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?

Bonsoir mathematically speaking, a relational understanding of maths is superior. It means a person has a good grasp of number, whereas someone who has a instrumental understanding can find an answer, but lacks the depth of understanding to know if the answer is correct.

For example (and sorry to pick on you witchwithallthetrimmings!) 44x5 could not have been 222 as any number in the 2x table has to end in a 0 or a 5. I'm not claiming witchwithallthetrimmings is a crap mathematician..!

The OP's child might be able to do bus stop method, but children need to be able to use a range of methods, and also explain their thinking.

If you look at this link, I've highlighted the bit that shows repeated subtraction is a level 2 requirement for maths
tinypic.com/r/addrh4/5

Incidentally, that's an APP assessment grid. A large number of schools use these as their assessment method. Certainly not all.

The bus stop method is, anyway, merely a contraction of what I have described:

7 bus stop 459:

I know that 7 goes into 450 60 times to the nearest 10 times, because 7 x 6 is 42 and 7 x 60 is 420. So I write the 6 tens of 60, then I find out what is left through subtraction (30) and attach those 3 tens to the remaining 9.

I know that 7 goes into 39 five times, so I write the 5 in the units space. I use subtraction to tell me that there is only 4 left, so that is the remainder.

A child with a clear understanding of chunking and place value will be able to 'contract' it into a bus stop method with understanding. A child who just learns the bus stop method as a 'black box' (you churn the handle, the answer comes out) will have no idea what to do if they can't produce 'the right answer' as they have no idea of the mechanism

teacherwithtwokids, I have greatly admired your scientific explanations of things this week! You make things very clear.

The fact is, if you don't understand chunking, it's not because chunking is nonsense, it's because you're not very good at it.

I think current maths teaching is much better at teaching a proper feel for numbers than the rote teaching I received in the seventies.

My father has quite a senior financial position. He says he quite often sends back work as wrong. People say "what is the answer then?" and he will say "I don't know, but I can tell it is wrong because..." e.g. it ends in a 2 when it needs to end in 5 or 0 (as above). Or nnn X nnn can't give me a number which is nnnn, that is too small, and so on. His staff have just used calculators or computers and so have no feel for the results they are producing.

And, in fact, in the work place the actual number may not be important. If I want to do something for each of my clients that will take 5 minutes, and I have 1823 clients, then thinking it will take around 9000 minutes is enough information for me to know whether it is a feasible idea or not.

I think children who are confident in today's teaching methods will be much more comfortable with things such as those. Would your DD understand division as repeated subtraction more if you did things like dealing decks of cards and counting for each time you go round?

44x5 is the same as 22x10 that makes it easy.

See I wasn't taught the 'bus stop method' as you describe it. I was taught as follows:

___
7|459

7s into 4 = can't be done. So carry the 4 and put a 0 on top

0_
7| (45)9

7s into 45 goes 6 with 3 left over. So put a 6 at the top and carry the 3 over to the 9.

06_
7|(45)(39)

7s into 39 goes 5 with 4 left over. So put a 5 at the top followed by a decimal point, a decimal point then 0 next to the 9 on the bottom and carry the 4 onto it.

065.___
7|(45)(39).(40)

7s into 40 goes 5 with 5 left over. So put a 5 at the top and carry the remainder over to the next 0 on the bottom.

065.5___
7|(45)(39).(40)(50)

7s into 50 goes 7 with 1 left over. So put a 7 at the top and carry the remainder over to the next 0 on the bottom.

065.57___
7|(45)(39).(40)(50)(10)

etc., etc. I won't continue as the answer is 65.(517428) recurring

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. And what is the 'left over'? Again there is no explicit understanding that what is happening is a subtraction to create this 'left over' number, not an understanding that its value is not 3, but 3 tens. It's fine if you can 'follow the black box instructions', but there is no understanding there, and therefore no way of 'unpicking' it if it goes wrong (e.g. putting the carried 3 to the right of the 9 not to the left - as there is no understanding of the value of that 3, the realisation that it HAS to go to the left because it is 3 tens, not 3 units or 3 tenths may not be there).

I think the subtraction thing is confusing too. But I can see how it could work if someone struggles with the rearranging concept. Perhaps one of those things where one method suits one person and another method another.

omigod, that all sounds complicated. I home ed so I taught my son the way I was taught, as all the rest seemed so unnecessarily complicated and he had NO trouble at all understanding, and neither did I when I was his age. If we can all do maths now, why was the old way wrong? It plainly didn't do us any harm or we would not now be discussing this subject so knowledgeably would we?
I'm a self confessed dinosaur.

"I think current maths teaching is much better at teaching a proper feel for numbers than the rote teaching I received in the seventies." You and me both Lottie!

I didn't until last week though - then school ran a session for parents who were all "Wtf?!" (me included) about the things their children were telling them they did in maths. Until they explained it to me it sounded so massively confusing and convaluted, but now it makes so much more sense, and I can see why the methods we learnt way back then work.

