Can someone please tell me why division is taught in such a confusing way in year 3?

[Divideandrule]

I have been polite in saying confusing, I really mean bloody stupidly. DD has come home confused as they have to use number lines and do repeated subtraction or something? She was trying to explain it to me and I didn't get it. (This is the child who enjoys and is good at maths, totally gets place value etc, in high group).

So, I showed her the bus-stop method for the same calculations. Eureka moment and a big smile on her face.

Do I have a word with her teacher about this? I get the need for some methods working for some children and different ones working for others. Surely if she understands the way I showed her - incidentally we had remainders with numbers so had to carry - she should go with that?

Anyone?

@ thetasigmamum

I agree with your comments. I know plenty of scientists and mathematicians at university who totally deplore the new primary school methods. For the reason that they are so limiting. They do (possibly) enable most children to get to basic level arithmetic (enough to reach level 4 at primary). But they are really limiting when you try to move on to higher order maths.

I fear this is why we get rave reviews about how good maths teaching is in our primaries, and yet we still get lousy results for most at the end of secondary school.

The problem is that primary (basic grounding in maths) is taught by people (in the main) who never really got maths at school themselves. They have now learnt these new methods that make sense to them (in their basic level) and help them do their job - which is to hit target x at age 10/11. They are busy making "able mathematicians" at 10 who are happily jumping along their number lines, and chunking and grid multiplying to their hearts content. But that does not really equip them with the higher (or more basic if you like) level of understanding how numbers work that you need to take maths to a deeper level. They do not (in my observation) learn how to play around with numbers and see how they relate (because the teachers have no concept of that themselves). Maths is not brought into everything they do - it is more compartmentalised into set times.

I tried to explain to a year 2/3 teacher that the grid method of multiplying is doing essentially the same thing as the old method she was saying was confusing. She just didn't get it and fell back on saying how brilliant the "new" methods are because they all understand it so well ! At which point I gave up.

"Lougle, but that is the 'black box' bus stop method - the 'crank a handle and something comes out' version. It carries no understanding that the 4 is not 4, but in fact has the value 400."

Is it?

I completely understand what I'm doing when I use the method. I understand why I am carrying the remainder, why I add the 0, why I put the decimal point where I do.

one of the problems with the bust stop method is that while it's easy to do on paper, it's difficult to do in your head.

Divideandrule as others have said if you encourage your DD to skip a step she may well be missing out on really getting to grips on what's going on here.

It's not just about churning out the right answer, it's about understanding how numbers relate to each other.

If she's finding the chunking method difficult then she heeds help understanding exactly that. Great if she can do bus stop, but that's a different thing, it's not just about getting the right answer, it's about understanding how you get there.

Bonsoir, yes I agree that practicing helps, but I think what I'm trying to say is even simpler than that.
Just occassionally I come across something in maths and think "I don't even understand that. How am I going to expain it to dd?"
However next day without dong anything about it I think, "sort of got it." Until in a few days I'm thinking, "How could I not have got it to begin with?"
I think the subconsoius works on it.
There is an argument that only if a child gets something straight away are they bright. I find they can progress further by being told, "Look, I'm just pointing out to you the next rung in the ladder. I don't expect you to understand it yet, I'm just telling you it's there." All the anxiety of whether or not your child is bright goes away.
Sorry I've digressed a bit from the opening question.

"To illustrate:
4)245

Start with 2/4. Does she know that this is 200/4? 4 actually does 'go into' 200 many times, does she understand why they don't include that and move on to looking at how many 4's in 24? (which is actually 240) Does she know what the 6 represents when she writes it on the top?

The repeated subtraction with number lines and chunking are important steps to understanding why this method works."

Start with 2/4 - yes it is actually 200. But if you do 'chunking' (I presume) you would say '4 into 200 is 50. 4 into 40 is 10. 4 into 5 is 1 but we have 1 left over. 4 into 1 is...hold on...I don't know what 4 into 1 is because I can only start with 1.

