opportunities and challenges for diagnosing teachers’ multiplicative reasoning

Opportunities and Challenges for Diagnosing Teachers’ Multiplicative

Reasoning

NSF DR-K12 PI Meeting

December 3, 2010

Andrew Izsák

University of Georgia

Diagnosing Teachers’ Multiplicative Reasoning

Andrew Izsák, Jonathan Templin, Allan Cohen

University of Georgia

Joanne Lobato

San Diego State University

Chandra Orrill

University of Massachusetts Dartmouth

Supported by the National Science Foundation under Grant No. DRL-0903411. The opinions expressed are those of the authors and do not necessarily reflect the views of NSF.

Existing Approaches to Assessing Teacher Knowledge

• Composite measures– Count college mathematics courses teachers

completed– Use item response theory to measure

Mathematical Knowledge for Teaching• Case studies

– Investigate topics such as subtraction with regrouping, arithmetic with fractions, and functions

Essential Features of DTMR• Assessing mathematical knowledge of middle grades

teachers• Emphasize knowledge needed for teaching:

– multiplication and division of fractions and decimals– proportional reasoning – using problem situations and drawn models to develop general

methods

• Combine mathematics education research with psychometric research on an emerging class of models called Diagnostic Classification Models (DCMs)

• One of the first projects to develop a test for DCMs using STEM education research

Opportunities• Curricular trends

– Reform-oriented curricula– Common Core Standards

• Need measures suitable for tracking growth and change in PD• Large literature

– Multiple components of reasoning – Glean attributes

• DCMs are multi-dimensional models that (compared to MIRT) can be reliably estimated with smaller samples and shorter tests

• Expand range of psychometric models applied to STEM education research

Fraction Attributes

A

B

1

B

• Referent Units– Understand units to which numbers refer

• Partitioning– Using whole-number multiplication to guide

partitioning

• Iterating– Interpret to mean A copies of

• Appropriateness– Identifying multiplication and division situations

Example: Referent Unit & Iterating

I. The diagram can show .

II. The diagram can show 1 .

III. The diagram can show .

3

5

2

3

5

2

Which of the following interpretations are sensible?

The Mastery Profile(Fractions)

• DCMs estimate an attribute mastery profile for each teacher:

Estimated Probability of Mastery Referent Units .3 Partitioning .5 Iterating .7 Appropriateness .8

0 0.5 1 Not Mastered Unsure Mastered

Anticipated Uses

• Use as formative assessment to inform PD• Detect growth and change in PD characterized

as increased proficiency with the attributes• Determine distribution of attribute patterns in

large samples of teachers• Examine relationships between attribute

patterns and enactment of curricular materials• Examine relationships between attribute

patterns and student achievement

Challenge: What are Workable Attributes?

• Existing examples from psychometrics – Steps in numeric algorithms– Branches of mathematics

• Our criteria for attributes– Written responses provide reliable information– Separable from one another – Separate teachers

– Cut across topics • Cannot translate research findings directly

– Multiple cycles of using attributes to write items and interviewing teachers

Example: Composed Unit Reasoning

One week Mr. Compton drove to a training course that required him to drive 8/3 the distance he usually drives to work. He noticed that 24 min. had passed when he had drive half way to the course. How long does it take Mr. Compton to drive to work?

Mr. Vargas gave the following task to his students:

One student tried to model the problem using two number lines as shown but is stuck. How could you help the student?

0 min

0 km

24 min

8

3

Challenge: Item Design

• Machine scoreable– Multiple-choice– Constructed response items

• Case studies often rely on observed strategies to make inferences– Composed unit reasoning– Fractions as multiplicative relationships

• Numeric computation should not help find correct response– Teachers’ use of computation obscures access to their

reasoning with quantities

Challenge: Item Design (Cont.)• Attention to pedagogy can drive responses

– Teachers are not always comfortable evaluating students

– How teachers would teach a topic and what they know about a topic are not the same

– Teachers can deflect mathematical issues by appealing to what their students can understand

• Drawings– Teachers do not always interpret diagrams in

ways that we intend

Challenge: Balancing Constraints

• Psychometric modeling and interviewing teachers to investigate constraints that include– Grain-size of attributes

– Item design

– Number of attributes

– Number of items per attribute

– Number of items (test length)

– Sample size

Proportional Reasoning Attributes

• Covariation and Invariance– Multiplicative relationship invariant as quantities co-

vary

• Connections between Ratios and Fractions– Conceptual links between ratios and fractions

• Appropriateness – Direct proportion vs inverse proportion – Direct proportion vs linear relationship

• Multiplicative Object

Summary

• Develop one of the first tests for use with Diagnostic Classification Models (DCMs) based on STEM education research

• Opportunities– Assess select aspects of teachers’ multiplicative reasoning

– Detect growth and change during professional development

• Challenges– Interpret the term “attribute” for mathematics education

research on multiplicative reasoning– Design items that get at fine-grained aspects of

multiplicative reasoning using drawn models

Expanded Fractions Attributes Attribute Sub Attributes

Norming Referent Units for Multiplication

Referent Units

Referent Units for Division Simple Partitioning Partitioning in Stages Partitioning Using Common Denominator

Partitioning

Partitioning Using Common Numerator

Iterating Unit Fractions Identifying Multiplication Situations Identifying Quotitive Division Situations

Appropriateness

Identifying Partitive Division Situations

Expanded Proportional Reasoning Attributes

Attribute Sub-Attributes

Iterating and Partitioning a Composed Unit Consolidating operations on composed units Making multiplicative comparisons within measure spaces

Covariation & Invariance

Making multiplicative comparisons across measure spaces Using composed unit reasoning to reinterpret a ratio as a fraction Using a multiplicative comparison to reinterpret a ratio as a fraction

Connections between Ratios & Fractions

Differentiating fraction and ratio operations

Appropriateness

Rate as an equivalence class of rat ios Meaning for equality in a proportion

Formation of a Multiplicative Object

Ratio-as-measure