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”:
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.
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).
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.
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.
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).
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 .
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.
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.
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
Gadanidis, G. (in press). Five affordances of computational thinking to support elementary mathematics education. Journal of Computers in Mathematics and Science Teaching36(2), 143-151.
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.
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.
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.
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.
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.
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:
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 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”.