the expression ―t‖ in the final answer. The students felt that an answer with a variable was still not the accepted final answer. This was due to an incorrect thinking of oversimplification.
  • 95. 79 The finding of the study corroborates with Egodawatte (2011) divulging that most students committed transformation and processing errors along word problems involving algebraic, polynomial and rational expressions. He explained that these errors were committed since the problems were too symbolic and the most challenging part for students was to find the correct method of solution or algorithm and making use of the algorithm to produce a correct answer. He further stressed that students had to choose the correct method from a wide range of possible strategies which include but were not limited to determining common denominators, common factors for cancellation, expansions using the patterns, building up expressions, simplifications and comparisons. Many of the incomplete answers of his students that were observed bear evidence that the students could not select and apply the correct strategy. He also explained that most students committed ―exhaustion errors‖ when dealing with rational expressions and simplifying answers in algebraic equations. Exhaustion errors are errors which were not made at the beginning of the problem where an opportunity for its commission existed. This type of Mathematising error may had existed due to the incomplete concept recall of the students. This error can also be attributed to the misleading background of the students pertaining to the subject at hand. This error can also be caused by misapplication of the algorithm learned.
  • 96. 80 It also agrees with Hall (2007) when he said that deletion and cancellation errors were prevalent among the respondents of the study as regards working on arithmetical equations and expressions, algebraic and rational equations and fractional expressions. He explained that ―overgeneralizing‖ was the main cause of this type of error. He also added that when students solve equations, they commit transposing errors such as forgetting the change in signs of quantities. Linear Equations in One Variable Table 15 reflects the error categories of the students on linear equations in one variable. It shows that 172.33 or 46.07% errors in linear equations in one variable were along reading, 99.33 or 26.56% were along comprehension, 55 or 15.71% were along Mathematising, 14 or 3.74% were along encoding and 10.67 or 2.85% were along processing. It can also be seen that 22.67 or 6.06% had completely answered the items with correct final answer. This implies that majority of the students committed reading errors. This means that the students failed to write any given data from the table; they left the item unanswered in linear equations. This was caused by poor mastery of the subject matter. It also shows that 186 errors in distance problem were along reading errors. This means that the students left the item unanswered.
  • 97. 81 Table 15. Error Categories in Linear Equations in One Variable Subtopic Error Categories R C M P E N Distance Problem 186 89 53 6 17 23 Money Problem 175 100 46 22 4 27 Age Problem 156 109 66 4 21 18 Average 172.33 99.33 55 10.67 14 22.67 Rate 46.07% 26.56% 14.71% 2.85% 3.74% 6.06% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error They failed to write even a single data from the problem. This was due to their lack of interest towards the problem. Blakelock (2013) asserts that students’ interest in math is high when they were still toddlers, but when they get older, this interest lowers down due to their experiences. This is the reason why most college students do not bother solving problems, especially so when such do not relate to their future profession. Further, 89 errors were along comprehension. This means that the students failed to fully understand the thrust of the problem. Most of them incompletely wrote the given data. Most of them did not present the data in a more comprehensible format, such as using a table. This was caused by poor skills. In addition, 53 errors were along Mathematising. This means that the students were able to present the data but failed to write the correct
  • 98. 82 formula, D = RT. Others wrote Vf2= Vo2 + 2 fusing Physics and College Algebra. Others wrote 440-220= 220 as their working equation. This was due to poor recall and deficient mastery of the subject matter. Also, 17 errors were along encoding. The students failed to write the correct unit of the final answer. They wrote 240 and 200 as their final answers. This was due to insufficient criticality and carelessness. Lastly, 6 errors were along processing. These were committed because of the incompleteness of the answers. The students failed to substitute the value of x, which was 2, to the data presentation for the covered distance. This was due to lack of criticality among the students. Moreover, 175 errors in money problem were committed along reading. This implies that most students left the item unanswered. This means that the students did not know what to do. This also means that the students failed to fully understand what the problem is talking about. Also, 100 errors were along comprehension errors. They failed to fully understand the problem; only partially indicating what the problem is giving. They also failed to understand that the problem data need to be presented in a more organized way, such as using a table or column. In addition, 46 errors were along mathematising. This implies that the students failed to correctly write the formula or working equation needed to solve the problem correctly, D x N = A or denomination multiplied to
  • 99. 83 the number of bills is equal to the OVERALL amount. Others added 1 and 20 and 1 and 50; then applied guess and check method for the two numbers. This means that the students cannot transfer their ideas into mathematical expressions. This is due to deficient mastery. Likewise, 22 errors were along processing errors. Students committed errors in multiplying 50 (27-x). Instead of writing 1350 – 50x, others wrote 1350 – x or 135 – 50x or worse, 135 –x. This was due to poor mastery and carelessness. Lastly, four errors were along encoding errors. These 4 errors were along writing the correct unit. Instead of writing km, they wrote kph; others left the answer with no unit. This was due to carelessness and lack of reflective ability to verify if the final answer is in its accepted form. It can also be seen that 156 errors in age problem were along reading. This means that majority of the students left the item unanswered. This implies that they did not know what to do to be able to get the correct answer needed by the problem. This is saddening since high school mathematics had taught them topics on applied problems in linear equations which started in first year, reinforced in the second year, enforced in their 4th year and repeated in their tertiary year. This was due to deficient skills in algebraic expressions and applied problems. Also, 109 errors were along comprehension. The students failed to completely present the data into tables. Others wrote in tables but failed
  • 100. 84 to represent the two time zones involved in the problem, the present and the future. This was due to lack of criticality. Also, 66 errors were along Mathematising errors. The students were unable to write the correct equation x+20+10 = 2(x+10). Others wrote 2x = 30 +10 and 30 + x = 2x. Others stopped when the data were already presented in correct tables. Others tried to solve using trial and error method by trying 2 numbers that fit the given categories. This was caused by poor mathematical skills. Lastly, 4 errors were committed along processing errors. The errors were along multiplication of constants and variables and transposition. Others wrote x+2x = 30+10 instead of x-2x=30-20. Others wrote x+20+10 = 2x +10 instead of x + 20+10 = 2x (20). This was incomplete distribution. This was due to deficient skills in handling algebraic expressions. The findings of the study run parallel to Clement (2002) divulging that most students’ errors on linear equations fall along transformation. He stressed that his respondents had difficulty in translating words to algebraic equations. He also expressed that analytical thinking falls short among his students which led them to an incorrect process. The findings of the study also run parallel to the study of Egodawatte (2011) revealing that in linear equations and systems of linear equations, most of the students got the correct answer; however, some committed transformation and processing errors. Students failed to
  • 101. 85 produce a correctly transformed equation. Students failed to form correct equations. The others failed to use correctly the methods of substitution, elimination and the working backward methods. He explained that students were unable to carry out these methods due to insufficient skills on the procedures. Students failed to use the standard mathematical practices. He also added that the number one problem of his students is on variables. The students misinterpreted the product of two variables. The students were not able to apply the laws of exponents. He explained that the students misjudged the magnitudes of the variables; he pointed out that the students lack the understanding of variables. Also, the findings agree with the study of Allen (2007) stressing that students had trouble solving such items. He stressed that students need to be skilled on fundamental principles pertaining to equalities. It can be deduced that insufficient background causes the predicament. Systems of Linear Equations in Two Variables Table 16 presents the errors of the students along linear equations in two variables. It shows that 119.5 or 31.95% errors in systems of linear equations in two variables were along reading, 95 or 25. 40% were along Mathematising, 72.5 or 19.39% were along comprehension, 14.5 or 3.88% were along encoding and 10 or 2.67% were along processing. It is
  • 102. 86 Table 16. Error Categories in Systems of Linear Equations in Two Variables Subtopic Error Categories R C M P E N Problems on fare/price 160 89 71 5 13 36 Problems on number relation 79 56 119 15 16 89 Average 119.5 72.5 95 10 14.5 62.5 Rate 31.95% 19.39% 25.40% 2.67% 3.88% 16.71% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error also good to note that 62.5 or 16.71% were not considered errors. Further, the errors imply that majority of the students committed reading errors. The students failed to fully understand the problem thereby leaving the item undealt. This further means that the students do not had the know-how in dealing with the given problems. This was due to poor mastery of the expected competencies. This is really saddening since this is not their first time to encounter such systems of linear equations. They were able to deal with these even during their secondary school days. The table also points out that 160 errors in applied fare/price problems were along reading. This means that majority of the students
  • 103. 87 left the items unanswered. The students failed to write even a single data deduced from the given problem. This is caused by insufficient exposure to such problem. It is true that when teachers facilitate the topic on systems of linear equations, majority of them focused on the methods of solving the value of x. Since this is one of the last topics offered in the course syllabus, most teachers fail to teach how such systems were transformed to applied problems due to lack of time. Also, 89 errors were along comprehension. This means that the students failed to completely and correctly understand the problem. They failed to completely present the data into a more fathomable way, using a tabular format. Also, they failed to correctly represent values for x and y. This is due to lack of organization and criticality. Further, 71 errors were along Mathematising. The students failed to correctly write the needed working equations 8x + 10y = 200 and 3x + 10y = 150. Others simply guessed and checked for 2 numbers that can satisfy the given conditions. This clearly pointed out to the fact that the students cannot transfer their ideas into mathematical expressions. Others did not write any working equation. This is due to poor mastery of the subject matter. Moreover, 13 errors were along encoding errors. The students forgot to write the answers in an unacceptable written form. Most of them failed to indicate the unit. The others were able to get the values for x and y, but failed to pinpoint which among the two values answer the
  • 104. 88 question of the problem. Lastly, 5 errors were along processing. Students failed to correctly apply substitution in the solution of the problem. Instead of writing (3(200-10y)/8) + 10y = 150, they wrote ((200-10y)/8) + 10y = 150. Others did not proceed with their solutions when they finished writing their working equation. This means that the students did not know how to deal with the formulated system of linear equations. This was due to carelessness and poor mathematical abilities. It is also reflected in the table that 119 errors in number relation were along Mathematising errors. The students were unable to correctly write the formula or the working equation. Others wrote ―x +x = 100 and x-x = 20‖ as their working equations. Most of the students applied trial and error in solving the correct 2 numbers. This means that the students were unable to correctly transform their ideas into mathematical expressions. They were able to guess and check their answers to the problem but find it hard to create a working solution to be able to get their ―theorized‖ answers. This is not surprising since Ashlock (2006) revealed the same finding in his study that students can jump into the answers without any working solution. They had their solution in their head but cannot write their solutions. Most instructors, even the researcher, often meet students who can give the answers right away but when asked of their solutions, fail to present any.
  • 105. 89 Further, 79 errors were along reading. This means that the students left the item unanswered. This implies that the students did not know what to do. Also, 56 errors were along comprehension. The students failed to fully understand the given problem. They failed to indicate representation of the given problem. This was due to lack of critical and analytical ability. Further, 16 errors were along encoding. The students were able to get the values for x and y but failed to indicate a final sentence to be able to correctly answer the thrust of the problem. Lastly, 15 errors were along processing errors. Students failed to solve the problem using a correct solution path. The students failed to substitute correctly the derived equation to the other equation such as y=100-x to x-y = 20. Others committed transposition errors in transposing y in x +y = 100. This is due to carelessness and low mastery of the subject matter. It is also good to note that 89 answered correctly the given problem. This means that some students correctly and completely answered the given problem. This contributed much on their level of performance, a satisfactory performance. The findings of the study agree with the study of Clement (2002) stressing that most of his students committed transformation and processing errors on systems of equations. He explained that these errors were caused by insufficiency of skill or knowledge pertaining to how
  • 106. 90 certain variables were handled or how certain equation algorithms were processed. He showed, too, that generally, students got the correct answers but failed to simplify the answers in problems that need simplification of answers. Others forgot to correctly indicate the unit for the answers to be accepted. These errors were due to carelessness. He explained that students forget to analyze their final answers. They did not verify their answers by some accepted means. Also, it agrees with Ashlock’s study (2006) divulging that students can even produce the correct answer even if the solution is incorrect. This situation abounds in problems involving numbers and number relations. With this situation at hand, teachers do not only need to correct the final answer but the process on how the answer is derived. He stated further that students commit what he calls as ―overgeneralizing‖. Students ―overgeneralize‖ data by jumping into the conclusions without adequate data at hand. This overgeneralizing error leads them to incorrect approach and answer. This error abounds in vast areas of mathematics especially on number problems, arithmetic and simplification problems. With this, he remarked that the students lacked the needed computational fluency.
  • 107. 91 Exponents and Radicals Table 17 presents the error categories in College Algebra, specifically along exponential and radical expressions. It shows that 297 errors were along reading. This means that the students did not understand the thrust of the problem. They really did not know what to do to be able to answer the problem. They left the item unanswered. This is so since the students had not touched this last topic of the course syllabus. Many schools had festivities on their foundation, intramurals, founder’s day and the like which limited the number of contact days for discussion. This means that the students failed to write the working equation. Others incorrectly wrote the working equation such as √(2x+7) + 3x = 90 despite the fact that the problem indicated the word ―angle bisector‖ and ―equal parts‖. Others combined trigonometric functions in the formula, including Sin x and Tan x. So, their working equation is A = ½ (√(2x+7) – Sin 3x)r2. This clearly pointed out that the students mixed up their concepts on College Algebra and Plane and Spherical Trigonometry. This is misassociation of concepts of two branches of College Mathematics. Further, they also wrote 90 = (√(2x+7)+3x, which pinpoints that the students jumped into the incorrect conclusion that the angle is a right angle even if there is no indication in the problem that the angle measures 90 degrees. Moreover, 29 errors were along comprehension errors. The students failed to
  • 108. 92 Table 17. Error Categories in Exponents and Radicals Subtopic Error Categories R C M P E N Exponential and Radical expressions or equations 297 29 35 8 1 4 Average 297 29 35 8 1 4 Rate 79.41% 7.75% 9.36% 2.14% 0.27% 1.07% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error indicate all the given data in the problem. Others wrote 2x- 7 only. Others wrote √(2x+7)/2and 3x/2 infusing the number 2 in the problem, a clear sign of misinterpretation. Also, 8 errors were along processing. The students just deleted the radical symbol in √(2x+7) = 3x without performing the squaring process. Others squared 3 but not x, resulting in the expression 9x instead of 9x2. Others chose the value -7/9˚ over 1˚ as the correct answer. This was due to carelessness and no criticality. Lastly, only one error is committed along encoding error. One did not indicate the unit for degrees ( ˚ ) in the final answer. This was due to carelessness. The findings agree with the study of Boon (2003) stressing that the high occurrence of errors in exponents and radicals is due to over- generalization. This over-generalization was due to carelessness and insufficient practice. It also appeared that such error existed due to
  • 109. 93 misconceptions that students had actively construed when they use their existing schema to interpret new ideas. He also explained that this error may be brought by deficient mastery of concepts, rules and pre-requisite skills which can be overcome by practice. He specifically stressed that most students misconnect the rule on √4 = 2 to be true to √16 =8 or worse, √6 =3. This was well explained by Allen (2007) when he enumerated some of the errors of the respondents of the study on error analysis in radical expressions and equations. He pointed out that students had incorrect interpretation and representation of radicals, especially on square roots. Students tend to divide the numbers when getting the square of 16. So, instead of 4, the students wrote 8. This was due to misalignment of rules. They applied the rule in √4 = 2 as true to all numbers being extracted. Summary on the Error Categories in College Algebra Table 18 shows the summary on the error categories in College Algebra. It reveals that 146.96 or 39.29% of errors in College Algebra were along reading, 69.38 or 18.55% were along Mathematising, 47.42 or 12.68% were along comprehension, 45.86 or 12.26% were along processing, and 30.29 or 8.10% were along encoding. This means that majority of the students committed reading errors. This means that most
  • 110. 94 Table 18. Summary Table of Error Categories in College Algebra Subtopic Error Categories R C M P E N Elementary Concepts 68 69.50 85 61 64 26.50 Special Product Patterns 78.33 16.67 47.67 151.33 36 44 Factoring 128.25 25.25 78 39 60.5 43 Rational Expressions 165.33 19.67 90 41 22 36 Linear Equation in One Variable 172.33 99.33 55 10.67 14 22.67 Systems of Linear Equation 119.5 72.5 95 10 14.5 62.5 Exponents and Radicals 297 29 35 8 1 4 Average 146.96 47.42 69.38 45.86 30.29 34.10 Rate 39.29% 12.68% 18.55% 12.26% 8.10% 9.12% Rank 1 3 2 4 6 5 Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error of the students failed to critically understand the problem which led them to leave the items unanswered. They never attempted to answer the given items. This is due to their fair performance. It can also be deduced and construed that students hardly can formulate the working equation or remember the formula to solve the given problem.
