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ACT-R Research Agenda: Learning to Solve Equations - Children and Adults - fMRI Data

Review of the ACT-R 5.0/6.0 architecture and its application to two experiments on learning to solve equations. Discusses the use of fMRI data to support architectural assumptions and presents parameter-free predictions. Covers future goals and compares ACT-R to ACT*.

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ACT-R Research Agenda: Learning to Solve Equations - Children and Adults - fMRI Data

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  1. Most of the Time State of ACT-R Research Agenda: Review the ACT-R 5.0/6.0 architecture. Illustrate its application to two experiments on learning to solve equations -- one with children and with adults. Show how fMRI data provide converging data for architectural assumptions. Show off nearly parameter-free predictions. Discuss future goals.

  2. ACT-R versus ACT* • Common: Cognition has declarative & procedural systems. • Common: Each system has subsymbolic & symbolic aspects. • ACT-R 2.0: Rational analysis guiding the subsymbolic level. • ACT-R 4.0: Central cognition integrated with perceptual-motor. 5. ACT-R 4.0/5.0: Different types of learning that really work. Knowledge Level Learning: Module products recorded as chunks -- focus on instruction and examples 6. ACT-R 5.0: has biologically-inspired module and buffer structure. 7. ACT-R 6.0: Use of this structure to foster cumulative science.

  3. ACT-R 5.0/6.0 Modules and Buffers ACT-R Parse 3x-5=7 Visual Perception Type x=4 Manual Control Production System Retrieve 7+5=12 Declarative Memory ProblemState Hold 3x=12 Control State “Unwinding” “Retrieving”

  4. The First Experiment Qin, Anderson, Silk, Stenger, & Carter (2004) 11-14 year-olds just about to start Algebra 1 Day 0: Instruction, paper & pencil practice, coaching Days 1 - 5: Computer-based practice 4.Student types answer by pressing finger in data glove. 5. Imaged in fMRI scanner on Day 1 and 5.

  5. Unwind Instructions that ACT-R Parses into an Internal Declarative Representation • 1. To solve an equation, encode it and • a. If the right side is a number then image that number as the result and then focus on the left side and unwind it. • b. If the left side is a number then .. • 2. To unwind • a. If the expression is the variable then the result is the answer. • b. If a number is on the right unwind-right • c. If a number is on the left unwind-left • 3. To unwind-right, encode the expression and • a. If the expression is _ + 0 then focus on the left part and unwind • b. Otherwise invert the operator, image it as the operator in the result, image the right part of the expression as the second argument in the result, evaluate the result, and then focus on the left part and unwind • 4. To unwind-left encode the expression and • If the expression is 1 * _ then focus on the right part and unwind • Otherwise check that the operator is symmetric, invert the operator, image it as the operator in the result, …

  6. ACT-R’s General Procedures for Interpreting and Following Declarative Representations of Procedures applied to 7x + 3 =38 Instruction 1a: Create image “=38” Instruction 2b: Unwind-right 7*x+3 Instruction 3b: Change image to “=38-3”, this to “=35”, and focus on 7*x Instruction 2c: Unwind-left 7*x Instruction 4b: Change image to “=35/7”, this to “=5”, and focus on x Instruction 2a: The answer is 5, key it. Initially instructions are retrieved and interpreted. Eventually production compilation produces task-specific production rules.

  7. Examples of Production Rules General Interpretive If one has retrieved an instruction for achieving a goal THEN retrieve the first step of that instruction Prior Knowledge IF one is evaluating the expression “a operator b” THEN try to retrieve a fact of the form “a operator b = ?” Acquired Task-Specific IF the goal is to unwind an expression and the expression is of the form “subexpression + 0” THEN focus on the subexpression

  8. Parallel module activity, limited by long encoding Production Compilation Production Compilation Retrieval Speed up ACT-R Modules: The first 2+ Seconds: 7x+3=38

  9. ACT-R Modules: The middle 2+ Seconds: 7x+3=38

  10. ACT-R Modules: The last 2+ Seconds: 7x+3=38

  11. Learning over 6 Days of Experiment

  12. Comments on the ACT-R Model Virtue: It actually does the task -- interacts with same software as subjects. Virtue: The model is not hand crafted but learns from instruction (albeit the instructions are a little hand-crafted to facilitate parsing). Fact: Two parameters were estimated to fit the latency data -- the latency scale for retrieval and the visual encoding time. Doubt: There is an great deal of theoretical complexity for a rather simple set of numbers. Resolution: We will use brain imaging to test for distinct patterns predicted by different modules in the model.

