The Calculus Bottleneck

What if someone told you that learners in high-school don’t actually need calculus as a compulsory subject for a career in STEM? Surely I would disagree. After all, without calculus how will they understand many of the topics in the STEM. For example basic Newtonian mechanics? Another line of thought that might be put forth is that calculus allows learners to develop an interest in mathematics and pursue it as a career. But swell, nothing could be farther from truth. From what I have experienced there are two major categories of students who take calculus in high school. The first category would be students who are just out of wits about calculus, its purpose and meaning. They just see it as another infliction upon them without any significance. They struggle with remembering the formulae and will just barely pass the course (and many times don’t). These students hate mathematics, calculus makes it worse. Integration is opposite of differentiation: but why teach it to us?

The other major category of students is the one who take on calculus but with a caveat. They are the ones who will score in the 80s and 90s in the examination, but they have cracked the exam system per se. And might not have any foundational knowledge of calculus. But someone might ask how can one score 95/100 and still not have foundational knowledge of the subject matter? This is the way to beat the system. These learners are usually drilled in solving problems of a particular type. It is no different than chug and slug. They see a particular problem – they apply a rote learned method to solve it and bingo there is a solution. I have seen students labour “problem sets” — typically hundreds of problems of a given type — to score in the 90s in the papers. This just gives them the ability to solve typical problems which are usually asked in the examinations. Since the examination does not ask for questions based on conceptual knowledge – it never gets tested. Perhaps even their teachers if asked conceptual questions will not be able to handle them — it will be treated like a radioactive waste and thrown out — since it will be out of syllabus.
There is a third minority (a real minority, and may not be real!, this might just be wishful thinking) who will actually understand the meaning and significance of the conceptual knowledge, and they might not score in the 90s. They might take a fancy for the subject due to calculus but the way syllabus is structured it is astonishing that any students have any fascination left for mathematics. Like someone had said: the fascination for mathematics cannot be taught it must be caught. And this is exactly what MAA and NCTM have said in their statement about dropping calculus from high-school.

What the members of the mathematical community—especially those in the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM)—have known for a long time is that the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. (emphasis added)

The problem stems from the fact that the foundational topics which are prerequisites for calculus are on shaky grounds. No wonder anything build on top of them is not solid. I remember having very rudimentary calculus in college chemistry, when it was not needed and high-flying into physical meaning of derivatives in physics which was not covered enough earlier. There is a certain mismatch between the expectations from the students and their actual knowledge of the discipline as they come to college from high-school.

Too many students are being accelerated, short-changing their preparation in and knowledge of algebra, geometry, trigonometry, and other precalculus topics. Too many students experience a secondary school calculus course that drills on the techniques and procedures that will enable them to successfully answer standard problems, but are never challenged to encounter and understand the conceptual foundations of calculus. Too many students arrive at college Calculus I and see a course that looks like a review of what they learned the year before. By the time they realize that the expectations of this course are very different from what they had previously experienced, it is often too late to get up to speed.

Though they conclude that with enough solid conceptual background in these prerequisites it might be beneficial for the students to have a calculus course in the highschool.

The problem with what is taught in schools

Many people have written on the problem of what is taught in schools and why children don’t like what they study. One of the major issue seems to be there is no direct relevance to what children are taught in the school and their own personal and social lives. The content in the school textbooks has been dissected of any meaningful connections that the children could make in their real lives. The school tasks are decontextualised so that they become insulated from the real world. The quote below very nicely captures what I wanted to say on this issue.

These kinds of situated-learning tasks are different from most school tasks, because school tasks are decontextualized. Imagine learning tennis by being told the rules and practicing the forehand, backhand, and serve without ever playing or seeing a tennis match. If tennis were taught that way, it would be hard to see the point of what you were learning. But in school, students are taught algebra and Shakespeare without cognitive apprenticeship being given any idea of how they might be useful in their lives. That is not how a coach would teach you to play tennis. A coach might first show you how to grip and swing the racket, but very soon you would be hitting the ball and playing games. A good coach would have you go back and forth between playing games and working on particular skills – combining global and situated learning with focused local knowledge.

