In our church we have a spot in our Sunday morning services where we invite children who have had a birthday that week to come to the front to receive a small gift. 'When was your birthday' said the pastor to young William. 'On Friday' he replied. 'And how old are you?' 'Four and a half', replied William!

Very interesting that reply, isn't it? On Friday he was four. But now he is more than four. Hence the 'four and a half'!

When we teach children simple fractions, like halves and quarters, we tend to major on the idea of a fraction as one or more equal portions of a whole unit. So we use images like a pizza or a rectangle cut up into a number of equal parts. But William's response reminds us that children's early experience of fractions also includes the idea of a fraction describing a point on a number line, a point 'lying between' one integer and the next. So 'four and a half' means a point on a time line 'somewhere between four and five'.

I suggest that we would do well to make much more of representing fractions as points on number lines, especially if we are teaching children about mixed numbers.

Thanks, William. And happy four and halfth birthday!

## Friday, 27 March 2015

## Thursday, 5 March 2015

### Yet another error in new Mathematics Primary Curriculum: 'irregular'

Call me pedantic if you wish, but I do think that mathematical terms used in the Mathematics National Curriculum should be used correctly. The misuse of the word 'irregular' is my latest find. This occurs in the measurement section of the Year 5 Mathematics Curriculum programme of study:

So, let's tidy this up. A regular two-dimensional shape is one where all the sides are equal in length and all the internal angles are equal. So, the only quadrilaterals that are regular are squares. All others are irregular, including non-square rectangles.

All these shapes, for example, are

The National Curriculum needs another way of describing the other kinds of 2-D shapes that it has in mind for which Year 5 children should learn to estimate the area. I usually call them non-standard shapes or non-geometric shapes.

*'Calculate and compare the area of rectangles (including squares) ... and estimate the area of irregular shapes.'**This statement seems to imply that a non-square rectangle is a 'regular' shape. The word 'irregular' is used here for two-dimensional shapes where you cannot calculate the area exactly using a formula, such as the outline of an island or that of a fried egg. These are irregular, but so are all rectangles that are not squares, all triangles that are not equilateral, and so on.*

So, let's tidy this up. A regular two-dimensional shape is one where all the sides are equal in length and all the internal angles are equal. So, the only quadrilaterals that are regular are squares. All others are irregular, including non-square rectangles.

All these shapes, for example, are

__irregular__polygons:The National Curriculum needs another way of describing the other kinds of 2-D shapes that it has in mind for which Year 5 children should learn to estimate the area. I usually call them non-standard shapes or non-geometric shapes.

## Thursday, 19 February 2015

### A further thought on composite numbers definition

Going back to the erroneous 'definition' of composite numbers in the Year 5 Programme of Study for the Mathematics National Curriculum for England ... where we read that pupils should learn about 'composite (non-prime) numbers'.

In my previous post I pointed out that you cannot use 'non-prime' as a synonym for 'composite' because the integer 1 is neither prime nor composite.

On reflection, I realised that the error here is much more substantial than this. The concept of 'prime' applies only to positive integers. So, a prime number can be defined as an

This means that 'non-prime numbers' would include all numbers that are not positive integers. So, the identification of composite numbers with non-prime numbers would imply that numbers such as –3, 2.4, ⅚ and √2, for example, are all composite!

Of course, they are not! 'Composite' is also a concept that applies only to positive integers – those integers with two or more factors greater than 1.

That was a sloppy bit of work by whoever wrote that programme of study.

---------------------------------------------------------------------------------------------

In my previous post I pointed out that you cannot use 'non-prime' as a synonym for 'composite' because the integer 1 is neither prime nor composite.

On reflection, I realised that the error here is much more substantial than this. The concept of 'prime' applies only to positive integers. So, a prime number can be defined as an

*integer*with precisely two factors (which will be 1 and itself).This means that 'non-prime numbers' would include all numbers that are not positive integers. So, the identification of composite numbers with non-prime numbers would imply that numbers such as –3, 2.4, ⅚ and √2, for example, are all composite!

Of course, they are not! 'Composite' is also a concept that applies only to positive integers – those integers with two or more factors greater than 1.

That was a sloppy bit of work by whoever wrote that programme of study.

---------------------------------------------------------------------------------------------

*Any reader looking for professional help in the area of primary mathematics education might like to take a look at the recently-launched website of my colleague, Ralph Manning: www.manningeducation.co.uk*

## Friday, 13 February 2015

### Error in Primary Mathematics National Curriculum

Come with me to the Year 5 Mathematics Programme of Study for England, where we read that pupils should be taught to ...

"know and use the vocabulary of prime numbers, prime factors and composite (non- prime) numbers"

This was clearly written by someone who doesn't!

'Composite number' is not synonymous with 'non-prime number'.

A composite number is a positive integer with two or more factors greater than 1.

So, composite numbers are: 4, 6, 8, 9, 10, 12, ...

The error in the National Curriculum is to overlook the fact that 1 is NOT a composite number. Of course, it is not a prime number either. So, 1 is the only integer that is neither prime nor composite.

OK, so it is only a small error. But it is still embarrassing to have a mathematical error in our National Curriculum: to add to all the pedagogical errors.

"know and use the vocabulary of prime numbers, prime factors and composite (non- prime) numbers"

This was clearly written by someone who doesn't!

'Composite number' is not synonymous with 'non-prime number'.

A composite number is a positive integer with two or more factors greater than 1.

So, composite numbers are: 4, 6, 8, 9, 10, 12, ...

The error in the National Curriculum is to overlook the fact that 1 is NOT a composite number. Of course, it is not a prime number either. So, 1 is the only integer that is neither prime nor composite.

OK, so it is only a small error. But it is still embarrassing to have a mathematical error in our National Curriculum: to add to all the pedagogical errors.

## 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 ...

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,

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.

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!

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 buy a packet of biscuits that costs £1.65. For convenience, I'll write this in pence, as 165p. So ...

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:

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:

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.

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 buy 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.

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