paper-and-pencil computation

[As our newly-elected Ontario government is considering how our mathematics curriculum might be improved, and as it may be considering an increased focus on basic math skills, including computation skills, below are some ideas to consider.]

Whatever paper-and-pencil methods we decide children should learn, we should ensure that they are understandable and have a mathematical future. Click To Tweet

In a previous post, I wrote: Children need practice with a wide variety of basic computation skills.

One of the basic skills listed was: Paper-and-pencil computation.

But which methods of paper-and-pencil computation?

Let’s look at multiplication, as an example.

Multiplication

Most parents I meet learned the multiplication method shown on the right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • write the 4
  • carry the 2
  • 6 times 2 is 12
  • add the 2 we carried to get 14
  • write the 14
This method has some advantages:
  • every multiplication we do involves only single digits
    • you can multiply very large numbers by only knowing your nines multiplication table
  • it always works the same way
    • regardless of how big the numbers are, the process is always the same
  • it’s concise and efficient
    • if you have a job where you have to do a lot of multiplications every day, and you don’t have access to a calculator, this method is an asset
It also has some disadvantages:
  • it un-teaches place value
    • before learning to multiply, children learn that 24 is 20 + 4 (or that 24 is 2 tens and 4 ones)
    • using this multiplication method, children practice calling numbers by the wrong names
      • for example, when we say “6 times 2 is 12”, the 2 is really 20 and the 12 is really 120
      • similarly, when we say “carry the 2”, the 2 is really 20
  • the method gives the correct answer even if you don’t understand
    • in other words, you don’t have to have good number sense to make it work
    • which also means that practicing this method does not help develop your number sense
    • for example, I once received a package where on it somebody had computed 10 x 12, as shown at right
    • this is why when you ask adults to explain this method, they often can’t: they learned it as a set of rules without conceptual understanding
  • it does not have a mathematical future
    • because the method breaks math rules (calling numbers by their wrong names), you can’t make connections to other math children have to learn (unless that math also breaks math rules the same way)

Because of the conceptual disadvantages, some education jurisdictions are hesitant to teach such paper-and-pencil computation methods to young children.

However, this does not mean that all paper-and-pencil computation methods have these disadvantages.

Let’s look at an example for multiplication:

Multiplication v.2

Let’s multiply 6 x 24 using a different computation method, as shown at right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • 6 times 20 is 120
  • 24 plus 120 is 144
How this method is different:
  • it reinforces place value understanding
  • it is similar to the way most people multiply in their head
  • it is easy to explain
  • it has a mathematical future, as it correctly uses place value, expanded notation, and the distributive property:
    • 6 x 24 = 6(20 + 4) = 120 + 24 = 144
  • it gets students ready for algebraic processes: 6(y + 4) = 6y + 24
  • it also requires students to know, and practice, how to multiply with multiples of 10 (as in 6 times 20 is 120)
What about bigger numbers?

Don’t we eventually have to carry?

Let’s take a look at 9 x 24.

Here’s the sequence of steps:

  • 9 times 4 is 36
  • 9 times 20 is 180
  • at this stage, if you can’t see that the sum is 216, you can use partial sums:
    • add the hundreds to get 100
    • add the tens to get 110
    • add the ones to get 6
    • altogether, that’s 216

Is there Multiplication v.3?

There are variations to the above method.

Whatever paper-and-pencil methods we decide children should learn, we should ensure that they are understandable and have a mathematical future.

That is:

  • paper-and-pencil computational methods should build on children’s previous knowledge
  • they should prepare children for more complex mathematics

 

kids need a wide variety of basic computation skills

In another life, I was a district math consultant. In my first month in the job, I received a call from an elementary school principal, inviting me to a parent council meeting for 10-15 minutes to talk about the new curriculum.

I asked, “Would it be possible to have an hour, or an hour and a half, so we can do some math together?” The principal checked with his parent council and we scheduled a math night for parents at the school library.

To make a long story short, word got around and I started receiving invitations for parent math nights across the district. Sometimes the turnout was 20 parents. Sometimes it was over 200, with tables and chairs filling a gym. It was one of my favourite things to do!

