So much of mathematics is focused on the end point: the answer, the solution, the proof. We attach so much emphasis on the destination — the more efficient the better. We place such high value on the final moment of the mathematical process; this is reflected in the language we choose to define one of the most important processes in mathematics: “problem solving”.

We must continually remind ourselves that students are completely new to the concepts we are presenting to them. Completion is expected, and correctness is mandatory. Problem solving signifies success—nothing less. The challenge that this presents our learners is that it leaves no space for divergent paths that might eventually find their way to a possible solution. It also paints a very distorted picture for a novice audience—one where the path to deep understanding is well-paved, and their struggle to stay on that path is due to their inability to walk in a straight line for long enough. We assume that the clarity of our explanations and assistance doesn’t diminish as it enters the minds of the student; we believe that we possess the key to unlock the shackles of the chains that are holding them back. The converse is also not an absolute truth. Problem solving is important, we do need to show students a path toward a deep understanding, and the clarity of our explanations does matter. But it isn’t all that matters.

We tend to see mathematics in its finished state—polished, complete. But that view obscures its unfinished, unresolved nature, the part students actually experience while learning. When we, as experts, are primarily concerned with discerning whether a student’s current understanding represents a complete or sound understanding of the newly learned content, we struggle to empathise with those who are not there yet. We focus on diagnosing what’s blocking them from the solution. But in doing so, we miss an opportunity to empathise with their current condition and their mathematical thinking so far. We support them to steady the ship and course correct toward the solution, often highlighting critical missteps that led them awry in the first place. This is all done with a pure heart—an insatiable appetite to be helpful and smooth the waters for struggling students. It unintentionally emphasises the importance of the arrival at a correct response. We might have gentle ways of framing this (e.g. “F.A.I.L. = First Attempt In Learning”), but despite our best efforts, we still imply one thing: you need to get the damn problem right.

An emphasis on “solving” impacts the way we support learners in very significant ways that are incredibly hard to argue against. We help students sharpen their skills towards the pointy conclusion of the problem solving process, but we’re less equipped to deal with students who are closer to the beginning than the end. The story we narrate to students about the problem solving process is more fiction than fact. We present a filtered view of what problem solving should look like: that it’s something that unfolds as you continue to make logical choices, but this simply isn’t true. The term “problem solving” implies that there is a problem to be solved—a conflict to be addressed. It implies that there is a grey area that needs to be parsed into pieces of black and white, something to be made sense of. This connects with how George Polya describes mathematical induction in How to Solve It:

Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further special cases.

We refrain from further remarks on the subject of induction about which there is wide disagreement among philosophers. But it should be added that many mathematical results were found by induction first and proved later. Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science.

How to Solve It: A New Aspect of Mathematical Method. (Polya, 1945)

This “experimental inductive science”—this targeted stabbing in the dark when the water is murky as a novice—speaks more truth to students than the clear picture we often try to convey. Students benefit from getting themselves “unstuck” and it’s in our best interest to provide them strategies to do that, rather than provide them with the resolution they seek. When we provide that clarity, we overlay a level of understanding on their problem solving process that is unrealistic and possibly even unhelpful. It’s possible that, by not supporting them to find a way out for themselves, that we are further emphasising that their current position is an unexpected path to a dead end. But the truth is quite the opposite—being lost is an essential part of the journey. It’s in this stage that they gather the information they need to move forward. Being stuck isn’t a bug; it’s a feature.

What I’m trying to convey here is that we do a great job at helping students with “problem solving” but I think we miss the mark with students who are “problem stalling”. Students who are trying to take off in second gear, instead of first. They’re trying to “jump” to the solution, trying to generalise before specialising, trying to solve a problem they haven’t made sense of.

This is a symptom of expecting mathematics to be clear and logical at all times, which is an impossibility while learning anything new in any domain. A requirement of being a learner is to “not know”, to have something to learn. Therefore, it could be said that a problem that is solved without conflict isn’t a solved problem at all, but rather an exercise to flex existing knowledge.

We need to do a better job at helping students with “Problem Starting” and be less obsessed with “Problem Solving”. We undoubtedly need to do a better job at normalising not knowing — that the feeling of the unfamiliarity is a prerequisite of experiencing something new. We need to give strategies for dealing with situations where you don’t know what to do. One of the simplest strategies I lean on is to specialise — to try specific examples to make sense of problems. I spend more time learning about the problem than I do solving the problem. Specialising slowly provides the resolution we seek, but it often illuminates unexpected structures and intriguing characteristics of the problem. Through specialising we essentially walk through the problem over and over until our journey represents a newly forged path through the unknown toward a generalisation or solution.

So, what does specialising actually look like? I’ve written about this many times before, but one of my favourites is when I tried solving the Palindromes problem from Thinking Mathematically. I’d hate to steal the opportunity for you to go through the process for yourself. You can check out my thinking if you really want to, but it’s your own experience that will matter more to you, so try it out for yourself instead — and specialise! Try some specific examples before immediately equipping your heavy artillery on this unassuming problem. Play with the problem—you won’t regret it.

12321 is called a palindrome because it reads the same backwards as forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by 11. Are they?

Thinking Mathematically 2nd Ed. (Mason, Burton & Stacey, 2010)