Standards in mathematics

Another sad reflection on declining standards, this time from an anonymous student.

Mathematical literacy and PISA revisited

Ireland's recent low place in the international "league tables" produced by PISA, especially for maths, has got a lot of attention. This example illustrates the problem nicely.

Fractals

Fractals are infinitely complex and beautiful patterns produced through the repetition of a simple formula or shape, patterns which often appear rough or chaotic and which can be found everywhere in nature (the surface of the sea, the edge of a cloud, the dancing flames in a wood fire). I have written about them briefly before.

What appeals to us in such patterns, perhaps, is the combination of simplicity and complexity. They allow our minds scope to expand, and our imaginations to take off in the direction of the infinite, but at the same time to rest in a unity. It is similar to the reason we love science. Scientists are seeking the simple secret at the heart of the complex - the formula or combination of universal laws that governs all of reality and explains why it works or appears the way it does.

Something similar is happening in art, when the artist seeks unity of concept or meaning or mood in a complex scene or sight or landscape.

Not all beauty is produced by these "recursive algorithms" or the repetition of self-similarity at different scales of magnitude. Sometimes a pattern is just there in the thing and does not repeat itself. But beauty always has something to do with order, which means the finding of a unity of form in something complex - a balance between the Many and the One. The finding of unity gives us joy (which is why we call it beautiful) because it enables us to recognise the Self in the Other, outside ourselves. It causes us to expand our boundaries to include the other thing as grasped and understood, or at least as situated in a relationship to us. Fractal patterns are a version of that experience. We sense the unity, but because it is expressing itself as never-ending complexity, it never gets boring.

Therefore all beauty, including fractal beauty, reminds us of God, who is both infinitely simple (in himself, as pure love) and yet infinitely complex (in what he contains and creates).

Fractals

Fractals are infinitely complex and beautiful patterns produced through the repetition of a simple formula or shape, patterns which often appear rough or chaotic and which can be found everywhere in nature (the surface of the sea, the edge of a cloud, the dancing flames in a wood fire). I have written about them briefly before.

What appeals to us in such patterns, perhaps, is the combination of simplicity and complexity. They allow our minds scope to expand, and our imaginations to take off in the direction of the infinite, but at the same time to rest in a unity. It is similar to the reason we love science. Scientists are seeking the simple secret at the heart of the complex - the formula or combination of universal laws that governs all of reality and explains why it works or appears the way it does.

Something similar is happening in art, when the artist seeks unity of concept or meaning or mood in a complex scene or sight or landscape.

Not all beauty is produced by these "recursive algorithms" or the repetition of self-similarity at different scales of magnitude. Sometimes a pattern is just there in the thing and does not repeat itself. But beauty always has something to do with order, which means the finding of a unity of form in something complex - a balance between the Many and the One. The finding of unity gives us joy (which is why we call it beautiful) because it enables us to recognise the Self in the Other, outside ourselves. It causes us to expand our boundaries to include the other thing as grasped and understood, or at least as situated in a relationship to us. Fractal patterns are a version of that experience. We sense the unity, but because it is expressing itself as never-ending complexity, it never gets boring.

Therefore all beauty, including fractal beauty, reminds us of God, who is both infinitely simple (in himself, as pure love) and yet infinitely complex (in what he contains and creates).

The height of intelligence

The positive correlation between height and intelligence has been documented many times. It is attributed generally to deficits in early life nutrition. According to some this explains much of the height/earnings premium. So is there any evidence on this correlation for Ireland? Using the Growing Up in Ireland data I graph the students mathematics score against their height (adjusting for sex, SES and birthweight) along with a 95% confidence interval.
So yes there is a slight upward gradient but it flattens out above about 1.25 metres, possibly even dipping at the very top of the range, the raw correlation is .114. But on the whole it looks fairly flat and shorter people who want to be mathematicians should not be discouraged.

Some Irish boys are better at maths than girls

Differences between the sexes in educational attainment are of interest to many people. In Ireland, as elsewhere, males are being left behind by females in key exams and university entrance. So where does it all start and is it the same for everyone?
Using the Growing up in Ireland data I look at differences in maths score: I estimate quantile regressions controlling for a bunch of chararacteristics (SES, birthweight, maternal smoking, income and more). These show the effect of being male on the maths score at different points of the conditional distribution : so high quantiles are not "high test scores" but "high test scores conditional on the covariates one has included". I interpret this as proxying unobserved ability, this could be cognitive ability but not necessarily. The outcome is scaled to have a mean of 100 and a std deviation of 15.
The results are striking. Boys do better on average so a linear regression gives a coefficient of about 1.2. By comparison, being right-handed or having been breastfed is worth an extra 1.4 points. But at lower quantiles the effects are smaller and are not statistically significant. At higher quantiles the effect is around 4 points.
So I interpret this as saying that at low levels of unobserved ability it doesn't matter if the child is a boy or girl. But for "smarter" children being a boy is an advantage i.e. being male and being "smart" are complements.

