Getting to the Heart of Measurement (When We Usually Don’t)

1. Getting to the Heart of Measurement(When We Usually Don’t) Jack Smith (a.k.a., John P. Smith III)

2. Session Overview • A research session • Focus first on the problem of teaching and learning of spatial measurement • STEM Project’s approach: Look carefully at the content of the elementary written curriculum • Examine all textbook pages that address spatial measurement • Do these materials provide sufficient “opportunity to learn”? • Main STEM message: Our elementary materials are not currently adequate • Too focused on the procedures of measurement • Neglect important conceptual issues • The solution means doing more/better with what we have • Goal: Finish in ≤ 40 minutes (balance of time for discussion) STEM Presentation, 2010 NCTM

3. The Problem(take #1: the surface) • U.S. students perform poorly on spatial measurement tasks (NAEP results, especially at 4th and 8th grade) • Performance declines as dimension increases (length > area > volume) • In 2-D and 3-D situations, confusions of different spatial quantities is a particular problem (e.g., perimeter & area) • Instruction focuses on procedures (ruler use & computational formulae) • We’re not teaching the conceptual principles, so students are learning by rote • BUT…. We can do better. STEM Presentation, 2010 NCTM

4. your Position(relative to the problem) • What brought you to this session? • Do you see the problem primarily in terms of • Poor annual performance results (state, district, school, classroom)? • Getting time (in the spring) to teach measurement? • Having to re-teach measurement? • Dissatisfaction with your current curriculum materials? • Listening to kids’ talk & work with measurement? • Won’t have a solution for you; will give you some tools STEM Presentation, 2010 NCTM

5. The Problem(take #2: A Little Deeper) • We spend a lot of time on length measurement with rulers • In Michigan, statewide performance looks good for Grade 2 and 3 content • Now consider the Toothpick problem on the NAEP • Haven’t yet found a compelling item for area (for many reasons) • There is no “ruler” for area • We aren’t asking equivalent questions for area, e.g., explain how multiplying lengths produces a collections of squares • 8th & 12th grade performance on surface area and volume is terrible STEM Presentation, 2010 NCTM

6. Problem Sources • Naming the problem year after year will not solve it • Many likely contributing factors • One basic one to explore: Do our written curricular materials contain the right content? • If not, we have one root cause AND we can work to address specific deficiencies • The STEM Project has identified specific conceptual deficits for length and area STEM Presentation, 2010 NCTM

7. The STEM Project(in brief) • Three elementary curricula • Everyday Mathematics • Scott-Foresman/Addison-Wesley’s Mathematics • Saxon Mathematics • If the problem exists in these materials, we have a national problem • Develop a systematic list of conceptual, procedural, and conventional knowledge for length, area, & volume • Code every sentence that addresses spatial measurement • Aggregate across pages to assess “opportunity to learn” specific elements of knowledge • Overarching question: Do we have the “right stuff”? STEM Presentation, 2010 NCTM

8. Conceptual Knowledge (length) • From a long list, here are two key examples • Unit-Measure Compensation: Smaller size units produce larger measures (of the same object) • A sense of identical units • An ability to fill the same space with two different units and compare the results • Unit Iteration: Measures of length are produced by tiling or iterating a length unit from one end of an object to the other, without gaps or overlaps, and counting the iterations. • A sense of identical units • Filling the space (by tiling or iterating) • The count represents the total space • Gaps and overlaps of units introduce error STEM Presentation, 2010 NCTM

9. Some length Results • Amount of content grows each elementary years • Conceptual foundation in Grades K-3 • All three curricula are heavily Procedural (more than 75% of all codes, all curricula, Grades K–3) • Central procedures • Direct Comparison • Visual & Indirect Comparison • Measure with non-standard units • Measure with rulers • Draw segments • Find perimeter STEM Presentation, 2010 NCTM

10. More Results (length) • Some attention to conceptual knowledge but attention is sparse and there are major gaps STEM Presentation, 2010 NCTM

11. Yet More Results (length) • Virtually no work with “broken rulers” • No attention to the fact that non-standard units (e.g., rectangular tiles) have multiple attributes (length, width, covering area) => sets up confusion in understanding and communication • The official terms for length are problematic • “Length” is the top-level quantity • “Length” is also a property of 2-D shapes and objects • What happens with the “length,” “width,” and “height” of objects and shapes when we rotate them? STEM Presentation, 2010 NCTM

12. Some AREA Results • Even more procedural, across curricula and grades (K–4); 88% or more of all codes • Procedural content (overview) • K-2: Emphasis on visual comparisons (which shape is larger/bigger) • Next, covering and counting • Finally, computational procedures, beginning with rectangles • Area is defined as a quantity in Grade 2 (all curricula) • Everyday Math emphasizes rectangular arrays in the service of both multiplication and area (Grades 3, 4) • Weaker attention to Unit Iteration for area than length across curricula STEM Presentation, 2010 NCTM

13. Some Volume Results(preliminary) • Long duration of development; weak conceptual clarity • “Capacity” (property of containers, continuous quantity) is interleafed with “volume” (filling and counting, discrete quantity) • But relation is never clarified • Qualitative work (more, less, equal) before quantitative • STEM has only examined Grades K–3 thus far; filling boxes begins to appear in Grade 3 STEM Presentation, 2010 NCTM

14. Resources • A solution to the problem is not yet at hand • But good teaching is possible with today’s resources • Understanding the problem is essential; watch and listen to your kids • Move away from a procedural focus • Dimensions of a solution • Ask why and why not: Make good tasks better • Violate standard tools and solutions • Listen carefully to the language of measurement discussions and support classroom communication • Make it dynamic; recover the motion in measurement STEM Presentation, 2010 NCTM

15. What you Can Do • Get into the data • National Assessment; rich site (Google “NAEP”) • Your statewide (& district, school, classroom) data • Read about kids’ thinking • Lehrer, Measurement chapter, Research Companion to PSSM (2003) • 2003 NCTM Yearbook, esp. chapter by Stephan & Clements • Target some key lessons in your materials and thinking critically about them • Grade 1 or 2 for length: Unit iteration => Ruler construction • Grade 4 or 5 for area: Why the L x W = A formula works? • Argue for the importance of measurement • Document and study your own teaching STEM Presentation, 2010 NCTM

16. A telling Contrast • Measurement competes with Number & Operations for time & attention in the elementary grades • Consider this contrast: STEM Presentation, 2010 NCTM

17. Future STEM work • We want to put our curricular knowledge to work • Lobby curriculum authors • Work with pre-service teachers (e.g., Lesson Study in measurement) • Work in professional development (e.g., experiment with one measurement lesson) • Complete our volume work in U.S. curricula • Develop some international curricular comparisons STEM Presentation, 2010 NCTM

18. IN closing • Welcome your comments & suggestions • Contact Jack at jsmith@msu.edu • Play with STEM’s simulations at https://www.msu.edu/~maleslor/STEM/simulations.html • Look for a vastly improved STEM web-site by the end of the summer; Google: “STEM Project, MSU” STEM Presentation, 2010 NCTM