Bibliographic record
Abstract
In her work concerning algebraic thinking, Kieran notes that students learning algebra tend to fall into two groups—“algebraic” thinkers who use undoing as a way to solve equations, and “arithmetic” thinkers who use trial-and-error substitution to solve equations. “Algebraic” thinkers rely on inverse operations; for example, this group would solve 5 + a = 12 by saying 12 – 5 = 7, ignoring the variable itself. When these students move on to more complex equations, such as 3a + 3 + 4a = 24, they tend to overgeneralize and get stuck (“24 divided by 4, minus 3, minus, um, no, divided by 3”). They are unable to balance the equation because they have not assigned enough significance to the role of the equal sign within the equation- solving process (Kieran 1988, p. 94). When arithmetic learners speak of their solutions, however, because they are using trial-and-error substitution, Kieran finds that they discuss the balance required between the two sides of the equation. She further states that of these two, “arithmetic” thinkers are using a method that “may provide a more intuitive basis for the more structural solving methods” (1992, p. 401). I was curious to see if an eighth-grade student whose thinking could be characterized as “arithmetic” would indeed find this type of thinking a help or a hindrance to her further development of algebraic concepts.
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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 | 0.003 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.006 | 0.013 |
| Scholarly communication | 0.006 | 0.011 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.004 | 0.009 |
| Insufficient payload (model declined to judge) | 0.010 | 0.004 |
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".