VRG and Teacher Moves
Why is it important for teachers to make groups visibly random?
So the students can perceive & believe they are really random.
What is a way to acknowledge a student without immediately giving the answer?
Listen and respond with a prompt such as “What have you tried?”
Students finish a task in two minutes by applying the same algorithm. What is one way to increase its cognitive demand?
Remove the prescribed method, ask for multiple representations or a justification, introduce a meaningful constraint, or ask students to generalize.
How can you make sure the students still know they have been heard?
Smile/look at them before walking away
A group used a diagram, another a table, and another an equation. What should determine the order in which their work is discussed?
The mathematical connections and accessibility of the representations—not speed, correctness alone, or which group finished first.
A student asks to work only with friends. How would you explain VRGs?
Explain that changing groups helps everyone share strategies and learn from different thinkers.
How could you tell whether VNPs are increasing thinking?
Look for more students discussing, trying approaches, revising, and explaining—not just completed answers.
What is the difference between a task that requires thinking and one that simply requires following a procedure?
A thinking task requires students to make decisions, explore strategies, reason, and justify. A procedural task primarily requires applying an already-known method.
Proximity, Stop-Thinking, & Keep-Thinking
Students have finished a task and are copying a teacher solution. Redesign the final five minutes as BTC consolidation.
Use selected student work, ask students to compare and connect strategies, identify the key mathematical idea, and record a meaningful synthesis.
Why should teachers avoid answering every question immediately during group work?
Immediate answers can transfer cognitive responsibility to the teacher and interrupt students’ reasoning and persistence.
What do you do to keep students on task while at the boards?
Thank you!
A task has a low floor but no meaningful extension. Design a high-ceiling follow-up that preserves the central idea.
Ask students to explore all cases, change a constraint, prove a pattern, compare methods, or generalize—rather than simply giving more of the same calculations.
What could be a response to a proximity question?
Isnt that interesting? Why do you think that is? Are you sure? Does that make sense? Are you asking me or telling em? Why don't you try something else?
A student offers an incorrect strategy that reveals a common misconception. How might you use it productively in consolidation?
Invite analysis of where the reasoning changes, compare with a representation, and guide students to revise without publicly labeling the student as wrong.
A group has been silent for two minutes at a vertical surface. What evidence would you gather before intervening?
Look for attempts, gestures, representations, and group talk; ask what they have noticed or tried before deciding on the lightest useful prompt.
One student is doing all the writing at the VNP while the others watch. What would you do to increase participation without taking over?
Encourage students to share responsibility for recording and explaining. Rotate the marker, ask another student to explain the group's reasoning, or have students build on one another's ideas.
A teacher launches a rich task by modeling the first two steps. Analyze the trade-off and propose a better launch.
The model may remove the need to think. Clarify context, goal, and materials without demonstrating a solution path; let groups select an entry point.
A student asks, “Can we use a table to test the next case?” Classify it and explain the response.
A keep-thinking question. Engage with it without taking over: “What might the table reveal?”
Three groups have an incorrect strategy, an efficient unexplained method, and a visual model. Sequence and facilitate a consolidation.
Start with an accessible representation, surface and resolve the misconception, connect it to the efficient method, and have students articulate the generalizable mathematics.
Two groups have very different strategies. What question could connect their reasoning without evaluating either one?
“Where do you see the same quantities or relationships in both approaches?” or “How could each group explain the other’s method?”
During a lesson, groups show strong thinking but the closing discussion is rushed. What would you prioritize and why?
Connect selected strategies and name the mathematics; activity alone does not ensure shared understanding.
A group finishes early and asks for the next worksheet. What BTC-aligned response could deepen thinking?
Ask them to justify, generalize, find another approach, test a boundary case, or create a related challenge.
A group asks: “What do we do next?” then “Is this right?” then “Would our pattern hold for 100?” Classify each and respond strategically.
“What next?” may be stop-thinking; “Is this right?” seeks validation (stop-thinking); “Would it hold for 100?” is keep-thinking. Redirect the first two to group reasoning; engage with the generalization question.
During a lesson, groups show strong thinking but the closing discussion/consolidation is rushed. What would you prioritize and why?
Protect a brief bottom-up consolidation that connects selected strategies and names the mathematics; activity alone does not ensure shared understanding.