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Finding and fixing misconceptions

on April 18, 2012

Right, I’m back.  It was a very restful Easter break, due to me managing to pull/sprain an intercostal (between your ribs) muscle from coughing too hard.  As it’s not the kind of muscle you can ‘rest’, it rather jolly hurt, and I didn’t get a lot done 😦  The worst bit is that the cough is still going! (new school = new germs) grrrr.


This week I have mostly been addressing misconceptions.  

I asked my Y9 (level 6-7) group to do the sum 700-182 on the board.  I even set it out for them.  It took 4 people, correcting each other’s mistakes, to finally get the correct answer.  They simply couldn’t cope with borrowing when there were two zeroes.

The original plan for the lesson had been to get them to look at any method for add, subtract, multiply or divide, and tell me WHY it works.  Why is that the algorithm that we follow, and can you explain to me why we do it that way?

The new plan obviously became for all of them to do borrowing as their method!  I gave the example of column addition and carrying to show them what I meant.

If you’re adding 312 + 59, imagine it’s in money.  HOWEVER there has been a new rule introduced that says that only pound coins, 10p pieces, and pennies are allowed.    So you have 3 pound coins, one 10p and 2 pennies.

If you add five 10p’s and 9 pennies to that, you’ll have 11 pennies, six 10p’s and 3 pounds.    Who wants to be given a whole load of pennies though?!?  It would be much better to change 10 of them for a 10p piece, and then it’s easier for everyone.  That’s where the little ‘carry 1’ comes in, as it reminds you about the extra 10p.

(This works better in numbers on my board…!)

The idea was to get them using the money analogy to identify the fact that borrowing in maths is just the equivalent of changing a 10p for 10 pennies, so you have enough pennies to ‘pay’ with.  

I told them they had the rest of the lesson to come up with their own explanation, and then the following day, I made them do a presentation in their pairs/threes to the rest of the class on what they had come up with.    I was pleasantly surprised by many of the presentations, as they had clearly ‘got’ it. Some obviously just got up and showed us the method (there’s always one) but at least they hopefully addressed their issues with the method in doing so!

I finished with showing them the original problem again:  700-182.

Imagine it’s £7 take £1.82.  How do we get change?

Lots still did borrow from H to put into T column, then borrow from T to put into U, but quite a few came up with the strategy of changing one pound coin into 9x10p and 10x1p  🙂

Y8 did a variety of methods, and we addressed lots of confused ideas about things like how to deal with remainders when you divide, so as to end up with a decimal answer instead of having to write ‘remainder 4’. Also,  I think I finally get Napier’s Bones now, so a useful lesson on lots of levels!

With my level 2-4 Y7 group, I showed them the money analogy, and got them to practice lots of sums until they were happy with it.  Some needed to have a different way of thinking about it, and my TA was explaining in terms of packets of sweets (10 in a pack – if you don’t have enough to give out, you have to open a new pack/box of packs) and ‘borrowing from your next-door-neighbour’.    They all had managed by the end to successfully answer questions, going up to 3 digit numbers, which for them is a big deal.  

It did scare me though, that my students could be so confused about a method they’ve been ‘doing’ since primary school.  Wonder what else I’m taking for granted that they can do, when they actually can’t….? 



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