I quickly noticed 2 things:

  1. Many of parents feared or even hated math.
  2. Quite a few commented on their children’s basic computation skills.

To get parents thinking about what basic computation skills might be, I did 3 warm-up activities with them:

1. Counting money

I brought some bills and coins and I laid them out on a table: a collection of twenties, tens, fives and coins.

I invited a parent volunteer to count the money. I asked them to touch the bills and coins as they counted.

Then I invited a second parent volunteer to check the first answer.

How would you count the bills and coins?

If you are a typical person (like the parents at the Math Nights), you would first count the $20 bills, then the $10 bills, then the $5 bills, and so on … finally counting the coins, with the smallest last.

I said: “Great! We agree on the answer.”

And I asked: “How many of you would add money this way?” Most parents raised their hands.

“But we have a problem,” I added. “The math is backwards.”

Most parents (if not all) learned in school to add numbers from the right, starting with the smallest place value.

But when asked to add bills and coins, their minds naturally added from the left, starting with the greatest place value.

This is interesting, isn’t it?

Parents noticed:

  • The calculation methods they learned in school are not always (actually, not often) the ones they naturally use in everyday situations.

2. All you need is a calculator (?)

Next, I pointed out that calculators were abundant and inexpensive. (Today, every cellphone and tablet has a built-in calculator. Or just type the question in your browser’s address bar and it will find you the answer.)

So I asked parents: “What do you think about not teaching children how to add, subtract, multiply and divide and just let them use a calculator?”

Their answer was that this would not be a good idea.

They explained that kids need to understand the math. With the calculator, they just push buttons.

So, I handed out calculators and asked them to solve 12 x 25.

I added, “But there’s a catch: you’re not allowed to use the #2 key. No cheating by doing this in your head. You have to get the answer using the calculator.”

Here are 3 of the calculator methods parents used.

  • 3x4x5x5 (multiply the factors of the numbers)
  • 6×50 (half the fist number, double the second one)
  • 6x5x5 + 6x5x5 (a combination of the two methods above)

Can you think of a different method?

Parents noticed:

  • They wanted their children to understand.
  • A calculator can be a tool to think-with.

3. On paper & in your head

Last, I presented parents with the problem, 16 x 24.

I asked 2 parents at each table to solve this using pencil and paper and the rest to solve it in their heads.

Before you read further, try calculating 16 x 24 in your head.

I asked the parents who used paper and pencil to share the methods they used. They hesitated to volunteer. When a few did share, we noticed that they used the same method.

When asked to explain the paper-and-pencil method, they had a very difficult time. They could describe how, but could not explain why. For example, “Why did you indent the second row?” Some parents became frustrated and said: “It’s just a rule!”

The atmosphere changed when I asked parents to share the mental methods they used to solve the same problem. There was a palpable energy in the room. Parents were eager to share their personal solutions. They praised others who came up with different methods. They expressed delight at methods that surprised them.

Here are some of the mental methods parents shared:

  • They multiplied 10 x 24, 6 x 20 and 6 x 4, and then added the three products.
  • Some multiplied 20 x 16 and then 4 x 16.
  • Some multiplied 16 x 25 and then subtracted the extra 16.
  • Some multiplied 4 x 4 x 4 x 6.
  • In some rare cases, parents used algebraic structures:
    • 16 x 24 became (20–4)(20+4) = 400 + 80– 80 – 16
    • or, more simply, (20–4)(20+4) = 400 – 16.

Parents noticed:

  • When using a mental method, they understood what they were doing and could easily explain it to others.
  • The paper-and-pencil method was more like using a calculator: pushing buttons to get an answer without much understanding.

Basic computation skills?

I’m old enough to know that I don’t have all the answers.

I do spend numerous days each year in elementary school classrooms, working with students and teachers. Here’s my opinion:

Children need practice with a wide variety of basic computation skills. Click To Tweet

Some basic computation skills:

  • Paper-and-pencil computation.
  • Mental computation, with ability to think flexibly, creatively and playfully about numbers and operations.
  • Solving number-based puzzles with and without tools (like calculators, concrete materials, games, coding, etc).
  • Making sense of numbers and operations in big idea contexts, such as the one below from Grades 1-2 classrooms.

The lyrics in this music video come from parent comments, after children shared their learning at home. From a project supported by KNAER.

