1. 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

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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

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

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

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).

13. 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?

14. 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

15. 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?

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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