Project Staff

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Project Staff Advisory Board

• Project Staff
• PI me
• Graduate Students Kuo-Liang Chang, Leslie
  Dietiker, Hanna Figueras, KoSze Lee, Lorraine
  Males, Aaron Mosier, Gulcin Sisman (METU)
• Undergraduates Patrick Greeley, Matthew Pahl
• Advisory Board
• Thomas Banchoff (Brown), Michael Battista (Ohio
  St.), Richard Lehrer (Vanderbilt), Gerald Ludden
  (MSU), Deborah Shifter (EDC), Nathalie Sinclair
  (Simon Fraser)

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Our Session Goals

• Motivate more research on learning and teaching
  spatial measurement
• Length, area, volume measurement
• Describe our STEM project (as one research
  effort)
• Enable and learn from discussion with you

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Session Overview

• Prior research (Kosze)
• STEM overview (Hanna)
• Locating the spatial measurement content
  (Lorraine)
• Our principal tool for assessing curricular
  capacity (Leslie)
• Results thus far (length primary grades) (Jack)
• Comments from a measurement expert (Rich)
• QA discussion (All of us)

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Prior Research

• Kosze Lee

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Prior Research Categories of Studies

• Students performance in spatial measurement from
  large scale studies
• NAEP
• TIMSS
• Smaller studies examining students solutions and
  reasoning on spatial measurement tasks
• Length
• Area and its relation to length

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Large Scale Assessments

• National and international studies indicated US
  students are weak in learning measurement
• NAEP (2003) Low Performance by 4th, 8th, and
  12th graders
• TIMSS (1997) gap between US 8th graders and
  their international peers is greatest in geometry
  measurement
• Minority students and girls face more struggle
  (Lubienski, 2003)

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Students struggles with length

• Unaware that any point on a scale can serve as
  the starting point. (Lehrer, 2003 NAEP, 2003)
• Count marks (vs interval) on the scale
  (Boulton-Lewis et al., 1996)

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Example (length)

• A large majority students fail to find the length
  of a segment in a broken ruler task. (NAEP,
  grade 4, 2003)

2.5 inch?
10.5 inch?
3.5 inch?
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Students struggles with area

• Conceptual challenges
• Square as a unit of measurement (Kamii and Kysh,
  2006)
• Visualizing the row-by-column structure of
  tiled rectangle as area measure (Battista,
  2004)
• Relating area and length
• Confusing area with perimeter (Kidman Cooper,
  1997 Moyer, 2001 Woodward Byrd, 1983)
• Difficulties in relating the length units with
  area units (Chappell Thompson, 1999 Battista,
  2004)

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But students can do better!

• Teaching experiments show that elementary
  students can learn to do and understand
  measurement (Lehrer et al., 1998 Stephan,
  Bowers, Cobb, Gravemeijer, 2003)
• Students progressively construct understanding of
  knowledge and measuring processes built into
  standard rulers
• Core units, unit iteration, how to deal with
  left-overs

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How can we explain the weaknesses?

• The weaknesses are systematic, fundamental, and
  pervasive
• No compelling explanations have been proposed
• Hunches only
• No strong empirical basis
• So.What are some possible explanations for
  students continuing struggles to learn spatial
  measurement?

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Possible Explanatory Factors

• 1) Weaknesses in the K-8 written curricula
• Procedurally-focused (Kamii and Kysh, 2006)
• 2) Insufficient instructional time
• Usually located at the end of textbooks and
  taught at the end of the school year (Tarr,
  Chavez, Reys, Reys, 2006)
• 3) Static representations of 2D 3D quantities
  (Sinclair Jackiw, 2002)
• Dynamic representations could help show how
  length units can compose area and volume

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More Explanatory Factors

• 4) Classroom discourse about measurement poses
  special challenges (Sfard Lavie, 2005)
• Ambiguous references to spatial quantities and
  numbers
• 5) General calculational orientation in
  classroom instruction and discourse (Thompson,
  Phillip, Thompson, Boyd, 1994)
• divorce the value of measure from its spatial
  conception
• 6) Weaknesses in teachers knowledge (Simon
  Blume, 1994)
• These factors likely influence and interact with
  each other

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So why target written curricula?

• Weakness in written curricula influence other
  factors
• Analysis of written curricula has national scope
• Large scale classroom studies are
  resource-intensive
• Analyzing widely-used curricula provide maximal
  access to problems faced by most parts of the
  nation
• Clarify the exact nature of curricular weaknesses
• More focused than general characterizations
  (procedural focus)
• Beyond the presence/absence of topics

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STEM Project Overview

• Hanna Figueras

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Research Question

• What is the capacity of U.S. K - 8 written and
  enacted curricula to support students learning
  and understanding of measurement?

