When I do have to pay I usually opt for the ‘cheap and cheerful’ meal option: pizza, burger, curry. Very occasionally there is cause for complaint, but only rarely and then with obvious cause: sorry, I ordered orange and this is banana.

It is much more common for people to return a very expensive meal: perhaps you ordered the steak rare, and it had arrived a bit too medium. Or, maybe, the white wine is warm. Or perhaps the presentation is a bit sloppy. And, if the food takes too long to arrive then you will certainly not give a tip and might even demand a reduction in the bill. In my more affluent days I was often surprised about just how much those people paying big bucks for food had to complain about. And the restaurants always obliged.

So, I fear, it will be with higher education. When it was free you got what you were given, said thank you (at least in public) and went your way after 3 years.

The same almost was the case when it was still reasonably cheap and just about managable. The odd person complained about this and that, but they were the ‘trouble makers’ or those who, clearly, ‘weren’t up to the course’.

Now, it will be the absolute right of students paying £9000 fees per year to complain if their received teaching is poor. And complain they should. In my travels I see too many examples of archaic chalk and talk lectures, filled with rows of bored or lost students frantically scribbling down obscure arguments or trying to decipher handwriting. I see too many examples of students in seminars nodding politely as a, doubtless clever, researcher simply writes out model solutions without pause for question. I see too many examples of tutors talking at their students at length, rather than talking with them. I wonder if some of these students are too intimidated, lost or jaded to simply say: hold on, this simply isn’t working for me. Or, perhaps, they don’t feel that it is their place to complain.

It strikes me that a very comfy and long-standing equilibrium is about to be challenged. I hope that the outcome will be a good one for the teaching and learning of higher mathematics.

Rather coincidentally, a few minutes later Joseph said:

“How much will it cost to go to university in the future when I go to university?”

A good question, I thought: “About £9000 per year. Well, probably about £10,000 or more by then.”

Joseph then replied “Well, I don’t think that I will go to university then. That sure is a lot to spend on learning.’

I was stunned by this: my 7-year-old deciding not to go to university on account of the future cost. The fact he had even asked the question, out of the blue, showed that it had been preying on his mind. Somewhat wrong-footed, I casually and encouragingly said

“Don’t worry: I might be able to help to pay for it.”

Joseph then replied.

“Why would you do that?”

I was even more stunned by this: clearly the cost, in Joseph’s mind, totally outweighed the value of any future education. Why, indeed, would you spend so much money on learning?

This starkly contrasts my history: I remember when I was 7 or 8 deciding quite firmly that I WAS going to go to (Cambridge) University. It was part of the vision for my life; and I took steps to make it happen. Sadly, Joseph’s instinctive vision now appears to be NOT TO GO to university, on account of the cost involved. Perhaps he has heard me or some newsreader talking about fees; perhaps he has heard me saying that going into debt is not a good thing to do if you can help it; he has certainly heard me saying that we cannot afford this and that; perhaps he has combined various facts in his mind to conclude this: I CANNOT POSSIBLY AFFORD TO GO TO UNIVERSITY; THEREFORE I WILL NOT GO.

Ah, such are the subtle mental barriers imposed on those to whom cost is necessarily an important factor in many day-to-day decisions. Of course, mental barriers can be overcome. And it might be more simple for Joseph to overcome these barriers than other children (his parents have 8 degrees and several professional qualifications between them and his dad elected to devote his energies to the cause of mathematics education). However, barriers still need to be overcome. And the earlier in life that a barrier is formed, the more impassable it becomes over time.

I suspect that those politicians and parents who do not hail from a standard, normal background can not fully understand the mental obstacle that a huge future financial burden places on people, regardless of the way in which repayment schemes are packaged up. Students from less well-educated backgrounds often need to work hard and with vision to achieve a university place. I fear that now that vision will have been removed from the future plans of many of our country’s bright young things.

Ability = Raw Talent + Motivation + Opportunity + Facilitation

A gifted student is one with a great potential ability. It is the job of the pupil to provide the raw talent; it is the job of the education system to provide the rest. But is this being done? Let’s think about this here.

Sadly, many educators believe that getting a fistful of grade As, or whatsoever letter/symbol combination represents the top offering, automatically determines a great success of the education system.

However, imagine an examination system whereby it is possible to get yourself a fistful of grade As, or whatsoever letter/symbol combination represents the top offering, by relying solely on the ‘Raw Talent’ in the equation for ‘Ability’.