  • 111. 95 It is also reflected that only 9.12% completed and correctly solved the given problems in College Algebra. This means that majority of the students really committed errors on the different categories. The findings run parallel to Hall (2007) divulging the following errors of his respondents in College Algebra: Computational Constraint. Many students while they understand mathematical concepts are inconsistent at computing mainly because they misread signs or carry out numbers incorrectly or may not write numerals in the correct column; Difficulty in transferring knowledge. Many students experience difficulty in mathematics because of their inability to connect abstract or conceptual aspects of mathematics with reality. Understanding what mathematical symbols represent in the physical world proves to be difficult to most students and this makes it common to find that some students cannot visualize an equilateral triangle; Making Connections. Some students cannot comprehend the relationship between numbers and the quantities they represent and this makes mathematical skills not to be anchored in any meaningful manner, making it harder for them to recall and apply mathematical knowledge in new situations; incomplete understanding of the language of mathematics. Further, for some students, mathematical disability is as a result of problems with the language of mathematics. Such students had difficulty with reading, writing and speaking mathematical terminologies
  • 112. 96 which normally were not used outside the mathematics lesson. They were unable to understand written or verbal mathematical explanations or questions and cannot relate mathematical knowledge to physical world; Difficulty in comprehending the visual and spatial aspects and perceptual difficulties. Many students had the inability to visualize the mathematical concepts. This makes students to memorize mathematical formulae and facts - the difficulty in applying such knowledge in solving unfamiliar mathematical problem. Validated Instructional Intervention Plan in College Algebra Rationale Mathematics has always been regarded as a very essential element in education for it does not only provide higher training for the human mind but it is life, itself. Everyone, whether consciously or not, uses mathematics in his daily life. College Algebra, one branch of Mathematics, deals with elementary topics, special products and factoring, rational expressions, linear equation in one unknown, systems of equations in two unknowns and exponents and radicals. It provides avenues for students to recall important concepts learned in the secondary school. It also provides a good foundation of readiness for students to hurdle the demands of
  • 113. 97 higher mathematics such as Trigonometery, Advanced Algebra, Geometry and the like. The noted dismal performance in this subject is caused by different factors such as negative attitude, misconceptions, misapplications, misalignmnet of rules, lack of criticality among others. With these presents, it is apt to look into the reasons behind these. One good mechanism that can address such dismal performance is an instructional intervention plan. The validated instructional intervention plan is based upon the identified students’ level of performance, their capabilities and constraints and the different error categories in College Algebra. All the error categories are addressed in the plan since all of them were considered constraints; but, more emphasis is given to a course content with a very poor to poor performance level - these were the areas on rational expressions, linear equations, systems of linear equations and radicals and exponents. The instructional plan also gives emphasis on addressing two (2) foremost error categories, reading and mathematising. Further, there are some instructional interventions that address two error categories. Further, the instructional intervention plan details the specific objectives; topics; level of performance, error categories and theoized causes (arranged according to degree of error); error samples;
  • 114. 98 interventions, process, activities; and assessment strategy. The contents serve as a comprehensive guide for teachers to improve performance and check on errors. General Objectives The instructional intervention plan is formulated to: 1. Improve performance in all the topics in College Algebra; and 2. Address the different errors of students in solving problems in College Algebra. Matrix of the Instructional Intervention Plan The Instructional Intervention Plan in College Algebra is detailed in matrix form in the succeeding pages (see pages 99 – 295).
  • 115. 99 Instructional Intervention Plan in College Algebra Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To use Venn Diagrams in representing sets and set relationships To utilize Venn Diagrams in solving applied problems. To present complete and accurate solutions involving A.Elemen- tary Topics Sets and the Venn Diagrams Poor Mathema- tising (dismal performance, insufficient recall) Incorrect working diagram- using tables as solution diagram. ―250-160- 150-180 = x‖ as the working equation. Visual-Spatial Processing This is a skill-based intervention that emphasizes the skill on visualizing the given problem. It makes use of diagrams and illustrations to show to students how a certain problem is translated into an illustration or diagram for easier understanding. It uses direct instruction and the instructor models how problems are illustrated. After, the students are given handouts and sample The instructor can check students’ learning during the solving process of the students. The learning is further assessed when the students explain their answers on the board.
  • 116. 100 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy sets and Venn Diagrams problems for them to demonstrate how such problems are illustrated and solved. Procedures: 1. The instructor presents a given problem and uses the Venn Diagram to illustrate. He has to emphasize why the Venn Diagram is the correct strategy to be used. 2. After the instructor models, he gives each student a handout that contains an empty Venn Diagram where students can write their answers.
  • 117. 101 Specific Objectives Topics Level of Perform- ance Error Catego- ries and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3. The students are still guided by the instructor as he roams around the classroom. The instructor checks students’ answers. 4. Students who are done with their answers and have presented their correct answers can be assigned to students who need assistance. 5. Presentation of problems will be done after. Priority of presentation is given to those who are assisted to check
  • 118. 102 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Think-Pair-Share- Explain Activity This is an interesting cooperative-learning- based activity for students who struggle to come up with correct answers. Procedures: 1. After the instructor finishes discussing the lesson, the instructor pairs the students. The pairing is done strategically pairing the fast and the struggling ones. 2. The students will be given a problem to solve. They will be given a problem to solve. They will help Students’ answers
  • 119. 103 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy each other in reading, analyzing and solving the problem. They are required to discuss the problem until the two are convinced of the solution. 3. The instructor monitors the students’ activity and checks for their answers. The struggling ones will be required to explain the solution to the instructor. Model-Matching Activity Sheets This is an interactive instructional activity that will lead students to match problems to The students explain their answers to the instructor and to the class.
  • 120. 104 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy their accomplished Venn Diagrams or to match Venn Diagrams to their corresponding working equations. Procedures: 1. Instructors provide activity sheets containing a matching type assessment. 2. The students match the items in column A to Column B. 3. After, they will be asked to craft their own solution without the models.
  • 121. 105 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) The students left the items unanswered *Note: If instructors do not want activity sheets, he can write the items to be match on the board and let a matching exist among students. Using this approach, the instructor can even ask the students to explain how the matching of concepts is done. Direct Instruction with Paired reading This is a type of instruction that focuses on the essentials or the specific skill that needs to be targeted. Students’ scores in the activity sheets.
  • 122. 106 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. The instructor focuses on how applied problems on sets and Venn Diagrams are understood or solved. 2. He can use technology in presenting the problem or hand-outs. 3. The instructor pairs the students for reading of the item assigned to them. 4. During the discussion of the item assigned to the students, the instructor asks questions on how students understood the problem. The
  • 123. 107 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Students incompletely indicated all data necessary to the solution of the problem students can switch to the vernacular when not comfortable in using English when explaining. Other students are asked to give comments regarding the understanding of the presenters. Conceptual Processing This standards-based mathematical inter- vention emphasizes the need to build a deeper understanding of concepts. This involves making ideas, facts and skills reflecting upon and refining Students explain their understanding of the problem. Students are asked to give comments.
  • 124. 108 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy one’s own under- standing. It utilizes concept-builder materials such as diagrams and other manipulative. Procedures: 1. After an interactive discussion, the instructor asks students to indicate all data from the problem. 2. After that, the students are asked to explain the meaning of each data, how the data must be sorted or how such data must be treated.
  • 125. 109 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness Incorrect difference of 180- 175 =10 Incorrect placement of data and difference in the Venn Diagrams. 3. The instructor gives redirection or gives clarifying questions if students are mislead. Trio Timed Drill This is a variant of group learning that creates groups of 3 students. Procedures: 1. The instructor assigns a student leader in a group of 2 students, making them a trio. 2. The instructor gives math worksheets that the students will solve. 3. The leader facilitates the solution process of Students explain their understanding of the problem.
  • 126. 110 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Students just left 5 inside the Venn diagram; thus, there is the given problem. The leader is given a copy of the correct answer for him to verify if his answer is correct and to check whether his group mates get the correct answer. * Note, the instructor gives only the copy of the answer if the leader gets the correct answer. 3. The student leader directs and redirects the students under him. Self-Check This is a strategy that directs and redirects students to check their personal work. Students’ answers on the drill exercises. Students’ answers on the
  • 127. 111 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To visualize correctly real numbers Real Number System Fair Encoding (carelessness, lack of criticality) no indication of final answers. Students just indicated 5, instead of 5 people Students just wrote 11 instead of 11 units as the final answer. Procedures: 1. The students are given worksheets with directing questions which include: Is your working solution correct? Is your final answer in its simplified form? Does your final answer address the question of the problem? Does it have a unit? 2. The instructor checks on the students’ answer. Answer-switch-verify This interactive activity asks for students to compare, contrast and give comments to the self-check exercises. Students’ answer to the given problem Seat works
  • 128. 112 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy using the number line. To perform operations involving real numbers. To solve applied problems on real numbers. Incorrect distance. Incorrect Counting answers of their fellow students. Procedures: 1. The instructor asks the students to answer a given problem. 2. The instructor sets the time for all the students to answer the given problem. 3. After the given time, the students exchange solution sheets with each other. The students give or write comments as to the completeness of the final answer, etc. 4. After the comment period, the students address the comments.
  • 129. 113 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness Students failed to indicate the correct formula or the working equation. They wrote ―9 + (-2) = 7‖ and ―10 + 8 = 18‖; thus the formula they used was D = P1 +P2. Others did not indicate any formula. Students left the item unanswered. Gallery Walk This is a post-teaching instructional strategy that assesses how students solve a given problem. Procedures: 1. The instructor divides the class into smaller learning groups. 2. Each group is assigned an item to solve. They are also given manila paper and markers to present their solutions. 3. The students are required to solve the items individually. They are only allowed Student solution presentations
  • 130. 114 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance deficient recall) Students incompletely wrote all the given in the problem to write their answer on the manila paper once everyone has solved the problem at hand. 4. The students are asked to present their answers to the class. The other groups can give reactions to their answers. Formula Match This is a strategy that involves the formula used in solving items. Students’ answers to the activity Recitation
  • 131. 115 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) Number line pinpointing to the distance of 10 and -2, as the largest and smallest coordinates from among the 4 coordinates Procedures: 1. The instructor presents the formulas or the working equation and the different problems. 2. The students are asked to match the needed formula to the respective problems. 3. The students will be asked to explain their choices. 4. The class is free to give comments. Round Robin Reading This is a reading improvement strategy that successively calls on students to read aloud a given problem. Student reads the given problem. Answered hand-outs
  • 132. 116 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Procedures: 1. The teacher gives handouts on different word problems or mathematical expressions. 2. The students will be asked to read on a round-robin basis. 3. After the reading sessions, the students are asked to explain what they read. The students are free to use the vernacular. Comprehension Checker This is self-check strategy that focuses on students’ understanding. Understanding checker sheets
  • 133. 117 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. The instructor gives a work sheet that contains word problems and comprehension question item checklist. 2. The students read the problem and check the item that corresponds to their understanding. 3. The instructor checks the items. If he sees that the students have low scores, he gives direct instruction or assigns him to someone with a perfect score. The instructor again gives another set
  • 134. 118 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To translate verbal expression to numerical expression and vice versa. To perform operations on algebraic expression To simplify all answers Algebraic Expres- sions Poor Comprehen- sion (poor exposure, lack of skills) Students did not understand well the term ―the height is (x+9) cm more than the base. Students just wrote (x+9) instead of (x+9) + (2x-5). of comprehension checker work sheets to the students to check on improved understanding. Explicit Instruction It is a dynamic, structured and systematic methodology for teaching academic skills. It is characterized by learning guides or scaffolds, whereby students are guided throughout the learning process. Procedures: 1. Focus instruction on critical content, com- Seatwork Assignments Student Board work
  • 135. 119 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy prehension, analysis, problem-solving strategies. 2. Break down the content on specific targets. 3. Tell students of what they need to learn before starting instruction. 4. Review prior knowledge and provide learning supports or guides for students to learn the rudiments of the lesson. 5. Break the class into smaller learning groups to check on the extent of attainment of the instructional objectives.
  • 136. 120 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Students left the items unanswered. They do not know what to do. 6. The rote classroom activities can be done to assess learning Systematic Instruction It means breaking down complex skills into smaller, manageable ―chunks‖ of learning and carefully considering how to best teach these discrete pieces to achieve the overall learning goal. Procedures: 1. Sequence learning chunks from easier to more difficult and providing scaffolding, Seatwork Assignments Student Boardwork
  • 137. 121 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy or temporary supports, to control the level of difficulty throughout the learning process. 2. Teachers break down a complex task, like analyzing and solving a math problem, into multiple steps or processes with manageable learning chunks and teach each chunk to mastery before bringing together the entire process. 3. In turn, the students do the same process independently or by pair.
  • 138. 122 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) Students failed to write the formula for the problem. Others wrote A = bh instead of A = ½ bh. Sustained Silent Reading (SSR) SSR is reading instructional strategy that gives students instructional time to read and analyze the problem. Procedures: 1. Students are given problem sets to be read silently. 2. The instructors give the instructional item for them to analyze and read the problem 3. After the SSR period, the teacher asks questions that students will answer. Seatwork Assignments Student Board work
  • 139. 123 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performanced eficient recall) Wrong addition of (2x+5) and (x- 9). Instead of writing (3x-4); others wrote The questions focus on how the students understood the problem, how they can deal with the problem, the strategy and the like. 4. The teacher again gives another problem using the SSR method, but the difference is the students will explain their understanding and method on the board. Quick Write It introduces a concept and connects this concept with prior knowledge or experiences and allows students to Students’ sharing of prior knowledge and responses.
  • 140. 124 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3x +4. discuss and learn from each other Procedures: 1. Introduce a single word, phrase formula to the class. 2. Students copy the concept on index cards. 3. Students are given two minutes to write whatever comes to their minds relative to the concept. They may write freely using single words, phrases, sentences, etc. 4. After time is called, students may volunteer to share their thoughts on the
  • 141. 125 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness Others overdid their solutions. They wrote 2x+5 <0 and 3x<4 Others failed to write the unit of measurement . subject. 5. The teacher gives direction, clarify or affirm the student’s answers Solve and React This allows students to solve whether independently or independently. The students will be asked to comment on the solutions to be presented as regards the procedures of the solution. Procedures: 1. The students will be asked to solve different items. Students’ answers and reactions.
