John Bald/language and literacy

Maths Made to Measure - top drawer HMI.

HMI's latest report on maths brings together the key points made, or sometimes hinted at, in their work over the past five years. The pattern emerging from their evidence is one of isolated examples of excellence in a sea of inconsistency in teaching, caused by staffing problems, poor national guidance and, above all, failure to match teaching to what children of all abilities most need to learn. 

Polite as ever, the criticisms are devastating. The national numeracy strategy has focused teachers attention on the topics required for each year, but not on the underlying concepts that are the foundation of progress in mathematics. Interim calculation methods rather than the more efficient traditional algorithms have a life of their own, leaving some children in secondary schools using multiple addition because they don't know  tables.  Some teachers simply don't know the subject well enough, and pupils not in top sets may well not see a qualified maths teacher at all. Schemes of work are taken off the shelf and not matched to the needs of pupils, even at GCSE and A level. Multiple resits are used to ensure C grades at examinations, and not enough pupils are reaching A and A*.

Read it and weep? No. In the words of the prayer book, read, mark, learn and inwardly digest. This report is the most important contribution to mathematics education I have seen since I started teaching in 1973, and should be used to rebuild the subject. Jane Jones and her specialist mathematics colleagues in Her Majesty's Inspectorate have done the nation an important service.

Understanding the two times table.

It may appear that there is not much to understand, or that understanding doesn't matter, provided you know it. The first statement is true, and this is an advantage. We can (when we are young) climb stairs one at a time or two at a time. We tend to do the latter to save time. Counting in twos is faster than counting in ones - ten steps get us to 20 instead of 20. That is the most important thing to understand, and it applies to other tables - they enable us to get around faster, and, once we have learned them, to go straight to the top of the stairs. Zoom me up, Scottie! We can also jump straight back down again without breaking our neck, or stop anywhere inbetween, like a lift.

These things are not hard to understand, and are obvious once we've thought of them. Explaining things does not have to be conplicated, but it makes work meaningful rather than mindless. 

 

 

"A brilliant tip..."

...said the mother of a genuinely brilliant six year old - six years three months, to be precise, and already reading Alice in Wonderland, though having trouble with tables. Splitting up the twos into groups of five at a time, and coming back to reinforce hesitations after an interval knocked out the twos and then we worked on threes. Met the parents again ten days later - "Had those tables stuck?"  Yes.

And my pupil who had severely injured himself is now doing well, reading on his own, and tackling words like exhausted with no problem. Exhausted - have to know not to pronouce the h, unlike exhale, and to put the stress on the second syllable, unlike excellent. Also the slightly difficult au pattern. Nice to see him putting all three together. 

National Numeracy - a solution, or perpetuating the problem?

I became involved with numeracy at the request of people who had trouble with it. My approach was to identify what they were having trouble with, think about why they were having the problem, and explain it to them so that they could do it. The problem was almost always with how to carry out a calculation, and the solution was to unpack the base-10 system, explain it and practise. I have a high success rate, though as all of my work is with individuals and as I don't have millions from a mega-charity, the work can be dismissed in various ways by people who have never seen it, most recently Derek Haylock.

Progressive maths teachers told me that the problem was not with doing the sum, but with finding the right sum to do. My students patently could not do the sum, and I believed them and not my maths colleagues. I still do.

National Numeracy, set up with more millions from a  mega charity, part of the pseudo-governmental "fourth sector",  is repeating the old argument, saying that the problem is with the emphasis on computation in schools, which requires people to pay attention but not to think. 

This is as wrong now as it was in the early eighties. Computation, calculation, sums, however they are termed, are not being over-taught, but are not taught well enough. The key systems, recognised by HMI as the most efficient, work perfectly well provided they are clearly explained so that children know what they are doing and above all don't lose their place. The progressives' error has been to take the problems  people find with calculation as an excuse to abandon teaching them, rather than to rethink teaching and tackle the problem. This is why the previous government's numeracy strategy was still having to put out materials to teach the 2 times table to 10 year olds, who should have been taught it three years earlier.

