Mathematics Should be Optional, not Compulsory in Schools
Mathematics in middle and high school should be an optional subject, only taught to those students who have a genuine taste for it. It should not be part of the core program as it happens now, imposed on everybody as some form of a ritual of passage, separating the virtuous from the unworthy in our society. It should not have a preferred place in the academic program, hovering above the rest of the subjects. Arts, literature, philosophy, physical education: they should all be regarded with just the same deference as math. But most important of all, mathematics should not be a weapon to terrorize millions of kids that grow up with some form of post-traumatic disorder, victims of our obsession with treating mathematics as some form of divine discipline.
Before moving on, I must clarify something. I am a mathematician, and I love mathematics. In college, I went through the full cycle, from undergrad studies to MSc and then to a Ph.D. I taught mathematics in universities, published articles in specialized journals and gave talks on my research. I left academia, and now I apply mathematics in the financial industry, where there is an abundance of complex, quantitative problems to solve. If you ask me, nothing is more enjoyable for a relaxing weekend than going through some theorems and solving a few equations. So, it should be clear that my opinion about mathematics in schools does not come from resentment or hate, but rather from appreciation.
Whenever I introduce myself as a mathematician to someone new, I often perceive in them a bit of a surprise, something along the lines of “Really? Did you willingly studied that!?” I always ask what their feelings and memories from learning mathematics are and in the vast majority of cases, the answer is far from positive. Math was complicated, the exams were stressful, the teachers were monsters, and all that is left is a phobia to anything involving numbers.
I don’t think anyone would find these remarks shocking, after all, being terrible at maths has become a universal punchline. You can hear celebrities, politicians, and journalists from all over the world bragging about their incompetence in mathematics, something they would never admit, say, in History or Geography. But this collective disdain clashes head-on with the litany of benefits that supposedly come with a solid education in mathematics. What´s the problem then?
Misconceptions of the benefits
There are various arguments people use to shove mathematics down the throat of harmless adolescents, but there are three that I hear repeatedly:
- Mathematics is everywhere: on your iPhone, in the weather, in the movement of the stars. Nature is expressed in mathematics, and perhaps more appealing, you might even have the power to manipulate it with mathematics.
- Mathematics gives you a better, more structured way of thinking. You will become more rational, more analytic; it will probably make you a better citizen.
- Mathematics opens the doors to profitable careers in Engineering, Computer Science, or Finance. No one makes money from studying Classics so, why not betting instead on a sure winner?
Let me then address these points one by one. It is true that mathematics is essential for the development of science and technology, but this message must be hard to believe for someone who is buried in trigonometric identities and evaluating limits. Instead of providing students with a panoramic view of the field, we insist on isolating the hardest bits of the discipline and provide them with little or no context.
I would be very supportive of a short math course for school students giving a very high-level, descriptive view of the main ideas, the history behind them, and the snazzy applications, but bypassing the technicalities completely. This proposal, of course, is unthinkable. The notion that in mathematics there is room only for solving the exercises in the textbook, but not for appreciating the bigger picture, is firmly ingrained in the education system. This view misses the point that, though you only learn how to do mathematics properly by working hard on the problems, you still can enrich your life by learning the big concepts and the challenges that mathematicians seek to overcome, even while avoiding the specific details. You may not be able to play an instrument, but that does not mean that you cannot have music in your life.
The second argument, that math strengthens your brain and makes you think differently, is correct to the extent that you need to shape your mind in a certain way to solve analytic problems. However, this is a side-effect of the overall experience and not the primary goal of doing mathematics. Before reaching the stage of developing your mind more rigorously, you will need to learn the long list of definitions, concepts, and techniques. A better, more direct way of sharpening those skills is, for example, by playing games that stimulate your brain and have fewer rules and prerequisites than doing mathematics. Chess, Sudoku, Rubik´s cubes, crosswords: the list is endless, and I would be an enthusiast of a school class in which all you do is playing those games.