  • 142. 126 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Others performed incorrect simplification of final answers. 2. A student will be asked to present solution on the board. 3. The students who are seated will be asked to comment on the solution procedures as regards their correctness. 4. The students take note of this for future use. Say Something This is a variant of solve and react that asks students to comment on the answers of the students. Students’ group work sheets
  • 143. 127 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. The students will be grouped into several small learning groups (LG). 2. They will be solving specific problem. 3. They will be exchanging and commenting on the answers of the students. 4. The teacher guides the students in the correct examination and scrutiny of the solution. Reflection Sheets Teachers provide reflection sheets that ask the following: Answered Reflection sheets
  • 144. 128 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To perform operations on polynomials To simplify polynomials accurately To solve problems involving polynomials Polyno- mials Fair Mathemati- sing (dismal performance, insufficient recall) Incorrect working equation such as ―(x4-1)- (x+1) No written working equation 1. Is my answer in its acceptable form? 2. Is my final answer simplified correctly? 3. Does my answer contain unit? 4. Does my answer have the correct unit of measurement? Five Word/ Formula Prediction Its purpose is to encourage students to make predictions about text, working equation or solution, to activate prior knowledge, to set purposes for reading, and to introduce new vocabulary Quizzes Seat works Assignment Recitation Group work
  • 145. 129 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, careless-ness) Incorrect factoring ―(x4- 1)‖ such as ―(x3)(x-1)‖ Incorrect cancellation (x4-1)/(x+1) Procedures: 1. Select five key math words/ working equa- tion from a set of problems that students are about to read. 2. List the words in order on the chalk- board. 3. Using Socratic Method, Clarify the meaning of any unfamiliar words. Carousel Brainstorm Purposes: This strategy can fit almost any purpose intended, especially when students find difficulty in understanding Class presenta- tions and reactions to solutions
  • 146. 130 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy problems or presenting solutions to problems. Procedures: 1. Teacher determines what problems will be placed on chart paper. 2. Chart paper is placed on walls around the room. 3. Teacher places students into groups of four. 4. Students begin at a designated chart. 5. They read the prompt, discuss with group, and respond directly on the chart. 6. After an allotted amount of time, students rotate to next
  • 147. 131 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy chart. 7. Students read next prompt and previous recordings, and then record any new discoveries or discussion points. 8. Continue until each group has responded to each prompt. 9. Teacher shares information from charts and conversations heard while responding. 10. Students will be asked to clarify points in the solution of the problem. ** This strategy can be modified by having the chart ―carousel‖ to
  • 148. 132 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Incorrect copying of signs in the final answer; but correct solution No unit. groups, rather than groups moving to chart. Say Something This encourages students to react on one’s work and then eventually to react on other’s work. Procedures: 1. Instructor asks the students to solve different problems. 2. The instructor gives direction and time frame for students to solve. 3. After the specific time, the instructor reminds the students Solution sheets
  • 149. 133 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) Students left the items unanswered. to finalize their answer. 4. After 2 problems, the students can exchange solution sheets and say something about the solution and final answer. GIST (Generating Interactions between Schemata and Text) It directs students’ reflection on the content of the lesson and leads them to summarize the problem and strategies to differentiate between essential and non- essential information. Students’ GIST sheets
  • 150. 134 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: The task is to write a summary of the keywords, the problem-solving strategy in groups. The words, the notes and strategies capture the ―gist‖ of the text. 1. The instructor models how to solve a certain problem. 2. Instructor models the procedures by drawing blanks or columns on the board. 3. Instructor thinks aloud as (s)he begins to facilitate the intervention activity.
  • 151. 135 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incorrect copying of given data as to the signs Incomplete representa- tion of data 4. Students work with a group or partner to complete a GIST for the next chunk of problem. Students will eventually be asked to create independent GISTs. Copy-Solve-Cover It arouses students’ keen observation and comprehension about a certain text or problem. Procedures: 1. Instructor sets the objectives of the class. 2. The instructor demonstrates how certain data are Solution sheets
  • 152. 136 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy organized and how certain mathematical expressions are properly understood. The students just copy. 3. On the succeeding items, the instructor covers the other half of the item, then students will continue. They will also be asked to explain their answers. 4. Gradually, students will do the same task independently.
  • 153. 137 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To get the product of two polynomials To solve problems involving product of two (2) polyno- mials. B. Special Product Patterns Product of two (2) polyno- mials Fair Processing (lack of practice, poor mastery, carelessness Incorrect multiplication of (3x2-5) to (3y+4) and (2x2+45) to (5y+2) Incorrect evaluation in ―(3(10)2-5 = 295)‖, they wrote ―900-5 = 895 Strategic Teaching It is a teaching focused on specific lesson contents. It is done after a diagnostic assessment is done. Procedures: 1. Administer a diagnostic test. For this study, the research tool served as the diagnostic assessment. 2. From the results of the assessment, plan or strategize the teaching based on the results. For this study, the focus is on product of two (2) polynomials. Quizzes Worksheets Recitation Group Presentation
  • 154. 138 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) Item unanswered 3. The teacher teachers using different approaches; then assesses after the instructional time. The Directed Reading-Thinking Activity (DRTA ) The DRTA is a discussion format that focuses on making problems more understandable. It requires students to use their background knowledge, make connections to what they know, make predictions about the text, set their own purpose for reading, Activity Sheets
  • 155. 139 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy use the information in the text and then make evaluative judgments. It can be used with nonfiction and fiction texts. FOCUS: Comprehension Strategies: Prediction, Inference and Setting Reading Purpose Procedures (begin by explaining and modeling): 1. The teacher divides the reading assign- ments into meaningful segments and plans the lesson around these segments.
  • 156. 140 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 2. In the class introduction, the teacher leads the students in thinking about what they already know about the topic. (―What do you know about ...? What connections can you make?) 3. The teacher then has the students preview the reading segment examining the illustrations, headings and other clues to the content. 4. The teacher asks students to make predictions about what they will learn.
  • 157. 141 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 5. Students may write individual predictions, write with a partner or contribute to an oral discussion creating a list of class predictions. 6. Students then read the selection and evaluate their predictions. Were their predictions verified? Were they on the wrong track? What evidence supported the predictions? Contradicted the predictions? 7. Students discuss their predictions and the content of the reading.
  • 158. 142 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Comprehen- sion (poor exposure, lack of skills) Incomplete solution. Correct solution but incorrect generalization 8. The teacher and students discuss how they can use this strategy on their own and how it facilitates understanding and critical thinking. 9. The teacher and students repeat the process with the next reading segment that the teacher has identified. Self-Verification Procedures: 1. The teacher guides the students in the reflection of final answers. Students’ comments and reactions Students’ work solutions
  • 159. 143 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 2. The students rectify their solutions based on the reflection directions. 3. The teacher gives another item to check on understanding Question-Answer Relationship (QAR) FOCUS: Comprehension Strategies: Determining Importance, Questioning and Synthesizing QAR is a strategy that targets the question ―Where is the answer?‖ by having the
  • 160. 144 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy classroom teacher and eventually the students create questions that fit into a four-level thinking guide. The level of questions requires students to use explicit and implicit information in the text: • First level: ―Right There!‖ answers. Answers that are directly answered in the text. • Second level: ―Think and Search.‖ This requires putting together information from the text and making an inference.
  • 161. 145 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy • Third level: ―You and the Author.‖ The answer might be found in the student’s background know- ledge, but would not make sense unless the student had read the text. • Fourth level: ―On Your Own.‖ Poses a question for which the answer must come from the student’s own background knowledge Procedure (begin by explaining and modeling): 1. The teacher makes up a series of QAR questions related to QAR Chart
  • 162. 146 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy the materials to known to the students and a series of QAR questions related to the next reading assignment. 2. The teacher introduces QAR and explains that there are two kinds of information in a book explicit and implicit. 3. The teacher explains the levels of questions and where the answers are found and gives examples that are appropriate for the age level and the content. A story like Cinderella that is known by most students usually works
  • 163. 147 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy well as an example, even in high school classes. 4. The teacher then assigns a reading and the QAR questions he/she has developed for the reading. Students read, answer the QAR questions and discuss their answers. 5. The teacher and students discuss how they can use this strategy on their own and how it facilitates understanding and critical thinking. 6. After using the QAR strategy several times, the students can begin to make up their own
  • 164. 148 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy QAR questions and in small groups share with their classmates. 7. The teacher closes this activity with a discussion of how students can use this strategy in their own reading and learning. The ultimate goal of this activity (and most of the activities presented here) is for students to become very proficient in using the activity and eventually use the activity automatically to help themselves comprehend text. QAR Chart
  • 165. 149 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Incomplete encoding of data from the problem No formula or working equation written KWL Chart and Demonstration The know/want-to- know/learned (KWL) chart guides students’ thinking as they begin reading and involves them in each step of the reading process. Students begin by identifying what they already know about the subject of the assigned reading topic, what they want to know about the topic and finally, after they have read the material, what they have learned as a result of reading. The strategy requires students to build on KWL Chart
  • 166. 150 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy past knowledge and is useful in making connections, setting a purpose for reading, and evaluating one’s own learning. FOCUS: Comprehension Strategies: Activating Background Knowledge, Questioning, Determining Importance Procedure (begin by explaining and modeling): 1. The teacher shows a blank KWL chart and explains what each
  • 167. 151 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incomplete encoding of data from the problem No formula or working equation written column requires. 2. The teacher, using a current reading assignment, demonstrates how to complete the columns and creates a class KWL chart. K W L • For the know column: As students brainstorm background knowledge, they should be encouraged to group or categorize the information they know about the topic. This step helps them get prepared to link KWL CHART
  • 168. 152 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy what they know with what they read. • For the want-to- know column: Students form questions about the topic in terms of what they want to know. The teacher decides whether students should preview the reading material before they begin to create questions; it depends on the reading materials and students’ background knowledge. Since the questions prepare the students to find information and set their purpose for
  • 169. 153 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy reading, previewing the material at this point often results in more relevant questions. Students should generate more questions as they read. • For the learned column: This step provides students with opportunities to make direct links among their purpose for reading, the questions they had as they read and the information they found. Here they identify what they have learned. It is a crucial step in helping students identify the
  • 170. 154 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy important information and summarize the important aspects of the text. During this step, students can be reflective about their process and make plans. 3. The teacher on the next reading assignments can ask students individually or in pairs to identify what they already know and then share with the class, create questions for the want to-know column either individually or in pairs and share with class, and finally after reading, complete the
  • 171. 155 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To square a trinomial correctly To solve applied problems using Square of a trinomial Fair Processing (lack of practice, poor mastery, carelessness Incorrect squaring of (2x-4y+6z)2 as (4x2+16y2+36 z2) learned column. 4. The teacher closes this activity with a discussion of how students can use KWL charts in their own reading and learning. 5. The teacher demonstrates the process in formulating working equations and deriving formulas. Error Bull’s-eye It directs students to target specific errors in presented solutions. Procedures: 1. The instructor presents different List of errors culled out from the solution. Students’ presentation of correct answers.
  • 172. 156 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy squaring a trinomial Reading (insufficient recall, deficient mastery, poor exposure) No answer solutions with errors. 2. The students will be given instructional time to study the solutions. 3. The students will be asked to identify the errors. They will be asked to explain why that certain part of the solution is wrong. 4. They will present the correct solution afterwards. Group Reading with Guide Sheets It directs reading comprehension by giving questions that cull out student understanding. Answers in Reading Guide sheets
  • 173. 157 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect formulas as A= 2∏r, and A= 2∏r2 Instead of writing Procedures: 1. The instructor groups the students and gives guide sheets in interpreting the applied problems. 2. The instructor checks the answers in the guide sheets. 3. The teacher gives comments and redirections if necessary. Comparison Matrix FOCUS: Comprehension Strategies: Recognizing Similarities and Differences Answers in the Comparison Matrix
  • 174. 158 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy (4x2+16y2+36 z2–16xy +24xz -48yz)∏ cm2, they wrote 4x2+16y2+36z 2–16xy +24xz -48yz∏ cm2 (parenthetical error) Procedures (begin by explaining and modeling): 1. The teacher writes the subjects/categories/to pics/etc. across the top row of boxes. 2. The teacher writes the attributes/characterist ics/details/etc. down the left column of boxes. 3. Use as few or many of rows and columns as necessary; there should be a specific reason students need to recognize the similarities and differences between the
  • 175. 159 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy provided topics and details. 4. Explain to and model for students what each column/row of the matrix requires. Expressions Given Mathemati- cal expresions Ans- wers Operations Related vocabulary Patterns
  • 176. 160 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Instead of writing (2x-4y + 6z), others even wrote (2x+4y+6z). Response Notes FOCUS: Comprehension Strategies: Questioning, Inferring, Activating Background Knowledge Procedures (begin by explaining and modeling): 1. The teacher introduces the response notes and models how to respond to open-ended questions, share understanding, make connections to background knowledge, share feelings, justify Students’ answers on their response notes
  • 177. 161 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To correctly perform cubing of a binomial. Cube of a binomial Fair Comprehen- sion (poor exposure, lack of skills) Processing (lack of practice, poor mastery, carelessness) Incorrect cubing of (2x+4). Their answers were (8x3+63) or pinions, etc. 2. Students then read and create their own responses in their notebooks or journals. 3. The teacher then asks students to share with the class and/or collects the notes. 4. The teacher and students discuss how they can use this strategy on their own and how it facilitates understanding and critical thinking. Solution Theater This will present different answers of students and will let them select the correct Presented Solution
  • 178. 162 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To apply the correct process of cubing a polynomial in word problems. worse (8x3+12) solution. Procedures: 1. The students will be presented with a problem. They will be asked to present solutions on the board. 2. After, the students, by group, shall be watching or observing (like in theater) all the solutions. 3. After, they will select which is a wrong solution and which is correct. 4. The students will explain the error of the solution and to correct the error they found out,
  • 179. 163 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Reading (insufficient recall, deficient mastery, poor exposure) No unit No answer Self-Verification (Please look at the details of the strategy in the earlier cells) Listening Teams – prior to the lesson, the class is divided into 4 groups/sectors of the class: FOCUS: CULLING OUT UNDERSTANDING OF WORD PROBLEMS Procedures: 1. The teacher classifies students into: * Readers – responsible for reading the applied problem Students’ work sheets Student responses
  • 180. 164 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy * Strategists – responsible for coming up with a solution strategy. * Questioners – responsible for coming up with 2 questions they have about the topic *Agreers – responsible for coming up with 2 points they agree about on the topic *Nay Sayers – 2 points about the lecture that they disagree with *Example Givers – 2 examples that are applicable to the topic. *Listeners – responsible to listen and list down key
  • 181. 165 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect formulas: V= 3s3 and V= 4s ideas. 2. The instructor facilitates the presentation. 3. After some time, students do it alone. Think-Alouds/ Metacognitive Process STRATEGY FOCUS: Comprehension Strategies: Monitoring for Meaning, Predicting, Making Connections Procedures (begin by explaining and modeling): 1. The teacher chooses Students’ answers in the instructional activity.
  • 182. 166 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy applied problems. 2. The teacher reads the text aloud and thinks aloud as he/she reads. 3. Read the text slowly and stop frequently to ―think-aloud‖ — reporting on the use of the targeted strategies — ―Hmmm….‖ can be used to signal the shift to a ―think-aloud‖ from reading. 4. Students underline the words and phrases that helped the teacher use a strategy. 5. The teacher and students list the strategies used.
