The Group Work That Isn't Working
You are teaching a 10th-grade geometry lesson on triangle congruence criteria. You designed a 25-minute group task in which groups of four must decide whether four given sets of triangle measurements determine a unique triangle, and justify each decision. You planned for groups to use patty paper and rulers. Ten minutes in, you observe: Group A has one student doing all the work while three watch; Group B is arguing about whether SSS and SAS are 'the same thing' and has stalled; Group C finished in four minutes with 'yes, yes, no, yes' and no justification; Group D is off-task, folding patty paper into origami. You have 15 minutes left before the exit ticket.
1. Diagnose what is going wrong in each group in terms of the mathematical demand of the task, not just behavior.
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The behaviors are symptoms; the diagnosis is about where the cognitive demand collapsed.
- Group A (one worker, three watchers): The task has no role structure that forces distributed thinking. Deciding uniqueness is a single-answer task, so one confident student can carry it. The demand collapsed to 'produce four answers.' There is no artifact that requires each member's reasoning to be visible.
- Group B (SSS vs. SAS argument): This is actually productive - they are wrestling with a real mathematical distinction (which combinations of parts force congruence). The stall is because they lack a shared representation. They're arguing in words about a spatial claim. The demand is high but the tool (patty paper) isn't being used to settle it.
- Group C (fast, unjustified): They answered the literal question ('does it determine a unique triangle?') but skipped the justification clause. The task's success criterion wasn't made visible to them, so they optimized for speed. Demand collapsed to guessing.
- Group D (off-task): The patty paper is a manipulative with no prescribed use, so it became a toy. Without a structure like 'trace, then try to build a different triangle with the same measurements,' the material has no mathematical job.
Common thread: the task asked for justification but didn't build in a mechanism that requires it.
2. Describe the specific moves you would make in the next 15 minutes, in order, and explain the reasoning behind each.
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I'd prioritize making justification unavoidable over covering all four groups equally.
Minutes 0–2 - Whole-class reset (brief). State the success criterion explicitly: 'A correct answer without a reason earns no credit today. Your group's job is to convince another group.' This fixes Group C's optimization target and reframes the task.
Minutes 2–6 - Targeted intervention, Group B. They're closest to the mathematical heart. Ask: 'Can you build a triangle with these three sides that is different from your partner's?' Hand them the patty paper as a test, not a craft supply. If they can't build a different one, that's the argument for uniqueness. This converts their verbal argument into a spatial proof.
Minutes 6–9 - Group A. Assign a reporting role: 'In two minutes, each of you will explain one of your four decisions to me, chosen at random.' This distributes accountability without singling anyone out. Don't lecture; just make the requirement visible.
Minutes 9–12 - Group D. Redirect the patty paper with a concrete job: 'Trace one triangle, then try to make a different triangle with the same two sides and included angle.' This is the same move as Group B and turns the off-task material into the task.
Minutes 12–15 - Group C. Ask them to justify just one of their 'yes' answers to a neighboring group. One is enough to reset the norm.
Reasoning: I'm triaging by proximity to the mathematical goal, not by behavior severity. Group B and D need the same conceptual tool (build a counterexample), so one move serves both. Group A needs structural accountability. Group C needs the success criterion restated. I'm not trying to 'finish' all groups - I'm trying to leave every group with a reason to justify.
3. How would you use the exit ticket to learn whether the lesson's conceptual goal was met, and what would you look for?
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The exit ticket should test the reasoning, not the recall of which criteria work. A recall item ('list the congruence criteria') would tell me nothing about whether students understand why some combinations force uniqueness.
Design: One item - 'Here are two triangles with side lengths 5, 6, and an angle of 40° that is not between them. Can you build two different triangles? Draw or explain.' This targets the SSA ambiguity, which is the conceptual crux of the lesson and the thing Group B was circling.
What I'd look for (a rubric, not a score):
- Level 3 (conceptual): Draws two non-congruent triangles with the given parts, or explains that the angle can swing, producing two possibilities.
- Level 2 (partial): Says 'no, it's not unique' but can't show why, or shows one triangle only.
- Level 1 (procedural): Recites a criterion ('SAS works') without connecting to the item.
What the data would tell me: If most students land at Level 1, the lesson produced label-matching, not reasoning - and I'd reteach with the patty-paper counterexample as the main task next class, not an extension. If Group C's members land at Level 2–3, my in-class redirect worked. If Group A's non-workers land at Level 1, the accountability move didn't transfer to individual reasoning, which is a design problem to fix next time (individual written justification before group reporting).
I'd also read the tickets by group to see whether the intervention moved the group or just the reporter.