1. 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,

2. 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
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3. 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
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4. 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)
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5. 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
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6. 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)
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7. 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”?
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8. 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)
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9. 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
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10. STEM Presentation, MEC, 9-07
Other Factors?
Write new factors on the board
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11. 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.
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12. STEM Presentation, MEC, 9-07
End of Part I
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13. 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)
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14. 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
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15. 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
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16. 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
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17. 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
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18. 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
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19. 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?
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20. STEM Presentation, MEC, 9-07
End of Part II
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21. 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
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22. 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.
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23. 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
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24. 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
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25. 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
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26. 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.”
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27. 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).”
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28. 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)
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29. 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.
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30. 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
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31. STEM Presentation, MEC, 9-07
End of Part III
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32. 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
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33. 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?
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34. 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
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35. 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
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36. 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!
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