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** Peter Bottomley (Worthing, West):**
The hon. Gentleman will have the support of everyone in the House in the tribute that he has paid to the Speaker's Secretary. He might also want to know that the Speaker's Secretary was a classicist and therefore rather better at Latin and Greek than he necessarily was at quadratic equations.

** Mr. McWalter: **
I am grateful to the hon. Gentleman for his intervention and his good wishes to Sir Nicolas. I take the liberty of dedicating a debate to Sir Nicolas because he has shown a strong interest in what one might call the non-conformist debates that have characterised the House from time to time. He has encouraged me to raise with the House the vital philosophical questions that Governments of all persuasions find it too easy to ignore. Despite the mathematical title of the debate, my aim is a philosophical one—it will be an essay in the philosophy of mathematics—and one main objective that I hope to secure is that Sir Nicolas will indeed find it enjoyable.

I put this matter on the agenda today because I have been troubled since the president of a teachers' union suggested a couple of months ago that mathematics might be dropped as a compulsory subject by pupils at the age of 14. Mr. Bladen of the National Association of Schoolmasters and Union of Women Teachers was given a lengthy slot on the "Today" programme to present his views. He cited the quadratic equation as an example of the sort of irrelevant topic that pupils study. I had hoped that the Government would make a robust rebuttal, but there was no defence either of mathematics in general or the quadratic equation in particular.

If such assertions are left unrebutted, what was an ignorant suggestion at one time can become received wisdom a very short time later and an article of educational faith a short time after that. I wish to short-circuit that process and provide a rebuttal of that union leader's suggestion. I note that he was a maths teacher, but I do not regard the desire for changes that simply make the teacher's job easier to be in the best union traditions. He was happy to teach maths to those who enjoyed it, but he wanted to stop teaching maths to those who did not. By defending the centrality of the quadratic equation to mathematical education, I hope also to submit some thoughts on what we would be missing if we allowed mathematics to be regarded as a subject of no greater worth than any other subject on the curriculum.

When I proposed this debate, it was arranged that the Minister for School Standards would respond. I submitted a draft to his office, but I note that he has taken flight, so I congratulate my hon. Friend the Minister for Lifelong Learning, Further and Higher Education on having stepped into the breach. I hope that the absence of the Minister for School Standards signifies nothing other than unavailability, as opposed to possible hostility to what I am about to say.

I hope that you will forgive me, Madam Deputy Speaker, if I remind the House what an equation is, and then what a quadratic equation is. Of course, we all know that there is a strong appetite for equations in the House—witness the large assembly gathered in the Chamber, as well as the large number of hon. Members who seem to have mastered the mathematical material employing the calculus in the 18 volumes of background support papers for the Chancellor's recent statement on the euro—but it will come as a surprise to hon. Members who are interested in these things to hear that not everyone has an appetite for them. Indeed, it is said that Sir Stephen Hawking was told not to put even a single equation anywhere early in a book of his that sought to popularise science on the ground that one equation would immediately halve its readership. Apparently, the casual reader flicking through the book in a shop would put it back on the shelf if he or she saw the offending line of print.

What are these equations? From an early stage in primary school, we were given problems such as "If *x* + 5 = 7, what is *x*?" You will notice, Madam Deputy Speaker, that I am not making the problems too difficult at this stage. Since the time of Descartes, it has been customary to use letters from towards the end of the alphabet for such unknown quantities. Later on, at about the age of 11, we will grapple with so-called simultaneous equations, where there are two or more unknown quantities and two or more equations.

Even at that stage, many people, whether old or young, feel bewilderment when such problems are posed, and once the going gets a bit complicated the person who does not want to jump through those hoops is liable to ask, "Why should I bother?" If the education environment is one that says to children, "Study only what interests you", then because the *x*s and *y*s look about as boring and detached from reality as anything could be, the pupil is more than pleased when someone in authority says, "If you really don't fancy getting your head around these things, you don't have to." I believe that there is an underlying tension about what we are doing in education, and that the prevailing model is that if someone finds something hard or uninteresting, they are more than welcome to drop it and to move on to something that they find much more tractable and believe, in their minority and youth, to be of much more practical and immediate relevance to their lives.

