Project Staff & Advisory Board

2. Project Staff & Advisory Board • Project Staff • PI: me • Graduate Students: Kuo-Liang Chang, Leslie Dietiker, Hanna Figueras, KoSze Lee, Lorraine Males, Aaron Mosier, Gulcin Sisman (METU) • Undergraduates: Patrick Greeley, Matthew Pahl • Advisory Board • Thomas Banchoff (Brown), Michael Battista (Ohio St.), Richard Lehrer (Vanderbilt), Gerald Ludden (MSU), Deborah Shifter (EDC), Nathalie Sinclair (Simon Fraser)

3. Our Session Goals • Motivate more research on learning and teaching spatial measurement • Length, area, & volume measurement • Describe our STEM project (as one research effort) • Enable and learn from discussion with you

4. Session Overview • Prior research (Kosze) • STEM overview (Hanna) • Locating the spatial measurement content (Lorraine) • Our principal tool for assessing curricular “capacity” (Leslie) • Results thus far (length; primary grades) (Jack) • Comments from a measurement expert (Rich) • Q&A discussion (All of us)

5. Prior Research Kosze Lee

6. Prior Research: Categories of Studies • Students’ performance in spatial measurement from large scale studies • NAEP • TIMSS • Smaller studies examining students’ solutions and reasoning on spatial measurement tasks • Length • Area and its relation to length

7. Large Scale Assessments • National and international studies indicated US students are weak in learning measurement • NAEP (2003): Low Performance by 4th, 8th, and 12th graders • TIMSS (1997) : gap between US 8th graders and their international peers is greatest in geometry & measurement • Minority students and girls face more struggle (Lubienski, 2003)

8. Students’ struggles with length • Unaware that any point on a scale can serve as the starting point. (Lehrer, 2003; NAEP, 2003) • Count marks (vs interval) on the scale (Boulton-Lewis et al., 1996)

9. Example (length) • A large majority students fail to find the length of a segment in a broken ruler task. (NAEP, grade 4, 2003) 2.5 inch? 10.5 inch? 3.5 inch?

10. Students’ struggles with area • Conceptual challenges • Square as a unit of measurement (Kamii and Kysh, 2006) • Visualizing the row-by-column structure of “tiled” rectangle as area measure (Battista, 2004) • Relating area and length • Confusing area with perimeter (Kidman & Cooper, 1997; Moyer, 2001; Woodward & Byrd, 1983) • Difficulties in relating the length units with area units (Chappell & Thompson, 1999; Battista, 2004)

11. But students can do better! • Teaching experiments show that elementary students can learn to do and understand measurement (Lehrer et al., 1998; Stephan, Bowers, Cobb, & Gravemeijer, 2003) • Students progressively construct understanding of knowledge and measuring processes built into standard rulers • Core: units, unit iteration, how to deal with left-overs

12. How can we explain the weaknesses? • The weaknesses are systematic, fundamental, and pervasive • No compelling explanations have been proposed • Hunches only • No strong empirical basis • So….What are some possible explanations for students’ continuing struggles to learn spatial measurement?

13. Possible Explanatory Factors 1) Weaknesses in the K-8 written curricula • Procedurally-focused (Kamii and Kysh, 2006) 2) Insufficient instructional time • Usually located at the end of textbooks and taught at the end of the school year (Tarr, Chavez, Reys, & Reys, 2006) 3) Static representations of 2D & 3D quantities (Sinclair & Jackiw, 2002) • Dynamic representations could help show how length units can compose area and volume

14. More Explanatory Factors 4) Classroom discourse about measurement poses special challenges (Sfard & Lavie, 2005) • Ambiguous references to spatial quantities and numbers 5) General “calculational” orientation in classroom instruction and discourse (Thompson, Phillip, Thompson, & Boyd, 1994) • divorce the value of measure from its spatial conception 6) Weaknesses in teachers’ knowledge (Simon & Blume, 1994) • These factors likely influence and interact with each other

15. So why target written curricula? • Weakness in written curricula influence other factors • Analysis of written curricula has national scope • Large scale classroom studies are resource-intensive • Analyzing widely-used curricula provide maximal access to problems faced by most parts of the nation • Clarify the exact nature of curricular weaknesses • More focused than general characterizations (“procedural focus”) • Beyond the presence/absence of topics

16. STEM Project Overview Hanna Figueras

17. Research Question What is the capacity of U.S. K - 8 written and enacted curricula to support students’ learning and understanding of measurement?

18. STEM Project Overview • Assess carefully the impact of Factor 1 (quality of written curricula) • Assess selectively Factors 3, 4 & 5 (nature of the enacted curriculum for specific lesson sequences) • Focus on spatial measurement in grades K-8 • length, area, & volume • Exclude measurement of angle • Draws on different roots than measurement of spatial extent (Lehrer et al., 1998) • Written curricula seemed like a good place to start

19. STEM Project Overview (cont’d) • How much of the problem can be attributed to the content of written curricula? • Develop an unbiased standard for evaluating the measurement content of select written curricula • Phase 1 - Analysis of written curricula • Phase 2 - Examination of enacted curricula • Start with length • Appears first, beginning in Kindergarten • Foundational for area and volume • Most extensive coverage and development

20. Which Curricula? Elementary School Curricula (K–6): • Everyday Mathematics • Scott Foresman-Addison Wesley Mathematics • Saxon Middle School Curricula (6–8): • Connected Mathematics Project • Glencoe’s Mathematics: Concepts & Applications • Saxon

21. Project Development Process Locating Measurement Content Creating Framework Generating Knowledge Elements Coding Content Analysis

22. Project Goals • Our goal is not to rank the three curricula at each level • National scorecard for written curricula in spatial measurement • Expect different patterns of strengths and weaknesses • Do we have common patterns of weakness (across curricula)?

