A Defensible Distractor in a Dot-Movement Analogy
An instructional coach is assembling a twelve-item analogy practice set for second graders. Her draft item pairs two figures. Figure A is a large open, unfilled equilateral triangle with a small solid dot resting on the top vertex. Figure B is the same large open triangle with the dot moved to the midpoint of the bottom edge. The second pair begins with Figure C, a large open square with the dot on its top-left corner, and the student must choose the figure that completes the pair. Option 1 puts the dot on the bottom-right corner. Option 2 puts it at the midpoint of the bottom edge. Option 3 puts it on the top-right corner. Option 4 puts it at the center of the square. A colleague warns that both Option 1 and Option 2 can be defended from the first pair. The set goes to the printer on Friday, and the coach has one afternoon to decide whether to keep, repair, or cut the item.
1. State the relation in the first pair as precisely as you can, then say what each of the four options would require you to believe.
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The clean description of A to B is: the dot starts on a vertex and moves to the midpoint of the side opposite that vertex, crossing the interior, while the triangle keeps its size, orientation and outline. The dot also stays on the boundary.
Applying that rule to a square with the dot on the top-left corner gives the midpoint of the bottom edge, Option 2.
Option 1 (bottom-right corner) requires a different relation: the dot moves to the point diametrically opposite through the figure's center. Option 3 (top-right corner) requires only that the dot slide along the boundary to another corner, which the first pair never shows, since B is not a vertex. Option 4 (center) requires the dot to leave the boundary, which the first pair also never shows.
So the item is not a four-way puzzle. It is a two-way puzzle between Options 1 and 2.
2. Explain why two rules survive the first pair, name the geometric accident responsible, and repair the item so only one survives.
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In an equilateral triangle, the top vertex, the center and the midpoint of the base lie on one line, and reflecting the vertex through the center lands exactly on the base midpoint. The two rules, vertex-to-opposite-midpoint and point-reflection-through-center, therefore produce the same figure. The triangle's symmetry hides the difference, and a square, which lacks that symmetry, exposes it.
The repair belongs in the pair, not the options: put the dot at the midpoint of the square's top edge in Figure C. Now point reflection through the center and midpoint-to-opposite-midpoint both yield the midpoint of the bottom edge, Option 2, and Option 1 no longer fits any rule the first pair supports. Deleting Options 1 and 2 would hide the flaw rather than fix it, because the ambiguity lives in the stimulus.
A useful design rule: pick a C figure with less symmetry than the A-B pair, never more.
3. Which option would a student who encoded only "the dot goes down" choose, and is that a reasoning error or an encoding problem?
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That student picks Option 3, the top-right corner. The dot is lower than it was and still on the boundary, so a direction-only reading is satisfied.
The failure is encoding, not logic. The student extracted one attribute, vertical displacement, and dropped the anchor: displacement measured from which point, and by how much. Once the anchor is missing, several options look valid, and the child is effectively guessing.
The coaching move is to make the anchor verbal, not to drill more items. Ask, "From where to where? Say the start point and the end point out loud before you look at the options." NNAT3-style analogy items are built so partial encodings land on a drawn distractor, so the habit of naming start and end is what protects the score.