ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER
EDUCATION INSTITUTIONS OF LA UNION
A Dissertation
Presented to
the Faculty...
ii
INDORSEMENT
This dissertation entitled, ―ERROR ANALYSIS IN COLLEGE
ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF
LA U...
iii
APPROVAL SHEET
Approved by the Committee on Oral Examination as PASSED with
a grade of 96% on January 11, 2014.
MARIA ...
iv
ACKNOWLEDGMENT
The researcher wishes to express his sincerest gratitude to the
following persons who contributed much i...
v
Mesdames Grace, Lea, Melody, Graziel, Jay Ann, Abegail, Sister
Grace, Mafe, and Sir Roghene, the researcher’s friends, w...
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 h...
vii
ABSTRACT
TITLE : ERROR ANALYSIS IN COLLEGE ALGEBRA IN
THE HIGHER EDUCATION INSTITUTIONS OF
LA UNION
Total Number of Pa...
viii
equations in one unknown; systems of linear equations in two
unknowns; and exponents and radicals; b) the capabilitie...
ix
found to have very high validity. Based on the findings, it was concluded
that the students cannot competently deal wit...
x
TABLE OF CONTENTS
Page
TITLE PAGE………………………………………………………………… i
INDORSEMENT…………………………………………………………… ii
APPROVAL SHEET…………......
xi
Page
Sources of Data………………………………………. 28
Locale and Population of the Study……………... 28
Instrumentation and Data Collecti...
xii
Page
Summary on the Level of Performance
of Students in College Algebra …………. 52
Capabilities and Constraints of Stude...
xiii
Page
Summary………………………………………………. 301
Findings………………………………………………… 302
Conclusions…………………………………………… 302
Recommendations…...
xiv
LIST OF TABLES
Table Page
1 Distribution of Respondents ………………………… 29
2 Level of Performance of Students in Elementary...
xv
15 Error Categories in Linear Equations in One
Variable…………………………………………….
Page
81
16 Error Categories in Systems of Lin...
xvi
LIST OF FIGURES
Figure Page
1 Ragma’s Error Intervention Model…………………………… 13
2 The Research Paradigm ……………………………………….....
1
CHAPTER I
INTRODUCTION
Background of the Study
Education, in its general sense, is a form of learning in which
knowledge...
2
discoveries have been at the forefront of every civilized society and in use
even in the most primitive of cultures. The...
3
It is undeniable that Mathematics expresses itself everywhere, in
almost every facet of life - in nature and in the tech...
4
income and expenditure. Various stores use algebra to predict the
demand of a particular product and subsequently place ...
5
Countries around the world are alarmed by the lowering
performance of their students, especially in College Algebra. In ...
6
of performance among students would not help much in Mathematics
Education; researches need to dig deeper into the reaso...
7
(2012), 40-50% of the students enrolled in College Algebra failed.
According to him, this performance is caused by poor ...
8
when presented with a word problem while many were not able to craft
their own procedures in solving the given problems
...
9
mathematising errors are committed when someone had understood
what the questions wanted him/her to find out but was una...
10
Dewey (1899) and Roger’s (1967) active learning and experiential
learning theories propose that students are able to le...
11
As applied in the study, the instructional interventions are student-
centered so that learning becomes more active.
In...
12
strategy. Moreover, according to Manitoba Education Website (2010), an
instructional intervention plan contains the pur...
13
Figure 1. Ragma’s Error Intervention Model
INSTRUCTIONAL
INTERVENTION
(Game-based,
visual/spatial-based,
motivational i...
14
14
structure the working equation, solve and then finalize the answer/s. In
each of these successive stages, errors can...
15
concept attainment and processing. Mathematising errors caused by
poor mastery and insufficient recall can be addressed...
16
overall meaning of the words; thus, he can only indicate partially what
are the given and what are the unknown in the p...
17
committed by the students when solving word problems can give baseline
data to teachers to help them improve on their m...
18
Patterns
PROCESS OUTPUTINPUT
Validated
Instructional
Intervention Plan
for College
Algebra in the
Higher
Education
Inst...
19
categories of the students along the specified topics in Math 1 or College
Algebra along reading, comprehension, mathem...
20
a.3. Algebraic Expressions
a.4. Polynomials
b. Special Products;
c. Factoring Patterns;
d. Rational Expressions;
e. Lin...
21
Assumptions
The researcher was guided with the following assumptions:
1. The level of performance of the students in Co...
22
The Mathematics department heads. This study will give them
insights about the performance and errors in College Algebr...
23
study. This can also give them an idea on how to structure their own
instructional plan based on their students’ needs ...
24
Venn diagrams. These are diagrams proposed by the
mathematician A. Venn, which are used to show relationships among
set...
25
that are used in this certain topics include graphical, substitution and
elimination methods.
Constraints. These refer ...
26
working equation, but did not know the procedures necessary to carry
out these operations or equation accurately
Transf...
27
CHAPTER II
METHOD AND PROCEDURES
This chapter presents the research design, sources of data, data
analysis, the parts o...
28
Sources of Data
Locale and Population of the Study. The population of this
study was composed of College Algebra studen...
29
Table 1. Distribution of Respondents
Respondent HEIs N n
Institution A 78 5
Institution B 482 31
Institution C 230 15
I...
30
aligned along the synthesis-evaluation/evaluating-creating level under
the Bloom’s Taxonomy. As such, the questions dug...
