Bye bye chunking?

[PastSellByDate]

Wow if I hadn't read about this in the Times education supplement, I wouldn't have believed it but it seems that chunking is being abandoned - officially - link here: www.tes.co.uk/article.aspx?storycode=6316142

Although I will concede that understanding division is multiple subtractions was good - I also know that many children were told they couldn't divide by old fashioned methods (long division - I guess now called short division) - and my children were certainly told their work was wrong.

Was the problem here that a method was adopted which parents were excluded from and which did not allow tried and true old fashioned methods to occur alongside them as well?

[alanyoung]

Hi Lougle. I think there is an additional problem in that you are implying that imperial units are a system and have a common base, but nothing could be further from the truth. The imperial measurements certainly don't form any kind of a proper system and are made up of a number of units that were either chosen randomly or have no relevance today. For example, the furlong was chosen as the average distance an ox could reasonably be expected to pull a plough before being made to turn round. Apparently oxen are very difficult to turn (I've never tried it myself) so it was a good idea to let them go a good distance before turning. The acre was the area of land that a farmer could reasonably be expected to plough in a day.

It is often said that the foot was the length of a king's foot, but that's nonsense. I take size eleven shoes and my feet are only 10.5 inches long. The inch is supposed to be the width of a man's thumb. When do we ever measure these days using our feet or our thumbs? It really is all mumbo jumbo.

Neither do the imperial units operate according to a base. A system using a base would have the same base throughout, not sixteen ounces in a pound, fourteen pounds in a stone, two stones in a quarter, four quarters in a hundredweight and twenty hundredweight in a ton. No mathematician would call that a 'base' system.

Compare that with the metric system:

1000 grams = 1 Kilogram
1000 Kilograms = 1 tonne

A hectare is the area of a rectangle 100m x 100m (i.e. 10000 square metres) and we all know what 100m looks like; we see athletes running and swimming that distance all too often. Without looking it up, do you know what an acre is and how many acres make a square mile? Not many people do (at least below a certain age).

[lougle]

But Alanyoung, the beauty of mathematics, is that once you know the multiplier, you can convert anything.

You don't need to restrict yourself to 10s 100s and 1000s

[alanyoung]

Merrymouse. Thanks for the support, but if I could be permitted to add a little to your first comment, almost all of the rest of the world is fully metric and has been for a long time. That's over six billion people!

Some people often reply to this by saying, 'Yes, but the French still use a pound (livre?)' Apparently they do, but they don't mean the pound we mean. For them a pound is half a kilogram (500 grams).

[alanyoung]

Lougle, apparently, based on your example above, you can't.

[lougle]

ok, alan

[HumphreyCobbler]

"I like the idea that chunking isn't just thrown out but is used in that early stage of explaining the concept of division (when not inverse multiplication).

Perhaps the half-way house solution is that chunking is used early in KS2 and gradually abandoned as skills develop?"

This is what happens

[alanyoung]

Lougle, I probably overdid it a bit in my last comment. I really don't want to make this a personal matter and I hope you are not upset (I'm a very nice guy really). Why then, you may be asking, do I keep pushing this issue. Well, the answer is that I am passionate about improving the standard of mathematics in this country and I know that children who use exclusively metric units have so many more opportunities to improve not only their measuring skills, but their number and calculation skills too. For example, as I am sure you agree, place value is a key issue in understanding the number system and this is so easily demonstrated with metric units:

E.g. 2.65 Kg = 2.65 x 1000 = 2650 g. Multiplying by a thousand just requires a left shift of three places. So, multiplying 2.65 Kg by 2000 means just left shifting three places and then doubling the answer, giving 5300 g. This improves mental arithmetic skills enormously and it is the sort of thing you can do while travelling in the car on a long journey. What a terrific use of your time.

You just can't do this sort of thing using imperial units and, as I said earlier, even if you could, nobody is going to bother.

