How do we learn mathematics?

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  • After examining how maths is used in everyday life, Dr Stefan Buijsman, Sweden’s youngest PhD, found that the philosophers’ theories do not apply to ordinary people’s use of mathematics. Currently a Postdoctoral Researcher at Stockholm University, Dr Buijsman explores the philosophy and pedagogy of mathematics in order to understand how we learn maths and uncover more effective teaching methods.

    Over 3000 years ago philosophers started thinking about mathematics, and what it is about. For some time, this philosophy has been led by two schools of thought: one based on the idea that numbers exist and that by doing mathematics we can ascertain information about them; the other founded on the notion that numbers do not exist and mathematics is ‘just made up’. Both concepts influence mathematical pedagogy and the perception of how we learn maths.

    Dr Buijsman signing books at a children’s books festival. Photo credit: Fred Ernst.

    Dr Buijsman’s research into the philosophy of mathematics examined how ordinary people use and learn maths in real life. He discovered that the philosophers’ theories don’t explain what happens when non-experts employ maths in everyday tasks, such as shopping, where proofs are not required to carry out arithmetic tasks. He concluded that, “some people are making it all more complicated than it needs to be – as a result, we don’t have a good understanding of how people learn maths”. He anticipates that his work will be beneficial to teachers and psychologists, as well as students. “It could be very helpful. Of course, telling people some philosophical story of what numbers are isn’t very useful, but right now we don’t have a story that everyone agrees on anyway. Understanding how we learn maths may lead to more effective teaching methods.”

    Dr Buijsman’s popular science book, which explains the value of mathematics, will be published this October.

    Learning to count
    Psychologists say that we are all born with some abilities to think about amounts, but it is how we develop this ability that matters for the learning of mathematics. Our mathematical journey starts with learning to count the natural (or whole) numbers one, two, three and so on. Not only do we all learn the same numbers, we also seem to learn them in the same way. Children start reciting the numbers without knowing what they mean – they are simply acquiring the vocabulary. Then, around 22 months, children begin to differentiate between one thing and many things. Before then, they don’t always manage: make them choose between one and four cookies, and they’ll pick at random! Once children can identify one object, as opposed to many, without guessing and understand that ‘one’ applies in those cases, they understand the number one.

    A continuing process
    How do you get beyond the number one? Well, children can repeat the trick, because they also know that adding something changes the number. Even if they do not know the actual number, they know they started with one object and added one object to it. If they hear people say that this new group has ‘two’ objects often enough, they can establish the meaning of ‘two’.

    They can go on, to three, four, and even higher numbers, in the same way. Adding one thing to a group of two gets you to the meaning of ‘three’. Three and one is enough to understand the number four, and so on.

    The structure of numbers: we understand that we can increase a number either by adding a zero to the right hand side (and therefore shifting each number up one level) or by increasing one of the digits, for example the left-hand digit.

    Learning larger numbers
    Even so, it’s unlikely that we continue to learn numbers this way when the numbers become larger. We seldom count groups of 50 objects and it is unlikely that we would ever be required to count a collection of, for example, 10,000 items. Yet we understand these numbers too. Evidence suggests that we process numbers with two or more digits differently from those with single digits. Considering that our decimal number system is structured with tens on the left of units, hundreds on the left of tens and so on, it is not surprising that our brain breaks up long numbers into separate digits (so 1265 is decomposed to get 1, 2, 6, and 5), rather than as complete numbers.

    This means some people are making it all more complicated than it needs to be.

    Unfortunately, we don’t yet know how children learn to do this. They probably rely on this structure of the larger numbers in order to understand numbers such as 10,000. How they do so, is still an open question.

    The successor axiom
    There’s one question left: why is it that we all learn the same numbers; why don’t we each learn slightly different numbers? The answer to that question lies with the mathematical definitions of numbers, or axioms. The way we learn numbers, in order and by adding one, perfectly matches these axioms. Except that they don’t explain why our numbers all fit the most important axiom of them all: that there are infinitely many numbers. This axiom, known as the successor axiom, says that ‘for every number there is a larger one’.

    The difference between cardinal and ordinal numbers: cardinal numbers indicate the amount of items, in this case, the number of people; ordinal numbers indicate in what order an item occurs, in this case, whether a person is first, second or third in the queue.

    Exploring the implications
    This is not only a question for psychologists; philosophers have also long been wondering how we learn something about infinity. So, how do children manage this feat?

    One suggestion is that our knowledge of the successor axiom is based on the process of adding one. Children at some point realize that you can always do ‘plus one’ to get a larger number. Dr Buijsman believes, however, that children learn the syntactic structure of the numeral system, the place-value system e.g. hundreds, tens and units, whereby the leftmost digit carries the greatest value. Once they can interpret each digit as an amount, the next step is to recognise that a larger number can be produced by fiddling with the syntax: add a zero to the right, and you get a number that is ten times larger.