In search of discovery learning

Does Ontario have a 'discovery math' curriculum? Click To Tweet

We’ve all been to school, we’ve spent a lot of time there studying the basics, and where most of us learned to dislike math. In fact, it’s not uncommon to hear adults pronounce with a sense of pride, “I’m not a math person.”

And now, as adults, we want the same for our children, which I think is really another issue (having to do with the “good old days” and the older generation underestimating the younger generation), but here are some of my thoughts on all the talk on “discovery learning”:

  1. Ontario education. Over the last 10 years I’ve spent on average 40-50 days each year in elementary classrooms, and I haven’t seen “discovery learning”. I’ve seen guided investigations, where the teachers prompt and scaffold student thinking to help them understand concepts in new ways. And I’ve seen a variety of other approaches too.
  2. Discovery learning. Years ago, Constance Kamii, who studied with Jean Piaget, did a lot of research on students discovering solutions rather than being told how to solve math problems. In one study, she compared Grade 3 children from three different classrooms, who learned how to add sums like 7 + 52 + 186, in three different ways: 1) by learning the addition algorithm you and I learned in school; 2) by not being told how to add and having to discover how to add by themselves; and 3) using a mix of methods 1) and 2) (for example, see this paper). Interestingly, on standard paper-and-pencil tests, the “discovery math” children did the best and the “let’s learn the algorithm” children did the worst. It was also interesting that when the “discovery math” children gave wrong answers, their answers were close, while the wrong answers of the “algorithm” children would sometimes be off by thousands (showing that they had not developed any number sense).
  3. Right vs left. You wouldn’t want to push analogies too far, but if we stereotype a bit, Kamii’s discovery learning approach can be seen as belonging to the right of the political spectrum. Like building a business from scratch, she asks children to construct their own, individual understanding of what addition is. The way I was taught math in school was more like a hand-out: my teacher gave us “the” way to add, ready-made.
  4. Some of this, and some of that. I think we need a balanced approach. Some of this, and some of that. Even if we think we know the “best” way (which I don’t). Children spend a lot of time in math class and need variety, across the spectrum of “tell” and “discover”. Most of all, we need to appreciate the wonderful minds of young children.

If you want to see some Ontario math classrooms in action, take a look at these lesson studies on Repeating Patterns in Grades 1-3.

To code, or to model, that is the question!

There's a gap between coding and authentic computational modelling practices Click To Tweet

A MOMENTUM AND A GAP

There is a growing momentum in education to engage K-12 students with computational thinking. At the same time, there is a gap between coding (as an end in itself) and authentic computational modelling practices of scientists and professionals to solve real-world problems and build knowledge – to learn – through computational “conversation” and “interaction” with their field (Barba, 2016), “with and across a variety of representational technologies” (Wilkerson-Jerde, Gravel and Macrander, 2015, p. 396).

COMPUTATIONAL MODELLING

Our societies are growing in complexity, in big part because of the intertwining connections afforded by new technologies. The use of computational tools to model phenomena, processes and relationships is becoming a prerequisite to scientific progress and economic success, as evidenced by the emergence of numerous computational modelling fields, such as computational biology, computational mathematics, computational finance, computational medicine, to name a few examples.

A POWERFUL LEARNING TOOL

A focus on computational modelling in education, which is not isolated but integrated with curricular subjects, not only prepares students for future success: it also provides students a powerful learning tool with which to design, test and refine conceptual models and build powerful understandings of what they are studying .

FIRST STEPS

The SSHRC Computational Thinking in Mathematics Education Research Partnership and the  Computational Thinking in Mathematics Education Community of Practice, which is part of KNAER‘s Mathematics Knowledge Network, have been working in the direction of computational modelling in collaboration with the Wellington Catholic District School Board, the Robertson Program at the Eric Jackman Institute of Child Study, and St Andrews Public School.

REFERENCES

Barba, L.A. (2016). Computational Thinking: I do not think it means what you think it means. Blog post, retrieved 6 January 2018 from http://lorenabarba.com/blog/computational-thinking-i-do-not-think-it-means-what-you-think-it-means.

Wilkerson-Jerde, M.H., Gravel, E.G. & Macrander, C.A. (2015). Exploring shifts in middle school learners’ modeling activity while generating drawings, animations, and computational solutions of molecular diffusion. Journal of Science Education and Technology 24, 396-415.