IRBMO,

The thing is, you learned, and I learned, but an awful lot of people DIDN'T learn to calculate effectively using the old calculation methods. My mum, for example, a supremely intelligent woman, cannot 'do maths' (though she is learning to, along with her grandchildren, and finds new methods magical in their encouragement of true understanding)

It's a bit like phonics - 80% of children learned to read using mixed methods, and those people reading and writing on here are from that 80%. The question for educators is how do we get that 80% to VERY much nearer 100%.

Same with maths methods - if the 'modern' methods (and the alternative models and images that are employed) allow e.g. 10% more children to learn to carry out calculations, then they are worthwhile. To an extent, the larger percentage who would have learned either way are not the 'target audience' - it's that percentage who will understand using the new methods but would not be able to use a 'black box' routine effectively who are the point.

Just to add to teacherwith2kids's post

Lots of people learnt one method when they themselves were in school, but might have benefited from knowing a range of methods. Having a choice of methods is fantastic. Say you temporarily forgot how to use the bus stop method, then knowing how to repeatedly subtract would still get you the right answer (hopefully!)

Like others, I was never taught to work on a number line or number square, yet different aspects of my mathematical understanding have clicked into place since I became a teacher and had to explain the 'new' methods to children and other adults.

I think the subtraction method is important to understand for when they start dealing with fractions.
To go back to Feenie's example. 18 sweets shared between 3 children is 6.
Imagine instead we asked how many children could get sweets if we gave them 3 sweets each? Same sum, but quite obviously the subtraction method.
Imagine instead the sum were 18 divided by a half. Children have a problem with dividing by fractions. If we asked, however, with 18 sweets how many children would get something if you gave them half a sweet each? They could then see it would be 36. So 18 divided by a half is 36.

Slowly then you can build the idea of dividing by a fraction being the same as multiplying by it's inverse.

Honestly, it's nothing to do with one method working for one person and one method working for another - everyone will have to learn the old fashioned method too, eventually. There's no point going on about one thing being superior and not needing to know something else, when you are simultaneously admitting you don't even understand what it is you don't know much about. It's like saying you don't need to be taught anything about how a computer works when you only need to know how to use one - not enough people now know enough about how computers work (which is harmful for the economy), even though we almost all know how to use one, and there are now urgent calls to change the way computer science is taught in schools. Times change and so do teaching methods and it is not always for the worse. So, until a university maths professor comes on here saying that chunking is causing huge harm to a generation of mathematicians, or is utterly pointless, I'm not going agree with people who don't understand it arguing that it therefore doesn't need to be taught to their children. They are just arguing that they think their children can get away with not understanding it, because they got away with it.

@rabbitstew where did you do your maths degree?

I wonder why they still teach the bus stop method at private schools?

AFAIK they use a combination of methods, but mainly the "old" ones.

I got an A in A-level maths (pre-A star days...) and my dh has a maths degree. And I have absolutely no problem understanding chunking or any of the other modern maths teaching methods - in fact, I find it quite interesting that many of the techniques I naturally always used for mental arithmetic but wasn't taught are now being taught in schools, so have decided to be entirely tolerant of chunking et al as a result. thetasigmamum - do you have a maths degree????

The question is does she understand WHY she is doing something? The 'bus stop' method is a process to learn, it doesn't mean she understands the maths behind it, when it is appropriate to use it and when there is a better method.
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.

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.

@rabbitstew yes I do. From Cambridge. And I got A's (pre A* obviously) in maths and further maths A level when that meant something. 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. I understand chunking just fine, obviously, it's just so pedestrian. It's the complete opposite of being able to play around with and have fun with numbers. It's a method of non explanation that was designed to help people who can't do maths and can't grasp figures try and get at least the basics. It's a method that was designed to enable maths teachers who can't in fact do maths to teach it. It's part of the problem that has seen the standard of the maths A levels we have now dip below the standard of maths O levels in the early 80s.

juniper904 - but is it right (in your analogy) to teach map reading skills first and think of them as superior? They are higher level skills that some people might never get to, or not get to without mastering the visual and physical landmark method first.

Remember New Math (Modern Mathematics) and the disaster that was!

I agree the understanding behind why a method works helps greatly as they progress.
I also think there is an expectation sometimes that a child should understand new things straight away. Maths needs time to sink in. No matter how clear the explaination, let them walk away from it for a while. When they come back to it again it's suprising how much easier they'll find it

"I wonder why they still teach the bus stop method at private schools?

AFAIK they use a combination of methods, but mainly the "old" ones."

Because the teachers in private schools think the children have a better understanding of numbers (for whatever reason) than do the teachers in state schools?

richmal - I think that children often go away from the classroom with semi-complete understanding of a topic, but enough knowledge to try it out in RL or practical situations eg they might go and play several games of Monopoly and revise their 4x or 10x table and how to add and subtract using pretend money. The skills they hone will then help them in the classroom.