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STEM Project Overview

• Assess carefully the impact of Factor 1 (quality
  of written curricula)
• Assess selectively Factors 3, 4 5 (nature of
  the enacted curriculum for specific lesson
  sequences)
• Focus on spatial measurement in grades K-8
• length, area, volume
• Exclude measurement of angle
• Draws on different roots than measurement of
  spatial extent (Lehrer et al., 1998)
• Written curricula seemed like a good place to
  start

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STEM Project Overview (contd)

• How much of the problem can be attributed to the
  content of written curricula?
• Develop an unbiased standard for evaluating the
  measurement content of select written curricula
• Phase 1 - Analysis of written curricula
• Phase 2 - Examination of enacted curricula
• Start with length
• Appears first, beginning in Kindergarten
• Foundational for area and volume
• Most extensive coverage and development

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Which Curricula?

• Elementary School Curricula (K6)
• Everyday Mathematics
• Scott Foresman-Addison Wesley Mathematics
• Saxon
• Middle School Curricula (68)
• Connected Mathematics Project
• Glencoes Mathematics Concepts Applications
• Saxon

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Project Development Process
Locating Measurement Content
Creating Framework
Generating Knowledge Elements
Coding Content
Analysis
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Project Goals

• Our goal is not to rank the three curricula at
  each level
• National scorecard for written curricula in
  spatial measurement
• Expect different patterns of strengths and
  weaknesses
• Do we have common patterns of weakness (across
  curricula)?

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Locating the Spatial Measurement Content

• Lorraine Males

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Finding Measurement Content

• The Task
• Compiling a list of all pages where measurement
  content (e.g., tasks) is found in each
  curriculum.
• Who Does It
• Lead coder for each curriculum with a secondary
  coder to verify their work.
• What It Means
• Reading through every page of each written
  curriculum and noting where spatial measurement
  concepts are utilized.

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Establishing Measurement Content

• Our Fundamental Principle
• We will count as "measurement" all lessons,
  problems, and activities where students are asked
  to complete some spatial measurement reasoning,
  either as the intended focus of study or in order
  to learn some other content.

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Finding Measurement Content

• All content designated as spatial measurement in
  the written curricula will be coded.
• However, every page does need to be examined, not
  just the measurement chapters.
• In the chapter Measurement and Basic Facts we
  have Measure your bed with your hand span (EM,
  1, p. 285).
• In the chapter entitled Addition and Subtraction
  (EM) we have Measure the length of this line
  segment. Circle the best answer (EM, 2, p. 281).

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Finding Measurement Content

• Difficulties
• Judging if the content is likely to engage
  measurement reasoning.
• Determining which spatial attribute is being
  addressed.

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Establishing Measurement Content

• Types of Measurement
• Pre-Measurement
• Measurement proper
• Reasoning with or about Measurement

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Pre-Measurement

• Reasoning about spatial measurements without
  appeal to units and enumeration
• Is your tower of cubes the same size as the
  persons next to you? How do you know? Hold it
  next to your neighbors tower. Is it the same?
  (Saxon, K, p. 8-2)

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Measurement Proper

• Partitioning and iterating a spatial unit to
  produce a spatial measure. This content is what
  is commonly classified as measurement.

(SFAW, 1, p .365)
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Reasoning with or about Measurements

• Using spatial measures to determine other
  quantities, spatial or non-spatial.
• It takes about 5 seconds for the sound of
  thunder to travel 1 mile. About how far can the
  sound of thunder travel in 1 minute? (EM, MinM
  1-3, p. 81)

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Lessons from Applying the Principle

• Determining the focal spatial quantity can be
  problematic.
• How is perimeter different from area? (SFAW, 2,
  p. 351A)
• Even if the focal spatial quantity can be
  determined, it is not trivial to determine if
  measurement reasoning will be utilized.

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Lessons from Applying the Principle

• We think there are topics that are not
  traditionally considered measurement content that
  utilize spatial measurement reasoning.
• Draw lines to show how to divide the square into
  fourths in two different ways (Saxon, 1, p.
  119-7).

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Our principal tool for assessing curricular
capacity

• Leslie Dietiker

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Start of Process

• Started with conceptual knowledge found in
  research
• Identified elements of knowledge that holds for
  quantities in general) before those that hold for
  spatial quantities specifically
• Transitivity The comparison of lengths is
  transitive. If length A gt length B, length B gt
  length C, then length A gt length C.
• Unit-measure compensation Larger units of
  length produce smaller measures of length.
• Additive composition The sum of two lengths is
  another length.
• Multiplicative composition The product of a
  length with any other quantity is not a length.

