The Tyranny of Numbers (And How to Break Free)

Launching Pad

I launched this blog on Friday, 1 March 2024. I knew that I wanted to have a post specifically for International Women’s Day on Friday, 8 March 2024, which became Primordial Prime Ordinals. So I had planned something else as my third post for this blog. However, after the first post, The Eye of the Beholder, I received many responses indicating that people were somewhat apprehensive about reading a blog on mathematics. A few confessed to having some kind of phobia about anything related to the subject. Others said that they just could not understand anything related to it. I am thankful to those who, despite their apprehensions or expected non-comprehension, went ahead and read the posts.

One of the purposes for this blog is to enable readers to appreciate – dare I say, grow to love? – mathematics. But if so many wrote back expressing apprehensions, I’m sure there were many more who remained silent about them. Having enjoyed and loved mathematics for as long as I can remember, I find it impossible to place myself in the shoes of someone who is wary of, or even hates, the subject.

However, I have encountered this before, most often from students who have been forced to take the subject even though it was an elective. For example, in one school I worked at, students had to choose between Art, Environmental Management (EVM) and Mathematics. Now Art is a skill based subject and caters to a small niche of students. And EVM, unfortunately, is not known as being a rigorous subject. Hence, many students who were not skilled enough to take Art and who were reluctant to take a not-so-rigorous subject like EVM ended up taking Mathematics! Given that there are constraints that schools face, such unlikely options are inevitable. Nevertheless, the bottom line remains – many, if not most, of the students just did not want to be in my class, primarily because they have developed an indifference to or an aversion for or hatred of the subject in their earlier years at school.

So I have often wondered, “Why is it that so many people actively dislike this subject that I love so much?” Granted that it is foolish to expect everyone to love what I love, is it too much to hope that there wouldn’t be as much indifference, aversion, and hatred going around for it?

An Important Insight

The first thing that we need to grasp before we can address the indifference, aversion, and hatred is that mathematics is not really about numbers. If this thought seems strange to you and you have done at least a little mathematics beyond 8^th grade, then I would request you to pause here and write to or call your mathematics teacher and ask her/him why she/he stiffed you when it came to teaching you. I’m serious. This post will still be on your screen when you return after writing or calling!

So why do I say that, if you do not know that mathematics is not really about numbers, you should confront your mathematics teacher? Well, first, as a teacher, I would have hoped your teacher knew more than you did! More not just about the procedural aspects of mathematics, about which I have no doubts she/he far outstripped your knowledge. What I’m more concerned about is communicating to the student the idea that, important as what we teach them may be, what they learn in school is not just barely scratching the surface of the subject but also, and more importantly, what they learn in school is largely misrepresentative of it! If you are surprised, prepare for more.

In his excellent online course Introduction to Mathematical Thinking, Keith Devlin asserts that mathematics is the study of patterns. The patterns could present themselves as patterns of chance or change or quantity or relationships or shape, etc. In other words, mathematics is concerned about the patterns that emerge when we focus on any aspect of the real, or even imagined, world. It is so much bigger than what we are introduced to in school.

Impediments to Learning

In school, what we are introduced to are procedures. We learn how to add, subtract, multiply or divide. We learn how to find the Least Common Multiple (LCM) or the Greatest Common Divisor (GCD). We are taught how to perform long division. In other words, we are given ‘recipes’ to carefully follow lest we flounder at the ‘cooked’ up problems we are given to solve.

Now, it is true that some students may be given some patterns to observe. For example, they may be told the rules for divisibility by 2, 3, 4, etc. But do we have the patience and do we allow the space for a journey of mathematical discovery that would lead the students to have their own eureka moments?

Unfortunately, we have crammed each year of mathematical ‘learning’ with endless ‘recipes’ and convince ourselves that we are giving the students a robust education. Due to this we do not have the ‘luxury’ of allowing the students to embark on a journey of discovery. More damaging, however, is the conclusion that students, and unfortunately many teachers, reach that mathematics is about mastering these ‘recipes’.

