Computational Discovery on Jupyter
Bibliographic record
Abstract
The authors have been using a largely algebraic form of "computational discovery" in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics.This paper describes what we mean by "computational discovery," what good it does for the students, and some specific techniques that we used.1 Setting the stage "The imparting of factual knowledge is for us a secondary consideration.Above all we aim to promote in the reader a correct attitude, a certain discipline of thought, which would appear to be of even more essential importance in mathematics than in other scientific disciplines."Pólya & Szegő vol I. [28, p. VII]The preface quoted above from the classic book cited, which is nearly a hundred years old now, opens with an epigraph which we further paraphrase, as follows: "What is good education?Giving students systematic opportunities to discover things for themselves."Indeed, Computational Discovery, also called "Experimental Mathematics," is also very familiar to the research mathematician, not just mathematics educators: nearly everyone uses it (even if they say that they don't, or don't say that they do).There can be no shame in it, if the likes of Gauss and Euler used the technique [6,7].See also the excellent book [14].The most basic idea is, after all, very simple: one computes a few cases, tries to guess a pattern, and if successful,
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.004 | 0.005 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.022 | 0.003 |
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".