• Activation-based declarative retrieval

• Computation and conflict resolution

• Error modeling

• The problem-size effect

• Errors in addition retrieval

• Errors in multiplication computation

• The problem-size effect over time

• Analysis of cognitive dynamics

• Power-law curve of retrieval odds

• Influence of various factors

• A model which learns (almost) everything

• Analysis tools

• Sensitivity to experimental conditions

• Extension of formal analysis

• Practical task

• One of the three basic Rs

• Not a toy problem

• Everybody does it

• Reliable data across the population spectrum

• Abstract task

• No built-in bias

• Hierarchical task

• Each skill is used to learn more complex skills

• Regular task

• Fewer rules/exceptions than natural language

• Statistics-friendly

• Very hard for humans but very easy for computers

• Insights into human architecture

• Lessons for AI

Bayes Equation

Activation Equation

Base-Level Equation

Associative Strength Equation

• Fact reinforcement upon retrieval

• Fact creation or reinforcement upon goal popping

Conflict Resolution Equation

Latency Equation

• P and C of retrieval and computation productions

• Recovery from subgoal failure

• Gradual switch from computation to retrieval

Full Activation Equation including mismatch penalty and noise:

Retrieval can only be completed if activation is above the retrieval threshold RT.

The mismatch measure encodes the similarity between memory elements such as numbers.

Larger problems take longer and produce more errors.

• Models: computation vs retrieval-based

• Assumption of frequency differential: external (presentation) vs internal (procedures) sources.

• Standard prbms: base level and associative learning

• Ties: associative boost

• Reencoding strategies & independence assumption

• Zeroes: proceduralized

• Error percentage increase with size

• Problems involving 1 have many fewer errors

• Tie problems have fewer errors (e.g. 4+4 vs 5+3 or 3+5)

• Problems with a first operand larger than the second have fewer errors than their symmetric counterpart

• Erroneous answers tend to be smaller than the correct answer.

• Larger problems have less activation and therefore are more likely to suffer from errors of commission

• Facts involving 1 can leverage the strength of counting

• Tie problems get extra activation boost

• Backup strategies (counting errors, swap strategy) explain the argument asymmetry

• Smaller problems are stronger, more likely to intrude

• Errors increase with problem size.

• Addition by 5s has very few errors.

• Latency and problem size effect decrease over time

• The decrease approximates a power-law across grades

Assuming a presentation of 100 problems a day:

• The difference between the base-level activation of two memories reflects the ratio of their frequencies:

Assuming constant frequencies, this implies that practice does not affect differences in activations and thus commission errors!?

• But while the environment may be fixed, performance and learning are a dynamic feedback loop. Performance, i.e. the retrieval odds, depends upon activation:

Boltzmann Equation

and, since the activation difference reflects the log of the ratio of past rehearsals, the odds of retrieval are a function of the odds of past experience:

Through learning, these odds of retrieval then become part of the history. The behavior of the system then fundamentally depends upon the noise level s:

• If s<1, the retrieval odds are more extreme than the past odds and the system becomes winner-take-all.

• If s>1, the retrieval odds are less extreme than the past odds and all alternatives become equally likely.

• If s=1, the retrieval odds reflect the past odds (probability matching) and the system drifts randomly with experience.

• Large differences in convergence time can be explained by small differences in initial performance.

• The noise slows down convergence but can keep an error from locking in early, which suggests a role similar to the temperature in simulated annealing and reflects the changing and uncertain nature of the environment in which human cognition evolved.

• Context: add spreading activation to frequency-based base level. The analysis still works, but the complexity of the context, i.e. the number of sources, acts as a noise multiplier because only one source may separate it from its competing neighbors, i.e the three sources of arithmetic facts effectively triple the noise.

• Partial matching: the mismatch penalty improves the performance by a constant factor of but does not affect the speed of convergence.

• Multiple alternatives: the odds of any one choice reflects the (harmonic) average of each pairwise odds (Luce’s choice axiom), which slows down the emergence of the ultimate winner.

• External feedback sources: feedback reinforces the retrieval odds of the correct answer by the feedback probability. This underscores the importance of early teaching, before odds have grown much larger than probabilities. It also raises the question of the impact of transient strenghening on long-term learning.