What Do We Mean When We Say, “Teaching Through Problem-Solving” In Mathematics Classrooms

For over 30 years, with NCTM reforms, we have been talking about “teaching through problem-solving”, rather than teaching *about* problem-solving. But what do we mean by that? I am not sure “teaching through problem-solving” is all that well-defined.

We who are, broadly speaking, constructivist teachers need to answer for how kids will do enough math, to be good at math, in problem-solving classrooms, but that’s a topic for another post.

Here’s how it used to be. Somewhere, on a dusty shelf, you might have an old binder full of problems. That binder might be organized by problem-solving strategies, like: working backwards; solve a simpler problem; or make a table or chart. Solving problems is not really something that can be taught like that. That approach could be characterized as “teaching about problems”, or “teaching how to solve problems”. The school mathematics we envision is more fulsome, and uses problems as the basis for exploring the mathematical ideas of the curriculum, building skills and understanding, and builds more flexible and capable mathematical thinkers.

As a child, perhaps you had a “problem of the week” to solve, perhaps on Fridays. Engaging with a word problem was your reward for getting through four days of worksheets and homework. You faithfully highlighted the key words, and went to work. You approached that problem as something mechanical, and summarized your work neatly in a sentence. Problem-solving as an event or reward is not what we have in mind for our school mathematics.

What do we mean when we say “mathematics problems”? Any worthwhile mathematics task or question could be considered a problem. A problem is something that gets students thinking. A problem is not just a one-off question. A problem sparks more questions, a problem sparks more questions until the sparks make a raging fire. Mathematics is about question asking as much as it is about question answering.

Worthwhile mathematics tasks are those that get students thinking about big and important mathematical ideas. Worthwhile mathematics tasks are those that get students to think about mathematics that is interesting, useful, or beautiful. They may or may not be “real world”. The mathematical world is interesting enough, sometimes. It is a world of thought, or visualization, and representation, and then bringing those representations into being, through speech, and on paper.

Our students are sensitive to pseudo-context, as they should be. Problems about pizza or cookies are far too common, and often not problematic enough. While you can get students thinking about division as fair sharing easily enough by posing a cookie problem, you could probably also activate their thinking about that same idea in a more interesting way. Sometimes we mathematics teachers treat our subject like parents treat cough medicine: something that goes down bitterly, but is necessary for our health. It need not be that way. Likewise, we are not talking about pointlessly sweetening or watering down our subject matter. The mathematical world is an interesting place to be, and to live. Our students need to be shown that it is our world, too.

Consider this definition of “problem-solving”:

“The term ‘problem solving’ refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development.”

Cai, J. & Lester, M., 2010

Do you like this definition? Does it encapsulate what problem-solving means to you?

How about this idea:

“by ‘problems’ in mathematics, we refer to mathematical tasks that are problematic.”

Wieman/Arbaugh

Sure, but many things in the world are “problematic”, the behavior of celebrities, and politicians, for example.Problems are “problematic”? A definition that perhaps begs the question: what is “problematic”?

A teaching through problem solving approach means using problems, questions, or tasks that are intellectually challenging and invite mathematical thinking through both mathematical content and mathematical processes in our students.

“It is important to understand that mathematics is to be taught through problem solving. That is, problem-based tasks or activities are the vehicle by which the desired curriculum is developed. The learning is an outcome of the problem-solving process.”

Van de Walle, 2007

Powerful words from the late Mr. van de Walle. Working on problems allows us to access the content, the curriculum, the ideas and concepts and skills we are working on.

A tentative definition then, of teaching through problem-solving:

Teaching through problem solving is an instructional approach that begins with a problem to be solved.

The pedagogical moves you make, the way you explain the concepts, the way you follow up this lesson, and the practice you give: that part is up to you. That’s the art and practice of teaching.