About two years ago I visited a local high school and on the Mathematics notice board was a sign congratulating last year's Year 11 students - apparently 63% of them passed with a grade C or higher. At first I thought that wasn't bad, and then I realised, of course, that meant that the other 37% 'failed'. Now I know there is no such thing as pass or fail at GCSE, but that's not the way employers and colleges see it. Having a Grade C opens doors that a Grade D doesn't.

When I got home I looked up the national figure and this is only 54%. So, over the whole country, 46% of our youngsters don't even get a grade C at GCSE. Now most people will tell you that GCSEs are two year courses, but maths is an eleven year course. And 46% of our youngsters (after hundreds and hundreds of hours in the classroom) don't even get a grade C.

I think this is an appalling situation which is why I am so passionate about trying to improve it, so I hope you will understand where I am coming from.

[PastSellByDate]

Very interesting all - and AlanYoung all I can say is that my DD1 (Y5) My Maths homework quite specifically has asked her to convert metric to imperial & visa versa. My problem is I can quite happily work in both scales separately but find making the conversion difficult - I'm still looking it up - and I fear I work by gas marks as well, so with newer recipes or foreign recipes have to convert to gasmark (yes I'm very ancient).

What I will say, is that imperial measurements have a long and proud history, and that is worth exploring as well, beyond numeracy. I personally find it fascinating that the pint volume has antecedents in pottery from the Roman and prehistoric past of Britain. I find it wonderful that DDs grandfather could run his finger down a 3 column ledger of pounds, shillings and pence and do the maths in his head.

I adore the base 10 simplicity of the metric system, but I understand the organic development of the imperial system and that history is worth appreciating as well, certainly for anyone interested in studying or working with objects/ buildings from our past.

So whilst I agree, at least whilst UK remains in Europe (and Scotland clearly will regardless), children should learn metric and the conversions between - ideally also appreciating 1 cubic centimeter = 1 ml, etc.... I would like a little time devoted to the imperial system and old money - because it does clarify so much in literature, history and indeed architecture.

[alanyoung]

Hi Pastsellbydate. Thanks for your comments. Naturally I don't agree with much of what you say. Firstly, when I was young the coal man brought sacks of coal in hundredweights, flour was sold in stone bags and you could even buy a pound of broken biscuits in Woolworths. These units had a relevance and what we saw in the world around us reinforced what we learned in school. Now children learn mainly metric units, but see largely imperial units in their everyday lives. The opportunity to reinforce school learning is no longer there. Because of this, many children believe we live in an imperial world, when in fact we live in a metric one.

Secondly, I take your point about the historical interest in imperial units, but this is rather like studying Latin or Ancient Greek - those that want to can do so, but most people are just not going to do it.

Thirdly, you refer to the imperial units as a system when they are no such thing as I explained above.

Fourthly, this is nothing to do with Europe. Metric units are used all over the world. We will still use metric units in this country whether we are part of Europe or not.

And fifthly, the fact that your young DD is having to convert from one to the other I find absolutely appalling and with Mr Gove in charge, this is only going to get worse. It is only British children that have to waste time doing this - time that could be far better spent.

If I could give you another metric example, a typical car takes about 10 000 measurements to make - every one of them done using metric units. But what do our children see? They see miles on the odometer and miles per hour on the speedometer. The funny thing is that these devices and particularly the numbers on them are all designed in metric units! If there was a label on a car for every measurement that had been made in its manufacturer, the metric ones would completely swamp the imperial ones about 5000 to 1.

[alanyoung]

With regard to the temperatures (I thought you said 'gas masks' at first - must get my glasses changed!), have you noticed how the temperatures are mostly given in Celcius (sometimes with accompanying Fahrenheit) during normal times, but when it gets hot newspapers (typically the Mail and Exress) give it in Fahrenheit. Suddenly it is 95 degrees, just at the time when children are learning that water boils at 100 degrees. If it gets much warmer, surely we will all evaporate! How children are ever supposed to succeed in maths and science in this country is a mystery to me.

[Elibean]

I totally agree with AlanYoung re metric v imperial.

I may feel nostalgic for shillings (and slide rules ) but please, schools, do not inflict this confusion on the next generation!

