The Matrix Item With a Colour Flip in the Bottom Row
The nonverbal item under review is a 3 × 3 matrix with eight cells filled and the bottom-right cell missing.
Row 1: a white circle containing one black dot; a white circle containing two black dots; a white circle containing three black dots. Row 2: a grey square containing one black dot; a grey square containing two black dots; a grey square containing three black dots. Row 3: a solid black triangle containing one white dot; a solid black triangle containing two white dots; then the missing cell.
The scoring key gives the missing cell as a solid black triangle containing three white dots. Eighteen students attempted the item: twelve produced the keyed cell, four produced a solid black triangle containing three black dots, and two produced a white triangle containing three black dots. A teacher on the review panel argues that dot colour is a drawing convention rather than a rule, so the four students using black dots should also be credited - which, she says, makes the item flawed.
1. Reconstruct the missing cell and show how each attribute is determined.
Read the worked answer
Four attributes are needed, each with its own source.
- Outer shape is constant across each row: circles, then squares, then triangles - so the missing cell is a triangle.
- Number of dots is constant down each column: 1, 2, 3 - so three dots.
- Fill is constant across each row: white, then grey, then solid black - so solid black.
- Dot colour is not free. Dots are black on the white and grey grounds and white on the black ground; black on black would be invisible. Two row-3 cells already show white dots, so the rule is evidenced twice rather than guessed once.
Combining: a solid black triangle containing three white dots, which is the keyed cell. Note that shape, count and fill are each forced by one axis; only dot colour depends on another attribute. That dependency is what makes the cell feel uncertain even though it is fully determined.
2. Diagnose the two wrong answers and say which one is the more diagnostic of a partial-rule strategy.
Read the worked answer
Solid black triangle, three black dots (4 students). Shape, count and fill are all correct; only the colour of the dots is treated as a fixed property of the whole matrix rather than as a consequence of the fill. This is a near-miss built on good multi-attribute tracking. A child who checks 'would I be able to see these dots?' recovers it. The useful prompt is not 'look again' but 'what colour is the paper under the dots?'
White triangle, three black dots (2 students). The count is right but the row rule for fill has been lost, probably because the first row's white fill was taken as a default and the triangle inherited it. Nothing in the matrix supports a white triangle in row 3, so this is an error and not an alternative reading.
The first error is the more diagnostic: three of four attributes are tracked correctly and only the dependent one is missed. The second shows an independent attribute being dropped, which usually means one feature is being tracked at a time.
3. The colleague wants to credit both answers. Write the argument you would make, and state the condition under which you would agree with her.
Read the worked answer
Argue from inferability, not authority. Dot colour is not a rendering accident here: it is determined by the fill and it is demonstrated in two of the nine cells before the missing one. A student can derive it from the given matrix without outside knowledge, which is the standard for any rule-based item.
The cost of crediting both is concrete: if dot colour is treated as free, the item stops measuring a dependent attribute and collapses into a simpler three-attribute item. You would be scoring a different construct and reporting it under the old label.
I would concede in two specified conditions: if only one row-3 exemplar existed, so the rule rested on a single instance and could plausibly be a printing choice; or if the dots had been drawn black on black, making contrast a matter of reproduction quality rather than reasoning. Neither holds.
Separate the uses: for formative work, credit the response and discuss it; keep the key for summative scoring.