For over 10 years we’ve been working in classrooms in Ontario and in Brazil, collaborating with teachers to develop activities that will engage young mathematicians with mathematical ideas that will surprise them, that capture their imagination.

Our starting point has been to ask the question, How can we help?

So, we start with teacher needs and student learning needs, and then we wrap around those more complex ideas of mathematics.

For example, when teachers in grade 3 asked us for ideas for area representations of fractions, and how to teach that in different ways, we added around that content the concept that I described in the previous video, of infinity and limit.

So, students had lots of practice representing fractions as areas, but also they had opportunities to see that you can hold infinity in your hand.

We also take the time in classrooms to prepare students to communicate these ideas to others.

Our success is dependent on how well students can share these ideas in ways that will capture their parents’ imagination, that will offer them mathematical surprise and insight. We want to see that in the feedback that we get back.

Following this, is the animated video of Grades 1-2 students singing parents’ comments sent to their teacher, after students shared at home what they learned about growing patterns.

There’s this idea that good teachers make math easy to learn. Seems logical and common sense. Let’s consider an example:

Imagine sitting in a movie theatre, watching a movie. And, typically what you do is you try to predict; you try to guess what might happen next. Now, if the movie is easy-to-learn, your predictions will be correct. That might feel good the first or second time, but eventually that movie isn’t going to work for you.

We don’t pay for movies to experience our guesses, our predictions, to be correct. We pay for movies to be surprised, to flex our imaginations, to see new things that aren’t expected, and to think hard about things that happen.

When was the last time you were surprised mathematically? I don’t mean that your teacher dressed up as Pythagoras, and that surprised you. I mean mathematically, conceptually.

Surprise

Surprise is not a frill. It’s not something we can ignore. Surprise is a biological necessity, actually. Humans enjoy creating, experiencing and learning from surprises. If that wasn’t the case, then our ancestors, a long time ago, would have been some wild animal’s lunch.

The word surprise would be a rare find in a math curriculum document. In fact, any curriculum document.

Let’s put it in there.

Although we don’t have a focus or a tradition in education of engaging young children with mathematical surprises, math is full of beautiful surprises.

Where odd numbers hide

For example, let’s take the first four odd numbers: 1, 3, 5 and 7. We can build them using block like these. There’s the first one, number 1. Number 3. Number 5. And number 7. Made out of 1, 3, 5 and 7 blocks.

If you ask children to make these and play with them, they will naturally put them together because of their sense of pattern and fit, and they will get something like this.

They’ll notice that this fits in there, like spoons, and this one fits in there, and this one fits in here. So, all those 4 odd numbers fit together to make this little square. That’s interesting, isn’t it? Because the first 4 odd numbers give you a 4 by 4 square. If we take the fourth odd number away, then the first 3 odd numbers, 1, 3 and 5, come together to give you a 3 by 3 square. The first two? The first 2 odd numbers give us a 2 by 2 square. So, there’s a surprise there, that odd numbers hide inside squares.

So, every time I see a square, let’s say this one, I wonder, how many numbers are hiding inside. What do you think?

This idea of odd numbers hiding in squares, meets the expectations in grades 1, 2 and 3, of patterning, of number sense. But really, it’s coming from grade 11 mathematics, where you study sequences and series, where you try to find the sum of the first n odd numbers or the first n even numbers, and you can see how this would help you to do that. So, what we’ve done, is we’ve taken those big ideas and offered a high ceiling and children can engage with them and develop abstractions. Because they can see that it’s not just the first 4: as long as you know how many of them you have, that’s really your answer. Ten odd numbers? It’s 10 by 10.

How to hold infinity in your hand

Another example would be fractions. So, working in a grade 3 classroom, if you want to give children a sense of fractions and their relationships, you might say, let’s try walking out the door.

And, walking out the door, well that’s quite easy, we’ve done it many times before. But what if we used fractions to walk out the door?

So, let’s say I start here, and I walk half way to the door first. And then, I walk half the remaining distance. That would be a quarter. And I walk half of the remaining distance. That would be one eighth. And then half the remaining distance, and half the remaining distance, and you can see that if I keep doing this it would go on forever, because no matter how many steps I take, there’s still a small distance, getting smaller but it’s still there, that I can take half of.

We can then ask children to use squares to represent those fractions using area representations. For example, using a square like this, you can divide it in half and shade this side one colour to represent one-half. Or you can divide it into 4 sections and shade this part to shade one-quarter, and so on. So now the question would be, what if you did that for say the first 5 fractions, 1/2, 1/4, 1/8, 1/16, 1/32.

And then you took those shaded parts and you cut them out with scissors. So you cut out the 1/2, and you cut out the 1/4, and you have those 5 fractions in front of you on your desk. Now imagine putting those fractions together, to make a new shape. Now remember, this is going on forever. So you’re constantly adding more pieces. How big would this shape be if you keep adding fractions to it? Would it fit in this room? Would it fit in your town or city? Would it fit on Earth? How can infinity fit in a finite space?

So what children say if you do this activity with them, is that the fractions fit in a whole. If you ask them what do you mean by a whole, they will say that if you take any one of those squares that you started with, the 1/2 fits in there, the 1/4 fits in there, the 1/8, the 1/16, the 1/32, and so on, all the fractions fit inside this square and they never leave outside, they never spill out. So there’s an understanding that an infinite number of fractions, fit in finite space, and their sum is 1.

This activity would cover the content of area representations of fractions, but it also gives you a sense of what you would study in Calculus, which is infinity and limit.

The mathematical surprise here is that you can hold infinity in your hand.

Video ends with Aboriginal recording artist Tracy Bone & Bob Hallett of Great Big Sea singing “Infinity in my Hand”.