  • 183. 167 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incorrect copying of the given (2x +4); some wrote (2x-4), (2+4), (x+4) 6. The teacher asks students to identify other situations in which they could use these strategies. 7. The teacher reinforces the process with additional demonstrations and follow-up lessons. COMPREHENSION CHECKER It helps teachers to check whether students have correct comprehension or not. It is a variety of anticipation-reaction guide. Student responses Recitation
  • 184. 168 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1.The instructor provides checklist on the key words of the applied problems. The students check the expressions that correspond to their understanding. The instructor directs the checking and redirects students who are misled. Daily re-looping of previously learned material It is a process of always bringing in previously learned
  • 185. 169 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy material to build on each day so that students have a base knowledge to start with and so that learned structures are constantly reinforced. This is for a topic that uses the same content area: linear equations, systems or rational expressions. Procedures: 1. Before beginning discussion, the teacher elicits prior knowledge on the previous but related topics. 2. The students will be directed to relate the lesson at hand to the
  • 186. 170 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To perform factoring involving difference of 2 perfect squares, To answer applied problems involving factoring of difference of 2 perfect squares. C. Factor- ing Patterns Difference of 2 Perfect Squares Pooor Reading (insufficient recall, deficient mastery, poor exposure) Item left unanswered previous knowledge. 3. Discussion begins after the above-cited processes. Structured Language Experiences It is a well-structured learning activity where students have abundant opportunities to use language to describe their mathematical understanding. It directs Students can verbally explain/describe their math understanding, they can write out their understanding, or Student presentations
  • 187. 171 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy they can draw pictures and then explain. Procedures: 1. Select a math concept/skill for which students have received prior instruction and for which they have demonstrated at least initial acquisition. 2. Develop a structured activity in which students can describe their math understanding. The activity should clearly relate the math concept/skill to the language activity (e.g. students should clearly "see" the relationship
  • 188. 172 Specific Objectives Topics Level of Perform- mance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy student elaborations in a systematic way, ensuring that every student receives feedback regarding their explanations (e.g. for smaller groups, the instructor does this individually; for larger groups, peer tutors evaluate each other’s explanations while instructor monitors tutor pairs). 6. Instructor has opportunity to evaluate at least one explanation/descriptio n for every student. 7. Review activity by modeling an accurate description of the math
  • 189. 173 Specific Objectives Topics Level of Perform- mance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy concept/skill, providing appropriate cueing (e.g. "think alouds," visual, auditory, kinesthetic, tactile modalities). Metacognitive strategy It is a memorable "plan of action" that provides students an easy to follow procedure for solving a particular math problem. It is taught using explicit teaching methods. It includes the student's thinking as well as their physical actions.
  • 190. 174 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 1. Provide ample opportunities for students to practice using the strategy. 2. Provide timely corrective feedback and remodel use of strategy as needed. 3. Provide students with strategy cue sheets (or post the strategy in the classroom) as students begin independently using the strategy. Fade the use of cues as students demonstrate they have memorized the strategy and how (as well as when) to
  • 191. 175 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy use it. (*Some students will benefit from a "strategy notebook" in which they keep both the strategies they have learned and the corresponding math skill they can use each strategy for.) 4. Make a point of reinforcing students for using the strategy appropriately. 5. Implicitly model using the strategy when performing the corresponding math skill in class.
  • 192. 176 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect formula for area of a square: A = s2 (A=2s) Paired Think tank It is variant of partner learning that uses recalling of formulas encountered or taught. Procedures: 1. The instructor asks the students to pair, pairing must be according to degree of mastery. 2. The teacher directs the recall of the formulas in math. 3. The students, in pair, will list down all the recalled formulas. 4. Checking of answers will be done afterwards. Student responses
  • 193. 177 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incorrect copying of data from the given Reading for Meaning Students become curious about printed symbols or mathema- tical expressions once they recognize that print, like talk, conveys meaningful messages that direct, inform or entertain people. One goal of this curriculum is to develop fluent and proficient readers who are knowledgeable about the reading process. Effective read- ing instruction should enable students to eventually become self- directed readers who can:  construct meaning Students answers and responses
  • 194. 178 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect and incomplete factoring of (2x2-162). They factored from various types of print material;  recognize that there are different kinds of reading materials and different purposes for reading;  select strategies appropriate for different reading activities; and,  develop a life-long interest and enjoy- ment in reading a variety of material for different purposes. Independent Study (Using Learning Activity Packages) This is a form of a seat work, using learning Students answers and responses
  • 195. 179 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy activity packages as learning materials. Procedures: 1. Students will be given some work to do, based on prepared learning activity packages or skill book. 2. The students will be asked to check on their answers by comparing with the answer key. 3. The students will be asked to continue solving. The target is for them to solve at least 5 items.
  • 196. 180 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Problem solving instruction: explicit instruction in the steps to solving a mathematical problem including understanding the question, identifying relevant and irrelevant information, choosing a plan to solve the problem, solving it, and checking answers. Procedures: 1. The teacher presents certain problems and how these items are solved with different solution strategies. Answers Sheets
  • 197. 181 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To factor PST. To differentiate a PST from other trinomials To solve applied problems Perfect Square Trinomial Fair Encoding (carelessness, lack of criticality) No unit in the answer. Wrong unit of measurement indicated: cm2, m instead of cm 2. The students chose which among the strategies they should use. 3. They solve individually but can compare answers with their seatmates. They discuss their answers, especially when the items Self-help and self- correcting materials Students practice a math concept/skill using materials that provide them both math concept/skill prompts (e.g. questions, math equations, word Students’ answers Quizzes
  • 198. 182 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy involving PSTs. problems, etc.) and the solutions to each prompt. Procedures: 1. Identify appropriate math skill for student practice. 2. Incorporate materials that include the features listed in Critical Components. 3. Model how to perform the math skill using each self- correcting material. 4. Ensure that students clearly understand how to use the self-correcting material. Be especially sensitive to individual
  • 199. 183 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy students who have difficulty with particular verbal or nonverbal response modes that are required when using each self-correcting material. Be especially sensitive to individual students who have difficulty with particular verbal or nonverbal response modes that are required when using each self-correcting material (e.g. for students who have significant writing problems, then materials that require writing responses may
  • 200. 184 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy produce student frustration and therefore would not be appropriate). 5. Periodically monitor students who are using self- correcting materials, providing them feedback about appropriate or inappropriate use of self-correcting materials. 6. Provide students with a way to record their responses (e.g. a sheet of paper on which they record their responses; have students record responses with dry-
  • 201. 185 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy erase-marker on laminated response cards/sheets that contain each math skill prompt). 7. Evaluate student responses by examining student response sheets. Provide students with corrective feedback regarding their performance as soon as possible. 8. Do not grade student performance using self-correcting materials! Grading performance will detract from the motivation self- correcting materials
  • 202. 186 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy can elicit from students and grading will inhibit student willingness to "take risks," a crucial behavior for learning. Scaffolding Instruction It provides students who have learning problems the crucial learning support they need to move from initial acquisition of a math concept/skill toward independent performance of the math concept/skill. Also referred to as "guided practice."
  • 203. 187 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. Begin scaffolding after you have first directly described and modeled the skill at least three times. 2. Perform the skill or learning task while asking questions aloud and answering them aloud (questions should pertain to specific essential features for specific problem solving steps). Choose one or two places during the problem solving process to question your students.
  • 204. 188 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3. Provide immediate and specific feedback as well as positive reinforcement with each student response. 4. When students answer incorrectly, praise the student for his/her risk-taking and effort while also describing and modeling the correct response. When students answer correctly, always provide positive reinforcement by specifically stating what it is they did correctly.
  • 205. 189 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 5. As your students demonstrate success in responding to one or two questions, then ask for an increased number of student responses with the next example. (Corrective and positive feedback continues as indicated by student responses). 6. When your students demonstrate increased competence, continue to fade your direction, prompting students to complete more and more of the problem solving process. Eventually, you only ask questions and
  • 206. 190 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy your students provide all the answers. 7. As your students demonstrate success in responding to one or two questions, then ask for an increased number of student responses with the next example. (Corrective and positive feedback continues as indicated by student responses). 8. When your students demonstrate increased competence, continue to fade your direction, prompting students to complete more and more of the problem solving
  • 207. 191 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy process. Eventually, you only ask questions and your students provide all the answers. 9. When you are confident that your students understand the problem-solving process, invite them to actively problem-solve with you (students direct problem-solving students ask question, then both students and you respond). 10. Let student accuracy of responses and student nonverbal behavior guide your decisions about when to continue fading your
  • 208. 192 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) No indication of solution direction. Brain Storming (Formulas) This is done by using learning circles. Procedures: 1. Students will be given different applied problems. 2. The task of the students is to give the corresponding working equations or formulas that are needed for the problems to be solved. 3. Presentation and critiquing of students’ answers will follow. Student responses Recitation
  • 209. 193 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect Formulas: A = 4s S = A/2 Curriculum-Based Probe It directs students to solve 2-3 sheets of problems in a set amount of time assessing the same skill. Instructor counts the number of correctly written digits, finds the median correct digits per minute and then determines whether the student is at frustration, instructional, or mastery level. Students’ responses Recitation
  • 210. 194 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. The instructor assesses the students’ mastery level through the quiz or seat work. 2. Based on the results, the instructor gives students differentiated student exercises based on their mastery level. 3. The instructor can focus on teaching the students under the frustration level. The students in the mastery level can facilitate drill for the instructional level.
  • 211. 195 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect factoring of PST. They simply divided by two. Assigned Questions (as Assignments) Focus: Reading, Comprehension, Content Procedures: 1. Students give assignments to students to read at home. 2. When they enter the class, the instructor asks them to present their work. 3. Other students will be asked to give reactions. 4. Discussion on critical concerns follow. Student Responses
  • 212. 196 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To factor General Quadratic Trinomials To solve applied problems using general quadratic trinomials. General Quadratic Trinomial Fair Comprehen- sion (poor exposure, lack of skills) Encoding (carelessness, lack of criticality) Reading (insufficient recall, deficient mastery, poor exposure) Wrong indication of sign of the copied data No unit of measurement No solution Catching Signs This is a strategy patterned sign mnemonics. Apply Self-Correcting Materials (See procedures above) Adjusted speech: instructor changes speech patterns to increase student comprehension. It includes facing the Student responses Solution Sheets Recitation Students’ Answers Recitation Students’ Answers Student responses Recitation Recitation
  • 213. 197 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy students, paraphrasing often, clearly indicating most important ideas, limiting asides, etc Procedures: (This is simply a variant of language switching) 1. Instructor can ask the students to read. 2. When the instructor directs students to understand the problem, he can switch to the students’ mother tongue to stress the essentials and to clarify vague thoughts, especially on mathematical expressions.
  • 214. 198 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Encoding Incorrect working equation: (x2+3x-40) - (x-8) instead of (x2+3x-40) / (x-8) No unit Puzzle Game This is a variant of instructional game, or another form of interactive worksheets. Procedures: 1. Instructor gives an activity sheet that has formulas and empty cells. 2. They match the formula and the letters to guess the magic word. Structured Peer Tutelage It is a well planned/ structured practice activities where students problem Solution sheets Activity sheets Students’ answers on the activity sheets
  • 215. 199 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy solve in pairs Procedures: 1. Determine goals for each peer tutelage activity. 2. Target specific math skills to be practiced. 3. Select appropriate materials that match learning objectives and that can be implemented within a peer tutoring format (i.e. provide both a prompt sheet that contains problems to be solved and an answer key that can be easily used by your students).
  • 216. 200 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 4. Design and teach procedures/behaviors for tutoring. 5. Review classroom rules and teach new rules when appropriate. 6. Pair students of varying achievement levels. 7. Practice peer- tutoring procedures before implementing them with academic tasks. 8. Divide peer- tutoring time into halves so each player has equal time as coach and as player. 9. Signal students when it is time to
  • 217. 201 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect cancellation in (x2+3x-40)/ (x-8;x2 an x were immediately cancelled. switch roles. 10. Set goals for tutoring pairs and provide positive reinforcement for tutoring pairs that meet goals. 11. Provide response record sheets so you can evaluate the performance of individual students. Authentic Contexts The purpose of Teaching Math Concepts/Skills within Authentic Contexts is to explicitly connect the target math Students’ answers on the activity sheets
  • 218. 202 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Adding of superfluous data in the given (x2+3x- 40)/2 instead of concept/skill to a relevant and meaningful context, promoting a deeper level of understanding for students Procedures: 1. Instructor chooses appropriate context within which to teach target math concept/skill. Refer to the assessment strategy Dynamic Assessment, for information about how to collect information about students' interests and to use this information Students’ answers
  • 219. 203 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy simply (x2+3x-40) to create authentic contexts for assessment and teaching Mathematics Student Interest Inventory 2. Instructor activates student prior knowledge of authentic context, identifies the math concept/skill students will learn, and explicitly relates the target math concept/skill to the meaningful context. 3. Instructor explicitly models math concept/skill within authentic context.
  • 220. 204 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 4. Instructor involves students by prompting student thinking about how the math concept/skill is relevant to the authentic context. 5. Instructor checks for student understanding. 6. Students receive opportunities to apply math concept or perform math skill within authentic context. Instructor monitors, provides specific corrective feedback, remodels math concept/skill as needed, and provides positive reinforcement.
  • 221. 205 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 7. Instructor provides review and closure, explicitly re-stating how the target math skill relates to the authentic context and remodeling the skill. 8. Students receive multiple opportunities to apply math concept or practice math skill after initial instruc- tional activity. Incor- porating the instructor instruction strategies, Building Meaningful Student Connections, Explicit Instructor Modeling, & Scaffold- ing Instruction when teaching within
  • 222. 206 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To apply the correct procedure in factoring by grouping To use the correct procedure in answering word problems. Factoring by grouping Poor Reading (insufficient recall, deficient mastery, poor exposure) Item left unanswered authentic contexts can be very effective. Focusing on "Big Ideas" or the essentials It facilitates student understanding by concentrating student attention on key concepts and procedures. The linkages and connections between math concepts are made explicit by linking previously learned big ideas to new concepts and problem solving situations. By emphasizing the big ideas in each lesson, instructors can build students' acquisition Student responses
  • 223. 207 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy and use of key conceptual knowledge across lesson content. Procedures: 1. Choose math big ideas that are foundational to the lesson and that represent understandings that can be applied across lessons (e.g. formula, mathematical expressions). 2. Explicitly teach the math big idea, linking it to previously learned information. 3. Explicitly teach the target math skill within the context of
  • 224. 208 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 4. The math big idea. 5. Provide multiple practice opportunities for students using the Big Idea with the new math skill you taught. 6. Apply the math big idea to the target math skill using a variety of problem solving situations. 7. Pair a visual cue with each math big idea (e.g. a picture of an array for the Big Idea of "area"). 8. Post the visual cue along with one sentence describing why the big idea is important.
  • 225. 209 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect working equation (x2+2xy+y2+x+ y) - (x+y) instead of (x2+2xy+y2+x+ y) / (x+y); Structured Cooperative Learning Groups Students practice math concepts/skills they have previously required with peers in teams or small groups. Procedures: 1. Determine goals for each cooperative learning activity. 2. Target specific academic skills to be learned/practiced. 3. Select appropriate materials that match learning objectives. 4. Design and teach procedures/behaviors for team members to Recitation
  • 226. 210 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy help each other. 5. Review classroom rules and teach new rules when appropriate. 6. Assign students of varying achievement levels to the same team. 7. Practice cooperative group procedures before implementing them with academic tasks. 8. Set team goals and provide positive reinforcement for teams that meet goals. 9. Evaluate success of cooperative learning activity.
