Assessing Curricular Contributions to Poor Measurement Learning The STEM Project Team Michigan State...

Assessing Curricular Contributions to Poor Measurement Learning

The STEM Project TeamMichigan State

University

Strengthening Tomorrow’s Education in MeasurementNCTM Pre-SessionApril 10, 2008

Strengthening Tomorrow’s Education in Measurement

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) Q&A 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) 10


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

numbers5) General “calculational” orientation in classroom

instruction and discourse (Thompson, Phillip, Thompson, & Boyd, 1994) divorce the value of measure from its spatial

conception6) Weaknesses in teachers’ knowledge (Simon &

Blume, 1994)

These factors likely influence and interact with each other 14


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 15


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

start18


STEM Project Overview (cont’d)

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 (K–6): Everyday Mathematics Scott Foresman-Addison Wesley Mathematics Saxon

Middle School Curricula (6–8): Connected Mathematics Project Glencoe’s 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

person’s next to you? How do you know? Hold it next to your neighbor’s 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 specificallyo Transitivity: “The comparison of lengths is transitive. If

length A > length B, length B > length C, then length A > length C.”

o Unit-measure compensation: “Larger units of length produce smaller measures of length.”

o Additive composition: “The sum of two lengths is another length.”

o Multiplicative composition: “The product of a length with any other quantity is not a length.” 35


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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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.”

Procedural Knowledge Elements

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Cultural conventions of representing measures; devoid of conceptual contento This is one inch:

Notations, features of tools (e.g., marks on rulers)o Rulers have inches on one side and centimeters

on the other.

Conventional Knowledge

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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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Question - Provided by Teacher

Direct Comparison x 2

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Worked Example - Student Text

Measurement with non-standard units

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Problem - Student Text

Measurement with non-standard units

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Questions Problems Worked Examples

Larger units of length produce smaller measures of length

Direct comparison

Measure length with Non-standard units

Sample Coding Sheet

1

2

1

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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

K–2 contains the foundation for length measurement Substantial content devoted to the topic Deficits may not get corrected in later grades

We’re 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) 54


So…Is 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.,

K–2) 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

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followed by Q & A

Assessing Curricular Contributions to Poor Measurement Learning The STEM Project Team Michigan State...

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