Then saying 50+10+1 = 61 plus 1 left over.

The reason you 'carry the 2' is because it makes no sense to look at the 200 in isolation. What you are doing, when you carry the 2, is to say, mathematically, 'wait a minute, there must be a number bigger than 200 that is divisible by the 4, because I can't get a whole number, or a whole number plus a remainder, when I divide the first number.' You now know that the answer doesn't exceed 99, and so you put a 0 in the '100s' column of your answer, because the 2 was in the '100s' column of your question.

So then, you look further. 4 into '24' goes 6, but we know that the 24 is in the 'tens' column so we know that although we have reduced the number to 24, the position of it gives the mathematician the knowledge that it is actually 240. Then you are saying '4 into 24 goes 6, so I know that the answer is at least 60. But, because you haven't used the whole number in your sum, you know it is more than 60. The 6 goes in the '10s' column of your answer, because the number you divided was in the '10s' column of your question.

So you move on to the 5, which is in the units column. 4 into 5 goes 1, with 1 left over. So you write the 1 in the units column of your answer.

Your question sum didn't have anything in your '1/10' column, but you know the answer MUST have something, because if it didn't, you wouldn't have a 1 left over. So you add the decimal point and a 0 to represent the '1/10' column and carry the 1 over to it. 4 into 10 goes 2 with 2 left over. Write the 2 in the tenths column in your answer.

Your question sum didn't have anything in the '1/100' column, but you know the answer must have, because you have a remainder. So write a 0 to represent the 1/100s and carry the 2 onto it. 4 into 20 goes 5 with no remainder. SO you write your 5 in the 1/100 column of your answer. You have nothing left to work with so you have your solution.

4)245= 61.25

Thank you to the teachers who posted explanations - I have learned loads about modern teaching of maths! I have an A-level in maths and a phd in a mathematical subject, but couldn't understand the jumble of numbers on the page when the DCs were being taught division. I now know they got taught the "chunking" method which is "relational" learning. Luckily for me, I was one for whom the "black box" "bus stop" method made sense, but I am aware that there were plenty of others who never got it.

Can anyone explain how you would get decimal places by the number lines method?

richmal - I sort of agree with the theory of the subconscious coming to the fore, but I also think that things in our environment trigger us to "practice" new concepts and skills - not always as obviously as a game of Monopoly (I was trying to find an easy illustration). For example, when DD was learning her numbers in English with me, she automatically started reading the 3-digit number plates on motorbikes in an obsessive sort of way. The practice jumped out at her from her environment, IYSWIM.

@ thetasigmasmum: "So, your condescending and rude statement about maths for unchallenging exams (although yes GCSE and A level are unchallenging) looks a bit silly now, doesn't it." No, it doesn't look silly. You have admitted you understand chunking just fine and that school exams are unchallenging, so you clearly agree with me - anyone naturally good at maths finds the current school exams unchallenging and chunking easy to understand - they don't need to be taught it, because the understanding is already there, not because it is confusing.

Yes, I think there isn't enough practice of the techniques to cement them into some children's understanding and too much time is spent on building up to the grand finale for children who actually are good at maths, but I don't think that means what is being taught is a load of unhelpful rubbish, or that you can miss bits out if you don't understand them. Why not just skip through them quickly, rather than claim they are confusing and pointless if you don't get what they are on about?

Bonsoir, it's probably then a mixture.
Cecily P, OK, the bus stop is the best method for decimals, but the number line will give fractions. Going back to this much passed around bag of sweets. If there were 19 sweets and 3 sweets to a child then 6 would get a whole portion and the 7th, a third. Hence 19/6=6&1/3.
This is perhaps a bit advanced for a child doing number lines, but this basic idea of division meaning so many lots of subtraction would help explain to an older child that if you have 1/2 a cake and you've decided one portion is 1/3 of a cake, you'll only get 1&1/2 portions from it.
Understanding ideas in maths from lots of different directions will help later.

