Exploring Spatial Measurement Through a Conceptual Lens Lorraine Males, Funda Gonulates, Shannon Sweeny, Nic Gilbertson MSU 2011 – Math in Action, Grand Valley State University Strengthening Tomorrow’s Education in Measurement (STEM) Project

Introductions Lorraine – 4 th yr doctoral student, working on the STEM project all 4 years, taught secondary methods, currently supervising interns, taught middle/high school mathematics for 8 years 2 MSU 2011 – Math in Action, Grand Valley State University Funda – 3 th yr doctoral student, working on the STEM project all 3 years, formerly taught middle school mathematics for 4 years Shannon – 4th yr doctoral student, taught elementary & middle school for 8 years (2 years as a math specialist), taught elementary math methods & currently supervise elementary MSU Nic – 2 nd year doctoral student, taught middle school & high school math for 6 years, currently teaching elementary methods courses at MSU

Agenda 3 MSU 2011 – Math in Action, Grand Valley State University Introductions Reflections on Measurement A Look at Some National Data Introduction to the STEM Project Measurement Tasks A Look at STEMS’s Curriculum Data Evaluation

Reflections on Measurement Take some time to think about and share your answer with a partner to the following: 4  What are the key ideas you want your students to know about measurement?  What do you find challenging about teaching length, area and/or volume? MSU 2011 – Math in Action, Grand Valley State University

The Toothpick (Broken Ruler) Problem “What is the length of the toothpick?” [ NAEP, Grade 4, 2003, Open response] MSU 2011 – Math in Action, Grand Valley State University 5

Toothpick Performance Data [Grade 4, 2003, open response] Response% Responding 2 ½ inches (correct) 10 ½ inches 3 ½ inches Other Omitted MSU 2011 – Math in Action, Grand Valley State University

Toothpick Performance Data Response% Responding 2 ½ inches (correct) ½ inches13 3 ½ inches20 8 ½ inches7 MSU 2011 – Math in Action, Grand Valley State University 7 Response% Responding 2 ½ inches (correct) ½ inches14 3 ½ inches23 Other42 Omitted2 [Grade 8, 2003, multiple choice][Grade 4, 2003, open response]

Toothpick Results Over Time MSU 2011 – Math in Action, Grand Valley State University 8 Percent Correct Assessment Year4 th Grade8 th Grade *58* * Statistically lower than 1996 and NAEP results across three assessments

The STEM Project Initial situation – Problem was recognized; no explanation – So no idea about where to invest in a “solution” STEM I: Examine the curricular contribution (elementary curricula) – Two years (Fall Fall 2009) – Do current US elementary mathematics provide sufficient “opportunity to learn” (OTL) spatial measurement STEM II: Put what we have learned to work – Three years (August 2009 – July 2012) – PD is one project component 9 MSU 2011 – Math in Action, Grand Valley State University

STEM – Three Curricula The three carefully chosen curricula are:  Scott Foresman- Addison Wesley Mathematics  UCSMP’s Everyday Mathematics  Saxon Math 10 MSU 2011 – Math in Action, Grand Valley State University

STEM – Our Analysis In our analysis we are looking at every lesson, problem, and activity of teaching curricula for two important aspects:  Knowledge elements - Spatial measurement knowledge (conceptual, procedural, conventional) [What content is in the textbook]  Textual elements - The ways in which this knowledge is expressed (statements, demonstrations, worked examples, questions, problems, games) [How the content is presented in the textbook] 11 MSU 2011 – Math in Action, Grand Valley State University

Measurement Tasks – Part I Length – Strange Rulers – original STEM task Area – Crazy Cakes – Investigations, grade 4/Developing Mathematical Ideas (DMI) Volume – The Wet Box Task – adapted from Dr. Michael Steele Pick one of these measures (length, area, or volume) and complete the task MSU 2011 – Math in Action, Grand Valley State University 12

Measurement Tasks – Part II What would a student need to understand about measurement in order to successfully complete this task? [Record on your poster paper] MSU 2011 – Math in Action, Grand Valley State University 13

Some Results (LENGTH) All three curricula are heavily procedural (more than 75% of all codes, all curricula, Grades K–3) Common procedures – Direct Comparison – Visual & Indirect Comparison – Measure with Rulers – Draw line segments of given lengths – Calculate perimeter 14 MSU 2011 – Math in Action, Grand Valley State University

More Results (LENGTH) Some conceptual knowledge is addressed ElementFrequency Definition of lengthUncommon; hard to do Greater means longerVery common Unit-measure compensationFairly common Unit IterationUncommon; gaps & overlaps 15 MSU 2011 – Math in Action, Grand Valley State University

Some Results (AREA) All three curricula are even more procedural with area than with length (more than 88% in grade K-4) Common procedures – Visual Comparison – Covering and counting to find area – Computing area with formulas (starting with rectangles) 16 MSU 2011 – Math in Action, Grand Valley State University K-1 K-2 2-3

More Results (AREA) Very little conceptual knowledge ElementFrequency Definition of areaUncommon Unit IterationUncommon Unit-measure compensationUncommon Area remains the same when partitioned (gr. 3 & 4) Very uncommon (1 in each curricula) Composition/Decomposition of Regular Polygons (gr. Gr. 4) Very uncommon (2 in EM) 17 MSU 201 – Math in Action, Grand Valley State University

Some Results (VOLUME) All three curricula are heavily procedural (more than 75% of all codes, all curricula, Grades K–1) Common procedures – Visual Comparison – Measure capacity/volume with standards and non-standard units – Estimating Capacity/Volume 18 MSU 2011 – Math in Action, Grand Valley State University

More Results (VOLUME) Some conceptual knowledge is addressed ElementFrequency Greater means largerUncommon Definition of volume/capacityVery Uncommon Unit-measure compensationVery Uncommon Unit IterationVery Uncommon Only units of vol/cap can be used to measure volume/capacity Uncommon The spatial structure of rectangular prisms Uncommon 19 MSU 2011 – Math in Action, Grand Valley State University

Major Lessons Conceptual foundations of measurement are weakly developed Weak attention to Unit Iteration (length, area, volume) Conjecture: The sheer extent of visual content on the page (esp. for EM & SFAW) may make it hard for teachers to find and focus on the conceptual content Implication: Teachers will need to enrich the written curriculum 20 MSU 2011 – Math in Action, Grand Valley State University

Thank you! We want to thank the National Science Foundation for funding this work We want to thank you for coming! Please take a few minute to fill out our evaluation. For more information : If you have any questions please us at: 21 MSU 2011 – Math in Action, Grand Valley State University