This is another post about a teaching book I’ve read recently. This one is about the Which One Doesn’t Belong Teacher Guide by Christopher Danielson.

It goes with a beautiful little picture book called “Which One Doesn’t Belong?”, which is a shapes book different from any you’ve ever seen before. In this book, each page has four pictures, and asks the readers to say which one doesn’t belong, and why. The fabulous thing about the book is that there is at least one reason why each of the four pictures doesn’t belong, and talking through these with children (or indeed anyone) is a rich conversation about the properties that shapes have and don’t have.

The Teacher Guide is all about these rich conversations: why it’s important to have them, what you and your students/children can learn through them, and how to facilitate them. Chris has a friendly and welcoming style which draws you easily into a new appreciation of the sophisticated thoughts of children as they make sense of geometry and the world.

There are a few key things Chris talks about that really impacted my thoughts about teaching and learning maths. I’ve organised them by quotes from the book:

Commonly in maths class, student responses are compared to a standard answer key – the measure of what’s right is what’s in the back of the book, or what the teacher has in mind. In a conversation about a well-designed Which one doesn’t belong? task, the measure of what’s right is what’s true. — page 3

I read this quote first when someone else tweeted it out of the book and it struck me as awesome then. In my job at the Maths Learning Centre, students are always asking me if things are right, as if the measure of rightness is if I say it is. But in most places in maths, correctness is measured by truth. Your vectors will either be an orthonormal basis for the subspace or not. A number is either prime or it’s not. You can tell if you’re right by thinking about whether it’s true. I very much want to see opportunities to talk about the truth of things with students, to put the measure of rightness outside an authority figure.

“The van Heiles haven’t argued that it is

difficultto go from level 1 thinking directly to secondary school geometry; they have argued that it isimpossible. If students don’t have experience and instruction building informal geometry arguments, they will not learn to write proofs.” — page 8.

Chris is referring to the van Hiele model of “how childrens’ geometric thinking develops over time”. In this model, there is a build-up from noticing that shapes look like things they’ve seen (level 0), to noticing properties that shapes have and don’t have (level 1), to relating properties between properties of shapes (level 2), to logically supporting claims about these relationships (level 3).

The thing in the above quote that really struck me is the idea that it’s impossible to learn to write proofs without experiencing informal arguments first. I see so many students at university every day who struggle with proofs, and it makes me wonder that they maybe need more experience with informal arguments. Indeed, it makes me wonder if they need more experience simply *noticing* properties, since that’s an even earlier level. This is essentially applying the van Hiele models to other types of maths, but certain aspects of the progression still feel right to me, especially for things vaguely geometrical like vectors or matrices or graphs of functions.

I wonder if a student struggling with proofs might benefit from talking through a progression like this, and then helping them have experiences at the earlier levels before helping them with proofs.

“Of course being able to state new facts is an aspect of learning, but much more important to me is being able to ask new questions.” — page 21

I had never thought of this idea explicitly before, but immediately I saw that new questions were important to me as well. I was reminded of the time someone asked me if my students were understanding my statstics lectures. I said that I wasn’t completely sure, but certainly the students were asking very deep and complex questions. Instinctively I knew that a new type of question indicated learning.

Also, in the Drop-In Centre, there’s a certain joy when a student asks new questions you’ve never thought of before. They are wondering about the connections between things, which means they are *learning*, because learning is all about connections.

I am excited to listen out for new questions as a sign of learning, and to *tell* the students that it’s a sign of learning to have new questons!

“… I hope you will begin to see geometry through children’s eyes as well as through the eyes of a mathematician. Mostly, I hope you will come to understand that these two views of geometry are not nearly so distant as the school curriculum might lead us to believe.” — page 37

Now, I already believe that children’s investigations and ideas are actually very close to the way mathematicians work. You can’t be married to a very excellent early childhood educator without coming to some appreciation of this! It’s so nice to have someone publish a book telling teachers and parents the same.

Even more, this whole section is all about noticing and naming things and their properties. It’s about whether properties need names at all, or whether the objects that share those properties need names. It’s about what properties are important to make a thing a special thing and what aren’t, and in what context. It’s about the relationships between things. All of these are the work of professional mathematicians both pure and applied. And they are the work of children sorting out how the world works.

The geometry of children and the geometry of mathematicians are definitely not so far removed.

“I have come to understand that talking about this difference is more important than defining it away.” — page 54

Along with the rest of this chapter, this quote got me thinking about a whole new way to approach definitions in mathematics. As a pure mathematician, definitions are very important to me, and I always used to start with the definition. But I know those very definitions took years and even centuries to come to their current forms, and I also know that humans don’t learn through definition but through comparison of things that do and do not fit an idea. I think this is precisely what Chris is getting at here.

By skipping straight to the definition, we’re robbing people of a key part of mathematical thought, and we’re skipping them through the van Hiele levels before they’re ready. You don’t need a definition until you have a need to distinguish a thing from the other things around it. You don’t need a definition until you’ve noticed the properties you can use to define something.

The classic example in my own teaching is subspaces in linear algebra. The properties used to define a subspace aren’t even discussed until the definition is given. Little wonder, then, that the definition is meaningless to students!

It’s not just definitions either. I help a lot of students learn statistics, and one of the things that is never explicitly taught in your traditional statistics course is how to choose what is the most appropriate statistical procedure for the situation. I have been teaching this by focussing on some specific aspects of these procedures that statisticians use to distinguish things. Reading this chapter and this quote in particular helped me realise what I was doing was exactly “talking about this difference”. To distinguish between things you need to notice the properties that make them different, and to notice them, you need to compare things. I now have a much clearer idea of what I’m doing when teaching in the way I do.

I want to spend more time putting students in situations where they *notice* the differences between things and have to talk about them, so that they can distinguish between things they need to, and so that the properties I use to define things make more sense.

Thanks Chris for a most thought-provoking book.