What does it mean to truly understand mathematics?
Is it being able to get the right answer? Explain your steps? Teach it to someone else? Apply it to a new situation?
This question has been central to how I think about teaching maths, not just what we teach, but how we know students actually understand it. It’s something I return to often, whether I’m in a classroom or a lecture theatre. And it's a key principle I want maths teachers that I teach to reflect on.
For my first ever blog post, I wanted to start with with one of the keystones of my approach to teaching maths, which any Education students I have ever taught will be familiar with, as I talk about it at the beginning of every maths content or pedagogy unit! This is based on a wonderful piece of research discussed in the book "Teaching Secondary School Mathematics" by Merrilyn Goos , Gloria Stillman & Colleen Vale. I begin these discussions in lectures by asking students to think about a single piece of mathematics, or single concept which they think they understand really well. I then ask the question (as suggested in the book), "How do you know that you really understand this? What kinds of things can you do which show that you understand?" I often get a bunch of really great answers from students like:
"I can explain it to somebody else"
"I can apply the concept to more difficult examples"
"It just makes sense"
"You know why the formula works"
We then look at the table published in the text:
This is a summary of the responses given by 329 secondary school students when asked how they knew they understood something in maths, coded into 5 different categories. Usually, the kind of ideas the pre-service teachers suggested earlier fit into these categories, but they are often very surprised to see the low percentage of students who equate "understanding maths" with any responses outside of the "Correct Answer" category.
The next question I like to ask is, "How could students possibly get the idea that maths is all about completing exercises and getting them right?" We reflect on the fact that a large number of maths lessons are still taught in a "traditional" way - "I'll show you how to do one example, then you go ahead and do 20 examples yourself by following the method. If you finish these, you can then do a further 20 questions". Thankfully, over the past 10 to 15 years, I am seeing an increasing number of pre-service teachers who’ve experienced more variety, but it is still common and widespread.
Then comes the (intentionally obvious) trick question:
"Which of these types of understanding would you like your future students to have?"
Unsurprisingly, we always agree: All of them.
So if we want students to apply concepts, explain ideas, solve problems, enjoy maths, and see connections between topics — we must give them opportunities to do these things.
This needs to be intentional in our planning.
If we want students to develop problem-solving skills, we must include authentic problem-solving tasks.
If we want them to apply concepts, we must design activities that let them apply what they’ve learned.
Yes — we want students to get the right answers. But if we want more than that, we must plan learning experiences that build deeper understanding.
Yes of course we want students to answer questions and get the right answers, but if we want them to be able to do more than this, we need to prepare learning activities to help them achieve this. This leads to a very natural discussion of what we can do differently, and exploration of the approaches discussed in e.g. "Teaching Mathematics: Using research-informed strategies" by Peter Sullivan.
Bibliography:
Goos, M., Stillman, G., & Vale, C. (2016). Teaching Secondary School Mathematics: Research and Practice for the 21st Century (2nd ed.). Australia: Allen and Unwin.
Sullivan, P. (2011). Teaching Mathematics: Using research-informed strategies. ACER Press.