Tuesday, 18 November 2014

Outrageous Performance Descriptor Maths KS1

The Department for Education in England is engaged at present in a consultation about their proposed performance descriptors for children in Key Stages 1 and 2 in relation to the new primary curriculum.

I have just been looking at the mathematics descriptors for Key Stage 1.

The proposal includes the requirement that children who are to be judged as working 'at the national standard' at the end of Year 2 will be able to ...

     add and subtract numbers using ... the written columnar methods ...

They can't be allowed to get away with that! Columnar methods for addition and subtraction are NOT in the curriculum for Key Stage 1 mathematics. They are mentioned in the non-statutory guidance for Year 2, but they are quite clearly not required in the actual curriculum.

Non-statutory guidance is non-statutory!

But these proposed performance indicators will be statutory. Key Stage 1 teachers will be requried to assess pupils against these criteria. So, de facto, they will become the key focus of the curriculum.

This is an outrage. We must protest strongly at what looks like a deliberate attempt to give non-statutory advice about formal traditional written calculation methods the same status as what is in the actual curriculum.

Sunday, 16 November 2014

It's a long way to 67P/Churyumov–Gerasimenko

As readers of this blog will know, I enjoy trying to find ways of getting our heads around large numbers and huge quantities. In my experience children in primary schools also find large numbers fascinating, and, with the help of a calculator and some approximate mental calculations, we can play with big numbers and do some interesting mathematics.

This last week saw the landing of 'Philae' on Comet 67P/Churyumov–Gerasimenko. The newspaper reports told us that this comet was 300 million miles away. That's a long way, I'm sure, but how can we understand a distance like this? I always try to connect these big numbers with our own experience.

So, let's imagine I start driving my car at an average speed of 45 mile per hour. How long is it going to take me to clock up 300 million miles, assuming I have a co-driver and we manage to drive night and day non-stop? 

Well, let's see 45 mph is 45 × 24 miles per day, which is 1080 miles per day.

That's about 1080 × 365.25 miles per year, which looks like approximately 400,000 miles per year.

So to find how many years it will take me ... careful with all these zeros ... we need 300,000,000 divided by 400,000, which is 750 years!

So, put it like this: imagine I had started this epic drive in 1264, back in the Middle Ages, during the reign of Henry III, in the middle of the Crusades, when Marco Polo, that intrepid traveller, was 10 years old;  I would have continued on driving through all the subsequent Henrys, through the reigns of the Tudors and Stewarts, and the Civil War; I would have still been driving when Beethoven was born and when he died; and when Queen Victoria came to the throne and when she died; and still driving during the two World wars, mysteriously passing the day of my own birth, and right through to the present day, and I would be now just closing in on the 300 million miles target!!

Yes, it's a long way to Comet 67P/Churyumov–Gerasimenko!

Friday, 7 November 2014

Rounding to the nearest

I occasionally find myself having a disagreement with teachers and others involved in assessment who want to set children a test question in which to get the mark you have to assume that, for example, 148.5  rounded to the nearest whole number is 149. My argument is that in this situation there is no nearest whole number. The number 148.5 is as near to 148 as it is to 149. The only way you could decide which to round it to (if at all) would be to look at the context that gave rise to the number 148.5.

There is a myth that the convention is that if it ends in a 5 you round it up. I won't go along with this, because it is unnecessary and unjustified. Here's an example that illustrates my point.

I have a discount card from Waitrose that allows me 10% off any purchase. I by a packet of biscuits that costs £1.65. For convenience, I'll write this in pence, as 165p. So ...

original price = 165p
discount = 16.5p
reduced price = 148.5p.

So let's assume we are to round the results to the nearest penny, because we can only deal in whole numbers of pence in Waitrose. If we use the rule of 'rounding up when it ends in a 5', we get:

original price = 165p
discount = 17p (to the nearest penny)
reduced price = 149p (to the nearest penny).

This is plainly impossible! A discount of 17p gives a reduced price of 148p.
So if the discount is rounded up, the reduced price has to be rounded down; or vice versa. So, assuming the generosity of Waitrose, I would expect:

original price = 165p
discount = 17p (rounded up)
reduced price = 148p (rounded down).

