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, 2010

Session Overview A research sessionA research session Focus first on the problem of teaching and learning of spatial measurementFocus 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 STEM Project’s approach: Look carefully at the content of the elementary 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 adequateMain STEM message: Our elementary materials are not currently adequate 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) 10/17/2015STEM Presentation, 2010 NCTM2

The Problem (take #1: the surface) U.S. students perform poorly on spatial measurement tasks (NAEP results, especially at 4 th and 8 th grade)U.S. students perform poorly on spatial measurement tasks (NAEP results, especially at 4 th and 8 th grade) Performance declines as dimension increases (length > area > volume)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)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)Instruction focuses on procedures (ruler use & computational formulae) We’re not teaching the conceptual principles, so students are learning by roteWe’re not teaching the conceptual principles, so students are learning by rote BUT …. We can do better. BUT …. We can do better. 10/17/2015STEM Presentation, 2010 NCTM3

your Position (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 performance results (state, district, school, classroom)?Poor annual performance results (state, district, school, classroom)? Getting time (in the spring) to teach measurement?Getting time (in the spring) to teach measurement? Having to re-teach measurement?Having to re-teach measurement? Dissatisfaction with your current curriculum materials?Dissatisfaction with your current curriculum materials? Listening to kids’ talk & work with measurement?Listening to kids’ talk & work with measurement? Won’t have a solution for you; will give you some toolsWon’t have a solution for you; will give you some tools 10/17/2015STEM Presentation, 2010 NCTM4

The Problem (take #2: A Little Deeper) We spend a lot of time on length measurement with rulersWe spend a lot of time on length measurement with rulers In Michigan, statewide performance looks good for Grade 2 and 3 contentIn Michigan, statewide performance looks good for Grade 2 and 3 content Now consider the Toothpick problem on the NAEPNow consider the Toothpick problem on the NAEP Haven’t yet found a compelling item for area (for many reasons)Haven’t yet found a compelling item for area (for many reasons) There is no “ruler” for areaThere is no “ruler” for area We aren’t asking equivalent questions for area, e.g., explain how multiplying lengths produces a collections of squaresWe aren’t asking equivalent questions for area, e.g., explain how multiplying lengths produces a collections of squares 8 th & 12 th grade performance on surface area and volume is terrible8 th & 12 th grade performance on surface area and volume is terrible 10/17/2015STEM Presentation, 2010 NCTM5

Problem Sources Naming the problem year after year will not solve itNaming the problem year after year will not solve it Many likely contributing factorsMany likely contributing factors One basic one to explore: Do our written curricular materials contain the right content?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 deficienciesIf 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 areaThe STEM Project has identified specific conceptual deficits for length and area 10/17/2015STEM Presentation, 2010 NCTM6

The STEM Project (in brief) Three elementary curricula Three elementary curricula Everyday Mathematics Everyday Mathematics Scott-Foresman/Addison-Wesley’s Mathematics Scott-Foresman/Addison-Wesley’s Mathematics Saxon Mathematics Saxon Mathematics If the problem exists in these materials, we have a national problem 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, & volumeDevelop a systematic list of conceptual, procedural, and conventional knowledge 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” specific elements of knowledgeAggregate across pages to assess “opportunity to learn” specific elements of knowledge Overarching question: Do we have the “right stuff”?Overarching question: Do we have the “right stuff”? 10/17/2015STEM Presentation, 2010 NCTM7

Conceptual Knowledge (length) From a long list, here are two key examplesFrom a long list, here are two key examples Unit-Measure Compensation : Smaller size units produce larger measures (of the same object) Unit-Measure Compensation : Smaller size units 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 and compare the resultsAn 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. 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 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 errorGaps and overlaps of units introduce error 10/17/2015STEM Presentation, 2010 NCTM8

Some 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 than 75% of all codes, all curricula, Grades K–3)All three curricula are heavily Procedural (more than 75% of all codes, all curricula, Grades K–3) Central proceduresCentral procedures Direct Comparison Direct Comparison Visual & Indirect Comparison Visual & Indirect Comparison Measure with non-standard units Measure with non-standard units Measure with rulers Measure with rulers Draw segments Draw segments Find perimeter Find perimeter 10/17/20159STEM Presentation, 2010 NCTM

