Quantitative Reasoning: Right Computation, Wrong Direction
A grade-7 math coach is reviewing a 24-item quantitative reasoning subtest. Items fall into four kinds: number series, short word problems, quantity comparison, and items whose correct response is that the answer cannot be determined. The student scored 13/24. The breakdown: 8/8 on series with a constant difference, 1/5 on series where the rule changes between terms, 2/4 on geometric series, 2/5 on one-step word problems, and 0/2 on cannot-be-determined items. On the series 3, 7, 15, 31, 63, ___ he wrote 95, having continued the differences as +32 rather than checking whether the differences themselves were changing. On a word problem - a shirt costs $40 after a 20 percent discount, what was the original price - he answered $48. He answered every item, including the two cannot-be-determined items, and averaged 55 seconds per item against a 60-second allowance.
1. Classify the error source using the item breakdown, and say what the cannot-be-determined result adds.
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Computation and reading are not the problem. 8/8 on constant-difference series shows he can compute differences accurately, and 2/5 on word problems with one-step translation shows he can read and set up familiar operations. Errors are not spread evenly across difficulty; they cluster at specific structural features.
Cluster 1: rules that change. Second-order series and geometric series both require the student to notice that the operation between terms is not fixed. On 3, 7, 15, 31, 63 he did compute first differences successfully (+4, +8, +16, +32) - that is real reasoning - but then carried the last difference forward as a constant instead of testing whether the differences form their own pattern. He stops modeling one level too early.
Cluster 2: inverted operations. The discount item is the same failure in numerical form: he applied the percentage to the given value rather than reversing the operation.
The cannot-be-determined result is diagnostic. Answering 0/2 while attempting all 24 items indicates a bias toward producing a number. He treats an unsolvable item as a computation he has not found yet, rather than a category of answer. That is metacognitive, and it is independent of the arithmetic clusters.
Treat this as a rule-depth and operation-direction weakness, not a number-sense deficit.
2. Give the full reasoning for the discount item, then name one follow-up question that separates a conceptual error from a procedural one.
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Reasoning. After a 20 percent discount, $40 represents 80 percent of the original price. So 0.80 × original = 40, and original = 40 ÷ 0.80 = $50. Check: 20 percent of $50 is $10, and $50 − $10 = $40.
What he did. He treated $40 as the original and added 20 percent of it ($8). That is two errors compounding: the wrong base, and the wrong direction of the operation.
The follow-up question. Ask, without computing: Is the original price more than $40 or less than $40?
- If he says less, or is unsure, the concept is missing - he does not yet hold that a discount reduces price and that recovering the original means going back uphill. Teach with a concrete anchor before any procedure.
- If he says more but cannot set up the division, the concept is intact and the gap is procedural. Teach the complement step and the division.
Second follow-up worth adding: have him check his own answer. Twenty percent of $48 is $9.60, and $48 − $9.60 = $38.40, not $40. A student who will run that check catches his own error without being told the rule, which is often the faster route to durable correction.
3. Design a 10-item practice set and a response requirement that addresses all three error patterns, and justify each component.
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Practice set
- 3 second-order series. Require two written lines: the first differences, then the pattern among them. Targets Cluster 1 by making the second level visible and non-skippable.
- 3 reverse-percentage items, at least one involving an increase rather than a discount. Require a written check step - apply the percentage back to the answer and confirm it returns to the given value. Targets Cluster 2 and builds self-correction.
- 3 cannot-be-determined items, written so that some are solvable and some are not, in mixed order. Targets the answer-production bias; a student who knows the category exists still has to decide, so mixed order is essential.
- 1 mixed item combining a reverse percentage with a comparison.
Response requirement. Before selecting, the student writes one justification sentence. This is the mechanism, not decoration: it forces the rule to be externalized, which is exactly what he skips on the 95 answer and on the discount item.
Format. Interleave the types rather than blocking them by kind, because the real subtest mixes them and the skill under test is choosing the right approach without a label telling you which one to use. Score by error category, not total, so improvement in one cluster is visible even if the overall count moves slowly.