Rong But Reasonable: Chapter 9
The Choodamani of Math
Welcome to the ninth post of the series Rong But Reasonable. In these essays, I share my thoughts on math education in schools. I hope these will be of interest to math learners, math practitioners, educators and anyone interested in critical thinking. Usually in each essay, I choose a common wrong answer to a math question and ponder over its reasonableness. Previous articles can be found here.
This week’s essay is broader in scope, in that it is a broadside against all subjects taught in school. This article is a critique on the fundamental approach to pedagogy.
A few years back, I discovered a neat trick to teach subtraction. Six-year-old Siddharth and I were playing with tamarind seeds. I told him I could see with my eyes closed. He didn’t believe me. To convince him, I took twelve seeds and laid them in front of him. I closed my eyes and ask him to hide a few seeds. Siddharth did so, all the while ensuring I didn’t peek. Then I opened my eyes. I noticed there were seven seeds still lying.
‘You have hidden five seeds,’ I declared.
Siddharth was astonished. We played this game a few times. Each time, I correctly told him how many seeds were hidden. He was mesmerized with each correct answer. Nevertheless, he refused to believe I had magical powers.
‘Then how can I tell each time?’ I asked.
‘Sir, you are clever. You are using maths.’
I told him that he too could perform this trick. This is how we got into subtraction. To Siddharth, math borders on magic. It allows one to be clever. This was the most fascinating aspect of math for him. As an adult, perhaps I would phrase it differently.
‘Maths helps us to know things which we cannot directly see.’
One of the fundamental tenets of our education system is that schools should produce well-rounded individuals. A student who spends certain number of years in school should eventually be able to function within a society. To achieve this, we teach all children basic literacy, numeracy, some science, civics, etc. After reaching a certain level, say Class 10 or Class 12, students can pursue further education or seek employment, based on their inclination and circumstance. This may be a well-intentioned objective. However, in practice, schools rarely achieve this goal.
A) Schools don’t produce well-rounded individuals: Our education system does not teach social and emotional skills, or other life skills. Policy documents unfailingly highlight the necessity to impart these skills. Schools acknowledge it too. Yet the emphasis is always on certain subjects. The activities students enjoy the most — sports or arts are relegated to the lower tiers of academic hierarchy as ‘extra-curricular activities.’
B) Schools don’t produce good citizens: We teach civics but not civic sense. We still don’t know how to live as a community. We still don’t know what it means to be a good citizen. One who drops out of school steals from the till. One who completes higher education steals from the treasury. Leaving aside the moral values, most students lack basic life skills. After years of schooling, most are unable to perform simple tasks, like buying vegetables or going to a bank.
C) Schools don’t produce good workers: By workers, I do not mean obedient drones, but individuals who are productive to the society. Employers bemoan the lack of necessary skills among college graduates. If this is the state of higher education, what can be said of secondary education? No one leaves school with the satisfaction of having learnt a craft. Almost no craftsman has learnt his trade in school. He has acquired it after school.
The current education model does not achieve any of the practical goals expected of it. My biggest lament, though, is that our education system does not foster a love of learning. The subjects are not studied for their own sake, but as a necessity for some future purpose. Knowledge is reduced to a functional level stripped of its beauty and higher purpose.
Instead of the current model, where we teach an imagined minimum level of many subjects, I suggest that we create a model where we showcase the best of each subject.
Instead of a ‘push’ strategy, we must adopt a ‘pull’ strategy which attracts the students based on their interests.
The Crown Jewel Of Each Subject
(What I Thought vs. What I Got)
In Indian culture, choodamani is an ornament worn on the head. Poetically, it represents the ‘best’ or ‘highest’. For example, Vivekachoodamani is a philosophical treatise attributed to Adi Shankara, whose title means ‘Crown Jewel of Discernment’.
What is the equivalent choodamani of each subject taught in school?
The highest achievement in a field of study could be a result or assertion that is incomprehensible for a child or a layperson. For example, General Theory of Relativity could arguably be the choodamani of physics. But only expert physicists can fully understand what it says. So, instead of seeking a specific achievement, let us inquire into ‘what is the best that each subject has to offer’.
In school, I was good at maths and physics. I was bad at biology. I positively hated social studies and languages. My ineptitude and aversion had a connection, although I cannot tell which is the cause and which is the consequence. I started liking these subjects only after they stopped teaching them in school. It was many years later, when I pursued these subjects out of my own interest, that I understood the true essence of these subjects.
Like running into an ex-girlfriend many years after a bad break-up. The failed relationship does not necessarily mean that they are a bad person. Just that we didn’t understand each other at the time.
Whatever I was taught in school had given me a skewed perspective of the subject.
