## the quadratic equation & hansard

I found this debate here

Quadratic Equations

3.43 pm

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

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

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

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

simple as it appears. There are two solutions:

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,

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,

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

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.

Someone who thinks that the quadratic equation is an empty manipulation, devoid of any other significance, is someone who is content with leaving the many in ignorance. I believe also that he or she is also pleading for the lowering of standards. A quadratic equation is not like a bleak room, devoid of furniture, in which one is asked to squat. It is a door to a room full of the unparalleled riches of human intellectual achievement. If you do not go through that door—or if it is said that it is an uninteresting thing to do—much that passes for human wisdom will be forever denied you.

Throughout human history, that door was locked to those of the working classes, to women and to those who come from nations that were enslaved. Now at last we have a society and culture that have made it possible for people on the largest scale to understand at a most fundamental level the culture they have inherited and the debts that they owe to their forebears. Now we have a society in which many citizens can be empowered to understand the natural world.

Siren voices still aver that many cannot cope with quadratic equations and similar structures, and with the worlds that they unlock. It is the Government's job to resist those who would devalue the educational currency in this way. An educational curriculum that is too undemanding cheats those who could have gained understanding, but who are denied that opportunity. Sadly, those who have been so cheated do not even know what it is that they do not know. If real education can be mentally taxing and painful, that is also one of its greatest values. Those who have profited from it are grateful to those who helped them to attain it for the

whole of their lives. To deny real education to the many on the dubious ground that they cannot cope with difficulty is to fail to grasp an historic opportunity for human liberation.

Remembering the point about Stephen Hawking, perhaps one day, books that feature equations will have their circulation enhanced by that feature. Perhaps we will know that then, we have an educated citizenry.

4.11 pm

I congratulate my hon. Friend the Member for Hemel Hempstead on securing this debate. He said that he was expecting my hon. Friend the Minister for School Standards to reply, and I should point out that there has been a fierce struggle among the seven Ministers in the Department for Education and Skills in that regard. In the end, my hon. Friend was thought to be far too junior. He has a good career in front of him, but he was part of the 2001 intake and is therefore far too young. So I have the honour to reply to today's debate.

I thank my hon. Friend the Member for Hemel Hempstead for defining what quadratic equations actually are. In fact, my Parliamentary Private Secretary provided me with one, and I shall check it with my hon. Friend afterwards to see whether my Parliamentary Private Secretary will continue to hold his post in future. I hope that I can provide my hon. Friend with a repudiation of the comments, made on the radio a couple of weeks ago, of the trade union leader to whom he referred.

Quadratic equations allow us to analyse the relationships between variable quantities, and they are the tool for understanding variable rates of change. It is in variable rates of change that quadratic equations are seen in economics, science and engineering. Examples of the use of quadratic equations include acceleration, ballistics and financial comparisons. Most drivers would feel capable of working out whether they can overtake the car in front, but do they realise that they are solving a quadratic equation in doing so? I dare say that many do not. In fact, it is claimed that the Babylonians, in 400BC, were the first to use the notion of quadratic equations in problem solving, although at the time they had no idea what an equation was.

In preparing for this debate, the DFES conducted a straw poll involving a 16-year-old who had just sat maths GCSE, a head of maths and an experienced chemical engineer. The 16-year-old thought that quadratic equations were logical and fairly straightforward because

"you substitute stuff into a formula".

He did say, however, that his opinion might have been influenced by having a good teacher. The head of maths said that quadratic equations formed an important step

in students' ability to solve equations, taking them from simple—one unknown—and simultaneous—two unknowns—and paving the way for more advanced work in mechanics and complex number theory. The engineer said that he did not use quadratic equations now, but had in the past in detailed design applications. Where he works, the chemists use them to explain multiple reactions.

