Is the draft maths curriculum good enough?

What next for maths

Recently, I had the privilege to observe a problemsolving lesson with a Primary 5 class (10 and 11-yearolds) in Singapore. Here is one of the problems they worked on:

The wood problem

Jo has two pieces of wood. One piece is five times as long as the other. When Jo cuts 87cm off the longer piece of wood it is then twice as long as the shorter piece. How long is the shorter piece of wood?

The children were using a computer program to create models along the lines of:

From this, using calculators if they wanted to, they figured out that:

• 3 units = 87 cm
• 1 unit = 87 ÷ 3 cm
• 1 unit = 29 cm
• Shorter piece = 29 ÷ 2 cm
• Shorter piece = 14.5 cm

One of the striking things about the lesson was the palpable sense of ‘can do’ the children displayed. The pairs worked well together, discussed their methods and, after each problem, confidently approached the board to explain their solutions. Their equally confident peers would then comment and build upon these solutions.

Only after the children had left the room did the teacher reveal this was a group of lower attaining children who had been put together because they needed extra support.

Y5 in England’s draft curriculum

It has been widely touted that Singapore’s curriculum is one of the inspirations for the latest incarnation of England’s national curriculum. But how does the type of mathematics described above compare with what Y5 children in England might look forward to if the proposals are accepted? Well, there are 11 bullet points in the calculation strand of the draft programme of study for maths. These are mainly about methods of calculation, finishing with:

Solve word problems involving addition and subtraction, multiplication and division.

To me, this reads like the sort of problem solving I did at school – always at the end of the chapter following pages and pages of calculations. Not real problem solving; just more calculations wrapped up in words.

Of course, this does not preclude the type of problem solving I saw in Singapore, but it doesn’t exactly encourage it either.

So is it just Y5 that gets a bum deal, or are the proposals generally not inspiring?

At first glance

The proposals start off promisingly. The ‘purpose of study’ described in the introductory pages states pupils need to learn about three distinct aspects of mathematics: being fluent in fundamental aspects, solving a variety of problems and reasoning mathematically.

Readers who have been following my writing in this magazine over the recent months will know that all three of these strands of mathematical activity are dear to my heart, so I am pleased to see these made explicit as curriculum aims.

These aims are followed by words on the significance of spoken language, and the importance of justification, argument and proof are also flagged, which is encouraging.

But the devil is in the detail, so how do these ideas play out in the specific programmes of study that follow the ‘purpose of study’?

Programmes of study

Reading through the programmes of study leaves me with a sense that more attention has been paid to fluency than to either problem solving or reasoning. There isn’t the space here to do a detailed analysis of the content, but a word count on key terms does reveal some interesting, and worrying, patterns.

Fluency (or fluent) comes up 16 times and recall (or recalling) 11 times. We know that fluency is important, as it frees up the shortterm memory so that attention can be given to rich problem solving and reasoning. But we also know that problem solving and reasoning play a big part in developing fluency – so I’d expect references to problem solving and reasoning to match references to fluency.

‘Problem solving’ comes up 32 times, but 14 of these refer to ‘word problems’ - all positioned at the end of lists of decontextualized number skills.

This reinforces my impression that it is not rich, challenging and engaging problem solving that is being encouraged, but more practice of calculations wrapped up in words.

The only place where ‘problems’ is not preceded by ‘word’ is in measures or data. And none of the statements about ‘shape’ make any reference to problems at all!

And reasoning? Well, the term pops up. Once.

Yes, ‘reasoning’ occurs in the programmes of study one time only, and this is in the context of reasoning about properties of shapes. There’s no reasoning to be done in number.

But let’s not be hasty, there’s all that stuff in the ‘purpose of study’ on ‘justification, argument and proof’, so how many times do these terms, or variations of them, crop up? Each romps home with zero references. Even ‘explain’ only crops up once and that’s in ‘explain how they derive unknown angles and lengths from known measurements’.

In short, the overall flavour of the programmes of study does not live up to the promise of the stated aims. The ‘purpose of study’ begins to look like something added at the end to give the impression this is a balanced mathematics curriculum, rather than something used as an organising set of principles on which to design a 21st century curriculum.

A curriculum of fluency without reasoning is an impoverished curriculum. Like fish without chips. Romeo without Juliet. Gove with Gibb (on second thoughts…).

But perhaps I’m missing something. Maybe progression between the years will reveal a carefully constructed curriculum.

What about progression?

Another claim for this curriculum has been the need to bring back rigour. Since calculation is at the heart of many people’s definition of rigour, how does progression in number pan out?

Here’s the progression in addition and subtraction from Y2, which is fairly typical:

• Y2: add and subtract numbers with up to two 2-digits (sic) including using column addition without carrying and column subtraction without borrowing
• Y3: add and subtract numbers with up to 3 digits, including using columnar (sic) addition and subtraction
• Y4: add and subtract numbers using formal written methods with up to 4 digits
• Y5: add and subtract whole numbers with up to 5 digits, including using formal written methods

Spot the pattern? I guess, by my great age, I must by now be able to add and subtract whole numbers with up to 53 digits. The strength of column arithmetic is that, once you’ve learned it, it works for whole numbers with any number of digits. Adding two 5-digit numbers may take a bit longer than adding two 3- digit numbers, but it is conceptually no harder

And we know that a solid grounding in being able to add or subtract 2-digit numbers mentally is a better foundation of ‘number sense’ than rushing to set these out in column fashion.

Sadly, this looks more like an arithmetic curriculum rather than a mathematics curriculum. It is biased towards fluency and has an implicit model of teaching as ‘delivery’ rather than helping learners develop a deep understanding.