In that sense, I contend that our educational system has become too focused on working with the current beliefs and enthusiasms of the pupil and insufficiently focused on ignorance. Since education is meant to dispel ignorance—and for all of us appreciating and overcoming our own tendency to ignorance is hard work—an educational model that moves only along the grooves of pupil preference must be deemed too soft. I contend that the soft model should be repudiated and that our model of education should explicitly

countenance how important it is for pupils and students to master skills that at first glance seem to them to be strange and uncongenial. Indeed, I might put it more strongly. An idea or a book that seems uncongenial or difficult can, if its subject matter is important, engender those profound changes of attitude that education at its best can precipitate. Education is about climbing mountains, not skipping molehills.
Why should anyone feel passionate about the *x*s and *y*s in systems of equations? One answer is this: because if one does not make the effort to see what those *x*s and *y*s conceal, one will be cut off from having any real understanding of science. My passion comes from a sense that our society eschews educational difficulty, and hence culturally directs people away from the sciences. What that means—here is the source of my passion—is that in my constituency of Hemel Hempstead, women must wait 18 weeks for a laboratory to process their cervical smear test because many more young people want to work in television than in science, so there are not enough people to work in the laboratory. We have a society that is founded on science, but educationally we provide a university system that offers far more scope for studying the media than for studying physics. I do not wish to engage in the fashionable castigation of media studies or business studies, as many excellent courses go by such names, but where a society provides a very large number of opportunities to study such subjects and a fast-diminishing set of opportunities to study engineering and the mainstream sciences, it makes sense to ask how we have arrived at such a peculiar juncture.

To reflect on that point—and to use an analogy that Sir Nicolas would like—we can observe that a society can regress from being scientifically and technologically cultured to being backward. In the Rome of 800 AD, anyone who wanted to use metal for any purpose would have to find some left by those who lived when the empire was at its zenith. The technical citizens of ancient Rome knew which rocks contained metal ore and developed a furnace technology to liberate the metal from its elemental attendance. Eight hundred years later, such knowledge had been entirely lost.

We live in a society that has inherited an extraordinary wealth of knowledge about the world. However, that wealth appears daunting to the pupil or student. To become a scientist appears to require a capacity not only to amass a huge amount of knowledge but to master some ideas, which, at first glance, seem difficult, confusing, remote and mentally too taxing. As David Hume observed, most people have a sufficient disposition toward idleness to want to avoid excessive labour if possible. Consequently, our science-dependent culture is not replenishing the scientific basis that is needed for its continued existence. That neglect has terrible consequences, not only in Hemel Hempstead hospital.

How do quadratic equations relate to all that? First, they are a little more complicated than the linear and simultaneous equations that I mentioned earlier. They have only one unknown expression but they allow it to be raised to a power of 2, for example *x*2 = 4. Of course, *x*2 means that *x* is multiplied by itself. Another example is 3*x*2 + *x*—10 = 0. I suppose that it comes as a shock to find that solving those equations requires some effort. Even the first one—*x* x *x =* 4, what is *x*?—is not as

Why should anyone try to understand quadratic equations and the principles that lie behind solving them? They underpin modern science as surely as the smelting methods of the Romans were the key to their building culture. Modern science dawns with the experiments of Galileo. To describe how bodies fall, he knew from Kepler that he had to use the precision of mathematics rather than the imprecise language of Aristotle. The equation that he used for the most fundamental laws of motion was a quadratic equation in time, *s* = *ut* + ½ *ft*2, in which *s* is distance travelled, *u* is the initial velocity *f* is the accelerating force—usually gravity—and *t* is time.

To tell students that quadratic equations are beyond them, that they are about nothing and that educated people need have no inkling of what they are is to say that it is all right if they are so ill equipped to understand modern science that they cannot even comprehend its starting point. Those who tell us that we need no familiarity with quadratic equations are telling us to ignore 400 years of intellectual, scientific and technological development. When educators tell us that we should do that, I rejoin that they have a strange view of education, which I should like the Government to repudiate.

If we forget straight lines, the second aspect of the quadratic expression is that it gives us the simplest example of a graph. All admit the utility and importance of that method of presenting information.