The place of quadratic equations in everyday life is pretty clear, but what are pupils taught at school? The national numeracy strategy has had a significant impact on raising the standards of mathematics in primary schools. Last year's key stage 2 results showed that 73 per cent. of pupils achieved the expected level for their age in mathematics, which is a 14 per cent. increase since 1998. We want to build on that impressive record, which is why we have launched "Excellence and Enjoyment—A Strategy for Primary Schools". Our vision for primary education is of excellence and enjoyment at the heart of a broad and rich curriculum.

The key stage 3 national strategy for 11 to 14-year-olds provides a comprehensive professional development programme for teachers, new materials and support from expert local consultants. The maths teaching framework emphasises the development of algebraic reasoning. It encourages pupils to develop an understanding of how algebra is a way of generalising from arithmetic, and to represent problems and solutions in a variety of forms.

Simple linear equations are taught from key stage 3. In 2002, key stage 3 results stand at their highest ever, with 67 per cent. of pupils achieving level 5 plus in the key stage 3 tests for both maths and science. Quadratic equations, with their more complex parabolic curves, are taught from late key stage 3 or key stage 4. Factorisation of algebra is a difficult concept to understand and students need plenty of practice. Teaching students to think logically and to analyse different problems is a very important skill, which is not only transferable to other areas of the curriculum but can be used beyond student life.

Pupils taking the intermediate tier GCSE mathematics study algebraic manipulation, including the solving of quadratic equations. They are covered in greater depth by pupils following the higher tier GCSE course and developed further for those going on to study maths on A and AS-level courses. It is at that level that students are taught the concepts that they will need, should they choose to do degrees in maths, sciences, engineering or economics.

There is a shortage of people nationally who can construct these mathematical models and who understand them enough to use them. The fact that an inquiry is taking place into post-14 mathematics is a testament to the importance of the subject. The aims of the post-14 inquiry, announced in July 2002, are to

"make recommendations on changes to the curriculum, qualifications and pedagogy for those aged 14 and over in schools, college and higher education institutions to enable those students to acquire mathematical knowledge and skills necessary to meet the requirements of employers and of further and higher education".

Professor Adrian Smith, the inquiry chairman, is due to report his findings this autumn.

The use of information and communications technology in schools and in teaching and demonstrating mathematics models is helping the understanding of all learners. With ICT becoming a more integral part of classroom teaching, students can visualise and problem-solve in more creative ways. Those, too, are lifelong skills that can be applied in daily life, not just as a student. ICT gives students confidence in their abilities and increases their eagerness to learn. Using graphical calculators to learn about quadratic functions, for example, helps pupils learn in a more innovative environment.

Several interesting initiatives support teachers in making maths lessons more challenging and exciting. I should like briefly to describe them. Census at school can be used in a number of curriculum subjects, particularly maths. Pupils fill out a questionnaire, see how a census works and are able to compare their school's results with those of other schools in the UK and elsewhere. That is particularly helpful for key stage 3 pupils encountering data and learning how to handle them.

The UK mathematics trust works with secondary school teachers and pupils to promote mathematics. It encourages all secondary schools to take part in competitions and events, including the team maths challenge. The trust also identifies and trains students for the international mathematics olympiad.

The work of the Cambridge university-based millennium maths project allows teachers and students to tap into resources over the internet. Students can ask university undergraduates for help with mathematical problems. The project also offers tailored and continuous professional development for teachers.

An increasing use of information and communications technology and innovative ways of teaching are both positive steps for the subject, as is the rise in the number of mathematics teachers.

The intake to initial teacher training courses in mathematics rose to 1,670 in 2002–03, an increase of 8 per cent., and an overall increase of 28.5 per cent. since 1999–2000. The number of graduates applying to train as teachers of mathematics on post-graduate certificates in education courses in 2003–04 was 35 per cent. higher than for the same period last year. In a recent Ofsted report, it was noted that today's newly qualified teachers are the best trained ever. We need not only to continue to increase the number of teachers, but to support mathematics subject specialism. We want teachers to maintain their enthusiasm for maths and to develop their expertise throughout their careers.

