JohnbaldJohn Bald says maths teaching is in a mess

Elizabeth Truss' speech to the North of England Education Conference  makes good reading.  Unlike many modern political speeches, it is not a collection of soundbites, but a wide-ranging, cogent argument, putting maths in a broad cultural and educational context, and supporting her central point, that we need to reintroduce the most efficient calculation methods into primary schools, with evidence from high performing jurisdictions such as Singapore, Japan and Hong Kong,
as well as from Her Majesty's Inspectors of Schools.

I will admit to a sigh of relief. Reading the comments in the Telegraph (remember  Yes Minister's  irregular verb, "I give confidential briefings, you leak, he is being prosecuted under S2 of the Official Secrets Act"), I rather feared a straight "back to basics" message, which is fine for those who agree with it, a red rag to those who don't.  

Instead, we had logic, consistency and reasoning, and I suspect the audience were rather nonplussed.  After all, HMI had stated unequivocally that the standard procedures (algorithms) for addition, subtraction, multiplication and division were the most efficient and Labour had fallen out with progressive education almost as much as we have. So, what was there to fight about?

The answer was foreshadowed at a fringe meeting at the Party Conference, attended by several maths consultants whose advice had not been taken. Ministers were entitled to ask "awkward questions", said one, but then should follow advice. Instead, they were listening to people (he said) he'd never heard of and "taking advice from Rasputin".  So, a large number of maths advisers are miffed. Having listened to them, at length, ministers have decided that they are wrong, and they don't like it. In fact, like a collective Victor Meldrew, they don't believe it.

I've been looking for evidence in favour of the methods of "chunking" (division) and "gridding" (multiplication)  that are advocated by the opponents of standard procedures, and until this week I had found none.  An interesting book, "Children's Errors in Mathematics", had analysed the causes of typical errors, such as always subtracting a smaller from a larger digit, no matter what its position, so that 12 minus 5 gives the answer 3 or 13 instead of 7.  It is, however, not news that children make mistakes. What evidence was there that they made fewer mistakes using extended methods?

Enter Norfolk maths teacher Tim Handley, writing as Tom Hanzley on his blog Classroom Tales, which, in addition to expressions of horror and accusations of authoritarianism has links to three studies of 10 year old (Year 5) children in Norfolk carried out at two-yearly intervals between 2006 and 2010  that, he says, show that chunking and gridding are more effective than traditional procedures, aka algorithms. Each study involved the same group of 22 Norfolk schools, with samples of around 1,000 children in each, and their answers to the same questions from the end of year AQA test – 546 +423,  317 – 180,  56×24  and  222÷3. All figures quoted have been rounded to nearest whole number.

In 2006, pupils using traditional columns did slightly better than the others on addition, 95% accurate against an average of 90%, but less well on subtraction (52%) multiplication (29% ) and division (48%) than the most common alternatives (respectively, 86-88%, 65% and 62-67%). These discrepancies are worrying, but the percentages of correct answers for the whole sample, 42% for subtraction, 22% for multiplication and 21% for division, are catastrophic. Over 600 of the pupils had no clear method for the multiplication task, and over 700 no clear method for division. Their respective success rates for these tasks were 6 and 4 per cent.

The 2008 sample saw similar results in addition, an improvement to an overall score of 54% in subtraction (standard algorithms 62-76%, numberline 86%), 28% in division (standard algorithm, used by under 2% of the pupils, 5%) and 29% in multiplication (standard algorithm 48%, grids 62-3%). Once again, over 70% of pupils could not complete the multiplication task or even the simple division task by 3.

The final paper, from 2010, continues the pattern of 90% accuracy in addition (with 97.1% for the standard algorithm), 58% in subtraction (57% standard algorithm, 84% number line), 36% multiplication (21.1% standard, though again 5% of pupils, 61.5% grid method), 33% division (57% standard, 62-65% chunking, 35% with no clear method, but 14-22% success in this category).

This is a complex picture. The 2006 results are the nadir of Labour's national strategies. Steady efforts by the local authority are given credit for the growth of the grid system for multiplication, and of the number line in subtraction.

Even so, over two-thirds of pupils being taught by these methods could not divide by 3, and only just over a third could multiply two-digit numbers. As the researchers conclude, "children are likely to choose their own strategy which is either inefficient, ineffective, or both", and have too little sense of number.

The research has some obvious limitations. It's not a fair test of the standard methods if so few schools are using them,  and if the local authority's efforts are all moving in a different direction. We don't know how well each approach has been taught, or how well children knew their tables, on which multiplication and division depend.  The research shows that any method is better than what a Norfolk guru has termed "adhocirithms",  but it is certainly not proof that the alternatives work better than standard methods.

Elizabeth Truss described her speech's reception as "mixed ", which is not bad going for a Conservative at this forum, and said that a teacher form Leeds made a point of coming up to her afterwards to say how strongly he agreed with what she had said.  So do I.  

The thirteen year old who came to me from the same area, not knowing any tables at all and unable to count without using his fingers, has learned a full set of tables in three months (at the same time as learning to use joined writing and to read to a proper level), and begun to handle all types of calculation, using standard methods.

The one new element I use is French lined paper, which has large squares, provides more guidance for handwriting and makes carrying or bridging much easier in maths.  The last thing I'm going to have him do is start dividing work up more than is necessary, and compensate for weak number skills by drawing lines and diagrams. But there is no hiding the fact that we all have a lot of work to do to get out of this mess.

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