Good times for the future of STEM?

It seems to me that good times are on the way for future STEM undergraduates. Since I became explicitly involved in STEM matters in 2009 I have noticed a growing awareness that a holisitic approach to a school STEM education can benefit, motivate and inspire school students who plan to move on to a university STEM course. Schools, universities, government and industry appear to be giving a united message: The UK needs a skilled STEM workforce; a workforce of creative, innovative problem solvers; a workforce from which world leaders in STEM will emerge. We ought not let this evolve by chance and, fortunately, there has been lots of work done recently which give us insights which allow us to grow a decent STEM workforce: Research on STEM education has been done; STEM resources have been invented, dusted off, and refined; important STEM collaborations between diverse groups have formed. As a result, students are now being given a fighting chance of seeing the bigger picture prior to embarking on their university course. As a teacher, you are the guardian of this bigger picture.

My involvement with STEM began with the creation of biology, chemistry, physics and engineering problems on NRICH and whilst engaged with this fascinating project met with people from all over the STEM community. The STEM material on NRICH is part of a wider offering in which teachers can help students to see connections between their school science, technology and mathematics. What do I hope arises for students as a result of this work? In a nutshell I want our students to receive an education in STEM, rather than learn to answer exam questions on topics in subject silos.

As the year drew to a close it seemed like a good time to collect my main thoughts on STEM together in a blog post. These are my own opinions based on thinking around the area for quite some time and I hope that they might prove useful more widely.

I’ll look at three matters: some Golden Rules concerning good STEM practice; some of the mathematical issues facing STEM students; and some tips and suggestions for teachers in schools who wish to help students in their overall STEM education.

Part 1: Key Elements of Good STEM Practice

Golden rule 1: Respect
Have respect for different departments and no sense of any subject being better or more important or more fundamental than any other: the differences are real and significant, but there are also commonalities. Talk to people in other departments so you know what the differences and commonalities are.

Golden rule 2: Use positive language when talking about mathematics
It is still unfortunately seen as OK to dismiss maths as un-cool, pointless, geeky or to confess to being pretty bad at it. This has a very negative impact on students who either wish to study mathematics or might encounter a lot of mathematics in their university course.

Golden rule 3: Build on learning from other subjects
Don’t try to teach things from scratch that students might have encountered elsewhere.

Golden rule 4: Be aware of the difficulties in applying mathematics
Don’t assume that an ‘easy’ piece of maths is easy for students when it is located in a context where they wouldn’t expect to find it. Don’t assume that the maths is easy for students just because you think it is obvious.

Golden rule 5: Be open-minded about STEM
Be aware that your particular interests are not necessarily the particular interests of all of your students. Try to find the hidden gems in any topic area, even if it is not your personal favourite. Good enthusiastic teaching is a wonderful device. No topic is really intrinsically dull or boring, even if you think it is. Who knows, you might find thinking about STEM re-energises your teaching on a jaded area of the curriculum.

Golden rule 6: Celebrate the subtle and complex skill of good teaching
Be aware that the role of the teacher is as a learning facilitator in many cross-curricular activities, rather than the transmitter of all of the knowledge. It is OK not to know ‘all’ the answers. In fact, it is desirable to provide contexts in which you do not know all the answers to all possible questions! How else are students to learn how to solve real problems?

Golden rule 7: Don’t try to force links where none meaningfully exist
I feel that a poorly conceived STEM task can cause more harm than good. The universe is wonderfully constructed. There are many brilliant STEM contexts around –  don’t try to make STEM links for the sake of it.

Golden rule 8: Make technology a fundamental part of learning
Whatever your students end up doing, they will end up doing a great deal of it using a very wide range of technologies (many of which won’t yet exist). Pencil and paper tasks are still very desirable, but so is good level of techno-literacy.

Golden rule 9Don’t forget the STEM history of the educators
Many teachers, lab assistants, TAs and others involved in education will have had experience of STEM matters in various contexts. Celebrate these! Don’t feel that as a maths teacher you have hide the fact that you did engineering or biochemistry – celebrate your quantitative past history and bring it into the education of your charges where appropriate

Part 2: Quantitative matters

Why would clever students struggle with the mathematical aspects of their university course in science, technology or engineering? There are several possible reasons which I have encountered many times.

