Thinking about numbers and quantity is at the heart of so many of the choices we make. When we look for the shortest line in the grocery store, choose to sit in the least busy part of a train or gauge how much wine in our glass is enough, we are relying on our innate ability for quantitative reasoning.
But despite how pervasive this kind of reasoning is in human cognition, scientists still know little about how — in combination with language — it brings about uniquely human concepts such as symbolic mathematics. Nor is it fully understood how young people acquire this way of thinking, or why they sometimes struggle to do so.
Researchers at the University of British Columbia’s Centre for Cognitive Development are investigating how human minds reason and decipher numbers and quantity, how these abilities develop from preschool onward, and how they relate to higher cognitive abilities.
Math can be tricky, but why?
“Why do kids learn certain things really quickly and really easily and yet struggle to learn other things?” asks Darko Odic, director of the Centre. One example that Odic’s team explores in depth is the contrast between how children acquire language compared to how they acquire math skills.
“Every kid, by the time they’re five years old, is pretty much an expert in knowing their first language in terms of the syntax and vocabulary,” Odic explains. “And yet, other concepts that are simpler than language in principal, like basic arithmetic, are something that a lot of us struggle with, even though we’ve had decades of explicit instruction on it.”
To get to the heart of why this is the case, Odic and his team work with children to better understand what we inherently understand about numbers, and what we need to go beyond our intuition to learn.
What they’ve found is that, although estimating quantity might be a basic survival skill we’ve acquired in our evolution, the math we use for more precise calculations isn’t nearly as ingrained.
“Our inherent number system is not particularly well designed for representing numbers explicitly and exactly,” says Odic. “When you’re using your intuitive number system, being able to tell apart 10 apples in a tree from 20 apples in a tree is pretty easy. But being able to tell apart 20 from 19? It’s almost impossible. The inherent imprecision in our perception of number doesn’t allow these nearby numbers to be detectable as different.” That can be a problem for kids learning math, especially since much of what is taught — addition, subtraction, even counting — is about exact numbers.
To guesstimate is human
Darko Odic explains how our inherent ability to estimate quantity is important in human evolution and survival — Is this band of charging animals bigger than my group? — but has its limits when it comes to more exact mathematics
3 minutes, 27 seconds to listen
[ODIC] One analogy that I like to draw on is that if you think about something like roman numerals. Roman numerals are fantastic for certain ways of representing number. They sort of, at least for the smaller numbers, especially visually, iconically represent a number.
But very famously, roman numerals are terrible for doing things like division. And part of the reason why historically they were abandoned by most cultures throughout the world is because the way in which roman numerals represent a number ends up being not so great for doing certain mathematical operations.
And in that same way, by analogy, the way in which our mind intuitively represents number is really, really amazing for certain things and really not so great for other ones.
So what we know from my work, and the work of others, is that this intuitive number system, which very often is called the Approximate Number System or ANS, is something that we’re born with. It’s something that we share with a lot of other non-human animals, and most researchers think that it’s sort of this evolved sense of number that maybe has played a very important role in our survival, like maybe helping us that we quickly appraise which tree has more fruit for foraging. Or is this band of charging animals bigger than my group? And should I fight them or should I flee? Or something like that. And what the sense of number is really good for, it’s for representing number approximately and for representing number relatively. What it really seems to be quite well designed for is telling you, is there more on this side or that side? Is there more fruit in this tree or that tree?
But what it’s not really quite well-designed for is representing numbers explicitly and exactly. So, to make it maybe a little bit more concrete, when you are using your intuitive number sense, being able to tell 10 from 20 apples on a tree or something like that, is pretty easy. But telling apart 20 from 19 is almost impossible, because the inherent sort of noise and imprecision in our perceptual representations of number, don’t allow these nearby numbers to really be detectable as different.
But when you think about mathematics, a lot of what we learn in mathematics is really not about sort of gist or approximations, it’s really about exact numbers, so the process of counting and differentiating numbers, one from another is about representing numbers precisely. You know, 19, that’s as different from 20, as 20 is from 10. And so precisely because of this sort of way in which the intuitive number sense is very much optimized for relative, but approximate comparison, it makes it quite good for when you need to do something like that. Like, when you’re trying to judge how many items are in your shopping bag or something like that, if you’re going to decide, can I go to the express checkout line or something like that.
