Recently I've been reflecting on a discussion that's playing out amongst my fellow pre-service teachers about project-based learning and whether we should be attempting it in our future classrooms. Without diving into the depths of the debate, something that has jumped out to me as (perhaps) being the crux of the issue is the fundamental question of what is mathematics? I believe how we answer this question is a key driver for how we go about mathematical teaching and learning.
In some ways, the answer to this question is simple. If we apply the "I know it when I see it" principle, then we can essentially define mathematics by pointing to everything that it clearly is. Adding up numbers is clearly mathematics. Using Pythagoras' theorem is clearly mathematics. And rinse and repeat ad infinitum. But when I ask the question: what is mathematics? I am thinking at a more abstract level of whether maths is a skill or a way of thinking. If math is a skill, arguably we can teach it in the same way we would teach any other skillset. If we take the example of driving a car (undeniably a skill), you may remember that to get your licence you needed to work through a carefully curated set of tests, and only through passing one test were you then given the freedom to move on to the next (with many hours of repetitive practice being the key to success). In many classrooms, this is how math is taught: we learn addition and subtraction so that we can then learn multiplication and division, so that we can then learn algebra, so that we can then learn calculus, and so on (and when the high-stakes tests come our way, the key to "success" is to drill the material into our brains through repetitive practice). In contrast, if math is a mindset then (so the argument goes) we need to move beyond the mere mechanical processes of mathematics and help students delve into the depths of what is really going on, so they can become "mathematicians" in the truest sense of the word.
I think one reason for the (seemingly) unending debates amongst math teachers about things like project-based learning (and the broader concept of inquiry based learning) is that math is both a set of skills and a way of thinking, and actually it can be different things to different people. To some, math will only ever amount to a skill that will help them on their tax returns (because who can deny that grappling with our marginal tax system needs a bit of mathematical prowess). To others, they need to see math as a way of thinking because they will be the ones who end up grappling with the Millennium Problems, which by their very nature will need innovators who can come up with new ways of thinking and new ways of doing math (although even these great thinkers will also need to be skilled in the lowly art of doing their tax, since they'll become millionaires when they finally succeed). I wonder whether those teachers who are fervently on the anti-inquiry side of the fence are simply placing more value on math as a skill (and vice-versa)? I also suspect teaching math as a mindset is a trickier proposition for teachers, and I wonder whether this is also a factor for those teachers who decide to stick with teaching math just as a skill?
Ultimately, as an aspiring mathematics teacher I need to decide what I will do in my classrooms. If I think my students will go on to be professional mathematicians, then maybe I need to prepare them for math as a way of thinking. But if my students simply need math to work out their tax, then maybe math as a skill is all I need to show them. But how can I know who is who? Who are the aspiring mathematics professors who will discover the next big thing, and who are the ones who will be happy just to know what they need to get by in their day-to-day life? Do I even have to know? Can I just teach math as both a skill and as a way of thinking, and let the kids decide their own destinies?