Tracing the Origins of Weak Learning of Spatial Measurement Jack Smith & STEM Team* Leslie Dietiker, Gulcin Tan (METU), KoSze Lee, Hanna Figueras, Aaron Mosier, Lorraine Males, Leo Chang, & Matt Pahl “Strengthening Tomorrow’s Education in Measurement” (NSF, REESE Program, 2 years & NSF via CSMC) MSU Mathematics Education Colloquium, 9-19-07

Presentation Overview (major chunks) The Problem: What is it and why address it? Project Design: STEM structure & logic “Results” #1: Our Curriculum Coding Scheme (in development) “Results” #2: Tough calls in deciding what is measurement content Wrap-Up and look ahead 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 The Problem (briefly) Students don’t seem to understand spatial measurement very well (length, area, & volume) Highly practiced, routine problems OK Any sort of non-routine or multi-step problem not OK Poor written explanations Not a problem of understanding the quantities and intuitively how to measure them (e.g., covering a surface) Students “confuse” measures, in 2-D and 3-D situations Length is poorly related to area and volume 2/28/2019 STEM Presentation, MEC, 9-07

NAEP Evidence (2003 Mathematics Assessment) 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) Gap between poor & minority students and majority students is greatest for measurement (4th & 8th) 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 Evidence from TIMSS Overall (and consistently) our 4th graders perform pretty well and our 8th graders lag The 8th grade lag was greatest for geometry and measurement Textbook analysis also showed U.S. texts included less geometry & measurement content in grades 5 to 8 Lag is not made up in 12th grade 2/28/2019 STEM Presentation, MEC, 9-07

Evidence from Empirical Research Common finding: Confusion of area and perimeter for simple 2-D figures (Woodward & Byrd, 1983; Chappelle & Thompson, 1999) Poor grasp of the relationship between length units and area units (e.g., inches and square inches) (Nunes, Light, & Mason, 1993; Kordaki & Portani, 1996) Weak understanding of how length is related to area & volume in computational formulas (Battista, 2004) Not all elementary students “see” rows and columns in a rectangular arrays (Battista 1998; Battista, et al. , 1998) 2/28/2019 STEM Presentation, MEC, 9-07

But… There is a Positive Side Young children (e.g., 2nd grade) can learn to do and understand spatial measurement (Lehrer & Schauble’s work at U. Wisconsin) Carefully designed tasks Expert guidance from thoughtful researchers Teachers who understand & question children Similar results for length from Stephan & Cobb (1st grade, 2005?) Common element: “Problematize unit”; build a kids’ theory of measurement Issue: What to do with these “existence proofs”? 2/28/2019 STEM Presentation, MEC, 9-07

Sharpening the Problem What explains the problem of learning & teaching spatial measurement (length, area, & volume) in ordinary American classrooms? Why is performance/understanding so low even with students’ extensive experience space and informal measurement outside of school? Lots of evidence OF the problem but no explanation of its nature & genesis Quandary for educators: What do we work on to help? Curriculum Pre-service teacher education Professional development Importance: Valuable knowledge for both college and non-college bound students; links to content in Algebra (analytic geometry) 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 Six Possible Factors Weaknesses in written curricula Too little instructional time on measurement The dominance of static representations of spatial quantities (esp. for area & volume) Problems specific to talk about spatial quantities in classrooms (common everyday vocabulary, speakers talking past one another) Instructional & assessment focus on numerical computation; numbers lose meaning as measures Weaknesses in teachers’ knowledge 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 Other Factors? Write new factors on the board 2/28/2019 STEM Presentation, MEC, 9-07

Commentary on the Factors These factors constitute a space of solutions Cartesian analogy: Solution is a region in 6-space, with a range of values on each dimension But… the dimensions are not independent; there are many relations of influence Our approach (analogy to statistical models) Look for “main effects” Expect large (massive?) “interactions” We start with Factor 1 (written curricula) because curricula are fundamental Examples of factor interactions: Support claim of fundamental: (1) textbooks influence teacher knowledge, (2) textbooks may help or hinder classroom discourse on spatial quantities and measurement, (3) textbooks can vary in their focus on computation relative to meaning of operations. 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 End of Part I 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 STEM Project Overview Goals: Assess impact of Factor 1 (quality of written curricula) carefully and Factors 4 & 5 selectively Focus exclusively on length, area, & volume, grades K–8 Develop an “objective” standard for evaluating the measurement content of select written curricula How much of the problem can be attributed to the content of written curricula? Prepare for next steps (pursue a program of research on this problem) 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 STEM Project Sequence 1. Pick a small number of representative elementary and middle school mathematics curricula 2. Locate the measurement content of these curricula 3. Develop an appropriate framework for evaluating the that content Mathematically accurate and deep Informed by existing research 4. Complete the evaluation 5. Report the evaluation, to the community & the authors 6. Examine some classroom “enactments” of specific measurement topics 2/28/2019 STEM Presentation, MEC, 9-07