Finally, the last point, that mathematics is the key to the door to a profitable career is not a favorable aspect but rather a problem, one that would require a revolution to change this paradigm of education. Mathematics has become a filter to go to college, and in many countries, you need to take Calculus in school even if you want to major in Law. I cannot think of a greater source of anxiety than having your future depending on how you perform on a subject for which you have no interest and which you inherently find difficult. This filter is as arbitrary and ridiculous as demanding everybody in school to pass cryptic exams on Theology, even to those who just want to study Engineering.
The problem is not pedagogical (though some think it is)
During the 1990s there was an intense debate between two lines of thinking about mathematics pedagogy that later would be referred to as the “Math Wars”. One camp advocated a traditional approach, with an emphasis on procedures and formulas, and the other one, the reformers, proposed a method that highlighted ideas by exploring them in real contexts. The discussion remains until our days, but certainly, the teaching of mathematics has evolved for good, emphasizing what matters.
We have come a long way from the days in which all you had for learning mathematics was a dense tome full of formulas with dry explanations, and intelligible figures. Nowadays there is a myriad of alternatives, with colorful books, engaging blogs, and charismatic YouTubers motivating and explaining every corner of school mathematics. All these efforts are brilliant and show that math educators have put all their energies to make the subject more palatable.
However, the debate around the Math Wars started already with the acknowledgment that schools must teach mathematics to everyone, but not with the more fundamental question of whether students in middle and high school should learn it at all. The pedagogical issues have been addressed extensively in recent decades, but the content of the subject and the suitability of making it mandatory have been mostly removed from the discussion.
Forget about Calculus
The path of mathematical education in school traces an arch that starts with basic algebra and finishes with Calculus. The end goal of all the efforts partaken by teachers and students is to reach that level in which the pupil commands the powers of the derivative and the integral. But making Calculus the pinnacle of high-school mathematics is a colossal mistake.
Calculus is a triumph of the human intellect, but it is also an extraordinarily challenging and almost counter-intuitive way of thinking. It deals with “infinity” and the “infinitesimally small”, concepts that are far from the intuition of anyone. It took the genius of the pioneers of Calculus during the 17th and 18th centuries (Newton, Leibnitz, the Bernoulli brothers, Euler, etc.) to construct the scaffolding of the subject, but not even them could properly define what they meant by “vanishing quantities”, nor escape to the inconsistencies that seem to arise when you manipulate those “fluxions”.
Calculus was a tremendous success since its earliest days because of the precision of its results, but the handwaving descriptions of its foundations made the mathematicians of that age, just as much as the modern student, nervous. There is probably no mention in schools today that less than 50 years after the work of Newton, Calculus was under open attack because of the vagueness of its fundamental concepts, perhaps the most eloquent made by George Berkeley, Bishop of Cloyne, in his 1734 book “The Analyst”:
“And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”
The lambasting of the good old Bishop could be repeated these days, word by word, by any modern teenager struggling with the exercises in her Calculus textbook.
It took most of the 19th century, and the work of an army of mathematicians following the lead of the great Cauchy, to address all the points and make Calculus perfectly consistent and free from ambiguity. But this conceptual framework is quite rigorous and takes a lot of work and mathematical maturity to be absorbed, not something that every single student in the school will be able to, nor should they, do. And there lies the false dilemma: should schools teach the operational, mechanical aspects of Calculus, sacrificing the profound meaning of what it really is, or should they try to go deeper in the material at the peril of losing everyone in the classroom? The answer is that you should simply not teach it all.
Structuring the academic curriculum in such a way that Calculus is the end of the journey is also an odd choice, as in fact, it is just the point of departure for a long list of other subjects. The first contact you make with the cool, powerful applications is in the course of Differential Equations, the powerhouse of applied mathematics, but it takes at least a couple of years of university education after you take Calculus to get there. The real fun of applying mathematics does not happen when you learn how to compute derivatives, but when you use them in optimization, probability or mathematical modeling. Finishing with Calculus is like going through all the effort of going to the airport, checking your bags, going through security, queuing in the gate, boarding the plane, taxiing on the runway and then, when the plane is about to take off, stopping abruptly and hearing the captain saying that she hopes you have enjoyed the ride.
