{
  "id": "math.fractions.division-is-sharing-only",
  "uri": "https://open-misconceptions.github.io/miscon-data/m/math.fractions.division-is-sharing-only",
  "uuid": "50e4d120-cd01-49b2-95e6-5ec8df0762d1",
  "version": "0.1.1",
  "status": "draft",
  "trust": "low",
  "title": "Division of fractions only means sharing",
  "statement": "Dividing means sharing a quantity equally among a number of people, so a division by a fraction has no meaning and must be a mistake or a different operation.",
  "kind": "undergeneralization",
  "domain": "math.fractions",
  "about": [
    {
      "scheme": "CCSS",
      "code": "6.NS.A.1",
      "uri": "http://corestandards.org/Math/Content/6/NS/A/1/"
    },
    {
      "scheme": "CCSS",
      "code": "5.NF.B.7",
      "uri": "http://corestandards.org/Math/Content/5/NF/B/7/"
    }
  ],
  "level_band": [
    "primary",
    "middle",
    "adult"
  ],
  "locale": "en",
  "evidence_patterns": [
    {
      "item_shape": "choose the operation for a measurement-division word problem",
      "signature": "rejects division and multiplies or divides by the wrong number, because 'you can't share among half a person'",
      "example": {
        "item": "How many 1/2-cup servings are in 3 cups?",
        "expected": "3 / 1/2 = 6",
        "response": "3 / 2 = 1.5 (or 3 x 2 = 6 'because it can't be division')"
      }
    },
    {
      "item_shape": "write a word problem for a given fraction division",
      "signature": "writes a sharing story that does not match the expression, or says it is impossible",
      "example": {
        "item": "Write a story problem for 3/4 / 1/4",
        "expected": "How many quarter-cups are in three quarters of a cup?",
        "response": "Three quarters shared between a quarter of a person (or: 'you can't')"
      }
    }
  ],
  "discriminators": {
    "vs_slip": "A slip is one wrong operation choice. The belief is revealed when the learner is asked to model or explain division and only ever produces sharing stories.",
    "vs": {
      "math.fractions.divide-makes-smaller": "Divide-makes-smaller computes an answer with the wrong direction; this belief refuses or reinterprets the operation."
    }
  },
  "relations": {
    "conflicts_with": [
      {
        "external": "http://corestandards.org/Math/Content/6/NS/A/1/",
        "label": "CCSS 6.NS.A.1: interpret and compute quotients of fractions"
      }
    ],
    "resolved_by": [
      {
        "external": "http://corestandards.org/Math/Content/6/NS/A/1/",
        "label": "CCSS 6.NS.A.1: interpret and compute quotients of fractions"
      },
      {
        "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"
      }
    ],
    "confusable_with": [
      "math.fractions.divide-makes-smaller"
    ]
  },
  "alignments": [
    {
      "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": "Partitive (sharing) model as the primitive model of division."
      },
      {
        "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": "book",
        "citation": "Sinicrope, R., Mick, H. W., & Kolb, J. R. (2002). Interpretations of fraction division. In B. Litwiller & G. Bright (Eds.), Making Sense of Fractions, Ratios, and Proportions: 2002 Yearbook (pp. 153-161). NCTM.",
        "note": "Measurement, partitive and other interpretations of fraction division."
      }
    ],
    "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"
}