Bonus points for maths

The Irish Independent has an interesting article on bonus points & not just because they quote me.

Handedness and ability at maths: evidence from Ireland

There is a great deal of interest both popularly and amongst scholars about whether cognitive ability is predicted by handedness. The literature contains many findings which cannot be simply summarized and there are many many myths. Evidence for Ireland has been non-existent, as far as I am aware, until now with the release of the Growing Up in Ireland data.
So what can we say? Below I plot the density of attainment at a maths test that the 8 year olds in GUI sat.
Sadly, if you are a ciotóg, you can see the distribution is shifted to the left - but not by much. The good news is that when you look at the distribution of reading ability, there is no difference at all.
In numerical terms, left-handers are about 8% of a standard deviation lower. By comparison girls are about 11.5% of one standard deviation lower.

Bonus points for maths: the new policy

It has just been reported that the Irish Universities Association have agreed a policy whereby students taking Higher Maths in the Leaving Certificate and getting at least a D will get an extra 25 points.
Aside from concerns about the possible effects on equality of access raised by Kathleen Lynch , I am curious about what incentives this new scheme provides. So the idea is to get more doing Higher Maths and presumably doing better all round. Often "non-linear pricing" generates perverse incentives.
So at the high end there is no additional incentive: an A is worth more than a B by the same amount. I would have thought there was an argument for increasing the bonus as one gets a higher grade. Say a student wants to get a certain amount of points from maths. In the past he could have got it from say a C-. Now a D will do (I'm not sure of the exact numbers). Might he be tempted to put in less effort, settle for a D instead, and re-allocate effort to other subjects? So one might predict a clumping of the distribution around D for this paper.
Take another student who is thinking of taking Higher Maths but is worried about failing. The relative penalty to failing has increased (the E-D gap in points) so a risk averse student might think "no thanks". There might be an argument for encouraging students to take the chance by giving some additional reward for getting an E (i.e. a smaller bonus).
Finally,lets say the policy is successful in attracting more students to doing higher Maths. Presumably these will be the people who are moderately good at maths. So on the lower paper we get fewer A's and B's and more D's on the higher paper. Why is this something to be so pleased about anyway?
When the distribution of grades is published next year, it will be a nice little project to compare before and after.

Bonus Points for Maths at UCD

As reported today in the Irish Times, and on the front-page of the UCD website, the UCD Academic Council have decided to introduce bonus CAO points for Leaving Certificate Higher Level Mathematics for a trial period of four years, commencing in 2012. The idea of a trial period is a welcome addition to the debate surrounding bonus CAO points for Leaving Certificate Higher Level Mathematics (henceforth "bonus points"). Instead of making a decision now in the absence of any (quasi-) experimental evidence, why not experiment with this measure and subsequently make a decison? Of course, the best we can hope for in this case is quasi-experimental evidence; one would imagine that randomising bonus points to different students (or schools) would be potentially very difficult. Not to mention the issue of fairness in the college admissions process. But maybe somebody can come up with a suggestion that would address all of these concerns. In any event, the precise scheme for the awarding of bonus points will be decided in the coming weeks; the objective is to have a single scheme for all institutions that will be awarding bonus points.

The UCD announcement contains a lot of additional content; it states that the introduction of bonus points will "only be successful if it is part of a suite of measures to interest students in mathematics, to ensure the best possible teaching and to support student learning... We will research the impact of bonus points to ensure it is equitable and effective." Specific research questions to be investigated are as follows:

* How much time, compared to other subjects and compared to other students
internationally do students spend on mathematics?

* What is the impact of bonus points on the uptake of higher level mathematics?

* Do bonus points have an effect on equity of access to third level?