The solution begins with practising addition combinations of single digit numbers, so that they become automatic, and then the two-times table, which must be taught very carefully to establish a pattern for the others. At this stage in particular, we do not move to the next step before the one we are teaching is fully understood and secure. Then other tables are relatively straightforward, and we can proceed to multiplication and division. Once these are in place, the remainder of numeracy up to GCSE is very simple indeed, as much of it has been designed to compensate for weaknesses in calculation skills. The alternative is to leave people with their fingers and thumbs, and to hope they press the right button on a calculator.

Modern maths and Derek Haylock. Jargon versus progress. Updated

Stanislas Dehaene, in The Number Sense, argues that infants have a primitive sense of number, which is reflected in undeveloped societies. He focuses very strongly on numbers 1 to 3, which seem virtually universal in ancient notation and even in their origins in arabic numerals - 2 is two horizontal lines joined, 3 is three horizontal lines joined, see the comparable Chinese notation.

Whatever goes beyond this is cultural, and the issue for schools is how to maximise this development. Numicon helps by consolidating the initial idea of numbers as something to be counted up into units themselves, so that 3 becomes recognised as 3 and not just as the third number as we count. This criterion of help to the learner should be the touchstone against which all teaching is judged. One of Dehaene's comments is that school "plays a crucial role not just because it teaches children new arithmetic techniques, but because it helps them to draw links between the mechanics of calculation and their meaning".

So far, perhaps, no argument. But the rub comes when we consider the techniques presented to teachers to enable them to do this. Very few primary teachers are graduate mathematicians, and mathematical terminology can either help or confuse them. Even the term algorithm, to denote a method, is not part of normal language - I've never heard a teacher use the word, though I've heard it often from specialist advisers. The distributive law, which allows calculations to be split up and carried out piecemeal, is the basis of much of arithmetic, but I've not yet known of success or failure in teaching calculation to be affected by knowledge of it or ignorance. To explain things clearly to children, we need first to explain them clearly to teachers, and blinding them with science does not do this.

So, I find myself somewhat indignant at the arrogance of Derek Haylock, author of the bestselling and overly influential Mathematics Explained for Primary Teachers, dismissing the efforts of "non-mathematicians such as Bald" to help children tackle problems that are making their lives difficult, and which can be solved with a little careful teaching. I was not born with a vocation to teach the two tmes table to ten year olds, or to teach adults to do basic arithmetic so that they could pass the police entrance examination. I started on the work, and will continue with it, because they need help and no-one else was prepared to provide it.

Mr Haylock complains at the following quotation comes from what he calls "an old edition" of his book

I do not intend…to explain long division. In my view it is a method that could well be laid to rest in the twenty-first century. To be honest, this is mainly because I have been singularly unsuccessful myself when I have attempted to teach the method. (3rd edition, P101)

His complaint is somewhat economical with the truth. This edition was replaced last year, and all of Cambridge University Education Department's ten copies of it are out on loan (it has four of the new one, also all out).  Mr Haylock has now included one example of long division in his new edition, and has, withdrawn the confession that he can't teach it.  His alternative, continued subtraction, takes 13 lines instead of 6 and then leaves a sum to do at the end. It also breaks down once he has to move into decimals, whereas the compact version does not.

In fact, all of his alternative calculation methods are longer than the original. If you don't know that 6 x 8 = 48, he suggests starting with 6 x 2, doubling it and doubling it again. Three steps - provided of course that the child has been taught the 2 times table, which we cannot assume. Mr Haylock pays lip service to tables, and says everyone should learn them, but his guidance on the matter is skimpy to put it mildly. If a child does not know what 6 x 8 are, at an age when he or she should know this, then they need to be taught their tables, and not to make do and mend with 2s.

Mr Haylock points out that he is a mathematician, and I am not. Neither am I going to defer to someone whose solution is more complicated than the problem it seeks to solve.  The neglect of calculation by progressive education needs to be put right. Developing an alternative to these parts of Mr Haylock's book would be a good start.

Maths - is long division easier than short? HMI, and how French lined paper helps with working figures.

HMI's recent paper on good practice in primary maths is less positive about division than about other aspects of maths. Long division is particularly controversial, and the leading textbook for primary teachers, Mathematics Explained for Primary Teachers, specifically and deliberately refuses to discuss the topic at all. There are something like twenty copies of this book in Cambridge University's education library. 

