Every Student Is a Math Thinker Part 2: Moving Beyond "Right Answers"
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Picture a student in your class who finally blurts out an answer. Maybe they hesitated, maybe they looked nervous, but at last, they say: “Is it… 42?”
If you’re like me, you’ve probably responded instinctively: “Yes, that’s right!” or “No, not quite.” Then you moved on.
But here’s the problem: if the only thing students are asked to think about is whether they’re right or wrong, then that’s what they’ll remember.
As we explored in Part 1 of this Every Student Is a Math Thinker series, cognitive psychologist Dan Willingham1 reminds us that we remember what we think about, and we master what we practice. If students mostly think about correctness, they’ll remember little beyond the relief of being right (or the sting of being wrong). But if they think about why an answer works, and practice reasoning strategies, they’ll remember mathematics and master these strategies.
More importantly, they’ll start to see themselves as math thinkers; people who do math, not just answer it.
The Trap of Correctness
Traditional math instruction has conditioned students to chase the right answer. This isn’t entirely bad because we all know that accuracy matters. But when correctness becomes the only goal, several unintended consequences emerge:
1. Shallow Memory - Students focus on procedures that “work” but don’t understand why they work. The result: fragile learning that collapses under new contexts.
2. Fixed Mindsets - Students who are often wrong may conclude they “aren’t math people,” while fast finishers equate speed with intelligence. Both beliefs are deceiving and undermine equity.
3. Fear of Risk-Taking - If the only acceptable contribution is the right answer, students won’t share partial ideas, conjectures, or questions, which is the foundation of mathematical thinking.
If every student is to become a math thinker, we must reframe the classroom so that reasoning matters more than correctness.
Strategy 1: Ask for Justification, Not Just Answers2
One of the simplest and most transformative shifts we can make as teachers is to ask for justification. Instead of asking for answers as a starting point, we can ask students to explain their thinking:
“How did you go about solving this problem?”
“Can someone add on to that idea?”
“Why does your method make sense?”
This change keeps the spotlight on the process of thinking, not just the product. Even incorrect answers become valuable fodder for classroom dialogue. Now the class is analyzing reasoning instead of labeling correctness.
Strategy 2: Use Rich, Low-Floor/High-Ceiling Tasks
To encourage thinking, we need tasks that invite all students to think. As Jo Boalar3 states: low-floor, high-ceiling tasks are accessible to all learners while offering opportunities for deep exploration.
Example: One of my favorite tasks to begin the year: The Four 4s Challenge - Make expressions equal to the integers 1 through 20 using four 4s in each expression and any operations.
One student might start with: 4 + 4 + 4 + 4 = 16.
Another might do 4 x 4 + 4 / 4 = 17.
Still another may try something like 4! - 4 - 4 / 4 = 19.
Every student is reasoning, but at different levels - low-floor. Students also have the opportunity to explore the task in depth - high-ceiling. And each student’s contribution is a valuable part of the collective mathematical conversation.
Strategy 3: Elevate Mistakes as Learning Opportunities
When correctness dominates, mistakes feel like failure. But when reasoning is valued, mistakes become springboards for deeper understanding.
Practical moves:
- Error Analysis: Present a worked solution with a mistake and ask students to find, explain, and fix it.
- My Favorite No: Display an incorrect student solution anonymously and celebrate the thinking behind it before addressing the error.
- Mistake Journals: Have students reflect weekly on an error they made and what they learned from it.
By normalizing mistakes as an essential part of reasoning, teachers create a classroom culture where students feel safe taking intellectual risks.
Strategy 4: Structure Peer-to-Peer Dialogue
Reasoning grows stronger when it’s verbalized. When students explain, justify, and critique ideas together, they deepen understanding and develop agency.
Structures to try:
- Think-Pair-Share: Students first consider individually, then discuss with a partner before sharing with the class.
- Sentence Frames: “I agree with ___ because…” or “I noticed something different…”
- Board Work Protocols: In Peter Liljedahl’s Building Thinking Classrooms,4 students work visibly on vertical whiteboards so reasoning is public, collaborative, and mobile.
These structures shift authority from the teacher as “answer-giver” to the classroom as a community of mathematicians.
Why This Matters for Equity
When correctness dominates, only the subset of students who typically are fast, confident, or already fluent feel successful. Others receive a steady stream of evidence that they don’t belong.
But when reasoning is the measure of success:
- All contributions have value - even partial ideas are worth discussing.
- Multiple ways of thinking are affirmed - this validates linguistic, cultural, and cognitive diversity.
- Belonging increases - students see that mathematical success comes from engaging with mathematics, not innate talent.
Equitable classrooms don’t remove challenges; instead, they redefine it. Challenge becomes about depth of reasoning, not speed or perfection.
Closing Thoughts
If we want every student to see themselves as a math thinker, we must expand what counts as “doing math.”
In Part 1, we explored how routines like Notice & Wonder and Which One Doesn’t Belong? help students get comfortable thinking aloud. Here, in Part 2, we’ve seen that shifting the focus from correctness to reasoning helps sustain that thinking over time.
When students reason, explain, and revise their ideas, they’re practicing the very habits that lead to mastery because, as Willingham reminds us, we remember what we think about and master what we practice.
And this naturally leads us to Part 3 of our Every Student Is a Math Thinker series, coming next month: Creative & Critical Thinking in Math. We’ll explore how open-ended and generative tasks push students to apply reasoning flexibly. In this environment, students create with mathematics, not just reproduce it.
By continuing to build these layers - curiosity, reasoning, creativity - we move closer to classrooms where every student doesn’t just learn math, but thinks mathematically.
Pete Grostic, Ph.D
Executive Director
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1Why Don't Students Like School? - Daniel T. Willingham
4Liljedahl, P. (2021). Building Thinking Classrooms in Mathematics, Grades K–12. Corwin.
11/03/2025