1. Problem Solving and the Development of Conceptual Understanding in the Middle Grades University of North Carolina at Chapel Hill Carol E. Malloy, Ph.D. Carol E. Malloy, Ph.D. Crystal Hill, Doctoral Student Crystal Hill, Doctoral Student California State University, Fullerton Mark Ellis, Ph.D. Mark Ellis, Ph.D.

2. Principal Investigators: Carol E. Malloy, Ph.D. Carol E. Malloy, Ph.D.cmalloy@email.unc.edu Jill V. Hamm, Ph.D. jhamm@email.unc.edu Judith L. Meece, Ph.D. meece@email.unc.edu University of North Carolina-Chapel Hill – NSF Grant REC 0125868 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l MIDDLE Research Associate: Mark W. Ellis m ellis@exchange.fullerton.edu

3. Purpose To better understand how mathematics reform affects students’ development as mathematics knowers and learners To better understand how mathematics reform affects students’ development as mathematics knowers and learners To provide a longitudinal analysis of students’ mathematical development during the middle school years To provide a longitudinal analysis of students’ mathematical development during the middle school years To identify the processes that explain changes in students’ mathematical learning and self- conceptions To identify the processes that explain changes in students’ mathematical learning and self- conceptions

4. Three-Year, Longitudinal Cross-Sectional Design 2002-03 6 th grade 2003-04 6 th grade 7 th grade 2004-05 7 th grade 8 th grade

5. Level I Data on Conceptual Understanding MIDDLE Assessment Items MIDDLE Assessment Items Fall4 grade level 2 developmental Fall4 grade level 2 developmental Spring4 grade level 6 developmental Spring4 grade level 6 developmental Strands included in MIDDLE assessment items Strands included in MIDDLE assessment items Number (esp. rational number) Number (esp. rational number) Geometry and Measurement Geometry and Measurement Proportional Reasoning Proportional Reasoning Problem Solving Problem Solving

6. Carpenter and Lehrer (1999) What is Conceptual Understanding? Type of knowledge needed for mathematical proficiency: Type of knowledge needed for mathematical proficiency:  Procedural  Conceptual Characteristics of conceptual understanding: Characteristics of conceptual understanding:  Generative  It’s NOT static  Student Involvement  Different Progression

7. Conceptual Understanding Problem Development Item Selection Item Selection –NAEP and TIMSS released items –“Open-middle” items  Allow for student explanations  Allow for focus on concept (s) –Prompt: ”Explain why you think your answer is correct.” –Within grade and longitudinal items  4 grade level items for within grade growth  6 items for assessment of growth over time –Reviewed by panel of experts

8. Rubric Development Overview “Conceptual Rubric”  0 points: There is no response.  1 point: The response demonstrated no conceptual understanding of the problem.  2 points: The response demonstrated limited conceptual understanding and/or had significant errors.  3 points: The response demonstrated some conceptual understanding but was incomplete.  4 points: The response demonstrated complete conceptual understanding of the problem.

9. Item-Level Rubric Development Conceptual Rubric Concepts Assessed Indicators Student Responses

10. Purpose of Secondary Analysis To understand students’ solution strategies and misconceptions To understand students’ solution strategies and misconceptions To understand how items asking for students’ thinking can be used as assessment tools for instruction To understand how items asking for students’ thinking can be used as assessment tools for instruction To understand instructional practices that support students’ development of conceptual understanding in rational number, proportional reasoning and measurement To understand instructional practices that support students’ development of conceptual understanding in rational number, proportional reasoning and measurement

11. Sample Information 5 Middle Schools 5 Middle Schools 51 Girls 51 Girls 28 Boys 28 Boys 40.5% African-American 40.5% African-American 50.6% White Students 50.6% White Students 3.8% Latino Students 3.8% Latino Students Total Scores at least 16 out of 32 Total Scores at least 16 out of 32

12. Methods and Analysis 3 items identified 3 items identified –Rational Number –Proportional Reasoning –Measurement Examined student papers Examined student papers –Recorded strategies, correct and incorrect –Highlighted “interesting” approaches Classified strategies and misconceptions Classified strategies and misconceptions –Examined prior research –Related to key concepts

13. Decimal Fractions  Which of these is the smallest number? Explain why you think your answer is correct. A. 0.625 B. 0.25 C. 0.375 D. 0.5 E. 0.125 Source: TIMSS 1999, Middle School, B-10, p-value = 46%

14. Score Distribution CUScore0 1234Total Wave312412231979 Wave 4 01623221879 For both waves combined, p-value = 52% (score 3 or 4).