And my mother is French. She only uses 'livre' when in the UK, referring to an Imperial pound. In France, she only uses kilos.

[alanyoung]

Elibean, thanks for your support. Does this mean they don't even use 'livre' to mean 500g any more?

[noblegiraffe]

Re chunking versus short division: rote learning of algorithms is all very well when it comes to answering the sorts of questions that come up in maths tests, but when it comes to mental maths and real life where you don't always need an exact answer, an understanding of what division actually means is also important.
I'm thinking of an exam paper my pupils sit in either Y7 or 8, I can't remember. It has a question on it that most pupils get wrong or don't even attempt. It's something like 'a lorry can carry 26 tonnes maximum. A log weighs 0.8 tonnes, how many logs can the lorry carry?'
Those that go for short division and attempt 26/0.8 usually come unstuck. Those that do repeated addition waste lots of time. The successful ones say '10 logs weigh 8 tonnes, so 30 weigh 24 tonnes, then you can fit on another couple' tend to get it right, but there aren't many of them.

[alanyoung]

To the supporters of imperial units, please let me put it this way. If you are arguing to keep both sets of units and do the conversions, you are obviously keen for your children to do well, in mathematics as well as other subjects, so let's take that as our starting point.

Every time you use a gas mark instead of Celsius temperature, you are closing the door on the opportunity for children to see how the temperature of cooking a cake compares with the temperature outside or the temperature of boiling water, for instance.

Every time you use feet and inches instead of metres, you are closing the door on the opportunity for children to see how much taller (or perhaps shorter) you are than they and to see how many times taller you are. For instance dad is 1.82 m, child is 1.43 m. Dad is 0.39 m or 39 cm taller. Dad is 1.82/1.43 = 1.27 times as tall as child. Try doing that when the parent is five foot seven and the child is 1.43 m. Yes, I know you can do it using those conversions you keep talking about, but people don't!

Every time you use pints instead of litres and millimetres, you are closing the door on the opportunity for children to learn about left shift and right shift of numbers when multiplying by 10, 100 etc. So 2.6 litres = 2.6 x 1000 = 2600 ml. Try doing that with pints, quarts, gallons etc. This is an extremely important idea in the understanding of maths today.

Every time you use acres instead of hectares, you are closing the door on the opportunity for children to understand that a hectare is 100m x 100m = 10000 square metres, so 4.6 hectares is 46 000 square metres. You just can't do that when an acre is the area of a rectangle a furlong long and a chain wide!

There are so many opportunities to help your children practise maths with metric that are not there with imperial. When I was at school we had to add, subtract and multiply (and the bright ones even had to divide) hundredweights, quarters, stones, pounds and ounces. At least it had some relevance in those days. But what a waste of time nowadays.

[alanyoung]

Noblegiraffe, my take on this question is to multiply both numbers by ten so the sum becomes 260/8 which is easier. Now this takes time to teach and for the children to understand why it works. All the more reason then to stop doing all these silly metric/imperial conversions and release time to explore this sort of idea. This is exactly the sort of thing I am talking about.

Have to rush now, just remembered I have to collect wife - she's great, but woe betide me if I'm late!

[noblegiraffe]

Alan, multiplying both numbers by 10 is what we would teach in order to do division by decimals using short division, however, I don't think that had been explicitly taught by the time the pupils did the test. However, that is another method that they need to remember, whereas if they understand what division is, they should be able to figure out the method I outlined intuitively without needing to be told what to do in that particular situation. The problem a lot of maths students have is seeing a question like the one I gave and thinking 'I've forgotten how to do that method' and so not attempting it, or applying the method incorrectly and not spotting a nonsensical answer, rather than figuring out the answer for themselves based on what they know about numbers and the operations.

It's why employers bemoan that young people can't work out percentages etc when that same young person passed a GCSE where percentages were correctly worked out. They learn the abstract method for the test, then immediately forget it.

[Shattereddreams]

I got a C in gcse maths, a scrape at my grammar school.