Now, there was a time when mathematics was about mastering these ‘recipes’. But that is simply because mathematics had not advanced as much then as it has now. In the 17^th century the pinnacle of mathematics was Newton and Leibniz‘s development of calculus. However, even those pioneers did not think their discoveries could be put to use in the study of economics, epidemics, and politics, as they are today. Mathematics as a subject has developed as much as any other field in the intervening three centuries. However, mathematics that we teach in school remains stuck in a time warp while other areas of knowledge at least include newer ideas.

Other areas of study also include the history of their subject. For example, chemistry students are taught the original Periodic Table developed by Dmitri Mendeleev based on atomic mass and how it differs from the modern Periodic Table based on atomic number. Similarly, physics students are taught Newton’s Law of Gravitation and are at least given an introduction to Einstein’s Theory of Gravitation with some discussion on the superiority of the latter, yet sufficiency of the former to land humans on the moon.

Despite this, mathematics is presented as ahistorical, with one formula after another, one ‘recipe’ after another thrown at the students. However, I believe that students have – and are right to have – a suspicion that an ahistorical body of knowledge is actually inhuman! Please read the previous sentence again. If all we have in mathematics are final results rather than at least some indication that there were human struggles behind the development of those results, students will not be motivated to learn because they know that all genuinely human knowledge is developed in and through time and space.

But somehow we have convinced ourselves that mastering the ‘recipes’ implies mastering the subject. And we teachers tell our students that. But not everyone is interested in numerical ‘recipes’, just as not everyone is actually interested in cooking!

In addition, we mathematics teachers have developed a strange version of the idea that ‘practice makes perfect’. However, it is only in the classroom that we are presented with 25 identical problems with only the numbers changed! Indeed, if the law of diminishing returns has anything to tell us it is that we can well overdo this ‘practice’ and make the students repeat things mindlessly just to say they have finished an assignment. In that case, with the students having the attitude of an automaton, it is quite likely that very little learning is actually happening.

It is my hypothesis that, because we tell students that learning mathematics involves the mastery of ‘recipes’, those students who find it easy to master these ‘recipes’ will float along under the illusion that they excel at mathematics, an illusion often shattered when they enter either Grade 11 or college, when the mathematics involved is more abstract and not based on ‘recipes’. And those students who do not care about numerical ‘recipes’ will disengage from the subject and become at best indifferent to it and at worst develop a hatred for it.

The Encroachment of Numbers

This is compounded by the fact that the ‘bread and butter’ of these ‘recipes’, that is, numbers, seem to be exceptionally aggressive entities that transcend the well defined silos our education systems hold sacrosanct. What do I mean?

Well, you have probably recited the alphabet and realized that ‘m’ is the 13^th letter of the English alphabet. But have you ever heard anyone say that ’13’ is the m^th counting number? Even the superscripted ‘th’ in the previous sentence refers back to ordinal numbers! What gives these ‘numbers’ the right to intrude where they don’t belong?

But that’s not even the worst. Every student has her performance in every subject indicated by a number. Of course, some schools or grading systems may try the deceptive path of assigning letter grades. But if you push them and ask, “Why did I get a B and not an A?” your teacher will probably come back with, “Well, you scored 87 and the grade boundary was 88” or something of the sort. So even though some of us pretend that we aren’t using numbers to evaluate the student’s performance, lurking behind our ‘recipe’ or algorithm for assigning even letter grades are those very numbers we pretended we weren’t using.

In other words, we have (ab)used numbers and enslaved them to do a task that makes assessment convenient for us rather than meaningful for the student. And as with all (ab)use, there is enough trauma to go around.

You see, it is my hypothesis that many students slowly develop a fear of and hatred for numbers precisely because, on the one hand, the algorithmic approach to mathematics leaves nothing to the imagination, thereby asking inherently inquisitive and curious young children to shut down those faculties, and because, on the other hand, these entities of cold rationalism, i.e. numbers, are nonetheless used to measure everything they do. It is as though a child was told that the playground bully would be writing up the report about every aspect of their lives!

