Damaging short cuts
Yesterday, my year ten class were doing Standard Form (Why did we drop the word index? Standard Index Form is much better!) During the lesson it became very apparent that they were nowhere near fluent in the use of negative numbers, so today I taught a lesson on directed number to fill in the gaps.
The first few examples were of the form – a + b, or a- b, and were easily dealt with, but then I gave them an example which blew their minds. The example in question was:
“-7 – 4”
I talked through it to nods of agreement but then I got this:
“It can’t be -11, there are two negatives and Miss so and so said that two negatives always make a positive”
AAAAAAAAARRRRRRRRGGGGGGGHHHHHHHHHHH!!!!! went my internal monologue. Others in the class murmured agreement. A long discussion ensued, about where this fallacy comes from and why it’s entirely wrong. We got there, and now the class have a deeper understanding of negative numbers and how to deal with them, however, I think this could have been entirely avoided if this short cut just wasn’t taught. If they had just been taught how to handle negatives in the first place.
This isn’t the only topic that falls foul of this. I’ve written before about “BIDMAS“, Tina Cardone (@crstn85) has put together a superb book on a variety of similar things and just yesterday Michael Tidd (@michaelt1979) tweeted “Can we all just agree to stop using the crocodile inequalities analogy”.
This is one that infuriates me. Every year I have to unteach this because a number of pupils have quite understandably changed the story in the head to “the big number eats the little number”. This seems sensible, as a big crocodile would certainly be more likely to eat a smaller crocodile than the other way round.
Why can’t we just teach the concepts and forget about the shortcuts? They are more of a hindrance than a help!
cavmaths 
Reblogged this on The Echo Chamber.
I see it both ways. Many children struggle with algorithms because of working memory issues. There’s no doubt that conceptual understanding leads to better retention and problem-solving skill but children with even mild learning disabilities need various accommodations to jog their memory. Sometimes I model a mnemonic for them and ask them to devise their own like a game. My credo in teaching is “whatever works!” Of course I completely agree that shortcuts without understanding what lies beneath will often fail but imo there’s a difference between teaching for understanding and teaching for performance. Sometimes reality dictates the emphasis.
By the way did your students respond better to rewriting it as (-7) + (-4) to eliminate the confusion between “–” as subtr vs “the opposite of”?
My pet hate is “half it and half it again” to find a quarter without understanding that “halving” is dividing by 2 … So they could just divide by 4 which would make finding 1/3 or 1/5 etc so much easier as they could see the link between fractions and division.
I agree wholeheartedly. I had my 4th graders read from left to right just like a regular sentence once they inserted their sign. It helped even if it didn’t undo the crocodiles.
At the very least we need to think very carefully about the glib phrases we choose to get them to memorize. But I agree completely with the negatives thing. Trying to teach multiplying negatives today, we wrote things like 1*-1=-1 and -1*-1=1, and I even said “I’m not going to write ‘two negatives make a positive’ because that’s just far too vague, and we already know it’s not true when adding”.
I quite agree, Cavmaths. Here’s another one for you: the triangle!!! You know the one I mean, for speed, density, etc etc… AAARRRGGHHHHH!
Yep, the formula triangle. The bane of my life.
I don’t think the formula triangle is like the other ‘tricks’ here, in that it’s a good stepping stone towards rearrangement. I still don’t see what the problem with it is. (Oo, this blue touchpaper seems to be alight. How did that happen?)
And that’s why its in the category “Colin gets it wrong”! I think its the fact that its often used instead of rearraging, and is part if the reason people can get good grades on GCSE without the algebraic knowledge required to tackle A level.
As dedicated math educators we naturally respond to mnemonics and ‘mindless’ shortcuts with disdain. We see the bigger picture and know that such expediencies do not lead to understanding mathematics and, in the end, these devices will hamper mathematical growth. But we also need to remind ourselves how WE survived in school with subjects that were NOT our strengths.
Balance is usually the best approach to these kinds of issues. I have no trouble allowing students to use well-established devices or even better to devise their own, provided they also demonstrate conceptual understanding and recognize the pitfalls in these strategies. For example, in formulas for slope, we might suggest ‘y’s rise” or “if you’re y’s you rise to the TOP’! These are somewhat different from the other ‘bad’ shortcuts discussed but they are survival techniques for students. We have to remind our ourselves that we are not only teachers of mathematics but also teachers of children and other humans!
So you’re saying students should only have one way to approach a problem of a particular form?