More Results (length) Some attention to conceptual knowledge but attention is sparse and there are major gapsSome attention to conceptual knowledge but attention is sparse and there are major gaps ElementEmphasis Definition of length Infrequent; hard to do Greater Longer Very frequent Unit-Measure Compensation Relatively frequent Unit Iteration Infrequent; focus: gaps & overlaps 10/17/201510STEM Presentation, 2010 NCTM

Yet More Results (length) Virtually no work with “broken rulers”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 communicationNo 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 problematicThe official terms for length are problematic “Length” is the top-level quantity“Length” is the top-level quantity “Length” is also a property of 2-D shapes and objects“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?What happens with the “length,” “width,” and “height” of objects and shapes when we rotate them? 10/17/2015STEM Presentation, 2010 NCTM11

Some AREA Results Even more procedural, across curricula and grades (K–4); 88% or more of all codesEven more procedural, across curricula and grades (K–4); 88% or more of all codes Procedural content (overview)Procedural content (overview) K-2: Emphasis on visual comparisons (which shape is larger/bigger)K-2: Emphasis on visual comparisons (which shape is larger/bigger) Next, covering and countingNext, covering and counting Finally, computational procedures, beginning with rectanglesFinally, computational procedures, beginning with rectangles Area is defined as a quantity in Grade 2 (all curricula)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) 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 curriculaWeaker attention to Unit Iteration for area than length across curricula 10/17/201512STEM Presentation, 2010 NCTM

Some Volume Results (preliminary) Long duration of development; weak conceptual clarityLong duration of development; weak conceptual clarity “Capacity” (property of containers, continuous quantity) is interleafed with “volume” (filling and counting, discrete quantity)“Capacity” (property of containers, continuous quantity) is interleafed with “volume” (filling and counting, discrete quantity) But relation is never clarifiedBut relation is never clarified Qualitative work (more, less, equal) before quantitativeQualitative work (more, less, equal) before quantitative STEM has only examined Grades K–3 thus far; filling boxes begins to appear in Grade 3 STEM has only examined Grades K–3 thus far; filling boxes begins to appear in Grade 3 10/17/201513STEM Presentation, 2010 NCTM

Resources 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 today’s resourcesBut good teaching is possible with today’s resources Understanding the problem is essential; watch and listen to your kidsUnderstanding the problem is essential; watch and listen to your kids Move away from a procedural focusMove away from a procedural focus Dimensions of a solutionDimensions of a solution 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 discussions and support classroom communicationListen carefully to the language of measurement discussions and support classroom communication Make it dynamic; recover the motion in measurementMake it dynamic; recover the motion in measurement 10/17/2015STEM Presentation, 2010 NCTM14

What 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) dataYour statewide (& district, school, classroom) data Read about kids’ thinkingRead about kids’ thinking Lehrer, Measurement chapter, Research Companion to PSSM (2003) Lehrer, Measurement chapter, Research Companion to PSSM (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 critically about themTarget some key lessons in your materials and thinking critically 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 measurement Argue for the importance of measurement Document and study your own teaching Document and study your own teaching 10/17/2015STEM Presentation, 2010 NCTM15

A telling Contrast Measurement competes with Number & Operations for time & attention in the elementary gradesMeasurement competes with Number & Operations for time & attention in the elementary grades Consider this contrast:Consider this contrast: 10/17/2015STEM Presentation, 2010 NCTM16 Procedural KnowledgeKey Conceptual Knowledge Number & Operation Algorithms for single & multi-digit arithmetic Place-value & composite units MeasurementProcedures (e.g., ruler use & computational formulae Unit Iteration

Future 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 Study in measurement)Work with pre-service teachers (e.g., Lesson Study in measurement) Work in professional development (e.g., experiment with one measurement lesson)Work in professional development (e.g., experiment with one measurement lesson) Complete our volume work in U.S. curriculaComplete our volume work in U.S. curricula Develop some international curricular comparisonsDevelop some international curricular comparisons 10/17/2015STEM Presentation, 2010 NCTM17

IN closing Welcome your comments & suggestionsWelcome your comments & suggestions Contact Jack at Jack at Play with STEM’s simulations at htmlPlay with STEM’s simulations at html html html Look for a vastly improved STEM web-site by the end of the summer; Google: “STEM Project, MSU”Look for a vastly improved STEM web-site by the end of the summer; Google: “STEM Project, MSU” 10/17/2015STEM Presentation, 2010 NCTM18