1. Participants 15 adults from the Pittsburgh community 18-33 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) 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 To From PFGVICFC Start 1061165 PF 405323 GV 65219 IC 62770 Difficulty: p <.0005 Behavioral Results Difficulty x Highlight 1-Inference3-Inferences Not Provable No 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. Conclusions Participants 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 Work Analyze 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 Manual Control Declarative Memory (Retrieval) Visual Perception Goal State Parse 5x+3=38 Retrieve 38-3=35 “Unwinding” “Retrieving” Hold 5x=35 Type x=7 5x + 3 = 38 Outside World Production System Imaginal (Problem State) Motor Parietal Prefrontal 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.