  • 227. 211 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Invalid cancellation in (x2+2xy+y2+x+ y) / (x+y); the expression ―x+y‖ was immediately cancelled. Think-Pair-Share Think-Pair-Share is a strategy designed to provide students with "food for thought" on a given topics enabling them to formulate individual ideas and share these ideas with another student. It is a learning strategy developed by Lyman and associates to encourage student classroom participation. Rather than using a basic recitation method in which a instructor poses a question and one student offers a response, Think-Pair- Students’ answers
  • 228. 212 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Share encourages a high degree of pupil response and can help keep students on task. Procedures:  With students seated in teams of 4, have them number them from 1 to 4.  Announce a discussion topic or problem to solve. (Example: Which room in our school is larger, the cafeteria or the gymnasium? How could we find out the answer?)  Give students at least 10 seconds of think time to THINK
  • 229. 213 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy  of their own answer. (Research shows that the quality of student responses goes up significantly when you allow "think time.")  Using student numbers, announce discussion partners. (Example: For this discussion, Student #1 and #2 will be partners. At the same time, Student #3 and #4 will talk over their ideas.)  Ask students to PAIR with their partner to discuss the topic or solution.
  • 230. 214 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy  Finally, randomly call on a few students to SHARE their ideas with the class. Instructors may also ask students to write or diagram their responses while doing the Think-Pair-Share activity. Think, Pair, Share helps students develop conceptual understanding of a topic, develop the ability to filter information and draw conclusions, and develop the ability to consider other points of view.
  • 231. 215 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No indication of unit. Structured Controversy Using structured controversy in the classroom can take many forms. In its most typical form, you select a specific problem. The closer the problem is to multiple issues central to the course the better. This strategy involves providing students with a limited amount of background information and asking them to construct an argument based on this information. This they do by working in groups. Solution sheets
  • 232. 216 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures:  Choose a discussion topic that has at least two well documented positions.  Prepare materials: o Clear expectations for the group task. o Define the positions to be advocated with a summary of the key arguments supporting the positions. o Provide reference materials including a bibliography that support and elaborate the
  • 233. 217 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy arguments for the positions to be advocated.  Structure the controversy: o Assign students to groups of four. o Divide each group into dyads who are assigned opposing positions on the topic. o Require each group to reach consensus on the issue and turn in a group report on which all members will be evaluated.
  • 234. 218 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incorrect data. Adding unnecessary data. Instead of (x2+2xy+y2+x+ y) only, they wrote (x2+2xy+y2+x+ y) / 2  Conduct the controversy: o Plan positions. o Present positions. o Argue the issue. o Reverse positions and argue the issue from those perspectives. o Reach a decision. Explicit Instructor Modeling The purpose of explicit instructor modeling is to provide students with a clear, multi- sensory model of a skill or concept. The instructor is the person best equipped Students’ responses
  • 235. 219 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy to provide such a model. Procedures: 1. Ensure that your students have the prerequisite skills to perform the skill. 2. Break down the skill into logical and learnable parts (Ask yourself, "what do I do and what do I think as I perform the skill?"). 3. Provide a meaningful context for the skill (e.g. word or story problem suited to the age & interests of your students).
  • 236. 220 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 4. Provide visual, auditory, kinesthetic (movement), and tactile means for illustrating important aspects of the concept/skill (e.g. visually display word problem and equation, orally cue students by varying vocal intonations, point, circle, highlight computation signs or important information in story problems). 5. "Think aloud" as you perform each step of the skill
  • 237. 221 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy (i.e. say aloud what you are thinking as you problem-solve). 6. Link each step of the problem solving process (e.g. restate what you did in the previous step, what you are going to do in the next step, and why the next step is important to the previous step). 7. Periodically check student understanding with questions, remodeling steps when there is confusion.
  • 238. 222 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To simplify correctly RAEs To apply simplificatio n of RAEs in D. Ration- al Expres- sions Simplifi- cation of RAEs Fair Reading (insufficient recall, deficient mastery, poor exposure) Item unsolved. 8. Maintain a lively pace while being conscious of student information processing difficulties (e.g. need additional time to process questions). 9. Model a concept/skill at least three times before beginning to scaffold your instruction. Assigned Questions (as Seat work) Focus: Reading and Formulas Students’ homework
  • 239. 223 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy applied problems. Procedures: 1. Students give assignments to students to read. They can be grouped and mixed according to degree of ability. 2. When they enter the class, the instructor asks them to present their work. 3. Other students will be asked to give reactions. 4. Discussion on critical concerns follow.
  • 240. 224 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect cancellation in (12x4y6/7xy) and (21/6x3y5). Incorrect placing of simplified form. Instead of 3 in the numerator, it was placed in the denominator. Planned Discovery Activities The purpose of Planned Discovery Activities is to provide students who have learning problems the opportunity to make meaningful connections between two or more math concepts for which they have previously received instruction which they have previously mastered. It is important to remember that this is a student practice strategy and it is not intended for initial instruction. Students’ solutions
  • 241. 225 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. Determine two or more concepts that have a mathematical relationship which students would benefit from understanding. These concepts must have already been taught and must have been already mastered by the students. 2. Develop a well organized/structure d activity that provides students the opportunity to understand the desired mathema-
  • 242. 226 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy tical relationship between the selected math concepts. 3. Provide explicit directions for completing the activity. 4. Develop and provide to students a structured learning sheet or other appropriate prompt that guides students toward the learning objective. 5. Monitor students as they participate in the activity. Circulate the classroom, provide specific corrective
  • 243. 227 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy feedback, model appropriate skills as needed, prompt student thinking, and provide positive reinforcement. 6. At the conclusion of the activity, provide students with the solutions to the structured learning sheet/prompt and explicitly state/ illustrate the desired mathema- tical relationship(s). 7. How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?
  • 244. 228 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect working equation such as (12x4y6/7xy) ÷ (21/6x3y5) or (12x4y6/7xy) - (21/6x3y5) instead of (12x4y6/7xy) x (21/6x3y5) Experiential Learning (focus: solving math problems) Experiential learning is inductive, learner centered, and activity oriented. Personalized reflection about an experience and the formulation of plans to apply learning to other contexts are critical factors in effective experiential learning. The emphasis in experiential learning is on the process of learning and not on the product. Experiential learning can be viewed as a Students’ answers Quizzes Recitations
  • 245. 229 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy cycle consisting of five phases, all of which are necessary:  experiencing (an activity occurs);  sharing or publishing (reactions and observations are shared);  analyzing or processing (patterns and dynamics are determined);  inferring or generalizing (principles are derived); and,  applying (plans are made to use learning in new situations).
  • 246. 230 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No unit. Answers were not simplified:6/1 Procedures: 1. Instructor presents solved problems. 2. Instructor leads the students to analyze the solved problems, to direct the students to analyze solution patterns. 3. Students are given items to solve. They can clarify misconceptions if necessary. Graphic organizers: visual displays to organize information from the problem. They help students to consolidate informa- Organizer sheets
  • 247. 231 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incomplete data tion into meaningful whole and they are used to improve comprehension of stories, organization of writing, and understanding of difficult concepts in word problems. Procedures: 1. Instructor presents problems. 2. Instructor Demonstrates how graphic organizers are used. 3. Understanding of students will be checked based on the teaching techniques of
  • 248. 232 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To perform the basic operations on RAEs To apply operations on RAEs to applied problems Operation on RAEs Poor Mathemati- sing (dismal performance, insufficient recall) Incorrect working equation (1/2x)(8x/2) instead of (5/2x)(80x/2) the teachers. Follow- up questions that will probe into the in-depth understanding of the students must be structured. 4. Students use the organizers independently. Think-Pair-Share Think-Pair-Share is a strategy designed to provide students with "food for thought" on a given topics enabling them to formulate individual ideas and share these ideas with another student. It is a learning strategy Students’ responses
  • 249. 233 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy developed by Lyman and associates to encourage student classroom participation. Rather than using a basic recitation method in which a instructor poses a question and one student offers a response, Think-Pair- Share encourages a high degree of pupil response and can help keep students on task. Procedures:  With students seated in teams of 4, have them number them from 1 to 4.
  • 250. 234 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy  Announce a discussion topic or problem to solve. (Example: Which room in our school is larger, the cafeteria or the gymnasium? How could we find out the answer?)  Give students at least 10 seconds of think time to THINK of their own answer. (Research shows that the quality of student responses goes up significantly when you allow "think time.")  Using student numbers, announce discussion partners.
  • 251. 235 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy  (Example: For this discussion, Student #1 and #2 will be partners. At the same time, Student #3 and #4 will talk over their ideas.)  Ask students to PAIR with their partner to discuss the topic or solution.  Finally, randomly call on a few students to SHARE their ideas with the class. Instructors may also ask students to write or diagram their responses while doing the Think-Pair-Share activity. Think, Pair,
  • 252. 236 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Share helps students develop conceptual understanding of a topic, develop the ability to filter information and draw conclusions, and develop the ability to consider other points of view. Learning Partners: discuss a document, interview each other for reactions to a document or presentation, critique or edit each others’ work, recap a lesson, develop a test question together, compare
  • 253. 237 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) Items unanswered notes, stump your partner. Procedures: 1. Instructor provides problems after the students are paired. 2. The students are asked to recall the correct formulas. 3. Critiquing of answers is done. Continuous Performance Charting The goal of continuous monitoring and charting of student performance is twofold. First, it provides you, the instructor, information about Student responses Recitation
  • 254. 238 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy student progress on discrete, short-term objectives. It enables you to adjust your instruction to review or re-teach concepts or skills immediately, rather than waiting until you've covered several topics to find out that one or more students didn't learn a particular skill or concept. Second, it provides your students with a visual represen- tation of their learning. Students can become more engaged in their learning by charting and graphing their own performance.
  • 255. 239 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. Determine a specific instructional objective (classify objects according to color, size, shape, pattern; add two digit numbers without regrouping, solve story problems with + and - fractions, select the relevant information in a story problem). 2. Design a "curriculum slice" using the C-R-A assessment strategy (see Additional Information for an example of a curriculum slice below.) Choose
  • 256. 240 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3. appropriate items that accurately reflect the target math skill at the appropriate level of understanding (concrete, representational, abstract) and that can be administered in a short time period (perhaps a 1-3 minute timing). Include more items than you think the student can complete within the designated time period so that you get an accurate indication of their optimal performance.
  • 257. 241 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. Determine a specific instructional objective (classify objects according to color, size, shape, pattern; add two digit numbers without regrouping, solve story problems with + and - fractions, select the relevant information in a story problem). 2. Design a "curriculum slice" using the C-R-A assessment strategy (see Additional Infor- mation for an example of a curriculum slice below.) Choose appro- priate items that
  • 258. 242 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3. accurately reflect the target math skill at the appropriate level of understanding (con- crete, representa- tional, abstract) and that can be administered in a short time period (perhaps a 1-3 minute timing). Include more items than you think the student can complete within the designated time period so that you get an accurate indication of their optimal performance. 4. Administer and score the assessment. 5. Have students plot incorrect and correct
  • 259. 243 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy responses on a graph (see Additional Information for an example of a graph/chart). 6. Discuss and draw goal lines on graph. 7. Repeat process. 8. Determine success of your instruction based on the "learning picture" depicted on the student's chart/ graph (see Additional Information for examples of different learning pictures and what they mean). 9. Make appropriate instructional decisions based on the student's learning picture.
  • 260. 244 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Others wrote (5/2x) ÷ (80/2x) instead of (5/2x) x (80x/2) Drill & Practice As an instructional strategy, drill & practice is familiar to all educators. It "promotes the acquisition of knowledge or skill through repetitive practice." It refers to small tasks such as the memorization of spelling or vocabulary words, or the practicing of arithmetic facts and may also be found in more sophisticated learning tasks or physical education games and Student responses Recitation Performance Chart
  • 261. 245 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy sports. Drill-and- practice, like memorization, involves repetition of specific skills, such as addition and subtraction, or spelling. To be meaningful to learners, the skills built through drill-and-practice should become the building blocks for more meaningful learning. Procedures: 1. Students will be given exercises to solve independently, by pair or by small groups. 2. The instructor roams around to check
  • 262. 246 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incomplete data. They missed out 5 and 10 for the denomination . No unit. on students’ answers. 3. The instructor assists students who are experiencing difficulty. 4. Students are asked to present their solutions. Drill & Practice As an instructional strategy, drill & practice is familiar to all educators. It "promotes the acquisition of knowledge or skill through repetitive practice." It refers to small tasks such as the memorization of spelling or vocabulary Student responses Recitation Solution sheets on drills
  • 263. 247 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy words, or the practicing of arithmetic facts and may also be found in more sophisticated learning tasks or physical education games and sports. Drill-and- practice, like memorization, involves repetition of specific skills, such as addition and subtraction, or spelling. To be meaningful to learners, the skills built through drill-and-practice should become the building blocks for more meaningful learning.
  • 264. 248 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) Procedures: 1. Students will be given exercises to solve independently, by pair or by small groups. 2. The instructor roams around to check on students’ answers. 3. The instructor assists students who are experiencing difficulty. 4. Students are asked to present their solutions. Didactic Questions This is variant of partner learning that focuses on students’ understanding. It focuses on how Student’s responses Recitation
  • 265. 249 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To simplify complex RAES. To use simplificatio n of RAEs in solving applied problems. Simplifi- cation of Complex RAEs/ fractions Very Poor Reading (insufficient recall, deficient mastery, poor exposure) Items left unanswered students should understand applied problems. They will focus on how answers are solved and simplified for acceptance. Apply SSR (see mechanics above) Response journal: Students record in a journal what they learned that day or strategies they learned or questions they have. Students can share their ideas in the class, Student responses Recitation Journals
  • 266. 250 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect formulas: I = 1 + PRT and I = PR with partners, and with the instructor. Procedures: 1. Students are given assignments. They are asked to write their questions pertinent their reading assignment. 2. The questions shall be shared and discussed in class. 3. The questions shall serve as the starting point of the instructor. Didactic Questions This is variant of partner learning that focuses on students’ understanding. It Student responses
  • 267. 251 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Time (t) was not represented and indicated. focuses on how students should understand applied problems. They will focus on how answers are solved and simplified for acceptance. Instructional Game The goal of each student practice strategy in this program is to provide students who have learning problems multiple opportunities to respond to a particular learning task. This is certainly true for Instructional Games as well. Recitation
  • 268. 252 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Instructional Games can also make practicing math skills fun due to its game format. Procedures: 1. Determine math skill(s) for which target students have received prior instruction and which they can perform with at least moderate success. 2. Select a student age/interest appro- priate game context in which the target math skill can be performed. 3. Develop game procedures that allow for many math-skill
  • 269. 253 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy practice opportuni- ties (complexity of game procedures should not detract from skill practice). 4. Provide students with necessary materials to play the game. 5. Model the math skill(s) to be practiced at least once in isolation and at least once within the game context before the game is played. 6. Provide explicit directions for playing the game and model game procedures.
