Critical Thinking in Mathematics: Tasks, Questions, and Evidence

Decision first: Mathematical critical thinking is visible when students make a claim, choose or create a representation, connect evidence to the claim, test an alternative, and revise. A hard problem alone does not produce reasoning; the task, questions, wait time, and evidence routine must make reasoning necessary.

What to plan for

The National Research Council’s Adding It Up describes mathematical proficiency as interwoven conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. A reasoning task should therefore do more than ask for an answer or a preferred trick.

• Claim: What conclusion, comparison, prediction, or generalization must students make?
• Evidence: Which quantities, cases, diagrams, definitions, or properties could support or challenge it?
• Connection: How must students explain why the evidence supports the claim?
• Alternative: Which counterexample, second method, or competing claim should students test?
• Revision: What opportunity will students have to change an answer after critique?

Task 1, grades 2–3: Is the addition argument valid?

Launch

Read the claim without announcing whether it is correct. Ask students to build or draw 38 and 27, combine like units, and record what happens when the ones exceed nine. Students may use an open number line, partial sums, place-value blocks, or an equation, but the explanation must track the value of the regrouped ten.

Teacher questioning moves

• “Where is the value of each digit represented?”
• “What does 15 ones equal in tens and ones?”
• “Point to the ten that the original explanation lost.”
• “Would the same reasoning fail for 31 + 22? Why or why not?”
• “How can the estimate 40 + 30 help check the answer?”

Weak and stronger reasoning

Weak: “It is wrong. The answer is 65 because that is what I got.” The answer is correct, but no evidence identifies the error.

Stronger: “The explanation loses value. Eight ones plus seven ones makes 15 ones, which is one new ten and five ones. The new ten must be added to the five tens, so there are six tens and five ones. An estimate of about 70 also makes 55 unreasonable.”

Extension and access

Ask students to create a different incorrect regrouping explanation and trade it for diagnosis. Keep place-value labels visible, allow oral or drawn explanation, and do not require copying the full prompt when that is not the objective.

Task 2, grades 4–6: All rectangles with 24 tiles

Launch

Do not give the factor pairs. Ask how the class can know the list is complete. Require a recording method that would reveal a missing rectangle, such as testing factors in order up to the point where the side lengths reverse.

Teacher questioning moves

• “How do you know there is no 5-by-something whole-number rectangle?”
• “Which quantities remain constant and which change?”
• “Where is the perimeter visible in your model or table?”
• “Does rotation create a new rectangle for this question?”
• “What could ‘longer, narrower’ mean in numbers rather than appearance?”
• “Would evidence from only 1×24 and 4×6 be enough to justify ‘always’?”

Weak and stronger reasoning

Weak: “Yes, skinny shapes have more perimeter because the sides are long.” This uses appearance, omits the complete cases, and does not define the comparison.

Stronger: “For the whole-number factor pairs of 24, the perimeters are 50, 28, 22, and 20. As the difference between side lengths decreases from 23 to 2, the perimeter decreases. The table supports the claim for all whole-number rectangles with area 24, but it does not by itself prove the claim for every possible fixed area or non-whole-number side length.”

Extension and access

Ask whether the pattern holds for area 36 and which cases are needed. Provide tiles, grid paper, a factor chart, or an accessible digital manipulative as appropriate. Students can dictate dimensions to a partner while still supplying the mathematical reasoning.

Task 3, grades 7–10: When does each pricing plan cost less?

Launch

Ask for a prediction at 0, 5, 8, and 12 miles before solving symbolically. Require students to label the independent variable, cost units, practical domain, intercepts, and meaning of the intersection. The graph is evidence only if scale and labels make the comparison interpretable.

Teacher questioning moves

• “What does each intercept mean in the situation?”
• “Which plan changes faster, and where is that visible in each representation?”
• “Why is the inequality direction different on either side of 8?”
• “Does the algebraic solution alone tell a customer which plan to choose?”
• “Which assumption is most likely to matter: distance rounding, minimum fare, wait time, taxes, service area, or another condition?”
• “What evidence would be needed before treating the simplified model as an actual purchasing recommendation?”

Weak and stronger reasoning

Weak: “Plan A is cheaper because 0.75 is less than 1.00.” This ignores the different booking fees and the distance range.

