Problem Posing

A framework to engage teachers’ mathematical thinking and communicating

Chris Bolognese and Mike Steward

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The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and makes real advance in science.
—Albert Einstein, 1938

Einstein recognized the fundamental importance of problem posing, not only as productive practice, but as a gateway to new understanding. The National Council of Teachers of Mathematics (2000) recommends the inclusion of problem posing in mathematics curricula. But what do we mean by problem posing, exactly?

Silver (1994) writes, “Problem posing refers to both the generation of new problems and the re-formulation of given problems. Thus, posing can occur before, during, or after the solution of a problem” (p. 19). Kilpatrick (1987) further suggests that “problem formulation should be viewed not only as a goal of instruction but also as a means of instruction” (p. 123).

In this way, problem posing is both a learning tool and an instructional tool. As a learning tool, it gives students a way to communicate their mathematical ideas and questions. As an instructional tool, it gives teachers insight into students’ thinking, and helps teachers guide classroom inquiry.

MTCs can support teachers in understanding and implementing problem posing. What better entry point into this mathematical activity, than for teachers to practice posing problems together? This experience of “thinking like a student” is an important step for teachers who want their students to develop a practice of problem posing.

The following vignettes illustrate the common opportunities and pitfalls of facilitating an MTC.

Kimberlee is a sixth grade math teacher attending her first MTC. Jenny, the facilitator, says that they will be exploring the brownie problem tonight.

Jenny gives the brownie problem: “After baking a tray of brownies for your students, you leave it on the counter to cool. During that time, a thief takes a piece. How can you share what’s left of the brownie between your two classes?”

Jenny distributes a handout for teachers to work on. The handout says:

Consider the brownie as a rectangle in the plane with a rectangular region removed. We define a “cut” to be a single line segment. Answer the following questions under these definitions:

Kimberlee hasn’t done proofs since tenth grade, so she is not even sure how to start. She draws a picture of the brownie pan on her paper and notices that the remaining brownie is three pieces. She wonders if this always happens, and what other situations are possible after the cut?

Kimberlee’s brownie pan drawing.

Kimberlee calls Jenny over to share her observation. Jenny says, “That’s interesting, but it’s not what we are thinking about tonight.”

The facilitator, Jermaine, introduces the same brownie problem to his group. “Play around with it,” he says. “Pay attention to any questions you have while you explore. Then share your thoughts with your neighbor.”

After ten minutes, Jermaine brings the participants back together to share their questions with the whole group. Eitan, a seventh grade math teacher, says, “I’m wondering how many different ways you can cut the brownie so everyone gets an equal share.” John, a veteran participant at another table, exclaims, “We were wondering the same thing!”

Jermaine reorganizes the participants into new groups to explore the questions they generated. Eitan and John start to collaborate on their mutual question. For the remainder of the session, Jermaine monitors their progress, occasionally asking tables to share their work with the larger group. When Eitan and John get frustrated with the question they chose to pursue, Jermaine has follow-up questions to help provide insight.

In the first MTC, Jenny follows a prescribed sequence of questions that may be unnatural or uninteresting to Kimberlee and other participants. In the second MTC, Jermaine allows Eitan and other participants to follow their own questions, thereby developing a sense of ownership in their pursuits.

How can we make our MTCs look more like the second than the first?

The Columbus MTC organically developed a three-level framework for understanding problem posing. The different levels do not pertain to the problem, but rather the solver’s relationship to the problem. In this way, problem posing is meant to be a personalized activity. For example, two teachers may rate the same question at different levels, or change their rating as their understanding of a question improves.

In the table below, an example for each level is provided in the context of arithmetic:

We shared this framework at the 2017 Association of Mathematics Teacher Educators conference. After a brief introduction to the brownie problem, participants generated questions and assigned levels to them:

• How can you cut the remaining brownie in half with a single, one segment cut? (Level 1)
• What happens if the thief removes the entire brownie? (Level 1)
• What if the thief took a piece that isn’t the same depth as the rest of the brownie tray? (Level 2)
• Under what conditions will the brownie pieces be
  mathematically congruent? (Level 2)
• What if the pan is hexagonal and the thief takes a
  hexagonal piece? (Level 2)
• What is the minimum number of cuts needed to
  create n pieces of equal area? (Level 2)
• Could you cut the remaining brownie into 3 equal
  pieces with just two cuts? (Level 3)

Level 2 questions predominated. Participants either did not think of Level 1 or 3 questions, or they were reluctant to share them. As one participant remarked, “Level 1 probably isn’t even worth saying, because if it’s kind of easy, then nobody is going to mess with it…. But distinguishing between Level 3 and Level 2 is important— An idea you want to know, but have no idea where to start; versus an idea that you have a semblance of, an idea that you get. That’s a good entry point.”

Level 1 questions may seem trivial, but an important part of mathematics is making sense of what is already known. Level 1 questions help us make sense of the current context, and can serve as reference points when exploring a bigger question.

Level 2 and Level 3 questions both expand the boundaries of the context of the problem. If one determines a strategy for solving the problem, one can begin to work on the problem. If all known strategies are exhausted, the question becomes a Level 3.

Before facilitating your next MTC, we ask that you consider these two thoughts:

First, thoughtful facilitation requires paying attention to participants’ inquiries. When MTC participants ask the questions, we open a problem up to different perspectives and broaden the range of valid approaches to the scenario. As one of our participants noted, “Problem posing humanizes mathematics.”

Second, MTCs can support teachers in facilitating problem posing experiences with their students. However, it’s not the full solution, as the dynamics of a classroom and an MTC can vary greatly.

We encourage you to use problem posing and this leveling framework to empower the mathematical thinking of your own participants, and to provide a means for teachers to empower their students.

• Kilpatrick, J. (1987). Problem formulating: Where do good problems come from. Cognitive science and mathematics education, 123-147.
• National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• Silver, E. A. (1994). On mathematical problem posing. For the learning of mathematics, 14(1), 19-28.

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This article originally appeared in the Summer/Autumn 2017 MTCircular.

Chris Bolognese is the K-12 math department chair at Columbus Academy, co-founder of the Columbus MTC, and current president of the Central Ohio Council of Teachers of Mathematics. Mike Steward is an assistant professor of mathematics at the United States Military Academy at West Point. He helped facilitate the Columbus MTC as a graduate student at the Ohio State University.