In March, my right hon. Friend the Secretary of State for Education and Skills announced that Adrian Smith would advise on options and costs for a national centre for excellence in mathematics later this year. The new centre should harness all the good work already under way and enable more teachers to tap into the resources and support that they need.

In conclusion, the teaching of quadratic equations, and of the mathematics curriculum overall, is key to a future work force that can develop and use mathematical models in daily life. As research in a book of quotations reveals, Napoleon said:

"The advancement and perfection of mathematics are intimately connected with the prosperity of the state."

We recognise the importance of mathematics at all stages of education, and we are committed to ensuring that all young people have the opportunity to acquire the skills that they need—as citizens, and as the mathematicians, scientists and engineers of the future.

Once again, I thank my hon. Friend the Member for Hemel Hempstead for securing this debate, and for making such an interesting and entertaining contribution.

Adjourned accordingly at twenty-three minutes past Four o'clock.

*Motion made, and Question proposed*, That this House do now adjourn.--*[Jim Fitzpatrick.]*3.43 pm

**Mr. Tony McWalter (Hemel Hempstead):**The subject of my debate may ensure that hon. Members will not want to stay for the whole of it, but I dedicate the debate to Sir Nicolas Bevan, who has been the Speaker's Secretary for 10 years. This is the last formal proceeding in the House before he leaves that position. I wanted to pay tribute to him and say that many hon. Members have valued greatly the service that he has given to the House. I know that he will be much missed.**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.

**26 Jun 2003 : Column 1260**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

**26 Jun 2003 : Column 1261**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**26 Jun 2003 : Column 1262**simple as it appears. There are two solutions:

*x*= 2 and*x =-*2. Most people in school learn a general formula to deal with more complicated problems. That is often done without understanding and the world appears to divide into sheep, who do not mind doing that sort of thing and goats, who view it as a pointless game that is less riveting than snap.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

**26 Jun 2003 : Column 1263**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

**26 Jun 2003 : Column 1264**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.

**Mrs. Eleanor Laing (Epping Forest):**Hear, hear.**Mr. McWalter:**Oh dear. I would like to have support from elsewhere as well.Someone who thinks that the quadratic equation is an empty manipulation, devoid of any other significance, is someone who is content with leaving the many in ignorance. I believe also that he or she is also pleading for the lowering of standards. A quadratic equation is not like a bleak room, devoid of furniture, in which one is asked to squat. It is a door to a room full of the unparalleled riches of human intellectual achievement. If you do not go through that door—or if it is said that it is an uninteresting thing to do—much that passes for human wisdom will be forever denied you.

Throughout human history, that door was locked to those of the working classes, to women and to those who come from nations that were enslaved. Now at last we have a society and culture that have made it possible for people on the largest scale to understand at a most fundamental level the culture they have inherited and the debts that they owe to their forebears. Now we have a society in which many citizens can be empowered to understand the natural world.

Siren voices still aver that many cannot cope with quadratic equations and similar structures, and with the worlds that they unlock. It is the Government's job to resist those who would devalue the educational currency in this way. An educational curriculum that is too undemanding cheats those who could have gained understanding, but who are denied that opportunity. Sadly, those who have been so cheated do not even know what it is that they do not know. If real education can be mentally taxing and painful, that is also one of its greatest values. Those who have profited from it are grateful to those who helped them to attain it for the

**26 Jun 2003 : Column 1265**whole of their lives. To deny real education to the many on the dubious ground that they cannot cope with difficulty is to fail to grasp an historic opportunity for human liberation.

Remembering the point about Stephen Hawking, perhaps one day, books that feature equations will have their circulation enhanced by that feature. Perhaps we will know that then, we have an educated citizenry.