This is why I refuse to accept the so-called convention! Context is everything.

I suggest that we just do not set a context-free maths assessment question about rounding a number ending in a 5 to the nearest something, if in fact there is no nearest something.

Bad assessment questions (but good for class discussion):
1. Round 6.05 to 1 decimal place.
2. Round 125 to the nearest ten.
3. Round 3500 g to the nearest kilogram.
4. Round 3 minutes 48.65 seconds to the nearest tenth of a second.

Monday, 3 November 2014

Numerous numbers processed

In my previous post I commented on the quantity of numerical data that we find ourselves processing all the time when driving a car these days. This shows both the importance of numeracy in real-life contexts, in two senses: (a) in a technological age a large proportion of communication is through numbers; (b) the principal reason for being 'numerate' is to ensure that we can handle with confidence information presented in numerical form. This almost always requires: a sense of number and an appreciation of the relationships between numbers; the ability to make estimates; understanding of measurements of various kinds and units in a practical context; confidence with concepts related to ratio, proportion and rate of change. It never seems to involve the need to perform paper and pencil formal, context-free calculations, to which those who designed our new curriculum have given so much prominence!

Anyway, here is the solution to the problem I set in my previous post:

15:35 was the current time
55 mph was my speed
80 kilometres per hour was the equivalent speed in metric
60 miles per hour was the speed limit displayed on the sat nav
3 × 1000 was the number shown on the rev counter
306 miles before I needed to top up with petrol
21 degrees C, the outside temperature
013399 the total mileage the car had done
¾ of a tank of petrol left
2 was the setting on the fan
R3 meant I was listening to Radio 3 (of course)
A17 was the road I was on
6.5 miles to the next junction
A1 the road ahead
5 indicating that I was in 5th gear
0:56 = 56 minutes estimated to reach my destination
16:31 estimated time of arrival
12V written on the adaptor for the satnav, for a 12 volt battery
120W likewise, indicating maximum 120 watts output

Tuesday, 21 October 2014

Processing numerous numbers

Driving from Norwich to Doncaster one warm, sunny afternoon recently, I was struck by the number of bits of numerical information a driver is expected to process simultaneously these days. I have a fairly modest car, with a basic, plug-in satnav, and air conditioning. At one point on my journey, as usual not breaking the speed limit, I found myself looking at all nineteen of these numbers:

3 × 1000

Can you work out what each one of these was telling me? The information given in the opening paragraph may help!

Looking at all this numerical data, I wondered how, say, some Year 6 children would get on trying to crack this? If anyone out there tries this as a problem-solving task with a class, I'd be interested to know how they do!

I'll give the solution in a week or so.

Tuesday, 7 October 2014

More bad statistics

Every time I open a newspaper I seem to come across badly presented or misleading statistics. Often the reporting of statistical findings is so bad that it is impossible to work out what is meant. Here are two examples I have noticed recently of dubious statistical inference.

1. House prices in National Parks in England and Wales.
A report in The Times yesterday was about the extra premium that people pay for a house in a national park, noting that house prices in Snowdonia have the lowest premium compared to houses in the surrounding area of all the National Parks in England and Wales. The findings were on the basis of the average prices of houses in various parks. I assume the 'average' being referred to is the arithmetic mean. The report concluded with the following puzzling statement:

Homes in the New Forest, Hampshire, were found to be the least affordable in the national parks, commanding the highest price premium ... (so far so good, I can understand that) ... with an average price starting at more than £500,000. 

What? An average price 'starting at'? Surely an average price is an average price; a set of data cannot have a range of averages!

So, do they actually just mean 'an average price of more than £500,000'? If so, then the 'starting at' is misleading. Or do they mean that they have worked out the average prices for a number of different categories of houses and the cheapest category has an average price of more than £500,000? That might be what is meant, but no other references to average prices in the article suggest this.

So, once again I am left not knowing what is being asserted by a statistical statement, other than a general sense that I probably could not afford a house in the New Forest!

