Student Engagement
Building Thinking Classroom Tasks: A Practical Guide
Short answer
A thinking task should require students to make and test a mathematical decision rather than merely repeat a demonstrated procedure. Selected routines associated with Peter Liljedahl’s Building Thinking Classrooms in Mathematics—including task choice, visibly shared work, grouping, and teacher questioning—can be considered as a coordinated design. Do not treat a single routine, standing position, whiteboard, random group, or “thinking” label as proof of engagement, equity, understanding, or achievement.
Use the model name precisely
Building Thinking Classrooms is a named instructional framework, not a generic label for every collaborative mathematics task. This guide uses selected task-planning ideas while keeping local implementation claims bounded. Teachers seeking model fidelity should consult the author’s published framework and applicable training materials. The research and professional sources below support task and discussion design; they do not establish that every routine or local adaptation produces a stated outcome.
Revise a routine problem into a testable task
| Planning field | Routine version | Revised task | Evidence to collect |
|---|---|---|---|
| Decision | Calculate 18% of 250 | A student says that increasing 250 by 18% and then decreasing the result by 18% returns to 250. Test the claim. | Calculation, counterexample or proof, and explanation of why the two changes do or do not cancel |
| Representation | Use the demonstrated percent formula | Use two representations, such as equations and a bar model or table, and explain how corresponding quantities match. | Correct links between quantities, not two unrelated displays |
| Variation | Complete ten similar items | Identify conditions under which successive percentage changes would return to the starting value. | A generalization tested against at least two cases |
| Individual capture | Submit the group board | Each student records the claim, one supporting step, and one limitation or unresolved question. | Individual reasoning distinct from the shared product |
Task-design checks
- Accessible entry: Students can begin from supplied information, a representation, or a small case without guessing an unstated trick.
- Mathematical demand: The task requires a decision, connection, explanation, test, or generalization tied to the intended content.
- Multiple routes: More than one valid representation or strategy is possible, but the task does not promise that every student will invent one without modeling or support.
- Visible revision: Students can change a claim, representation, or method as evidence develops.
- Individual evidence: The plan includes a brief response showing what each student can explain or apply.
- Teacher decision: The collected work will support a named action—continue, model, compare, reteach, or change the task.
Plan grouping as one variable, not an outcome
Random or visibly random grouping is one routine associated with the framework. Before using it, define the instructional reason, group size, duration, regrouping method, accessible roles, and individual evidence. Random assignment does not guarantee equitable participation, psychological safety, collaboration quality, or mathematical learning. Do not use a public process that discloses ability labels, services, behavior records, or other protected information.
A teacher may instead use pairs, stable groups, individual work followed by comparison, or another structure when access, safety, communication, sensory, mobility, language, or documented-support needs make that route more appropriate. The grouping routine remains subordinate to the objective and required accommodations.
Use shared workspaces without requiring standing
Vertical non-permanent surfaces are another model-associated routine. The relevant planning question is whether a shared, revisable workspace makes mathematical reasoning available for discussion and feedback. A wall-mounted surface, table whiteboard, paper chart, document camera, accessible digital canvas, or individual board can serve that function. Standing, reaching, rapid movement, handwriting, or public visibility should not become unplanned assessment criteria.
- Provide seated, low-reach, large-print, tactile, communication, and approved digital routes as needed.
- Ensure that glare, contrast, marker fumes, congestion, and trip hazards are addressed.
- Do not photograph or publish names, work, or faces without the approved process and required permission.
- Keep an individual response route so the visible group record is not treated as evidence for every member.
Ask questions that preserve the mathematics
| When students… | Ask… | Avoid assuming… |
|---|---|---|
| have not started | What quantity or case can you represent first? What information is fixed? | that hesitation means low motivation or inability |
| have a numerical answer | What claim does that example support? What second case could challenge it? | that an answer demonstrates the intended reasoning |
| have two representations | Where is this quantity in the other representation? Which relationship is preserved? | that producing two diagrams establishes a connection |
| disagree | Which statement differs, and what evidence would distinguish the two claims? | that disagreement itself is productive discourse |
| finish one case | What changes if this value varies? Which part of your explanation still holds? | that a harder number creates a meaningful extension |
Worked example: successive percentage changes
Task: “A store raises a price by 20% and later lowers the new price by 20%. A student says the final price must equal the original price. Test the claim, show the relationship in two representations, and state when two successive percentage changes would cancel.”
Launch
- Confirm prerequisite meanings for percent increase, percent decrease, and multiplicative factor.
- Present the claim without describing a preferred method.
- State the evidence requirements: two tested starting values, two linked representations, and an individual explanation.
- Provide calculators, reference materials, language supports, and accessible workspace routes according to the task and documented needs.
Monitor
- Record which groups use additive reasoning, multiplicative factors, tables, bar models, or equations.
- Ask students to identify the base quantity for each percentage rather than telling them whether the claim is true.
- Choose examples for discussion because they expose a relevant relation or misconception, not because one group appears quickest or most polished.
- Intervene directly when prerequisite knowledge, unsafe interaction, exclusion, or an access barrier prevents participation.
Consolidate and capture
Compare an additive argument with a multiplicative-factor representation. Ask students to map the original price, increased price, and final price across both. Each student then writes whether the claim holds, supplies one calculation or representation, and explains why equal percentage labels use different bases. That response is evidence for this task; it does not establish a stable identity as a mathematical thinker or transfer to every percent problem.
Decide whether to keep the routine
| Evidence | Possible interpretation | Next decision |
|---|---|---|
| Students copy a group answer but individual explanations omit the changing base | The shared board concealed an unresolved content issue | Model the base quantity explicitly and use a new individual check |
| One workspace route prevents a student from contributing or reading the work | The routine created an access barrier | Change the surface, position, tool, role, or response route before repeating |
| Groups finish examples but do not test a general claim | The prompt or prior instruction did not make generalization usable | Add a structured comparison and model one generalization step |
| Discussion is active but reasoning is not traceable in the work | Talk volume or frequency is not sufficient evidence | Require claims, representations, and revisions to be recorded |
Implementation boundaries
- Introduce one routine with a defined objective and review period rather than claiming full-model implementation from a single task.
- Preserve explicit teaching, worked examples, guided practice, and feedback when the evidence shows they are needed.
- Use task-specific participation evidence; do not diagnose confidence, mindset, anxiety, ability, or belonging from visible behavior.
- Apply documented accommodations and consult the responsible specialist when generic routines conflict with an individualized plan.
- Follow school rules for student records, photography, digital tools, safeguarding, physical safety, and public display.
- Report local findings as local findings. Engagement observations, completion rates, and teacher impressions do not establish achievement, equity, or causal effects.
Sources and further reading
- Peter Liljedahl, published books including Building Thinking Classrooms in Mathematics, Grades K–12 — primary model-specific source used to identify the named framework and its coordinated classroom practices.
- National Council of Teachers of Mathematics, Principles to Actions — used for mathematical task, discourse, representation, questioning, and evidence principles; it is not a Building Thinking Classrooms fidelity source.
- Institute of Education Sciences, Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students — used for worked examples, representations, solved-problem analysis, and attention to algebraic structure; it does not validate a branded routine or local outcome claim.
The model source defines the framework. The additional mathematics sources support particular instructional decisions. Local evidence is still required for claims about engagement, equity, understanding, or achievement.