4.11 pm

**The Minister for Lifelong Learning, Further and Higher Education (Alan Johnson):**I begin by echoing the comments of my hon. Friend the Member for Hemel Hempstead (Mr. McWalter) on Sir Nicolas Bevan's retirement. I should also like to associate myself with remarks made by Members on both sides of the House throughout the day, and with the sentiments of the well-supported early-day motion. It is not often that a Government representative supports an early-day motion, but I do so on this occasion because it recognises Sir Nicolas's long and distinguished service. I wish him well in his retirement.I congratulate my hon. Friend the Member for Hemel Hempstead on securing this debate. He said that he was expecting my hon. Friend the Minister for School Standards to reply, and I should point out that there has been a fierce struggle among the seven Ministers in the Department for Education and Skills in that regard. In the end, my hon. Friend was thought to be far too junior. He has a good career in front of him, but he was part of the 2001 intake and is therefore far too young. So I have the honour to reply to today's debate.

I thank my hon. Friend the Member for Hemel Hempstead for defining what quadratic equations actually are. In fact, my Parliamentary Private Secretary provided me with one, and I shall check it with my hon. Friend afterwards to see whether my Parliamentary Private Secretary will continue to hold his post in future. I hope that I can provide my hon. Friend with a repudiation of the comments, made on the radio a couple of weeks ago, of the trade union leader to whom he referred.

Quadratic equations allow us to analyse the relationships between variable quantities, and they are the tool for understanding variable rates of change. It is in variable rates of change that quadratic equations are seen in economics, science and engineering. Examples of the use of quadratic equations include acceleration, ballistics and financial comparisons. Most drivers would feel capable of working out whether they can overtake the car in front, but do they realise that they are solving a quadratic equation in doing so? I dare say that many do not. In fact, it is claimed that the Babylonians, in 400BC, were the first to use the notion of quadratic equations in problem solving, although at the time they had no idea what an equation was.

In preparing for this debate, the DFES conducted a straw poll involving a 16-year-old who had just sat maths GCSE, a head of maths and an experienced chemical engineer. The 16-year-old thought that quadratic equations were logical and fairly straightforward because

"you substitute stuff into a formula".

He did say, however, that his opinion might have been influenced by having a good teacher. The head of maths said that quadratic equations formed an important step

**26 Jun 2003 : Column 1266**in students' ability to solve equations, taking them from simple—one unknown—and simultaneous—two unknowns—and paving the way for more advanced work in mechanics and complex number theory. The engineer said that he did not use quadratic equations now, but had in the past in detailed design applications. Where he works, the chemists use them to explain multiple reactions.

The place of quadratic equations in everyday life is pretty clear, but what are pupils taught at school? The national numeracy strategy has had a significant impact on raising the standards of mathematics in primary schools. Last year's key stage 2 results showed that 73 per cent. of pupils achieved the expected level for their age in mathematics, which is a 14 per cent. increase since 1998. We want to build on that impressive record, which is why we have launched "Excellence and Enjoyment—A Strategy for Primary Schools". Our vision for primary education is of excellence and enjoyment at the heart of a broad and rich curriculum.

The key stage 3 national strategy for 11 to 14-year-olds provides a comprehensive professional development programme for teachers, new materials and support from expert local consultants. The maths teaching framework emphasises the development of algebraic reasoning. It encourages pupils to develop an understanding of how algebra is a way of generalising from arithmetic, and to represent problems and solutions in a variety of forms.

Simple linear equations are taught from key stage 3. In 2002, key stage 3 results stand at their highest ever, with 67 per cent. of pupils achieving level 5 plus in the key stage 3 tests for both maths and science. Quadratic equations, with their more complex parabolic curves, are taught from late key stage 3 or key stage 4. Factorisation of algebra is a difficult concept to understand and students need plenty of practice. Teaching students to think logically and to analyse different problems is a very important skill, which is not only transferable to other areas of the curriculum but can be used beyond student life.