If I imagine the simplest quadratic expression, *x*2, and I ask of it what values it takes when *x* assumes different values—when *x* is 1, *x*2 is 1; when *x* is 2, *x*2 is 4, and so on—I get a beautiful elementary curve: a parabola. Galileo used the properties of the parabola to analyse the motion of a falling body. He was able to do so because, long before him, Archimedes had identified some of the properties of the parabola. He knew, for instance, that it was impossible to measure exactly the long side of a unit triangle—a triangle with two sides of length 1, and the longest side, the hypotenuse. It is an extraordinary fact, however, that if such a triangle has a parabola—a curved side—it is possible to measure its area exactly. For example, when the parabola is defined by *x*2, when *x* goes from nought to six, the two straight sides and one curved side will form an area of exactly 72 units.

The mathematical materials of modern science and engineering were laid down by the ancient Greeks, and to tell students that they need not attend to any of these ideas is not merely to deprive them of the ideas that predate Galileo, it is to provide them with an education that neglects entirely the whole post-Hellenic edifice of human scientific culture. The Greeks thought that people were divided into those who could understand at least as far as proposition 47 of book I of Euclid's "Elements". Anyone who could get beyond that was not an ass. They called that proposition the pons

asinorum—the asses' bridge. One way of looking at the quadratic equation might be to say that it is the pons asinorum of modern science.I have two further observations on quadratic equations. First, it is powerful educational medicine to come to understand that something that can be expressed very simply can be extraordinarily difficult to solve. Much of modern culture tends the other way. People are presented with enormously difficult problems in politics or economics, for example—I have already mentioned the euro debate—and they assume that such problems have a simple and comprehensible solution. The quadratic equation can teach us to be humble.

Secondly, I have said that to solve such problems one has to make certain moves. In schools, pupils sometimes learn those moves without much understanding of what lies behind them. This is not the place to describe those moves, although I expect that the Minister will be able to remind us of what the generalised solution to a quadratic equation is, because I am sure that his team has equipped him to do so. He probably remembers it anyway from his own schooldays; it is the sort of thing that tends to stick. I am not sure how he is responding to that idea, but I shall persist with the thought.

It is quite extraordinary that one of the outcomes of making efforts to solve these equations is that we seem to have to expand the number system. For example, the solution of the humble-looking equation 2*x*2+2*x*+1=0, a very basic quadratic equation with no hard numbers, seems to require that there be a square root of -1. Since, when we multiply a negative by a negative, we get a positive, it is hard to see how a negative number could have a square root, but the humble quadratic equation suggests that there should be such numbers.

Most people think they know what "number" means; but, in reality, a substantial strand of human intellectual development has involved thinking of how to overcome the limitations of the elementary idea of "number" that we started with, and this rich heritage has been truly a world effort, whether it took place in Iraq, India or China. Knowledge of these things makes people less Anglocentric than they otherwise might be.

Most recently, this bizarre number—the square root of –1—has, since the work of de Broglie in 1923, played a key role in the equations that define quantum theory and which help us to understand our world in its microstructure. I might add to that that the structures that help us understand the other form of equation, simultaneous equations—structures called matrices—are also what are needed in the wave equations of quantum mechanics, since the work of Schrodinger, also in the mid-1920s. If we are to develop nanotechnology, for instance, it will be important that our students are at home with these ideas. Nanotechnology depends upon quantum effects.

I have the honour to serve on the Science and Technology Committee here in the House—you will have probably guessed that by now, Madam Deputy Speaker—and one important aim of that Committee has been to ask the Government to think again about their educational strategy. Some of the concepts and skills that we ask our children and our university

students to develop are regarded as "difficult." Mathematics lies in that realm, but so also do other activities such as learning foreign languages, mastering counterpoint and imagining biochemical structures in three dimensions. I submit that such activities, demanding as they do that the students make a real effort to change their perspective, are at the core of education; education as mountain climbing and not molehill jumping.Students and pupils are told too often that if they find ideas difficult, they can still attain high levels of educational qualification by avoiding such demanding materials. That is actually to do a disservice to those pupils. A key role for education is to help students understand, in all its richness and complexity, the world they have inherited, and perhaps it is also important that they understand the debt they owe to previous generations of many nations and cultures.

A second key role for education in a science-based culture is to equip a significant number of people with the skills to be able to transform that culture for the better. A Government who aspire to have 50 per cent. of school leavers in higher education but who are content for most of those students to have not an inkling of the science and technology that underpins the culture is a Government who are willing to preside over cultural and educational decline, whatever the statistics look like.

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