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Realization 1

• We cannot just analyze the measurement knowledge
  we need analysis of textual forms
• Why do you get different answers when you
  measure the same object using cubes and paper
  clips? SFAW, grade 2, p. 341
• When changing from larger units to smaller
  units, there will be a greater number of smaller
  units than larger units. Glencoe, Course 1, p.
  465

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Textual Elements

• Statements
• Questions
• Problems
• Demonstrations
• Worked Examples
• Games

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Realization 2

• We cannot focus solely on conceptual knowledge
  we need to capture procedural knowledge
• General processes for determining measures
• Broad interpretation of process
• Generally, PK elements are distinct from CK (with
  some exceptions unit conversion, perimeter, and
  Pythagorean Theorem)

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Procedural Knowledge Elements

• Visual Estimation Use imagined unit of length,
  standard or non-standard, to estimate the length
  of a segment, object, or distance.
• Draw Segment of X units with Ruler Draw a line
  segment from zero to X on the ruler.
• Unit Conversion To convert a length measure
  from one unit to another, multiply the given
  length by a ratio of the two length units.

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Conventional Knowledge

• Cultural conventions of representing measures
  devoid of conceptual content
• This is one inch
• Notations, features of tools (e.g., marks on
  rulers)
• Rulers have inches on one side and centimeters on
  the other.

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Realization 3

• We need to attend to curricular voice (who
  speaks to students)
• Teacher
• Textbook or other written materials
• Others (in case of Demonstrations)

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Coding Measurement Content
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Coding Measurement Content
Question - Provided by Teacher Direct Comparison
x 2
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Coding Measurement Content
Worked Example - Student Text Measurement with
non-standard units
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Coding Measurement Content
Problem - Student Text Measurement with
non-standard units
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Sample Coding Sheet





Problems
Worked Examples
Questions
Larger units of length produce smaller measures of length
Measure length with Non-standard units
1
1
Direct comparison
2
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Coding Scheme
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Length Results for Grades K 1

• Jack Smith

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Some Generalities

• An intermediate view of key spatial measurement
  topics in each curriculum (STEM Top 10)
• Continuous quantity (e.g., strings of cubes) site
  for both number ( operation) and length
  measurement
• Saxon SFAW
• Tough coding decisions for us
• K2 contains the foundation for length
  measurement
• Substantial content devoted to the topic
• Deficits may not get corrected in later grades
• Were short of our conference goal Grade 2 in
  process

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Density of Length Content
EM SFAW Saxon
Pages (K) 60 62 31
Total Codes (K) 210 261 131
Codes/Page (K) 3.5 4.2 4.2

Pages (1) 108 141 67
Total Codes (1) 479 893 150
Codes/Page (1) 4.4 6.3 2.5

Pages (2) 88 140 76
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Overview of K Results

• Textual presentation
• Problems, Questions, Demonstrations dominate
• Few Statements more in EM
• Knowledge content
• 80 of all knowledge element codes were
  Procedural (EM, 82, SFAW, 98, Saxon, 95)
• Matches the procedurally-focused attribution
• EM 13 Conceptual knowledge codes (n 28)

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Procedural Conceptual (K)

• Common procedures
• Measure with non-standard units (body parts,
  paper clips, linking cubes)
• Direct Comparison (align judge relative length)
• Visual Comparison (same for non-adjacent objects)
• Measure with a ruler (Saxon only)
• Common conceptual knowledge (caution small
  numbers!)
• Unit-Measure Compensation (EM)
• Greater measure means longer length (EM, SFAW)

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Overview of Grade 1 Results

• Recall the increase in pages and codes/page
• Attention gets more serious in Grade 1 (and
  continues in Grade 2)
• Textual presentation
• Problems and Questions dominate
• Drop in Demonstrations from K
• Increase in Statements from K (absolute )
• Knowledge content
• Procedural focus remains (EM SFAW, 78, Saxon,
  91)
• SFAW added conceptual content EM retained it

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Procedural Conceptual (Gr. 1)

• Common procedures
• Direct Visual Comparison
• Visual Estimation (new in Grade 1)
• Measure with a ruler
• Measure with non-standard units
• Common conceptual knowledge (larger numbers)
• Unit-Measure Compensation (SFAW, Saxon)
• Greater measure means longer length (EM, SFAW)
• Standard vs. non-standard units (EM)
• Rulers measure length (SFAW)

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SoIs the Analysis Promising?

• Not surprisingly, we think Yes
• Finding in more detail what others have reported
  (procedural focus)
• But we are much more specific about
• What that means (which procedures?)
• Differences across curricula
• Grade level and grade band patterns (e.g., K2)
• Tracking conceptual knowledge (present and
  absent) in a very specific way

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Challenges Ahead

• Careful analysis is costly (in human time)
• Choice between more extensive analysis of written
  curriculum and examination of some enacted
  lessons
• Both are important How to choose?
• Other limitations
• Can this serve as a national report card (on our
  written curricula)?
• Have not even started area and volume

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Comments from Richard Lehrer
followed by Q A