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Strategies to Help Your Measurement Instruction Measure Up Jack Smith & Lorraine Males STEM Project, College of Education, Michigan State Math In Action Conference February 25, 2009

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Introduction & Overview Welcome! Your presenters & you This session: A problem & one good idea Math focus: Units of length measure Teaching focus: Doing more with assigned textbook materials Disclaimer: One hour is not enough! 2

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Into the Content The Toothpick Problem [ NAEP, Grade 4, 2003, Open response] “What is the length of the toothpick?” 3

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Toothpick Performance Data [Grade 4, large national sample] Response% Responding 2.5 inches (correct) inches inches23 Other42 Omitted2 4

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Lessons from Toothpick We are not doing so well nationally Too many kids don’t understand length measurement, rulers, or both Some errors are sensible; some remain mysterious Not obvious what we are doing wrong => Not obvious what we should change 5

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A Bit About Our Project Premise: Part of the problem may be our curriculum materials Carefully examine 3 elementary programs –Everyday Mathematics –Scott-Foresman/Addison-Wesley Mathematics –Saxon Mathematics Look for every instance of measurement content Code each as an “opportunity” to learn some bit of conceptual or procedural knowledge Length analysis is complete for K through Grade 3 6

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A Bit More (attention to Conceptual Knowledge) Lots of different conceptual and procedural elements Focus on two conceptual elements –Unit-Measure Compensation –Unit Iteration Both important; Unit Iteration appears less frequently than we would like 7

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Unit-Measure Compensation Larger units of length produce smaller measures of length; smaller units of length produce larger measures. All curriculum mention this idea with some frequency & in different grades Could do more but not the heart of the problem 8

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Unit Iteration Measures of length are produced by iterating a length unit (repeatedly adjoining) from one end of an object, segment, or distance to the other and then finding the number of iterations (e.g., by counting). Iterated units may not overlap or leave gaps. Insufficient attention in all 3 curricula N = 19 instances from K to Grade 3!! Many are partial statements 9

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Unit Iteration (idea by idea) Measures of length are produced by iterating a length unit (repeatedly adjoining) 10

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Unit Iteration (idea by idea) Measures of length are produced by iterating a length unit (repeatedly adjoining) from one end of an object, segment, or distance to the other 11

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Unit Iteration (idea by idea) Measures of length are produced by iterating a length unit (repeatedly adjoining) from one end of an object, segment, or distance to the other and then finding the number of iterations (e.g., by counting). 12

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Unit Iteration (idea by idea) Measures of length are produced by iterating a length unit (repeatedly adjoining) from one end of an object, segment, or distance to the other and then finding the number of iterations (e.g., by counting). Iterated units may not overlap or leave gaps. How does this relate to Toothpick? 13

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Some Work with Activities in Existing Curricula Case 1: EM, grade 2 Case 2: SFAW, grade 1 May not be your curricula, but We hope you will see how to adapt our main points to your situation 14

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Some Issues in the Activities Tiling is not the same as Iterating Both depend on Partitioning (conservation) Tiling and Iterating are closely related, but Do young children see that? Shouldn’t we do more to help them? Tiling and Iterating are “built into” Rulers Shouldn’t we help them there too? 15

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Seeing Tiling & Iterating Besides using bodies, what else? We developed some animations –Measuring a pencil –Non-standard unit (a rectangular tile) Seek your reactions & judgments –Where you see differences –How your kids would react (same/different) Differences that arise could lead to good discussion and learning 16

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Tiling Sufficient number of tiles - numbered 17

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Iterating I Insufficient number of tiles – numbered with trace 18

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Iterating II Insufficient number of tiles – not numbered with tick marks 19

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Iterating III Insufficient number of tiles – not numbered with tick marks and alignment marks 20

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Closing Thank you for coming & engaging You are essential for our work to have meaning & impact Engage with us –Feedback form –Indicate an interest in measurement –Are you using one of our target curricula? –Look for us around the state, at NCTM, and in NCTM’s journals –Jack Smith Lorraine Males 21

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