I do not like the way statistics is explained in the MEI‘s draft “Critical Maths” Curriculum.

It appears from “Mathematical ideas which students should encounter through discussion of problems” (p. 8 of the document) that students are supposed to learn, understand, and apply the three basic limit theorems of probability theory/theoretical statistics:

* the (strong) law of large numbers — “average in time converges to average in space”;

* the central limit theorem — “all roads lead to Rome” (where “Rome” is the standard normal distribution);

* empirical distribution converges to theoretical distribution (Glivenko-Cantelli theorem).

I would prefer these three goals being presented in a more definite way, and then reformulated in practical terms, with a number of applications. At the moment, there is no any structure here.

A few more comments.

Page 9: Graph of a normal distribution is wrong: it must be strictly

positive everywhere!

“Very unlikely” does not mean “never happens”.

By the way, why there is 2\sqrt{n}, and not 3\sqrt{n}?

]]>*Children who enter school at six or seven – after several years of high quality nursery education – consistently achieve better educational results as well as higher levels of wellbeing. The success of Scandinavian systems suggests that many intractable problems in English education – such as the widening gap in achievement between rich and poor, problems with boys’ literacy, and the ‘summerborns’ issue – could be addressed by fundamentally re-thinking our early years policies.*

It’s a learned behavior because at some point of your life , not even knowing ‘when’ you decided you’re not good at mathematics. And so, your math phobia was born.

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?

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