simulating problem difficulty in arithmetic cognition...

Simulating Problem Difficulty in Arithmetic Cognition

Through Dynamic Connectionist Models+ Cognitive Modeling

Sungjae Cho

Interdisciplinary Program in Cognitive Science

Seoul National University

Ver. 6/7/2019 1BI Lab Seminar 20190607

Addition and subtraction for the binary numeral system

• Two operators: +,−

• Binary numeral system

• The number of carry operations ↑ ⇒ RT ↑ = Problem difficulty ↑

BI Lab Seminar 20190607 2Ver. 6/7/2019

Ver. 6/7/2019 BI Lab Seminar 20190607 3

Modeling diagram


Problem sets

Experiments on humans



Human experiment: Guiding examples


Human experiment: Results

Experiments on connectionist models



Connectionist model experiment: Confidence threshold 𝜃𝑐

𝜃𝑐 = .9

𝜃𝑐 = .9

Confident!Decide the digit to be 1.

Uncertain!Don’t decide the digit.

Confident!Decide the digit to be 0.

110 + 1101 = 10011

“Decide all digits = Answer the problem”


Connectionist model experiment: Confidence threshold 𝜃𝑐

𝜃𝑐 = .9

𝜃𝑐 = .8

𝜃𝑐 = .7

𝜃𝑐 = .7

𝜃𝑐 = .8

𝜃𝑐 = .9

110 + 1101 = 10011


Connectionist model experiment: The Jordan networks with confidence threshold


C’model experiment: Results (y: problem difficulty, x: #carries)


C’model experiment: Results (y: problem difficulty, x: hyperparameter)

Summary

Human experiments

• mean_RT(#carries): strictly increasing

•𝑑

𝑑 #carriesmean_RT(+) <

𝑑

𝑑 #carriesmean_RT(−)

Connectionist models (Jordan networks)

• mean_AS(#carries): strictly increasing

•𝑑

𝑑 #carriesmean_AS(+) <

𝑑

𝑑 #carriesmean_AS(−)

• (confidence threshold) ↑ ⇒ mean_AS ↑: a discernible increase

• (hidden units) ↑ ⇒ mean_AS ↑: negligible

BI Lab Seminar 20190607 14Ver. 6/7/2019

CogSci/MathPsy/ICCM

Sungjae Cho


MathPsy/ICCM 2019

• Host: Society for Mathematical Psychology

• Date: July 19th – 22nd

• MathPsy = Annual Meeting of the Society for Mathematical Psychology

• ICCM = International Conference on Cognitive Modelling

CogSci 2019

• Host: Cognitive Science Society

• Date: July 24th – 27nd

• CogSci = Annual Meeting of the Cognitive Science Society

MathPsy/ICCM 2019 & CogSci 2019



Mathematical Psychology & Cognitive Modeling

Cognitive architectures= Mathematical + Computational Models= Structure of mathematical models

Examples• ACT-R: cognitive neuroscience• Soar: cognitive psychology

Mathematical models• Mathematical modeling of cognitive processes• Establishing law-like rules that relate quantifiable stimulus

characteristics with quantifiable behavior• One or two equations that focus on a single phenomenon • Comparing models by looking at the number of parameters

these models have

Examples• Signal detection theory• Accumulator models• Diffusion models• Neural network/connectionist models• Race models• Random walk models

CognitiveModelingMathematical

Psychology

Derbinsky, N., & Essl, G. (2011). Cognitive Architecture in Mobile Music

Interactions. In NIME (pp. 104-107).

Soar


Cognitive Modeling & AI (+ Cognitive Science)

CognitiveModeling AI

What they are doing

“Modeling the human cognition”

Purpose

“Predict the human cognition”↓

“Explain the human cognition”

Good model

Human accuracy = Model accuracyHuman latency = Model latency

Purpose

“Over the human-level performance”↓

“Outsource the human cognition”

Good model

Human accuracy < Model accuracy

Cognitive Science

simulating problem difficulty in arithmetic cognition...

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