In this case, the education system need not worry about attempting to enhance the combination (Motivation + Opportunity + Facilitation). It is often very convenient to forget that it is the duty of the education system to do this for inconveniently talented students.

To do this is to fail the student, who does not cease to exist once the fistful of grade As, or whatsoever letter/symbol combination represents the top offering, is achieved. They are passed onto the next phase of education and life.

No wonder many students with good potential ability struggle at university, with their great deficit of received (Motivation + Opportunity + Facilitation).

Consider this chart for a moment:

A summative report of a secondary maths education

This is a chart of ALL maths bookwork and homework that received a score over the course of many years of secondary schooling for a particular student whom I studied. What perception might you form of such a student?

It would be all too easy to form an opinion which said:

“Wow – this guy/gal must really love their maths and work hard at it, and, I would imagine, read widely about the subject that they love so much. Just imagine how wonderful school must have been for somebody so successful!”

However … this opinion is very wrong.

In fact the guy in question found maths really boring, almost always did all of the homework during lesson time and never did a stroke more maths than was required. In fact, he probably spent less time doing maths at school than any of his peers, simply because he got it done and out-of-the-way so quickly.

And why was this? Because he could, and nobody thought at any time to show him anything else.

What is the purpose of a secondary education today and why should mathematics necessarily be a part of this education?

To better understand the complexity of this question, we need to uncover some sense of what we mean by mathematics. It is ironic that mathematics, which deals largely with precision and clarity, defies a simple classification. Is it an art-form or merely a tool? Is it to be studied for pleasure or because of its utility? Is it essentially useful or somewhat pointless? In short, there is no standard accepted answer to the question of ‘What is mathematics and why should it be taught’ (Ernst [1]). This state of affairs leads to particular difficulties in selecting and justifying a secondary curriculum of mathematics. Should only practical mathematics be included? How much emphasis should there be on proof as opposed to procedure? Does a rounded mathematics education necessarily include any instruction on ‘higher’ mathematics? And who should be taught what?

A compulsory education requires a curriculum and if the contents of this curriculum are to be determined rationally then we require a set of guiding principles on which to base our decisions. We shall consider 5 principles that Beck [2] suggests have been historically significant:

• Children’s and students’ own interest and choices
• Economic relevance
• Vocational relevance
• Shaping national identity and allegiance
• A humanistic conception of liberal education, emphasising the value of knowledge and understanding for its own sake

Whilst the relative importance of these five elements has varied across cultures, times and governments, mathematics has almost universally been considered a critical and necessary component of a well-rounded education.  But is this still the case in 21^st century England in the secondary school setting?

I shall argue that the compulsory inclusion of mathematics in the English secondary school from age 11 to 16 is justified according to each of Beck’s guiding principles.

Children’s and students’ own interest and choices

There is always a tension between child choice and adult compulsion, especially when the goal is to allow children to pursue their interests. Being realistic, there are many 11 year olds who claim to have no interest in mathematics. Why, then, should we still force them to study it? The answer is to be found hidden in the oft-uttered phrase in mathematics classrooms ‘I don’t get it.’: children who really don’t get what mathematics is all about are not in a good position to make an informed decision about whether they are, or might become, interested in it. And there is a good cognitive reason why this informed decision may not be made by a typical 11 year old: he or she will just be entering Piaget’s period of formal operations [3] in which the development and maturation of logical reasoning occurs. Prior to this phase of development is the period of concrete operations where learners can reason about events for which they have a concrete hook onto which to hang the reasoning. To stop learning mathematics before the abstract phase of reasoning is solidly reached would be similar to stopping reading once individual sentences could be sounded out; to allow an 11 year old to drop mathematics based on lack of interest would be as inconceivable as allowing a 5 year old to drop reading as they struggled to come to terms with its complexities.

In short, genuine interest in a subject cannot be gauged until some measure of comprehension of a subject is achieved. The pleasures in mathematics are developed steadily and slowly over time. Once an informed choice can be made, children might not wish to continue the study of mathematics, but with good teaching and a well-chosen set of activities, perhaps a love, or at least a quiet appreciation of mathematics, might grow in more of our children.

Another compelling argument for the compulsory inclusion of secondary mathematics is to be found  when mathematics is seen in the wider cross-curricular context: in school, mathematics has a significant role to play in the developing study of a very wide range of other subjects: the sciences, IT, geography, economics, business and sport science to name a few. Children who might be very interested in such subjects might not realise the importance of mathematics in their present or future subjects of interest, particularly for the subjects, remote to an 11 year old, which will be studied at Post-16 level. To allow children to properly study their subjects of interest requires the continued study of mathematics: a solid grasp of mathematics facilitates choice, and as such mathematics is necessarily a cornerstone of compulsory secondary education right through to the age of 16.

