Helping Every Student Think Deeply About Math

Mathematical understanding should be a goal for every student and something they feel capable of achieving. But translating that belief into classroom practice remains one of the central challenges in mathematics education.

“We really want students to sink their teeth into the math and really deeply understand it — all of our students,” says Professor Jon Star. “All students are capable of really wrestling with some fundamental deep issues and questions about mathematics.” 

The issue is not a lack of commitment among teachers, but the difficulty of making abstract mathematical ideas accessible in real time, for all learners. Star, who, in addition to his role at HGSE, teaches middle school math in a nearby K–12 school, shares a small set of practical strategies for teachers designed to deepen student understanding, shift classroom culture, and move beyond simply getting the “right answer.”

Four approaches to building deeper mathematical understanding:

1. Ask more open-ended questions
   Rather than relying on yes-or-no or purely numerical answers, Star encourages teachers to use questions such as, “Why do you think that?” or “How do you know that’s true?” These prompts require students to explain their reasoning, make sense of mathematical ideas, and engage in shared thinking with classmates. “By asking these kinds of questions in my class, I'm trying to create a culture in my classroom where everyone is expected to do a lot of thinking, that this is a place where we think together,” Star says.
2. Prepare extension and “back-pocket” tasks
   Teachers are urged to plan for students who finish quickly by having follow-up questions or variations ready. “It’s always going to be the case that some students are going to finish faster than other students,” Star says. Acceleration — having those students move on to the next lesson — isn’t always the best approach, he says, and instead suggests teachers think about additional ways to challenge students such as asking extension questions. These might include, “Can you solve it another way?” or “If the problem changed slightly, would your answer still be true?” The aim is to deepen thinking rather than accelerate ahead. He advises trying to enter each day’s math lesson with some ideas about extension questions and having tasks — in your back pocket — to use when a student accomplishes the daily objective.
3. Talk less, listen more
   A key shift in classroom practice is giving students more space to explain their thinking, even when it is incorrect. By listening carefully to student reasoning and encouraging peer discussion, teachers can surface underlying misconceptions and support more meaningful mathematical conversations. “I'm trying to get students to engage in mathematical conversations where they can share their thinking and they can listen to each other, they can challenge each other, they can share their thinking,” he says, noting that this is a key way to develop mathematical understanding.
4. Encourage multiple problem-solving strategies
   Students benefit from seeing that there are often many ways to solve the same problem. Discussing different approaches — and comparing their strengths, limitations, and underlying logic — helps students move beyond superficial knowledge to genuine deep understanding. Star notes that this also challenges the idea that the only thing that matters is getting the right answer. “Math problems can always be solved in a lot of different ways,” he says. “And I have found in my work and in my teaching that students benefit from the opportunity to produce and discuss the different ways that a problem can be solved. …Too often teachers implicitly or explicitly communicate to their students that the most important thing is you get the right answer.”