This ‘tyranny of numbers’ is ubiquitous and is, in my view, what causes many students to develop an aversion toward numbers, just as any people group under the rule of a tyrant would develop a hatred for the tyrant.

Does that mean, then, that we are stuck with mathematics being really enjoyable for a few students while being irrelevant to or despised by most others? Can we not find a better way to teach mathematics, one that will enable more young students to appreciate it, rather than live in a world in which high school mathematics teachers constantly bemoan the fact that our classes are full of students who just don’t want to be there? I believe there is a better way. Of course, I do! Otherwise, I would not be writing this post. So what could a better way be? I will delineate this in five parts.

Rejecting the Tyranny of Procedures

First, we need to move our teaching away from the procedural. Does it really matter how a student adds or subtracts, multiplies or divides? I mean, suppose she was told to add 14 and 29. Does it matter how she arrives at the answer 43? She could stack them one on top of the other and add using ‘carried’ numbers. Or she could count up from either of the numbers. Or she could split them into units and tens, add them and then combine. But does it really matter how the addition was done?

While I think that the above five pronged approach will lead to a situation in which more students will enjoy mathematics and less students will hate it or be apprehensive of it, I am not hopeful that any of them will be adopted. From schools to assessment boards, from individuals to nations, from teachers to students, we have become accustomed to what we are currently doing. And we can barely see any alternative to the tried and true. However, I put this five pronged approach forward with the slimmest of hope that perhaps somewhere down the line we will humanize mathematics and break free from the current cycle of tyranny.

Quite obviously some methods are more efficient than others. But can we leave the discovery of the pros and cons to the students? When one student realizes she is slower than her classmates, then, if she is interested, she will try to learn a new, more efficient method. Note the words ‘if she is interested’. It could be that the student is content with a much slower method. Does that mean she does not know how to add? Absolutely not! She is just content with an inefficient method.

But trust me, at a later date, when she realizes that the inefficient method is holding her back, she may want to learn a more efficient method. But what we are doing is linking the understanding of the operation, which is the crucial element, to the speed with which the student is able to execute the operation, which is really not all that important.

Don’t misunderstand me. Procedures are important. And efficiency is desirable. However, when we elevate the procedural above the understanding of how the various operations work, we make our students begin to believe that memorizing the procedures is crucial for their mathematical understanding. Then, when they find some procedure difficult they erroneously conclude that they do not understand the underlying concept that the procedure was designed to serve.

Avoiding the Tyranny of Time

Second, we need to move away from timed assessments. I claim that speed is not important because it is we who have artificially made speed important by assessing students in contrived timed situations the like of which they will never face when they hold down jobs of their own. We have done this because it is convenient for us to have a closed form, tightly structured environment for assessing students, rather than because it is actually beneficial to the students. We may say that this allows for uniform testing conditions. But who said that uniform testing conditions are the holy grail of student assessments?

Suppose one student solves a problem in 5 minutes while another takes 15 minutes. Does the difference in time actually reflect a difference in understanding? Apart from a life or death situation, in which I am confident one’s ability to use the quadratic formula will never play a role, I cannot fathom why the first student is considered better. All we are doing is privileging speed for, if the exam were an hour long, the first student would solve 12 questions while the second would solve only 4. In fact, in a timed context, we are specifically sacrificing our ability to assess the student’s understanding of the subject involved because we are placing an additional constraint of time that has nothing to do with the subject. This is true of most subjects, of course, except perhaps a course in bomb diffusing! Hence, it would require complete overhaul of the way we conduct assessments in school. The tyranny of time is rendered all the more acute if all we are testing is procedural proficiency, for then we are assessing the speed of execution of algorithms rather than conceptual understanding.

Now don’t get me wrong. This is not a case of sour grapes for I am an excellent test taker. I am able to function really well in high stress time situations. However, I really do not think timed assessments should have any role to play in assessing students. Just because the number of students is inordinately large does not justify making assessments easier for the teacher or assessing body. If we are truly concerned for assessments to reflect student learning then we need to find ways of assessing them that does not artificially introduce a constraint that has absolutely nothing to do with what the student is learning.