15. Key Knowledge for Understanding Decimal Fractions Place value system Place value system –Partitioning numbers –Representing values less than a whole Fraction Fraction –Meaning of denominator and numerator (unitizing) –Equivalent fractions (reunitizing) Strategies showing conceptual understanding Strategies showing conceptual understanding –Common unit (use smallest place value) –Composite units (use each place value separately)

16. Students’ Decimal Fraction Strategies Adapted from Stacey & Steinle (1999) Shorter is Smaller Shorter is Smaller –Using whole number reasoning –Decimal-Fraction connection not established Longer is Smaller Longer is Smaller –Misunderstanding of decimal-fraction connection, particularly denominator and numerator relationship (e.g., 0.35 means 1/35) Apparent-Expert Behavior Apparent-Expert Behavior –Follow correct rules without understanding why: Equalizing with zeros  CAUTION: Reinforces whole number reasoning Equalizing with zeros  CAUTION: Reinforces whole number reasoning Comparing digits from left to right Comparing digits from left to right

17. Shorter is Smaller (1)

18. Shorter is Smaller (2)

19. Longer is Smaller (1) Always give answer of 0.625

20. Longer is Smaller (2) “The squares in the 2 nd one are much smaller than in the first.”

21. Equalizing Length with Zeros Always give correct answer of 0.125

22. Comparing Digits Always give correct answer of 0.125

23. Place Value Understanding Always give correct answer of 0.125 and talk about comparing place values or relative size of numbers.

24. Strategies for Developing Decimal Fraction Understanding Use multiple representations Use multiple representations –Number line model (placement and reading) –Fraction notation –Real-life context (e.g., money; volume) Emphasize fraction-decimal connection Emphasize fraction-decimal connection Use diagnostic items Use diagnostic items –Which is smaller, 0. or 0. ? Discuss role of zero Discuss role of zero –When does zero affect a number’s value?

25. Proportional Reasoning If, then n = Explain why you think your answer is correct. Source: NAEP Released Item, p-value = 48%

26. Score Distribution CUScore0 1234Total Wave52126302979 Wave 6 138382373 For both waves combined, p-value = 79% (score 3 or For both waves combined, p-value = 79% (score 3 or 4).

27. Key Knowledge for Understanding Proportional Relationships Firm grasp of various rational number concepts Firm grasp of various rational number concepts Understand multiplicative relationships that exist between and within problem situations Understand multiplicative relationships that exist between and within problem situations Ability to discriminate between proportional and non-proportional problem situations Ability to discriminate between proportional and non-proportional problem situations

28. Students’ Solution Strategies Unit-rate Unit-rate –Finding the multiplicative relationship between ratios –“how many for one” Factor-of-Change Factor-of-Change –Finding the multiplicative relationship within ratios –“times as many” Cross Multiplication Cross Multiplication Build Up Build Up

29. Students’ Solutions—Writing in Mathematics 1. Understood the multiplicative relationship between ratios 2. Students used additive strategies 3. Students used cross multiplication method correctly 4. Students unable to carry out cross multiplication method because of computational errors

30. Student Work - Unit-Rate

31. Student Work - From Unit- Rate to Cross Multiplication

32. Student Work – Cross Multiplication and Unit-Rate

33. Student Work – Build Up

34. Strategies to Promote Understanding of Proportional Relationships Develop relationship between multiplication and proportionality Develop relationship between multiplication and proportionality Delay the introduction of the cross multiplication algorithm Delay the introduction of the cross multiplication algorithm Provide students with examples of proportional and non-proportional problem situations Provide students with examples of proportional and non-proportional problem situations

35. Measurement A certain rectangle has its area equal to the sum of the areas of the four rectangles shown below. If its length is 4, what is its width? A certain rectangle has its area equal to the sum of the areas of the four rectangles shown below. If its length is 4, what is its width? Source: NAEP released item, p-value = 17%

36. Score Distribution CUScore0 1234Total Wave59301232579 Wave 6 631852575 For both waves combined, p-value = 38% (score 3 or 4).

