My son is just sitting GCSE maths now, so we compared this to contemprary papers.

This is a bit harder (differentiation but also integration to find area under the graph etc) but crucially you only have to do three of the five questions in section b, so you can for example get through with an A without knowing any calculus, whereas with modern papers you have to answer EVERY question, so there’s no ecape, if you don’t udertand a particular topic you’re doomed

So I think the modern paper is possibly the tougher

Many people develop a phobia about math because of certain events in their past…
…it could be any teacher who have criticized you or embarrassed you in front of the class when you made some mistakes solving a math problem…
…it could be your lack of understanding of some basic concepts of mathematics which you never caught up with everyone else…
…It could be your parents also who would have scolded you and discouraged you and have put a question mark on your ability to take up mathematics for further studies…
GENDER DIFFERENCE
Another important reason behind Mathematics phobia is believed to be gender difference. A general trend is often seen that girls are more phobic to Mathematics than boys. Girls are basically anxious in nature .Besides most of our elementary school teachers are women and it is seen that they often pass their anxiety or phobia towards the subject to their students and gradually a girl also starts believing that she cannot excel in the subject . From a research conducted on 566 male and 567 female students aged between 11 – 13 years in both single sex and co-educational school it has been found that Humanities and English are mostly preferred by girls while boys are more inclined towards Physical Education , Science and Mathematics .
TEACHER’S BEHAVIOUR
Teachers behavior overtly or covertly influences the classroom atmosphere by affecting their emotional responses, beliefs and behavior related towards Mathematics . Teachers serves as mentor and they are role model to their students .They should be careful enough about how their attitude towards Mathematics can affect a student. Teachers who belief about utility of Mathematics are often found to correlate with either a more positive or negative attitude towards the subject .If a teacher believes Mathematics have little utility in our daily life she will have a negative attitude towards the subject . And this negative attitude will be inculcated to their students. Teachers have to enhance their understanding the subject . Only sound knowledge in the subject will not be enough for a teacher to motivate his or her students. A teacher has to remember, Mathematics anxiety does not come from the subject itself but from the way the subject is presented in the school.

So you get the best of both worlds!

“Here is a fraction with big numbers that you have never seen before: 36/54. Do you think it could be equal to a simpler fraction which you have come across before? Which one?”

The food is now in the child’s stomach, which contains, however, no knowledge of the 6 times table. How will the child digest this food?

This is you a very good problem for a [suitable] child because it is Richly Complex rather than Falsely Simple.

David Tall (27 May 2012) implicitly queried this claim, and I agree with him. It would be better to say, “The act of devising a curriculum is inevitably a top-down process AND a bottom-up process, as drafters select and interpret certain higher objectives and experts in children’s mathematical development relate them to children’s capacities at different ages.”

Next, two related points of criticism. Under the heading, “Learning by heart, fluency, automaticity”, he writes that, “Pupils certainly need to be on top of that limited collection of basic facts and techniques in terms which most elementary mathematics can be understood.” [Page 19]

My immediate reaction is to wonder why he prefers this assertion to the ‘dual’ assertion: “Pupils certainly need to be on top of that limited collection of basic insights and understandings in terms which most basic facts and techniques can be understood.”

I have tutored many pupils from private schools, preparing for the 13+ or for the examinations for school such as City of London or Westminster, who are perfectly capable of dividing one fraction by another (for example) by the usual rote rule but have no understanding whatsoever of why the rule works. Have they by learning the rule by heart and subsequently displaying fluency and automaticity, made any worthwhile step towards being the young mathematicians that I’m sure Tony Gardiner would like them to be? I think not.

Shortly afterwards he remarks, discussing the tackling of unfamiliar problems, how, “On a mundane level, when faced with routine inverse problems … such as simplify 36/54 […] one cannot begin unless THE relevant direct facts are known by heart so that we have a chance of recognising that they are needed (e.g. 36 = 4 x 9; 54 = 6 x 9 …).” [My emphasis]

I will query the definite article and refer back to his earlier admirable statement about ‘levels’: “This is a natural consequence of the recognition that each statement or idea can be mastered to different depths, and our consequent rejection of the interpretation of the statements as if they were rungs in some shared ‘ladder’ up which each pupil climbs at his or her preferred rate.”

Indeed. Strong pupils, of whatever age, may well simplify 36/54 instantly by recognising that both numbers appear in their 9 times table – or indeed by recognising that 36 and 54 and multiples of 18. Being ‘strong’, we hope that behind their automaticity lies a genuine understanding of fractions and ratios.

Weaker pupils however, for example primary pupils, might consult quite a different set of facts. For example, they might notice that 36 and 54 are both even, so they can cancel the fraction, once. They might also notice, by digit sum, that both numbers are divisible by 3; or having already halved both they might notice that 18 and 27 are both multiples of 3 and so they work their way towards the answer 2/3.

It is then a matter of insight – not easily achieved by most pupils – that because both numbers are multiples of both 2 and 3, they must be multiples of 6 also. By no means all of my tutees whom I mentioned earlier have achieved this fundamental insight – in which case, what is the mathematical value of their fluency in cancelling by rote by a factor of 6?

Agreed 100%. It seems to me that, to use an evolutionary metaphor, there are a very large number of beasts in the maths educational jungle, but there is very little interaction between them. Most of them exist very comfortably in their individual ecological niches and do very little to engage with other animals. The consequence is, inevitably, a lack of evolution and an overall stasis, only marginally disturbed as one animal after another opens its jaws to roar or shriek or howl. The other animals at best nod or paw the ground – but then return to their comfort zone.

There is very little argument, indeed, within the mathematics education community, or rather within and between its various parts. Teachers engage in little argument with each other, and are ignored by the academic research community, which ignores them in turn. Academic departments continue with their research, usually pursuing their own particular interests, also with very little interaction with those who disagree with them. The larger beasts in the academic sub-jungle attend yearly meetings of PME and other prestigious organisations, but these do little more than allow them to report on their current research agenda prior to returning to their lair to continue on the same lines as before. Occasionally a committee is formed by plucking one designated member for each of half a dozen or a dozen organisations, which committee then deliberates, once or twice a year, its deliberations are usually secret, and anything finally published is more or less ignored.

The only active beast in the jungle is Michael Gove, the Minister of Education, who like most ministers possesses as one of his qualifications for the job, no qualification whatsoever in education, apart from the fact that he was once a boy himself.

We all know that the modern mathematics education ‘community’ cannot possibly resemble the Athenian agora in which, according at least to caricature, great minds strolled in the sunshine disputing over the deepest questions in philosophy while their disciples hung on their every word. The opposite situation, as I have sketched it, however, is all too easy to achieve.

What we need – and what we must hope can surely be created – but how? – is an institutional and social means by which mathematics educators can actually behave like the scientists they are supposed to be and test their theories, vigourously and penetratingly, against the theories of others.

That way progress lies and potential evolution. If we fail to get our act together, then big-game hunting. Michael Gove will continue striding through the jungle taking potshots at whatever target takes his fancy, and we shall have no one to blame but ourselves.