Bibliographic record
Abstract
The purpose of this chapter is to describe how computer-based tools can help students in the doing and learning of mathematics, and to provide specific examples that illustrate the way in which well-designed technologies can support mathematical discovery and understanding. I begin with an example. Task . Take the three vertices of a triangle ABC and reflect them each across the opposite side of the triangle to obtain a new “reflex” triangle DEF (convince yourself you always do indeed get a new triangle!). Repeat the process. Most people who have seen this problem conjecture that the reflex triangle, after several iterations, converges to being equilateral. But I thought it was a perfect problem for The Geometer's Sketchpad , which effortlessly allows an arbitrary triangle to be iteratively “reflexed” and its measurements to be computed. I dragged vertex A after producing DEF and quickly realized that the equilateral conjecture was false, for I could produce a DEF that was a straight line! And more: DEF seemed to change in a very chaotic way as I continuously dragged vertex A . When, if ever, would the figure become equilateral? When would it not? How did the “function” behave? At this point, I realized that I needed some kind of measure of the degree to which the triangle had become equilateral, especially since the reflex triangles were getting increasingly large–exploding off the screen–as the iterations increased.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".