But that same sense of number is really not well-suited for thinking about numbers precisely and exactly. And so when the thing that kids are trying to learn about is something about exact precise numbers, their intuitive number sense is not really going to help them and in fact, it might actually hurt them. And when kids are trying to reason about number in a very relative gist kind of way, then they can really fall back on these intuitions and learn quite a bit from that.
Critical insights for changing curricula
Odic thinks our preoccupation with teaching kids abstract ideas like exact numbers could amount to a missed opportunity in their education. “We wait to teach kids a lot of concepts because we think they’re too advanced. We have this idea that division is a more advanced concept than counting, but my work is showing that in fact kids have many more rich intuitions about how things like division work that we can capitalize on a lot sooner than they do about counting and exact numbers,” he explains. “The curriculum that a lot of kids go through abstracts away from kids’ understanding and ignores a very valuable set of intuitions kids bring into the classroom.”
Although Odic says his research is still in early stages, it could eventually help shape educational policies and curricula, for both children and adults. As provinces like Ontario invest millions in revising math curricula, understanding the origins and structure of our numeric thoughts is an important pursuit.
Learning math, naturally
Darko Odic reflects on how math curricula might be more successful if they took advantage of kids’ intuitive sense of quantity rather than getting hung up on precision concepts, like counting, from an early age
2 minutes, 5 seconds to listen
[ODIC] Although there are these important shortcomings in intuitive number sense and how it connects and relates to our sense of exact number, there’s actually far more usefulness in it then we might initially anticipate.
So for example, what we know is that the intuitive number sense is capable of doing things like addition and subtraction. You’re not just able to represent the number of objects in a scene, but you can also mentally manipulate those approximate quantities through a lot of different mental computations, and again, not only addition and subtraction, even things like division.
And, I think that a lot of the curriculum on math is focused quite a bit on abstracting math away from children’s intuitions a lot of the time. Focusing in on things like multiplication tables or on trying to get kids really to master counting before concepts like addition and subtraction are introduced. But again, what my work is really suggesting is that there are a lot of really rich intuitions about a lot of different mathematical concepts that can be drawn out well before kids have started to master exact number concepts and things like that.
I think that one of the major hurdles that kids encounter when they’re learning symbolic mathematics is much more in sort of understanding what the symbols correspond to, rather than understanding the basic principle of addition or division or something like that, if that makes sense? So that it’s much more, but the fact that they already have an intuitive understanding of what division is, and what they’re really struggling to accomplish is to learn how to map that intuition to these exact, precise things that we’re teaching them.
The work begins with play
To pursue this understanding, Odic created a child‑friendly facility, with support from the CFI, which feels more like a place to play than a research lab.
Many children participate in testing, which consists of a series of games, over a period of years, which is a testament to both the Centre’s design and the researchers’ ability to make sure “science is not scary,” says Denitza Dramkin, a PhD student at the Centre. “The atmosphere is relaxed and fun.”
The children, who are between two and 12 years old, choose a T-shirt, book or stuffed animal as a prize for participating. They also get a diploma starting with a bachelor’s degree and progress from there. “I work with kids who come in with their master’s and walk out with their PhD in 15 minutes. I wish it were that easy,” jokes Dramkin.
An atmosphere for learning
Darko Odic discusses how developmental psychology research depends on having a welcoming research space where parents and kids can feel at ease
1 minute, 56 seconds to listen
[ODIC] The entire research program for my lab and virtually every other developmental lab in Canada really depends on this research space being extremely optimized for having parents take time out of their day, and very often take kids out of the daycares, to physically come to the space in order to do research with us. And if the parents are not willing to do so, and especially if the space and the entire experience is not positive enough, the research and developmental psychology will just stop.
Parents are not paid for this. We give little prizes and shirts and books to the kids as a token but this is something that parents just do because they want to be part of developmental science. They want to learn something more about their kids and how they are developing, and I think that parents recognize the value in the kind of discoveries that my lab and other developmental labs are finding for our understanding of how children’s minds develop as well.
When I got funding from CFI, I spent an enormous amount of time really trying to design our space in a way that it would be the most welcoming space that parents would really want to come back to again and again and again.