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 distributio...
32
Tools for Data Analysis
The data gathered, collated and tabulated were subjected for
analysis and interpretation using ...
33
The MS Excel Worksheet and StaText were employed in treating
the data.
Data Categorization
For the scoring/checking of ...
34
Special Products and Patterns/Rational Expressions/Linear
Equations in One Variable
Score Range Level of Performance De...
35
Score Range Level of Performance Descriptive Equiva-
lent Rating
1.00-1.99 Poor Performance (PP) Constraint
0-0.99 Very...
36
For the validity of the College Algebra test and the Instructional
Intervention Plan, the scale below was used:
Points ...
37
interventions. There are still interventions for those considered as
capabilities for sustainability.
Ethical Considera...
38
The research instrument was subjected to validity and reliability.
Their suggestions were incorporated in the instrumen...
39
CHAPTER III
RESULTS AND DISCUSSION
This chapter presents the statistical analysis and interpretation of
gathered data o...
40
Table 2. Level of Performance of Students in
Elementary Topics
Subtopic Mean Score Rate Descriptive
Equivalent
Sets and...
41
the fullest the needed competence in elementary topics in College
Algebra.
Further, the findings of the study conform t...
42
Table 3. Level of Performance of Students in
Special Product Patterns
Subtopic Mean Score Rate Descriptive
Equivalent
P...
43
The findings of the study adhere to the study of Wood (2003)
emphasizing that the students performed fairly in College ...
44
Factoring Patterns
Table 4 illustrates the performance of the students in College
Algebra along factoring patterns. It ...
45
Table 4. Level of Performance of Students in Factoring Patterns
Subtopic Mean Score Rate Descriptive
Equivalent
Differe...
46
Rational Expressions
Table 5 shows the performance of the students in College Algebra
along rational expressions. It sh...
47
Table 5. Level of Performance of Students in
Rational Expressions (RAEs)
Subtopic Mean
Score
Rate Descriptive
Equivalen...
48
Linear Equations in One Variable
Table 6 shows the performance of the students in College Algebra
along linear equation...
49
Table 6. Level of Performance of Students in Linear
Equations in One Variable
Subtopic Mean Score Rate Descriptive
Equi...
50
Systems of Linear Equations in Two Variables
Table 7 shows the performance of the students in College Algebra
along sys...
51
Table 7. Level of Performance of Students in Systems of
Linear Equations in Two Variables
Subtopic Mean Score Rate Desc...
52
Table 8. Level of Performance of Students in
Exponents and Radicals
Subtopic Mean
Score
Rate Descriptive
Equivalent
Exp...
53
Table 9. Summary Table on the Level of Performance of
Students in College Algebra
TOPIC Mean
Score
Rate Descriptive
Equ...
54
word problems where students are able to apply all the necessary
competencies learned to a situation that requires high...
55
Table 10. Capabilities and Constraints of Students
in College Algebra
TOPIC Mean Score Rate Classification
Elementary C...
56
Error Categories in College Algebra
The third problem considered in this study is on the error
categories of the studen...
57
Table 11. Error Categories in Elementary Topics
Subtopic Error Categories
R C M P E N
Sets and Venn
Diagram
94 51 149 3...
58
completely write the needed data. They missed out writing data such as
―20 customers chose all the brands‖. This was ca...
59
10 from 8. Instead of writing ―9- (-2) = 11‖ and ―10 -8 = 2‖, students
wrote ―9- (-2) = 7‖ and ―10 + 8 = 18‖. Others al...
60
incomplete representation of the phrase ―the height is (x+9) cm more
than the base‖. Instead of writing ―(x+9) + (2x-5)...
61
write the unit of measurement. This was due to lack of critical thinking
and carelessness.
Moreover, 89 errors along po...
62
These results agree with the study of White (2007) revealing that
most misconceptions of his respondents along College ...
63
students had difficulty with reading, writing and speaking mathematical
terminologies which normally were not used outs...
64
Table 12. Error Categories in Special Product Patterns
Subtopic Error Categories
R C M P E N
Product of Two
Binomials
6...
65
trinomial. Most of them answered (2x-4y+6z)2 as (4x2+16y2+36z2), worse
(4x2-8xy2+12y2) instead of 4x2+16y2+36z2+16xy+24...
66
The table also shows that 107 errors in cube of a binomial were
along processing errors. The students failed to correct...
67
remember and apply perfectly the special product and factoring patterns.
He further stressed that the students committe...
68
Table 13. Error Categories in Factoring Patterns
Subtopic Error Categories
R C M P E N
Difference of two
Perfect Square...
69
correctly write the correct formula or working equation demanded by the
problem. They failed to write the formula for t...
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
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Error analysis in college algebra in the higher education institutions in la union
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Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
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Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
Error analysis in college algebra in the higher education institutions in la union
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

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

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 16. xvi LIST OF FIGURES Figure Page 1 Ragma’s Error Intervention Model…………………………… 13 2 The Research Paradigm ……………………………………….. 18
  17. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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,

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