A few years ago, I read a book on misconceptions in mathematics. The authors, Julie Ryan and Julian Williams, contend that many misconceptions result from overgeneralising. Ever since, I have noticed these kinds of errors on an almost daily basis in my teaching.

What do they look like? I might be teaching function identities, for example, and demonstrate that for the function

The following is true

However, a student might then assume that this is true *for all functions* or for a larger class of functions.

Similarly, all maths teachers have encountered students making the following mistake

And this can be understood as an overgeneralisation of the distributive property of multiplication, something that *is* true i.e.

I have recently noticed a similar misconception. The distributive property is only true for a linear factor, yet some students may write e.g.

I used to deal with these through explanation alone. I would discuss with a student why the maths does not work, maybe by expanding out the terms. I would focus, as many maths teachers would, on students understanding the mistake. Usually, when confronted with the logic of the error, most students recognise it as an error pretty quickly.

I still do this, but I have come to realise that I am perhaps engaging the wrong part of my students’ minds. Let’s borrow the model of ‘System 1’ and ‘System 2’ thinking that Daniel Kahneman outlines in *Thinking Fast and Slow. *The kinds of errors listed above are usually made in the middle of a more complex maths problem that is consuming attention and so it is likely that students are conserving limited resources and using instinctive System 1 thinking when they make these mistakes. By appealing to a logic argument, I am engaging their System 2 thinking, but this is not the kind of thinking that is making the errors.

The solution is partly lots of practice. However, drawing on the work of Engelmann and others in the way they structure units of Direct Instruction, I think a key part of the solution is also the use of non-examples. By explicitly and repeatedly teaching what is, *and what is not,* we give students a more instinctive feel for avoiding these misconceptions. And the easiest way to build up a bank of these non-examples is to look at the mistakes students make.

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I’m totally with you on this! As a psychologist, I have often discussed “concept formation” (often in the context of reasoning/classification/categorization tasks). When children have lower skills with these types of reasoning tasks, I know they will be likely to have difficulty handling certain abstract academic concepts. When teachers ask, I suggest defining the concept, describing the concept (characteristics/attributes), and providing examples and non-examples. I stress the importance of the non-examples. Otherwise we are keeping secret half the information that helps to understand the concept fully! Now I will use your excellent simple illustration (I love the labeling “this is not the thing”). I appreciate seeing you come at it from math curriculum examples that show where teachers need this in somewhat higher level material. Too often my examples come from much lower down in the curriculum / earlier stages of cognitive development. As we know, it’s particularly critical for some students to have this direct instruction. Best chance at helping disadvantaged student (perhaps lower ability or a specific problem with this type of reasoning) to manage challenging topics. Thanks!

I wasn’t familiar with the term “non-examples”, but it’s a good name for it. Common technique for language teachers as well (for obvious reasons).

There is a lovely serious of books for undergraduate uni maths. Counter examples in Analysis and so on for topology and probability. (https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=counter+examples+in)

Perhaps Greg needs to write counter examples in arithmetic and high school algebra.

I’m not generally too fond of teaching non-examples, because for the strugglers you are likely to merely overload them with quantity and variety. I am always afraid that if I write (a + b)² = a² + b² that students won’t hear the injunction to not do it. I’ve tried giving lists of mistakes and they seem to make little difference.

So I’ve moved to explicitly stating the line. So when I teach logs I hammer it in explicitly that log a + log b = log ab is not a rule under *any* other circumstances. Then their minds are “this is a log rule only”.

Recently I’ve tried a different tack, by explicitly teaching functions at Year 12. That gives me the language to describe the issues involved. I can actually say that f(x) + f(y) = f(x + y) for no cases at all. It’s a detour, but a powerful one.

As is usual, those who are on their own declarations huge on “understanding” mathematics avoid putting functions into their curriculum or their teaching because they are “too abstract”. Despite them being a key tool in actually understanding the leap from addition and multiplication to other functions.

“I am always afraid that if I write (a + b)² = a² + b² that students won’t hear the injunction to not do it.”

I used to have a head of department who agreed with you. He would shudder at this blog post because he cannot stand to see incorrect statements with an equals sign and not a not-equals sign.

I am more relaxed, but I do think it is a non-ideological point on which maths teachers reasonably disagree. Wouldn’t it be great if education researchers would investigate some of the genuine questions that maths teachers have?

I couldn’t find the “not equals” on my phone Greg.

However, my experience is that students have not been taught it anyway, so writing it is to write something they don’t understand. And my version of Explicit Instruction does not allow me to write things they don’t understand.

I’m not that fussed, since I did what your blog suggested for a decade, but I thought I would offer a mildly contradictory position.

The main thing is, and this is where I fall out with discovery or indirect methods, is that teachers need to explicitly work at correcting known problems. And do so at the earliest opportunity. Students do not work them out by themselves. They will make false conclusions unless steered away. Allowing them to “discover” their errors later is to waste good teaching time.

Brilliant article! I teach just about all the maths from prep to senior calculus to pre-service teachers (PST), and often this is by exploring non-examples. It is very important that the PST are clear where the failure in the logic is. With kids you also have to be careful that they do not learn the non example.

For the past 7 years I have had the enormous privilege of observing practitioners such as Bernard Murphy and Simon Clay teach yearly 8-day courses titled: Teaching A-level Mathematics (TAM). Their approaches to teaching are always to cause their ‘students’ (i.e. Maths teachers) to derive or make sense of formulae for logs, trig, identities, calculus etc. They do this through both the lo-tech and hi-tech resources they use, the discussions they organise, the questions they ask, the different groupings and the tasks and problems they pose. If they see a mistake they never re-teach anything but instead pose other questions or ask the person who has made a mistake to either explain their working or justify in a pair or a small group an incorrect result.

I offer this reply because I wonder how practices such as these fit within the domains of Big DI or little di.

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Reblogged this on The Echo Chamber.

When I was a young mathematician I used to practice for speed. It wasn’t something anyone taught me, I just did it. I’d take a question – often one I had done before, and sometimes one I knew quite well. I would try to do it as quickly as possible. There were two different outcomes which helped me improve significantly – the first was that I explicitly noted the kinds of mistakes I made when I was working quickly. These did not come in huge numbers, so I was not dealing with five or six things, but one or two. These were things I needed to eliminate – I knew I had the knowledge to do it correctly, and I had become conscious that if I was working fast there was a risk of missing the application. I came to the (personal, anecdotal) conclusion that the mistakes characteristic of speed were rather different from other kinds of mistakes I made. The second was that I very occasionally saw a different approach to a familiar problem, and that helped too. Practice with familiar questions also helped to embed patterns of reasoning. I sometimes wonder whether there are quite a lot of pupils who work more slowly than they might for fear of getting things wrong, and whether explicit practice for speed might be done in a way which would improve confidence – and whether naming errors as (likely) errors of speed would help learning.