# Questioning for Deeper Mathematical Understanding

06/25/2024
Learn how to gently challenge students beyond what they think they can do with deep-thinking questions.

“I’m finished!” says a voice behind you, only moments after you have settled everyone into the task. A quick assessment can tell you all you need to know: It looks like you’ve rushed. Let’s look at it together and try again. Or: I think you’re missing something here. Let’s make sense of the problem together.

But what if your student has completed the problem accurately and thoroughly? Where do you go from there? How do you make sure you are reaching all your students in the moment, including those students who want—and are ready for—more?

## Developing Deep-Thinking Questions for Students

Some teachers seem to have those “back-pocket” questions at the ready—a just-right, rich mathematical query at the appropriate level of accessibility for their deepest thinkers. But how do you come up with those questions? How do you know what the right level is? And how can you be prepared with those kinds of deep-thinking questions when you need them?

Not surprisingly, the answer isn’t simple, and the job isn’t easy. Posing Purposeful Questions is one of the National Council of Teachers of Mathematics’ Effective Mathematical Teaching Practices. While we often focus on encouraging students to explain, reflect, clarify, and make the math visible, one of the more difficult and powerful types of questioning is that which helps students reveal a deeper understanding. You can practice doing this in your classroom by keeping these ideas in mind:

Help them discover for themselves why the math you are asking them to do is valuable.

Student: 5 + 4 = 9. That’s a fact I know.
Teacher: That’s great. What other problems become easier to solve because you know 5 + 4 = 9?

Student: I broke these numbers into tens and ones to multiply them.
Teacher: Are there other ways you could have broken them apart? Will any way work? Are some ways better than others?

Student: I solved it two different ways.
Teacher: What kinds of problems might prompt you to choose one strategy over the other?

## Listen Carefully to Your Students

Determine where their ideas came from, and let that take you down the path to the next steps.

Student: I found all the factors of 48 by starting at 1 and testing all the numbers for divisibility, up to 48.
Teacher: Do you actually have to test every number? Is there a place you could stop and feel confident that you didn’t miss any? Could you come up with a rule for this?

Student: I used this hundred grid to model my fraction, and now I can see it as a percentage and a decimal as well!
Teacher: Do you think you can have half of a percentage? How could you model that on the grid? How would you translate that into a decimal?

Get them to push their own thinking.

Student: I can put zeros at the end of a decimal to help me compare and compute with different-length decimals.
Teacher: What is a comparable action for fractions? Whole numbers?

Student: I noticed that every number that has a factor of 8 also has factors of 2 and 4.
Teacher: What other factors have a similar pattern? Why?

To feel successful asking students deep-thinking questions, it helps to practice deep thinking yourself. The more confident you are in how mathematical concepts develop across the grades, the more material you will have to pull from. But don’t underestimate the value of what you already know—try to recall how you made sense of the concept, perhaps many years ago. Your own personal “aha!” moments can provide fodder for your students as well.

Above all, remember to:

• Go for Depth
The questions shouldn’t have one answer or be answerable without significant work or thought.

There can be a fine line between pushing thinking to a new level and crossing over into another grade’s content. We want students to think deeply about the math that’s in front of them, not race ahead to something completely new.

• Aim to Stimulate Thinking and Reasoning
Ask open questions and questions requiring higher-level thinking.

• Encourage Students to Work in Pairs or Small Groups
Great ideas can be sparked through good discourse.

• Be Ready with These Question Starters

• How do you know . . .
• Will it always . . .
• Could you ever . . .
• What if . . .

It takes thought, time, and practice to be a teacher who always has the perfect question at the ready. But if you use these ideas to get started, you will find that it is a skill you can hone to help you quickly reach all your students.

Want to read more blogs from Laurie? Check out Make Building Math Fluency Fun and 5 Tips for Keeping Students Actively Participating and Engaged.