  • 270. 254 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 7. State behavioral expectations and model essential game- playing behaviors (e.g. turn-taking, respond- ing appropriately when I am not the winner, etc.) 8. Invite several students to model playing the game before game begins. Provide an opportunity for students to ask questions and to clarify misconceptions. 9. Monitor students as they play the game, providing specific corrective feedback, modeling skills when appropriate, and
  • 271. 255 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect substitution: P = I/RT as (1/6 x 12,000) =P/[(1- 1t/3)]. providing positive reinforcement. Demonstrate enthusiasm for game as students play Provide a way for students to show their work so that you can evaluate their performance after the game is completed. Procedures: Model-lead-test strategy instruction (MLT): 3 stage process for teaching students to independently use learning strategies: 1) instructor models correct use of strategy; 2) instructor leads
  • 272. 256 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No indication of ―t‖ in the final answer. No unit. students to practice correct use; 3) instructor tests’ students’ independent use of it. Once students attain a score of 80% correct on two consecutive tests, instruction on the strategy stops. Apply self-help
  • 273. 257 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To solve applied problems on distance using linear equations. To solve the value of x E. Linear Equations in One Unknown Applied Problems on Distance Poor Reading (insufficient recall, deficient mastery, poor exposure) No answer. Explicit vocabulary building through random recurrent assessments: Using brief assessments to help students build basic subject-specific vocabulary and also gauge student retention of subject- specific vocabulary. Procedures: 1. Instructor asks students to read certain problems. 2. They will be asked to share their ideas and even problems regarding the problem. 3. Discussions follow. Student responses Recitation
  • 274. 258 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incomplete indication of data. Native language support: providing auditory or written content input to students in their native language. Procedures: 1. Using GLCs, students will be redirected to understand the problem. 2. They are allowed to explain using the vernacular. Student developed glossary: Students keep track of key content and concept words and define them Student responses Recitation
  • 275. 259 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) They failed to write the correct formula, D = RT. Others wrote Vf2= Vo2 + 2 fusing Physics and College Algebra. Others wrote 440-220= 220 in a log or series of worksheets that they keep with their text to refer to. Correcting formula mismatch A variant of formula matching Procedures: 1. Students will be presented with columns of problems and formulas. The items are already matched, but incorrectly done. 2. The instructor will then give the task to correct the incorrect matching. Student responses Recitation Quizzes
  • 276. 260 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No or incorrect unit. Unit Matching This will let students match the conditions of the problem to its corresponding unit of measurement. Procedures: 1. Instructor provides activity sheets that contain a column for problem conditions and a column for unit of measurement. 2. After answering, discussion of answers follows. 3. Correction and redirection of misconceptions shall be a follow-up teaching procedure. Student responses Recitation
  • 277. 261 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incomplete solution. After getting the value of x, they did not go back to the tabular repre- sentation. Columnar Battle This is an instructional game that integrates group learning circles (GLCs) Procedures: 1. Students will be grouped into 3-4. 2. Leaders and assistant leaders will be assigned. 3. A representative per group shall be called to solve given items. 4. Students who are seated will also be asked. Monitoring by the instructor and assistant leader must be done. Student responses Recitation
  • 278. 262 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To represent data correctly To interpret problems correctly Applied Problems on Money/ Amount Poor Reading (insufficient recall, deficient mastery, poor exposure) Not answered. 5. This continues until majority of students have gone to the board. 6. Discussion of misconceptions will be done. Timed Reading This is a variant of structured reading strategy. Procedures: 1. The instructor gives students applied problems to solve. 2. After several minutes or the instructional time for reading, the instructor gives guide questions for students to be led Student responses Recitation
  • 279. 263 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Partial indication of data. towards the correct understanding. 3. The students are asked to present their answers highlighting keywords and translations. Student-led discussions This is a variant of cooperative learning groups. Its focus is to hasten reading, comprehension and mathematical skills. Procedures: 1. Using clustered groups, a leader will be assigned to facilitate the analysis and Seatwork
  • 280. 264 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Others added 1 and 20 and 1 and 50 as the working equation. solution of certain problems. 2. The leader will ask his group members to read. The leader, in turn, will give directions as to how the problem should be hurdled. 3. Checking of answers will be done. 4. Critiquing will also be done. Table Completion This is a form of data collection strategy. Completed Table Student responses Recitation
  • 281. 265 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Instead of writing 1350 – 50x, others wrote 1350 – x or 135 – Procedures: 1. The instructor provided supplemental materials on applied problems and tables where headings for all required data are indicated. 2. After silent reading, the students will be asked to fill in the table. 3. The teacher checks and redirects students when necessary. Equation Generator This will ask students to present solutions on the board. Generated equations Quiz
  • 282. 266 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 50x or worse, 135 –x Procedures: 1. After teaching, students will be called to go in front to generate equations or to recall formulas. 2. Students will be asked to write answers on the board. Students who are seated will be asked to verify written formulas. 3. The students will explain whether a generated equation is correct or not. It should be clear when an equation is correct or not. 4. A seat work may follow for assessment.
  • 283. 267 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No unit. Incorrect unit of KPH instead of KM only. Quiz bees This is a form of instructional activity that tests students’ mastery of the subject matter. Procedures: 1. The instructor divides the class into three. The instructor provides flaps for students to answer. 2. The flaps will be raised once the bell is signaled. 3. Checking of answers follow. 4. If an item is not solved by majority of the groups, the teacher stops temporarily the Student responses Recitation Answers
  • 284. 268 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy game and discusses with the class the correct mechanical procedures. Fishbowl of Units This is another form of Q&A technique. Procedures. 1. The teacher provides a bowl/ box where units of measures are indicated. 2. Students will be called to pick a strip paper from the bowl/box. 3. The students are asked to explain when they should use the unit of measure. Students’ explanation
  • 285. 269 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To formulate working equations on age problems To find the value of x To represent data correctly To solve age problems Applied Problems on Age Poor Reading (insufficient recall, deficient mastery, poor exposure) Item unanswered. Reciprocal peer tutoring (RPT) to improve math achievement This is a general strategy in improving performance. Focus: How to deal with problems - 1. Have students pair, choose a team 2. Explain to the students the goal of the activity. 3. Let the fast learners tutor on math problems, and then individually work a sheet of drill problems. Students get points for correct problems and work toward a goal. Student responses Recitation
  • 286. 270 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Send me a problem This is a reading game that leads students to read and solve certain problems. Procedures: 1. 10 students shall be selected to write certain mathematical expressions or problems. 2. The problems or mathematical expressions shall be sent to certain students. 3. The students, in turn, will solve the problem and will give the answer to the students who gave
  • 287. 271 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) No represen- tation for the age of two persons involved. them the question. The student-sender determines if the answer is correct or not. If incorrect, the sender will give guides. Question Generation This lets students to write questions and give their corresponding understanding as regards the given and the requirement of the problem. Procedures: 1. Students are asked to create five types of questions from a reading assignment, Generated Questions
  • 288. 272 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy with each question moving to a higher level of thinking. Place the questions on note cards to be passed and discussed or handed in. 2. Students are then asked to write their opinions regarding the thrusts of the problem: the given and the required. 3. The instructor checks and reinforces topics not understood by majority of the class.
  • 289. 273 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) No written formulas or working equations. Others wrote 30 +x = 2x as the working equation. Concentric Circles – small circle forms inside a larger one, smaller circle discusses while the larger circle listens and then roles are reversed. Procedures: 1. The students shall form 2 concentric circles. 2. On the first round, the inner circle shall compose a problem. The outer circle shall write the corresponding equation or formula. Student responses Recitation
  • 290. 274 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No unit. 3. On the 2nd round, the tasks are reversed. 4. Discussion of answers is a follow-up procedure. Deck of Cards This is an interactive seatwork. Procedures: 1. Students are asked to fold a sheet of paper, creating a card. 2. On the left part, a problem is written. 3. The students will exchange cards. 4. After, they will be solving the given problems. The solutions will be placed Student responses Recitation
  • 291. 275 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To formulate expressions To apply concepts on linear equation in Applied Problems on Fare Poor Processing (lack of practice, poor mastery, carelessness) Reading (insufficient recall, deficient mastery, poor exposure) Incorrect transposition; Others wrote x+2x = 30+10 instead of x- 2x=30-20 Items unanswered. on the right portion of the card. 5. Then, the card shall be given back to the owner for checking. 6. The students will determine if the answers are correct or not. 7. Students who did not get the answer correctly shall be helped by the person who gave the problem. Re-teaching This is a form of instruction where the instructor re-teaches the topic but with a different strategy. Student responses Recitation Seatwork
  • 292. 276 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy word problem Procedures: 1. The teacher, after assessing students’ difficulties, plans for re-teaching. 2. The teacher has to alter the strategies like using math websites or instructional games. 3. The target of the intervention is to refocus the skill and target the difficulties.
  • 293. 277 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) No tabular representa- tions or representatio n of the missing data. Idea Spinner This is an interesting way to elicit students’ knowhow and knowwhy of the subject matter Procedures: 1. instructor creates a spinner marked into 4 quadrants and labeled: Predict, Explain, Summarize and Evaluate. 2. After new material is presented, the instructor spins and asks a student to respond accordingly. 3. Redirection is done when necessary. Student responses Recitation Seat work
  • 294. 278 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) No working equation. Majority applied guess and check. Comprehension Builder This is an instructional strategy that helps students to structure their understanding of the applied problems. Procedures: 1. Table utilization shall be demonstrated to the class – how it is filled up completed. 2. After, the students will be asked to do it independently or by pair. 3. Redirection is done when necessary. Completed tables
  • 295. 279 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No unit. Find the Rule – students are given sets of examples that demonstrate a single rule and are asked to find and state the rule. Procedures: 1. Students will be presented with PowerPoint presentations on algebraic expressions, formulas and working equations per each problem. 2. The instructor models how each equation is derived for the working equation. 3. After ample items, the students are asked Student responses
  • 296. 280 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect substitution; Instead of writing (3(200- 10y)/8) + 10y = 150, they wrote ((200- 10y)/8) + 10y = 150. to do it themselves. 4. Units of measure can also be included in the presentations. Solution Inspector This is an instructional activity that will lead students to be critical about presented problems. Procedures: 1. The instructor presents the objectives of the activity and to elicit from the class the job of an inspector. 2. The instructor presents to the class solutions to applied problems. Inspectors’ reports
  • 297. 281 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To formulate equations involving verbal expressions To solve problems in number relation Applied Problems on Number Relation Fair Mathemati- sing (dismal performance, insufficient recall) Incorrect Working Equation. Others wrote ―x +x = 100 and x-x = 20 as the working equation. 3. As inspectors, they will look into the errors of the solutions. They will explain why that is an error and where did it start. 4. The students are asked to provide the necessary corrections. Jumbled Equations This activity elicits students’ prior knowledge on formulas and working equation. Procedures: 1. Instructor provides a randomly ordered set of equations. Student responses Recitation
  • 298. 282 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Reading (insufficient recall, deficient mastery, poor exposure) No working equation. Others applied trial and error No solution. No representa- tions for x and y. 2. The students will be asked if the equation is correct or not. 3. They will be asked to reorder the jumbled formulas and equations based on the dictates of the word problems. 4. They will be asked to explain their answers. Apply Structured Reading Variable Basket This is a variant of fishbowl method for formula selection but used in improving comprehension. Student responses
  • 299. 283 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) They committed transposition errors in transposing y in x +y = 100 No unit. No indication of final answer in the conclusion. Procedures: 1. The instructor presents the objectives of the instructional activity. 2. In the basket are variables and the expressions where the variables are culled out. 3. They will explain if the representing variables are correctly written or not. They will be asked to explain their answers and correct the representations when necessary. Recitation
  • 300. 284 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Encoding (carelessness, lack of criticality) Problem solving instruction: explicit instruction in the steps to solving a mathematical problem including understanding the question, identifying relevant and irrelevant information, choosing a plan to solve the problem, solving it, and checking answers. Procedures: 1. The teacher presents certain problems and how these items are solved with different solution strategies. Student responses Recitation Solution sheets
  • 301. 285 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To simplify exponents and radicals correctly G. Expo- nents and Radicals Very Poor Reading (insufficient recall, deficient mastery, poor exposure) Item not answered. 2. The students chose which among the strategies should they use. 3. They solve individually but can compare answers with their seatmates. They discuss their answers, especially when their answers are different. 4. Board work can be a good assessment strategy. Direct Instruction with Paired reading This is a type of instruction that focuses on the essentials or the specific skill that needs Student responses Recitation
  • 302. 286 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy To solve exponential and radical equations To solve problems involving radical and exponential expressions and equa- tions. to be targeted. Procedures: 1. The instructor focuses on how applied problems on sets and Venn Diagrams are understood or solved. 2. He can use technology in presenting the problem or hand-outs. 3. The instructor pairs the students for reading of the item assigned to them. 4. During the discussion of the item assigned to the students, the instructor asks
  • 303. 287 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Mathemati- sing (dismal performance, insufficient recall) Incorrect formulas/ working equation: √(2x+7) + 3x = 90 A = ½ (√(2x+7) – Sin questions on how students understood the problem. The students can switch to the vernacular when not comfortable in using English when explaining. Other students are asked to give comments regarding the understanding of the presenters. Graffiti This is an instructional activity that elicits students ideas on a certain problem. Student responses Recitation Board work
  • 304. 288 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy 3x)r2 Procedures: 1. An issue/question/ problem is indicated on flipchart paper and there may be many in the room on tables. 2. As individuals or groups (with different colored markers) the students visit each station and write their opinions/answers/que stions. 3. Sharing of ideas is done. Redirecting and processing of answers will also be done.
  • 305. 289 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Comprehen- sion (poor exposure, lack of skills) Incorrect data representatio n Others wrote √(2x+7)/2 and 3x/2 instead of √(2x+7) and 3x only. Reading Corners This is a variant of reading stations or centers. Procedures: 1. Students are presented with word problems and different options of solving the problem. Theses sets are placed on the corners of the room. 2. Groups or batches of students go to the corners and read the problem. They will also select the best strategy listed in the options column. Student responses Recitation
  • 306. 290 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Processing (lack of practice, poor mastery, carelessness) Incorrect deletion of the radical symbol in √(2x+7) = 3x without squaring. 3. After, the students will be asked to share answers. 4. Discussion of answers will be done Formula Derivation and Analysis This is a form of direct instruction. Procedures: 1. The instructor directly teaches students on how to generate working equation out of the given applied problems. 2. Demonstration and chunking of data will be done. Student work sheets
  • 307. 291 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Encoding (carelessness, lack of criticality) No unit Wrong selection of the value of x. 3. Instructor asks the students the area of the process where they find difficulty. 4. Essentials will be stressed on the area of difficulty. 5. Analysis and derivation of formulas will be done. 6. Pairing can be done so the students can help each other. 7. Activity sheets are good supplemental materials. Using data diagram This is a form of tabular data collection. Completed diagram
  • 308. 292 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. Instead of tables, the teacher demonstrates how to use diagrams in collecting data from the given problem. 2. The teacher explain how the items are translated into workable expressions. 3. The students are asked to do it. Policy Recall This is an interactive strategy that will lead students to critique their own work. Student responses Recitation Student responses
  • 309. 293 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy Procedures: 1. The students will be presented with rules when square root is done, deleting radical symbols, squaring, etc. 2. The students will also be presented with correct and incorrect application of policies. 3. In the correction of policies, students will be asked to give comments on what policy is violated. 4. They can also be paired for supplement or assistance. 5. The students will be given task to perform Recitation Quiz
  • 310. 294 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy operations involving radicals and exponents. Direct Instruction focusing on the essentials This is an integrative approach of teaching students emphasizing on the essential aspects, specifically their point of error. Procedures: 1. Using the results of the assessment, the instructor plans direct instruction focusing on key elements in data
  • 311. 295 Specific Objectives Topics Level of Perform- ance Error Cate- gories and Theorized Causes (arranged according to degree of error) Sample Error Interventions, Process, Activities Assessment Strategy selection, conclusion, simplification and unit utilization. 2. The students will be given the task to solve something. 3. The instructor monitors and checks immediately on students’ errors.