Division amounts to five steps:

• estimate the number of times the divisor will go into the dividend, beginning at the left, and continuing until it will go at least once 
• enter this estimate in the answer area
• multiply the estimate by the divisor
• subtract from the dividend
• move to or bring down the next figure

The process requires fast accurate knowledge of tables, accurate subtraction and good co-ordination so that you don't lose your place. HMI list short and long division. In short division, pupils hold everything in their head, except for figures they carry over to the new number. In long division, everything is written out.

The paradox is that long division can be easier than short. As everything is written down, pupils have less to hold in their head, and can therefore concentrate on doing one thing at a time and not losing their place.  The pupil whose earlier work is described in this link has just got right his first long division calculation dividing a three-digit number by a two-digit number, to two places of decimals. 

He could not have done this without the guidance provided by French lined paper. It has fairly large squares, with three narrow, feint horizontal lines within each square. These allow working figures to be inserted that are clear enough to read, but small enough not to cause confusion with the main figures of the calculation. So "carrying" becomes much more manageable.

HMI note in their favourable report on maths in Finland, here, that pupils' books often include squares.The Finnish squares hold only a single digit.  Download Finnish squares. The French paper's capacity to include working figures is a key advantage. Children for whom calculation is not straightforward often have less neat and well formed handwriting and figures. A little extra space, combined with the extra guidance of the feint lines, can be a great help.

PS - a senior mathematician has gently corrected my earlier use of the term "sum" in relation to division rather than addition. I accept that Macavity, The Mystery Cat, who was to be found doing complicated long division sums, is not good authority for mathematical terminology...

Place Value - or losing your place? French-ruled paper can help.

Since decimalisation, place value to two places of decimals has become easy. Everyone knows that 1p is not much or 10p either these days, though it is of course better than 1p, which you might not bother to pick up. A pound has become the unit, and everyone knows that £10 is a lot and £100 a great amount. So, we work to two decimal places all the time, and HMI have quite rightly commended money as a way of teaching place value. Children understand it.

But move from this to putting figures on paper, and we have a different problem. The range of problems that children have as they do this, and the errors they make, encompass virtually any way you can think of to get something wrong . I have personal experience of this. In my  Year 7 arithmetic exam, I interpreted my father's advice that it was more important to show your working than to get the right answer, by carefully and neatly writing out every single sum and then not actually doing any of them. It took a concerned visit from a parent to persuade the school that I really could add up. They, alas, then took that too literally by putting me in a top set, doing work I could not understand (in geometry. I  understood it in my thirties).  This propensity to get things wrong has, my piano teacher tells me, been very valuable to me as a teacher, as it enables me to understand my pupils' problems. The poor man has to think of something positive to say. My wife sympathises with him.

Well, I've hit on the idea that the sums involved in arithmetic to two places of decimals are not that difficult, provided children know their tables, and so I spend a great deal of time teaching these and practising them. But then the problem seems not so much to be in the maths, which are often based on single digits, but in losing track of the sum, losing your place, leaving out a step, forgetting to put in a 0, etc.  Pupil in Y8 (with statement for dyslexia, in my view unjustifiably) is prone to these errors, and I've developed for him a series of steps for division which I've posted below and which he finds helpful

French ruled paper (a sample here) helps him to keep on track better than anything else. The main squares are big, so his largeish figures don't throw him out of track with columns, and the series of steps and improving tables mean that he is beginning to handle division by a single digit to two places of decimals  accurately. Rounding up was a problem though. P understood that you rounded up the second digit, or left it untouched. He did not, though, delete the third placed digit. Another small step, quite a lot of practice, and he'd got it by the end of our session. Maybe doing things like that myself helped me to understand how other people do them, and to adapt the explanation accordingly. 

French ruled paper is equally useful for helping people with weak handwriting - the feint narrow lines between the main squares are very good for establishing consistent size in the body of letters.