  • Procedural thinking
    Mathematics exams can often be passed by learning the content procedurally. This means that students can answer certain types of question by following a recipe. The problems in applying mathematics arise because even minor deviations from the precise recipe cause the student to fail to know what to do.
  • Inability to translate mathematical meaning to scientific meaning
    Students who are very skilled at mathematics might have trouble seeing how to relate a mathematical process to a real-world context; this hampers the use of common sense, so valuable in quantitative science.
  • Inability to make estimates or approximations
    Mathematical contexts in real applications are rarely simple. In order to apply mathematics predictively, approximations will need to be made. To make approximations requires the student to really understand the meaning and structure of the mathematics.
  • Poor problem solving skills
    Mathematical issues in applied mathematics problems are not usually clearly ‘signposted’ from a mathematical point of view. The student must (perhaps subconsciously) assess the situation, decide how to represent it mathematically, decide what needs to be solved and then solve the problem. Students who are not well versed in solving ‘multi-step’ problems in mathematics are very likely to struggle with the application of their mathematical knowledge.
  • Lack of practice
    There are two ways in which lack of practice can impact mathematical activity in the sciences. First is a lack of skill at basic numerical manipulation. This leads to errors and hold-ups regardless of whether the student understands what they are trying to do. Second is a lack of practice at thinking mathematically in the applied context.
  • Lack of confidence
    Lack of confidence builds with uncertainty and failure, leading to more problems. Students who freeze at the sight of numbers or equations will most certainly under-perform.
  • Lack of mathematical interest
    Students are hopefully strongly driven by their interest in science. If mathematics is studied in an environment independent of this then mathematics often never finds meaning and remains abstract, dull and difficult.

Part 3: Some quick tips for the classroom

There are various ways in which you can bring STEM into the classroom, from the light-touch reference here and there to the full off-curriculum week. Here I list some of my favourite quick ways to get STEM moving in your schools

  • Data collection in SET then analysis of the data in maths; possibly with a feedback into SET.
  • Create an equation/make a prediction/do a calculation for a physical process in a maths lesson and then test out the prediction by performing the experiment in science.
  • Introduce a task in a SET lesson (to ensure the pupils all have the SET knowledge required), continuing to work on it in a maths lesson (using maths skills) and then complete it for a double homework – marked by both maths and SET teacher with their particular subject focus.
  • Introduce a short, regular and scheduled ‘discussion’ or circle time in both maths and SET lessons (perhaps 10 minutes a fortnight). Give students a chance to comment on the SET that they have noted in maths and vice versa and also to ask teachers/other students of wider questions concerning the mathematics they have seen in SET or the SET that they have seen in maths.
  • Set a half-termly cross-curricular review homework:
  • Maths: Review SET books for mathematical content; make links with the mathematics that they have learnt during the term.
  • SET: Review maths books and suggest SET links or connections with the material covered. Look for parts of maths which have the most common uses.
  • Joint poster project: Choose a theme and work on the same poster in both a maths and SET lesson, where the focus is finding the maths/SET related to a big theme, such as global warming.
  • Use snippets of problems/weekly challenges as starters when students enter room. Choose topic to give a flavour of the main theme of the day and plan for appropriate content in schemes of work.
  • Include some cross-curricular display material in corridors and classrooms.
  • Ask student newspaper to interview members of science and maths departments to find out their ‘STEM history’ in terms of the STEM experience at university, hobbies or other jobs.
  • Build on the clear links between ‘investigation cycles’ in science, ‘design pentagon’ in D&T and ‘data handling cycles’ in maths.
  • Use data loggers, light sensors, paper and other patterns, and other hands-on technology in maths.
  • Show a picture of an experiment/activity as used in SET to act as stimulated recall in maths at the appropriate time.
  • Have AsSTEMblies – assemblies which focus on some aspect of STEM
  • Form a STEM club or a maths club or both. Form links between these clubs and clubs from other schools.

General references

Prepare for university

Mathematical Preparation for the Cambridge Natural Sciences Tripos

Interactive Workout – Mathmo

Maths in the Undergraduate Physical Sciences

Algebraic Fluency of Advanced Students

The NRICH – Transkills Project


stemNRICH – advanced

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Superficial contexts in STEM problems

A couple of years ago I was given the wonderful brief of devising a set of mathematical science tasks to help scientists and engineers in the mathematical aspects of the  transition from school to university. Rather than rattle something off in a week or two, I had luxury of thinking about this over an extended period of time, during which I worked with school and university students and lecturers from different faculties.

I think that the concept of the superficial context proved to be the most significant issue arising and the most challenging one to overcome. Superficial contexts are tasks in which there is superficial or artificial reference to science in a maths question or superficial or artificial use of mathematics in a science question. In both of these cases, the importance of either the real-world part or the mathematics part is overstated. And students see straight through this deception.

