ACED Proficiency Model
Graphical Structure
The full ACED proficiency model has three separate branches for arithmetic, geometric and other recursive sequences. Only the middle branch was used in the Evaluation. Thus although ‘Sequences’ is the top node in the model ‘Solve Geometric Problems’ was treated as if it was the top.
This is a hierarchical breakdown, and a reasonable approximation to this model could be made by reducing the number of levels in each hierarchy.
This was realized in ACED as a Bayesian network, where each variable can take on three states: High, Medium or Low. It is a tree structured model where each variable has exactly one parent.
Note that there were a series of tasks tapping the ‘Solve Geometric Problems’ task that turned out to really be about recognizing geometric sequences. This indicates that there may be a missing node in the model.
Joint probability distribution
Unconditional Probabilities
| Variable | High | Medium | Low |
|---|---|---|---|
| Sequences | 0.20 | 0.50 | 0.30 |
Note that the conditional probability tables were created through a number of “regressions” in which the Math Expert specified the correlation and intercept for the regressions. These are presented below. (I’ve lost the original numbers and need to back translate them from the conditional probability tables. This is a relatively straightforward process, simply treat it as weighted regression with the conditional probabilities as the weights, but I haven’t had a chance to do that yet.)
Conditional Probabilities
| Parent | Child | Correlation | Intercept |
|---|---|---|---|
| Sequences | Recognize Sequences | ||
| Recognize Sequences | Distinguish Types | ||
| Sequences | Solve Sequence Problems | ||
| Solve Sequence Problems | Solve Geometric Problems | ||
| Solve Geometric Problems | Common Ratio | ||
| Solve Geometric Problems | Examples Geometric | ||
| Solve Geometric Problems | Extend Geometric | ||
| Solve Geometric Problems | Model Geometric | ||
| Solve Geometric Problems | Table Geometric | ||
| Solve Geometric Problems | Visual Geometric | ||
| Solve Geometric Problems | Induce Rules Geometric | ||
| Induce Rules Geometric | Verbal Rule Geometric | ||
| Induce Rules Geometric | Algebra Rule Geometric | ||
| Algebra Rule Geometric | Explicit Geometric | ||
| Algebra Rule Geometric | Recursive Rule Geometric |
The complete set of conditional probability tables for this model are given in The Proficiency Model.
ACED development and data collection was sponsored by National Science Foundation Grant No. 0313202.