miscon:math.fractions.compare-whole-number-components · draft · trust low · v0.1.1 · kind overgeneralization

Comparing fractions by their whole-number parts

To compare two fractions you compare the numerators and the denominators as whole numbers; a fraction with bigger numbers is a bigger fraction.

Stable URI
https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-whole-number-components
UUID
210d5db3-daa8-49be-aaf9-aa453ea6b234
Domain
math.fractions
Level band
primary, middle, secondary
Locale
en
About
CCSS 4.NF.A.2 http://corestandards.org/Math/Content/4/NF/A/2/

Evidence patterns

Pattern 0: compare two fractions where the larger fraction has the smaller numerator and denominator

chooses the fraction whose numerator and denominator are both larger

ItemWhich is larger, 4/5 or 5/9?
Expected4/5
Response5/9

Incongruent item: whole-number ordering and fraction ordering disagree, so the error is diagnostic. Pair with a congruent item (2/7 vs 3/8) which this belief answers correctly; near-perfect congruent accuracy with systematic incongruent errors is the signature.

Discriminators

vs a slip. Compare accuracy on congruent and incongruent items. A holder of this belief is near-perfect on congruent items and systematically wrong on incongruent ones; a slip does not follow that split.

vs math.fractions.bigger-denominator-bigger-fraction. Unit-fraction items isolate the denominator rule. Component comparison also covers non-unit fractions such as 4/5 vs 5/9.

vs math.fractions.compare-by-gap. On 5/6 vs 7/8 gap thinking says they are equal; component comparison picks 7/8.

Relations

conflicts_with
CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators http://corestandards.org/Math/Content/4/NF/A/2/
resolved_by
CCSS 3.NF.A.3: explain equivalence of fractions and compare fractions by reasoning about their size http://corestandards.org/Math/Content/3/NF/A/3/
CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators http://corestandards.org/Math/Content/4/NF/A/2/
confusable_with
math.fractions.compare-by-gap

Alignments

Provenance

Origin: llm-drafted

  1. Obersteiner, A., Van Dooren, W., Van Hoof, J., & Verschaffel, L. (2013). The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction, 28, 64-72. doi:10.1016/j.learninstruc.2013.05.003
    Congruent/incongruent comparison paradigm; the bias persists in reaction times even for experts.
  2. Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2015). Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education. Educational Studies in Mathematics, 90(1), 39-56. doi:10.1007/s10649-015-9613-3
  3. DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39-49. doi:10.1016/j.learninstruc.2014.07.002
  4. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. doi:10.1207/s15326985ep4001_3

Drafted from the cited literature for the Open Misconceptions seed pack; statement, examples and discriminators are original text. Awaiting maintainer review.

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Formats

Cite

miscon:math.fractions.compare-whole-number-components
https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-whole-number-components