It might seem clear that there will be many fascinating scientific-mathematics questions out there. After all, fundamental to the process of all (and I really do mean all) scientific activity is quantification of the world. This quantification may be more or less explicit; hidden or overt. But it is there. Mathematics is the study of quantification in its purest state. Thus, mathematics is essential to the activity of the scientists.

In my opinion, a good scientific mathematics question is one in which the level of the mathematics and the science are comparable. Moreover, it is a question in which both the mathematical and scientific parts of the question are meaningful to the solver at their particular stage of education. Ideally, both a scientist and a mathematician would declare the task to be meaningful from the context of their individual disciplines.

Unfortunately, I soon found that good tasks for the classroom which included both scientific and mathematical elements were hard to come by, although there were plenty of tasks which made use of superficial contexts. It is easy to describe the usual suspects in the world of superficial contexts:

1) The cut and shut: A science task at a certain level and a mathematics task at a certain level are welded together because they are, on the surface, on the same topic. These are easy to create, but make meaningless links.

2) The post-script: A mathematics task or a science task is bolted onto the end of a science task or mathematics task respectively, often with a throw-away remark such as ‘something to think about’, ‘extension’ etc. Post-scripts almost always result in a mismatch of levels and styles, such as a hopelessly advanced investigation appended to something rather procedural and routine.

3)The reach for the stars: In the interest of being interesting, tasks refer to ultra-high level mathematics or science, such as string theory or DNA folding. Such tasks revolve around the sort of content that students might encounter should they elect to study for a PhD in a very specialised niche topic. Not only is this entirely irrelevant to the students, the task will necessarily touch the motivating topics in utterly superficial ways.

4) The neither one thing nor another: In the interests of making a tractable problem the resulting task is neither mathematically nor scientifically interesting or relevant. Such tasks are quite easy to make, and will never be used as intended.

5) The mutton dressed as lamb: A simple mathematical task is dressed up in pointless and inappropriate real-world clothing. This degrades the mathematics and fools nobody.

I have no straight-forward resolution of these difficult issues. However, I found that making the simple split between basic quantitative reasoning and mathematical modelling was useful in thinking around the issues:

Basic quantitative reasoning: counting, ratios, proportional reasoning, estimations, approximations, use of numbers and units. You simply cannot have a sensible conversation about science without being able to speak the language of quantification. ‘Basic’ quantitative reasoning might involve many subtleties and actually be very difficult, but is thoroughly grounded in school mathematics.

Mathematical modelling: Using formulae or mathematical techniques to answer scientific questions or to make scientific predictions.

In general the importance of the basic quantitative reasoning was often underplayed by the scientists and the importance of the mathematical modelling often overplayed by the mathematicians: scientists commonly assumed that the mathematicians would find the basic quantitative reasoning, well, basic; mathematicians often lacked appreciation of the scientific complexity underlying the application in which genuine modelling could only really occur at university.

From my work, it seems that the area of basic quantitative reasoning is the best source of many meaningful contexts in which the science and mathematics support each other in a genuine context. Such contexts might well lack the ‘reach for the stars’ aspects but are intellectually meaningful and satisfying to a learner at a particular phase of education.

The area of mathematical modelling is far more challenging. I found few, if any, areas where it was possible to create a task in which the mathematics and the science were genuinely equal partners. Historically, the one area of the school curriculum where mathematical modelling meaningfully took place was in physics, with its essential ties to mathematics. Sadly, many recent approaches to physics have removed the serious parts of mathematics from the subject. Were it to be reinstated students could experience some meaningful understanding of mathematical modelling whilst at school. As it stands, students need to look elsewhere for this stimulation.

So, in conclusion, I found that contexts in which classroom mathematics can be meaningfully linked to classroom science are difficult to come by. Moreover, I have come to the firm conclusion that superficial attempts to dress up mathematics and science in each others’ clothing degrades both subjects. However, I have learned that elevating the intellectual status of the quantitative reasoning in science can be of benefit in both the mathematics and science classrooms and, therefore, the students. I find this both exciting and liberating in an age of targets and subjects silos.


You can see some of my attempts at creating problems in meaningful contexts at

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C vs PC

I am a white, middleish class, successfully educated to the highest of levels, man. Perhaps an obvious point, but one worth making clearly at the outset. As a well trained mathematician, I know the importance of stating starting points clearly.

I am also a white, middleish class, successfully educated to the highest of levels, man who wishes to improve access to higher education to people who might share all, some or none of my defining traits.

I will be discussing three points in this post:

1) There are not enough young adults from ‘not-middleish class’ backgrounds reaching the highest levels in mathematics.

