# What determines how good we are with numbers?

Back when I was a psychology researcher, I used to successfully use maths quite a lot, largely for statistics. However, I also recall teaching a class where we’d been counting up answers to some kind of questionnaire. I had to do a fairly basic sum like 63 + 74 = ? and I suddenly couldn’t do it. This has happened to me on and off throughout my life. Some other people experience this kind of difficulty with numbers constantly and at a much greater depth than I did, and some people will never have a moment’s hesitation even with complex mental arithmetic. Why?

*It me, in front of a class*

The answer is a two-parter. All of us exist on a continuum from *maths genius* to *I don’t get numbers at all*, because humans tend to vary in their capacity to do intellectual tasks like mathematics, reading and drawing. There are also some people who have dyscalculia, a learning difficulty which makes it extremely hard to understand numbers.

# Why is there a continuum?

This is a very complex question because there are a lot of different factors involved.

These might be characteristics inherent to you, for example how good your working memory is. Working memory is the part of your memory that holds and manipulates temporarily-needed information. When you’re making a cake, your working memory might contain the thoughts: the recipe says I need the baking powder, where do we keep the baking powder, I’m sure this shelf didn’t have tentacles a moment ago, oh damn I forgot to close the hell portal.

While I am closing the hell portal, please do the sum 842 - 57 in your head.

…You used your working memory to do this sum. You might have pictured the numbers written in front of you, as though you were writing the sum on paper, or you might have said the numbers to yourself as you were calculating, or if you’re like me, you picture the numbers as a line in space and kinda shuffle along the line.

Imagine if you had a much better working memory than you do – you’d be quicker, right? Or if you had a much worse working memory, you might have to write down the sum so you had an external reminder of what step you were on. This tells us that because we differ in how good our working memories are, we also differ in how easily we can do maths.

However, factors like these that are to do with deep individual differences are probably not the main driving force – Tosto and colleagues [i] looked at how similar identical and fraternal twins are in their basic number abilities, and found that most of the variation in how good an individual is at these things can be explained by environmental factors, not genetic ones.

So let’s consider those environmental factors, like your age, your gender, your socio-economic status and your ethnicity. Please note that it’s not that *being* a certain age, gender, class or ethnicity explains your maths ability, but rather the *social context* of belonging to a particular group, e.g. sexism and racism preventing access to learning – it’s not the genes that explain maths ability, but the environment that you (and your genes) exist in. Let’s have a look at age and gender as examples.

Since 1995, the International Association for the Evaluation of Educational Achievement (IEA) has done worldwide studies (Trends in International Mathematics and Science Study, or TIMSS) on fourth-graders’ and eighth-graders’ mathematical abilities every four years. All the results I discuss here come from the 2015 study [ii]. First off, children typically scored higher on the tests in 2015 than they did in 1995. It’s hard to say what might be causing this – perhaps teaching styles are changing [iii], or caregivers are have changed how (much) they talk about maths with children [iv].

*“Do you know how many hours of uninterrupted sleep Mummy has had since you were born, Celeste?”*

So, the specific time period in which you went to school probably contributes to where you are on the continuum. What about gender?

The IEA reports that in most countries in 2015, fourth-grade boys tended to score higher than fourth-grade girls, but this gender difference disappears by the eighth grade. Though women used to be worse at maths than men, there are no longer any gender differences in maths abilities for adults [v], which is great news considering that in many countries there’s been a long and sexist history of it being “improper” for anyone other than men to be good at maths and consequently a long and sexist history of men being “better” at maths than anyone else when it would have been more accurate to say they were “getting more education” in maths.

However, there is still the problem of stereotype threat: when you are reminded that you belong to a group that is stereotypically bad at a particular task (like women are “bad” at maths), then you’ll respond by becoming worse at that task [vi]. This is one of the reasons that it’s really hard to pick apart why the continuum exists – when you do a maths test, you might not be performing to the best of your ability because of things like stereotype threat. Ideally, we’d measure everyone’s maths performance on their absolute best day, but it’s very hard to tell what’s going to be a good day and even if it were easy to tell, what a logistical nightmare that would be.

# Dyscalculia

Some people – around 7% of us – find maths difficult because of a developmental disorder called dyscalculia (you might also see it called *mathematical learning disability*, but they’re the same thing). Dyscalculia is specific to maths ability: you might be otherwise very intelligent and have access to good all-round teaching, but struggle to learn maths. In this way, it’s very like dyslexia, where it can be very difficult to read not only compared to other people but also *compared to any other activity you are doing*.

It’s not really clear why dyscalculia happens. Part of the problem with figuring out what’s going on is that there seem to be lots of different systems that deal with numbers in the human mind. Let’s take a quick look at how Butterworth [vii] breaks these down:

## The approximate number system

This allows us to estimate roughly how many of something we can perceive. For example, you can probably make a guess at approximately how many piranhas are in this picture without counting.

*I counted and the number of piranhas is Too Many*

Interestingly, the piranhas themselves can probably also estimate numbers in the same way as we do [viii], though they can’t communicate that estimation to us (I think (what if they can talk and we just don’t know (oh wow, maybe there are dyscalculic piranhas))).