The UCD statement recommends the introduction of two examinations; one testing basic mathematical competency, which if passed would secure a pass overall and entry to third level, and another to test advanced mathematics ability. This is as much of a distinct recommendation as introducing bonus points, and arguably a measure that could encourage take-up of Higher Level Maths in the absence of a bonus points scheme. Behavioural economists will recognise the potential role of loss aversion in how Leaving Cert. students decide whether or not to persist with Higher Level Maths. Only 16% of Leaving Certificate students take the higher level paper on the day of the examination but almost 40% of students register with the State Examination Commission to take the paper.

Also, the UCD statement identified three possible dangers:

* That bonus points may contribute to increased competition, or a worsening
points race

* That students who do not require high points might not see bonus points as
much of an incentive to persist with higher maths

* That higher level mathematics would not be available to some students,
particularly in schools in poorer areas, and that this would worsen issues of
access to university. (In 2009, 79 schools had no higher-level candidates
sitting mathematics in the Leaving Certificate. According to UCC Registrar
Professor Paul Giller, about half of second-level maths teachers do not have
maths as a major subject in their degree).

Finally, some outstanding issues in the debate on bonus points are as follows:

(i) Tom Boland, the chief executive of the Higher Education Authority (HEA), has asked whether the bonus points would be awarded only to those who are going on to take a third-level course that requires maths. (According to David Quinn from the Irish Independent: "Those who are good at more literary subjects like History or English will be effectively penalised because they'll find it even harder to compete for university places with those who are good at Maths").

(ii) A HEA report on Career Opportunities in Computing & Technology mentions that Higher level maths and certain science subjects (i.e. physics and chemistry) are seen by many students to be particularly difficult and requiring a level of work that is not conducive to the objective of maximising CAO points. Is bonus points for Maths enough, or should there be bonus points for certain science subjects aswell?

(iii) According to the Irish Independent, the existing courses for which bonus points are available -- including all courses at UL and some courses at DIT -- do not have a higher rate of application from higher-level students, nor do they have a higher number of applicants than courses in the same disciplines where the bonus is not available.

(iv) Also according to the Irish Independent, when bonus points for higher-level maths were removed on foot of curriculum reform in 1993, participation in the subject at higher-level actually increased.

(v) According to David Quinn, journalist at the Irish Independent, "those who are already good at the subject will simply get more points without helping those who are currently failing it".

(vi) Kevin Denny has some comments related to bonus points on Ferdinand von Prondzynski's University Blog: "Is the idea that this initiative will propel more students into technical subjects like science and engineering at third level, thus helping the smart economy etc? This might be a laudable objective but it seems terribly naive to think that having effectively bribed students to do more maths in secondary school that they will just keep going i.e. that this will make them change what they wanted to study at university forgetting that they weren’t that crazy about maths in the first place. Maybe, maybe not but it seems more likely to me that they will take the points and run i.e. study what they wanted to study anyway."

(vii) A separate comment by Kevin Denny on Ferdinand von Prondzynski's University Blog: "There is a good basis for bonus points for maths unrelated to these arguments namely that ability at maths is a better proxy for general cognitive ability. So if we think that more able students should get priority, as the points system does, then a student who does better at Leaving Cert maths than another student is better in general even if they have the same total points (i.e. that student will do significantly better in university)."

(viii) A survey conducted by Engineers Ireland found nearly two-thirds of ordinary-level maths students said they would not opt for higher-level maths even if offered bonus points. The online survey was completed by 122 students who sat higher- or ordinary-level Leaving Cert maths this year.

(ix) On a separate but related point, it has been recommended that Maths should be compulsory for CAO points purposes to ensure students persevere with the subject, according to a report submitted to the Minister for Education Mary Coughlan.

The Golden Circle

In chapter 4 of my book I talk about a rectangle inscribed within a circle. Naturally there are an indefinite number of such figures. Take the diagram on the right, kindly produced by Michael Schneider. Look at the outermost circle, and the largest rectangle that lies inside it, touching its circumference at A, B and C. You could move points A and B nearer to the left-hand end of the horizontal diameter of the large circle, or else push them further apart towards the two ends of the vertical diameter, producing an ever-thinner oblong shape. Halfway between these  extremes the rectangle would become a square. But the shape Michael has drawn is a Golden Rectangle, so we can call the whole figure a Golden Circle ("Golden" because of the presence of the Rectangle). The G.R. is famous for being the "most beautiful" of rectangles, possessing the peculiar property that its sides are in the ratio of 1 to Phi (1.618...), so that if you cut off a square portion what remains is a smaller Golden Rectangle - and so forth, forming a logarithmic spiral, as in the following image.