Like many psychology labs, we’re in a very old building that is not exactly pleasing to the eye and we in many ways, wanted our lab to be a kind of oasis where a parent can walk in and can really feel that they are in a place that’s fun and a place that’s safe and a place where they could feel very comfortable with having their kids be in where they can really transparently see the kind of work that goes on.
More than 5,000 children have participated since the Centre opened. Odic’s team hopes to one day be able to create tasks and tools to help adults take advantage of their intuitive skills in ways that could enhance their math abilities.
Some parents are already putting the research to good use. The centre doesn’t release data, but parents are able to observe their children taking the tests, giving them insight into how their child learns.
New insights about math-related learning disabilities
Darko Odic describes “dyscalculia,” the mathematics equivalent of dyslexia
1 minute, 16 seconds to listen
[ODIC] There’s a subset of children, for whom it’s not just the case that they’re on the lower end of a normal distribution for their number sense, but who actively seem to have a kind of math selective learning disability. And this, many people have called dyscalculia. So it’s meant to be a sort of math specific version of something like dyslexia. And this is something that my lab is much more recently starting to recruit a larger sample of children from Vancouver over to really begin understanding a lot more about this population.
It’s very rare. I think the typical estimates are that it’s less than five percent of the population, but for children who are diagnosed with dyscalculia, this does result in lifelong challenges with mathematics, both intuitive and more symbolic and formal.
And what we already know from other work is that these kids are extremely impaired on their intuitive number system, to the point that I think some researchers really believe that the best way to conceptualize of this disorder is that it’s a very extremely impoverished intuitive number sense that really sort of propagates onward as kids start learning math more formally.
Fundamental questions driving a young research leader
Odic, 32, has always been fascinated by what makes human minds unique and the origins of much of human knowledge. After finishing his PhD at Johns Hopkins University in Baltimore, Maryland, Odic was drawn to UBC in large part to work at the UBC Early Development Research Group. At that time, it was a group of six research centres interested in cognitive and social development from birth to adulthood. The Centre for Cognitive Development became the seventh after Odic became an associate professor in 2014.
“One of the really rewarding things is that the intuitive number sense has reached a broader consciousness with the psychology community, and now we’re attacking the problem from many different ways,” he says.
Neuroscience, comparative psychology, linguistics and computer science are just four of the other disciplines Odic believes could help unlock the mysteries of learning for future generations.
One, two, three — what does that really mean?
Darko Odic describes a simple game to play with young kids to assess how well they understand counting
2 minutes, 8 seconds to listen
[ODIC] Parents spend a long time with their kids teaching them the count sequence: one, two, three, four, five and you know, many kids by the time they’re three, they will go: one, two, three, four, five, right? And they’ll sometimes even point at things in the world. And many parents come really believing: hey, my kid knows how to count. You know, they know how to count to 10. They know that they’re supposed to point to things, but it turns out that kids don’t actually know, most three-year-olds at least, don’t know what these words actually mean.
And the way that you can really assess this, and it’s one of my favourite things, I think one of those experiences that really challenges the parents’ understanding and sort of sometimes shocks them is, you know, take a pile of anything. It can be coins, it can be candy, it can be toys, and you can ask the three-year-olds, hey, can you give me five of them? Can you give me five of the coins?
And what you will find is that kids will very often just take a completely random number of coins and give them to you. And you can double down and you can be like but I wanted five. Can you count and make sure there are five? And the kid will go: one, two, three, four, five, six, seven, eight and then you’ll be like well, can you make it five? And a lot of times they’ll just sort of look at you, think about it for a bit and then they’ll add like three more coins to the batch because ultimately, while they know that we do this thing called counting, most three-year-olds are really treating it like a little sing-a-long song. They’re treating it like Twinkle, Twinkle Little Star or something like that.
They don’t actually understand the concept of these numbers, even though they seem to go through the routine in the way that they’ve been taught again and again from just observing their parents. And it’s not until most kids are somewhere between four and four and half that they really begin to understand what these words actually mean and are capable of actually showing you that. If you ask them for exactly five things, they’ll give you five. No more, no less. And if you ask them for six, they’ll give you exactly six and no more and no less.
And so there’s this very long protracted period of time during which kids know the words, they know the sequence, they know they’re supposed to point at things and yet they don’t actually know what these words mean.