There has been much chatter recently about early life experiencs of mathematics and the effect it has on subsequent learning and understanding.

Hence this research is rather well timed.

It would be interesting to survey how wide spread an early introduction to the subject correlates with later academic achievment in the subject.

(Personally I certainly remeber at comparatively young age forcing my dad to break a sweat struggling to remeber school-boy trigonometry)

Comments, Commendations, Castigations…

First my apologies for havnig been so quiet as of late – I’ve been busy.

The status of the subject area known as combinatorics within wider mathematics is a bit of a strange one. It seems to exists in a sort of ‘glorious isolation’ very much seperated from other subjects.

Whilst most mathematicians would be happy to attend the British Mathematics Colloquium combinatorialists seem to insist on holding an alternative event of their own, the British Combinatorial Conference, instead. In the UK most mathematics events are funded by the London Mathematical Society whilst combinatorics event tend to be funded by the British Combinatorial Commitee.

So what is the cause of this oasis in mathematics?

Well from the outside there appears to be a certain level of snootyness – my supervisor has certainly aired the view that in any other area of mathematics one frequently encounters problems in combinatorics, but when this happens you simply solve the problem and move on. There is a view that many people doing combinatorics are simply doing the mathematics that remains when the real meat of an impotant problem is removed.

From within there appears to be a very insular attitude. The combinatorialists, certainly within my university, appear to actively discourage their students from taking any sort of interest in anything (lecture courses seminars etc) outside combinatorics.

This cannot be healthy. It is well known that many of the most fruitful pieces of mathematics come from taking the ideas and techniques from one area and applying them to another. Indeed had combinatorialists acted as they do now fifty years ago then the probablistic method, a powerful means of proving results in combinatorics, may never have been discovered.

This is particularly worrying given the applicability of combinatorics to real world situations such as problems of experiment design encountered in statistics or combinatorial optimisation as well its many uses within pure mathematics.

What should be done about this and how?

Comment, Commendation, Castigations…

Whilst the older generation like to hark back to more traditional 3Rs teaching the simple fact is that it is simply not appropriate for the modern world (oldstyle teaching itself not necessarily better than current practices). (The same recent “Now we are 50” report has already been discussed on Mathematics under the Microscope, so I won’t go into it here.) The teaching of mathematical ideas such as the rudiments of statistics being a case in point. The current media hyperbole about knife crime – worsened by the stabbing of a relative of a celebrity – is a prime example. A recent piece on the BBC news website, Safety in numbers, reveals official statistics showing that violent crime is actually decreasing! Clearly a good example of why mathematics teaching in the UK needs to be improved!

It has just been reported on Mathematics in Australia that in true Aussie style, they have just won the inaugural Mathematics Ashes.

Unlike the cricketing tradition, however, its interesting to note that history shows the United Kingdom would generally have won most years had the ashes been contested over the past couple of decades.

Having helped teach linear algebra classes to undergraduates several times I am often faced with a basic lack of understanding of some basic concepts. I have often found that a real “penny dropping” moment comes when understanding the following elementary exercise:

Let be a vector space and let be a linear map. Let denote the eigenspace for the eigenvalue and denote the kernel of . Show that .

If a student knows that this basic fact is true, then many other things become much easier to eaplain. I would seriously recommend more lecturers setting this as an exercises often.

Comments, Commendations, Castigations…