Exploring Spatial Measurement by Attending to Core Conceptual Principles Funda Gonulates Lorraine Males Dan Clark MSU 2011 – NCTM Research Session, Indianapolis, IN Strengthening Tomorrow’s Education in Measurement (STEM) Project

Session Overview 2 Introductions The Problem STEM Project Overview Results and So What? Discussion [last 20 minutes] MSU 2011 – NCTM Research Session, Indianapolis, IN

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 3 MSU 2011 – NCTM Gallery Session, Indianapolis, IN Funda – 3 rd yr doctoral student, working on the STEM project all 3 years, taught elementary content course, formerly taught middle school mathematics for 4 years Dan – 3 rd yr doctoral student, working on the STEM project for 2 years, taught mathematics in college for 5 years, currently teaching mathematics content courses for aspiring elementary teachers

The Toothpick (Broken Ruler) Problem “What is the length of the toothpick?” [ NAEP, Grade 4, 2003, Open response] 4 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Toothpick Performance Data [Grade 4, 2003, open response] Response% Responding 2 ½ inches (correct) 10 ½ inches 3 ½ inches Other Omitted MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Toothpick Performance Data Response% Responding 2 ½ inches (correct) ½ inches13 3 ½ inches20 8 ½ inches7 6 Response% Responding 2 ½ inches (correct) ½ inches14 3 ½ inches23 Other42 Omitted2 [Grade 8, 2003, multiple choice][Grade 4, 2003, open response] MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Toothpick Results Over Time 7 Percent Correct Assessment Year4 th Grade8 th Grade *58* * Statistically lower than 1996 and NAEP results across three assessments MSU 2011 – NCTM Gallery Session, Indianapolis, IN

‘ Possible Contributing Factors 8 MSU 2011 – NCTM Gallery Session, Indianapolis, IN The Problem of Learning Spatial Measurement The Problem of Learning Spatial Measurement Time & Timing of Instruction Time & Timing of Instruction “procedural” “calculational” Discourse Challenges Teachers’ Understandings Limitations of Written Curricula Limitations of Written Curricula State Standards & Assessments Static Representations ambiguous terms ambiguous reference (2D, 3D) essential content student challenges content presentation in text Focus of instruction

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 – NCTM Gallery Session, Indianapolis, IN

STEM – Three Curricula The three carefully chosen curricula are:  Scott Foresman- Addison Wesley Mathematics  UCSMP’s Everyday Mathematics  Saxon Math 10 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

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 – NCTM Gallery Session, Indianapolis, IN

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 12 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

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 13 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Unit Iteration 14 MSU 2011 – NCTM Gallery Session, Indianapolis, IN. Conceptual principles? Units cannot have gaps or overlaps Units must lie on the path being measured Conceptual principles? Units must have equal length. Units cannot have gaps/overlaps Units must lie on the path being measured Ruler is a collection of iterated units of length Possible improvements Change unit so students need to recognize units must have equal length Make part of instruction not only assessment Ask student to explain the principle violations made in each of the “wrong” answers Possible improvements Clarify attribute being measured In demonstration, use opportunity to help students see that the ruler could be used to measure from points other than zero. (Everyday Mathematics, Grade 1, Assessment Handbook, P. 152) (SFAW, Grade 1, Teacher’s Guide, P. 371B)

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) 15 K-1 K MSU 2011 – NCTM Gallery Session, Indianapolis, IN

More Results (AREA) Very little conceptual knowledge ElementFrequency Definition of area (gr. 2 and up) Relatively common 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. 4) Very uncommon (2 in EM) 16 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Definition of Area 17 MSU 2011 – NCTM Gallery Session, Indianapolis, IN. Conceptual principles? Covering a region inside a figure Counting number of square units Conceptual principles? Area quantifies the space inside a shape. Multiple methods- units of measure – not just square units Unit measure compensation Possible improvements Area quantifies the space enclosed in a region. Counting square units is one way. Regions does not have to be rectangular. Provide non-conventional shapes like a leaf Measurement without a grid paper Possible improvements Regions does not have to be rectangular Provide non-conventional shapes like a leaf Advantages of square units over non- square units in area measurement (Everyday Mathematics, Grade 3, Teacher’s Guide, P. 203) (SFAW, Grade 3, Teacher’s Guide, P. 468)

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 – NCTM Gallery Session, Indianapolis, IN

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 – NCTM Gallery Session, Indianapolis, IN

More Results (Volume) Distinctions between volume and capacity are often unclear 50 total definitions of the two terms across the three curricula Volume tends to be thought of as displacement, capacity as a property of containers Volume tends to be discussed in discrete units (eg.: cm 3 ), capacity in continuous units (eg.: mL) These are just general trends. No overarching statements of difference between volume and capacity can be given that we found to be universally agreed upon. STEM MCTM Conference20

Major Lessons Written curriculum is procedurally focused Conceptual foundations of measurement are weakly developed Weak attention to important conceptual content across all three measures (e.g., Unit Iteration, Unit- Measure Compensation) Implication: Teachers will need to enrich the written curriculum 21 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Discussion 22 MSU 2011 – NCTM Gallery Session, Indianapolis, IN

Thank you! We want to thank the National Science Foundation for funding this work We want to thank you for coming! For more information : Please fill out our contact form on our website If you have any questions please us at: 23 MSU 2011 – NCTM Gallery Session, Indianapolis, IN