Step 1: Choose the Curricula Elementary (K–6): Everyday Mathematics Scott Foresman-Addison Wesley’s Mathematics Saxon Mathematics Middle School (6–8): Connected Mathematics Project Glencoe’s Mathematics: Concepts & Applications Criteria for choice Market-share Standards-based vs. publisher developed Saxon is different from both Representativeness argument 2/28/2019 STEM Presentation, MEC, 9-07

Step 2: Find the Measurement Content Should be easy, right? Just look for the “measurement”units In fact, has not been so easy We include “measurement” content, but also other content that looks like measurement to us Units of text: units, lessons, problems We include measurement lessons & problems (in non-measurement lessons) Our criterion: Does this content very likely require reasoning with/about measures of length, area, or volume? If so, it is “in” If we get there in the talk, there are interesting issues related to specific choices, some that are pending for us right now. A lot more to say about this step, and if time allows we will come back to this step 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 Step 3: Develop the Framework (henceforth, Curriculum Coding Scheme [CCS]) Quality of the analysis depends directly on the validity & applicability of the CCS Core STEM question: Do students have sufficient opportunity to learn the mathematics of spatial measure? Validity of the CCS depends on: Mathematical completeness & depth Learning from the empirical research literature Review by “experts” Applicability of the CCS depends on: Match to textual types in written curricula Appropriate grain-size of measurement knowledge 2/28/2019 STEM Presentation, MEC, 9-07

Step 4: Code the Curricula (i.e., the spatial measurement content) Our current state: Step 2: 90% complete, some thorny issues in middle school Step 3: Detailed CCS for length, 80% complete Have “test-driven” versions of the CCS This semester: Code the elementary length content Some content explicitly involves multiple measures Still need to decide which of “length & area” and “length & volume” will included in the length analysis Shape of the analysis (some options): Results for length, for area, and for volume OR Length, area, length & area, volume, length & volume, surface area & volume 2/28/2019 STEM Presentation, MEC, 9-07

Step 5: Explore Some Classroom Enactments “Some enactments”: Limited time & resources Want to extend the use of the CCS to classroom lessons Same question: Do students in this classroom have sufficient opportunity to learn this measurement topic? Our target lesson segments: Introduction to length Complex lengths Introduction to area Area & perimeter Surface area & volume Videotape & analyze lessons; Not an evaluation of teachers Focus: How do teachers who are using their curricula seriously transform it in their teaching? What effect on OTL? 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 End of Part II 2/28/2019 STEM Presentation, MEC, 9-07

Overview of Development Process (Curriculum Coding Scheme [CCS]) Initial focus was on conceptual knowledge, because research suggested doing without understanding Identified elements of knowledge that holds for quantities in general (e.g., transitivity) before those that hold for spatial quantities specifically Realization #1: Can’t just analyze the measurement knowledge; Need analysis of textual forms (e.g., statements vs. questions vs. demonstrations) Realization #2: Can’t focus solely on conceptual knowledge Realization #3: Need to attend to curricular voice, who speaks to students (teacher vs. text) Examples of quantity knowledge generally and specifically spatial knowledge Realization #2: Conceptual knowledge is sparse in curricula; can’t ignore procedural knowledge; adding PK takes nothing away from analysis of CK Realization #3: Lessons from curriculum theorists; text as a form of communication 2/28/2019 STEM Presentation, MEC, 9-07

Curriculum Coding Scheme Conceptual Procedural Conventional Text 3 feet = 1 yard Statements Teacher Text Questions “What happens to the measure when the unit is changed?” Teacher By teacher Demos By others Worked Examples Text Text Convert 9 ft. to yds. Problems Teacher Text Games Teacher One unit of length is equivalent to some number of a different unit of length. …multiply the given length by a ratio of the two length units. Table of numerical conversion ratios. 2/28/2019 STEM Presentation, MEC, 9-07

Overview: Curriculum Coding Scheme Textual Elements Conceptual Knowledge (40 elements) Procedural Knowledge (25 elements) Conventional Knowledge (9 elements) Statements  Questions Demonstrations ? Worked Examples Ø Problems Games Need to distinguish Questions from Problems 2/28/2019 STEM Presentation, MEC, 9-07

Focus on Length First in CCS Length is fundamental spatially Length gets lots of curricular attention (e.g., measured in sheer number of pages & problems) Introduced early in elementary grades, still part of the middle school curriculum 2/28/2019 STEM Presentation, MEC, 9-07

