## 17 September 2016

### The Deadly Sins of Maths

A Twitter conversation with Ben Ward got me thinking about a display idea...

There are some misconceptions and mistakes that come up time and time again. Last week I set a lovely Don Steward quadratics homework for my Year 11s and was dismayed to see a number of them do this:
(x + 2)2 =  x2 + 4
When I returned the homeworks I talked to the whole class about this common mistake. I told them that it makes me cry every time I see it so they must never ever do it again! When I asked them why it was wrong, they were able to explain what had happened and tell me the correct way to do it (phew!).

When I next spot common misconceptions in my students' work, I'm going to tell the perpetrators that they've committed one of the deadly sins of maths, and point them in the direction of a classroom display. Making a big deal about these mistakes (in a half-jokey way that will stick in their heads) is a good way to lower the risk of students repeating the same mistake in an exam!
You can download my posters here (there are 20 posters to pick from - you may not agree with all my choices). I've also made a US-friendly version (using the word math instead of maths) which you can download here.

Some of these posters will make students stop and think - it might not be obvious to them what the mistake is.

In our teaching we should try to preempt common misconceptions - this is where clever questioning and well-written resources come into play. But sometimes we can't see mistakes coming - this is why effective marking and feedback are essential.
These days marking doesn't have to be handwritten - websites like Hegarty Maths can do it for you.

If you're looking for resources for highlighting and tackling common misconceptions, here are a few starting points:

I have a misconceptions page here that links to various websites and resources relating to common misconceptions.

If you use the 'deadly sins' idea, I'd love to hear how it goes!

Two days after creating this display, it was up on @jase_jwanner's classroom wall.  He has also suggested creating a students' display called 'Maths Heaven', where they write down the correct versions eg (-5)2=25.

1. Oh, some great mistakes here!

Lisa at eatplaymath made an awesome sheet for basic operation sins http://eatplaymath.blogspot.com/2015/09/dont-kill-kitten-please.html

I have one for fractions/trig identities
http://www.megcraig.org/2015/10/24/evolution-of-a-theorem-or-how-to-save-all-the-animals/
and exponents
http://www.megcraig.org/2014/10/05/mtboschallenge-1-2-3-sunday-summary/

My favorite is you kill a unicorn if 2^-3 = -8. "But Ms Craig, there are no unicorns." "I KNOW. It's all because of math students who wouldn't learn negative exponenets!! We must be diligent and save the 3 remaining ones that live in a secluded meadow in Ireland." ;)

2. I like the idea of highlighting common errors, but I don't like the sin terminology, nor do I like the "this kills a kitten/unicorn" part.

I'm not one of the "we only learn through mistakes, so let's make as many as possible" brigade, but I do think that what is more important is to learn how to catch ones errors than to try to never make any at all. And the "sin" and "this kills a kitten" language puts the emphasis on never making a mistake rather than on catching them when they are made.

So I call these "gotchas" and describe them as traps for the unwary. Thing is, eventually we all get caught out by some trap or other. So I'd rather use these to learn about traps, how to spot them, yes how to avoid them, but also how to get out of them when you fall in.

With that in mind, I encourage students to learn their own predilections. One student might find that they are a bit slap-dash with denominators and that they keep creeping into the numerator. So while I encourage them to not make that mistake, I also encourage them to look back over their work looking for that particular mistake.

I actually think that this is a more reliable strategy for learning not to make that mistake. Once they get into the habit of looking back over their work to see if they've made a particular error, they start catching it earlier and earlier, until they effectively aren't making the mistake at all. It's not that they *aren't* making the mistake, but just that their internal error-correcting code catches it before it gets onto paper.

But if we say "mistakes are evil", we don't learn how to deal with them when they inevitably occur. There's no such thing as bug-free code, and even published mathematics contains errors (I know, I've contributed a few myself!). Learning how to deal with ones own errors is very helpful for learning how to deal with the mistakes of others.

1. Thanks for your comment. I'm certainly not trying to say 'mistakes are evil' - we all make mistakes. As I say in the post, it's a jokey way of trying to get students to stop and think when they make one of these common mistakes. I've found this kind of thing (eg the kitten killing joke) effective in the past for engaging students in conversation about misconceptions. Of course it may not work for all students (and teachers).

2. I get that it's meant in a jokey way. But sometimes jokes don't come across how they're meant. I think that Maths has an image problem (and I will admit that I may be hyper aware of it) and part of it is that Maths is all about "right answers". I really want my students to see the messy side of maths, and part of that is being able to make mistakes and tidy up afterwards.

Real Mathematicians^TM don't worry about signs until they need to. Many, many calculations are done with "plus/minus" signs to avoid having to think about them. Knowing when it's okay to say "I'll sort that out later" and when it's time to do that sorting out are all key skills for mathematicians. I tell my Y10s that they must *never* divide by zero, but just today let my Y12s in on the secret that sometimes you can (as in, taking derivatives). I just think that "sin" carries with it the implication that we should put "Stay away!" signs, but each of these common misconceptions should really be provoking us and the students to examine it carefully - and from what I've read here and on the linked sites then that's what you and the rest want.

I really am just quibbling about terminology, but again that's a key skill for a mathematician! Names are important because they convey meaning[1].

[1] Although whoever introduced the term "hedgehog" into the mathematical literature clearly was having a bad day.

3. AAARRRRRGGHHH!!!! You don't divide by zero when taking terivatives. you deal with limits approaching zero!!!!!!!

4. I'm guessing this was a reply to my comment above.

Sorry, I forgot to mention that I was working in the ring R[ϵ]/ϵ^2 so that Newton and Leibniz's ad hoc approach to calculus works pretty much as-is. I find this far more accessible to students than Cauchy's ϵ-δ definition which, if I were to use the language of limits, I would of course spend the required five weeks to ensure that my students understood all its intricacies and the subtleties of removable limits.

5. These minor mathematics mistakes are actually disastrous. And most of the students commit such mistakes. The core reason behind this is unclear concepts and understanding which lays the base. Hence strong concept base is very much required to escape from these blunders.