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# Assessment in mathematics

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### Assessment in mathematics

1. 1. Assessment in Junior High School Mathematics Dr. Carlo Magno De La Salle University, Manila 1
2. 2. Answer the following questions: • What is assessment for you? • When do you conduct assessment? • How do you assess math skills? 2
3. 3. Advance Organizer • Standards in Math (Junior HS) • Sources of Assessment Information – Assessment literacy – Reading Assessment results – Assessment for Learning • Formative Assessment • Guidelines in Developing Items 3
4. 4. Why do we need standards? • To make sure that everyone delivers quality work • To produce quality students • To deliver quality programs • Basis on what to assess 4
5. 5. Mathematics Standards for Junior HS • Algebra – explore the concepts involving a quadratic function and its graph and solve problems involving quadratic functions and equations. – solve equations involving rational expressions – explore relationships of quantities that involve variation and solve problems involving direct, indirect and joint variation 5
6. 6. Mathematics Standards for Junior HS – simplify expressions with rational exponents and solve problems involving them. – perform fundamental operations on expressions involving radicals and solve problems involving expressions and equations with radicals. 6
7. 7. Mathematics Standards for Junior HS • Trigonometry – explore the concept of trigonometric ratios and use these to solve problems on angles of elevation and depression and navigation. – generate an arithmetic and a geometric sequence, find the sums of the terms in the sequence and solve problems involving these sequences. – explore polynomial functions 7
8. 8. Mathematics Competencies for Junior HS • Geometry – use the fundamental theorems of proportionality – prove and use concepts on triangle similarity, particularly on similarity of right triangles to solve problems. – prove and use theorems involving quadrilaterals. – find parts of a circle and solve problems involving the circle and its parts. 8
9. 9. Mathematics Competencies for Junior HS – explore geometric figures on the rectangular coordinate plane. • Statistics – describe a set of data using measures of position. – count occurrences of an event and arrangements using the Fundamental Counting Principle, Permutations and Combinations. – find the probability of compound events. 9
10. 10. DepEd Taxonomy • real-life application of understanding • enduring big ideas, principles, and generalizations inherent to the discipline • cognitive operations that the student performs • content of the curriculum, the facts and information that the student acquires Knowledge Process Product/PerformanceUnderstanding
11. 11. Explain Which of the following statements of the relationship between market price and normal price is true? a. Over a short period of time, market price varies directly with changes in normal price. b. Over a long period of time, market price tends to equal normal price. c. Market price is usually lower than normal price. d. Over a long period of time, market price determines normal price.
12. 12. Translation from symbolic form to another form, or vice versa Which of the graphs below best represent the supply situation where a monopolist maintains a uniform price regardless of the amounts which people buy? A B C D S Price Quantity S Price Quantity S S Price Quantity S S Price Quantity S Interpret
13. 13. Apply In the following items (4-8) you are to judge the effects of a particular policy on the distribution of income. In each case assume that there are no other changes in policy that would counteract the effect of the policy described in the item. Mark the item: A. If the policy described would tend to reduce the existing degree of inequality in the distribution of income, B. If the policy described would tend to increase the existing degree of inequality in the distribution of income, or C. If the policy described would have no effect, or an indeterminate effect, on the distribution of income. __ 4. Increasingly progressive income taxes. __ 5. Confiscation of rent on unimproved __ 6. Introduction of a national sales tax __ 7. Increasing the personal exemptions from income taxes __ 8. Distributing a subsidy to sharecroppers on southern farms
14. 14. Have perspective After reading the passage answer the following questions… 1. Where was Carol walking? a. park b. beach c. mall d. city hall 2. How did she feel on this walk? a. envied b. sad c. relaxed d. happy
15. 15. Have perspective 3. Carol envied the people around her because they _____________________. a. were sad and lonely b. love the city life c. were laughing and joking d. don’t like the city
16. 16. Empathize • Your new maid from the mountain destroyed your very expensive Narra door and she used it as firewood and cooked rice in your newly landscaped garden. How should you react? • A… • B… • C… • D…
17. 17. Sources of Assessment Information • Assessment Results – Classroom Assessment: Quarterly Test, Quizzes – National Assessment: NAT Results (Grade 6) 17
18. 18. Sources of Information on Student Mastery • Forms – Formative – Summative • Types – Paper and Pencil – Alternative forms: Performance, authentic, Portfolio • Approaches – Assessment “of” learning – Assessment “for” learning 18
19. 19. Assessment Literacy • (1) Assessment comes with a clear purpose • (2) focusing on achievement targets • (3) selecting proper assessment methods • (4) sampling student achievement 19
20. 20. Reading Assessment Results Levels of Proficiency 20
