Calculations: foundations of mathematics?

The myth that the mastery of the processes of written calculations, particularly long multiplication and division, is fundamental to doing mathematics continues to be perpetrated by people with political influence and control of our school curriculum. This myth was exposed for me recently in the experience of helping one of my grandsons prepare for his A-level mathematics examination.

We worked together through loads of questions from past examination papers – pure maths, mechanics and statistics. I made the following observations.

Not once in doing A-level mathematics was he required to do a written calculation, since he always had a calculator to hand. His calculator skills were stunning and showed real mathematical understanding, in terms of processing the steps of a complex calculation in the appropriate order, in selecting the correct function keys and handling brackets, and doing this with speed and accuracy.

More important than written calculation skills were the ability to interpret the calculator answer and checking whether it looked reasonable. Additionally, with a little encouragement from me, he improved markedly in using mental strategies for calculations that could be done more efficiently that way than by resorting to the calculator.

But, I repeat, not once did he use a formal written calculation procedure. Yet, there he was doing advanced level mathematics! If he had had to take his eye off the structure of the problem to do a written calculation it is very likely that he would have lost his grasp on where he was going.

For centuries mathematicians have devised ways of avoiding or reducing the demand of written calculations, simply because they get in the way of the real mathematics and effective problem-solving, and take up too much of your precious time. So, we had Napier's bones, and logarithm tables and slide rules, and so on. Now we have modern technology, so please let's give younger children the chance to use it and start doing real mathematics.

The heart of mathematics:

In an interview about the new primary mathematics curriculum, Debbie Morgan, Director of Primary at the National Centre for Excellence in Teaching Mathematics, has stressed that mathematical reasoning and problem-solving are to be at the heart of children's experience of mathematics.

http://www.youtube.com/watch?v=I4aDZEZaF_A&feature=youtu.be

She argues for the prominence of the three aims given at the start of the mathematics section of the document in determining what children learn and how they learn it. These three aims are, in summary, (1) fluency based on conceptual understanding; (2) reasoning mathematically; and (3) problem-solving.

All this would be great, if it were not for the fact that the detail in the subsequent pages and pages of statutory requirements focusses almost entirely on the first of these and contain very little to indicate how precisely the other two are to be developed.

Two factors that will be crucial in this are the end-of-key-stage 2 national tests (the so-called SATs) and Ofsted's approach to inspection.

Over the years those responsible for the national tests have at least identified aspects of 'using and applying mathematics' as defined in the current curriculum that can be assessed in the context of written tests. This has been possible because, for all its deficiencies, the current curriculum does actually  contain specific statutory requirements for children's learning in this respect. I fear that the likelihood is that there will be political pressure on test developers for the 2016 tests onwards – when Gove's new curriculum will start to be assessed – to emphasise disproportionately the assessment of written, formal arithmetic skills and to assess only the detailed statements in the programmes of study. Is there any hope that aims 2 and 3 will be not be overlooked entirely in the national tests? If they are then teachers will overlook them in their classrooms as well. And Debbie Morgan's laudable aspirations will prove to be a fantasy.

And is there any hope at all that Ofsted will give the highest endorsements to those schools who seek to embrace all three of the aims in the experience they provide for children? Will they be checking that children are engaged in genuine mathematical reasoning, following a line of enquiry, conjecturing relationships and generalisations, developing arguments, applying their mathematics to non-routine problems? Or will they just be checking that Year 4 children can multiply a 3-digit number by a single-digit number using the formal layout, that Year 5 children can multiply a mixed number by a whole number and that Year 6 children can divide a fraction by a whole number?

Generalising Luke's problem

I have just had a wonderful week walking in the area of Agen in southwest France – with some time to relax. Spurred on by my energetic and delightful sister, Brenda, I have been reflecting on how to generalise Luke's problem (see my two previous blogs). This is what I have come up with.

There are no two- or three-digit solutions.

There are solutions for any other number of digits, but only for multiplying by 4 (I am not including the trivial solution of multiplying a number like 7777 by 1).

The solutions are as follows:

2178 × 4 = 8712

21978 × 4 = 87912

219978 × 4 = 879912

2199978 × 4 = 8799912

21999978 × 4 = 87999912

219999978 × 4 = 879999912

...

and so on, ad infinitum!

What a nice problem that proved to be! Thank you, grandson Luke.

King James Bible becomes a mathematical problem

To celebrate the 400th anniversary of its publication in 1611, I am organising, in association with the Millennium Library in the centre of Norwich, a public reading of the King James Bible – an astounding work of literature and scholarship that has shaped the language, culture and values of this nation.

Planning the schedule for reading this has proved to be an interesting experience of mathematical problem-solving. First there was a clear goal: to read the entire Bible in public, starting at 12 noon on Sunday 10 April and concluding at 4 pm on Sunday 17 April. And there are some givens: the number of words in the Bible and the fact that we could do this only during Library opening hours. The whole thing will have to fit into 72 hours. There are other constraints, such as making breaks from one day to the next at appropriate points.

A key variable is the number of words of 17th century English per minute that can be read on average by experienced readers nominated by local church leaders. After trials I decided on 170 words per minute. Lots of sampling involved in this! Multiply 170 by 60, that's 10,200 words per hour, which gives 734, 400 words in 72 hours. That's not allowing anything for time lost in changeovers between readers.

This raises a problem: there are actually 783,137 words in the KJB! (No, I didn't count them myself!) Solution: read it faster or change the goal? I decided to change the goal, so I have edited out the repeated text and some of the more baffling passages in the Old Testament, to get it down to around 700,000 words, which gives us a bit of flexibility.

The rate of 170 words per minute allowed me to estimate (more sampling) that on average we would read 17 pages of my copy in an hour – which is 3.53 minutes per page. This became the key number in solving the problem, which enabled me to determine how many pages should be read each day and to calculate an approximate time for the start of each section of the reading. To avoid awkward breaks between days, I used the 150 psalms fairly flexibly to fill in gaps!

The problem-solving involved a process of iteration, using 'trial and improvement': with changes being made gradually here and there, with the plan day by day getting closer and closer to a solution that satisfied all the constraints.

Well, the problem is solved, the schedule is done and it's now all safely saved on a spreadsheet! If you happen to be in Norwich that week (10–17 April), pop into the Library in the Forum and have a listen.

Anniversaries

Christina and I celebrated on Saturday our 45th wedding anniversary. Yes, 45! We married very young, of course.

Is it unusual that a 45th anniversary should fall on the same day of the week as the original event?

I'll leave that as a little problem for readers to ponder as in ten minutes time I head for Stansted to fly to Bergerac for a short walking holiday with some of my family. You'll need to find a way of coping with those annoying leap years.

Au revoir.