There might indeed be dyscalculic piranhas, if Mazzocco and colleagues [ix] are correct and what is underlying dyscalculia is a problem with the approximate number system. They asked their participants (14- to 15-year old humans, not piranhas) to do a bunch of different maths tasks, including one which taps the approximate number system: looking quickly at a group of yellow dots and blue dots, then saying whether there are more yellow dots or blue dots. The teenagers who had dyscalculia also did poorly on this task. However, both Mazzocco and colleagues and Butterworth point out that not everyone who has tested the approximate number systems of people with dyscalculia has found this association, and of course having difficulties with the approximate number system might be a *consequence* of dyscalculia, not a *cause*, because if you have dyscalculia then you might well have less practice with numbers than someone who doesn’t have dyscalculia. Well, OK then, let’s look at another system that humans use to think about numbers…

## The small number system

You know how I was saying that you can guess the approximate number of piranhas or soft toys or, I don’t know, grapes in the office fruit bowl? If it’s a very small number – under about 5 – you can actually tell *exactly* how many things there are at a glance. This is called subitizing, and it’s pretty cool because it tells us that humans deal with small numbers and large numbers in different ways. Do people with dyscalculia have trouble with *this* system for thinking about numbers?

Butterworth points out that it’s kind of hard to tell, because children with dyscalculia have problems with learning the whole range 1-9, not just numbers up to 5 – but if you can’t count up to 5 because your small number system is having trouble, you’re probably not going to be able to count up to 9 anyway. So that’s probably not going to get us anywhere. Let’s try one last system!

## Numerosity coding

There’s a picture of some pencils below. I’d like you to count how many there are.

Great! You have just found out the *numerosity* of the pencils – that is, the number of items in the set of things you are counting. You might remember learning to do this as a toddler, or have recently taught a toddler to do it. As an adult, you know that numerosity can come ‘unstuck’ from the set – for example, it’s perfectly possible to use that number again to describe how old someone is, how many days of holiday you have left to take this year, or how many dogs is the perfect number of dogs (though you would be wrong about the last one because the perfect number of dogs is infinity).

*Impromptu dog montaaaaaage!*

Now, imagine you couldn’t figure out numerosity correctly, no matter how many or how few items you had. It would disrupt even the fundamentals of maths, right? You wouldn’t be able to tell how many of anything you had, and you’d also get knock-on effects when you tried to do maths: 2 + 3 could just as easily be 17 as it could 5. This is what Butterworth thinks is the core problem for people with dyscalculia, and there is a decent amount of evidence to support him (for example, children with dyscalculia have trouble with figuring out numerosity in a way that children without dyscalculia do not [x]).

Hang on, hang on, hang on. I feel like there might actually be more than one thing that’s causing dyscalculia, because in psychology when you ask, “What causes this behaviour/change/difficulty?” the answer is almost always, “Many different things!”

Sure enough, children with dyscalculia have problems doing tasks that are related to the approximate number system, but also the small number system and to working memory [xi]. On top of that, it seems that if you have dyscalculia, it may well be possible to reduce its effects through games that train you to be better at counting and manipulating numbers [xii]. (e.g. Butterworth & Laurillard, 2010).

So, if you are bad with numbers, don’t despair: it *is* possible to get better. If you’re struggling with numbers, or you’re teaching or parenting someone who is, Butterworth and Laurillard have made their number games available for free here. And if you are good at maths, don’t be too smug: it’s largely the environment you’re in that’s made you that way!

# References

[i] Tosto, M. G., Petrill, S. A., Halberda, J., Trzaskowski, M., Tikhomirova, T. N., Bogdanova, O. Y., ... & Plomin, R. (2014). Why do we differ in number sense? Evidence from a genetically sensitive investigation. *Intelligence*, *43*, 35-46.

[ii] Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). *TIMSS 2015 International Results in Mathematics.*

[iii] Mercer, N., & Sams, C. (2006). Teaching children how to use language to solve maths problems. *Language and Education*, *20*(6), 507-528.

[iv] Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: relations between parents’ input and children’s cardinal‐number knowledge. *Developmental Science*, *14*(5), 1021-1032.

[v] Hyde, J. S. (2014). Gender similarities and differences. *Annual Review of Psychology*, *65*, 373-398.

[vi] Spencer, S. J., Steele, C. M., & Quinn, D. M. (1999). Stereotype threat and women's math performance. *Journal of Experimental Social Psychology*, *35*(1), 4-28.

[vii] Butterworth, B. (2011). Foundational numerical capacities and the origins of dyscalculia. In S. Dehaene & E. Spelke (Eds.), *Space, Time and Number in the Brain* (pp. 249-265). London: Academic Press.

[viii] Piffer, L., Agrillo, C., & Hyde, D. C. (2012). Small and large number discrimination in guppies. *Animal Cognition*, *15*(2), 215-221.

[ix] Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). *Child Development*, *82*(4), 1224-1237.

[x] Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., ... & Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. *Cognition*, *116*(1), 33-41.

[xi] Andersson, U., & Östergren, R. (2012). Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities. *Learning and Individual Differences*, *22*(6), 701-714.

[xii] Butterworth, B., & Laurillard, D. (2010). Low numeracy and dyscalculia: identification and intervention. *ZDM Mathematics Education*, *42*(6), 527-539.