A quintessential discussion in mathematics education, the field within which I mainly operate, concerns whether mathematical skills or understanding are more important to build mathematical competency. I have always found debates about this topic needlessly emotional and combative. After all, the obvious answer to me has always been: both are crucial.

The research is pretty clear on the issue. Work by Rittle-Johnson and colleagues has indicated that procedural knowledge (eg, knowing the procedure to solve an equation) and conceptual knowledge (eg, knowing what the role of a variable is) go “hand in hand” and both are developed iteratively.

But despite this “commonsense” view, educators have spent (wasted?) decades arguing about the issue. In some parts of the world, it has even been referred to as the “Math Wars”.

These “wars” run deep; in some circles even positioning these discussions as a waste of time provokes feverish reactions.

It is not helpful to remain stuck in these endless arguments, especially because both sides have plenty to offer.

I, for example, am a big fan of algorithms, something I have written about in an article with my former colleague Lianghuo Fan. Algorithms are seen by some as rules that need to be followed slavishly in order to get to a solution. This is not a bad thing. Algorithms often work because the building of maths is rigorous and structured, because - for example - we have a robust number system underpinning our algebra. But it is also important that students know these rules and over time also understand why they work.

So algorithms require procedural knowledge and conceptual knowledge. Without one another, there is an evidenced risk of making more mistakes. For example, a risk with an over-reliance on just following the algorithm is that you apply the recipe in situations where it’s not suitable. But the converse, knowing all the intricacies of variables and concepts but not being able to swiftly solve an equation, isn’t desirable either.

Even if we finally manage to agree on both procedural and conceptual knowledge being important, though, we find ourselves in disputes on which has to come first.

For me, this always depended on what students I had in front of me and what topic was being taught. In some cases, I would start with more conceptual and historical explanations, sometimes realising that not everybody would “get it” immediately, after which we would do loads of practice. In other cases, it was the other way round: first practice, then practice and some more practice, and after that tasks more focused on deeper understanding.

But in most cases, I just used a mix of both during lessons, right through the voyage of students going from novice to expert.

Saying that, I fear, may prompt many more of those endless arguments.

Christian Bokhove is a lecturer in mathematics education at the University of Southampton

This article originally appeared in the 27 September 2019 issue under the headline “When two tribes go to maths war”