Economic relevance

Over the past 20 years a utilitarian, employment-focussed view of education, in which the purpose of the system is to produce a productive and competitive workforce, appears to have grown. Indeed, the first paragraph of the influential Tomlinson [4] report has a very utilitarian feel:

This report was published 18th October 2004 and is concerned primarily with ensuring that ‘all young people are equipped with the skills and knowledge they need to succeed and progress in education, employment and adult life.’

This utilitarian view is mirrored in the main official aims of Key Stages 3 and 4 Mathematics [5], in which it is clear that we are to prepare students for participation in the knowledge economy:

Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making.  Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world.

The policy makers are not misguided:  the world is rapidly becoming more complex, quantitative, technological and mathematised; to understand this world requires a sound grasp of mathematics and mathematical thinking. The government rightly aims to place the UK as a leader in this knowledge economy, and this requires a workforce which is highly mathematically literate relative to our international competitors. And this requires that the future workforce as a whole studies mathematics throughout their time at secondary school.

Vocational relevance

Some children enter the secondary school with a very clear sense of their vocation in life. Many of these views of vocation change over time and many children only make a choice concerning career towards the end of their time at secondary school. But, at some stage in their schooling it is hoped that each child will say: This is what I want to do. Choice and desire are important in education: they create motivated children who are prepared to work to succeed. This in turn makes it more likely that they will enjoy and achieve and make a positive contribution, as required by the every child matters agenda [6]

Regarding vocation, the utilitarian aspects of mathematics are not to be underestimated: mathematical content and mathematics thinking is useful or crucial in very many vocations. For example, the traditional vocations of medicine, law, teaching, nursing, veterinary science and business all rely on quantitative and logical elements which need to be carefully developed over time. It is no surprise that a Grade C GCSE pass is an entry-card to very many vocational courses and careers; the large number of adults who failed to engage with mathematics whilst at school and are now enrolled in functional mathematics classes at evening school are a testament to the vocational relevance and importance of mathematics across a great range of careers.

To allow children to opt out of mathematics before the age of 16 would be to allow them to opt out of many future vocations, reduce their choices and reduce their chances for a happy and productive career.

Shaping national identity and allegiance

At a basic level, in England the role of the mathematics qualification at O-level and then GCSE has taken on a notable significance. Achieving the grade C in mathematics is of great cultural importance, often seen as a black and white line between success and failure, and parents who failed to achieve this grade in mathematics are often determined that their children do not do the same. Whilst parental views of a good mathematical education might not necessarily fit precisely with those of mathematics educationalists, to many of the carers of the children in secondary school education it would be culturally unthinkable to allow mathematics to become optional before the age of 16.

More generally the national identity of the English is in large part founded upon its tremendous historical success and influence in the development of mathematics, science and engineering. The English feel a great pride at laying claim to Sir Isaac Newton and Sir Stephen Hawking, and the events at Bletchley Park in which the Enigma machine was cracked by Alan Turing and his team of mathematicians are singularly English in feel. Furthermore, the City of London is synonymous with financial power throughout the world, a power built on mathematics. It is crucial that our education system celebrates these successes and gives children an insight into their foundations and national importance. The English history is littered with world-changing breakthroughs in mathematically based activity. What better way to build a sense of national pride and allegiance than to study the successes of the past?

Recent governments have quite rightly wished for this success to continue, hoping that England will play a leading role in the emerging technological economy. For this to happen, mathematics must be studied, applied and celebrated for its own sake. The English must fundamentally view themselves as mathematically literate, and this requires the compulsory, positive and enriched study of mathematics to the age of 16.

A humanistic conception of liberal education, emphasising the value of knowledge and understanding for its own sake

Mathematics is at the pinnacle of human achievement. It has many facets and applications, many of which have been alluded to in earlier points in this essay.  Although knowledge of various pieces of mathematics is undoubtedly useful, even fundamental, in a great many areas of endeavour, to attempt to justify the compulsory inclusion of mathematics in the school curriculum simply based on its utility as a set of tools is to miss much of the point of mathematics; it is far more than the sum of its uses and applications. The pursuit of mathematics isn’t simply pursing knowledge for its own sake: mathematics at its purest is the search for perpetual, universal and unalterable logical truth.