Educating young mathematicians (#4): Our greatest asset

Our greatest asset

Young children thirst for big math ideas. Click To Tweet Math is big - children's minds are bigger. Click To Tweet

Trigonometry in Grade 3

Related resources

Gadanidis, G. (2012). Trigonometry in grade 3? What Works: Research into Practice, Research Monograph #42.

Video Transcript

Our greatest asset, and our greatest opportunity in mathematics education, are the young mathematicians that we work with, and their amazing minds.

Chances are that we all carry some negative experiences with us from our mathematics learning in school. We have to be careful not to pass this on to young children.

Underestimating children

It’s quite common in our society for adults to underestimate young children. In fact, this is a historical pattern, dating back thousands of years.

The ancient Sumerians wrote on clay tablets to complain about their new generation: that young kids didn’t pay attention, they didn’t have a work ethic, they weren’t polite, and generally were a disappointment. Every generation that followed has said something similar, including our generation.

It’s not easy for us to create an education system worthy of children’s incredible minds, if we underestimate them.

Can young children think abstractly?

Jean Piaget brought to us the wonderful idea of constructivism: that young children develop an understanding of mathematics and other ideas from the inside out. They develop their own understanding as opposed to having it ready-made, transferred to them.

Jean Piaget also brought to us the idea of stages of cognitive development. And he said that young children are concrete thinkers and they develop their capability to abstract later on, maybe around age 12.

Seymour Papert who worked with Jean Piaget disagreed. He said these stages you’ve identified are not in children’s minds, they are in the learning cultures that we create in classrooms. They are a symptom of the way we educate them, and not their potential.

Kieran Egan said, young children can’t abstract? How could that be possible? Because if they can’t abstract, how would they ever develop language?

When we develop understandings of words like table, chair, dog, we look at a variety of these objects in the world around us, and extract the essential characteristics.

For example, there are dogs that are big, small, different colours, different dispositions, and we extract the essential characteristics, and with that abstraction that we create, we then look around the world and identify them, and distinguish them from other things that look like them.

So, young children naturally abstract at a young age, to be able to develop understanding of language.

A thirst for mathematics

Young mathematicians thirst for mathematical ideas, for big ideas, for surprises, and for mathematical insights.

And when they share these ideas with their parents, we get incredibly positive feedback.

For example, working in a classroom where the teacher wanted to cover the content of bar graphing, we developed an activity where students would measure the height of time, measuring the height of hours on a clock and graphing those as a bar graph, and noticing the patterns in them.

And a related activity is having a yellow dot on a car tire, and imagining what path that dot would travel as the car moves forward. What would that graph look like?

Children created comic strips of how they might share these ideas with parents. And then they shared those when they went home, along with some of the other artefacts.

We took parent comments, we put them into a Word document, we organized them into themes, we took out any repetition, and then we made that into a song. And then children sang that song back to their parents.

And we’ll end this series of videos with giving you an opportunity to listen to the children, singing to their parents.

The music video, Dots, Clocks & Waves, plays at the end of this video. Here are the song lyrics:

Dots, clocks and waves

my daughter explained
how to conduct experiments
and make bar graphs
plotting the results

she was amazed
by the wave pattern
excited to explain it
to her brothers at home

a dot on a car tire
makes a wave pattern
at first I thought
it would be a spiral

the wave pattern
is still there
even if the wheels
even if they are square

it’s great to see my son excited 
about school and about math
it’s great to see enthusiasm 
and interest in school math

 

my son enjoyed
testing his hypothesis
he was surprised
surprised by the result

he shared his comics
of what he learned
about math waves
on tires and clocks

the height of every hour
on a grandfather clock
plotted on a bar graph
makes a wave shape

like the height of a dot
on a rolling tire
or seasonal temperatures
or sunrise and sunset times

it’s great to see my daughter excited 
about school and about math
it’s great to see enthusiasm 
and interest in school math

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.

 

 

10 affordances of computational modelling

Components of the above 5 affordances described above may be subcategorized to create a list of 10 affordances that come into play when computational tools are used to model mathematical concepts and relationships:

 

 

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 surprises. 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”.