Thursday, 16 July 2015

Australian mathematics curriculum

My apologies to anyone out there who might read this blog from time to time. I have sadly neglected you for the last three months or so, such is the busy nature of my life as a 'retired' academic! I am now back from a wonderful holiday in Provence and able to give some time to writing the occasional post.

At my publisher's request I have spent quite a bit of time mapping the Australian primary mathematics curriculum to the content of the 5th edition of Mathematics Explained for Primary Teachers, to help them with marketing the book in that country. I must say I found the curriculum considerably more intelligent in its construction and content than the shoddy and embarrassing curriculum that we now have in place in England. It has quite clearly been written by people who actually know something about mathematics and mathematical pedagogy.

I drew on this Australian material when talking about algebra at a conference for primary school teachers in Cambridgeshire a few weeks ago. We noted that the English curriculum does not recognise algebra until Year 6! By contrast the Australian curriculum has an algebra strand all the way through from the start of primary education.

For example, all these are given as examples of algebraic reasoning developing in earlier primary years:


·      Sort and classify familiar objects and explain the basis for these classifications
·      Copy, continue and create patterns with objects and drawings
·      Describe patterns with numbers and identify missing elements
·      Solve problems by using number sentences for addition or subtraction
·      Describe, continue, and create number patterns resulting from performing addition or subtraction
·      Explore and describe number patterns resulting from performing multiplication
·      Solve word problems by using number sentences involving multiplication or division
·      Use equivalent number sentences involving addition and subtraction to find unknown quantities
·      Use equivalent number sentences involving multiplication and division to find unknown quantities
·      Continue and create sequences involving whole numbers, fractions and decimals
·      Describe the rule used to create the sequence

My contention has always been that the most fundamental component of mathematics education is not doing harder and harder calculations (the message in the English curriculum) but algebraic thinking and reasoning. This is where real mathematics emerges, powerful, widely applicable and creative. So, a proper mathematics curriculum must recognise and seek to develop algebraic reasoning right through the primary years; and not see it just as an add-on in Year 6. So, well done, Australia!

But I still hope you lose the Ashes.



Wednesday, 22 April 2015

Solution for Cheryl's birthday

It is 16 July! Below is the explanation of this very pleasing problem.

Cheryl has told A and B that her birthday is one of these ten dates:

May 15, 16, 19

June 17, 18
July 14, 16
August 14, 15, 17

She has told A the month and B the number of the day, and they both know this.

A says, 'I do not know C's birthday.

This tells us nothing, because whatever month he had been told A would not be able yet to deduce the birthday.

But B would know C's birthday if he had been told that the day was, say, 19: because 19 occurs in the list of options only in May. Likewise B would know the birthday if had been told that the day was 18: because this occurs only in June.

So, then A says:

'... but I know for sure that B does not know either.'


A would not be able to say this, if the month he had been told was May or June, because for each of these two months there is the possibility that B actually knows the birthday already.

From this B (and we) can deduce that the month of C's birthday is neither May nor June.

B replies: 

'At first I did not know C's birthday, but I do now.'

This means that the number of the day (which B knows) must occur in either July or August but NOT in both these months. For example, if B knew the day was 17 then he would know it had to be August 17. Likewise, B would know the birthday if the day was 16 (July) or 15 (July).

A has deduced all this as well! He knows that the day is 15, 16 or 17. But A also knows the month. And he then replies:

'Now I know as well!

Now A could only know the birthday if the month (which he knows) has only one of these three options in the list: 15, 16 or 17. That month is July.

This is how we (and B presumably now) know that Cheryl's birthday must be 16 July.



Tuesday, 21 April 2015

Solution to the hats logic problem

Here is the solution to the easier logic problem in my previous post:

A, B and C are told that in a bag there are 3 red hats and 2 blue hats. B is blindfold, so is unable to see anything. One hat is put on each person's head and they have to work out what colour hat they are wearing. A and C can see the hats on the other two, but none of them can see their own hat.
A says: I do not know what colour my hat is.
C says: Nor do I.
B says: Then I am wearing a red hat.
How did B work that out?


B considers the possibility that the hat on her head is blue.
Assuming this, when A says, I do not know what colour hat I am wearing, it follows that C must be wearing a red hat: because if C were wearing a blue hat, A would see two blue hats and know that her own hat would have to be red.
But C would also work this out! So C would be able to deduce that she is wearing a red hat.
But, even after A has spoken, C does not know what colour her hat is.
So B deduces that she cannot be wearing a blue hat.
Hence she knows that she is wearing a red hat.

Cheryl's birthday to be disclosed in my next post ...

Saturday, 18 April 2015

Cheryl's birthday

I'll give my solution to this logical reasoning problem in my next post, so anyone reading this has the chance to solve it themselves first. This is the problem from a Singapore mathematics test that has apparently 'gone viral'. It was constructed by Dr Joseph Yeo Boon Wool, a mathematics professor at the Singapore National Institute of Education.

So here is the actual problem.

Albert and Bernard want to know the birthday of their new friend Cheryl. She tells them that it is one of the following ten options:

May 15, 16, 19
June 17, 18
July 14, 16
August 14, 15, 17

She whispers in A's ear the month of her birthday and tells B that she has done this.
She then whispers in B's ear the day of her birthday and tells A that she has done this.

Then A says, 'I do not know C's birthday; but I know for sure that B does not know either.'
B replies: 'At first I did not know C's birthday, but I do now.'
A replies: 'Now I know as well!'

This is an excellent example of a logical reasoning problem that involves making deductions from what people say about what they know or do not know. These puzzles always assume that all the people involved have high powers of deductive reasoning, so you can assume that if something can be deduced they will deduce it!

Here is another example, much easier than finding Cheryl's birthday!

A, B and C are told that in a bag there are 3 red hats and 2 blue hats. B is blindfold, so is unable to see anything. One hat is put on each person's head and they have to work out what colour hat they are wearing. A and C can see the hats on the other two, but none of them can see their own hat.
A says: I do not know what colour my hat is.
C says: Nor do I.
B says: Then I am wearing a red hat.
How did B work that out?

Solutions in my next post.

Friday, 27 March 2015

How old are you?

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!


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:

'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 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.

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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