A few months ago, my wife and I introduced our 2 year old son to his first bike. It’s different to my first bike in a lot of ways: it’s brand new #middlechildlyf, weighs nearly nothing, and has no pedals or brakes. It also has no bell, which is a genius omission. My first bike had pedals, brakes, and was about double the size of my son’s bike. My first bike also had two tiny “training wheels” bolted on, which prevented the bike from falling over.

My son’s bike is called a “Balance Bike”. As the name suggests, the emphasis is on learning how to balance. Toddlers normally start off by walking around on these before developing to push off the ground, lift their feet and glide (often at speeds not approved by their parents).

On my first bike, the emphasis was on pushing the pedals. When I was comfortable with that, my parents removed the training wheels, held the seat and promised they would hold it until the end of time. They’d follow behind me as I built up speed and let go without letting me notice. I’d then ride by myself, until I realised they weren’t there anymore, at which time I’d turn my head, lose control and slowly crash, reacting very dramatically.

These two experiences couldn’t be any more different. My son learned that to balance on two wheels, you must maintain movement. His body learned the little movements and countermovements required to maintain balance, developing a sense of what it feels like to ride a bike, despite not pedalling. It was unconventional in the sense that, when an adult rides a bike, their feet don’t touch the ground at all. Pedalling is required to maintain movement and, subsequently, balance. My training wheels allowed me to bypass the lessons my son learned and move straight to pedalling. I could ride incredibly slow or even come to a complete stop and I still wouldn’t fall over, thanks to my extra two tiny wheels. If the main lesson of learning how to ride a bike is how to balance, I was faking it until I made it. My son however, would’ve ridden rings around me, at half the age.

So, what does this have anything to do with teaching mathematics? Everything.

How I was taught mathematics is synonymous with how I was taught to ride a bike. My training wheels were the worked examples from my teacher and the hand on my seat was the exercises in the textbook that got progressively more difficult. Sure, I ended up learning how to ride a bike, just as I learned how to use the mathematics I was taught. What I’m less convinced about, however, is whether I learned how to balance, or in terms of mathematics, problem solve.

My son learned something about balance with every step he took. Every movement led to the next one, slowly learning more and more without any shortcuts. Kids who learn how to ride on a Balance Bike first, find riding bikes with pedals a breeze in comparison to those who don’t. With pedals, they can use gears to get them around faster and more efficiently. This is the essence of how I want my students to learn mathematics.

What does it actually look like though?

Toddlers use their ability to walk to actually get them moving on the bike. They solve the movement problem by using something they already know how to do. In mathematics, I think the same can be done. Stick with me here.

When my students would encounter a problem, they often either went down one of three paths:

1. It was familiar enough to apply a method they’ve used before.
2. It was unfamiliar, but they’d try a method for a different problem and hope for the best.
3. They’d do nothing.

Paths 1 and 3, in my opinion, are not problem solving. Path 2 could be, but students either get it right by chance and earn some undue confidence or get it wrong and head down path 3.

My strategy as a teacher was often to whip up a similar worked example so they could make their merry way down path 1. In reality, I was merely bolting two little training wheels onto their bike. They’d pedal for a bit and I’d cheer because they looked like they were riding a bike. They looked like they were problem solving.

So, what would a fourth path look like? A Balance Bike.

Problem solving means actually working through something you don’t currently know what to do. If you know what to do, hakuna matata – there’s no problem to solve! It’s the initial stages we need to work with students to become more comfortable with – when they don’t know what to do or how to start. They need to make sense of the problem before they try to solve it. They need to take steps before they start pedalling. To do this, we need to normalise not knowing. It’s ok to not know the answer, that’s expected. It just means that there’s a problem to solve.

John Mason describes two distinct phases of problem solving: Specialising and Generalising. Put simply, specialising is our attempts to understand the problem, to get a grip of it, and make sense of it, whereas generalising is the formulation of a conjecture or the discovery of a relationship. To specialise, students try out several examples, which help them to consider the general case. In my experience, we’re all so eager to jump to a solution, we attempt to generalise before specialising. It’s through specialising that students make sense of the problem and notice patterns. It’s through specialising that they learn how to balance. They might spend longer specialising than you anticipate and even pursue something they’ve noticed that has piqued their interest, which in my opinion, is the sweet spot of doing mathematics.

Try this problem for yourself, but instead of jumping to answer it, try some specific examples for yourself first:

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)

To truly normalise not knowing, we need to provide time, space, and strategies for students to make sense of unfamiliar problems. We need to step back, let them wobble, and learn to balance. We need to let students know that specialising is a prerequisite of generalising. It’s easy to find comfort in training wheels, but when students realise that we’re no longer holding onto their seat, it’s no surprise they don’t know what to do.