I'm 37. At work I use percentages, which I can do in my head. Division which I can't. I looked at the you tube video linked above, if I had been taught that at school instead of long division I reckon I would have been a far more confident mathematician.

I was rubbish at maths at school. It just wasn't relevant to my life and I was poorly taught. I taught myself number bonds in my twenties. Learnt my tables too. My maths improved. There was only ever one technique taught in my day.

Alan I agree wholeheartedly with everything you have said.

[PastSellByDate]

Hi AlanYoung

Can I just say that I wasn't asking for imperial to be taught for maths - I was saying that it should move across to 'history' perhaps as part of a Victorian history unit. I find it a huge mistake that children aren't exposed to history of mathematics as part of the primary curriculum introductions to Egyptians, Romans and Greeks.

The Egyptian numbering system, which heavily relies on counting by fives, turned out to be a breakthrough for DD1, who is particularly visual in her learning. She was suddenly able to subtract because she could cross out upside down U symbols or hand symbols - and she got that before she could master doing the same with digits.

Roman numerals. Greek Golden Mean or Pi (sorry can't make symbol) are all fun. Archimedes displacing mass with water to work out density (linke to TED Ed video here: ed.ted.com/videos?q=how+taking+a+bath+lead+to+Archimedes%E2%80%99+Principle - are great examples of logic and problem solving (ye olde lateral thinking) and a lot of fun.

Again Alan, I think you need to update yourself on the latest Gove plans - because he is signalling that knowledge of the imperial system will be a requirement in the primary curriculum www.telegraph.co.uk/education/educationopinion/9790670/Modern-schools-must-teach-imperial-measurements.html.

And just to be persnickety Alan - time is not base 10 but based on multiples of 12 (24 hrs in a day/ 60 seconds in a minute). And time fundamentally underlies the history of the metre (brief history video here: ).

Can I also add that I work in gas marks because even the brand new cooker delivered to us this week is labelled in gas marks. You tend to work with what you're given. It would be lovely to have celsius on the knob as well - but I suppose space was the issue - and yes I basically know gas mark 4 = 250C because I use it so much - but have to look up the rest.

[alanyoung]

PastSellByDate, I don't think we are disagreeing about very much at all. I am not against studying old number systems. On the contrary, I very much welcome it as it helps children appreciate how powerful our place value system is. Try multiplying MMDCCXLII by CXXIV without converting to our number system first, which is what the Romans had to do.

In a historical context, of course we can take a look at imperial units as we can most other things, but that is not what is happening at present. We are in a crazy situation as I have described earlier in that we actually live in a metric world, but mostly the children see imperial units and they are expected to convert between one and the other. I am aware of Mr Goves new emphasis on conversion and even more familiarisation with imperial units and that is what I find so horrifying. Andrew Percy and his ilk just don't get it, to use a modern political phrase. What a coincidence that he also mentioned the car as an example, but did he tell us about the 10000 metric measurements that go into making a car? Not likely! All he said is that children see miles and miles/hour, which is exactly my point ...Why?

He also mentions pints for beer, but does he mention that the glasses that hold the beer and designed and manufactured in metric units, and the barrels that hold the beer, and the equipment that pumps the beer, and the equipment used to produce the beer in the first place and so and so on? Not likely!

And as to road signs, well the changeover to metric is long, long overdue and even the Government knows this. I have written to them several times to ask why this has not been done and every time they give a different answer ranging from the cost to the fact that they are waiting for at least 50% of drivers to have learnt metric units at school. We passed that stage years ago.

As to the cost, the Government estimates £800 million to change the signs; the UK Metric Association estimates slightly less based on the Irish experience, but even if we are generous and allow for £1 billion, that only works out at less than £3 per person per year if we spread the cost over five years. In Government terms that's absolutely peanuts and this is a once only cost - it will not have to be repeated year on year like most Government expenditure. By comparison, the O2 (formerly the Millennium Dome) cost £800 million and the tunnel they were thinking of building around Stone Henge a few years ago (now abandoned) was to cost £500 million.