Amongst all of human endeavour mathematics is uniquely deductive in character. An education which places any value on knowledge for its own sake must surely include exploration of the structures of mathematics. To not expose children to the beauty and delight of the study of mathematics for its own sake throughout their secondary schooling would be akin to not continually exposing children to the greatest works of art or literature as their minds matured. This should not be choice: it should be necessary.

Summary

Mathematics is so important and far-ranging a subject of study that the argument for its compulsory inclusion in the secondary curriculum may be made from a variety of viewpoints. It should necessarily be studied throughout the secondary school for many reasons: It is beautiful, develops the mind, underpins the study of other subjects, is a necessary component of many jobs, is necessary to be a successful citizen; it is historically of great significance; and, finally, it is unique amongst human intellectual development. Perhaps any one of these points would be sufficient reason alone for the compulsory study of mathematics. Collectively, they make it clear that compulsory mathematics education is here to stay.

References

[1]Paul Ernst, “Why Learn Maths?”, edited by John White and Steve Bramall, London: London University Institute of Education, 2000.

[2] Beck, J., The School Curriculum The National Curriculum and New Labour Reforms, p15 ‘Key Issues in Secondary Education’ Ed. J. Beck, M. Earl, Continuum Press

[3] Walford, R., Classroom Teaching and Learning, p54 ‘Key Issues in Secondary Education’ Ed. J. Beck, M. Earl, Continuum Press

[4] The Tomlinson Report – 14-19 Curriculum and Qualifications Reform

http://www.heacademy.ac.uk/assets/ps/documents/briefing_papers/ps0007_the_tomlinson_report_14_19_curriculum_and_qualifications_reform_feb_2005.pdf

[5]The National curriculum http://curriculum.qcda.gov.uk/key-stages-3-and-4/subjects/key-stage-3/mathematics/index.aspx

[6] Every child matters, http://www.dcsf.gov.uk/everychildmatters/about/aims/aims/

Damning words indeed, especially if I was speaking within earshot of Homer, Ted Hughes, Prokofiev or Sir Peter Maxwell Davis. They might rightly be upset, or even incensed about such comments; for such people devoted their entire creative lives in pursuit of that perfect line or phrase. What a boring waste of time, I thought, and then thought about it no more. More fool them for doing something so pointless. And, as a teenager I was surely correct.

The list continues: modern art was  stupid; football is just kicking a ball around; history is just a dull list of facts; French a list of verbs. All of these things that I do not or did not understand were dismissed out of hand.

I am now somewhat older than I was when I formed those statements and look back with interest. It is so easy to pre-judge based on ones own personal experience, especially if the subtleties of the art-form in question are difficult or time-consuming to acquire. And even more so if you are unlucky enough to have been subjected to dull, procedural, lifeless teaching in which poetry is reduces to syllable counting, music to score analysis or art to mucking about at the back of the classroom.

I now listen regularly to modern classical music, enjoy (watching) a decent game of football and find the messages and mysteries of history fascinating and illuminating. But why is this? It is because I have come to understand what these things are about, to some extent. I can now appreciate why these human achievements are interesting, worthwhile and, at times, shocking or breathtaking. And once this appreciation is there, genuine love and enjoyment soon follow.

Poetry is somewhat different: I don’t particularly have the patience for poetry. Does that make it rubbish? The fact that I have thought about it little, and poets a great deal, would suggest that there is far more to it than I am aware. And so I respect it as an art form, even though I choose not to pursue it. Thus, if I met a poet I would be polite and genuinely interested.

Now let’s move to the subject of mathematics. THIS is something to which  I have thus far devoted my working life. I have complex relationship relation with mathematics, but it is fair to say that I love the subject. Many people say that they hate maths or that it is pointless. And, in no uncertain terms, they tell me about it. Dare I say, even, rudely? How should a mathematician react? Apologetically? With shame? With anger? No: I react as a mathematician should – logically. I know that either they are wrong, or I am blindly misguided in my rich appreciation of the subject.

So, where do you stand on the maths fence? For or against?

Perhaps if you encounter someone who is negative towards mathematics you might suggest that they might not have been exposed to its beauty and wonder.

Perhaps if you have a hostile relationship with numbers and mathematics you might wonder ‘is there something that I am missing?’ or ask yourself ‘could I have received more eye-opening teaching?’

At NRICH we are passionate about showing people what mathematics is really all about, and WHY people devote their lives to thinking about it. It is fun, intriguing, inspiring and addictive and ALL people can be shown how to understand this. Perhaps it still might not light your personal fire, but at least you will appreciate the art form of mathematics and respect those children and adults who like mathematics.