One challenge in mathematics education is that many of the commonly used terms mean different things to different people. Problem solving is one example. The lack of a shared understanding complicates discussions at all levels, including national curriculum^{1} stoushes like the recent ones in Australia.

Conflicting definitions of problem solving have plagued us for decades. Alan Schoenfeld in his seminal 1992 paper *‘Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics*’ suggested that time was overdue for a consensus on definitions of the various aspects of problem solving as ‘great confusion arises when the same term refers to a multitude of sometimes contradictory and typically underspecified behaviours’. Thirty years on, we still seem to be in a state of collective misunderstanding.

Problems are,* *in the words of Paul Halmos,* *‘the heart of mathematics’. Halmos goes on to say that ‘a mathematician’s main reason for existence is to solve problems and that, therefore, what mathematics really consists of is problems and solutions’. So before we examine problem solving in more detail, we need to first consider what we mean by problems themselves.

### Routine exercises and problematic tasks

To categorise problems, I like to think of a line representing the level of perplexity presented by the problem. At one end we have problems as **routine exercises**. These are typically practice questions designed to test mastery of a particular technique, usually one that was recently taught. In the preface to his book ‘*The Art and Craft of Problem Solving*’, Paul Zeitz distinguishes between perplexity and difficulty, saying that ‘an exercise may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed’ and that ‘the path towards solution is always apparent’. The majority of problems found in school textbooks are of this type, particularly when they are located in chapters and sections with helpful titles as to the technique that should be practiced. At the other extreme we have **problematic tasks. **These are problems that are perplexing, challenging and difficult, and cannot be answered immediately as the way forward is typically unclear. Research mathematicians spend most of their time untangling problems of this nature.

Can each problem be characterised as either routine or problematic? In his 2010 blog post ‘The problem of non-problems in maths programs’, Grant Wiggins has used Zeitz’s definitions to give examples of each. But a simple delineation is, pardon the pun, problematic. Consider the following, taken from Siemon and Booker:

‘A farmer had sheep and emus together in a paddock. If the farmer counted 34 heads and 88 legs, how many sheep and how many emus were in the paddock?’

What might a Year 4 student do? What might a Year 11 student do? The former might draw diagrams, use manipulatives, try trial and error; the latter would likely apply a learned technique such as solving via simultaneous equations. A task can be problematic at first and then become routine as knowledge and experience grows. In other words, tasks can move along our own individual line of perplexity. And this is exactly what we want to have happen in education as students learn.

Should we focus our time on routine exercises or on problematic tasks? Zeitz gives a useful analogy by comparing a gym-goer with a mountain climber. The gym-goer does lots of repetitions on various weight machines, perhaps in the basement of a building under artificial lights (at least, that’s my experience!). The mountain climber goes on a long, arduous trip in nature. They might suffer blisters and scrapes. They might lose their way and need to retrace their steps. But eventually the mountain climber arrives at the top of the mountain, and experiences the amazing vistas and the thrill of accomplishment.

The parallels to solving routine exercises and problematic tasks are clear. From the description above, it is tempting to dismiss routine exercises in favour of problematic tasks, but both have their place in learning mathematics. Both the gym-goer and the mountain climber get stronger. And I know from my own experience that my hard slog up the mountain (or down a ski slope) is made vastly easier and more enjoyable when I am in good shape having spent time strengthening my muscles and improving my cardio fitness.

### Three flavours of problem solving

Given that mathematics consists of problems, it seems obvious the problem solving should be the focus of school mathematics. And it is here that we often talk at cross purposes, as some teach *for* problem solving, some teach *about *problem solving, and some teach *through *problem solving. (I first encountered this distinction in Siemon and Booker (1990) but it traces back through Schroeder and Lester (1989) to Hatfield (1978) and possibly even earlier.)

The missing preposition—*for, about, through*—makes it clearer what problem solving could entail. Kaye Stacey (2005) describes the differences between them as follows:

- Teaching
*for*problem solving: teaching mathematical content for later use in solving problems - Teaching
*about*problem solving: teaching heuristic strategies to improve generic ability to solve problems - Teaching
*through*problem solving: teaching mathematical content by presenting non-routine problems involving this content.

**Teaching for problem solving** typically follows a ‘direction instruction’ or ‘explicit teaching’ approach. A technique is introduced and demonstrated by the teacher. More exercises are provided for practicing the technique. The student then tries to apply that knowledge to solve other problems. Ideally the student is eventually able to transfer from one problem context to another. Traditional maths classrooms, and certainly those of my formative years, almost exclusively follow this model. A strength of this approach is the focus on mastery of knowledge and skills. A shortcoming is that problem solving is typically only considered after the acquisition of all required knowledge, and so the problem is unlikely to be problematic in the sense I described earlier.

**Teaching through problem solving** turns this upside down. Students start with a rich problem that has been carefully chosen by the teacher to embody the key aspects of the topic. Knowledge or skills to be learned emerge from students exploring the problem, which is sometimes referred to as ‘inquiry-based learning’. (The Three-Act Task format is one example of this teaching method.) A strength of this approach is the focus on engaging students in contexualised, self-generated learning. A shortcoming is that with insufficient consideration given to task selection and learner support, students can struggle unproductively and aimlessly without learning anything.