In my opinion, people like Andrew Percy are doing untold damage to our children's maths education and because of his status as an MP, people listen to him.

Coming to your point about time, the first agreed definition of a metre was based on the distance from the North Pole to the Equator. The only reason for linking it to time via the caesium atom oscillations was to fix it more accurately and that is what modern scientists need to do to calibrate their instruments accurately, of course. I don't think I have ever said that time is based on 10. What I said is that the only metric unit for time measurement is the second (subdivided when necessary into milliseconds etc). We certainly use 12, 24 and 60 in everyday time, but that's a matter of convenience and not something that damages our children's maths.

Some people seem to think that because I advocate complete metrication in all aspects of everyday measurement, I think children should only be multiplying and dividing by 10. That's just nonsense, of course. What I am saying is that multiplying and dividing by powers of ten using metric units helps children to understand left and right shifting so that they can then go on to multiply by many other numbers. For example, a book weighs 456 grams, how much would 300 similar books weigh? Here we multiply by 100 by left shifting two places giving 45600 grams and then we multiply by 3, giving 136800 grams. If we wish to have this in kilograms, all we have to do is right shift three places to divide by 1000, giving 136.8 Kg. If we want this in tonnes, we divide by 1000 again (right shift of 3 places), giving 0.1368 t.

Finally, let me put it this way. Supposing we had been using metric units for hundreds of years and someone came along and said, 'Hey guys, I've just invented a new measuring system. We start with a unit of one inch which I randomly chose as the average width of some guys' thumbs and if we put twelve of these together we get a foot, which doesn't bear much resemblance to a real foot, but don't worry about that. Next, we fit three of these feet together and call that a yard. Are you with me so far? Next we put 22 yards together and that makes a chain, and ten of these make a furlong. Finally eight furlongs make a mile, and just to make it interesting we will put tenths of a mile on our car odometers that don't bear any resemblance to those furlongs I was just telling you about. Area, of course, will be the acre (lovely name, don't you think) and that will be the area of a rectangle one furlong long and one chain wide, giving 4840 square yards in an acre.

Now let me tell you about weight. There will be something called the ounce...'

Do you think it would catch on?

[alanyoung]

Coming back to chunking as per the original posting, I have the question, 'What comes after chunking?' and no-one I have read has answered that yet. The point here is what do you do about decimals. Suppose, for example you want to do the sum 785/8, giving the answer to 3 decimal places. You will get to the point where you have the answer 98 remainder 1. What do you do then?

[Haberdashery]

If you understand decimals and fractions, surely you will know that 1/8 is 0.125? And if you don't, you are likely to know that a half is 0.5 or a quarter is 0.25 and can just halve them until you get it right? As long as you understand what you are doing, there won't be a problem.

[alanyoung]

Haberdashery, okay, perhaps I chose an easy example. What about dividing by 7 or 13 etc? Or supposing when dividing by 8 the remainder was 7?

[Haberdashery]

supposing when dividing by 8 the remainder was 7?

This one would obviously be very easy because you'd be able to take 1/8 away from 8/8 in order to get 7/8.

As for the others, I can't see what would be wrong with doing 7 or 13 into 100 or 1000 or any multiple of those and then moving the decimal place if you only want a few decimal places. If you are getting into non-terminating decimals and want a lot of decimal places, I'd imagine that you would have sufficient understanding of what you are doing to either use a calculator or another method apart from chunking.

[alanyoung]

I would suggest then, that if you can handle the complexities of what you have described, you should be able to divide in the traditional long/short division layout. What do you think?

[Haberdashery]

It doesn't sound at all complex to me! It sounds like common sense, given a basic understanding of decimals and fractions.

I'm sure someone who was able to do that would also be able to handle the traditional layout and method but I don't think anyone who was at a point where the above was confusing or complex would really benefit from it that much. I'd far rather see them using whatever strategies they had available to try and understand what they are actually doing than blindly applying a method (which I think is what often used to happen when the traditional methods were the only ones taught).