Teaching the brighter end of Year 8 for many years, I learned of some basic errors in mathematical understanding. I share those here in the hope that this will reduce the incidence.

Many of the problems seem in Y7-9 are of course created earlier, which menas that the primary teaching failed. I have no doubt at all that there are similar failures at secondary level, but we generally fail to see those continue past Y11. There is a fuindamental difficulty with Maths (and 1st language English) taht we must continue having what amounts to a lesson per day until KS4 is complete. Considering the wealth of lessons, it is a continual source of embarrassment that so many points of failed teaching (failed learning) exist.

One such extended example can be found in the handling of maths operators. At the simplest level we work with four: add, subtract, multiply divide. We may well teach that these are pairs of opposites. However, both addition and multiplication are commutative [a+b=b+a and a*b=b*a] while subtraction and division are not commutative.

**Addition/Subtratction**

Since we seem to like teaching about the number line, it seems to me that we could cure much of the apparent dyslexia that confuses a-b with b-a by declaring that subtraction is the addition of a negative number, i.e. that a-b means a + ^{-}b. then the order of terms in a mixed addition such as 2-3+7-11+15 can be juggled to 2+7+15 + ^{-}3+ ^{-}11 = 24+^{-}14 = 10.I think the cause of the confusion is that those of us comfortable with such arithmetic have not recognised what it is that we are doing, so the kids see that the numbers can be taken out of order, which actually requires an understanding of why that is permissible.

Incidentally, wouldn’t the number line be better vertically?

Of course, it is when we start using algebra that those who have used calculatores to fix their difficulties reveal their problem. The longer these misunderstandings go unrecognised, the harder they are to fix. Un-learning is difficult. I think that the cures depend upon the student seeing that their learned technique needs adjustment, and then seeing that they can fix this themselves. Open discussion about how the problem occurs (without picking on someone with a problem, but instead discussing how such a confusion can occur) avoids all the embarrassment issues and provides a safe way to fix a problem. On the way, I often fouund that someone with no problem began to see a better explanation for whatever the topic was (subtraction, ordering terms, juggling operators, whatever).

**Multiplication/Division**

The equivalent confusion, before we get to fractions, is caused by teachers using the phrases ‘divided by’ and ‘divided into’ interchangeably, when they are in many ways opposites. a/b is 'a divided by b' and is 'b divided into a’. Since one way of typing this division is a/b, we should (I strongly suggest) keep the left to right concept and ALWAYS choose to say 'divided by’. I go further: ’divided into’ needs to be removed from the lexicon; when children say it one would nicely ask to not use that phrase but to say it the ‘better,' more helpful way. If you are dealing with Chiinese (and I suspeect several other Asian) children, be aware that their language says b\a, ‘b under a’, where English says 'a over b’. This causes an exception to discuss directly with those students and not in front of the very children whose confusion you are trying to eliminate.

**Fractions**

Fractions are difficult if the only previously understood numbers are integers. That is because they don’t fit the framework (schema in demagoguery) that the students have built - you’re moving to rationals after integers. I still say theses can be called ‘unfinished’ while also helpful and useful. I do not see benefit in conversion to a decimal form, though I can tell that this is what many have been taught, and I do see that decimals allow one to recognise continuity For those pupils, fractions break the model that fits decimal reals quite thoroughly. Which seems to me to make the topic really rather difficult. I therefore question whether fractions belong in KS2 at all. See here. I read that the limit is dividing a fraction by an integer. Why go further, then? Issues here about what ‘extension’ is.

I am pretty sure that all the confusion over fractions has, at root, one or both of the earlier described problems above to be cured first. As I have written on my other pages on fractions, only multiplication of fractions is easy. Even then, why that is so is a topic to discuss with a class. Which would you pick? Many of the remaining problems with fractions and other topics have at root the sloppy use of language. It may help to persuade children to see a fraction line as a sort of bracket (which it is) while at the same time it is a result. ¾ is both a value and a calculation to be worked - one that you can choose to leave until later because it may well cancel out with other terms. See Lower School Numeracy, link near the top of the sidebar.

When handling fractions, I have often started with handling reciprocals, 1/n, integer n, treating this as a bit of history from before the ability to do division. Those who think they hate maths can be persuaded to play at some historical role modelling (“Game on: We don’t know how to do division, only how to write down the problem”). So ¾ becomes 3 x ¼, for example, and problems with fractions can be broken down into a series of (commutative) multiplications, which can then be shuffled to best effect, especially if the resulting ‘impossible’ divisions are eliminated. I say that we can recognise a problem and play at avoiding it for a while, putting off the evil moment when we must remove the problem properly.

How do fractions ‘break the rules’? How do you put them in order? That is sufficiently difficult to belong at KS4 - quite right too, as a non-calculator task. Half of something requires the something to be specified - and I bet most of us fudge that detail. As the link points out I can have half of £10 andf you can have half of £100 - we each have half but they’re different somethings. The word ‘half’ is helpful in not have any two-ness, but by the time we get to fifth, sixth and so on, the word does double duty as fraction and as cardinal; more scope for confusion.

**BODMAS**

My test for success with operators is 2+3x5 (any three relative primes, within a spontaneous ‘test’ used to fill a pause in a lesson, such as people turning up late). Those with an issue solve (2+3)x5=25; those with no issue reach 17. This tells you, obviously, that there is some explaining to be done.

In general, my approach to teaching was always to question the thinking. That might be called challenge, but I’d like to say taht was couched as a challenge to do thinking, not having one’;s thinking challenged. I certainly tried to persuade students to explain how they reached an answer (right or wrong). A wrong answer, where possible, needs to be seen as unsatisfactory becasue it fails to check out. In arithmetic, a product with an even number will be even, a porduct with only odds must be odd. Estimation is therefore also a valuable skill; indeed, as teh use of electronic substitutes for arithmetic become more prevalent, so teh ability to check an answer for sense becomes ever more vital and may be the thoing we need to teach most thoroughly.

Examples:

70x1.6 = a) 11.2 b) 112 c) 1120 d) something else (11.20, for instance)

3+5x7 = a) 56 b) 38 c) 15 d) something else (3.5^{7}, for example)

DJS 20180410

Thanks to Sally Prout (as was) for the prompt to write about this. Responses will cause extension of this page and possibly alterations or additions to the Lower School collection. Write to me at 1st name at 2nd name dot net.

At secondary level I remain concerned that we systematically failed to ‘fix’ the equivalent problems we created by poor explanations. While I recognise the spiral technique to teaching, revisiting topics periodically and adding to the total, where a student has misunderstood, the extension lesson tends to drive the misunderstanding out of sight until much later.

For example we teach the suvat equations in some form very early in science. As a mechanics teacher I was forever moaning that they applied this whether or not acceleratrion was known to be constant, that is, the initial test (is the acceleration a constant?) was missed out. The same occurs in trigonometry with triangles having no right-angle. I say this latter could be checked with a scale drawing drawn at speed (but few can do this well enough to help, which strikes me as an invitation for some practical lessons as opposed to theoretical lessons). Similarly, science teachers moaned that the maths dept failed to tyeach aboput graphs, when it would be nearer a truth to say that the needs of these disciplines is different - and that both departments need to discuss what those needs are to mutual benefit; it would be much better teaching to be able to say “In physics we would draw this like so, but in maths we do it like this” and with the right class, give a credible explanation. There may even be a common ground that causes all the departments to agree on a minimum common standard, [axes will be labelled].