The Choodamani of Biology
I used to think that biology was about understanding the living world. What could be the best of biology? Darwin’s Origin of Species, perhaps. It captures the essence of biology. The book describes a natural law explaining much of the living world. Like most children, I too was curious about the world around me. How can a giant tree grow from a small seed? How can a cat walk across a narrow plank effortlessly? After studying biology till Class 10, I still had no answers to these questions. Instead, all that I remember is that mitochondria is the powerhouse of the cell!
To me, biology at school meant…
- Memorizing lots of information, most of which was in Latin. These included parts of a cell, names of organs, names of animals etc.
- Being able to draw well. It was an axiom that only those who could draw well should study biology. I was not good at drawing at that time.
- Witnessing animal cruelty. One of the experiments in respiration was to drown a cockroach sideways instead of headlong, because insects breathe through trachea along the sides of their body. Class 11 onwards we no longer need to be mute spectators, we can become active perpetrators. We can look forward to crucifying and disemboweling a frog.
- Gross icky stuff. This includes accurate description of digestion, vivid symptoms of diseases, graphic explanation of animal behaviour, and uncensored details about human reproduction. The omitted chapter on reproduction taught me a great truth about our contemporary society. There is no need to ban or burn books. Whatever offends you, put it in a textbook and mark it ‘Optional’ or ‘Not for Evaluation’. No one will ever read it. Suffices to say that most kids do not learn about reproduction from the textbook.
I write this not to denigrate biology. Without doubt, biology underpins modern medicine, ecology, agriculture and much more. My issue is not with biology, but the way it is presented. Whatever was taught in the syllabus did not inspire me to study biology. To be inspired about the natural world, watch anything with David Attenborough in it.
The Choodamani of English
Shakespeare could be a worthy contender. To me, English is a means to communicate. A good grasp of English allows us to articulate our thoughts clearly, perhaps cleverly. The greatness of Shakespeare is that he was able to convey complex thoughts and emotions through words. Of course, we expose children to Shakespeare. But now it is an exercise in memorizing lines of Middle English, a language nobody speaks today. These days I am interested to know more about the language. For example, how the printing press caused the spelling and pronunciation to diverge.
But English at school meant…
a) Prestige: English speakers are regarded as classy and sophisticated. It is a necessity to climb the socio-economic ladder. In school, we were expected to speak in English. Somehow, we were embarrassed to speak our native tongue. Now, I feel this is nothing more than a colonial hangover. For me, English is a tool, not my identity.
b) Correct grammar: English grammar is an arbitrary illogical set of rules that even native speakers cannot comprehend. ‘I read a book.’ Have you done it or are you doing it? Pronunciation of ‘read’ depends on the tense. When I grew up, ‘invite’ used to be a verb, ‘impact’ used to be a noun. Now it is all jumbled and I spent years learning parts of speech for nothing. It’s always ‘the big bad wolf’ and never ‘the bad big wolf’ although nobody knows why. I laugh at those who are proud of mastering English grammar. It’s like being certified the sanest inmate in a lunatic asylum.
c) Format of a formal letter: To this day, the most maddening lesson in English is how to write a formal letter. The insistence on commas at the end of each line of address, ‘yours sincerely’ without the apostrophe, ‘thanking you’ instead of ‘thank you’. In an entire sheet of paper, the actual content forms only three lines. And those three lines are written in a decorative convoluted language.
The formal letter represents the polar opposite of clear and concise communication. All I remember from middle school English is signing off each letter with ‘yours sincerely’ when I was least sincere about anything.
History
What is history? It is the study of patterns in our past. History, when taught properly, could exceed even science or math in making students think critically. It provides opportunities to propose hypothesis, uncover evidence, assess credibility of information, advance alternate hypothesis, etc. History is replete with innumerable cause-effect connections, correlations, and coincidences. The essence of history is that it is a story, a narrative about humanity.
Instead, history has become glorification of war and mass murder. It is presented as a series of conquests, achieved by a handful of ‘great men’. History worships war and historians worship war criminals who were personally responsible for the death and destruction of millions. And yet, historians have the temerity to mock the masses who worship prophets, who have caused far less destruction.
In addition, history is conflated with mythology and propaganda. Academics do not study history to understand the past, but to find evidence to strengthen their pre-existing convictions. It would be dangerous to feed this toxic cocktail to young impressionable minds. That is why we present a diluted homeopathic version of this cocktail in our textbooks. The history taught in schools is just a list of names and dates. This is good because
- It is harmless and inoffensive
- It is easier to test in an exam
The most profound meditation on history I have read is not from a textbook, but in the epilogue of Tolstoy’s magnum opus War and Peace.
As with other subjects, the school curriculum did not inspire me to study history. In fact, I hated social studies so much, that I slogged to become the school topper in Class 10 board exam. Out of sheer spite.
