{
  "id": "math.fractions.compare-whole-number-components",
  "uri": "https://open-misconceptions.github.io/miscon-data/m/math.fractions.compare-whole-number-components",
  "uuid": "210d5db3-daa8-49be-aaf9-aa453ea6b234",
  "version": "0.1.1",
  "status": "draft",
  "trust": "low",
  "title": "Comparing fractions by their whole-number parts",
  "statement": "To compare two fractions you compare the numerators and the denominators as whole numbers; a fraction with bigger numbers is a bigger fraction.",
  "kind": "overgeneralization",
  "domain": "math.fractions",
  "about": [
    {
      "scheme": "CCSS",
      "code": "4.NF.A.2",
      "uri": "http://corestandards.org/Math/Content/4/NF/A/2/"
    }
  ],
  "level_band": [
    "primary",
    "middle",
    "secondary"
  ],
  "locale": "en",
  "evidence_patterns": [
    {
      "item_shape": "compare two fractions where the larger fraction has the smaller numerator and denominator",
      "signature": "chooses the fraction whose numerator and denominator are both larger",
      "example": {
        "item": "Which is larger, 4/5 or 5/9?",
        "expected": "4/5",
        "response": "5/9"
      },
      "notes": "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_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.",
      "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": [
      {
        "external": "http://corestandards.org/Math/Content/4/NF/A/2/",
        "label": "CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators"
      }
    ],
    "resolved_by": [
      {
        "external": "http://corestandards.org/Math/Content/3/NF/A/3/",
        "label": "CCSS 3.NF.A.3: explain equivalence of fractions and compare fractions by reasoning about their size"
      },
      {
        "external": "http://corestandards.org/Math/Content/4/NF/A/2/",
        "label": "CCSS 4.NF.A.2: compare two fractions with different numerators and different denominators"
      }
    ],
    "confusable_with": [
      "math.fractions.compare-by-gap"
    ]
  },
  "alignments": [
    {
      "scheme": "CCSS",
      "code": "4.NF.A.2",
      "uri": "http://corestandards.org/Math/Content/4/NF/A/2/",
      "relation": "about"
    }
  ],
  "provenance": {
    "sources": [
      {
        "type": "paper",
        "citation": "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",
        "note": "Congruent/incongruent comparison paradigm; the bias persists in reaction times even for experts."
      },
      {
        "type": "paper",
        "citation": "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"
      },
      {
        "type": "paper",
        "citation": "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"
      },
      {
        "type": "paper",
        "citation": "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"
      }
    ],
    "origin": "llm-drafted",
    "notes": "Drafted from the cited literature for the Open Misconceptions seed pack; statement, examples and discriminators are original text. Awaiting maintainer review."
  },
  "history": {
    "changelog": [
      {
        "version": "0.1.1",
        "date": "2026-09-05",
        "change": "Schema 0.2 migration: about/alignments carry a free-string scheme (corestandards.org URLs relabelled CCSS where present); trust computed from reviews[] (none yet, so low); no change to the statement or evidence."
      }
    ]
  },
  "license": "CC-BY-4.0"
}