Project based learning holds a lot of promise. However, that would actually render large assessment bodies like the CISCE, CBSE, CAIE, and IB obsolete since they would not be able to administer such assessments. It would increase the burden for fair assessments on local schools, which could prove to be a problem. Also, given the prestige that association with some of these bodies gives schools, thereby also allowing them to artificially inflate fees, this will prove to be one of the biggest roadblocks should what I am saying hold any water.

Shunning the Tyranny of Quantity – Aspect 1

Third, we need to streamline our curriculum. Since it is relatively easy for motivated students to learn how to replicate algorithms and since we have convinced ourselves that memorizing algorithms is what mathematics is about, we are unable to think of reasons for which another algorithm should not be added to the curriculum.

However, many things we teach are just different ways of combining previously memorized algorithms rather than being cases of new knowledge for the students. For example, once a student knows that the angles around a point add up to 360° and understands how to apply the sine and cosine rules to a triangle, does she need to be burdened with learning how to ‘apply’ this to a situation with bearings? Unless she is planning a career in aviation or maritime navigation it is unlikely she will ever come across the idea of a bearing. Given that there are only about 350,000 commercial airline pilots in the world, this would seem to be quite a niche requirement. Yet, we ask most of our students to learn how to use bearings.

Another example is the early inclusion of statistics. In many schools we introduce students to ideas of mean and median from Grade 1, perhaps earlier. However, mathematics teachers know that, at that age, we cannot really explain the nuanced differences between the mean and the median nor when one would be more appropriate to use than the other. Yet, all we are doing is bringing in another context within which we ask the students to repeat previously memorized algorithms – addition and division in the case of the mean and ordering in case of the median. But these algorithms are almost trivial to learn. Why can we not introduce the ideas of mean and median at a stage when we can also explain at least some of the nuances?

What we do when we artificially bloat our curriculum with inconsequential, unimportant, or niche concepts is communicate the idea that the subject is beyond the grasp of most students. This is exacerbated with our textbooks, which enable our students to give professional weightlifters a run for their money! When we have to complete more than 20 chapters in a year, we only create a high stress atmosphere for the teacher – stress that then permeates into the subconsciousness of the students. The students have no time to stop and appreciate the mathematics they have learned because we are dragging them immediately to the next item on our ever burgeoning checklist.

Refusing the Tyranny of Calculators

Fourth, we should move away from the use of calculators. If we are honest, any motivated student will be able to learn how to use any calculator in a matter of days or, at worst, weeks. Just look at how adept at using personal computers and smartphones the young people are. They can master a calculator, if needed, quite quickly.

However, once you expect students to have a calculator, you need to justify this requirement. This is evident most clearly in the area of probability and statistics. For instance, if a student understands what a discrete probability distribution is, for example, the binomial distribution with small numbers, how much additional learning is needed before she  can apply what she learnt to the context of the Poisson distribution or the geometric distribution or the binomial distribution itself but with larger numbers? The inclusion of other distributions only changes which keys on the calculator the student has to press. Is that really how pathetic our idea of learning has become?

Another area where this shows up is the use of exponential and logarithmic functions in questions that require calculation of derivatives or integrals. For example, it is well known that a closed form anti-derivative of e^x2 does not exist. What then is gained by a student being asked to find the area under this curve when it requires the use of a calculator? What is lost if instead the question used the function e^x for which a clear anti-derivative exists and is known? Conceptually, nothing. But the latter can be done without a calculator while the former requires it. With no new conceptual understanding being granted to the student, the inclusion of the former can only be to justify the requirement of a calculator.

It is my hypothesis that there is a nexus between assessment boards and calculator manufacturers that ensures the unnecessary requirement of calculators from too early an age. And I am confident that the use of calculators actually hinders rather than furthers student understanding.