Common Length Topics (by grade band) Grades K-2 Grades 3-5 Grades 6-8 Estimate & Measure objects; non-stan. units Measure with rulers Perimeter formulas Estimate & Measure objects; standard units English & metric systems & unit conversions Scaling & similarity Draw segments of given length Find perimeters of polygons Pythagorean Theorem Find perimeters of polygons Estimate lengths Slope 2/28/2019 STEM Presentation, MEC, 9-07

Conceptual Knowledge (length) General truths about length & the measurement of length Some examples of “deep” conceptual knowledge: Transitivity: “The comparison of lengths is transitive. If length A > length B, length B > length C, then length A > length C.” Unit-measure compensation: “Larger units of length produce smaller measures of length.” Additive composition: “The sum of two lengths is another length.” Multiplicative composition: “The product of a length with any other quantity is not a length.” 2/28/2019 STEM Presentation, MEC, 9-07

More Conceptual Knowledge Examples (length) Midpoint Definition: The midpoint of a segment is the point that divides the segment into two equal lengths. Pythagorean Theorem: “In right triangles, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on other two sides.” Circumference to radius: “The circumference of any circle is proportional to the length of the radius (or diameter).” 2/28/2019 STEM Presentation, MEC, 9-07

Procedural Knowledge (length) General processes for determining measures Broad interpretation of “process” Visual, e.g., comparison, estimation Physical, e.g., using a ruler Numerical, e.g., computations with measures Visual as well as physical & numerical processes Generally, elements of PK are not procedural images of CK Two instances were the match is close: Perimeter: Meaning/definition (CK); How to compute (PK) Pythagorean Theorem: The relationship (CK); How to compute missing sides (PK) 2/28/2019 STEM Presentation, MEC, 9-07

Some Procedural Knowledge Examples (length) Visual Estimation: Use imagined unit of length, standard or non-standard, to estimate the length of a segment, object, or distance. Draw Segment of X units with Ruler: Draw a line segment from zero to X on the ruler. Unit Conversion: To convert a length measure from one unit to another, multiply the given length by a ratio of the two length units. 2/28/2019 STEM Presentation, MEC, 9-07

Conventional Knowledge (length) Cultural conventions of representing measures; devoid of conceptual content Notations, features of tools (e.g., marks on rulers), numerical ratios in English system 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 End of Part III 2/28/2019 STEM Presentation, MEC, 9-07

Back to Step 2: Is it Measurement? Recall criterion: “…very likely involves measurement reasoning” Process: Team discussion toward consensus; time intensive Face validity of the process: We have excluded nothing that curriculum authors present as measurement Four coding categories: ** “very likely measurement reasoning required” ?? “measurement reasoning possible” P “pre-measurement” No code Only ** content will be analyzed Want to show/discuss some surprising results & problematic choices 2/28/2019 STEM Presentation, MEC, 9-07

Interesting Result: Fractions via Partitioned Regions Many ways to introduce fractions: Positions on the number line, parts of sets, parts of 2-D shapes The last may be the most common Construct an equal partition of a shape (“a whole”) Quantify a subset of the resulting parts Initial view: Fractions is a number/operations topic But when criterion is applied, we include some partitioning and some fraction problems b/c they entail measurement reasoning (i.e., visual comparison of areas) Consider two examples Vote1: How many agree with our coding decision? How many disagree? How many are not sure? Vote2: If visual comparison of areas is accepted as measurement reasoning, how many agree, disagree, are not sure? 2/28/2019 STEM Presentation, MEC, 9-07

Problematic Topic: Ratios of Lengths Ratio and related topics are important measurement content in middle school Lengths can be arguments in ratios; numbers (quantities) to be related/compared Lengths can also be found by reasoning with ratios (similarity, trig) Our struggle: In a variety of ratio contexts, when is measurement reasoning very likely? Consider two examples Example 1: Measurement, not measurement, not sure Same for the other examples 2/28/2019 STEM Presentation, MEC, 9-07

Sum Up: Is it Measurement? Much of the K–8 spatial measurement content is unproblematic to identify & include But some has not been; Surprises & problems Must get this step right; consequences of mistakes at this step are large and negative If not coded as measurement, will not be analyzed 2/28/2019 STEM Presentation, MEC, 9-07

STEM Presentation, MEC, 9-07 Conclusion We hope that we have convinced you of the importance of the problem In search of an explanation, we must explore a complex space (main effects & interactions) We hope to be back next year with real results But we have miles to go before we rest Thank you! 2/28/2019 STEM Presentation, MEC, 9-07