Fourteen years ago I was starting my teacher training course. I had already been teaching a little (two undergraduate courses at university) and been to a mathematics education conference and numerous seminars. But I felt as though I knew nothing about teaching as a discipline and a career. After leaving my teacher training there were still so many things I did not know!

It’s impossible for teacher training courses to completely prepare teachers for what lies ahead. Mostly I hear other teachers say that they learned nothing of use in their training. But there are two things I still remember and draw from daily or weekly, fourteen years later.

**1. Students need to experience an idea in order to learn it.**

I was greatly impacted by reading *Experience and Education* by John Dewey. (It was a set text in my Philosophy of Education class.) In it, I learned that students need to play around with ideas in their pursuit of understanding. They need to create, investigate, take apart, put together, describe, and explain. Students shouldn’t be told what to think but come to think it for themselves. Additionally, students need to come at a concept from many angles before they gain a deep understanding of it. (Some of the other ideas in *Experience and Education* don’t sit so well with me anymore. That’s another blog post.)

For several years after first becoming a teacher I was struggling to keep pace with my workload and found it really hard to help students experience ideas. But I decided to make sure I thought about this in detail during my planning time once a week, or twice a week, or more often as I became able. Now it is the founding statement of my education platform: students must experience mathematics in order to learn it. I ask myself, how could a student come across this idea? How could them come to investigate it?

Recently I’ve added a new mantra to this one: the first time students see an idea they don’t have to produce exam-ready solutions. Getting to grips with an idea is important first, then coming to a deeper understanding, then being able to apply that learning. Exam-ready answers to exam-style questions on that topic can be introduced along the way, but aren’t needed at first.

**2. The context for a mathematics problem can be “real world” or just plain “intriguing”.**

My mentor and biggest influence during teacher training was a professor who studied and celebrated problem solving in the classroom. He didn’t care if a problem was based on some real life context or just really interesting.

In fact, working with him led me to prefer a puzzling situation over an over-simplified, meaningless “context”.

Thanks, John Grant McLoughlin, for embedding puzzling problems into my teaching. (He is still at the University of New Brunswick. See him in action in this video.) John is a huge fan of problem solving and non-routine questions and caused me to not really worry about real world connections; my students are intrigued by the problems and they are engaged in solving them, so I’m happy they are thinking about the concepts that I want them to learn.

Recently, I’ve been revisiting this idea thanks to reading about Peter Liljedahl‘s work on Building Thinking Classrooms. One of his recommended practices is to start each lesson with a quality problem.

I reinvigorated my love for problems this year; for the last term I set myself a goal of using a problem in every year 8 lesson that I taught. Here’s one of the problems I used. (It’s from a collection of problems devised for AQA GCSE [the link is a pdf].)

My students loved this problem and we spent most of a lesson talking about it. I used my visualiser to show several of the students’ solutions on the interactive whiteboard. They had lots of ideas about how to show the required idea – and this was one of the first times they used algebra to prove something. It was thrilling!

**Do you still use ideas that you first encountered in your teacher training?**