  • 312. Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan I. Rationale The challenges of teaching mathematics to students of the 21st century are not easy. One challenge is the lowering performance of students in Mathematics. But, the most pressing according to Egodawatte (2010) are the causes why students really fail in their performance in mathematics. Educational experts contend that teachers, for them to be effective, must not only know what to teach; they must also know how to teach. Instructional strategies and principles of teaching are very necessary in the life of the teacher. But, how can these instructional strategies or interventions be known and mastered by the instructors, especially if they are not sent to seminars and are attending graduate school, more so with instructors who are not graduates of a teaching course? Indeed, seminar-workshop is necessary. Seminar-workshop is the process of acquiring specific skills to perform a better job. It helps people to become familiar with essential tools and elements necessary for them to be more effective. Through seminar-workshop, people’s behavior towards a task becomes modified. Such modified behavior contributes to the successful attainment of goals and objectives. 296
  • 313. 297 The proposed two-day seminar-workshop is based upon the foremost constraints and error categories of the students in College Algebra. Their foremost error categories are along reading and mathematising. II. General Objectives 1. Improve instructional pedagogical competencies; and, 2. Apply and adopt the different instructional interventions. III. Seminar-Workshop Course Contents Instructional Interventions on the different error categories IV. Methodologies Participative Lectures and demonstration will be the main methodologies of the seminar-workshop. V. Facilitators The facilitators for the proposed seminar-workshop were chosen based on their extent of involvement in the research, qualifications, trainings and seminars attended and organized. Name Position/Extent of Involvement in the Research Qualifications Feljone G. Ragma Instructor Proponent Ed.D
  • 314. 298 Nora A. Oredina Professor Proponent’s Adviser Ed.D Lea L. de Guzman Professor Ed.D Jovencio T. Balino Professor External Evaluator Ed.D. VI. Participants All mathematics instructors in the Higher Education Institutions (HEIs) of La Union VII. Duration Two consecutive Saturdays: April 14 and 21, 2014 (refer to the proposed program of activities) VIII. Logistics Registration fee (P500 per participant) e.g. 50 participants P 25,000.00 Expenses Honoraria for speakers (2,500/speaker) P 10,000.00 Meals/Snacks for the speakers (P250/speaker) P 1,000.00 Meals/Snacks for the speakers (P250/participant) P 12,500.00 Certificates and kits P 1,500.00 IX. Success Indicator The mathematics instructors of the Higher Education Institutions of La Union shall be able to utilize the instructional interventions.
  • 315. 299 SAMPLE FLYER OF THE TWO-DAY SEMINAR-WORKSHOP
  • 316. 300 Level of Validity of the Instructional Intervention Plan Table 19 shows the level of validity of the instructional intervention plan. It shows that the level of validity of the intervention plan is 4.51, interpreted as very high validity. This means that the instructional intervention plan is very highly functional, acceptable, appropriate, timely, implementable and sustainable. It further implies that the plan is a very good material that can address the dismal performance and the different error categories. Table 19. Level of Validity of the Instructional Intervention Plan Criteria Validators Mean A B C D E I. Face 3 5 4 4 4 4.0 II. Content a. Functionality 5 4 5 5 5 4.80 b. Acceptability 4 5 5 4 4 4.40 c. Appropriateness 5 5 5 4 5 4.80 d. Timeliness 3.5 5 5 4 5 4.50 e. Implementability 3.5 5 5 5 4 4.50 f. Sustainability 4 5 5 4 4 4.60 Average 4.0 4.86 4.86 4.29 4.43 4.51
  • 317. 301 CHAPTER IV SUMMARY, CONCLUSIONS AND RECOMMENDATIONS This chapter shows the summary, findings, conclusions and recommendations of the study. Summary The study identified and analyzed the error categories of students in College Algebra in the Higher Education Institutions of La Union as basis for formulating a Validated Instructional Intervention Plan. Specifically, it determined the level of performance of the students in College Algebra along elementary topics, special products patterns, factoring patterns, rational expression, linear equations in one unknown, systems of linear equations in two unknowns and exponents and radicals; the capabilities and constraints of the students in College Algebra; the error categories of the students along reading, comprehension, mathematising, processing and encoding errors; and the validated instructional intervention plan. The study is descriptive with a researcher-made College Algebra test as the instrument of the study. The test was administered to 374 students of the HEIs in the province of La Union for 1st semester of the school year 2013-2014. The data collected were treated using frequency count, mean, rate and the Newmann’s error analysis tool.
  • 318. 302 Findings The researcher found out the following: 1. The students had fair performance in elementary topics, special products and factoring while poor performance in rational expressions, linear equations and systems of linear equations and very poor performance in exponents and radicals. The students had a general performance of poor. 2. The performances of the student in the specified topics were all considered as constraints. 3. Mathematising and comprehension were the major error categories of the students in elementary topics, processing and reading errors in special products, reading and Mathematising in factoring, reading and Mathematising in rational expressions, reading and comprehension in linear equations, and reading and Mathematising in systems of linear equations and exponents and radicals. In general, their major error categories in College Algebra were along reading and Mathematising. 4. The instructional intervention plan is very highly valid. Conclusions Based on the findings of the study, the following are concluded:
  • 319. 303 1. The students cannot competently deal with elementary topics, special product and factoring patterns, rational expressions, linear equations, systems of linear equations and radicals and exponents. 2. The students are deficient in terms of their skills of the topics in College Algebra. 3. Majority of the students cannot start the problem-solving process which leads them not to successfully finish all the stages of problem solving. 4. The instructional intervention plan is a very good material that can address the dismal performance and errors of the students. Recommendations Based on the conclusions of the study, the following are humbly recommended: 1. The schools should adopt the Instructional Intervention Plan and let their mathematics instructors attend the two-day seminar- workshop. 2. The students should exert more effort in understanding the different concepts in their College Algebra course. They should spend more time in dealing with drills and exercises rather than dealing with social media and entertainment.
  • 320. 304 3. The instructional interventions plan should be used not only in the province of La Union but in all schools experiencing the same student error patterns in College Algebra. 4. The mathematics teachers should suit their teaching strategies based on their students’ needs and interests. 5. The English teachers should intensify the development of the students’ skill of reading with comprehension in their classes. 6. A study should be conducted to determine the effectiveness of the instructional intervention plan. 7. A similar study should be conducted in other branches of Mathematics, applied sciences and English, especially in the subjects where percentage of failures is high.
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  • 323. 307 %20Mathematics %20Proficiency%20And%20Algebra%20Assistance%20Class%20For %20Freshman%20College%20Students.pdf Serna, Andrea (2011). Remediation to college algebra: factors affecting persistence and success in underprepared students. Retrieved August 24, 2013 from http://udini. proquest. com/view /remediation-to-college-algebra-pqid:2353236201/ Wells, Richard B. (2006). Mathematics and mathematical axioms. Retrieved August 23, 2013 from http://www.mrc. uidaho. edu/~rwells/Critical%20Philosophy%20and%20Mind/Chapter%20 23.pdf Manitoba Education,2010. (n.d.). Retrieved August 5, 2013, from http://www.edu.gov.mb.ca/k12/specedu/bip/sample.html "New online algebra tutorial." Practical Homeschooling Nov.-Dec. 2004: 8. Infotrac Custom 1000. Web. (retrieved 25 June 2013) http://go.galegroup.com/ps/i.do?id=GALE%7CA139835396&v=2.1 &u=phcicm&it=r&p=GPS&sw=w "New algebra courses from chalk dust. (News Shorts)." Practical Homeschooling Jan.-Feb. 2001: 14. Infotrac Custom 1000. Web. (retrieved 25 June 2013).http:// go.galegroup.com /ps/i.do? id= GALE% 7CA104 031518&v=2.1 &u=phcicm&it=r&p=GPS&sw=w The National Center for Academic Transformation (NCAT). (2009). NCAT Resources. Retrieved January 15, 2009, from http:// www.center .rpi.edu/. http://www.mathsisfun.com/sets/sets-introduction.html (retrieved 31 July 2013) http://www.jamesbrennan.org/algebra/numbers/real_number_system.h tm (retrieved 1 August 2013) http://otec.uregon.edu/learning_theory.htm#situated learning (retrieved December 04, 2013) http://olc.spsd.sk.ca/DE/PD/instr/experi.html (retrieved 17 December 2013)
  • 324. 308 www.rti4success.org/instructiontools (retrieved 11 December 2013) www.doe.virginia.gov/instructional_interventions (retrieved 13 November 2013) www.intensiveintervention.org (retrieved November 11, 2013) www.carlscorner.us.com/intervention/reading (retrieved December 01, 2013) www.carlscorner.us.com/intervention/mathematics (retrieved December 01, 2013) www.gcsnc.com/educational_intervention_strategies (retrieved December 01, 2013) www.readingrockets.org (retrieved November 11, 2013) www.edutopia.org/scaffolding_instruction (retrieved December 01, 2013) www.niu.edu?instructional_games (retrieved December 01, 2013) epitt.coe.uga.edu?index.php/think_pair_share (retrieved December 11, 2013) C. Magazines, Journals and Other Materials Saint Louis College Faculty Research Journal (2009). Vol VI, No.1. City of San Fernando, La Union. Blakelock, Clara (2013). Seminar hand-outs on engaging learners in the mathematics classroom. Saint Louis College, City of San Fernando, La Union. D. Researches Accessed from the Net Allen, D. (2007). ―Misconception analysis in algebra.” Dissertation. Texas A & M University. Retrieved August 12, 2013, from http://www.math.tamu.edu/~snite/MisMath.pdf Ashlock, R. B. (2006). Error patterns in computation. Retrieved August 08, 2013, from Pearson; Merrill Prentice Hall, OHio:
  • 325. 309 http://www.pearsonhighered.com/assets/hip/us/hip_us_pearson highered/samplechapter/0135009103.pdf Carbonel, Loneza (2013). ―Learning styles, study habits and academic performance of college students at Kalinga-Apayao state college, Philippines.” Retrived on January 13, 2014 from https://www.academia.edu/5450640/LEARNING_STYLES_STUDY _HABITS_AND_ACADEMIC_PERFORMANCE_OF_COLLEGE_STUD ENTS_AT_KALINGAPAYAO_STATE_COLLEGE_PHILIPPINES_Lonez a_Gas-ib_Carbonel_INTRODUCTION Clement, J. (2002). Algebra word problem solutions: thought process underlying a common misconception. Retrieved August 08, 2013, from Journal for Research in Mathematics Education: http://links.jstor.org/sici?sici?=00218251%281928201%3A1%216 %3AAWPSTP%3E2.0.CO%3B2-P Denley, T. (2009). ―Assessing the performance of college algebra students using modularity and technology: A two-year case study.” Retrieved on January 12, 2014 from http://www.hawkeslearning .com/Documents/SITE%202010%20Paper.pdf Elis, Jhemson (2013). Diagnostic test in college algebra for freshman non-education students of Westmead International School: Input to proposed remedial activities. Retrieved January 12, 2014 from https://www.academia.edu/5463264/Diagnostic_Test_in_College_ Algebra_for_Freshman_NonEducation_Students_of_Westmead_Inte rnational_School_Input_to_Proposed_Remedial_Activities Egodawatte, G. (2011). “Secondary school students’ misconceptions in algebra.” Dissertation, Ph.D in Curriculum Development . Canada. Retrieved August 1, 2013 from https://tspace.library.utoronto. ca/bitstream/1807/29712/1/EgodawatteArachchigeDon_Gunawa rdena_201106_PhD_thesis.pdf.pdf Gordon, Sheldon (2008). What’s wrong with college algebra? Retrieved January 12, 2014 from http://www.contemporarycolleg ealgebra.org/national_movement/a_course_in_crisis.html Hohmann, U. (2006). Quantitative methods in education research. Retrieved August 4, 2013, from http://www.edu.plymouth. ac.uk/resined/Quantitative/quanthme.htm
  • 326. 310 Howell, K. (2009). Retrieved August 5, 2013, from http://www.wce.wwu.edu/Depts/SPED/Forms/Resources%20and %20Readings/H-CEC%20Prob-solving%204-09%20final%20.pdf Ketterlin-Geller, L. R. (2009). ―Diagnostic assessments in mathematics to support instructional decision making.” Practical Assessment, Research, a peer-reviewed electronic journal . Retrieved July 31, 2013 from http://pareonline.net/pdf/v14n16.pdf Kuiyuan, L. P. (2007). “A study of college readiness in college algebra.” The e-Journal of Mathematical Sciences and Mathematics Education Retrieved July 29, 2013 from http://www.uwf. edu/mathstat/Technical%20Reports/Assestment2%202010-1- 6.pdf Lambitco, B. (2009). ―Determinants of college algebra performance.” Retrieved August 12 2013 from http://ejournals.ph/index.php? journal=hdjskandkjahewkuhri&page=article&op=viewArticle&path %5B%5D=3242 Laura, P (2009). Student preferences of learning college algebra. Retrieved on January 12, 2013 from http://www.aabri.com /manuscripts/09203.pdf Leary, M. R. (2010). Introduction to behavioral research methods. Retrieved August 5, 2013, from http://wps.ablongman.com/ ab_leary_resmethod_4/11/2989/765402.cw/index.html Leongson, J. A. (2001). Assessing the mathematics achievement of college freshmen using Piaget’s logical operations. Bataan, Philippines. Retrived August 11, 2013 from www.cimt.plymouth.ac.uk/journal/limjap.pdf Li, Kuiyan, et al (2007). A study of college readiness for college algebra. Retrieved January 12, 2013 from http://uwf.edu/cutla/ publications/Study_of_College_Readiness_for_College_Algebra.pdf McIntyre, Z. S. (2005). An analysis of variable misconceptions before and after various collegiate level mathematics courses. Retrieved August 2013, 2, from Master's Thesis. University of Maine: http://www.umaine.edu/center/files/2009/12/McIntyre_Thesis.p df
  • 327. 311 Mohammad A. Yazdani, P. (2000). The exclusion of students' dynamic misconceptions in college aLgebra: A paradign of diagnosis and treatment. Retrieved August 07, 2013, from Journal for Mathematcial Science and Mathematics Education: http://msme.us/2006-2-6.pdf Newmann, M. (1977). ―Strategies for diagnosis and remediation.‖ Victorian Institute for Educational Research Bulletin. Sydney: Harcourt, Brace Jovanovich. Retrieved July 5, 2013 from http://www.compasstech.com.au/ARNOLD/PAGES/newman.htm Okello, N. (2010). ―Learning and teaching college algebra at university level: challenges and opportunities.” A Case Study of USIU. Africa. Retrieved July 29, 2013 from www.ajol.info/…p /jolte/ article/download/51999/40634 Peng, A. (2007). ―Teacher knowledge of students’ mathematical misconceptions.” Thesis. M.S. in . Sweden. Retrieved July 23, 2013 from http:// math. coe. uga. edu/tme/Issues/v21n2/4-21.2_ Cheng%20&%20Yee.pdf Silva, Dante M. C. (2006). Factors Associated with Non-Performing Filipino Students in Mathematics. Proceedings of the IMT-GT Regional Conference on Mathematics, Statistics and Applications, Univeristy Sains Malaysia, Penang. Retrieved August 1, 2013 from http://math.usm.my/research/OnlineProc/ED12.pdf Small, Don (2005). College algebra: A course in crisis. Retrieved on January 12, 2013 from http://www.contemporary collegealgebra. org/national _ movement/ a_course_in_crisis.html Weins, A. (2007, July). ―An investigation into careless errors made by 7th grade mathematics students. Master's Thesis. University of Nebraska . Lincoln, NE. Retrieved July 24, 2013 from http://scimath.unl.edu/MIM/files/research/WeinsA.pdf White, A. L. (2007). A Re-evaluation of Newman’s error analysis. Sydney. Retrieved August 1, 2013 from http://www.mav.vic. edu.au/files/ conferences/2009/08White.pdf Wood, C. B. (2003). Working with logarithms: Students' misconceptions and errors. Retrieved August 01, 2013, from
  • 328. 312 http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_2/Final%20C hua%20Wood.pdf E. Theses and Dissertations Espe-Bucsit, M. E. (2009). ―Determinants of math I (college algebra) performance of freshmen computer science of private schools in San Fernando City.” Master’s Thesis. Don Mariano Marcos Memorial State University-Mid-La Union Campus, San Fernando City, La Union. Nisperos-Pamani, M. D. (2006). ―Mathematics I (college algebra) competencies of college algebra freshmen of NCMST.” Master's Thesis. Don Mariano Marcos Memorial State University-Mid-La Union Campus, San Fernando City, La Union. Oredina, Nora A. (2010). ―A validated worktext in college algebra.” Institutional Research. Saint Louis College, City of San Fernando, La Union. Picar, O. (2009). ―Performance in college algebra.” Master’s Thesis. Cavite, Philippines. Subala, F. G. (2006). ―Competence of graduating mathematics majors in teacher-taining institutions in Region I.” Dissertation. Saint Louis College, City of San Fernando, La Union.
