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 23 rd,

Getting to the Heart of Getting to the Heart of MeasurementMeasurement

(When We Usually Don’t) (When We Usually Don’t)

Jack SmithJack Smith(a.k.a., John P. Smith III)(a.k.a., John P. Smith III)

2010 NCTM Annual MeetingSan Diego, CAApril 23rd, 2010

Session OverviewSession Overview

• A research sessionA research session

• Focus first on the Focus first on the problem problem of teaching and learning of of teaching and learning of spatial spatial measurementmeasurement

• STEM Project’s STEM Project’s approach: Look carefully at the content of the approach: Look carefully at the content of the elementary written curriculumelementary written curriculum• Examine all textbook pages that address spatial measurementExamine all textbook pages that address spatial measurement• Do these materials provide sufficient “opportunity to learn”?Do these materials provide sufficient “opportunity to learn”?

• Main STEM message: Our elementary materials are not currently Main STEM message: Our elementary materials are not currently adequateadequate• Too focused on the procedures of measurementToo focused on the procedures of measurement• Neglect important conceptual issuesNeglect important conceptual issues• The solution means doing more/better with what we haveThe solution means doing more/better with what we have

• Goal: Finish in ≤ 40 minutes (balance of time for discussion)Goal: Finish in ≤ 40 minutes (balance of time for discussion)

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The ProblemThe Problem(take #1: the surface)(take #1: the surface)

• U.S. students perform poorly on spatial measurement tasks U.S. students perform poorly on spatial measurement tasks (NAEP results, especially at 4(NAEP results, especially at 4thth and 8 and 8thth grade) grade)

• Performance declines as dimension increases (length > area Performance declines as dimension increases (length > area > volume)> volume)

• In 2-D and 3-D situations, confusions of different spatial In 2-D and 3-D situations, confusions of different spatial quantities is a particular problem (e.g., perimeter & area)quantities is a particular problem (e.g., perimeter & area)

• Instruction focuses on procedures (ruler use & computational Instruction focuses on procedures (ruler use & computational formulae)formulae)

• We’re not teaching the conceptual principles, so students We’re not teaching the conceptual principles, so students are learning by roteare learning by rote

• BUTBUT…. We can do better.…. We can do better.


your Positionyour Position(relative to the problem)(relative to the problem)

• What brought you to this session?What brought you to this session?

• Do you see the problem primarily in terms ofDo you see the problem primarily in terms of• Poor annual Poor annual performance performance results (state, district, results (state, district,

school, classroom)?school, classroom)?• Getting Getting time time (in the spring) to teach measurement?(in the spring) to teach measurement?• Having to Having to re-teach re-teach measurement?measurement?• Dissatisfaction with your current Dissatisfaction with your current curriculum curriculum

materials?materials?• Listening to Listening to kids’ talk & work kids’ talk & work with measurement?with measurement?

• Won’t have a Won’t have a solution solution for you; will give you some for you; will give you some toolstools


The ProblemThe Problem(take #2: A Little Deeper)(take #2: A Little Deeper)

• We spend a lot of time on length measurement with We spend a lot of time on length measurement with rulersrulers

• In Michigan, statewide performance looks good for In Michigan, statewide performance looks good for Grade 2 and 3 contentGrade 2 and 3 content

• Now consider the Now consider the Toothpick Toothpick problem on the NAEPproblem on the NAEP

• Haven’t yet found a compelling item for area (for Haven’t yet found a compelling item for area (for many reasons)many reasons)• There is no “ruler” for areaThere is no “ruler” for area• We aren’t asking equivalent questions for area, e.g., We aren’t asking equivalent questions for area, e.g.,

explain how multiplying lengths produces a collections explain how multiplying lengths produces a collections of squares of squares

• 88thth & 12 & 12thth grade performance on surface area and grade performance on surface area and volume is terriblevolume is terrible


Problem SourcesProblem Sources

• Naming the problem year after year will not Naming the problem year after year will not solve itsolve it

• Many likely contributing factorsMany likely contributing factors

• One basic one to explore: Do our written One basic one to explore: Do our written curricular materials contain the right content?curricular materials contain the right content?