Stronger: “Plan A has the lower rate but the higher fixed fee. Solving the equality gives 8 miles. Checking 5 miles gives A = $7.75 and B = $7.00, while 12 miles gives A = $13.00 and B = $14.00. Therefore B is less below 8 miles and A is less above 8 miles, if the published linear rules include all charges. A minimum fare or distance rounding could move the break-even point.”

Extension and access

Add a third plan, a fixed budget, or a piecewise minimum fare. Provide a structured table, calculator, or accessible graphing method when permitted. Keep the reasoning target separate from motor, vision, reading, or technology barriers.

Questioning moves across grade bands

On a small screen, scroll this table horizontally to see all columns.

Questioning moves across grade bands table

Purpose	Teacher move	Avoid
Clarify the claim	“What exactly are you claiming, and for which cases?”	Replacing the student’s claim with the teacher’s wording before hearing it
Press for evidence	“Which number, diagram feature, definition, or property supports that step?”	“Are you sure?” without identifying what evidence is missing
Connect representations	“Where is that quantity in the equation, table, graph, or model?”	Collecting several representations without requiring a connection
Test generality	“Would it still work if…? Find a case that could disprove it.”	Treating two supporting examples as proof of an unrestricted claim
Support critique	“Restate the reasoning before naming the point you question.”	Turning critique into a vote for a student or answer
Invite revision	“What will you keep, change, or qualify after hearing the evidence?”	Ending the task as soon as the correct answer appears

Short reasoning rubric

On a small screen, scroll this table horizontally to see all columns.

Short reasoning rubric table

Level	Claim and representation	Evidence and connection	Alternative and revision
0 — Not yet interpretable	No clear claim or representation of the task	Evidence is absent or unrelated	No response to an alternative
1 — Emerging	A claim or method is present but incomplete	A relevant example appears without explaining why it supports the claim	Acknowledges a question but does not test or revise
2 — Reasoned	Clear claim with an appropriate representation	Relevant evidence is connected to the claim	Tests a serious alternative or corrects a limitation
3 — Generalized	Precise claim with domain or conditions	Multiple representations or properties are connected coherently	Uses a counterexample, boundary, or critique to refine the conclusion

Use the rubric formatively. A concise level-2 explanation may be stronger evidence than a long response filled with unrelated vocabulary.

Collect evidence without grading every discussion

1. Select one focus. Listen for one feature, such as connecting a representation to a claim.
2. Sample deliberately. Each lesson, collect three written responses and listen closely to two different pairs. Rotate the sample so every student is represented across the cycle.
3. Use a roster code. Mark C for clear claim, E for evidence, L for linked reasoning, A for tested alternative, and R for revision. Record only observed evidence; a blank is “not sampled,” not “cannot reason.”
4. End with an individual capture. Ask every student for a two-minute claim-evidence sentence, corrected example, or annotated representation.
5. Choose the next move. Use class patterns to select one question, model, or small group for the next lesson.

Do not assign a score to every spoken contribution. Public discussion provides partial, uneven evidence, and some students reason more clearly after preparation or in writing. Combine samples over time and provide required access routes.

Common mistakes

• Praising an answer as “creative” without checking mathematical validity.
• Asking “why?” repeatedly without naming the missing link.
• Calling a task open ended when only one teacher-approved method receives credit.
• Using citizenship or future-career claims in place of a mathematical purpose.
• Giving sources about reasoning that do not support the actual mathematical claim in the task.
• Grading speed, confidence, or amount of talk as though it were reasoning quality.

Sources and scope

• National Research Council, Adding It Up: Helping Children Learn Mathematics.
• National Council of Teachers of Mathematics, Principles to Actions.
• Standards for Mathematical Practice.
• Institute of Education Sciences, Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.

These sources support reasoning, representation, discussion, and evidence-informed instruction. They do not establish that one task, question stem, or rubric works in every curriculum or grade.

Related guides

• Lesson Plan Goals and Objectives: Examples and a Writing Formula
• From Assessment Data to Instruction: A Teacher Decision Guide
• Visual Aids in the Classroom: Design, Access, and Review Checklist

Next action: Select one grade-band task, write the claim and evidence students must produce, and choose two questioning moves before deciding which representations to provide.