When I wrote the book I was intending to use the Golden Circle as a way of exploring the relationship between Pi and Phi, but I never got around to it. My reason for being intrigued is simple. What we learn from Simone Weil - and what she learned from the Greeks - is that geometry is full of theological meaning. We have forgotten how to make those connections. It is not that we can prove the Trinity or the Incarnation with diagrams, but that the mathematical world is full of analogies that echo theological and spiritual truth. You might even say that mathematical necessities are a portrait of divine freedom, since in God freedom and necessity coincide. The beauties of geometry and arithmetic are a world of metaphors and help to raise our minds towards the contemplation of divine truth. My book only touches on this, but a much fuller and richer account is given by Vance G. Morgan of Providence College in his book Weaving the World: Simone Weil on Science, Mathematics and Love, reviewed here.

The Golden Circle

In chapter 4 of my book I talk about a rectangle inscribed within a circle. Naturally there are an indefinite number of such figures. Take the diagram on the right, kindly produced by Michael Schneider. Look at the outermost circle, and the largest rectangle that lies inside it, touching its circumference at A, B and C. You could move points A and B nearer to the left-hand end of the horizontal diameter of the large circle, or else push them further apart towards the two ends of the vertical diameter, producing an ever-thinner oblong shape. Halfway between these  extremes the rectangle would become a square. But the shape Michael has drawn is a Golden Rectangle, so we can call the whole figure a Golden Circle ("Golden" because of the presence of the Rectangle). The G.R. is famous for being the "most beautiful" of rectangles, possessing the peculiar property that its sides are in the ratio of 1 to Phi (1.618...), so that if you cut off a square portion what remains is a smaller Golden Rectangle - and so forth, forming a logarithmic spiral, as in the following image.

When I wrote the book I was intending to use the Golden Circle as a way of exploring the relationship between Pi and Phi, but I never got around to it. My reason for being intrigued is simple. What we learn from Simone Weil - and what she learned from the Greeks - is that geometry is full of theological meaning. We have forgotten how to make those connections. It is not that we can prove the Trinity or the Incarnation with diagrams, but that the mathematical world is full of analogies that echo theological and spiritual truth. You might even say that mathematical necessities are a portrait of divine freedom, since in God freedom and necessity coincide. The beauties of geometry and arithmetic are a world of metaphors and help to raise our minds towards the contemplation of divine truth. My book only touches on this, but a much fuller and richer account is given by Vance G. Morgan of Providence College in his book Weaving the World: Simone Weil on Science, Mathematics and Love, reviewed here.

Help in teaching math

I have come across a number of books and websites that math teachers may find helpful - or, come to that, teachers of other subjects who want to build bridges for their students to the mathematical aspects of their own topics. There are the classics, such as Constance Reid's From Zero to Infinity: What Makes Numbers Interesting, and H.E. Huntley's The Divine Proportion: A Study in Mathematical Beauty. Several others are mentioned in my bibliography, including Michael S. Schneider's and Clifford A. Pickover's. These books are full of exercises, drawings, puzzles and anecdotes. One book that isn't in my Bibliography because I only just heard about it is Alex's Adventures in Numberland, by Alex Bellos, but it looks fun. Another is 50 Mathematical Ideas You Really Need to Know by Tony Crilly - highly recommended by several readers.

Help in teaching math

I have come across a number of books and websites that math teachers may find helpful - or, come to that, teachers of other subjects who want to build bridges for their students to the mathematical aspects of their own topics. There are the classics, such as Constance Reid's From Zero to Infinity: What Makes Numbers Interesting, and H.E. Huntley's The Divine Proportion: A Study in Mathematical Beauty. Several others are mentioned in my bibliography, including Michael S. Schneider's and Clifford A. Pickover's. These books are full of exercises, drawings, puzzles and anecdotes. One book that isn't in my Bibliography because I only just heard about it is Alex's Adventures in Numberland, by Alex Bellos, but it looks fun. Another is 50 Mathematical Ideas You Really Need to Know by Tony Crilly - highly recommended by several readers.

It Doesn't Add Up

Every couple of months it seems a new (inter)national report is released bemoaning the performance of American students in mathematics or recommending improvements. But what is clear from today's Washington Post editorial on John McCain's voodoo economics is that math lessons didn't always 'take' for past generations of students either. McCain - a self-professed economics neophyte - either doesn't know that 2+2=4 or refuses to acknowledge it. I'm not sure which is worse.