Defining Problem Solving
 Introduction | Introducing New Content Via Problems | A Positive Problem-Solving Disposition | Problem-Solving Techniques and Examples | Working Backwards | Technology | The Teachers' Role | Your Journal

Earlier in this session, we've referred to the goal of fostering a positive problem-solving disposition. But what does this mean, and how can it be achieved?

Human beings are problem solvers. Research shows that young children enter school with considerable problem-solving skills, but for some, these skills and the disposition to solve problems are not sustained by their school mathematics experience. By the time they have arrived in high school, some students, their parents, and sometimes even their teachers have decided that "mathematics is not for them," and that is that.

Creating a problem-solving disposition in a classroom means arousing interest and excitement about worthwhile mathematical tasks. This will help both those predisposed towards problem solving and others who have "checked out" from mathematics. How can this be achieved?

Choose "low-threshold, high-ceiling" problems. By this we mean problems that all students can understand and solve in part, but that offer extensions and connections that may be more demanding. One familiar example is dot-paper line segments. How many different line segments can be created on dot paper in a 3-by-3, 4-by-4, or 5-by-five dot grid? What is the length of each of these segments? Students can enter this problem from a range of levels. Some will be able to collect the data, while others will see a pattern and make a conjecture; still others will be able to see trigonometric connections. It connects to both geometric and algebraic concepts, and many students will be intrigued by the idea of the square root of a number being represented by a line segment on the grid. In talking about this phenomenon with students, teachers can explore and informally assess if students understand the concepts of natural, real, and irrational numbers.

As complexity is added, students can explore how the Pythagorean theorem is represented here, how many Pythagorean triples (figures showing 32 + 42 = 52 squared or constant multiples of 3, 4, and 5), and later begin to make a conjecture about how many segments are possible for a grid of n squares.

This is a rich problem with layers of complexity, but with an understandable point of entry.

After we have selected good problems, we should attend to the atmosphere we create around them as teachers. In the classroom context, often problems as presented and solved by the teacher can seem like a befuddling series of magic tricks, particularly to the mathematically wary. The teacher -- the magician -- presents material in the form of a seemingly impossible challenge, provides an answer that reveals the technique -- the trick -- of how a solution can be found, and then sets students to drill on that technique. To stay with our analogy, the students are then learning to pull rabbits out of hats on their own.

In such cases, too much effort on the students' part is focused on following the teacher's performance as a master problem solver rather than on building their own understanding and learning strategies from the teacher. An approach that convinces students that problems can only be solved by a tricky insight or a completely unexpected fact or characteristic can distort understanding of the problem-solving practice. Although there certainly are such occasions in problem solving, patient, step-wise work that does not turn on a showy mathematical trick is much more representative actual mathematical work.

Instead of "performing" problems, we should model the use of strategies successfully. And as we model and teach specific strategies, it's important to remain mindful that problems in their "natural environment" seldom rely on tricks of any kind. Instead, they call for the systematic application and development of sound mathematical techniques. One famous formulation of problem-solving techniques comes from the mathematician and pedagogue George Polya. His steps are 1. understand the problem; 2. identify the unknown; 3. create a plan; 4. execute your plan; 5. review and examine. This is not the only way to think of problem solving, but it is a helpful and rational approach.

Learning from Error

The final point in fostering a problem-solving disposition is creating an environment in which it is okay to be wrong. In fact, we should aim to make a space where making errors and learning from them are seen as a valuable part of the classroom experience. No student should be singled out for ridicule because of errors in problem solving, and students and teachers alike should consider problem solving a shared activity, whenever possible, and learn to review, correct, and build on their own and others' work in a constructive manner. This includes the work of teacher, for in years of teaching, it is inevitable that even the most skillful and scrupulous teacher will make an error in his or her work with students. Students will notice and correct these errors! In an environment where such occurrences are opportunities to learn and move on, rather than punitive or embarrassing, good problem solving will happen.

"We're asking them to look in their toolbox of problem-solving strategies, choose those that they believe are most likely to help make sense of the problem. One may not work, but they have another one that they can try." (Martha Brown, Problem Solving video)

Watch the video segment (duration 0:36) at left to hear reflections from teachers Henry Kepner and Martha Brown.

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