What's Problem Solving Got to Do With It?

By Bill Kring, Mathematics Classroom Instructor and A Pass Educational Group Associate

When I was a child, my dad would give me problems to solve on our fishing trips. I would hunker down in the back window of our old Hudson and try to figure out what the answer was to the challenge he had given me. I thought for a long time I had been given a truly thoughtful father! I found out much later that he was using that strategy just to keep me quiet while he was driving us to the fishing spot. However, what it spawned was a joy and desire to challenge my mind to solve problems.

When I started teaching, problem solving often meant the word problems in the book. I didn’t see the connection between learning mathematics and the joy I found solving problems. It wasn't until the mid 1980s, after teaching for more than 15 years, that I had my eyes and heart opened to the power of teaching through problem solving. I had been the "Sage on the Stage" during the early part of my career, honing my skill as a presenter of mathematics. Most of my students were successful in learning what I had shown them, but there were some who had never made it to the level of learning I wanted for them. Fortunately, I had my style turned upside down through some training in how to use problem solving in the classroom.

So, what is this Problem Solving anyway?

In How To Solve It, George Polya proposed that problem solving could be taught as an art, an act of discovery, and encouraged presenting mathematics as an experimental and inductive science. This view of problem solving develops students’ abilities to become skillful and enthusiastic problem solvers, independent thinkers who are capable of dealing with open-ended problems. My dad had started me on this path, and I have followed it for most of my life.

Often, problem solving skills are taught as a separate topic in the curriculum. Teachers show students a set of general procedures for solving problems—drawing a picture, working backwards, making a list, etc. Students practice using these procedures to solve routine problems before attempting open-ended problems. Consequently, in many mathematics classroom settings, only advanced students get to open-ended problem solving, rather than offering it to allstudents.

I prefer to think of problem solving as the vehicle with which I teach mathematics. When problem solving is used this way, the emphasis is on finding interesting and engaging tasks or problems that help introduce or develop a mathematical concept or procedure. For example, a teacher could assign groups of students the problem of dividing pieces of licorice so that each student gets an equal share. This can allow students to get motivated (make discoveries about fraction concepts using a familiar and desirable object), to practice their learning (connect the concepts to skills), and to see relevance(fractions are worth learning).

One of the biggest challenges is finding the right problems, ones that have multiple possible answers found by various solution methods, while addressing important mathematical concepts. The right problems should connect to students’ previous learning, yet challenge and interest them. The focus is not on the answer to the problem, but on the methods for arriving at an answer. I like the analogy of the mule with a carrot dangling in front of it. If the carrot is too close, the mule eats it and goes nowhere. If it’s too far away, the mule sits down and goes nowhere. However, if the carrot is just far enough away, the mule will try to get it and make some progress, not minding that the carrot moves as the mule does. In a problem-solving classroom, the student perceives the problem as challenging (not too easy), yet not insurmountable (not so far out there that it is impossible).

Through open-ended problems, students make many of the decisions, not the teacher or textbook. They draw on previous knowledge and experience with related problems, constructing individual procedures before arriving at a solution. After the solution is found, the next step is crucial. The students must reflect on the problem-solving experience, trace their thinking process, and review the strategies attempted, determining why some worked and others didn't. This period of reflection deepens understanding of the problem, helps to clarify thinking about effective solution methods, and enables students to explain their reasoning to others. Thus, students can connect the problems and methods used to other areas of mathematics.

A couple experiences stick with me. I was able to be a guest teacher in a third grade classroom ready to study area and perimeter. I brought in some paper one-inch square regions and challenged the students to find as many polygonal regions made up of five pieces of paper. I was particularly struck by the enthusiasm and questions of a girl in the front row. She made wonderful discoveries about the area and perimeter and was willing to help others around her. I remarked to the teacher about the performance of that young lady. The teacher expressed surprise and delight. The girl was one of her special-education students who seldom was genuinely involved in the work of the class. The girl had been able to develop her own understanding of the mathematics and enjoy the work.

Another experience occurred in my Advanced Placement Calculus early on in my implementation of a problem-solving approach. We had been studying the calculation of area by taking thin "slices" and adding them. I asked my students to consider how they could find the volume of an onion I brought in to class. One of my struggling students volunteered to share his idea. I was pleased to see him get involved. He explained that he could peel the outer layer and think of it as a "ring", do the same for the next layer, and so on down to the core. If he could find the volume of each "ring", he could add them to get the total volume. His classmates, listening intently to his explanation, helped figure out how to do just that. He returned to his seat exuding a feeling of accomplishment he had rarely experienced before.

So, what's problem solving got to do with it? Nearly everything! Carefully chosen problems, implemented in conjunction with skill development, can get students excited about learning mathematics. They learn, too, that problem solving is not something they do every once in a while. It's a regular occurrence. Also, they develop persistence as they learn that what seems to be a new kind of problem is just an entry to new understanding. And, for me, it's more fun being a "Guide on the Side".

Bill Kring is a classroom veteran of over 30 years. He received the 1992 Presidential Award for Excellence in Mathematics Teaching. He was selected as the Contractor of the Month for February, 2014 by the A Pass staff.