    Maybe children even use both methods. For example, they might employ ‘adding on one’ for numbers up to 100 and the structure of the numeral for numbers greater than 100. We’ll need more research to find out what goes on in the minds of children struggling with the unending numbers, but we now have two viable hypotheses to explain how it happens. And thus how we all manage to learn the same numbers.

    What are numbers?
    What, however, does this have to do with philosophy? One thing it tells us, is what kinds of things numbers are. Whether they are real or made up, you can wonder what a number is. And so philosophers have long wondered whether these numbers are first and foremost ordinal, describing order, for example first, second and third, or cardinal, describing amounts, such as one thing, two things, three things and so on.

    Mathematically, you can use these ordinal numbers to define the cardinal numbers: you have three people standing in a queue if the last person you counted was the third in line. Some philosophers have gone on to say that numbers are, above all else, ordinal numbers: a number is nothing but a position in an order. If this is true, then you can’t learn about those cardinal numbers without understanding order first.

    Dr Buijsman is passionate about public outreach.

    That’s something you can test, by looking closely at what we have already discovered about the way children learn mathematics. The numbers that we use can only be fundamentally ordinal numbers if children always learn the order before they learn amounts. In other words, we can conclude that if children don’t understand ordinal numbers when they learn the cardinal numbers then numbers we use are fundamentally cardinal numbers. And that’s what seems to happen: as far as we can tell, children don’t understand order when they learn the number one. A tricky philosophical question can, to some extent, be answered with empirical data.

    Understanding how we learn maths may help give us more effective
    teaching methods.

    Going back to school
    Many children and young adults think that maths is tedious, difficult and tiresome. They do not understand the value of being able to do some maths, particularly when the advent of the smartphone means that everyone has instant access to a calculator. Dr Buijsman’s educational outreach builds on his philosophical research and demonstrates how maths is increasingly important with advances in technology. He has already published a book (in the Netherlands) so that children can learn more about mathematics and how it is useful in real life through the characters’ adventures. A popular science book where he uses his own research to demonstrate why everyone should know some mathematics, even if they don’t use it every day, will appear in October. With Dr Buijsman as a guide, readers of all ages can learn the true value and wonder of mathematics.

    Why do some children struggle to understand numbers when others make it seem easy?
    Above all, numbers are these abstract things governed by strict rules. While I’m convinced that everyone can understand them, it seems that it’s easier for some to grapple with the abstractness and rigour that comes with numbers – they pick up strategies to work with them faster, or make the leap to the underlying structures more easily. It would be great if we can find out what it is exactly that makes things easy for these children, but for now it seems that practice, simple explanations and a clear sense of why we’re learning math helps best.

    You have already made several remarkable achievements. What should we expect from you next?
    There’s still a lot of philosophy to be done in this area, and I am now looking for a good place to continue that work. Next to that, I always keep an open mind for other opportunities – I’ll keep working with companies on our mathematics eduction and aim to continue my public outreach work. As for other activities, who knows. I read quite a lot of fiction, so I’d like to try to write a novel sometime, but there are no concrete plans for anything yet.

    References

    • Buijsman, S. (forthcoming). Learning the natural numbers as a child. Noûs, 1–20.
    • Buijsman, S. (2018) (forthcoming). Two roads to the successor axiom. Synthese. Available at: [Accessed 18/06/2018].
    • Assadian, B. and Buijsman, S. (2018) (forthcoming). Are the natural numbers fundamentally ordinals? Philosophy and Phenomenological Research. Available at: [Accessed 18/06/2018].
  • Research Objectives
    Dr Buijsman’s work looks at how we learn mathematics and our understanding of the philosophy of mathematics.

    Collaborators

    • Prof. Peter Pagin, Stockholm University (PhD supervisor)
    • Prof. Dag Westerståhl, Stockholm University (PhD supervisor)
    • Dr Bahram Assadian (co-author)

    Bio
    Stefan Buijsman (1995) obtained a degree in philosophy from Leiden University at eighteen. By the time he was twenty, he had a PhD from Stockholm University – the youngest person to receive one in Swedish history. His dissertation is about how people learn mathematics, even if they are not expert mathematicians. In collaboration with psychologists and mathematics educators he still works on these questions. He also develops new tools for mathematics education with different companies and has written two books about the use of mathematics in daily life.

    Contact
    Dr Stefan Buijsman, PhD
    Stockholm University
    Universitetsvägen 10D, 114 19 Stockholm,
    Sweden

    E: Stefan.buijsman@philosophy.su.se
    W:
    Twitter:

  • How do we learn mathematics?

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