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

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! Welcome: Intro Jack and Lorraine, STEM; want to support teachers Participants: If large, show of hands for: (1) elementary focus, (2) primary focus, (3) practicing, (4) pre-service, (5) PD

Into the Content The Toothpick Problem [NAEP, Grade 4, 2003, Open response] “What is the length of the toothpick?” Given lots of times in NAEP; national sample of thousands of 4th graders

Toothpick Performance Data [Grade 4, large national sample] Response % Responding 2.5 inches (correct) 20 10.5 inches 14 3.5 inches 23 Other 42 Omitted 2 Lots of 4th graders; 20% (one in five) are correct, after instruction in the basics of length measurement is over Few omitted the item; 35% are reasoning incorrectly but in ways we understand More than 40% are responding in ways that NAEP could not easily interpret

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 Remember the 42% of “Other” With a good idea of the problem (what children are missing), hard to improve what we do STEM charges itself with finding some answers

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 Choice of three and only three was big: Give brief rationale Analysis moves grade by grade; find the measurement content first, then code it Coding involves tracking what is expressed (knowledge) and how it is expressed (statement, question, demonstration, worked example, prpblem, game)

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 For length, currently C = 64; P = 54 Focus on conceptual because that is where we think the problem and the answer lie

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

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

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

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

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).

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?

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 EM, grade 2, P. 661, kids laying on the floor head to toe SFAW, grade 1, P. 365

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?

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 be lead to good discussion and learning Select some number of animations Show them in order Write about it on the back of your feedback sheet: You/kids; what is different? Discuss, as time allows, and view more simulations if that seems useful

Sufficient number of tiles - numbered Tiling Select some number of animations Show them in order Write about it on the back of your feedback sheet: You/kids; what is different? Discuss, as time allows, and view more simulations if that seems useful Sufficient number of tiles - numbered

Iterating I Insufficient number of tiles – numbered with tiled trace Select some number of animations Show them in order Write about it on the back of your feedback sheet: You/kids; what is different? Discuss, as time allows, and view more simulations if that seems useful Insufficient number of tiles – numbered with tiled trace

Iterating II Insufficient number of tiles – Select some number of animations Show them in order Write about it on the back of your feedback sheet: You/kids; what is different? Discuss, as time allows, and view more simulations if that seems useful Insufficient number of tiles – not numbered with tick marks

Iterating III Insufficient number of tiles – not Select some number of animations Show them in order Write about it on the back of your feedback sheet: You/kids; what is different? Discuss, as time allows, and view more simulations if that seems useful Insufficient number of tiles – not numbered with tick marks and alignment marks

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 (jsmith@msu.edu) Lorraine Males (maleslor@msu.edu)