 

 

An interesting Saturday morning

To take the last pupil first, the parent who wrote me the kind email two postings below brought her daughter, aged nine, one of the quickest learning pupils I've known. Explaining English spelling took about twenty minutes, and she was able to apply the principles immediately. The pupil had been taught little or no phonics, and was using her high intelligence to guess at things. She had worked out some patterns - eg ea in sea, tea etc, but not in head, bread etc, and did now know ou (ground, found) or ow. own. The explanation that voice letters - vowels -are overworked, and that we need to know what they are telling us in particular words, did the trick, and we worked on each pattern that she did not know as they arose in the text - this is a perfectly reliable way of covering phonics, as the patterns are so imbued in the language that they will all turn up in a few pages of text. In this case, we used the first book of the Narnia series.

The approach also avoids a huge snag that arises in using simplified texts that focus on just one pattern at a time - when we read, we need to recognise and use each pattern quickly, with immediate support of a whole series of the same words. Too many phonics schemes do not enable the learner to do this.

Mother and daughter set off on their two-hour drive home with a happy wave, and left us with a request to see her nephew, and a cake.

Pupil 2 was having trouble with arithmetic, his teacher no doubt having seen that moving him from simple division to three places of decimals last week was too big a jump. We went back over smple division, and worked out the steps he needed to take, as follows (in shorthand):

To divide, say 162 by 8:

Set out the bus stop

Try - find the first number 8 will go into - in practice, move along the line till you reach a number that is 8 or over. It helps to add a zero each time you can't divide a number - doing this in the hundreds column is redundant, but helps you remember to do it later on, when it is important.

Add to the answer (in the correct column, on the bus stop roof)

Multiply what you've added to the answer by 8

Subtract from the number we are dividing

Bring down the next digit

Repeat

So, dividing even by a single digit number takes five steps that have to be repeated accurately and in order. Understanding the importance of columns in denoting tens, hundreds etc is important, but you also need to make sure you don't miss a step, put a figure in the wrong place, or take away the wrong number. These are matters of co-ordination rather than maths - we have to know in our mind exactly which stage of the process we are at, and not lose our place.

We made some progress - the pupil can use his tables, if hesitantly, and understands correcting to to places of decimals. We will see if he manages the half dozen I left him for homework, using the steps as set out on a card. One thing is certain - if you don't know your tables, and can't subtract accurately, you can't do division at all, which may be why it has been abandoned in so much of primary mathematics.

Pupil one is preparing for entry to a local private school, and is not being taught to write properly at his primary school - in effect, he is doing very little writing at all. I'm using the Galore Park English books with him, selectively - even they are capable of occasional improvement - and he is beginning to control punctuation, though he slips from past to present time without knowing what he's doing. He is ten, and the Galore Park book is designed for seven year olds, but we have to start with what he already understands. My thoughts on presenting grammar in ways that are easier for children to understand are here. 

Carol Vorderman - one cheer.

Carol Vorderman's report on maths has the excellent idea of separating the arithmetic needed for adult life and work from the more abstract pure maths that is needed for other purposes. Unfortunately, it doesn't say enough about how skills in arithmetic develop, or about how they are best taught.

Above all, I can find no mention of multiplication tables. After accurate addition and subtraction, multiplication tables are the key to accurate calculation. Without them, people can only add and subtract one digit at a time. How tables are best learned, and their contribution to calculation, requires research. The reason we  have no research evidence is that large numbers of academics don't consider arithmetic to be the basis of mathematics, and don't consider tables to be important. They are wrong, and we need some proper investigation of the issue. 

Tables update 3.

Scroll down below for an account of Peter's progress in learning tables over the past few weeks. Today, he arrived and had worked on his 7s. I asked him which one he thought I was likely to ask for first, and he said four sevens were 28. I then asked him all of the other items in the table in no particular order, and he got them all right. Excellent - in one week.

So, eights, I said, were easy, as the last digit went down by two each time. We said the eights, and he agreed that they were easy. Nines were easier, as the last digit went down by one each time. He picked up on this quickly, with only some hesitation on twelve nines are a hundred and eight. So tens were easy too, 11s were, in effect, the one times table said twice, and twelves, the one times table with a two. So, five new tables in a little over five minutes, with no stress, and to be practised for next week.

...lemon squeezy!