23. Locating the Spatial Measurement Content Lorraine Males

24. Finding Measurement Content • The Task: • Compiling a list of all pages where measurement content (e.g., tasks) is found in each curriculum. • Who Does It: • Lead coder for each curriculum with a secondary coder to verify their work. • What It Means: • Reading through every page of each written curriculum and noting where spatial measurement concepts are utilized.

25. Establishing Measurement Content Our Fundamental Principle We will count as "measurement" all lessons, problems, and activities where students are asked to complete some spatial measurement reasoning, either as the intended focus of study or in order to learn some other content.

26. Finding Measurement Content • All content designated as spatial measurement in the written curricula will be coded. • However, every page does need to be examined, not just the measurement chapters. • In the chapter Measurement and Basic Facts we have “Measure your bed with your hand span” (EM, 1, p. 285). • In the chapter entitled Addition and Subtraction (EM) we have “Measure the length of this line segment. Circle the best answer” (EM, 2, p. 281).

27. Finding Measurement Content • Difficulties • Judging if the content is likely to engage measurement reasoning. • Determining which spatial attribute is being addressed.

28. Establishing Measurement Content • Types of Measurement • Pre-Measurement • Measurement proper • Reasoning with or about Measurement

29. Pre-Measurement • Reasoning about spatial measurements without appeal to units and enumeration • Is your tower of cubes the same size as the person’s next to you? How do you know? Hold it next to your neighbor’s tower. Is it the same? (Saxon, K, p. 8-2)

30. Measurement Proper • Partitioning and iterating a spatial unit to produce a spatial measure. This content is what is commonly classified as measurement. • (SFAW, 1, p .365)

31. Reasoning with or about Measurements • Using spatial measures to determine other quantities, spatial or non-spatial. • “It takes about 5 seconds for the sound of thunder to travel 1 mile. About how far can the sound of thunder travel in 1 minute?” (EM, MinM 1-3, p. 81)

32. Lessons from Applying the Principle • Determining the focal spatial quantity can be problematic. • How is perimeter different from area? (SFAW, 2, p. 351A) • Even if the focal spatial quantity can be determined, it is not trivial to determine if measurement reasoning will be utilized.

33. Lessons from Applying the Principle • We think there are topics that are not traditionally considered measurement content that utilize spatial measurement reasoning. • “Draw lines to show how to divide the square into fourths in two different ways” (Saxon, 1, p. 119-7).

34. Our principal tool for assessing curricular “capacity” Leslie Dietiker

35. Start of Process • Started with conceptual knowledge found in research • Identified elements of knowledge that holds for quantities in general) before those that hold for spatial quantities specifically • Transitivity: “The comparison of lengths is transitive. If length A > length B, length B > length C, then length A > length C.” • Unit-measure compensation: “Larger units of length produce smaller measures of length.” • Additive composition: “The sum of two lengths is another length.” • Multiplicative composition: “The product of a length with any other quantity is not a length.”

36. Realization #1 • We cannot just analyze the measurement knowledge… we need analysis of textual forms • “Why do you get different answers when you measure the same object using cubes and paper clips?” [SFAW, grade 2, p. 341] • “When changing from larger units to smaller units, there will be a greater number of smaller units than larger units.” [Glencoe, Course 1, p. 465]

37. Textual Elements • Statements • Questions • Problems • Demonstrations • Worked Examples • Games

38. Realization #2 • We cannot focus solely on conceptual knowledge; we need to capture procedural knowledge • General processes for determining measures • Broad interpretation of “process” • Generally, PK elements are distinct from CK (with some exceptions: unit conversion, perimeter, and Pythagorean Theorem)

39. Procedural Knowledge Elements • Visual Estimation: “Use imagined unit of length, standard or non-standard, to estimate the length of a segment, object, or distance.” • Draw Segment of X units with Ruler: “Draw a line segment from zero to X on the ruler.” • Unit Conversion: “To convert a length measure from one unit to another, multiply the given length by a ratio of the two length units.”

40. Conventional Knowledge • Cultural conventions of representing measures; devoid of conceptual content • This is one inch: • Notations, features of tools (e.g., marks on rulers) • Rulers have inches on one side and centimeters on the other.

41. Realization #3 • We need to attend to curricular voice(who speaks to students) • Teacher • Textbook or other written materials • Others (in case of Demonstrations)

42. Coding Measurement Content

43. Coding Measurement Content Question - Provided by Teacher Direct Comparison x 2

44. Coding Measurement Content Worked Example - Student Text Measurement with non-standard units

45. Coding Measurement Content Problem - Student Text Measurement with non-standard units

46. Sample Coding Sheet 1 1 2

47. Coding Scheme

48. Length Results for Grades K & 1 Jack Smith

49. Some Generalities • An intermediate view of key spatial measurement topics in each curriculum (STEM Top 10) • Continuous quantity (e.g., strings of cubes) site for both number (& operation) and length measurement • Saxon & SFAW • Tough coding decisions for us • K–2 contains the foundation for length measurement • Substantial content devoted to the topic • Deficits may not get corrected in later grades • We’re short of our conference goal; Grade 2 in process

50. Density of Length Content