Fractions and Math Anxiety in Fifth Grade
Devon is a fifth grader with a specific learning disability in mathematics. He avoids math, erases frequently, and says, 'I'm dumb at math.' He can add fractions with like denominators when he draws fraction bars, but on a recent probe he wrote 1/2 + 1/3 = 2/5 and explained, 'You add the tops and the bottoms.' He cannot yet compare 3/4 and 5/6, saying 5/6 is smaller because sixths are small. His IEP goal is 70% accuracy on grade-level fraction problems; his current accuracy is 45%. He receives Tier 3 math support for 30 minutes four times weekly in a small group. His teacher reports that he understands multiplication facts and place value and that he shuts down when timed. The team wants to revise his fraction instruction and address his anxiety without lowering expectations.
1. Diagnose Devon's fraction misconceptions. What evidence supports each, and what prior knowledge should the team check?
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Devon shows two related misconceptions.
- Adding numerators and denominators separately (1/2 + 1/3 = 2/5). He is treating fractions as two whole numbers rather than quantities. The evidence is his verbal rule and his correct work with like denominators, where the rule happens to work. He may not understand that denominators name the size of the parts and must be common before joining.
- Whole-number reasoning about denominators (5/6 is smaller because sixths are small). He knows sixths are small but does not coordinate the number of parts with the size of the parts. He may also lack a robust unit-fraction concept (1/6, 1/4) and the idea that 5/6 is five 1/6s. Prior knowledge to check: unit fractions, equivalence, comparing same numerators or same denominators, and the meaning of the denominator as a divisor. His multiplication and place-value strengths are assets; the team should use them. Timed tasks may depress performance, so the diagnostic should be untimed and allow drawings.
2. Outline a concrete-representational-abstract sequence for teaching addition of unlike fractions and comparing fractions. Include two specific instructional activities and how you would fade supports.
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CRA sequence for unlike denominators and comparison: Concrete: Use fraction circles or bars. For 1/2 + 1/3, have Devon cover half with halves and thirds with thirds, then replace both with sixths so he can see 3/6 + 2/6 = 5/6. For comparing 3/4 and 5/6, build both with fraction bars and lay them side by side. He says which is larger and why, using the same-size whole. Representational: Move to drawings of fraction bars and number lines. He partitions the same whole into fourths and sixths, shades 3/4 and 5/6, and writes the equivalence. For addition, he draws both fractions on one number line and finds a common denominator by partitioning. Abstract: Teach the procedure with language: make the parts the same size; multiply the numerator and denominator by the same factor; then add the number of parts. Use worked examples and faded prompts. Fading: Start with 100% concrete, then have him draw after one modeled example, then solve with a number line only, then solve abstractly and check with a drawing. Do not remove drawings until he consistently explains why the denominator changes. For comparison, teach benchmark 1/2 and same-denominator strategies before the cross-multiplication shortcut.
3. How should the team assess Devon's progress and address math anxiety while maintaining high expectations?
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Assess with untimed curriculum-based measures on the IEP goal, plus a monthly cumulative review. Use items that require explanation, not only answers: 'Show how you know 3/4 is greater than 5/6.' Score accuracy and the quality of the representation. Keep a weekly 5-item probe and a biweekly interview. To address anxiety:
- Separate fluency timing from concept tasks; do not use timed drills for the fraction goal until accuracy is stable.
- Give a private signal for help and a 'start with a drawing' card.
- Use error analysis rather than red marks; have Devon correct one item and explain the revision.
- Preteach the vocabulary and provide a worked-example bank.
- Set short, attainable session goals, for example three accurate items with drawings, and build to 70% over three consecutive probes. Maintain high expectations by keeping grade-level fraction content and the 70% goal. Anxiety is not a reason to reduce the goal; it is a reason to change the conditions and build self-efficacy through evidence of progress. The team should review the goal if the data show the current goal is not ambitious or not measurable.