21. 21. Reading Assessment Results 21
22. 22. Reading Assessment Results Beginning Developing Approaching Proficiency Proficient Advanced 22
23. 23. Reading Assessment Results 23
24. 24. Reading Assessment Results • Mathematics NAT • 15 items – Place value = 1 – Fraction = 2 – Measurement = 2 – Multiplication = 1 – Division = 2 – Lines = 1 – Addition = 3 (Problem solving) – Ratio and proportion = 1 – Statistics = 2 (interpreting graphs) 24
25. 25. Approach in Assessment Assessment of Learning Assessment for Learning Assessment as learning 25
26. 26. ASSESSMENT OF LEARNING AND ASSESSMENT FOR LEARNING Effect of Previous Practices: rank students on achievement by graduation New Expectation: Assure competence in Math, Reading, Writing, etc. • Implications? Assessment and grading procedures should help students succeed. 26
27. 27. ASSESSMENT FOR LEARNING • We need to close the gap between standards and students competencies • Risk: our society will be unable to productively evolve in social and economic sense. • Assessment is a tool to ensure student mastery of essential standards. 27
28. 28. ASSESSMENT FOR LEARNING • Mistaken beliefs about how to use assessment to support school improvement: 1.High-stakes tests are good for all students because they motivate learning 2.If I threaten to fail you, it will cause you to try harder 3.If a little intimidation doesn’t work, use a lot of intimidation 28
29. 29. MISTAKEN BELIEFS 4. The way to maximize learning is to maximize anxiety 5. It is the adults who use assessment results to make the most important instructional decision. 29
30. 30. MISTAKEN BELIEFS PROFOUND MISTAKE Teachers and leaders don’t need to understand sound assessment practices – the testing people will take care of us. COUNTER BELIEF They do need to understand sound assessment practices. 30
31. 31. Assessment “for” Learning • Assessment Crisis: The Absence Of Assessment FOR Learning • By Rick Stiggins 31
32. 32. Assessment “for” Learning • School improvement requires: – the articulation of higher achievement standards, – the transformation of those expectations into rigorous assessments, and – the expectation of accountability on the part of educators for student achievement, as reflected in test scores. 32
33. 33. Assessment “for” Learning • When we assess for learning, teachers use the classroom assessment process and the continuous flow of information about student achievement that it provides in order to advance, not merely check on, student learning. 33
34. 34. Assessment “for” Learning • understanding and articulating in advance of teaching the achievement targets that their students are to hit; • informing their students about those learning goals, in terms that students understand, from the very beginning of the teaching and learning process; • becoming assessment literate and thus able to transform their expectations into assessment exercises and scoring procedures that accurately reflect student achievement; 34
35. 35. Assessment “for” Learning • using classroom assessments to build students’ confidence in themselves as learners and help them take responsibility for their own learning, so as to lay a foundation for lifelong learning; • translating classroom assessment results into frequent descriptive feedback (versus judgmental feedback) for students, providing them with specific insights as to how to improve; 35
36. 36. Assessment “for” Learning • continuously adjusting instruction based on the results of classroom assessments; • engaging students in regular self-assessment, with standards held constant so that students can watch themselves grow over time and thus feel in charge of their own success; and • actively involving students in communicating with their teacher and their families about their achievement status and improvement. 36
37. 37. Formative Assessment • Need not be graded as summative assessments (end-of-unit exams or quarterlies, for example) are. • They serve as practice for students • They check for understanding along the way and guide teacher decision making about future instruction; • they also provide feedback to students so they can improve their performance 37
38. 38. Formative Assessment • For assessments to be accurate, teachers need multiple measures of student understanding. • Teachers need evidence gathered over time in different ways to evaluate how effective the teaching and learning process has been. • Tomlinson and McTighe (2006) suggest that when teachers gather a "photo album" rather than a "snapshot" of our students, we can differentiate instruction based on a more accurate evaluation of our students' learning needs. 38
39. 39. Formative Assessment • 1. Student friendly targets from the beginning • 2. Models of strong and weak work • 3. Continuous descriptive feedback • 4. Teach self-assessment and goal setting • 5. Teach one facet at a time. • 6. Teach focused revision. • 7. Teach self-reflection to track growth 39
40. 40. Formative Assessment • Group Assessment - allows you to quickly identify problems or misconceptions, which you can address immediately. • Individual assessment - Provide some feedback to the learner, perhaps in the form of a brief comment or, at the very least, a check, check-plus or check-minus, with a brief verbal explanation about what each symbol indicates 40