At times it feels like the mathematics community consists of only these two camps, where people have declared themselves either ‘Team Explicit Teaching’ or ‘Team Inquiry’. But just like with the gym-goer and the mountain climber, we need to experience both types to be strong problem solvers. While a snapshot of our teaching at any given point in time might look like explicit teaching or inquiry-based learning, a combination of the two is needed over time to ensure that students become both confident and creative users of mathematics^{2}. Failing to acknowledge or incorporate the merits of the other camp does our students a disservice and fans the flames during public discourse around mathematics education.

I haven’t yet mentioned **teaching about problem solving**. This approach brings awareness to the act of problem solving by articulating, experiencing, and reflecting on mathematical processes. Proponents will typically invoke a framework like Pólya’s four principles of problem solving (understanding the problem, devising a plan, carrying out the plan, looking back) or the three phases of work (entry, attack, review) of Mason, Burton and Stacey’s *Thinking Mathematically*, along with a raft of different strategies or ‘heuristics’ such as trying particular cases, being systematic, looking for patterns, and working backwards.

Teaching about problem solving gets very little traction in curriculum documents. While the Australian Curriculum: Mathematics indicates that the four proficiency strands (understanding, fluency, problem solving and reasoning) describe how the content is to be explored or developed—‘the thinking and doing of mathematics’—no guidance is given as to what skills should be developed in students to help them in this endeavour.

I’d argue that teaching about problem solving is a powerful, but forgotten, member of the family. And rather than suggest that we pitch yet a third camp, we need to find a way to cohabit in a communal settlement. As Siemon and Booker concluded their 1990 article: ‘each approach has a vital and critical role to play in the acquisition of application of mathematical thinking at all levels. We simply cannot afford to concentrate on any one or two at the expense of all three.’

### Where to from here?

You might notice that almost all of the references in this blog post are at least thirty years old, which means that this is not a new problem. But given it has sustained our attention for decades, we must know it to be important. So, there is work to do. How do we move on from what seems to be an ever-growing divide? What are the impediments? What actions have failed in the past that we can learn from in the present? What new ideas are out there that we can implement for the future? A new Australian Curriculum: Mathematics will be released imminently. With it, I hope we can revive the heart of mathematics.

### References

Halmos, P R. 1980. “The Heart of Mathematics.” *The American Mathematical Monthly* 87 (7): 519. https://doi.org/10.2307/2321415.

Hatfield, Larry. 1978. “Heuristical Emphases in the Instruction of Mathematical Problem Solving: Rationales and Research.” In *Mathematical Problem Solving: Papers from a Research Workshop* edited by Larry L. Hatfield and David A. Bradbard, 21–42.

Mason, John, Leone Burton, and Kaye Stacey. 2010. *Thinking Mathematically*. 2nd ed. Harlow: Pearson. (The first edition was published in 1982.)

Polya, G. 1945. *How to Solve It: a New Aspect of Mathematical Method.* Princeton, NJ, US: Princeton University Press.

Schoenfeld, Alan H. 2016 (originally published in 1992). “Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint).” *Journal of Education* 196 (2): 1–38.

Schroeder, Thomas, L., and Frank Lester F. 1989. “Developing Understanding in Mathematics via Problem Solving.” In P. Trafton & A Schulte (Eds.), *New Directions for Elementary School Mathematics (1989 Yearbook)*, 31–42. Reston, VA: NCTM.

Siemon, Dianne, and George Booker. 1990. “Teaching and Learning FOR, ABOUT and THROUGH Problem Solving.” *Vinculum* 27 (2): 4–12. (Downloadable here: https://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/assessment/ppprobsolving.pdf)

Stacey, Kaye. 2005. “The Place of Problem Solving in Contemporary Mathematics Curriculum Documents.” *The Journal of Mathematical Behavior* 24 (3–4): 341–50. https://doi.org/10.1016/j.jmathb.2005.09.004.

Zeitz, Paul. 2007. *The Art and Craft of Problem Solving*. 2nd ed. Hoboken, NJ: John Wiley.

[1] In Australia, ‘curriculum’ refers to the standards of what should be taught. The Australian Curriculum: Mathematics is overseen by the Australian Curriculum, Assessment and Reporting Authority (ACARA).

[2] ‘Confident, creative users and communicators of mathematics’ is one of the aims of the Australian Curriculum: Mathematics

Brilliant article. Clear and concise explanation of the topic. I only wish that I had written it!

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I appreciate the feedback, Paul!

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I really like your analogy of Gym workouts and Mountain climbing. Your not going to necessarily mountain climb everyday but if you prepare well for the climb, make a plan, build your fitness, when you do engage in the experience it is one of true accomplishment, but more importantly one you will never forget. Really like the conversation starter.

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Thanks for this Amie. Totally in the ‘Team foot in both camps’ here. I’m currently in the middle of teaching about problem solving using teaching through problem solving in our MATH2920 course at Newcastle, and I’ll be recommending this to the students – mostly pre-service teachers.

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Brilliant. I have been looking for an article on this topic. You nailed it Amie.

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Thanks Stuart. Glad it’s useful!

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