37. Thinking About Finding Area Relationships Visualize area using manipulatives Visualize area using manipulatives Realize area can be a decimal or a fraction Realize area can be a decimal or a fraction Transform relationship of Area=width  length as width=Area/length Transform relationship of Area=width  length as width=Area/length No one method--Variable solution strategies No one method--Variable solution strategies

38. Students’ Solutions— Writing in Mathematics 1. Understood w = A/l and could complete division 2. Understood w = A/l and approximated 3. Found the length but not find width and stated “the problem does not make sense” or “is impossible” 4. Lack of conceptual understanding of relationship among length, width, and area—Students could not undo the formula, A = lw, but may have been able to use formula to find area 5. Faulty calculations and no reflection on the correctness of procedures or answers 6. Confusion between area and perimeter

39. Student Work— Guessing/Estimating FROM

40. Guessing/Estimating TO

41. Student Work – Questioning/Estimating FROM

42. Questioning /Estimating TO

43. Assessment/Instruction Provide students with tiles to cover the rectangles. This way they can see that the area is 9 tiles. Provide students with tiles to cover the rectangles. This way they can see that the area is 9 tiles. Ask students to develop a second method to find the area of the rectangles without using tiles. The idea here is to get them to think about length times width. Ask students to develop a second method to find the area of the rectangles without using tiles. The idea here is to get them to think about length times width.

44. Assessment/Instruction Have students try to construct a rectangle with all of the tiles. For those who could compute the length of 4, ask them to use the tiles to try to construct that rectangle. Have students work in groups use paper models and scissors to try to make the material in to a rectangle Have students work in groups use paper models and scissors to try to make the material in to a rectangle.

45. Strategies to Promote Understanding of Area Covering, measuring, transforming Covering, measuring, transforming Transforming relationships in measurements, i.e. using area or perimeter to find length or width Transforming relationships in measurements, i.e. using area or perimeter to find length or width Selecting and using figure attributes to measure area Selecting and using figure attributes to measure area Developing formula shortcuts for finding area Developing formula shortcuts for finding area

46. Why Use Items to Assess Students’ Thinking? As students organize, interpret and explain their knowledge through the writing process, their conceptual understanding and retention increase. As students organize, interpret and explain their knowledge through the writing process, their conceptual understanding and retention increase. Reading students’ explanations gives teachers insight into students’ understandings and misconceptions. Reading students’ explanations gives teachers insight into students’ understandings and misconceptions. Without explanations, student understanding can be misjudged when solutions are correct but for the wrong reasons. Without explanations, student understanding can be misjudged when solutions are correct but for the wrong reasons. When provided with feedback on their writing, students gain a sense that learning is a work in process, that they can grow in understanding. When provided with feedback on their writing, students gain a sense that learning is a work in process, that they can grow in understanding.

47. Conclusions Importance of understanding students’ thinking for assessment and instruction Importance of understanding students’ thinking for assessment and instruction Importance of instructional practices that support students’ development of conceptual understanding Importance of instructional practices that support students’ development of conceptual understanding

48. References Abrahamson, D., & Cigan, C. (2003). A Design for Ratio and Proportion Instruction. Mathematics Teaching in the Middle School, 8(9), 493-501. Boston, M., Smith, M., & Hillen, A. (2003). Building on Students’ Intuitive Strategies to Make Sense of Cross Multiplication. Mathematics Teacher in the Middle School. 9(3), 150-155. Burns, M. (1995). Writing in Math Class. Math Solutions Publications. Carpenter, T., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Classrooms that Promote Mathematical Understanding (pp. 19-32). Mahwah, NJ: Erlbaum. Countrymann, J. (1992). Writing to Learn Mathematics. Portsmouth, NH: Heinemann. Cramer, K., & Post, L. (1993). Connecting Research to Teaching Proportional Reasoning. Mathematics Teacher, 86(5), 404-407. Martine, S. L., Bay-Williams, J. M. (2003). Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244-247.

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