1. How Well Do Our Texts Introduce and Define Area?
Jack Smith, KoSze Lee, Leslie Dietiker, Hanna Figueras, Aaron Mosier, & Lorraine Males
The STEM project at MSU
(“Strengthening Tomorrow’s Education in Measurement”)
2007 MiCTM Conference
Understandings for Teachings, Teaching for Understanding
Holt High School; Holt, MI
Changes in title: “Texts” replaces “Curricula”; Cut out “the [Area] Concept”

2. 2007 MiCTM Presentation, STEM Project
Session Overview
Review Evidence of “The Problem”
Examine Some Potential Causes
Describe the STEM Project
Focus on One Corner of “The Problem”
Why area?
Examine some definitions
Discuss their similarities and differences
Consider their implications
How many came to last year’s conference? Was did anyone come to my session last year?
11/9/2018
2007 MiCTM Presentation, STEM Project

3. Please Help Us with Our Work
We would like to know about:
Where and what you teach
Your assigned curriculum
Your best guess at your text’s definition of area
Your hunches about what makes area difficult for students to understand
Why We Want to Know
Our Follow-up Request
11/9/2018
2007 MiCTM Presentation, STEM Project

4. 2007 MiCTM Presentation, STEM Project
The Problem
Students don’t seem to understand spatial measurement very well (length, area, & volume)
Not a problem of knowing what each measure is
But measures get confused, in 2-D and 3-D situations
Performance on all but very routine problems is low
11/9/2018
2007 MiCTM Presentation, STEM Project

5. Evidence of “The Problem” (2003 National Assessment data)
Only about half of 8th graders solved the “broken ruler” problem correctly (L)
Less than half of 4th graders measured a segment with a metric ruler correctly (L)
Only 2% of 8th graders found a figure’s area on a geoboard and constructed another figure with the same area (A)
Only 39% of 8th graders found the length of a rectangle, given its perimeter and width (L)
11/9/2018
2007 MiCTM Presentation, STEM Project

6. More Evidence (TIMSS data)
Generally (and consistently) our 4th graders perform pretty well and our 8th graders lag
But the 8th grade lag is greatest for geometry and measurement
Their textbook analysis also showed U.S. texts includes less content in that strand from grades 5 to 8
Lag is not made up in 12th grade
11/9/2018
2007 MiCTM Presentation, STEM Project

7. More Evidence (smaller-scale research studies)
Most common result: Confusion of area and perimeter for simple 2-D figures
Poor grasp of the relationship between length units and area units (e.g., inches and square inches)
Weak understanding of computational formulas (esp. area and volume), e.g. why they work
Not all elementary students “see” rows and columns in a rectangular array
11/9/2018
2007 MiCTM Presentation, STEM Project

8. Where Does This Leave Us?
Students don’t seem to understand what they are doing when they measure length, area, & volume.
If so, Why Do They Struggle? What’s So Hard?
No Explanation Out There
Is this an important question to tackle?
We don’t know what to do to address the problem
Importance: Valuable knowledge for both college and non-college bound students; links to content in Algebra (analytic geometry)
11/9/2018
2007 MiCTM Presentation, STEM Project

9. 2007 MiCTM Presentation, STEM Project
Six Possible Factors
Weaknesses in written curricula
Too little time spent teaching measurement
A dominant focus on static representations (esp. for area & volume)
Problems in talking about spatial quantities in classrooms (e.g., talking past one another)
Too much focus on numerical computation
Weaknesses in teachers’ knowledge
11/9/2018
2007 MiCTM Presentation, STEM Project

10. Other Factors?
Did you think of any different factors that might be involved?
Are they related to any of our six?
Write new factors on the board
11/9/2018
2007 MiCTM Presentation, STEM Project

11. 2007 MiCTM Presentation, STEM Project
Overview of STEM
STEM addresses factor (1); quality of written curricula
Focus on length, area, & volume
Written curricula seem like a good place to start
Very careful examination of the measurement content of 6 curricula (3 elementary, 3 middle school)
Elementary (K–6): Everyday Mathematics, Scott Foresman-Addison Wesley Mathematics, Saxon
Middle School (6–8): Connected Mathematics Project, Glencoe’s Mathematics: Concepts & Applications, Saxon
11/9/2018
2007 MiCTM Presentation, STEM Project

