We math teachers tend to ask a lot of questions as we work with students. These questions have various purposes, including building relationships, establishing and strengthening norms, communicating processes, and managing student activities. Many, of course, focus on mathematics content. Here we explore teacher questions that support student higher order thinking about mathematics content. These questions are designed to develop facility with those mathematical practices necessary for complex problem-solving, including connecting concepts and approaches, extending students’ thinking by making and testing conjectures, making generalizations, being creative, and applying critical thinking.

Checking for Content Knowledge

When we work with students as they learn about a topic, we often begin with questions designed to assess what they already know and can do. For example, while modeling a procedure for the class, we might ask students, “What is the next step?” Or, when discussing subtraction, we might ask an example problem in a context such as “How many pencils will I have left from a box of 10 if I give away 3?” We might ask a more open question such as, “What defines a rectangle?” When a student answers accurately, we learn something about what they know.

However, to know best how to proceed it is important to know more about how students arrived at their answers. Did a correct response result from an in-depth conceptual understanding, from following a rote procedure with little understanding, or perhaps from using a creative way to figure out a solution? We might probe an inaccurate answer or an unexpected response to find out whether the student understood the question. We might want to know if there was specific vocabulary or a phrase related to the problem that they didn’t know. Was there a misconception or a computational error (or both)? Did they know the answer but not feel comfortable saying it? Answers to these types of follow-up questions can provide more detailed information about student thinking and inform our efforts to advance their higher order thinking.

Supporting Higher Order Thinking During Problem-Solving

We can use our knowledge of students’ understandings and ways of thinking to help us ask questions that are less about assessment and more about prompting them to apply higher order thinking throughout their problem-solving process. Using questions when students start a task, while they continue to work, and toward the end of the process may, over time, promote student autonomy and persistence, provide opportunities for them to reflect on their work, help them develop a repertoire of problem-solving strategies and behaviors, and support their identities as capable mathematicians.

Questions such as the following can support students as they begin their work on a task:

• What do you understand about this problem?
• What are you not sure about or are wondering about?
• What is something you have done before that seems like this?
• What do you know already?
• Can you make a rough estimate of the answer, such as an “ish” number?
• What is one resource or tool that could help you?
• What would a diagram or visual look like that shows this information?

Productive struggle is a crucial part of the problem-solving process and important for students to experience as they develop their higher order thinking skills and their identities as mathematicians. It can be helpful to remind students that feeling stuck is a normal part of the problem-solving process: if they already knew how to do it, it wouldn’t be as helpful for their learning or their development as mathematical thinkers.

Questions such as the ones below can be used to support students during work, including when they might feel stuck or uncertain about a path forward:

• What have you tried so far?
• What has you feeling stuck?
• What would be helpful for you to get unstuck?
• Do you think this approach will still work, or should you try another approach?
• Is there a way to break this next part down into smaller steps?
• Where could you look for fresh ideas?

As students reach partial or final solutions, questions can help them learn to assess their work, articulate their thinking, and reflect on themselves as problem-solvers. Their responses can give us teachers valuable information about their thinking about the mathematics. For example:

• How do you feel your problem-solving process went?
• What does your answer tell you in the context of the problem?
• What has been the biggest challenge for you while working on this? What did you do to keep making progress?
• Was there anything you did while working on this that you want to remember for the future?
• What did you learn while working on this?
• How could you improve your problem-solving process or your work next time?
• How confident are you in your solution? Why?

Connecting Approaches and Concepts

Questions that prompt students to think across multiple problem-solving approaches and concepts help deepen their conceptual understanding and gain flexibility for problem-solving—both important aspects of higher order thinking. Listening to students’ responses to these kinds of questions can also help us focus our teaching moves on what students need to better understand. Connecting questions can be used across a variety of contexts; for example, we can ask students to analyze and explain two worked examples, to connect the approaches of multiple students in the class, or to compare their own work to an example with a different approach. Questions such as the following can prompt students to make connections across concepts and approaches:

• How does this concept or approach compare to that concept or approach?
• What connections can you make between these concepts or approaches?
• How did other members of your group solve it?
• What about this other approach makes sense to you? What doesn’t make sense to you right now?
• What do you notice or wonder about these approaches or representations?
  • What is similar?
  • What is different?
  • Where do you see the same quantities or relationships in each of them?
• When might this approach be more useful or efficient than that one?

Extending Thinking

We can also ask questions that extend student thinking about the mathematics they are learning. These types of questions ask students to search for and make sense of patterns as well as to generalize, think creatively, apply their thinking in a new context, and think about a context or task from a different point of view. Some extend questions may require advanced planning, such as making changes to values or contexts, conceptualizing an idea for a non-routine task, or gaining knowledge about how people in other cultures solve similar problems. Example questions we can ask students that may extend their thinking include the following:

Predict and check changes to problem values and relationships

• What if this value is instead…?
• How would this be different if, instead, the relationship between the quantities is…?

Generalize

• Is there a pattern you can use? If so, what is the pattern? Why does it work?
• Can you find a rule?

Determine constraints

• Does this always work? If not, when does it work?
• What are the necessary conditions for the answer to make sense?

Apply an approach to a new context

• Can you use this approach for this [non-routine] problem?
• What if you apply this approach in this new context?

Examine from a different perspective

• What is a different way to think about your answer?
• How might this be different from the point of view of…?

Creating Contexts for Higher Order Thinking

Using teacher questions such as those above requires both a classroom culture and math problems that make space for students to be creative, to make conjectures, to examine a range of approaches and partial ideas, and to evaluate the accuracy of their solutions.

A safe, collaborative classroom culture in which students feel they belong is important for students to take the risks associated with answering these types of questions. Students must feel safe to express their thinking with the knowledge that what they have to say is important and will be heard. This requires that we as their teachers are curious about and have respect for all kinds of mathematical thinking and that we and our students listen carefully to each other. It is our responsibility to find meaning in student expression—including when it is not well formed or articulated. It is also important for students to believe that creating, testing, revising, and, when necessary, abandoning approaches are all normal and expected mathematical practices.

The types of problems we present to students also impact their ability to use higher order thinking. Tasks that provide contexts for higher order thinking can vary significantly in scope, including short open problems, longer word problems, 3-act tasks, and multi-lesson projects. They may be presented in a variety of forms, such as simple statements, sets of data that students interrogate, and worked example diagrams. It can be helpful if they are “low floor, high ceiling,” providing an easy entry for all levels of student capabilities and an opportunity for multiple levels of reasoning. There are many sources of rich and open tasks, including Open Middle and other websites listed on this Math for All post on rich tasks. Consider adapting your curricular resources to make them richer and/or more open or making your own rich math problems.

Final Thoughts

We encourage you to be creative with how you might best advance your students’ higher order thinking in your mathematics classes. For example, although our focus here has been about teacher questions, these and similar questions could be adapted to use in different ways, such as on anchor charts, as sentence starters, embedded in handouts, as student affirmations, and as student notebook reflection questions. Perhaps you and your students have conjectures about what questions and formats may work best, and you can test multiple approaches together.