{
  "id": "math.fractions.divide-makes-smaller",
  "uri": "https://open-misconceptions.github.io/miscon-data/m/math.fractions.divide-makes-smaller",
  "uuid": "72884d31-1a9a-418f-92ee-f073eef02008",
  "version": "0.1.1",
  "status": "draft",
  "trust": "low",
  "title": "Division always makes smaller",
  "statement": "Dividing always gives a result smaller than the number you started with, so dividing by a fraction cannot make a number bigger.",
  "kind": "overgeneralization",
  "domain": "math.fractions",
  "about": [
    {
      "scheme": "CCSS",
      "code": "5.NF.B.7",
      "uri": "http://corestandards.org/Math/Content/5/NF/B/7/"
    },
    {
      "scheme": "CCSS",
      "code": "6.NS.A.1",
      "uri": "http://corestandards.org/Math/Content/6/NS/A/1/"
    }
  ],
  "level_band": [
    "primary",
    "middle",
    "secondary",
    "adult"
  ],
  "locale": "en",
  "evidence_patterns": [
    {
      "item_shape": "predict whether a quotient will be larger or smaller than the dividend, without calculating",
      "signature": "predicts the quotient of a number divided by a proper fraction is smaller than the number",
      "example": {
        "item": "Without calculating, is 6 / 1/2 more or less than 6?",
        "expected": "More (it is 12)",
        "response": "Less"
      }
    },
    {
      "item_shape": "evaluate a whole number divided by a unit fraction",
      "signature": "halves the number instead of doubling it",
      "example": {
        "item": "6 / 1/2 = ?",
        "expected": "12",
        "response": "3"
      }
    }
  ],
  "discriminators": {
    "vs_slip": "A slip is one wrong prediction. The belief is stated as a rule ('dividing makes it smaller') and holds for decimals too; it often coexists with math.fractions.multiply-makes-bigger.",
    "vs": {
      "math.fractions.multiply-makes-bigger": "Mirror belief; test each separately.",
      "math.fractions.invert-dividend": "Ask the learner to predict before calculating. Procedure bugs give a wrong number but often a correct prediction of direction; this belief gets the direction wrong.",
      "math.fractions.division-is-sharing-only": "Sharing-only believers say 6 / 1/2 'makes no sense' or reinterpret it; this belief answers confidently with a smaller number."
    }
  },
  "relations": {
    "conflicts_with": [
      {
        "external": "http://corestandards.org/Math/Content/5/NF/B/7/",
        "label": "CCSS 5.NF.B.7: divide unit fractions by whole numbers and whole numbers by unit fractions"
      }
    ],
    "resolved_by": [
      {
        "external": "http://corestandards.org/Math/Content/5/NF/B/7/",
        "label": "CCSS 5.NF.B.7: divide unit fractions by whole numbers and whole numbers by unit fractions"
      },
      {
        "external": "http://corestandards.org/Math/Content/6/NS/A/1/",
        "label": "CCSS 6.NS.A.1: interpret and compute quotients of fractions"
      }
    ],
    "confusable_with": [
      "math.fractions.invert-dividend",
      "math.fractions.division-is-sharing-only"
    ]
  },
  "alignments": [
    {
      "scheme": "CCSS",
      "code": "5.NF.B.7",
      "uri": "http://corestandards.org/Math/Content/5/NF/B/7/",
      "relation": "about"
    },
    {
      "scheme": "CCSS",
      "code": "6.NS.A.1",
      "uri": "http://corestandards.org/Math/Content/6/NS/A/1/",
      "relation": "about"
    }
  ],
  "provenance": {
    "sources": [
      {
        "type": "paper",
        "citation": "Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17.",
        "doi": "10.5951/jresematheduc.16.1.0003",
        "note": "Implicit partitive model of division: the quotient must be smaller than the dividend."
      },
      {
        "type": "paper",
        "citation": "Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.",
        "doi": "10.2307/749817"
      },
      {
        "type": "paper",
        "citation": "Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservice teachers' misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 20(1), 95-102.",
        "doi": "10.5951/jresematheduc.20.1.0095"
      },
      {
        "type": "paper",
        "citation": "Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909-918.",
        "doi": "10.1037/edu0000025"
      }
    ],
    "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"
}