Strategies to Help Your Measurement Instruction Measure Up

Jack Smith & Lorraine MalesSTEM 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!

2

Into the Content

The Toothpick Problem[NAEP, Grade 4, 2003, Open response]

“What is the length of the toothpick?”

3

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 24

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

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

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

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

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

Unit Iteration (idea by idea)

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

10

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

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

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

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

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

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

Tiling

Sufficient number of tiles - numbered17

Iterating I

Insufficient number of tiles – numbered with trace 18

Iterating II

Insufficient number of tiles – not numbered with tick marks 19

Iterating III

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

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) 21