Lougle, when I was trying to teach DD (age 10) the 'bus stop' method I did it like this:

To use your eg 4)245

Q1 How many lots of 400 in 200?
A zero

Q2 How many lots of 40 in 240?
A 6

Q3 How many lots of 4 in 5?
A 1 ( + carry one to make 10 tenths)

Q4 How many lots of 4/10 in 10/10?
A 2 ( + carry two to make 20/100)

Q5 How many lots of 4/100 in 20/100?
A 5

Answer = 61.25

[All of the above requires a v. secure knowledge of place value and an understanding of the laws of multiplication i.e. knowing that 40 x 6 is the same as 4 x 60 and of course knowing that multipication is the inverse of division]

My DD does maths in the French system and number lines are not a feature of maths teaching - she is in the equivalent of Y3.

My query about number lines is that it encourages children to think of numbers as a sequence rather than an expression of increasing quantity.

I know very little of maths, but am saddened by the way teaching & learning is encouraged, nay required to plod along in sequential units, when sometimes using an advanced concept and working backwards to see why it works can be more beneficial and certainly a lot more interesting.

I think, perhaps, thetasigmamum doesn't understand that primary school teachers have a responsibility to teach maths to people who don't understand it, not just to those who do. And those who do have a natural understanding of numbers may find chunking very, very boring, tiresome and easy, but they don't find it confusing. I know I would never make a good primary school teacher precisely because I have a limited understanding of why other people find things like reading and arithmetic difficult.

"As an example I have seen children who have been taught the column method for addition do this

33
+9
__

Clearly this method is not appropriate and they simply use it because they lack understanding. "

Why?

33
+9
__

9+3 = 12

write the units down and carry the 10s, then (mentally) erase the sum that replaces

30
__
02
10

3+0+1=4
0+2+0=2

answer:
42

Now obviously, I wouldn't write the sum like that. I would use a little 1 under the 10s column, etc. But mentally that is the sum that this method produces.

Not inappropriate at all.

'As an example I have seen children who have been taught the column method for addition do this

33
+9
__

Clearly this method is not appropriate and they simply use it because they lack understanding. '

Why is that not appropriate?

I think that method is entirely appropriate for doing a sum on a piece of paper. However, when I add very large numbers in my head, I favour doing them the opposite way round - starting by adding the thousands together, then the hundreds, then the tens, then the units. If I try to do mental arithmetic by holding an image of me doing a sum on a piece of paper in my head, I start to forget what I've carried over and find it harder to remember what the original numbers were.

And 33+9 done in my head is easier done as 33+10-1.

Although I know that 3+9=12... so probably wouldn't actually break it down in that instance, unless I was having a bad day.

"And those who do have a natural understanding of numbers may find chunking very, very boring, tiresome and easy, but they don't find it confusing."

And do you think that "not confusing" a bright child is sufficient justification for making him or her plod through school, never needing to apply him or herself?

Nope - I think it's justification for enabling him to flit through the easy bits and go on to something more interesting, though. So, if the OP's daughter found the method boring, silly and easy, I would have agreed that she go into school and ask that she move on to something more interesting. A school that gets children naturally able at arithmetic to plod through all the methods at the same speed as the children who find it difficult is letting the brighter children down. But the current Government is of the view that it is better to make all children plod through at the same pace, to ensure the weakest have gained a proper understanding of the subject, which is a bit concerning.

Yes, I take the 'in-your-head' point, rabbitstew. It was just the blanket statement (not yours!) that it was wrong to add in columns.

:)

I don't think more arithmetic, done slightly differently, is necessarily hugely more interesting and it's only one small part of maths...

Primary schools now and primary schools in the past have always focused one hell of a lot on arithmetic. There is very little else in the maths curriculum that children appear to be required to have a very secure understanding of prior to secondary school. I don't think that is more the case now than it was 100 years ago. For those children who find arithmetic incredibly easy, the result is several years of doing something you can already do.

Life is several years of doing something you can already do.