2) There are not enough young adults from ‘ethnic minority’ backgrounds reaching the highest levels in mathematics.

3) There are not enough young adults from ‘non-male’ backgrounds reaching the highest levels in mathematics.

Now, if I say that I am interested in helping those from the ‘not-middleish-class’  get into university to do maths then nobody bats an eyelid. Good job someone’s doing this, keep up the good work. Everyone knows this is the right thing to do.

Now, if I say that I am interested in helping those from ‘minorities’ get into university to do maths, then nobody bats an eyelid. Good job someone’s doing this, keep up the good work. Everyone knows this is the right thing to do.

Now, if I say that I am interested in helping women get into university to study maths then the reaction is quite different, and batted eyelids are almost certain. Why? they often ask. You are not a woman. Or, they might suspect that I might be on the femaler side of male, perhaps. Maybe even a bit gay. Or maybe I’m just being politically correct. Why else would I have an interest in this issue?

Often, I have found, people will have an opinion on the issue of women in mathematics. Usually quite culturally deep seated. Perhaps that women just aren’t as good at maths as men; it’s biological. Perhaps women who do maths are more manly than most women; certainly not feminine. Perhaps that they actually know one or more women who are really great at maths, but that (presumably), these are the exception. Or, most frustratingly, I hear the opinion: universities should just take the best people (with the hidden view that these are, surely, men). It’s just PC or positive discrimination to try and get a certain proportion of women into the maths lecture theatre. And people say this to me as though they assume that I agree with them.

It strikes me rather clearly that if you replace the words ‘women’ and ‘men’ with ‘black people’ and ‘white people’ then you have a situation which, quite rightly, was declared Wrong some years ago. Not Politically Wrong. Just Wrong full stop, from the point of view of basic humanity. And the arguments used to promote such arguments are, as we have clearly discovered over time, intellectually wrong. At every level. Try mentally making this replacement, slowly reading the paragraph and see how you get on. Don’t forget: be honest with your reactions.

Rather wonderfully, the issue of gender is not at all an issue for many people who are immersed in mathematics. There are simply people who are really into maths (PRIMs). Sadly, however, in classrooms and homes across the country you will find people who have many deep-seated, culturally reinforced attitudes towards women in mathematics.

Large swathes of our culture have now accepted that divisions between black and white are not relevant in education. Providing an education in a way which doesn’t particularly help or hinder children from different racial backgrounds isn’t being Politically Correct. It is just Correct.

So, if everything is equal, why spend time working on access in this way? Because it takes an extra layer of toughness and determination to success in any situation where you are challenging cultural norms. Male ballerinas, female bus drivers, male nurses, female bouncers, stay at home dads, female chief executives. Men who care and cry, outstanding mathematicians who are female.

Whilst I can only indirectly relate to people from non-white-middleish class-male backgrounds, I can relate to such people because I know plenty of them. Moreover, I listen to what they have to say. And because I listen and pay attention to it, I can see this mathematical bias in favour of men all over the place. And I can attempt to empathise with the people I am trying to help.

So, why do I actively support the issues of women in mathematics? Because there are existing cultural norms which get in the way of women and mathematics. And because it is Correct to challenge this. Not Politically Correct, just Correct. 

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Could you reheat this? The teaching is cold.

It is very rare for people to return a free meal to the chef. Free meals are exciting; you often don’t really know what to expect and the variety is fun. Some parts you don’t like, and others give you ideas of what to order next time you go out to pay for food. Sometimes free meals are strange, exotic or unidentifiable and others constitute good old favourites, like tuna or cheese sandwiches.

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.

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The subtle, insidious effects of non-free higher education

I was recently enjoying watching a fantasy film with my 7-year-old son. We were chatting, as we often do, about all aspects of the film such as location, computer graphics, numbers of orcs, the physics of catapults, the implications of seeing into the future and so on; Joseph is full of a delightful stream of thought-provoking questions and comments. As we spoke, in my mind I was wondering, as I often do, what Joseph might end up doing at university: maths? physics? computing? something else?

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.

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Is an A enough?

I have long held the view that the needs of the most capable school-aged mathematicians are very often neglected. I have also long held the view that a simple equation holds:

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.

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Should maths in secondary schools be compulsory?

I was recently considering the question of why mathematics should be compulsory after the age of 13, and spurred on by Alison’s question of what we mean by maths thought that I’d share my thoughts here. It’s somewhat of a more formal post than usual, but life’s like that sometimes. Note that this post shares a collection of views; they are not necessarily my views!


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 21st 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.


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.


[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

[5]The National curriculum

[6] Every child matters,

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