Of course, I am not the only one who has pointed out that teaching Calculus in schools is a terrible idea. The report “The Role of Calculus in the Transition from High School to College Mathematics”, organized by the Mathematical Association of America, and the National Council of Teachers of Mathematics points out: “Calculus may not be an appropriate goal of the high school curriculum for all students, especially for those students who will not require it for their post-secondary plans“. Other voices, from teachers, students, parents and members of the mathematical community come to the same conclusion. See for example the engaging talk by math teacher John Bennet, “Why Math Instruction Is Unnecessary”.
Teaching what matters
Mathematics should not be a compulsory course in middle and high-school. And if we can’t agree on that, at least we should reconsider Calculus as the cornerstone of the academic curriculum. Schools would serve society much better by focusing on other mathematical-related material, that is accessible to everyone, and that is far more important for students to learn. In my opinion, the following are the basic mathematical skills that high school graduates must have, but that the current program doesn’t address adequately:
- Basic arithmetic calculations, computation of percentages, and exponentiation: most of this is the material of primary school but is staggering how many adults have zero understanding of this.
- Basic probability, statistics and making sense of data in general: not data science or machine learning, but just the basic skills that allow someone to navigate a world that is becoming more and more immersed in seas of data.
- Numerical literacy: fluency in the language of quantities and measurements. I always find it shocking when I see professionals stumbling with the difference between millions and billions, or incapable of making basic rough estimates of populations, times or distances.
Throwing on top of that a couple of modules describing the main ideas of mathematics, and adding a regular schedule of games that can boost your cognitive powers should probably be enough for the vast majority of kids attending school. And for that minority that finds enjoyment in mathematics, let’s dazzle and challenge them with serious Calculus and as much mathematics as possible. Surely such an arrangement will benefit everyone and will dispel that silly but frightening aura that mathematics carries these days.
In my opinion, mathematics is the root of all subjects and it should be learnt in object language. Learning transfer in mathematics becomes difficult in human language. It’s amazing that differential learnography is described in the working mechanism of our brain and this is the manifestation of calculus. Thanks for the descriptive writing
OMG ! “Math difficult in human language” – thank goodness, someone realizes it. I recall Euclid’s complicated construction of sentences for theorem – I simply cannot memorize them !
Construction and theorems in geometry are the best examples of object language and we don’t need to memorize them. But we have to make brainpage for knowledge transfer.
Yes, math is not a subject, subject to memorization, thereby adding to its ease.
Lengthy post. So without going below the 3 points in post, your observations are correct. Moreover, in my opinion Maths actually makes people more irrational (√2) and that’s where the difficulty begins as humans want to be rational. However, I’d not like to loose hope in trying to teach them math, even if I fail trying.
“Calculus was under attack” (oh, I couldn’t quit reading the post) – even the magic number, zero was also under attack because humans have always wanted to think rational (well, 1/0 is unmeasurable).
Calculus can be made additional subject of choice, but should not be eliminated from high school completely.
Thought-provoking. I’m not sure I agree really. I used to really love math, per se, when younger. I couldn’t sleep for the first week after I heard about the four-color theorem (around 1958). I was sure I could solve it if I just put my mind to it. But I decided I would become too obsessed with it for my health if I went into math. I still like math, but find it hard to spend much time on something unless I see a practical benefit. Anyway, I think algebra, geometry, simple trig and basics of stat — those are within most people’s grasp BUT such subjects should always be taught from the standpoint of mastery — not from the standpoint of a “competition” in which only the smartest kids can really understand what is going on and everyone else gets further and further behind.
Are you familiar with John Anderson’s AI-based geometry tutor, for instance? It’s almost as good as a real human tutor. Here’s a link to a post about what I mean about a statistics concept that nearly everyone needs to understand. https://petersironwood.com/2019/07/25/thinking-tools-the-whole-enchilada/
Thanks a lot for your comment Peter, and also for sharing your own experience with the subject. I like your view about mastery vs competition when learning Maths, something to think about. I haven’t heard about John Anderson’s tutor but I’ll check it out, as well as your blog. Wishing you a happy new year!