• If not, we have one root cause AND we can If not, we have one root cause AND we can work to address specific deficiencieswork to address specific deficiencies

• The The STEM STEM Project has identified specific Project has identified specific conceptual deficits for length and areaconceptual deficits for length and area


The STEM ProjectThe STEM Project(in brief)(in brief)

• Three elementary curriculaThree elementary curricula• Everyday MathematicsEveryday Mathematics• Scott-Foresman/Addison-Wesley’s Scott-Foresman/Addison-Wesley’s MathematicsMathematics• Saxon Saxon MathematicsMathematics

• If the problem exists in these materials, we have a national If the problem exists in these materials, we have a national problemproblem

• Develop a systematic list of Develop a systematic list of conceptualconceptual, , proceduralprocedural, and , and conventional conventional knowledge for length, area, & volumeknowledge for length, area, & volume

• Code every sentence that addresses spatial measurementCode every sentence that addresses spatial measurement

• Aggregate across pages to assess “opportunity to learn” Aggregate across pages to assess “opportunity to learn” specific elements of knowledgespecific elements of knowledge

• Overarching question: Do we have the “right stuff”?Overarching question: Do we have the “right stuff”?


Conceptual Knowledge Conceptual Knowledge (length)(length)

• From a long list, here are two key examplesFrom a long list, here are two key examples

• Unit-Measure CompensationUnit-Measure Compensation: Smaller size units : Smaller size units produce larger measures (of the same object)produce larger measures (of the same object)• A sense of identical unitsA sense of identical units• An ability to fill the same space with two different units An ability to fill the same space with two different units

and compare the resultsand compare the results

• Unit IterationUnit Iteration: Measures of length are produced by : Measures of length are produced by tiling or iterating a length unit from one end of an tiling or iterating a length unit from one end of an object to the other, without gaps or overlaps, and object to the other, without gaps or overlaps, and counting the iterations.counting the iterations.• A sense of identical unitsA sense of identical units• Filling the space (by tiling or iterating)Filling the space (by tiling or iterating)• The count represents the total spaceThe count represents the total space• Gaps and overlaps of units introduce error Gaps and overlaps of units introduce error


Some length ResultsSome length Results

• Amount of content grows each elementary yearsAmount of content grows each elementary years

• Conceptual foundation in Grades K-3Conceptual foundation in Grades K-3

• All three curricula are heavily Procedural (more All three curricula are heavily Procedural (more than 75% of all codes, all curricula, Grades K–3)than 75% of all codes, all curricula, Grades K–3)

• Central proceduresCentral procedures• Direct ComparisonDirect Comparison• Visual & Indirect ComparisonVisual & Indirect Comparison• Measure with non-standard unitsMeasure with non-standard units• Measure with rulersMeasure with rulers• Draw segmentsDraw segments• Find perimeterFind perimeter

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More Results More Results (length)(length)

• Some attention to conceptual knowledge but Some attention to conceptual knowledge but attention is sparse and there are major gapsattention 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 Yet More Results (length)(length)

• Virtually no work with “broken rulers”Virtually no work with “broken rulers”

• No attention to the fact that non-standard units No attention to the fact that non-standard units (e.g., rectangular tiles) have multiple attributes (e.g., rectangular tiles) have multiple attributes (length, width, covering area) => sets up (length, width, covering area) => sets up confusion in understanding and communicationconfusion in understanding and communication

• The official terms for length are problematicThe official terms for length are problematic• ““Length” is the top-level quantityLength” is the top-level quantity• ““Length” is also a property of 2-D shapes and Length” is also a property of 2-D shapes and

objectsobjects• What happens with the “length,” “width,” and What happens with the “length,” “width,” and

“height” of objects and shapes when we rotate “height” of objects and shapes when we rotate them?them?


Some AREA ResultsSome AREA Results

• Even more procedural, across curricula and grades Even more procedural, across curricula and grades (K–4); 88% or more of all codes(K–4); 88% or more of all codes

• Procedural content (overview)Procedural content (overview)• K-2: Emphasis on visual comparisons (which shape is K-2: Emphasis on visual comparisons (which shape is

larger/bigger)larger/bigger)• Next, covering and countingNext, covering and counting• Finally, computational procedures, beginning with Finally, computational procedures, beginning with

rectanglesrectangles

• Area is defined as a quantity in Grade 2 (all curricula)Area is defined as a quantity in Grade 2 (all curricula)

• Everyday Math Everyday Math emphasizes rectangular arrays in the emphasizes rectangular arrays in the service of both multiplication and area (Grades 3, 4)service of both multiplication and area (Grades 3, 4)