Shunning the Tyranny of Quantity – Aspect 2

Fifth, we need to move away from counting how many problems we solve in a given class. At one institute, my boss expected me to complete 25 questions in a 90 minute class. Could I do it? Sure! I knew all the problems and the solutions. I could certainly have written all the solutions on the board. And the students could have been faithful simian stenographers, mindlessly copying the scribbles on the board to their notebooks!

But this does not mean that any learning has happened. And this was a bone of contention with my former boss all through my tenure at that institute.

To the contrary, I remember one year when we unpacked a particularly gnarly probability problem over the course of many classes! With each suggestion the students made, we looked at the implications and whether it actually solved the problem. When we finally solved the problem, it was the students who had done it, not me. And though they were exhausted after the marathon, they were immensely satisfied. And I knew they had learned not just what the correct answer was but, more importantly, why the approaches we discarded didn’t work and why the one we finally settled for was more efficient than the discarded ones and other, more tedious, but viable approaches.

Some may say I wasted precious time by devoting so much to one problem. However, if Keith Devlin is right that mathematics is the study of patterns, then those students learned fruitful and unfruitful patterns of thought during those classes. I still think it was worth the time investment. But probably more to the point those students themselves remembered this years later and told me that they not only learned how to think but developed perseverance.

However, if we insist that quantity rules quality, we will never be able to ensure that any valuable and long-lasting learning happens. This ties back to the previous point about calculators. A calculator can help a student reach the answer with a few button presses in a much shorter time than if the student did not have a calculator. And a mathematically adept student will certainly know why she pressed those particular buttons. So nothing would seem to be lost by allowing a calculator and increasing the number of problems solved by the mathematically adept student.

However, we do not cater only to mathematically adept students! There are students who struggle with mathematical concepts. To them a calculator is just a magic box that churns out answers. Such a student may memorize which buttons to press for certain kinds of problems but she would not be learning anything because she is recognising not the mathematical patterns, but the types of problems. She may be able to even keep up with a mathematically adept student because she has memorized the problem types and the button presses for each type. But she is learning nothing because the trusted genie inside this lamp gives her the answers on demand.

In other words, increasing the number of problems solved based on the premise that calculators speed up the work hinders the learning precisely of those students for whom the subject is proving to be particularly difficult. This is a travesty!

However, if we expect students to think through their approach, we will, as teachers, have the opportunity to address any misconceptions or misunderstandings that would otherwise have been hidden inside the wirings of the magic box!

Breaking the Cycle

In conclusion, I propose a five pronged approach to addressing the malaise most students experience at the thought of mathematics. First, we need to embark on the much needed process of stripping down the school mathematics curriculum. The aim should be to include only topics that introduce new mathematical insights rather than combinations of old ones, since mathematical concepts can be combined in endless ways. Rather than teach students specific combinations of concepts, we should teach them how concepts can be combined.

Second, the focus should be on furthering mathematical thinking rather than memorization of procedures. Students should be asked and engaged to ask questions like, “Why does this procedure actually work?” and “How could this procedure be made more efficient?” This kind of reflection on the mathematics they have employed will enable them to recognize future dead ends before they proceed down those unfruitful roads.

Third, we should move away from giving students problem sets or worksheets that involve mindless repetition of memorized procedures. Rather, the problems we give them should be carefully curated with follow up ‘thought provoking’ questions asked that would further the development of mathematical thinking. We should also give them underspecified problems in which, by trying to solve the questions, they realize they need more information, which they can then obtain either from the teacher or through research.

Fourth, we should not fall prey to the lure of calculators. While I personally find that no significant mathematical insight is gained by allowing students up to high school access to calculators, others may certainly differ. However, at the very least we should limit access to calculators to the final two years of high school than allow it too early, thereby actually giving students a crutch on which they grow ever more dependent.

Fifth, as mentioned in the previous post and here, we should introduce our students to the historical development of mathematics. This will allow them to see that it was humans like us who struggled with problems and who used their imaginations and intuitions to break new mathematical ground. Students will appreciate that mathematics is not some esoteric body of knowledge that just dropped from the sky, but rather the fruit of human imagination and intuition.