Getting to the Heart of Measurement (When We Usually...

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

• 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)

8/3/2012 STEM Presentation, 2010 NCTM 2

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.


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 8/3/2012 STEM Presentation, 2010 NCTM 4

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


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


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”?


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


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

8/3/2012 9 STEM Presentation, 2010 NCTM

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


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?


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


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


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


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


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

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


IN closing

• Welcome your comments & suggestions

• Contact Jack at [email protected]

• Play with STEM’s simulations at

https://www.msu.edu/~maleslor/STEM/simulations.ht

ml

• Look for a vastly improved STEM web-site by the

end of the summer; Google: “STEM Project, MSU”


mailto:[email protected]

https://www.msu.edu/~maleslor/STEM/simulations.html

https://www.msu.edu/~maleslor/STEM/simulations.html

Getting to the Heart of Measurement (When We Usually...

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