Educating young mathematicians (#3): Five As for coding + math

Five As for coding + math

Coding is not new in education. Click To Tweet Coding and math are complementary subjects, and easy to integrate. Click To Tweet Coding can help us teach math better. Click To Tweet

Related

Computational Thinking in Math Education Community of Practice @ Math Knowledge Network

Gadanidis, G., Brodie, I., Minniti, L & Silver, B. (2017). Computer coding in the K-8 mathematics curriculum? What Works? Research into Practice: Research Monograph #69.

Gadanidis, G. (in press). Five affordances of computational thinking to support elementary mathematics education. Journal of Computers in Mathematics and Science Teaching 36(2), 143-151.

Video transcript

Today, there is a lot of pressure in our society to have young children to learn to code: from industry, from government, from academics in computer science, from non-profit and from for-profit organizations.

Coding is not new in education.

Forty, fifty years ago, Papert and his team developed Logo, which was an environment for young children to learn to code.

The difference between today and what we did in the past, is that we used to have a strong connection to mathematics. When Papert developed Logo, he said, learning math in Logo is like learning French by going to France.

Coding and math are complementary subjects, and they are easy to integrate.

Coding on its own, is one more thing to teach in an already crowded curriculum.

Coding and math are a natural fit. In fact, coding offers 5 important affordances that help us teach mathematics better.

Agency

The first affordance is agency.

When children are working in a coding environment, the world is open to them. They can explore the activity that the teacher gave to them, but they can also ask what-if questions and try other things.

For example, when I was working in a grade 3 classroom, and this was the first time that this classroom did coding and mathematics, one of the young students came up to me and said, I’m not going to do what you did on the screen,

I’m going to do something of my own, and then I’m going to come afterwards and say to you, watch this! That’s agency.

Access

The second affordance is access.

The beauty of the coding environments we have today, and the one that Papert developed with Logo, is that it has a low floor and a high ceiling.

You can engage with coding, with minimal prerequisite knowledge. At the same time, you have the whole coding environment available to you, so you can do very complex programming as well.

So you can explore mathematics ideas that are quite simple, or quite complex. This is what we mean by differentiated instruction. Students can enter at their level, and work to their potential.

Abstraction

The third affordance is abstraction.

When you want, for example, to teach the computer to draw a square on the screen, you need to know what a square is, you need to know its essential characteristics.

You can put the block in of walk forward 100 steps, turn right, and put a loop around it, do this 4 times, and that would create a square. So what you’ve done there, you created the essential characteristics of a square, and you can even take that and create a new block called draw square.

And then you can take this block and use it as an object of other things you write, so you can put it in another repeat loop, you can make it turn around, you can make it spin, you can make it do a variety of patterns. So what coding does is not only does it help you create abstractions, but those abstractions have a concrete feel, because they are objects on the screen that you can manipulate.

Making mathematical abstractions have a concrete feel, also makes them more accessible to young children.

Automation

The fourth affordance is automation.

Normally what you would do if you were working with pencil and paper, you would draw one sketch out and then you want another variation of that, well you have to draw it again.

But in a coding environment, you have your code, and you can say well I would to do this a little bit different, and change one little parameter, and then all you have to do is press on start and then it does it for you.

You’ve automated that process, you can try it over and over, you can show it to others, you can ask what-if questions, change this, see what happens, all very quickly. So it allows for that dynamic modeling, and that exploration, that normally you wouldn’t be able to do.

Audience

The last affordance is that of audience.

Audience is very important for learning for young students because they love to share ideas, they love to see the ideas of others, and you can do this very easily in a coding environment. It’s digital so you can share with parents at home, you can share with peers, and you can access the ideas of others and build on them.

Video ends with animated clip of Talk Math to your Computer.

 

 

Educating young mathematicians (#2): What did you do in math today?

Our starting point with reform has been to ask: How can we help? Click To Tweet Our success is dependent on how well students can share their learning with others. Click To Tweet

Related

Math that feels good: A model for math education reform

Interview with Western applied mathematician Lindi Wahl

Video transcript

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.

Educating young mathematicians (#1): Surprise me!

When was the last time you were surprised mathematically? Click To Tweet Surprise is a biological necessity. Humans enjoy creating, experiencing and learning from… Click To Tweet

The first in a series on Educating Young Mathematicians.

Related

“Infinity” interview with Western mathematician Graham Denham.

Gadanidis, G. (2012). Why can’t I be a mathematician? For the Learning of Mathematics 32(2), 20-26.

Video transcript

Easy-to-learn

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”.

Imagine this!

Several years ago, I attended a presentation by Nathalie Sinclair, who commented on seeing a professional violinist performing in the subway.
Nathalie wondered, what might it be like to perform mathematics in the subway?
So I wrote this song. Get the mp3 for your next math party 🙂
What do you imagine about mathematics? What might mathematics be for young mathematicians? Click To Tweet

Music by Nisha Bedi, Maurice Charland & Bryan Williston. Illustration & animation by Jenn Skerratt.

The last time I saw Molly
She was rocking on the subway
Strumming and singing her new math song
As the passengers sang along

The last time I saw Timothy
He was mixing at a dinner party
Telling a story of complex numbers
And circle wrapping functions

The next time that I see you
Help me laugh and sing and tell a story
Help flex my mind to a new height
See mathematics in a fresh new light

La la la-la-la-la, la la la-la-la-la
La la la-la-la-la, la la la-la-la-la

The last time I saw Alex
He was flexing his imagination
Picturing the sums of odd numbers
Seeing them as square tile patterns

The last time I saw Janette
She was designing a pen for her pet
Maximizing the area it bounded
Minimizing the perimeter all around it

The next time that I see you
I’ll laugh and sing and tell a story
Help flex your mind to a new height
See mathematics in a fresh new light

La la la-la-la-la, la la la-la-la-la
La la la-la-la-la, la la la-la-la-la