An anecdote demonstrates the perils of using a spreadsheet instead. A friend at a US investment bank contacted me with a ‘tricky problem’, namely solving this (in the context of bonds), then computing the yield (interest rate) from the coupon (repayments). Their software was taking appreciable time to do this, because it was solving the latter problem iteratively — it was iterating an iteration.

I sent him the solution by return of email and he responded that it gave _nearly_ the same result as his numerical solution. I waited a day or so and in his next email he reported finding his numerical error…. They ended up being very pleased with the speed with which their new software put answers on the dealers’ screens.

Anyway, I agree with Tim about the UoM sample papers, which are quite dreadful. However, the _syllabus_ is not bad. If you had to construct a UoM A-level you might well do what they have done: take all the calculus from the A-level core (C1-C4), right up to first-order separable DEs, and then apply it, to modelling problems of which this would be a perfect example.

My (personal) perfect framework for A-level maths qualifications would drop decision maths [modules] entirely, keep the statistics, revise mechanics to emphasize modelling and physical thinking (especially via dimensional analysis and scaling arguments), and apply calculus in two new ‘Modelling’ modules. If there is to be a UoM A-level then I would tweak the syllabus to include some dimensional analysis and scaling, and some discrete equations (following Robert May’s advice) — and replace the crap models with some decent ones. There’s so much you could do with this syllabus – finance (as here), population growth and logistic limits, reaction kinetics, epidemiology,…

In the end I think the argument is for giving mathematicians from HE a greater role in producing the papers (and perhaps the syllabuses). These really need a leavening of mathematical intelligence, experience and good ideas.

]]>The people that seem to retain math skills had some interest in math and took the time to think and learn and to try it out in real life scenarios. They had (and probably still have) an interest in mathematics beyond the need for good marks, beyond the goal of satisfying some academic requirements.

This is one point that ivory tower types do need to remember when debating a math curriculum. An interested person will learn and retain math, even if they start with a high school curriculum that isn’t perfect. Enlarging the base of people who become interested in math is a very noble goal. Nothing would increase the level of math skill in the population like a broad interest in math.

Maybe this updated test is a good way to interest more kids and maybe it isn’t, but you have to admit that it would be great if it did actually manage to interest more kids in math. So in addition to knocking it down, maybe it would be good to come up with ideas for interesting kids while keeping the kinds of skill training that you find important.

Also, anyone with a graduate degree in math (such as myself) must take care to remember that they do not really understand how most people see math. Most people approach math in the way that a mathematician may approach running or learning to dance or mastering the art of the salesman – these are things that give great benefit to those who master them, and yet hold zero interest to the average mathematician. So we must remember that we are talking about the masses here – the kids who start loving math in high school will not be hindered by this test, they will excel anyway. The proponents of this test are talking about taking some of those kids who don’t love math in high school and maybe helping them see why they should get to know it a little better in college. That’s a noble goal, even if they may not be approaching it the right way.

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]]>Algorithms are a funny case, in that students believe they understand a topic if they know an algorithm (and sometimes it may be good enough). The problem is when algorithms start to pile on algorithms and they hit a topic which requires an actual understanding of prior material — say, logrithms — and then their world crumbles around them. (Since we’re talking solely of motivation, having them believe they understand is enough.)

The earliest place this occurs is fractions. Legions of students get the two fraction cases I mention above mixed up, causing them to hit the “stupid arbitrary rule” threshold. This is not helped by some teachers at that level who believe the same thing.

]]>Probably I am overstating my case in my above post. To clarify, I am not opposed to real-life examples. What I am opposed to is the philosophy that teaching of mathematics should involve “relevance.”

It is also possible that I am taking too narrow a definition of “relevant.” As I read the term “relevant,” Gowers’s mortgage derivation above is certainly a real-life example, but is not at all relevant to anyone’s life.

Mathematics as a modeling tool is a wonderful topic, and should be included in the curriculum at all levels; but I believe it is a mistake to claim the models are directly “relevant” to the students. [Are we modeling the physics of the local bridge they cross every day? Well (they say), who cares? The bridge is already built, and even those who designed and built it probably didn’t do the calculations we’re doing!]

I agree with you that understanding is key in interesting the students. I also think surprise and mystery are powerful tools (of course, they can only be used once in a while).

In my experience the students are comfortable with *some* arbitrary rules. If too many manipulations are arbitrary, they certainly disengage (quickly!). However, it seems that most of them also do not wish for *everything* to be explained. Some of the more ruthlessly pragmatic students, in fact, want to distill everything down to an algorithm, and want to understand the underlying reasons only insofar as it helps them to execute the algorithm correctly.

The right balance between when to ask them to understand and when to ask them to accept is a constant challenge, I think.

By the way, I believe your water-pouring link is broken.

]]>Whoa there. Just because it is possible to handle real-life examples badly doesn’t mean they need to be jettisoned altogether.

It it true that relevance does not equal interest. People (often inexperienced with actual classroom experience) will often reach for real-life examples as an easy panacea to classroom boredom, but just like anything else it is possible to have interesting examples and boring examples.

