A simple tweet recently sparked a series of articles about how we visualise methods of addition.
“What happens in your head when you do 27+89?” was the question, and over 14,000 people responded, often surprised by what others had to say. “I’m amazed at how differently people experience mathematics,” was one comment.
The first thing to say here is the lens with which we view this is crucial. Spinning it as funny to say “I would literally never do this” or “I would always use a calculator” or just presenting it as a conflict between people on the internet is one way to look at it; another would be to see the joy in the diversity of approaches, with no one way being necessarily privileged over another. If you think about giving people the task of drawing a self-portrait, or explaining how it rains, you would expect a real range of ways of approaching those problems – some simple, some complex, some using pictures, some using words and so on. As mathematicians, we come across stories like this all the time and often greet them with a wry smile – it is no surprise to us at all that “people have so many different ways of doing it.” In fact, it is a central part of mathematics that multiple routes lead to the same solution and that people think about the same sorts of structures very differently (because human brains are very different from one another, hurrah!).
Despite people often saying “maths is so black and white” – and often, though not always, there is only one correct answer – the coexistence of different ways to think about maths is pretty normal. Mathematical methods are like tools in a kit – part of learning maths is collecting them and storing them neatly away in your toolbox, so that when you come across a new problem you have different options to choose that might fit the particular situation. You can also try one method but use another to check it – like using both glue and nails together on a wood joint.
In this way, we can see maths is really quite creative. One of the most powerful things you can do with anyone learning maths is to ask them to try and come up with lots of ways of finding an answer, and of course many maths teachers do this sort of thing regularly. Helping students see the structural similarities but the differences in notation and computation – and which features of the maths they help bring forward or push back – can be an enjoyable and key part of maths teaching. It can also be wonderful (or frightening) when a new method is shown to us that we have never seen before!
Often when we see maths in the media there seems to be a misunderstanding around the value of using different methods – some people respond with comments like “Everyone should just use column addition; I don’t know what the fuss is all about.” While it’s true that there are often methods that seem more efficient or obvious, actually many of the different approaches that may be explored in maths lessons are about exposing structure and are intended to complement rather than replace. We all recognise the frustration of learning to use something without understanding why it works – like an Excel formula in a spreadsheet, for example – because when it breaks it’s very difficult to see what’s wrong or how to fix it. Using different methods flexibly can really help people “get under the hood” of what’s happening mathematically, so they can spot pretty quickly if something’s gone wrong or switch to another way of “seeing” the problem if it’s helpful to them to do so.
For those who fear these sorts of sums and certainly would not want to expose their thinking publicly, it is worth saying that you can break down this question into parts to make it less terrifying. Here, adding these two numbers involves a collection of important skills:
- decoding written numbers (reading the sum aloud as “twenty-seven plus eighty-nine”)
- understanding place value (that the “2” is actually standing for “twenty”, for instance)
- understanding that the written sum is asking you to do something, which is add the two numbers together (as opposed to some written mathematics which is about simply making a statement instead)
- Seeing that when we add these particular two numbers, the units or ones are going to add to more than ten, which means some kind of movement across a boundary into the tens is going to happen (compared with 31+22, which might feel more straightforward), and also that this is going to cross over into the hundreds, too, because even with just the tens to add they already make one hundred. It’s not crucial to spot this beforehand but with practice you get to see it more easily and it helps with the decision about which strategy to choose.
And that’s even before we get to the adding. Then, how you actually perform the adding is often about how you “see” numbers and their building blocks. If you were asked to draw 27, what would it look like? When you put the 7 and the 9 together, how do they “fit”? All these questions start to touch on the way we visualise numbers, which is a fascinating thing to research (for example, when you think about numbers on a line, which way does it go in term of smaller numbers to bigger?). Some people, in response to the original question, changed the numbers before they even began, happily using an idea of “compensating” which comes from understanding that adding and subtracting are the same type of thing, and you can use them flexibly together as long as everything “balances” in the end. This is one of the fundamental ideas of algebra, too!
Finally, there are also some interesting things happening with the question “What happens in your head when…” We could equally apply this to talking about seeing shapes (“What happens in your head when you visualise “triangle”?”) or language (“What happens in your head when you translate “hello” into French?”). The way people “see” things in their mind’s eye is very varied and the way they communicate this process to other people is imperfect – sometimes it can be quite hard to describe a dream to someone else, for example. Another recent viral tweet asked people whether they actually see colours behind their eyes when they think about them, and many people were surprised that other people did not share what they assumed to be the experience of every other human!
How would you do 27+89…
- Without writing anything down?
- In front of a class?
- If you could only draw pictures?
- With a learner who felt anxious?
- If the context was money?
- In an assessment?
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