  13. ACT-R Modules and Buffers ACT-R Parse 3x-5=7 Fusiform Gyrus Visual Perception Type x=4 Manual Control Motor Cortex Production System Basal Ganglia Retrieve 7+5=12 Declarative Memory Prefrontal Cortex ProblemState Parietal Cortex Hold 3x=12 Anterior Cingulate Control State “Unwinding” “Retrieving”

  14. Motor/Manual: BA 3/4 (x = -37, y = -25, z = 47) Parietal/Imaginal: BA 39/40 (x = -23, y = -64, z = 34) Prefrontal/Retrieval: BA 45/46 (x = -40, y = 21, z = 21) Our Modules (all left lateralized) as 100 (5x5x4) Voxel Regions

  15. Ant Cing/Goal:BA 24/32 (x = -5, y = 10, z = 38) Caudate/Procedural: (x = -5, y = 9, z = 2) Actually 4 x 4 x 4 Our Modules (all left lateralized) as 100 (5x5x4) Voxel Regions

  16. 21.6 Second Structure of fMRI Trial

  17. Module Activity Mapping Module Activity onto the BOLD Response Activation

  18. Day 1: f(t) 2 steps Day 5: f(t) 2 steps

  19. BOLD Function Predicted BOLD Response Module Demand Function Day 5: f(t) 2 steps

  20. 0 1 2 Response Delay Motor/Manual: BA 3/4 (x = -37, y = -25, z = 47)

  21. Almost no Effect Prefrontal/Retrieval: BA 45/46 (x = -40, y = 21, z = 21)

  22. Ant Cing/Goal:BA 24/32 (x = -5, y = 10, z = 38) Almost noLearning

  23. Rather directly reflects time because of skipping steps in equation representation Parietal/Imaginal: BA 39/40 (x = -23, y = -64, z = 34)

  24. Caudate/Procedural: (x = -5, y = 9, z = 2) Rather directly reflects time because of production rule collapsing

  25. c2 measures of Match between Regions and Modules--small is good (<130 nonsignificant) Identical Response Different Peaks Little Response in 0 Operation Operation Large Learning Weak Operation Medium Learning Medium Operation Medium Learning Medium

  26. Observations about fMRI and Modeling While the analysis has been about ACT-R fitting the learning of algebra the same methods can be used to relate many different information-processing theories to many tasks. The unifying concept in all cases is that the BOLD response in a region reflects time a module is engaged. This allows us to map between an information-processing model and the BOLD response and so to track individual components of the model. The same prespecified areas behave as predicted in many adult studies. There is no claim one way or another about whether the modules are implemented in these regions. The critical fact is that we have a measure of the activity of specific modules rather than just the overall behavior. Challenge: Can we take this same model and fit it to another experiment.

  27. The Second Experiment-- Qin, Sohn, Anderson, Stenger, Fissel, Goode, & Carter (2003) Adults Day 0: Instruction and general practice Days 1 - 5: Computer-based practice Subject types answer by pressing thumb and then quickly keying 4 terms. Scanned on Days 1 & 5.

  28. 18 Second Structure of fMRI Trial Blank Give Prior Equation Period Answer Px4<->5 1-3-5-3-4 1.5 Second Scans

  29. Instructions for ACT-R • To solve an equation, first find the “<->”, then encode the first pair that follows, then shift attention to the next pair if there is one, then encode the second pair. • If this is a simple equation output it; otherwise process the left side. • To process the left side, first find the “P”. • If “<->” immediately follows then work on the operator that precedes the P; otherwise first encode the pair that follows, then invert the operator, and then work on the operator that precedes the P. • To process the operator that preceded the P, first retrieve the transformation associated with that operator, then apply the transformation, and then output. • To output press 1, then output the first, then output the next, then output the next, and then output the next + Knowledge of inverses (2-3, 4-5) and transformation rules for getting rid of 2,3,4, & 5 prefixes.

  30. ACT-R Modules: 2 P 3 4 <-> 2 5 Encoding

  31. ACT-R Modules: 2 P 3 4 <-> 2 5 Transforming

  32. ACT-R Modules: 2 P 3 4 <-> 2 5 Output

  33. Learning over 6 Days of Experiment

  34. Motor/Manual: BA 3/4 (x = -37, y = -25, z = 47) As before, BOLD response tracks response timing

  35. Almost no Effect As before, large effects of both factors -- weak response for 0 Prefrontal/Retrieval: BA 45/46 (x = -40, y = 21, z = 21)

  36. Ant Cing/Goal:BA 24/32 (x = -5, y = 10, z = 38) Almost noLearning As before, large effect for complexity, little for learning

  37. Large effect complexity, learning largely complete by Day 1 Parietal/Imaginal: BA 39/40 (x = -23, y = -64, z = 34)

  38. Caudate/Procedural: (x = -5, y = 9, z = 2) Very Weak Response in this Experiment -- yielding poor signal to noise ratio.

  39. c2 measures of Match between Regions and Modules--small is good (<90 nonsignificant) Identical Response Different Peaks Little Response in 0 Operation Operation Large Learning Near Zero Operation Medium Learning Weak Poor Signal to Noise Weak Day 5 Response

  40. Shape Magnitude Time to Peak -- a x s Larger Operations? Motor Saturation?

  41. 10 Future Directions for ACT-R Increased stress on parameter-free predictions. Increased effort to anchor the module structure of ACT-R with brain correlations. Focus on instruction -- starting our models from the beginning. Goal of producing a simulated student. Focus on reasoning and metacognitive processing. Continued effort at community support. Greater emphasis on re-use of components/models Including making knowledge basis available to community -- for instance, a middle-school math module. Finally get concerned with representational assumptions. Facilitate exchange of components between architectures. Local Science Shared Science

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