How is knowing you can use the triangle for a particular kind of equation different from learning a bunch of derivatives rather than doing proper calculus with limits?
It’s the ones don’t know they can use a triangle to solve a particular type of equation that i have a problem with.
So, if algebra was ever misapplied, we should ban algebra?
That’s a ridiculous analogy. Algebra isnt a one function trick that means you can avoid understanding. The analogy doesn’t even make sense.
Did you actually read the post on formula triangles? As i clearly stated in it that its their misuse I’d like to ban, and I have no problem with their use as an aid to understanding.
Yes, I did. And you keep blaming the triangle rather than its misuse, which is like blaming petrol for car accidents.
“I am fine with people who understand algebra using them as a shortcut to save time (although how long does it take to rearrange them properly? You must save milliseconds!) , I’m fine with teaching them to weaker students who have tried to learn algebra but are prone to mistakes after they’ve been shown how to rearrange them algebraically.
What I’m not fine with is the “do it like this and don’t worry about how it works” use of them. Especially when the learners in question want to go on to study Maths at A Level and beyond, it could damage their chances.” – I think you must have either missed, or purposely ignored, these paragraphs then….
OK – so a smallish subset of your A-level students (presumably a smallish subset of all of the maths students at your school) have a problem because of it. I agree, in an ideal world, everyone gets algebra straight away, but I’d be surprised if you lived in such a world; people come at it from different directions, and – as I keep saying – the triangle can (and should) be a step towards rearranging.
Meanwhile, if people are getting onto your A-level courses without adequate algebra skills, I think that’s a problem with the course entry requirements (or possibly with the transition period) rather than any particular method.
The “do it like this and don’t worry about why” method is pretty much how calculus is taught at A-level (at least in my experience of students coming to me). That’s adequate for A-level, unless someone’s especially interested in taking it further – in which case, the memorised methods can be used as a peg to hang the limit definition on.
As I alluded to earlier, i do think there is a wider problem with the gcse and how you can gain a good pass while not being competent at algebra. I hope the changes to the curriculum address this. I dislike the do it this way end of. Method. And i wouldn’t use it for anything calculus included.
Interesting, how do you teach the quotient rule?
(Oh – and I completely agree about GCSE, especially with the two-qualifications point. If you’ve got a qualification that’s supposed to measure how competent someone is as a numerate member of society, how capable they are of advanced maths, and how good the school is, it’s no surprise that it fails at all three.)
OK, I can come up with better examples, if you like: if it was ever misapplied, we shouldn’t learn to multiply by 5 by halving and multiplying by 10. If it was ever misapplied, we shouldn’t learn to get rid of a square by taking the square root of both sides. If it was ever misapplied, we shouldn’t learn that the angle opposite a diameter is a right angle. If it was ever misapplied, we shouldn’t learn to check for divisibility by 9 by adding up the digits. If it was ever misapplied, we shouldn’t learn Pythagoras,because it’s just a special case of the cosine rule. I could go on…
I don’t really see how they can be misapplied in this context? I am talking about peopke using them ti get the right answer without understanding what they are doing, i can’t imagine that applying to any of your examples.
I’m completely perplexed. You seem to be happy with people using the wrong method to get the wrong answer, but not with them using a ‘wrong’ method (which I contend is a perfectly good method) to get a right answer.
Students misapply – say – Pythagoras all the time. They apply it to non-right triangles. They add when they should subtract. They don’t know why Pythagoras works, they just know that a^2 + b^2 = c^2.
Students being wrong, ie applying Pythagoras’s theorem to a non-RAT, is part if learning, you explain their error and its a learning point. Students scoring a B at GCSE without understanding algebra because they’ve used this type if short cut means they struggle heavily when they attempt a level maths. It begs the wider question, “Whats the point in teaching them maths?” your thinking would imply “to get a piece if paper with a letter a,b or c next to the word maths on it”, which sadly is the view of a lot of people. I would argue it should be to teaxh an understanding of maths and inspire the next generation of mathematicians. This debate in itself is a particularly good argument for splitting the gcse into two. A functional, numeracy qualification for the “get a bit of paper to show you’re prepared” side, then maths for the fun stuff.
I also try to avoid “a^2 + b^2 = c^2” for tge very reason you imply here.
I think the gist of my point is a) the triangle isn’t bad in itself, and b) students coming to A-level generally have MUCH bigger problems than the triangle (e.g. I grovel pitiful thanks to any student who deals with fractions properly first time without asking me “do I need to…”)