2. Incidents of sexual abuse on trains in the UK.
A recent report in my daily newspaper invited me to be horrified at the rise in incidents of sexual abuse on trains. I am aware that rightly this is a sensitive subject, so let me state quite clearly that, of course, even a single case of sexual abuse on a train journey is one too many and should be condemned; and steps should taken to prevent such a thing happening. But this report claimed that there had been a 20% increase of such cases over five years from 2008 to 2013. It was at this huge increase in such cases that the reader was being encouraged to be horrified.

But the data provided in the article did not support the conclusion that there was a huge increase in such incidents. There may well have been, but there was no way of actually drawing this conclusion from the information given. For a start, there was insufficient detail about how the data was collected to know whether the comparison being used was valid. It could be, for example, that there have been social or procedural changes since 2008 that make it easier for victims of sexual abuse to report what had happened. 'More reported incidents' is not the same as 'more incidents'. An increase in the number of incidents being reported could be perceived as a good result because it could lead to a decrease in the number of incidents.

But there was also an inbuilt misunderstanding of the idea of a statistical variable in this newspaper report. Closer reading of the article revealed that they were just comparing the number of incidents reported in 2008 with the number reported in 2013. Is a rise of 20% from one to the other such a dramatic event as the headline suggested? Purely in statistical terms, no ... it is not, for two reasons.

First, no information was provided about the actual numbers being compared. So this means we have no idea as to whether this rise of 20% is significant or not. For example, if there had been only 5 incidents in 2008 and then 6 in 2013, that would have been a 20% increase, but somehow just one more incident does not seem like a dramatic increase requiring a headline.

Second, we are not told anything about how much variation there is in this statistic (the number of incidents per year). If, for example, there happens to be a very high variance in the number of incidents, then it could be that 2008 was one of the years towards the lower end of the range of values of this statistic and 2013 was one of the years towards the upper end. In the years in between the statistic might have gone up and down quite a bit. The article asserted that there had been an increase over the five years, giving the impression that the number of incidents has been gradually going up year by year. But that conclusion cannot be drawn from result of comparing 2008 with 2013. A cynical reader (what me?) might even wonder if the two years being used for the comparison had been chosen to generate the greatest possible difference and therefore the most dramatic headline.

Saturday, 20 September 2014

Scotland referendum results

The reporting of the victory of the Nosers over the Yessers in the referendum about Scottish independence provides another example of the potential confusion when people compare percentages.

The result was reported as 55% for No and 45% for Yes. So was the Nosers vote 10% higher than the Yessers, as I heard someone say? Well, no! There are two errors here. Let me explain.

When you compare two quantities you can do this either by using the difference between them or the ratio of one to the other. For example, let's say that I earn £45 an hour and you earn £55 an hour. Using difference I could say that you earn £10 an hour more than me. But using ratio, I could say that your hourly rate is 22.2% higher than mine (to one decimal place). This is because that extra £10 you earn is 22.2% of what I earn (22.2% of £45). This is similar to saying that when a price goes up from £45 to £55 that is a 22.2% increase.

This gets tricky when you are comparing percentages using percentages!

Let's look at the actual data, reported in The Times this morning:

No votes:     2 001 926
Yes votes:    1 617 989
Total votes:  3 619 915

Calculate the No vote as a percentage of the Total vote:   2001926 ÷ 3619915 × 100 = 55.30%
Calculate the Yes vote as a percentage of the Total vote:  1617989 ÷ 3619915 × 100 = 44.70%
(percentages given to 2 decimal places).

The difference between these two percentages is 10.6%, Note that this is closer to 11% than the 10% that was reported. So, that's the first error: a classic rounding error!

But it would still be confusing to report this by saying that the No vote was 10.6% higher than the Yes vote. What we can say, correctly, is:

'The No vote was 10.6 percentage points greater than the Yes vote.'

This is understood to mean that the comparison being used is the difference between the two percentages.

If we compare the actual figures, using ratio, by what percentage is 2001926 greater than 1617989? (Compare my example above for comparing two hourly rates). The No vote is 383937 more than the 1617989 votes for Yes. As a percentage that is 23.73% higher than the Yes vote (383937 ÷ 1617989 × 100). So a correct reporting would be:

'The number of people who voted No was 23.73% greater than than the number who voted Yes.'

That makes the extent of the victory clearer!