12. 2007 MiCTM Presentation, STEM Project
More About STEM
Step 1: Find the measurement content in the 6 curricula
Step 2: Evaluate the measurement content for its capacity to support deep learning (“how” and “why”)
Challenge of Step 1: Measurement appears in some surprising places
Challenge of Step 2: Fair evaluation of the curricula requires a sound and objective standard for judging
That measurement framework must
Be mathematically accurate and deep
Reflect what we know from research
11/9/2018
2007 MiCTM Presentation, STEM Project

13. 2007 MiCTM Presentation, STEM Project
Results?
Nothing we are ready to call “Results” yet
Have completed Step 1
Working hard on Step 2
Stay tuned; We’ll be back next year!!
Caution: Written curricula may be part of the problem; unlikely to be all of the problem
11/9/2018
2007 MiCTM Presentation, STEM Project

14. Why Focus on Definitions of Area?
Definitions are important in mathematics
Focus students’ attention on mathematical objects
Anchor the development of concepts
Influence how teachers talk about concepts
Definitions are important in measurement—to know which measure are we talking about
Definitions of spatial measures do differ across texts
Definitions of area influence how area and length (perimeter) are distinguished in 2-D figures
Definitions get students’ attention when they are first introduced and they also orient the textbook’s approach and discussion
11/9/2018
2007 MiCTM Presentation, STEM Project

15. Some Elementary Definitions from student materials (grade)
Children cover the surface and count the units. Tell them the result is called the area of the surface. (1)
The amount of surface inside a 2-D figure. (1)
Area is the measure of a surface or region bounded by a border, as opposed to the perimeter which is the length of the border. (2)
The space inside a shape. (2)
The number of square tiles you need to cover a shape. (2)
The number of units needed to cover the region inside a figure. (3)
11/9/2018
2007 MiCTM Presentation, STEM Project

16. Some Middle School Definitions from student materials (grade)
The number of squares needed to cover an object. (6)
How much surface is enclosed by the sides of a shape. (6)
The number of sq. inches needed to cover a surface. (6)
The number of sq. inches needed to cover a rectangle. (7)
The floor tiles cover the area of the hallway. Area is an amount of surface. (7)
The area of the broom sweep is equal to the width of the broom times the distance it is pushed. (8)
The number of square units needed to cover the surface enclosed by a geometric figure. (8)
11/9/2018
2007 MiCTM Presentation, STEM Project

17. 2007 MiCTM Presentation, STEM Project
Discussion
What do you see in these definitions?
Differences that matter?
Wording differences that don’t add up to much?
Are these definitions mathematically equivalent?
For kids?
For adults?
Are they equivalent as far as learning is concerned?
Should we start with one definition and finish with another?
Is your definition on this list?
When speaking the third question, make sure the teachers understand that an exact wording match is not required.
11/9/2018
2007 MiCTM Presentation, STEM Project

18. More About Definitions
Appear in multiple locations in curricula
Body of student materials
Glossaries in student materials
Teacher materials
Vary in number and frequency across curricula
Evidence of change (development?) over time, within curricula
11/9/2018
2007 MiCTM Presentation, STEM Project

19. One Important Difference for Us
Focus on a quantity to be measured, e.g., the amount of surface, vs. the number of square units to be counted/determined
Measurement is the union of space and number (the quantification of space)
Do definitions push the teaching of area to enumeration (counting & multiplication) too quickly?
Do our students know what they are computing?
11/9/2018
2007 MiCTM Presentation, STEM Project

20. 2007 MiCTM Presentation, STEM Project
Closing
Thanks for engaging with us!
Come back next year for some “results”
Turn in your sheet so we have your input
Look for our in September when your textbook is in front of you
11/9/2018
2007 MiCTM Presentation, STEM Project