41. 41. Formative Assessment • Summaries and Reflections .Students stop and reflect, make sense of what they have heard or read, derive personal meaning from their learning experiences, and/or increase their metacognitive skills. These require that students use content-specific language. • Lists, Charts, and Graphic Organizers Students will organize information, make connections, and note relationships through the use of various graphic organizers. • Visual Representations of Information Students will use both words and pictures to make connections and increase memory, facilitating retrieval of information later on. This "dual coding" helps teachers address classroom diversity, preferences in learning style, and different ways of "knowing." • Collaborative Activities Students have the opportunity to move and/or communicate with others as they develop and demonstrate their understanding of concepts. 41
42. 42. Formative Assessment • Formative Assessment can be an integral part of instruction (Guskey, 2007): • (1) use assessments as sources of information for both students and teachers, • (2) follow assessments with high-quality corrective instruction, and • (3) give students second chances to demonstrate success 42
43. 43. Formative Assessment • By varying the type of assessment you use over the week, you can get a more accurate picture of what students know and understand, obtaining a "multiple- measure assessment ‘window' into student understanding" (Ainsworth & Viegut, 2006). • Using at least one formative assessment daily enables you to evaluate and assess the quality of the learning that is taking place in your classroom and answer these driving questions: How is this student evolving as a learner? What can I do to assist this learner on his path to mastery? 43
44. 44. Formative Assessment • Response to Intervention (RTI) model – Tier 1 interventions include monitoring at-risk students within the general education classroom, ensuring that each student has access to a high- quality education that is matched to his or her needs. – RTI focuses on improving academic achievement by using scientifically based instructional practices. – Use alternative assessment which utilizes quality interventions matched to student needs, coupled with formative evaluation to obtain data over time to make critical educational decisions. 44
45. 45. Techniques in Writing Items 45
46. 46. Multiple Choice 1. Rene Descartes is a. a famous Italian. b. important in mathematics. c. known for his analytical geometry. d. the author of many books. •It is recommended that the stem be a direct question. •The stem should pose a clear, define, explicit, and singular problem. Why is the item faulty?
47. 47. Multiple Choice IMPROVED: With which one of the mathematics field is Rene Descartes associated? a. Analytical geometry b. Differential calculus c. Discrete mathematics d. Computational mathematics
48. 48. Multiple Choice 2. Milk can be pasteurized at home by a. heating it to a temperature of 130o b. Heating it to a temperature of 145o c. Heating it to a temperature of 160o d. Heating it to a temperature of 175o •Include in the stem any words that might otherwise be repeated in each response. Why is the item faulty?
49. 49. Multiple Choice IMPROVED: The minimum temperature that can be used to pasteurize milk at home is: a. 130o b. 145o c. 160o d. 175o
50. 50. Multiple Choice 3. Although the experimental research, particularly that by Hansmocker must be considered equivocal and assumptions viewed as too restrictive, most testing experts would recommend as the easiest method of significantly improving paper-and-pencil achievement test reliability to a. increase the size of the group being tested. b. increase the differential weighting of items. c. increase the objective of scoring. d. increase the number of items. e. increase the amount of testing time. Items should be stated simply and understandably, excluding all nonfunctional words from stem and alternatives. Why is the item faulty?
51. 51. Multiple Choice IMPROVED: Assume a 10-item, 10-minute paper- and-pencil multiple choice achievement test has a reliability of .40. The easiest way of increasing the reliability to .80 would be to increased a. group size b. scoring objectivity c. differential item scoring weights d. the number of items e. testing time
52. 52. Multiple Choice 4. None of the following is a prime number except a. 17 b. 64 c. 96 d. 98 •Avoid negatively stated items Why is the item faulty?
53. 53. Multiple Choice IMPROVED: Which of the following is a prime number? a. 17 b. 64 c. 96 d. 98
54. 54. Multiple Choice 5. Who is the primary proponent of the Field theory in solving equations? a. Ernst Steinitz b. Manny Paquiao c. Lea Salonga d. Mark Twain •If possible the alternatives should be presented in some logical, numerical, or systematic order. •Response alternatives should be mutually exclusive. Why is the item faulty?
55. 55. Multiple Choice IMPROVED: Who is the primary proponent of the Field theory in solving equations? a. Ernst Steinitz b. Richard Dedekind c. Leopold Kronecker d. Heinrich M. Weber
56. 56. Multiple Choice 6. Which of the following statements makes clear the meaning of the word “electron”? a. An electronic tool b. Neutral particles c. Negative particles d. A voting machine e. The nuclei of atoms •Make all responses plausible and attractive to the less knowledgeable and skillful student. Why is the item faulty?
57. 57. Multiple Choice IMPROVED: Which of the following phrases is a description of an “electron”? a. Neutral particle b. Negative particle c. Neutralized proton d. Radiated particle e. Atom nucleus
58. 58. Multiple Choice 7. What is the area of a right triangle whose sides adjacent to the right angle are 4 inches long respectively? a. 7 b. 12 c. 25 d. None of the above •The response alternative “None of the above” should be used with caution, if at all. Why is the item faulty?
