Participants

15 adults from the Pittsburgh community18-33 yrs; M = 24.4, SD = 5.3; 6 females

Design

2 x 3 within-subjects full factorial design

Task

Determine if the minimum number of logical inferences needed to complete the proof is 1 inference, 3 inferences, or if the proof is not provable. The goal statement is marked at the bottom of the diagram.

What are they thinking? An Information Processing Approach to Deconstructing a Complex Task • • • • • • •

Model

1) Goal stage (2.1 seconds) – encode the statement to be proven

2) Inference stage (2.1, 6.3, or 10.5 seconds) – integrate givens and intermediate inferences to formulate the proof

3) Decision stage (2.1 seconds) – make the response

4) Post-decision reflection (time not modeled)

Latency Distribution by Problem Type

0.00

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0 2 4 6 8 10 12 14 16 18 20 22 24 26

Time (sec) from Stimulus Onset

Pro

po

rtio

n R

esp

on

ses

Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Left Parietal

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0.80

1.00

1.20

1.40

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Time (sec) from Stimulus Onset

% B

OL

D

Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Left Motor

-0.10

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Time (sec) from Stimulus Onset

% B

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Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Yvonne Kao (ykao@andrew.cmu.edu) and John Anderson (ja+@cmu.edu), Department of Psychology, Carnegie Mellon University

Background

Goals

To use a variety of converging methods to understand the underlying cognitive processes that support geometry proof.

• Verbal protocol analysis

• Eyetracking

• Neuroimaging

• Cognitive modeling

To use our cognitive model to design an educational intervention to improve teaching and learning of geometry in the United States.

Motivation

Geometry and geometry proof are important• “Geometric thinking is an absolute necessity in every branch of mathematics.” (Cuoco, Goldenberg, & Mark, 1996, p.389)

• Formal proof is an important part of mathematical discovery and problem-solving. (Schoenfeld, 1992)

• Geometry and geometry proof are difficult for American students. (Martin & McCrone, 2001)

Geometry needs to become a more visible part of the middle grades curriculum. Too often it is ignored until high school, and delays students’ experience in a highly valuable and applicable part of mathematics. Geometric thinking and spatial visualization are linked to many other areas of mathematics, such as algebra, fractions, data, and chance. Teachers should spend more time developing geometric concepts (concretely and with many different representations) and principles (in varied settings) and not merely focus on practice involving algorithmic properties. (Wilson & Blank, 1999, p.13)

ACT-R

ACT-R is a cognitive architecture that defines specific cognitive processes and maps them to specific regions in the brain. ACT-R can provide a framework for predicting, interpreting, fitting, and discussing behavioral and neuroimaging data (Anderson, Bothell, Byrne, Douglass, Lebiere, & Qin, 2004).

fMRI

Verbal Protocol Results

BD, CD.BDCD.These two the same,And angle the same,ok, it is congruent,three step.

Codes

PF: Encode proof statement

GV: Encode givens

IC: Intermediate conclusion

FC: Final conclusion

ToFro

m PF GV IC FC

Start106 11 6 5

PF40 53 23

GV6 52 19

IC6 27 70

Left Caudate

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0 2 4 6 8 10 12 14 16 18 20 22

Time (sec) from Stimulus Onset

% B

OL

D

Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Right Anterior Prefrontal

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Time (sec) from Stimulus Onset

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Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Right Insula

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Time (sec) from Stimulus Onset

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Model 1-Inference Model 3-Inferences Model Not Provable

Data 1-Inference Data 3-Inferences Data Not Provable

Difficulty: p < .0005

Mean Latency by Difficulty

0

2500

5000

7500

10000

12500

15000

17500

1-Inference 3-Inferences Cannot be Proven

Difficulty

Late

ncy (m

s)

Difficulty: p < .0005

Mean Proportion Correct by Difficulty

0.00.10.20.30.40.50.60.70.80.91.0

1-Inference 3-Inferences Cannot be Proven

Difficulty

Pro

port

ion C

orr

ect

Behavioral Results

Mean Proportion of Errors by Type

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1.00

1 -> 3 1 -> NotProvable

3 -> 1 3 -> NotProvable

NotProvable ->

1

NotProvable ->

3

NR

Error Type

Pro

po

rtio

n o

f A

ll E

rro

rs

Difficulty x Highlight

1-Inference 3-InferencesNot

Provable

No

Highlight

Highlight

Procedure

Geometry Review

Participants were given a self-paced review of basic plane geometry theorems and concepts:

• Reflexivity

• Vertical angles

• Parallel lines

• Isosceles triangles

• Triangle congruence

Task Training

Participants were trained to a minimum of 75% accuracy on the geometry proof task.

fMRI Scanning

Participants performed 120 proofs while being scanned in a fMRI machine (TR = 2 seconds).

Verbal Protocol

6 participants returned for an additional verbal protocol and eyetracking session.

ConclusionsParticipants reliably encode the goal statement at the beginning of each proof. This is counter to findings from Koedinger & Anderson’s (1990) study of expert geometry problem solvers. This could be due to the different nature of the task, or the different populations from which the participants were drawn.

The critical process to understand appears to be how proficient problem-solvers integrate problem givens and diagram information to support their logical inferences, and how this process differs in experts, proficient problem solvers, and novices.

Ongoing and Future WorkAnalyze the eye movement data.

Re-run this study with the portion of the population that couldn’t be trained to proficiency on the task in the time provided and compare the results.

Method

ManualControl

DeclarativeMemory

(Retrieval)

VisualPerception

GoalState

Parse5x+3=38

Retrieve38-3=35

“Unwinding”“Retrieving”

Hold5x=35

Typex=7

5x + 3 = 38Outside

World

ProductionSystem Imaginal

(ProblemState)

Motor

ParietalPrefrontal

Anterior Cingulate

Caudate

References

Anderson, J. R., Bothell, D., Byrne, M. D., Douglass, S., Lebiere, C., & Qin. Y. (2004). An integrated theory of mind. Psychological Review, 111(4), 1036-1060.

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15(4), 375-402.

Koedinger, K. R. & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.

Martin, T. S. & McCrone, S. S. (2001). Investigating the teaching and learning of proof: First year results. In Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 585-94.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp.334-370). New York: MacMillan.

Wilson, L. D. & Blank, R. K. (1999). Improving mathematics education using results from NAEP and TIMSS (ISBN-1-884037-56-9). Washington, DC: Council of Chief State School Officers.