Tracing the Origins of Weak Learning of Spatial Measurement

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 Tomorrows Education in
  Measurement
• (NSF, REESE Program, 2 years NSF via CSMC)
• MSU Mathematics Education Colloquium, 9-19-07

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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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The Problem (briefly)

• Students dont 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

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 figures 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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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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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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But There is a Positive Side

• Young children (e.g., 2nd grade) can learn to do
  and understand spatial measurement (Lehrer
  Schaubles 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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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

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Six Possible Factors

1. Weaknesses in written curricula
2. Too little instructional time on measurement
3. The dominance of static representations of
   spatial quantities (esp. for area volume)
4. Problems specific to talk about spatial
   quantities in classrooms (common everyday
   vocabulary, speakers talking past one another)
5. Instructional assessment focus on numerical
   computation numbers lose meaning as measures
6. Weaknesses in teachers knowledge

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Other Factors?
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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

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End of Part I
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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 K8
• 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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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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Step 1 Choose the Curricula

• Elementary (K6)
• Everyday Mathematics
• Scott Foresman-Addison Wesleys Mathematics
• Saxon Mathematics
• Middle School (68)
• Connected Mathematics Project
• Glencoes Mathematics Concepts Applications
• Saxon Mathematics
• Criteria for choice
• Market-share
• Standards-based vs. publisher developed
• Saxon is different from both
• Representativeness argument

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Step 2 Find the Measurement Content

• Should be easy, right? Just look for the
  measurementunits
• 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

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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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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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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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End of Part II
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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 Cant just analyze the
  measurement knowledge Need analysis of textual
  forms (e.g., statements vs. questions vs.
  demonstrations)
• Realization 2 Cant focus solely on conceptual
  knowledge
• Realization 3 Need to attend to curricular
  voice, who speaks to students (teacher vs. text)

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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
Convert 9 ft. to yds.
Text
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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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 ? ? Ø
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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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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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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 gt length B, length B gt
  length C, then length A gt 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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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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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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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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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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End of Part III
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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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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

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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

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Sum Up Is it Measurement?

• Much of the K8 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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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!