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

Getting to the Heart
of Measurement
(When We Usually Don’t)
Jack Smith
(a.k.a., John P. Smith III)
2010 NCTM Annual Meeting
San Diego, CA
April 23rd, 2010
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
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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
•
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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)
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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.
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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
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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
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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
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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 ”?
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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.
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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
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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
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Direct Comparison
Visual & Indirect Comparison
Measure with non-standard units
Measure with rulers
Draw segments
Find perimeter
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More Results
(length)
• Some attention to conceptual knowledge but attention
is sparse and there are major gaps
Element
Emphasis
Definition of length
Infrequent; hard to do
Greater <=> Longer
Very frequent
Unit-Measure Compensation
Relatively frequent
Unit Iteration
Infrequent; focus: gaps &
overlaps
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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?
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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
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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
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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
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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
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A telling Contrast
• Measurement competes with Number & Operations
for time & attention in the elementary grades
• Consider this contrast:
Procedural Knowledge
Key Conceptual
Knowledge
Number &
Operation
Algorithms for single &
multi-digit arithmetic
Place-value &
composite units
Measurement
Procedures (e.g., ruler
use & computational
formulae
Unit Iteration
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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
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IN closing
• Welcome your comments & suggestions
• Contact Jack at [email protected]
• 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”
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