The Choodamani of Math
Mathematics offers many benefits. For one, it is essential to a lot of our day-to-day activities like shopping or cooking. Another aspect is that math is also a language to communicate and calculate. This is true in commerce, engineering or natural sciences. We are able to describe physical laws better using math. Math also fosters logical thinking.
While all of the above are important, the essence of math, at least to me, is its abstraction. An abstraction that is true and beautiful.
A mathematical truth is independent of the observer or even the physical reality. The beauty of math is that the truth reveals itself in unexpected forms. In math, we are free to choose the rules of the game (axioms and definitions), but we don’t know the consequences (theorems and conjectures). Even if we design an entire world according to our whims, there is still much to explore. We have to explore the territory inch by inch, uncovering truth after truth.
Being a mathematician is like being a detective, an artist, a magician, and a philosopher, all through the power of one’s mind.
The crown jewel of math should capture the beauty and truth. That honour goes to Euler’s identity, often described as the most beautiful result in math.
Research shows that when presented with this equation, a mathematician’s brain responds the same way a layperson’s brain responds to a painting or music. Why do mathematicians find it beautiful? Because it is an unexpected revelation of truth. The number e = 2.71828… is the limit of an exponential function, π = 3.14159… is the ratio of circumference to diameter of a circle. We don’t even know their exact value in decimal system. The letter i is the square root of -1, which doesn’t make sense intuitively. Yet these mathematical constants assemble themselves into a simple but elegant relation. This represents the best of mathematics. And we must show this to students of math.
‘But we can’t teach Euler’s identity to kids!’
This is a common wail from schoolteachers all over the world. How can we teach something so technical to students who do not yet know the basics of a subject? We cannot teach relativity to students until they have mastered Newtonian mechanics. A person needs an expansive knowledge of the field before they can understand the highest achievement in it.
I would argue that even extensive knowledge is insufficient. One needs immersive experience before they can appreciate the beauty of a subject.
This appears to contradict my argument calling for us to present the best of every subject to the student. There is no contradiction.
By ‘best of the subject’ I do not refer to the best product of the subject. I refer to the best feature of the process involved in the subject.
This is definitely accessible to school kids. There are countless examples within the curriculum to showcase the best that each subject has to offer. These results provide the same magical experience to the student as the highest results offer to a scholar.
Let me give a few examples.
a) The sum of angles in every triangle is 180°. Is this a coincidence? Why is it this particular number? How is this true for all triangles irrespective of their shape or size?
b) Every prime number is one away from a multiple of six, i.e. a prime is always either one more or one less than a multiple of six. Why? How can we know for sure? After all, there are infinitely many primes.
c) Joining the midpoints of any 4-sided polygon gives a parallelogram. Why?
d) Why is it colder at the top of a mountain? It should be hotter since it is closer to the sun!
e) Look at the state borders in India and the US. In India, all state borders are squiggly, whereas in the US, most state borders are straight. Why?
f) Small mountainous countries remain independent, even when surrounded by bigger powerful nations who could easily annex them. E.g. Nepal, Bhutan, Switzerland. How?
g) Why do all animals have even number of hands and legs?
We do not have to wait for students to come to university to see the beauty of a subject. It can be seen even in primary school. Think of the generations of students who have been put off by a subject before they even had a chance to know what it is about. We are still obsessed with the idea that ‘everyone should know a little bit of everything’. And as teachers, we want every student to love our subject. We consider it a personal insult if a student feels otherwise.
As a math educator, I come across children and adults who fear or hate maths. I reassure them that it’s okay to not like math. I only ask them, ‘Have a taste of real math before you reject it.’
As a student of English, if I decided that the language does not appeal to me after having read Keats or Wordsworth, that would be a fair call. Imagine if I came to that decision, not after reading literature but the Income Tax Act of 1961.
To summarize, instead of teaching a fixed quantum of superficial facts and methods to students, we must showcase the best aspect of the subject we teach. We need to be better ambassadors of our field of study.
We must teach not to inform, but to inspire.
After having seen the best of a discipline, students must be free to choose which subjects they wish to study. To reiterate, the key phrase is ‘after they have seen the best’. Such a pull strategy is far more effective than the push strategy currently employed in our education system.
To quote Antoine de Saint-Exupery: If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.
What is the best aspect of each subject according to you? Share your thoughts as comments below.
Big thanks to all the readers. Your encouragement means a lot.
Extras:
History Matters is an excellent YouTube channel that offers short and hilarious history lessons.
Map Men is another YouTube channel that talks about interesting aspects of geography.
David Attenborough is 97 years old. Planet Earth (2006) was the best programme to appear on TV. Until Planet Earth 2 (2016). His latest series Planet Earth 3 is out now.
Here is Quentin Tarantino talking about the best aspect of cinema.