*[Interruption.]*The hon. Member for Epping Forest (Mrs. Laing) from a sedentary position reminds us of that as I speak.Pupils taking the intermediate tier GCSE mathematics study algebraic manipulation, including the solving of quadratic equations. They are covered in greater depth by pupils following the higher tier GCSE course and developed further for those going on to study maths on A and AS-level courses. It is at that level that students are taught the concepts that they will need, should they choose to do degrees in maths, sciences, engineering or economics.

There is a shortage of people nationally who can construct these mathematical models and who understand them enough to use them. The fact that an inquiry is taking place into post-14 mathematics is a testament to the importance of the subject. The aims of the post-14 inquiry, announced in July 2002, are to

"make recommendations on changes to the curriculum, qualifications and pedagogy for those aged 14 and over in schools, college and higher education institutions to enable those students to acquire mathematical knowledge and skills necessary to meet the requirements of employers and of further and higher education".

Professor Adrian Smith, the inquiry chairman, is due to report his findings this autumn.

**26 Jun 2003 : Column 1267**The use of information and communications technology in schools and in teaching and demonstrating mathematics models is helping the understanding of all learners. With ICT becoming a more integral part of classroom teaching, students can visualise and problem-solve in more creative ways. Those, too, are lifelong skills that can be applied in daily life, not just as a student. ICT gives students confidence in their abilities and increases their eagerness to learn. Using graphical calculators to learn about quadratic functions, for example, helps pupils learn in a more innovative environment.

Several interesting initiatives support teachers in making maths lessons more challenging and exciting. I should like briefly to describe them. Census at school can be used in a number of curriculum subjects, particularly maths. Pupils fill out a questionnaire, see how a census works and are able to compare their school's results with those of other schools in the UK and elsewhere. That is particularly helpful for key stage 3 pupils encountering data and learning how to handle them.

The UK mathematics trust works with secondary school teachers and pupils to promote mathematics. It encourages all secondary schools to take part in competitions and events, including the team maths challenge. The trust also identifies and trains students for the international mathematics olympiad.

The work of the Cambridge university-based millennium maths project allows teachers and students to tap into resources over the internet. Students can ask university undergraduates for help with mathematical problems. The project also offers tailored and continuous professional development for teachers.

An increasing use of information and communications technology and innovative ways of teaching are both positive steps for the subject, as is the rise in the number of mathematics teachers.

**26 Jun 2003 : Column 1268**The intake to initial teacher training courses in mathematics rose to 1,670 in 2002–03, an increase of 8 per cent., and an overall increase of 28.5 per cent. since 1999–2000. The number of graduates applying to train as teachers of mathematics on post-graduate certificates in education courses in 2003–04 was 35 per cent. higher than for the same period last year. In a recent Ofsted report, it was noted that today's newly qualified teachers are the best trained ever. We need not only to continue to increase the number of teachers, but to support mathematics subject specialism. We want teachers to maintain their enthusiasm for maths and to develop their expertise throughout their careers.

In March, my right hon. Friend the Secretary of State for Education and Skills announced that Adrian Smith would advise on options and costs for a national centre for excellence in mathematics later this year. The new centre should harness all the good work already under way and enable more teachers to tap into the resources and support that they need.

In conclusion, the teaching of quadratic equations, and of the mathematics curriculum overall, is key to a future work force that can develop and use mathematical models in daily life. As research in a book of quotations reveals, Napoleon said:

"The advancement and perfection of mathematics are intimately connected with the prosperity of the state."

We recognise the importance of mathematics at all stages of education, and we are committed to ensuring that all young people have the opportunity to acquire the skills that they need—as citizens, and as the mathematicians, scientists and engineers of the future.

Once again, I thank my hon. Friend the Member for Hemel Hempstead for securing this debate, and for making such an interesting and entertaining contribution.

*Question put and agreed to.*Adjourned accordingly at twenty-three minutes past Four o'clock.