Cracking the code of place value: The relationship between place and value takes years to master
Bibliographic record
Abstract
Place value, which underlies the meanings of multi-digits, encompasses the principle of position and base-10 rules. To understand “65,” one needs to know that the digits “6” and “5” occupy different positions and thus represent ordered values of different magnitudes (the principle of position), and that the value of each position is determined by base-10 rules (e.g., the rightmost position is 10^0, followed by 10^1, 10^2, etc.). Without the principle of position, children cannot construct meanings for multi-digits. Previous studies show that children do not know the exact value of digit positions until early elementary school years, but less is known about the acquisition of positional knowledge for multi-digits. To study when and how children construct a relationship between position and value, this study explored when children begin to know that the leftmost digit represents the largest value, and whether such knowledge relates to learning number names. Four to 7-year-olds from primarily Caucasian, middle-class families were asked to compare different pairs of multi-digits. Some comparisons (e.g., 12 vs. 21) required knowledge of positional property, and some did not (e.g., 35 vs. 36). We found that as a group, 6-year-olds could recruit positional knowledge to compare multi-digits. We also found that children who knew the number names of both of the multi-digits in a comparison pair were above chance on multi-digit comparison. Our results shed light on the developmental steps towards acquiring place-value notation, and highlight a role of learning number names for learning positional property of the place-value notation.
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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 | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".