Perhaps most important in interesting the students is that they *understand* the material. I found more than anything boredom comes from a feeling that students are performing an arbitrary series of manipulations; in particular they often attempt things in mathematics they have no natural intuition for. The students need “buy-in”; note the quote above, “internal framework for processing the world”. This does *not* necessarily mean turning 5 and 7 into 5 apples and 7 oranges, but it does mean nothing should appear to be arbitrary. Why is it that 4/5 * 2/3 is (4*2) / (5*3) but that 4/5 + 2/3 isn’t (4+2)/(5+3)? If your students can’t explain on some level, then it is an arbitrary rule.

There was an article in Science last year which travelled the blogwaves which ostensibly demonstrated the advantage of abstract examples in learning mathematics. Unfortunately, the example chosen was horribly confusing and actually worked *against* natural intuition. If the experimenters had instead used a clock example rather than the water pouring one (see the link above) it would go with the inuition of the learners rather than against it.

Avoiding the sense of being arbitrary is sometimes easier to illustrate with the abstract, sometimes with the concrete. In both cases in education they are rhetorical devices and must be treated with the same pragmatism as rhetorical points in a speech; whether they work may depend on context, audience, sequencing, culture, and unfortunately for educators random cosmic rays. Most teachers are familiar with a lesson that comes off as crystal clear in one period, but repeating the same lesson again for a different group somehow turns into mud.

Note also that examples as quoted from the paper above are hardly the only use of models and relevance. One may approach a particular model scientifically, gathering data, and then trying out various models of best fit. One may simply use a model as a demonstration of a concept, like having students build a reflective parabola out of aluminum foil and file folders to illustrate the focus. One can approach a problem as an engineer: we need to build this thing, what mathematical tools do we need? — in some cases this kind of question can be used to have the students themselves discover new mathematics.

One last point is that “fixing” a problem from boring to interesting can hinge on things that might seem trivial to a mathematician but are essential to the psychology of a child. I can guarantee none of my students would be interested in the house example, but a great many are interested in a (roughly equivalent) example involving buying a car. Buying a house is an activity way off in adult-land, whereas buying a car is something they can relate to. Of course, note the caveat above on context; perhaps in New York (where less people own cars) students can relate more to the woes of apartment buying.

]]>1. We humans are not infallible when deciding what we dislike, particularly when we are young. For instance, when I was in university I would have declared without equivocation that I disliked potatoes and writing, and I would have been wrong on both counts. My mistake with respect to potatoes was due to ignorance — I simply wasn’t very familiar with their taste. Once I seriously tried them, I liked them. My mistake with respect to writing was more subtle. In retrospect, I have always liked writing in the sense of converting ideas into words (hopefully presentable, interesting words). What I disliked at the time were the particular writing *assignments* I was being asked to do. In my youth and inexperience I did not distinguish between the two, and the distinction is important.

2. Nevertheless, let us stipulate that humans are at least reasonably good at declaring what they dislike. In particular, let us assume that a large portion of students genuinely dislike mathematics, as it has been taught them. Far more problematic is that we are very bad at declaring *why* we dislike something. In the U.S. (and, based on the blog-post, seemingly in Britain) we have latched onto this concept of “relevance” as the reason we dislike maths. I do not know where it originated, but educators and students alike use this idea as a primary reason for students’ lack of enthusiasm towards the subject. “I do not enjoy this, because I do not see how it is useful in my life.”

I submit that it is FAR from proven that this is the reason for our dislike of maths. In fact, I have become convinced that it is decidedly NOT the reason.

3. It seems to me that “relevance” and “academic study” do not mix well. If a topic is relevant to a person, will they then be inclined to study that topic in an abstract, academic way? I think they will not — UNLESS they are already given to abstraction and generalization by nature. Only someone who thinks like a mathematician will enjoy the abstract modeling of even an IMMEDIATELY relevant topic! I like the kind of calculation that Professor Gowers performed in the above post, not because I think it is useful, but because I think it is FUN. More useful (as has been pointed out) would be a spreadsheet.

4. If we artificially force “relevance,” the students are not fooled. In my experience, all we do is propagate the myth that maths SHOULD have been relevant, while continuing to teach courses that are NOT relevant. And, in the meantime, we are no longer teaching the fundamentals.

5. The DETAILS are not relevant in ANY course (mathematics or otherwise), unless one aims to be a professional in that field. Take, for instance, a history class in which one studies the causes of World War I. Are the specific details of the stories surrounding that topic relevant? Absolutely not! What IS important is the study of human behavior, on both the small scale and the large — THAT is why we study history. The DETAILS, though, are not relevant. One can make such analogies with any subject. Likewise, the *details* of mathematics are not relevant at all, for most students. We should not apologize for (or worse, deny) this fact. Forcing “relevance” in mathematics is doomed to fail, and misses the point besides.

