Designing Cases that Help Teachers Learn about Children’s

Mathematical Thinking

Matthew J. Koehler

Michigan State University

AERA, April 12, 2001

Context for the Work

This work is about the design of tools for mathematics professional development in K-6.

These tools are part of a larger professional context from the work of Lehrer and Schauble.

Up to 40 practicing teachers at one point.

Regular teacher meetings that use text, classroom video, examples of children’s work, and teacher writings.

The challenge was to design materials that conveyed the richness of these data and the relationships between them ….. ----> CASE

Core Research Questions

What makes for a good case?

How do you design a case that meets your goals?

How do different case designs relate to what teachers see and learn?

What makes for a good case?

Some caveats

Primarily talking about what makes a good case in K-6 mathematics teaching

Keeping in mind the kind of materials that are used in the professional development communities that I described (video,children’s work, texts, etc.)

Five general principles to guide development. Good cases are:

Situated in Practice

Layered with Annotation

Annotated with Big Ideas

“Criss-cross” the domain

Anchor exploration

Elements of good casesSituated in practice

Since teaching is situated in classroom practice, cases of teaching should also be situated in classroom practices.

Advocate use of classroom video

Video is more engaging and facilitates remembering.

Video is more like “being there” than text

Written accounts of a classroom assume that textual expression can completely express the dynamics of classroom activity.

Elements of good casesLayered with annotation

Video does not speak for itself.

Any two viewers of a classroom video are likely to see different things, especially if they differ in experience, perspective, or expertise.

Classroom events are often subtle and difficult to interpret.

Therefore, video cases should be layered with annotation that helps teachers interpret classroom situations, so that teachers understand what the video is “a case of.”

Elements of good casesAnnotated with “Big Ideas”

Big ideas in mathematics are important landmarks in teaching based on models of student thinking (Schifter, 1996; Lehrer & Schauble, in press).

Accordingly, annotation should help teachers “lift out” and interpret the “big ideas” of the domain as they occur in the case.

Like big mathematical ideas

norms for argument (Yackel & Cobb, 1996)

general trajectories of student thinking (Carpenter & Fennema, 1992).

Elements of good cases“Criss-cross” the domain

Teaching and learning comprises a complex, ill-structured domain, cases often embody more than one “big idea.”

The same episode can be related to the “big ideas” in mathematics, children’s thinking, the use of tools and notations, and the classroom norms of teaching.

Good teaching requires not only understanding these ideas in isolation, but also orchestrating them to design effective classroom environments.

Cognitive Flexibility Theory (Spiro, Coulson, Feltovich, & Anderson, 1988) suggests that cases should “criss-cross” the conceptual landscape.

Elements of good casesAnchor Exploration

Cases that portray complex, ill-structured classroom situations often raise several important issues

For example, the same episode bring up “big ideas” in mathematics, children’s thinking, the use of tools and notations, and the classroom norms of teaching.

Cases should situate, or anchor (CTGV, 1990), explorations into these important ideas by providing access to further information (e.g., text, interpretation, related case, etc.) as issues arise in the case.

In contrast, if cases only represent the main story line, teachers may come to understand children’s development as a fixed progression through stages.

The Domain of Measurement

Tried to use these ideas in the development of a case-based tool for teachers about length and area Measurement (based on the work of Lehrer et. al).

Often taught and understood procedurally

Instead, instruction should help children to understand the mathematical ideas that underlie measurement (e.g., all the units are the same size)

Goal was to build a case-based hypermedia that emphasized 6 strands of teaching and learning:

Key Mathematical Ideas (e.g., Identical Units)

Classroom Norms (e.g., make thinking visible)

Children’s thinking (e.g., measurement = rulers)

Connections to other Math ideas (e.g., fractions)

Classroom activities (e.g., building tape measures)

Tools and Notations (e.g., graph paper)

Two types of casesExemplification

One emphasized exemplification

Mini-Demo of this type of case

Uses all five principles

Situated in practice

Layered with Annotation

Annotated with “Big Ideas”

“Criss-crosses” the domain

Anchors exploration

Two types of casesNarrative

The other type of case emphasizes narrative structure

Mini-Demo of this type of case

Uses all five principles

Situated in practice

Layered with Annotation

Annotated with “Big Ideas”

“Criss-crosses” the domain

Anchors exploration

An ExperimentRationale

Wanted to contrast the type of learning afforded by these two types of cases

Believed that the advantage of narrative cases lies in the causal structure that ties stories together (van den Broek & Trabasso, 1998).

The ability to apply knowledge relies, in part, on understanding the causal relationships between situations and actions that need to be taken (Eylon & Reif, 1984).

Therefore, I expected that narrative cases would be more likely to lead to knowledge that could be applied.

An ExperimentProcedure

Made two versions of the hypermedia tool

One version had exemplification cases only.

The other version had exemplification AND narrative cases.

Twenty-four pre-service teachers, randomly assigned to study with one version of the hypermedia tool.

Measures before study, after study, and 6 weeks after study

Speak aloud to video - participants saw short classroom segments. Following each clip, participants were asked to identify any important elements of teaching or learning about measurement that they saw. This was used to track the type of knowledge that participants acquired.

Analysis of student work - Participants were asked to apply their knowledge to an analysis of student work. Interviews addressed what the sample student understood (or did not), what the student needed to understand, and what classroom activities would most likely help this student gain understanding. This measured the ability to apply knowledge.

An ExperimentResults

Speak aloud video interviews showed that:

Both groups gained knowledge about the mathematics of measurement and about the teaching norms in place in the classes illustrated in the hypermedia.

No group differences before, after, or six weeks after instruction

Analyses of student work showed that:

The group who had access to narrative cases did better at applying their knowledge to their analyses of student work

More about this ...

An ExperimentAnalysis of student work

What does the following student understand about measurement?

Before instructionBoth groups tended to give procedural explanations

After instructionBoth groups improved … more so for the narrative group

Procedural Measurement Other

61% 20% 19%

Procedural Measurement Other

Exemplification only 18 % 33% 41 %

Exemp and Narrative

17 % 73 % 10 %

An ExperimentOther findings

This trend towards better application of knowledge by the narrative group shows up in other questions of the student work interview

Better at listing all the requisite knowledge a student would need to understand the problem

Better at suggesting appropriate follow-up activities

Have better memory for the classroom activities outlined in the hypermedia tool

Analysis of their time allocation during study supports the view that the narrative cases were responsible for these differences

Tended to read less text than their exemplar-only counterparts

Tended to watch less exemplar cases

The more time spent watching narrative cases was predictive of better analysis of student work (up to a point).

Conclusion

The nature of cases, how they should be crafted, and the consequences of different knowledge structuring are all important questions to investigate.

This work shows that even given some guiding principles for design (the five), competing designs have different affordances for learning

Cases used for exemplification are pretty good at helping students acquire declarative knowledge.

Cases organized around narrative of classroom events has some potential for fostering the application of that knowledge.

Future Work

Inservice teachers

Different domains other than measurement

Other designs for cases

The Paper

There is not a separate paper to handout.

The complete paper has been accepted for publication in Cognition and Instruction

Send me email (mkoehler@msu.edu) if you wish to be notified when it is published.