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Assessing Curricular Contributions to Poor Measurement Learning The STEM Project Team Michigan State University Strengthening Tomorrow’s Education in Measurement NCTM Pre-Session April 10, 2008

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Strengthening Tomorrow’s Education in Measurement 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) 2

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Strengthening Tomorrow’s Education in Measurement 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 3

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Strengthening Tomorrow’s Education in Measurement 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) 4

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Strengthening Tomorrow’s Education in Measurement Prior Research Kosze Lee 5

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Strengthening Tomorrow’s Education in Measurement 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 6

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

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Strengthening Tomorrow’s Education in Measurement 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) 8

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Strengthening Tomorrow’s Education in Measurement 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? 9

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Strengthening Tomorrow’s Education in Measurement 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) 10

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Strengthening Tomorrow’s Education in Measurement 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 11

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Strengthening Tomorrow’s Education in Measurement 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? 12

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Strengthening Tomorrow’s Education in Measurement 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 13

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Strengthening Tomorrow’s Education in Measurement 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 14

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Strengthening Tomorrow’s Education in Measurement 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 15

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Strengthening Tomorrow’s Education in Measurement STEM Project Overview Hanna Figueras 16

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

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Strengthening Tomorrow’s Education in Measurement 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 18

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Strengthening Tomorrow’s Education in Measurement 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 19

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Strengthening Tomorrow’s Education in Measurement 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 20

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Strengthening Tomorrow’s Education in Measurement Project Development Process Locating Measurement Content Creating Framework Generating Knowledge Elements Coding Content Analysis 21

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Strengthening Tomorrow’s Education in Measurement 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)? 22

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Strengthening Tomorrow’s Education in Measurement Locating the Spatial Measurement Content Lorraine Males 23

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Strengthening Tomorrow’s Education in Measurement 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. 24

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Strengthening Tomorrow’s Education in Measurement 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. 25

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Strengthening Tomorrow’s Education in Measurement 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). 26

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Strengthening Tomorrow’s Education in Measurement Finding Measurement Content Difficulties Judging if the content is likely to engage measurement reasoning. Determining which spatial attribute is being addressed. 27

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Strengthening Tomorrow’s Education in Measurement Establishing Measurement Content Types of Measurement Pre-Measurement Measurement proper Reasoning with or about Measurement 28

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Strengthening Tomorrow’s Education in Measurement 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) 29

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Strengthening Tomorrow’s Education in Measurement 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) 30

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Strengthening Tomorrow’s Education in Measurement 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) 31

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Strengthening Tomorrow’s Education in Measurement 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. 32

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Strengthening Tomorrow’s Education in Measurement 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 ). 33

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Strengthening Tomorrow’s Education in Measurement Our principal tool for assessing curricular “capacity” Leslie Dietiker 34

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Strengthening Tomorrow’s Education in Measurement 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 o Transitivity: “The comparison of lengths is transitive. If length A > length B, length B > length C, then length A > length C.” o Unit-measure compensation: “Larger units of length produce smaller measures of length.” o Additive composition: “The sum of two lengths is another length.” o Multiplicative composition: “The product of a length with any other quantity is not a length.” 35

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Strengthening Tomorrow’s Education in Measurement 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] 36

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Strengthening Tomorrow’s Education in Measurement Textual Elements Statements Questions Problems Demonstrations Worked Examples Games 37

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Strengthening Tomorrow’s Education in Measurement 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) 38

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Strengthening Tomorrow’s Education in Measurement 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.” Procedural Knowledge Elements 39

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

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Strengthening Tomorrow’s Education in Measurement Realization #3 We need to attend to curricular voice (who speaks to students) Teacher Textbook or other written materials Others (in case of Demonstrations) 41

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Strengthening Tomorrow’s Education in Measurement Coding Measurement Content 42

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Strengthening Tomorrow’s Education in Measurement Coding Measurement Content Question - Provided by Teacher Direct Comparison x 2 43

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Strengthening Tomorrow’s Education in Measurement Coding Measurement Content Worked Example - Student Text Measurement with non-standard units 44

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Strengthening Tomorrow’s Education in Measurement Coding Measurement Content Problem - Student Text Measurement with non-standard units 45

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Strengthening Tomorrow’s Education in Measurement Questions ProblemsWorked Examples Larger units of length produce smaller measures of length Direct comparison Measure length with Non- standard units Sample Coding Sheet

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Strengthening Tomorrow’s Education in Measurement Coding Scheme 47

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Strengthening Tomorrow’s Education in Measurement Length Results for Grades K & 1 Jack Smith 48

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Strengthening Tomorrow’s Education in Measurement 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 49

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Strengthening Tomorrow’s Education in Measurement Density of Length Content EMSFAWSaxon Pages (K) Total Codes (K) Codes/Page (K) Pages (1) Total Codes (1) Codes/Page (1) Pages (2)

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Strengthening Tomorrow’s Education in Measurement Overview of K Results Textual presentation Problems, Questions, Demonstrations dominate Few Statements; more in EM Knowledge content 80% of all knowledge element codes were Procedural (EM, 82%, SFAW, 98%, Saxon, 95%) Matches the “procedurally-focused” attribution EM: 13% Conceptual knowledge codes (n = 28) 51

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Strengthening Tomorrow’s Education in Measurement Procedural & Conceptual (K) Common procedures Measure with non-standard units (body parts, paper clips, linking cubes) Direct Comparison (align & judge relative length) Visual Comparison (same for non-adjacent objects) Measure with a ruler (Saxon only) Common conceptual knowledge (caution: small numbers!) Unit-Measure Compensation (EM) Greater measure means longer length (EM, SFAW) 52

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Strengthening Tomorrow’s Education in Measurement Overview of Grade 1 Results Recall the increase in pages and codes/page Attention gets more serious in Grade 1 (and continues in Grade 2) Textual presentation Problems and Questions dominate Drop in Demonstrations from K Increase in Statements from K (absolute & %) Knowledge content Procedural focus remains (EM & SFAW, 78%, Saxon, 91%) SFAW added conceptual content; EM retained it 53

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Strengthening Tomorrow’s Education in Measurement Procedural & Conceptual (Gr. 1) Common procedures Direct & Visual Comparison Visual Estimation (new in Grade 1) Measure with a ruler Measure with non-standard units Common conceptual knowledge (larger numbers) Unit-Measure Compensation (SFAW, Saxon) Greater measure means longer length (EM, SFAW) Standard vs. non-standard units (EM) Rulers measure length (SFAW) 54

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Strengthening Tomorrow’s Education in Measurement So…Is the Analysis Promising? Not surprisingly, we think Yes Finding in more detail what others have reported (procedural focus) But we are much more specific about What that means (which procedures?) Differences across curricula Grade level and grade band patterns (e.g., K–2) Tracking conceptual knowledge (present and absent) in a very specific way 55

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Strengthening Tomorrow’s Education in Measurement Challenges Ahead Careful analysis is costly (in human time) Choice between more extensive analysis of written curriculum and examination of some “enacted” lessons Both are important; How to choose? Other limitations Can this serve as a national report card (on our written curricula)? Have not even started area and volume 56

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Strengthening Tomorrow’s Education in Measurement Comments from Richard Lehrer 57 followed by Q & A

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