The 3x3 Figural Matrix: Nia Picks the Dark One
A three-by-three figural matrix is printed on one page, and the bottom-right cell is empty. The first row shows three circles: the first outlined only, the second half shaded, the third completely shaded. The second row shows three triangles in the same three shading states, in the same order. The third row shows an outlined square, then a half-shaded square, then the empty cell.
The four answer choices are: (A) a completely shaded square; (B) a half-shaded square; (C) a completely shaded circle; (D) a completely shaded square tilted to stand on one corner, with a small dot in the centre.
Nia, age eight, chooses C. She says, 'The last one in each row is always the dark one, and I picked the dark one.' She answers in about fifteen seconds and does not look at the third row before choosing. Asked what stays the same across the top row she says, 'They are all round.' Asked what changes across the top row she says, 'Nothing.'
1. Solve the matrix, state both rules, and explain what each distractor is testing.
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Answer: A.
Rule 1, read across a row: the outer shape does not change. Row 1 is all circles, row 2 all triangles, row 3 all squares. So the missing cell must be a square.
Rule 2, read down a column (equivalently, left to right within a row): shading increases. The first column is outline only, the second is half shaded, the third is fully shaded. The missing cell sits in the third column, so it must be fully shaded.
Combining: a fully shaded square.
Distractors. B tests whether the child tracked shading at all - it is the middle column's shading in the right shape. C tests whether the child tracked shape - it is the right shading in the wrong family, and it is the choice of a child who has only one rule. D tests whether the child invents a rule: the tilt and the dot appear nowhere in the grid. A figural matrix never asks you to add a feature that no cell contains, which is why D can be eliminated without deciding anything else.
2. Diagnose Nia's choice: which rule did she use, which did she miss, and what error pattern does her explanation reveal?
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She used the shading rule correctly - she knew that the third cell in a row is the dark one - and applied it to the wrong shape family. She did not use the shape rule at all.
Her own words confirm this. 'What stays the same across the top row?' - 'They are all round.' She can see the constancy when asked directly, so the perception is available; she simply does not use it as a rule when choosing. 'What changes?' - 'Nothing' shows she treated each row as one object repeated three times rather than as a shape carrying a changing feature.
Error pattern: one-feature dominance, or partial-rule satisfaction. The correct answer is the only choice that satisfies both rules, and a child who checks one rule can still land on a fully consistent-looking answer. Fifteen seconds and no glance at row 3 suggest she matched to a pattern remembered from the first two rows instead of scanning the whole grid.
The teaching target is not 'learn the two rules' - she has one of them. It is a checking routine: before choosing, confirm the answer against the row it sits in and the column it sits in, and reject anything that satisfies only one.
3. Design the next matrix item to test whether the checking routine has taken hold, and say how you would score it.
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Build an item where the shading rule alone cannot choose, so the shape rule has to be used.
Design: row 1 is outlined, half, full circles; row 2 is outlined, half, full triangles; row 3 begins outlined pentagon, half pentagon, blank. Choices: (A) fully shaded pentagon; (B) fully shaded hexagon; (C) half-shaded pentagon; (D) fully shaded circle; (E) fully shaded pentagon drawn slightly larger.
Why this works: three of the five choices are fully shaded, so 'pick the dark one' no longer produces an answer. The child must notice that the row is pentagons and that the third column is fully shaded. The size variation in E tests whether she is also over-weighting an irrelevant feature.
Scoring: count the item plus the explanation. A correct choice with 'because the row is pentagons and the last column is dark' is strong evidence. A correct choice with no explanation is weak - she may have eliminated on shading alone. An incorrect choice on this item after succeeding on the original tells you she is still matching one feature and needs the routine made physical, for example pointing to the row and column before each choice.