**What’s Your Problem?**

Of all types of questions a student will be called upon to answer during a course of study in mathematics, by far the most challenging and troublesome involve what is known as mathematical *problem solving*. These questions are assessed in their own separate criterion in high school mathematics and are often a source of anxiety and frustration for many students, even those who have a reasonably thorough understanding of mathematical techniques. So what exactly constitutes mathematical problem solving, and how can students best prepare themselves for the challenges of this process? This article will explore the constituent elements of problem solving and argue that with practice, perseverance and the correct mindset, problem solving is accessible for all students of mathematics.

Defining the essence of problem solving is a problem in itself, because the term has been used in so many different contexts in the last hundred years. Some theorists argue that problem solving is a discrete skill; others look upon it more as an art than a science (Schoenfeld, 1992). It may be more helpful to examine the characteristics of questions which assess problem solving in the mathematics curriculum to determine how they differ from other kinds of questions. Firstly, instead of dealing with a single concept, it is often the case that several distinct areas of mathematics must be utilised to achieve a solution. Secondly, questions are typically articulated in word form, which means that careful reading and interpretation are required — this is the source of most of the literacy demands in mathematics. Thirdly, and most importantly, problem solving will usually require the application of learned techniques in unfamiliar or abstract contexts. This characteristic, referred to as *initiative* in the senior curriculum documents (Queensland Studies Authority, 2008), means that problem solving strategies cannot be improved by rote learning or exhaustive practice alone. Once a particular problem is practised and rehearsed, its context is no longer unfamiliar, and this means that the question can no longer be used to assess initiative. So if practice alone will not guarantee success, what strategies can be used to improve students’ problem solving techniques?

The first ingredient for successful problem solving is a *positive, confident attitude*. This might sound clichéd and unimportant compared to the more tangible mathematical knowledge and skills that are required, but it is impossible to overstate the importance of high self-efficacy when attacking a problem. Every challenge involves a goal, and a problem solver must accept that goal with confidence. After all, you can be absolutely assured that you will never encounter a problem in high school mathematics that requires techniques or skills beyond what has been taught in class. The answer to every question will be within reach, although the paths to that solution may be difficult to navigate.

Unfortunately, when it comes to confidence and self-belief in mathematics, gender seems to be a very influential factor. There has been a great deal of research devoted to this particular topic and to elaborate on the many, complex factors involved would be beyond the scope of this article, but it is clearly the case that male students have significantly higher confidence in their own mathematical abilities than female students, and that this is a worldwide phenomenon (OECD 2004, p. 133). Another factor at work here is the notion of “stereotype threat”, in which female students are negatively impacted by persistent cultural stereotypes to the effect that boys are better at “math and science domains” and girls are superior at “English and reading domains” (Quinn and Spencer, 2001, p. 56). Poor confidence and self-efficacy are debilitating factors when it comes to problem solving, which is why a positive attitude is essential.

Confidence alone, however, will not guarantee success. The second ingredient for successful problem solving is *preparation*. This means that students must be intimately familiar with every mathematical technique studied in class, to the extent that that technique can be applied with speed and precision. As an analogy, think of the problem solving process as the construction of a house, and think of the range of mathematical techniques as the tools in your toolbox. If a builder arrives at a construction site and finds that they are missing half of their tools (and don’t know how to operate the ones they do have), building the house will be simply impossible. In contrast, the well-prepared builder who has ready access to their drill of division, their paintbrush of Pythagoras, and their bandsaw of binomial probability will find the construction process far more straightforward.

The third important ingredient is taking the time at the beginning of a question to *interpret and visualise the problem*. Don’t start furiously writing down everything that comes into your head. Instead, take a couple of minutes of deep thinking time without putting pen to paper. Think about what the question is asking and what your answer might look like. Are you asked to find a length, a mass, a dollar value, an algebraic expression? Estimate what a sensible answer might be, and let this guide you as you develop your solution. Highlight the key terms in the question and pay close attention to the phrases that link these words together.

When it comes time to begin work on the problem, the fourth ingredient of *representation* becomes important. Mathematics is a language and quite often the success of a particular strategy depends on how accurately verbal or written information is encoded or represented using the language of mathematics. Problems can be represented in various ways — algebra can be used to represent unknown quantities and the relationships between these quantities. Diagrams can also represent mathematical relationships. Often the most difficult step in problem solving is representing a written problem using mathematical language, so it is important to take sufficient time and caution in this step.

As you progress through the problem, the ingredient of *creativity* becomes vitally important. Mathematics is often misunderstood as being a linear, procedural discipline, when in fact, creative thought is essential to successful problem solving. Of course, this creativity is not absolute and must be confined to the accepted laws of mathematics (it isn’t wise, for example, to invent formulas at random). In terms of selecting which techniques to use, however, a creative problem solver can identify and exploit opportunities which may unlock the path to a solution.

*Persistence* is another important ingredient, especially for very challenging problems. Perspiration is often just as important as inspiration, and problem solvers must not expect a solution to come quickly or easily. This skill can be practised by adopting the right mindset during lessons. Students who ask their friends or teacher for help the instant they first encounter difficulty will find it almost impossible to cope when help is not available; conversely, students who take the time to re-evaluate their working, try different approaches, backtrack if necessary and think their way out of a difficult situation will be much better prepared.

The final ingredient is *evaluation*. Don’t start congratulating yourself the minute you write your final line of working. Instead, think carefully about the reasonableness of your answer in the context of the question. Does it resemble your initial estimate? Often there are methods you can use to validate your solution, for example, by substituting into an initial equation or graphing the solution on a calculator. The small investment of time spent evaluating your answer can be very worthwhile.

By concentrating on each of these key ingredients, it is possible to become a more accomplished problem solver. Many opportunities are available for students to practise these skills, but it is the environment in which the skills are practised which will be most important. Questions should be mixed up and attempted out of sequence to increase the initiative required. Distractions should be minimised to enable more single-minded persistence. Start with easier questions and progress gradually to the more complex and abstract. You will soon discover that problem solving is not only accessible; it can also be immensely enjoyable and rewarding. Few experiences can match the sense of excitement and accomplishment that results when persistence and deep logical thought overcome the challenges of a difficult mathematical problem.

**Mr G Bland**