• Weaker attention to Weaker attention to Unit Iteration Unit Iteration for area than for area than length across curriculalength across curricula


Some Volume ResultsSome Volume Results(preliminary)(preliminary)

• Long duration of development; weak conceptual clarityLong duration of development; weak conceptual clarity

• ““Capacity” (property of containers, continuous Capacity” (property of containers, continuous quantity) is interleafed with “volume” (filling and quantity) is interleafed with “volume” (filling and counting, discrete quantity)counting, discrete quantity)

• But relation is never clarifiedBut relation is never clarified

• Qualitative work (more, less, equal) before Qualitative work (more, less, equal) before quantitative quantitative

• STEM STEM has only examined Grades K–3 thus far; filling has only examined Grades K–3 thus far; filling boxes begins to appear in Grade 3boxes begins to appear in Grade 3


ResourcesResources

• A solution to the problem is not yet at handA solution to the problem is not yet at hand

• But good teaching is possible with But good teaching is possible with today’s today’s resourcesresources

• Understanding the problem is essential; watch and Understanding the problem is essential; watch and listen to your kidslisten to your kids

• Move away from a procedural focusMove away from a procedural focus

• Dimensions of a Dimensions of a solutionsolution• Ask why and why not: Make good tasks betterAsk why and why not: Make good tasks better• Violate standard tools and solutionsViolate standard tools and solutions• Listen carefully to the language of measurement Listen carefully to the language of measurement

discussions and support classroom communicationdiscussions and support classroom communication• Make it dynamic; recover the motion in measurementMake it dynamic; recover the motion in measurement


What you Can DoWhat you Can Do

• Get into the dataGet into the data• National Assessment; rich site (Google “NAEP”)National Assessment; rich site (Google “NAEP”)• Your statewide (& district, school, classroom) Your statewide (& district, school, classroom) datadata

• Read about kids’ thinkingRead about kids’ thinking• Lehrer, Measurement chapter, Research Companion to PSSM Lehrer, Measurement chapter, Research Companion to PSSM

(2003)(2003)• 2003 NCTM Yearbook, esp. chapter by Stephan & Clements2003 NCTM Yearbook, esp. chapter by Stephan & Clements

• Target some key lessons in your materials and thinking Target some key lessons in your materials and thinking critically about themcritically about them• Grade 1 or 2 for length: Unit iteration => Ruler constructionGrade 1 or 2 for length: Unit iteration => Ruler construction• Grade 4 or 5 for area: Why the L x W = A formula works?Grade 4 or 5 for area: Why the L x W = A formula works?

• Argue for the importance of measurementArgue for the importance of measurement

• Document and study your own teachingDocument and study your own teaching04/20/2304/20/23STEM Presentation, 2010 NCTMSTEM Presentation, 2010 NCTM 1515

A telling ContrastA telling Contrast

• Measurement competes with Number & Measurement competes with Number & Operations for time & attention in the Operations for time & attention in the elementary gradeselementary grades

• Consider this contrast: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 workFuture STEM work

• We want to put our curricular knowledge to workWe want to put our curricular knowledge to work

• Lobby curriculum authorsLobby curriculum authors

• Work with pre-service teachers (e.g., Lesson Work with pre-service teachers (e.g., Lesson Study in measurement)Study in measurement)

• Work in professional development (e.g., Work in professional development (e.g., experiment with one measurement lesson)experiment with one measurement lesson)

• Complete our volume work in U.S. curriculaComplete our volume work in U.S. curricula

• Develop some international curricular Develop some international curricular comparisonscomparisons


IN closingIN closing

• Welcome your comments & suggestionsWelcome your comments & suggestions

• Contact Jack at Contact Jack at [email protected]

• Play with STEM’s simulations at Play with STEM’s simulations at https://www.msu.edu/~maleslor/STEM/simulations.html

• Look for a vastly improved STEM web-site by Look for a vastly improved STEM web-site by the end of the summer; Google: “STEM the end of the summer; Google: “STEM Project, MSU”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 Don’t) Jack Smith (a.k.a., John P. Smith III) 2010 NCTM Annual Meeting San Diego, CA April 23 rd,

Documents

stem presentation

main stem message

length measurement

elementary materials

problem year

problem of teaching

length area volumein

surface area