59. 59. Multiple Choice IMPROVED: What is the area of a right triangle whose sides adjacent to the right angle are 4 inches and 3 inches respectively? a. 6 sq. inches b. 7 sq. inches c. 12 sq. inches d. 25 sq. inches e. None of the above
60. 60. Multiple Choice 8. If the sum of 70 and 60 is 130, then the sum of 700 and 600 is a. was greater than 1500. b. to be a whole number. c. was less than 900. d. to be approximately 1300. Make options grammatically parallel to each other and consistent with the stem. Why is the item faulty?
61. 61. Multiple Choice IMPROVED: If the sum of 70 and 60 is 130, then the sum of 700 and 600 is… a. 1300. b. 1400. c. 1500. d. 1600.
62. 62. Multiple Choice 9. The “standard error of estimate’ refer to a. the objectivity of scoring. b. the percentage of reduced error variance. c. an absolute amount of possible error. d. the amount of error in estimating criterion scores. Avoid such irrelevant cues as “common elements” and “pat verbal associations.” Why is the item faulty?
63. 63. Multiple Choice IMPROVED: The “standard error of estimate” is most directly related to which of the following test characteristic? a. Objectivity b. Reliability c. Validity d. Usability e. Specificity
64. 64. Multiple Choice 10. What is the number that raises a base to a certain power? a. exponent b. logarithm c. tangent d. cosecant In testing for understanding of a term or concept, it is generally preferable to present the term in the stem and alternative definitions in the options. Why is the item faulty?
65. 65. Multiple Choice IMPROVED: Which of the following statements is the best description of a logarithm?
66. 66. Reference Magno, C., & Ouano, J. (2010). Designing Written Assessment for student learning. Manila: Phoenix.
67. 67. • Ability to Recognize the Relevance of Information 67
68. 68. • Ability to Recognize Warranted and Unwarranted Generalizations 68
69. 69. • Ability to Recognize Inferences 69
70. 70. • Ability to Interpret Experimental Findings 70
71. 71. • Ability to Apply Principles 71
72. 72. • Ability to Recognize Assumptions 72
73. 73. Reading comprehension • Bem (1975) has argued that androgynous people are “better off” than their sex-typed counterparts because they are not constrained by rigid sex-role concepts and are freer to respond to a wider variety of situations. Seeking to test this hypothesis, Bem exposed masculine, feminine, and androgynous men and women to situations that called for independence (a masculine attribute) or nurturance (a feminine attribute). The test for masculine independence assessed the subject’s willingness to resist social pressure by refusing to agree with peers who gave bogus judgments when rating cartoons for funniness (for example, several peers might say that a very funny cartoon was hilarious). Nurturance or feminine expressiveness, was measured by observing the behavior of the subject when left alone for ten minutes with a 5-month old baby. The result confirmed Bem’s hypothesis. Both the masculine sex- typed and the androgynous subjects were more independent (less conforming) on the ‘independence” test than feminine sex-typed individuals. Furthermore, both the feminine and the androgynous subjects were more “nurturant” than the masculine sex-typed individuals when interacting with the baby. Thus, the androgynous subjects were quite flexible, they performed as masculine subjects did on the “feminine” task. 73 35. What is the independent variable in the study? a. Situations calling for independence and nurturance b. Situation to make the sex type react c. Situations to make the androgynous be flexible d. Situations like sex type, androgynous and sex role concepts 36. What are the levels of the IV? a. masculine attribute and feminine attribute b. rating cartoons and taking care of a baby c. independence and nurturance d. flexibility and rigidity
74. 74. Interpreting Diagrams Instruction. Study the following illustrations and answer the following questions. Figure 1 74 Pretest Posttest 101. Which group received the treatment? a. group A b. group B b. c. none of the above 102. Why did group B remain stable across the experiment? a. there is an Extraneous Variable b. There was no treatment c. ceiling effect occured 103. What is the problem during the pretest phase of the experiment? a. the two groups are nonequivalent b. the groups are competing with each other c. the treatment took place immediately Group B Group A
75. 75. Activity 1 • Groupwork • Form 6 groups and each group is assigned with one facet of understanding • Write 4 multiple choice items for each facet of understanding • Group presentation will follow • Working time is only 25 minutes
76. 76. Activity 1 • Group 1: Evaluating functions • Group 2: Identifying linear functions • Group 3: Identifying Quadratic functions • Group 4: Finding the Properties of a circle. • Group 5: Evaluating Trigonometric functions 76
77. 77. Insights on Assessment 1. After the activities, I realized that my test … 2. After the activity, now I know that … 3. After the activity, every time I construct test, I will …