Why do you teach through Problems?

This is not the way I was taught Math.

This essay is dedicated to John Van de Walle who died suddenly in December 2006. While I never personally met Mr. Van de Walle, his kindness and writings have greatly influenced my philosophy towards the teaching of mathematics.

Most, if not all, important mathematics concepts and procedures can best be taught through problem solving.

- John Van de Walle

Lots of parents ask me about teaching with problems and are concerned that their children will fall behind if they are not taught the standard algorithms in the same way they were taught when they were going to school. Sometimes, teachers and parents tell me that they are afraid using a problem-based approach will be too frustrating for the students. But it is the richness of the problem solving experience that allows students to develop skills in a meaningful way. If we walk them through every step of a problem so they are not frustrated, what will they do when we are no longer around? If we spoon-feed information to our students so that they can have quick and immediate success on a given task we may feel good about our teaching in the short term, but in reality we are only fooling ourselves that we are doing a good job and in the process are handicapping our students. In time those same students will come to resent us for not letting them struggle and figure out how to learn for themselves. The truth is that we will not always be around to help them so unless they have learned how to deal with adversity and to persevere for themselves they will quit. I would much rather have a student fail a problem in second grade and learn that to solve problems takes flexibility, hard work and persistence than to help them so much that when they get older they do not have those qualities and fail in life. We have no way of knowing what their lives will be like in the future, so there is no way we can anticipate and prepare them for every possible scenario. Instead, we need to teach them how to fend for themselves in an ever-changing world and solving problems is one way to do that. As John Van de Walle says in the forward of his book, Elementary and Middle School Mathematics: Teaching Developmentally,

“Believe in kids! Allow them to think, to struggle, and to reason with new ideas as together you find the excitement that happens when mathematics makes sense.”

Here are a few reasons I think teaching with problems is the best way to teach:

Solving problems is fun! This is first on my list because when something is fun, people tend to stick with it longer. If we can teach math in a way that is effective and the students like it, why wouldn’t we?

Solving problems takes persistence. In a study comparing U.S. and Japanese first-grade students working on a difficult task, U.S. students gave up in about 15 seconds, while Japanese students didn’t stop until the class came to an end an hour later (Stigler, 1999). Persistence can be learned if students are given enough opportunities to stick with a problem even when it is hard and that even when they solve it in one way, they should try to solve it in another way if they have time to.

Problem Solving is student-centered. Students of varying abilities can work on the same problem, but they will work on it in a variety of ways. Some will work concretely using the actual objects, others will draw diagrams or graphics, and others will use numbers and other math symbols to solve the problem abstractly. As long as the students are allowed to solve the problem in ways they choose, they will naturally choose a method that makes sense to them. This shifts the focus from the teacher to the student. Instead of the teacher trying to show math to the student, the student tries to figure out the math through the problem. This type of learning environment is often called student-centered. This makes this approach especially good for an inclusion model because students build off of what they know rather than trying to understand what the teacher is showing or telling.

Problem Solving enhances communication and reflection. Students often work with partners or small teams to solve problems, so as they are working they are sharing ideas and trying to explain their thinking to each other. When the class shares the various solutions, the students are exposed to other points of view helping them develop flexibility and an appreciation for the fact that there is more than one way to solve a problem. The first way they think of to solve a problem may not be the best. Listening to other perspectives helps problem solvers develop their empathy and appreciation for elegant insights.

Solving problems helps students develop connections between various math topics. When a student learns math by grappling with difficult and absorbing problems... rather than by simply memorizing and practicing predetermined procedures — she is free to "wonder why things are, to inquire, to search for solutions, and to resolve incongruities. This approach yields deep understandings of the kinds that we value" (Hiebert, Carpenter, Fennema, et al., 1996).

Problem solving enhances computational fluency. Students are motivated to learn facts and other mental math strategies because students see that they are valuable skills to a problem solver.

Problem solving provides great opportunities for assessment. When students solve problems, teachers have the opportunity to learn a lot about their thinking and what they understand. This article is one of the best in terms of explaining the benefits of formative assessment. Inside the Black Box: Raising Standards Through Classroom Assessment by Paul Black and Dylan Wiliam. A great professional development program that works with teachers to develop the skills to use formative assessment effectively is called, Dynamic Classroom Assessment. Kathy Richardson and her group at Math Perspectives developed another good example of using formative assessment strategies for younger students learning about counting, number relationships, addition, subtraction, and place value.

A great resource for reading more about teaching math with problems (and teaching science with inquiry) is The Mathematics and Science Education Center at North West Regional Educational Laboratory (NWREL).

The NWREL Mathematics Problem-Solving Model helps educators meet the challenges of teaching and assessing open-ended problem solving. The model includes a scoring guide for problem solving, open-ended tasks, and examples of student work for practice in scoring. Explore this site to learn more about the components of the model and teaching strategies for mathematics problem solving.
- In particular, I would suggest reading the Mathematics Problem Solving Monograph, which can be downloaded as a PDF file from a link at the bottom of the Components page. This document summarizes the research and literature on teaching and learning mathematics through open-ended problem solving that served as a foundation for the development of the NWREL model. The quote from Kyle Forman below comes from this monograph.

Their first issue of their Northwest Teacher Magazine was focused on the topic of problem solving. Problem Solving: Getting to the Heart of Mathematics.

It has three really good feature articles, which I used as I created my list of reasons I think teaching with problems is the best way to teach. Click on the title to get an on-line copy of it. To read the articles, you will need to click on the titles when the index page opens.

Fosnot and Dolk in their books talk about learning math as a journey, often involving side trips and backtracking but eventually reaching the horizon only to realize that there is another horizon in the distance. Kyle Forman, an eighth-grade student in Portland, Oregon, who is working in a problem-centered curriculum, also uses this metaphor as he expresses the benefits of learning math using the problem-solving approach:

Learning math is a journey full of freeways, detours, dead ends, and side trips. The journey is what we learn from, where we gain the confidence and knowledge needed to succeed as mathematicians. The journey is more important than the destination. In fact, the journey never ends. Any idea can be expanded and we can always search further. Our teacher trusts us as capable mathematical thinkers who can find our way... She knows that difficulties can temporarily impede our progress and she knows that we learn from the experience. The next time we venture down a path may be quicker or may take longer because of new side trips and discoveries (Foreman, 1998).

The reason I teach with problems is because I respect the students too much to teach any other way.

Definition: a problem is any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that is a specific “correct” solution method (Hiebert et al., 1997). A problem does not have to be a word problem it can be any task or activity that fits the above definition