]]>It’s really quite amazing how many people have no sense of order of magnitude, even those who are reasonably capable of calculation. Perhaps we should be teaching Fermi calculations, dimensional analysis, and the use of extremal cases instead.

]]>I apologize for having used the word “context” in “the model you are discussing in the context of the “use-of-maths A’ level ” “, because it may suggest that “the mortgage question would be a suitable one for a use-of-maths A’level”. It was a fault of mine and I did not mean that.

Further, I am now aware that my first two comments, as formulated, are irrelevant, misleading and off the central issues of your topic.

Please excuse me, Professor Gowers.

1. I wasn’t suggesting that the mortgage question would be a suitable one for a use-of-maths A’level — it was more like the first question that popped into my head that might come up in real life and involve some mathematics that was less trivial than addition and multiplication, say. The real point, which was explicitly made by Phil H above, is that it is very hard to think of *any* real-world problem that involves both modelling and mathematics of the right level of difficulty. It seems to me that the sample papers are dishonest: they are pretending to have found appropriate questions, but are in fact not testing modelling skills at all and are making the mathematics very easy.

2. The result is that the use-of-maths A’level is teaching a strict subset of the conventional maths A’level. If I were teaching use-of-maths A’level and wanted my pupils to do as well as possible, I’d teach them the conventional A’level maths syllabus and perhaps give them a quick practice paper just before they took the exam (so that they could get used to the peculiarities of the data sheets — no other reason). And as Anonymous says, someone who takes a strictly easier course ends up strictly worse off. (There is another obvious possibility though, which is to have an easier course that you don’t call an A’level. But that already exists, and the point of the campaign against the new A’level is to keep things as they are.)

]]>I am not against the use of spread sheets. I myself have used a few, but only after being able to do the calculations by hand at least two or three times. I even programmed my pocket calculator for these and similar type of things.

To me, the main point for an engineer is to guess the order of magnitude of the solution, so that he can detect faulty results due to data inputted incorrectly, which is something that may occur.

How can one be better prepared to do that? I prefer the better understanding provided by hand made calculations, during the learning phase. But others may rather go directly to spreadsheets.

Is this really a problem? I assume that many of these friends (like myself) would find an online “Mortgage calculator” to do the computation. In fact, this is probably a better solution then the by-hand computation, since it will be able to solve the discrete problem exactly, factor in nuances such as leap years, and output cool visuals with the numeric results.

Of course, this doesn’t address the group of people who wouldn’t even care to investigate these issues before taking out a mortgage. However, I think that this is a failure of even the current education standards, and goes much more to issues of teaching effectiveness than curriculum.

I would include some of the brightest people I know among the the subset of my friends who I wouldn’t expect to be able to do this calculation. In fact, the only friends of mine whose mathematical abilities present problems for in their work or daily life, are the mathematician.

This said, let me now make a case for the stronger/theoretical math curriculum. First, I think most people would agree that the current curriculum / philosophy is almost useless. We spend 4 years of high school teaching students the (1) field operations of the real numbers (2) the quadratic formula (3) the geometry of the unit circle (aka trigonometry). Despite spending 4 years teaching what I can rigorously derive on two/three sheets of paper, we do it in a way that the people who care about the theory don’t get to see it, and the people who don’t care about the theory forget it a year out of high school.

It is probably true that even in this (basic) form, the current setup adequately serves 95% of students. However, the issue is that if a student wants to go on to become an engineer or scientist then he needs a much better background in math. One approach to this is to simply offer multiple math programs of different rigour. The students who want to become engineers takes the “hard” one and the students who don’t take the easy one. For the most part this is the current system. However, I am slightly against it. First of all, I think there should be a bit of “eat your vegetables”-ness about your education. If a high school freshman comes to you and says he wants to be a ditch-digger and thus doesn’t need to be in high school, you don’t let him quite. So why should we let a 14 year-old decided to pass on math because he doesn’t think he will need it in the future?

Another argument against the two option approach is that the student who takes the “easier” path is strictly worse off. High school isn’t like college where you don’t take class X so that you can take class Y. When there are two math options, the students who elect the easier path don’t get less math in exchange for more Y. They just get less math. Additionally if there is a choice between a theoretical math course and say a “real world math course.” I would make a significant wager that the top student in the theoretical math class would be able to ace the “real world math course exam.” Things like spreadsheet programing are important, but there is no reason they can’t be integrated into a theoretical math curriculum. (When I needed to use a spreadsheet for the first time, I figured it out quickly via the help menu and google. I suspect my ability to do this was rooted in the problem solving abilities I learned while training in mathematics.)

A final (and only half-serious) argument against not having a rigorous (or only elective) math curriculum is based on the assumption that there is some innate component to math/scientific talent. If you assume that great scientists are randomly distributed in our population, then the quality of science drastically drops if a large portion of the population never gets introduced to the subject that they might be exceptional at.

]]>@chunter, the post above I linked to at Dan Meyer’s blog may interest you, as well as this one from my own:

http://numberwarrior.wordpress.com/2009/01/19/hint-tokens-getting-students-to-struggle/

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