  • 329. APPENDIX A Sample Computations  Sample Computations on the:  Reliability of the College Algebra Test of College Algebra Test  Validity of College Algebra Test  List of Suggestions Made by the Validators and the Corresponding Action/s by the Researcher
  • 330. 313 Sample Computation of Reliability of the College Algebra Test For the College Algebra Test Scores:{38,39,29,25,28,38,41,37,33,36,28,9,9,30,34,34,24,29,20,19,41,2 2,27,27,21,22,19,26,27,24} Data from StaText: k=100 k-1=99 𝑥 = 27.87 𝜎2 =69.77 𝐾𝑅21 = 𝑘 𝑘 − 1 1 − 𝑥 𝑘 − 𝑥 𝑘𝜎2 𝐾𝑟21 = 100 99 1 − 27.87(100 − 27.87) 100 (69.77) KR21= 0.71906 KR21 ≈ 0.72;high reliability
  • 331. 314 Sample Computation on the Validity of College Algebra Test Criteria Validators Mean 1 2 3 4 5 6 1. The directions are clear and specific and do not warrant misconceptions among students. 3 5 5 5 3 4 4.17 2. The sentences are free from grammatical errors and other construction lapses. 5 4 4 5 2 3 3.83 3. The test items are clearly and specifically formulated based on student’s level of understanding. 5 4 4 5 4 4 4.33 4. Mathematical expressions and equations are encoded clearly to avoid student misunderstanding. 4 4 4 5 5 3 4.17 5. There are provisions for students to show their solutions. 4 5 5 4 5 4 4.50 6. The test items cover the course content as indicated by the table of specifications. 4 5 5 5 5 4 4.67 7. The test items are written to cull out the specific errors of students in College Algebra. 4 5 5 5 4 3 4.33 8. Generally, the test items represent what they ought to measure. 4 5 5 5 4 4 4.50 Overall 4.13 4.63 4.63 4.88 4.00 3.63 4.32
  • 332. 315 List of Suggestions Made by the Validators and the Corresponding Action/s by the Researcher Suggestions Remarks The 30-item test cannot be accomplished by the students in the specified time frame. Incorporated The 30 items were reduced to 20 items only; however, the researcher saw to it that the scope of the College Algebra test still covered the specified scope of the syllabi. For example, instead of separate items for sets and Venn Diagrams, an item was constructed to deal with these 2 related topics; instead of separate items for addition, subtraction, multiplication and division of polynomials and rational expressions, an item that conglomerates the four basic operations was formulated. An item is solved by a student in at most 3 minutes. Provide more space for the students to show their answers. Incorporated More spaces were provided for the students to clearly and completely show their solutions. Delete the line for working equation and illustration since it will take much of the student’s time; anyways, these will be reflected when they start writing their preliminary steps for the solutions. This will also give the students the freedom of what specific strategy to use in solving the given word problems. Incorporated The provisions for working equations and illustration were deleted.
  • 333. 316 Suggestions Remarks Emphasize on the instructions that the students need to show their complete solutions. Incorporated The instruction on showing the complete solutions and the non- utilization of calculators was made bold and of bigger font size. Check on some lapses on grammar. Incorporated Grammar lapses were checked. Add more spaces between and among numerical coefficients, variables and their exponents for clarity. Incorporated Spaces were provided between and among the numerals, variables and their exponents. Some data need to be more realistic. Incorporated Some data were changed to be more realistic. Instead of a problem on a concert, a problem on fare in a jeep was written to replace the said item. The scoring scheme should be revised, in consultation with the adviser, so that an item will not just be 1 point. The points should be distributed along the different levels specified along the error categories. Incorporated. The scoring scheme was revised. Please see data categorization.
  • 334. APPENDIX B Research Tool  Letter to Students-Respondents to Administer the College Algebra Test  The College Algebra Test  Math I - College Algebra Test (Table of Specifications)
  • 335. 317 SAINT LOUIS COLLEGE City of San Fernando, La Union GRADUATE SCHOOL September 2013 My dearest students, The undersigned is a Doctor of Education Major in Educational Management (Ed.D-EdM) student of Saint Louis College undertaking the study entitled, ―Error Analysis in College Algebra in the HEIs in La Union.‖ It is with this cause that your support is sincerely solicited so that this study can be carried out and may greatly contribute to the improvement of the teaching-learning process. Please lend an hour to answer this word problems set. It may take much of your precious time but your answers to these problems will contribute much to the success of this study. Rest assured that all information obtained herein will be held strictly confidential. Your immediate attention to this request is highly cherished. Thank you so much! Sincerely yours, Mr. Feljone G. Ragma Researcher/ Ed.D. student
  • 336. 318 COLLEGE ALGEBRA TEST Name (optional)_______________________________School:____________________________________ INSTRUCTIONS: Please read, analyze and solve the problems that follow. Please indicate all information being asked in the given problems on the test sheets. PLEASE SHOW ALL SOLUTIONS. NO USING OF CALCULATORS! 1. 250 customers were asked in a survey as to what cell phone brands they like the most. The results reveal that 160 chose Samsung, 150 chose Nokia and 180 chose iPhone, 75 chose Samsung and Nokia, 100 chose Samsung and iPhone, 90 chose Nokia and iPhone. 20 customers choose all the 3 brands. How many love other brands? Given data: Solution: 2. What is the sum of the distance of 7 from -2 and 10 from 8 on the number line? Given data: Solution: 3. The base of a right triangle is expressed as (2x-5) cm and its height is (x+9) cm more than the base, what is its area in cm2? Given: Solution:
  • 337. 319 4. Juan de la Cruz finds out that his money is expressed in (x4-1) pesos. If he wantsto buy (x+1)ice cream, how many ice cream can he buy? Given: Solution: 5. Don Mario is choosing between lots A and B. Lot A is (3x2-5) square meters sold at P (3y+4) per square meter while lot B is (2x2+45) square meters sold at P (5y+2). If x = 10 and y = 2, which is cheaper? Given: Solution: 6. The radius of a circular table is expressed as (2x-4y+6z) cm, what is its area in cm2? Given: Solution: 7. The side of a cube measures (2x+4) cm, what is its volume? Given: Solution:
  • 338. 320 8. The area of a rhombus is (2x2-162) square units. If one of the diagonals measures (x-9), what is the measure of the other diagonal? Given: Solution: 9. The area of a square garden is expressed as (4x2-20x+25) meters2. What is the measure of its side? Given: Solution: 10. A string measuring (x2+3x-40) cm is divided into 2 parts. If one part measures (x+8)cm, what is the measure of the other part? Given: Solution: 11. A truck has (x2+2xy+y2+x+y) loads of stone to be delivered to 2 customers. If the first customer shall be delivered (x+y) loads only, what is the share of the second customer? Given: Solution:
  • 339. 321 12. Aling Maria wishes to buy 12𝑥4 𝑦6 7𝑥𝑦 kilo of tomatoes for 21 6𝑥3 𝑦5 pesos per kilo. How much will she pay? Given: Solution: 13. Agnes has 1 2𝑥 pieces of 5-peso coin and 8𝑥 2 of 10-peso coin. What is the product of the 5-peso and 10- peso amounts? Given: Solution: 14. The interest of an amount invested in a bank at simple interest is 1/6 of 12,000. If the rate is at (1-1/3), how much is the principal investment? Given: Solution: 15. Two vehicles travel at the same time but in opposite directions. Vehicle A runs at 120 kph while vehicle B runs at 100kph. After some time, their distance from each other is calculated to be 440 km. What is the distance traveled by each of the two vehicles? Given: Solution:
  • 340. 322 16. Feljone has 27 bills consisting of 20-peso and 50-peso bills, If he has a total of 990 pesos, how many 20-peso bills does he have? Given: Solution: 17. Lorna is 20 years older than her daughter, Rudylyn. In ten years, she will be twice as old as her daughter, how old is Rudylyn? Given: Solution: 18. The fare for a jeepney was P200 for 8students and 10regular passengers. The fare, on another day, was P150 for 3students and 10regular passengers. How much was the fare for a regular passenger? Given: Solution:
  • 341. 323 19. The sum of 2 numbers is 100 while their difference is 20. What are the two numbers? Given: Solution: 20. An angle bisector divides an angle into 2 equal parts. If one of the equal angles measures ( 2𝑥 + 7)˚ while the other measures (3x)˚, what is the measure of one the smaller angles? Given: Solution:
  • 342. MATH 1 - COLLEGE ALGEBRA TEST TABLE OF SPECIFICATIONS TOPICS TOTAL HOURS KNOWLEDGE COMPREHENSION REMEMBERING UNDERSTANDING ANALYSIS APPLICATION ANALYZING APPLYING SYNTHESIS EVALUATION EVALUATING CREATING ITEM PLACEMENT TOTAL ITEMS PRELIMS 15 7 7 Elementary Topics - Sets and Venn Diagrams - Real Number System - Algebraic Expressions - Polynomials 8 4 1-4 4 Special Products and Patterns - Product of 2 polynomials - Square of a Trinomial - Cube of a Binomial 7 3 5-7 3 MIDTERMS 15 7 7 Factoring - Difference of 2 Perfect Squares - Perfect Square Trinomial - General Trinomial 8 4 8-11 4 324
  • 343. 313 - Factoring by grouping Rational Expressions - Simplification of RAEs - Operation on RAEs - Simplification of Complex RAEs/ fractions 7 3 12-14 3 FINALS 15 6 6 Linear Equations in One Variable Applied Problems on: - Distance - Mixture/Money/Coin - Age 6 3 15-17 3 Systems of Linear Equations Applied Problems on: - Fare/Price - Number Relation 6 2 18-19 2 Exponents and Radicals - Exponential and Radical expressions and equations 3 1 20 1 45 30 20 325
  • 344. APPENDIX C Communications
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  • 355. APPENDIX D Sample of Corrected College Algebra Test
  • 356. 336 Legend: Guide to Checking 5 pts. – No Error 4 pts. - Encoding Error (EE) 3 pts. – Processing Error (PE) 2 pts. – Mathematising (ME) 1 pt. – Comprehension Error (CE) 0 pt. – Reading Error (RE)
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  • 374. 354 CURRICULUM VITAE PERSONAL DATA Name: Feljone Galima Ragma Date of Birth: July 31, 1986 Place of Birth: San Isidro, Candon City, Ilocos Sur Home Address: San Isidro, Candon City, Ilocos Sur e-mail Address: feljrhone@yahoo.com Civil status: single EDUCATIONAL ATTAINMENT Pre-Elementary: UCCP Candon City, Ilocos Sur Graduated 1991 With honors Elementary: Candon South Central School Candon City, Ilocos Sur Graduated 1997 With honors Secondary: Santa Lucia Academy Santa Lucia, Ilocos Sur Graduated 2003 With honors Tertiary: Saint Louis College San Fernando City, La Union Graduated 2007 Bachelor in Secondary Education Cum Laude Major in Mathematics Recognition Award Graduate Studies:Saint Louis College San Fernando City, La Union Graduated 2011
  • 375. 355 Master of Arts in Education Cum Laude Major in Mathematics Best in Research Post-Graduate Studies: Saint Louis College San Fernando City La Union Graduated 2014 Doctor of Education Magna Cum Laude Major in Educational Best in Research Management BOARD EXAMINATION/ CIVIL SERVICE ELIGIBILITY  Licensure Examination for teachers (LET) 2007  P.D. 907 Civil Service Eligible WORK EXPERIENCE, POSITIONS/SPECIAL ASSIGNMENTS School/Institution Position Inclusive Dates Saint Christopher Academy Classroom Teacher 2007-2008 Bangar, La Union Christ the King College Classroom Teacher 2008-2013 San Fernando City, La Union Subject Area Coordi- 2010-2013 nator Saint Louis College Instructor I 2013-2014 City of San Fernando, La Union OTHER WORK-RELATED EXPERIENCES  Adviser and Panelist, Graduate School Researches Saint Louis College City of San Fernando, La Union  External Evaluator, Undergraduate Researches Saint Louis College City of San Fernando, La Union
  • 376. 356  Review Facilitator, Civil Service Exam Dacanay Hall San Fernando City September-October, 2013 TRAININGS/SEMINAR-WORKSHOPS FACILITATED January 2014 Giving Feedback to Improve Student’s Learning and Behavior Christ the King College City of San Fernando, La Union 2013 Seminar on How to Love and Like Mathematics Saint Louis College City of San Fernando, La Union Back to Basics of Test Construction Christ the King College City of San Fernando, La Union June 29, 2012 Understanding by Design and K-12 Christ the King College May 20, 2012 Problem-Solving Techniques in Secondary Mathematics Association of Private Schools City of San Fernando, La Union July, 2010 Seminar-Workshop on Creating Gradebooks through MS EXCEL Christ the King College City of San Fernando, La Union 2009 Seminar-Workshop on Creating Interactive Slides through MS PowerPoint Christ the King College City of San Fernando, La Union 2009
  • 377. 357 CONFERENCES/ SEMINARS PARTICIPATED Engaging Learners in the Mathematics Classroom Saint Louis College September, 2013 Sustainability in the Classroom Saint Louis College September, 2013 Colloquy in Thesis Advising Saint Louis College September, 2013 International Education Conference: How to be an Effective and Successful Teacher SMX Conventional Hall, Pasay City August, 2012 Formative Assessment in the K-12 Curriculum Christ the King College August 3-4, 2012 Mathematics Trainer’s Guild Seminar on Singaporean Math Association of Private Schools July 6-7, 2012 Moving Forward with Backward Design: A Deeper look at UBD Saint Louis University Laboratory Elementary School January, 2011 Understanding and Planning for the 2010 SEC for Mathematics Phoenix Hall, Quezon City November, 2010 Training Program for Mathematics Teachers University of the Cordilleras September, 2010
  • 378. 358 Seminar on Yoga and Relaxation Christ the King College August, 2010 Critical Questions to Elicit Critical Thinking Christ the King College July, 2010 Seminar-Workshop on Homeroom Guidance and Counseling Techniques Christ the King College June, 2010 Utilizing and Interpreting CEM Test Data University of Baguio May, 2010 In-Service Training and Workshop on Curriculum Programs and Teaching Strategies Christ the King College November, 2009 Seminar on Innovations in Teaching and Learning Approaches Christ the King College July, 2009 Understanding and Planning for the SEC Phoenix Hall, Pangasinan September, 2009 PROFESSIONAL ORGANIZATION National Organization for Professional Teachers (NOPTI) PAFTE