How Well Do Our Texts Introduce and Define Area?

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

2. 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 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 2007 MiCTM Presentation, STEM Project

4. 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 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) 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 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 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 2007 MiCTM Presentation, STEM Project

9. 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 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? 2007 MiCTM Presentation, STEM Project

11. 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 2007 MiCTM Presentation, STEM Project

12. 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 2007 MiCTM Presentation, STEM Project

13. 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 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 2007 MiCTM Presentation, STEM Project

15. Some Elementary Definitionsfrom 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) 2007 MiCTM Presentation, STEM Project

16. Some Middle School Definitionsfrom 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) 2007 MiCTM Presentation, STEM Project

17. 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? 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 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? 2007 MiCTM Presentation, STEM Project

20. 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 e-mail in September when your textbook is in front of you 2007 MiCTM Presentation, STEM Project