1. Getting Down to the Basics with Length Measurement
Lorraine Males, Funda Gonulates,
& Jack Smith
STEM Project,
Michigan State University
MiCTM Conference
August 6, 2009

2. Introduction and Overview
Welcome!
Your presenters & you
A bit about the STEM project
A problem & NAEP data
Three Activities
Putting together the key length measurement ideas
Feedback
Welcome: Intro Lorraine, Funda
Participants: If large, show of hands for: (1) elementary focus, (2) primary focus, (3) practicing, (4) pre-service, (5) PD

3. A Bit About Our Project
STEM – Strengthening Tomorrow's Education in Measurement
Premise: Part of the problem may be our curriculum materials
Carefully examine 3 elementary programs
Everyday Mathematics
Scott-Foresman/Addison-Wesley Mathematics
Saxon Mathematics
We are not aiming to evaluate these curricula – we are looking to provide a description of the opportunity to learn measurement
Lorraine
Choice of three and only three was big: Give brief rationale

4. A Bit About Our Project 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
Lorraine
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, propblem, game)

5. Some Results Lots of different conceptual and procedural elements
We found that these curricula provide tasks that mostly have a procedural focus
We found some examples that may be used to introduce some key ideas related with length measurement
Funda
We will focus on these “examples” in this session.

6. The Toothpick Problem “What is the length of the toothpick?”
Into the Content
The Toothpick Problem “What is the length of the toothpick?”
Lorraine
Given lots of times in NAEP; national sample of thousands of 4th graders
How do you think your students might answer this?
Response
% Responding
2.5 inches (correct) 20%
10.5 inches 14%
3.5 inches 23%
Other 42%
Omitted 2%
[NAEP, Grade 4, 2003, Open response]

7. [Grade 4, large national sample]
The Toothpick Problem Data
Response
% Responding
2.5 inches (correct) 20
10.5 inches 14
3.5 inches 23
Other 42
Omitted 2
Lorraine
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% do not fall into these three categories (i.e., miscalculation, not able to interpret the question)

8. What lessons can we take from the toothpick problem?
students
teaching
content
curriculum
Funda
Small Group (5 min) Large Group (2)
What do you think is happening with students in our classrooms when answering this question?
What do you see in the curricula you think helps a lot into students’ understanding of measurement? Or you think less emphasized?
Is there something about concepts that build upon each other in order for students to learn to measurement with a ruler?
What may be the prerequisite concepts for supporting this learning?
What may be the conceptual basis as teachers we need to provide to our students besides presenting them some procedural activities?
What do you think is missing in students’ understanding measurement?

9. Come Potentially Useful Activities
We will do three length measurement activities
Buttons
Footprint
Funny Rulers
As we do these activities think about:
the key length measurement ideas
possible connections between these activities and ideas
Lorraine introduce
We are going to do each of these tasks and have a small discussion after each task about that task
We’d like to have the discussion connecting all three activities at the end
We will be collecting your work from these tasks so that we can look at the tasks as a group, but we will give you copies of all the tasks at the end of the session so you can use them or modify them for your own classroom
Funda pass out bags

10. Buttons
Question: What is the width of the paper you are given in buttons? 1. Sketch possible ways that a student may solve this task.
Funda
About 10 minutes
Small size paper
Worksheet has the directions
We will be walking around if you have questions
Don’t forget to [point to questions on board]
We will just have a small discussion
What ideas did this activity help you to think about?
If teachers don’t have much too say:
In terms of the math content you are teaching, what concepts may
come up with such an activity in the classroom?
Lorraine – collecting work
Funda – put work on board

11. Footprint
Question: What is the height of the desk you are sitting at? How long is it? 1. Trace you footprint on the paper provided in your bag, cut it and determine the height and length of your desk. 2. Sketch possible ways students might come up with.
Lorraine
About 15 minutes
Larger paper
Worksheet has the directions
We will be walking around if you have questions
Don’t forget to [point to questions on board]
We will just have a small discussion
What ideas did this activity help you to think about?
If teachers don’t have much too say:
In terms of the math content you are teaching, what concepts may
come up with such an activity in the classroom?
Funda – collect work
Lorraine – put work on board

12. Funny Rulers Question: How long is the paperclip provided in your bag?
1. Examine the rulers in your envelope to answer the following questions:
Which rulers could give you an accurate measure?
Which rulers you should not use?  
For each, explain why the ruler will not give an accurate measure.  
Funda
About 10 minutes
Envelope with rulers and 1 paperclip
Worksheet has the directions
We will be walking around if you have questions
Don’t forget to [point to questions on board]
We will just have a small discussion
What ideas did this activity help you to think about?
If teachers don’t have much too say:
In terms of the math content you are teaching, what concepts may
come up with such an activity in the classroom?
Lorraine – collect work
Funda – put work on board

13. Reflecting on Activities
Think back to the following questions: What were some of the key length measurement ideas in these activities? What are some possible connections between these activities and ideas?
Lorraine
As you can see we put all of the work on the board. We did this so we could see all the activities at once.
We’d like you to think back to these questions and use the work on the board to help you reflect. If you need to get up to see the work, feel free
We’d just like you to take a few minutes to think….

14. Key Ideas
In all those three activities there is an exhaustion of the space but
If we skip tiling ……………….
If we always provide sufficient materials but do not provide opportunities for unit iteration …………
If we start with measurement with ruler without providing an enough experience with tiling and unit iteration ………..
Rulers embed units
Lorraine facilitate discussion
Funda take notes on board
If we skip tiling the space exhausted may not be well-attended
Units may not be well attended
Emphasis on leaving no gaps or not overlapping in both tiling and iterating
Kids may take measurement with ruler as a tool that was always there without attention to how the ruler constructed and what it says in terms of length measurement
Why we need ruler?
What makes it more practical than using distinct standard units?
Rulers embed units
Different rulers will make those units more explicit
Rulers that do not start from zero will make kids to concentrate on
The exhaustion of the indicated attribute
The number of units that exhaust the space

15. Percentages of Selected Conceptual Knowledge Elements Across Grades
Funda
We want to take some time to connect what we did here to what we found in our project
First, we took two tasks from the curricula we are studying, so they have some useful tasks, but when we look at the overall distribution there is less emphasis in all three curricula on unit iteration and no explicit mention of ruler represents iterated units conceptually
We may need to modify our curriculum materials to address some of these key ideas

16. Percentages of Selected Procedural Knowledge Elements Across Grades
Lorraine
Here is a graph of some selected procedural knowledge elements
The footprint task that we just did is an example of measurement with insufficient non-standard units
As you can see there is relatively less emphasis on this concept compared to visual estimation of length and there is even much less emphasis on measurement with ruler by counting
This does not mean that the curricula do not include measuring with rulers, but there is less emphasis on the idea that a ruler is made up of units

17. Overall What happens if we don’t show strange and
incorrect ways of doing something?
Is it good if we just show students correct
procedures?
Think back to the toothpick problem
Some things may be worth investing more time
on than others because of the benefit these
things may provide
Funda
These are some questions we would like to leave you with.
Is there anything you want to say about these questions?
Is there anything you would like to add?

18. Closing
Thank you for coming & engaging Please let us know if you are interested in free professional development around measurement ideas

19. Closing 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
Lorraine Males
Funda Gonulates
Jack Smith