Allan Collins – Cognitive Apprenticeship (The Cambridge Handbook of the Learning Sciences)
Papert too has some nice metaphors for this, and constructionism hence includes problems or projects which are personally meaningful to the learner so that they are contextualised withing the lives of the learners..

Conditioning hatred for books

The Director opened a door. They were in a large bare room, very bright and sunny; for the whole of the southern wall was a single win-dow. Half a dozen nurses, trousered and jacketed in the regulation white viscose-linen uniform, their hair aseptically hidden under white caps, were engaged in setting out bowls of roses in a long row across the floor. Big bowls, packed tight with blossom. Thousands of petals, ripe-blown and silkily smooth, like the cheeks of innumerable little cherubs, but of cherubs, in that bright light, not exclusively pink and Aryan, but also luminously Chinese, also Mexican, also apoplectic with too much blowing of celestial trumpets, also pale as death, pale with the posthumous whiteness of marble.
The nurses stiffened to attention as the D.H.C. came in.
“Set out the books,” he said curtly.
In silence the nurses obeyed his command. Between the rose bowls the books were duly set out-a row of nursery quartos opened invitingly each at some gaily coloured image of beast or fish or bird.
“Now bring in the children.”
They hurried out of the room and returned in a minute or two, each
pushing a kind of tall dumb-waiter laden, on all its four wire-netted
shelves, with eight-month-old babies, all exactly alike (a Bokanovsky
Group, it was evident) and all (since their caste was Delta) dressed in
“Put them down on the floor.” The infants were unloaded.
“Now turn them so that they can see the flowers and books.”
Turned, the babies at once fell silent, then began to crawl towards those clusters of sleek colours, those shapes so gay and brilliant on the white pages. As they approached, the sun came out of a momentary eclipse behind a cloud. The roses flamed up as though with a sudden passion from within; a new and profound significance seemed to suffuse the shining pages of the books. From the ranks of the crawling babies came little squeals of excitement, gurgles and twitterings of pleasure.
The Director rubbed his hands. “Excellent!” he said. “It might almost have been done on purpose.”
The swiftest crawlers were already at their goal. Small hands reached out uncertainly, touched, grasped, unpetaling the transfigured roses, crumpling the illuminated pages of the books. The Director waited until all were happily busy. Then, “Watch carefully,” he said. And, lifting his hand, he gave the signal.
The Head Nurse, who was standing by a switchboard at the other end of the room, pressed down a little lever.
There was a violent explosion. Shriller and ever shriller, a siren shrieked. Alarm bells maddeningly sounded.
The children started, screamed; their faces were distorted with terror.
“And now,” the Director shouted (for the noise was deafening), “now we proceed to rub in the lesson with a mild electric shock.”
He waved his hand again, and the Head Nurse pressed a second lever. The screaming of the babies suddenly changed its tone. There was something desperate, almost insane, about the sharp spasmodic yelps to which they now gave utterance. Their little bodies twitched and stiffened; their limbs moved jerkily as if to the tug of unseen wires.
“We can electrify that whole strip of floor,” bawled the Director in explanation. “But that’s enough,” he signalled to the nurse.
The explosions ceased, the bells stopped ringing, the shriek of the siren died down from tone to tone into silence. The stiffly twitching bodies relaxed, and what had become the sob and yelp of infant maniacs broadened out once more into a normal howl of ordinary terror.
“Offer them the flowers and the books again.”
The nurses obeyed; but at the approach of the roses, at the mere sight of those gaily-coloured images of pussy and cock-a-doodle-doo and baa-baa black sheep, the infants shrank away in horror, the volume of their howling suddenly increased.
“Observe,” said the Director triumphantly, “observe.”
Books and loud noises, flowers and electric shocks-already in the infant mind these couples were compromisingly linked; and after two hundred repetitions of the same or a similar lesson would be wedded indissolubly. What man has joined, nature is powerless to put asunder.
“They’ll grow up with what the psychologists used to call an ‘instinctive’ hatred of books and flowers. Reflexes unalterably conditioned. They’ll be safe from books and botany all their lives.” The Director turned to his nurses. “Take them away again.”
Aldous Huxley, Brave New World

Though fictionalised the above passages capture what makes people hate books in general. The conditioning happens in reality in a more subtle manner. The conditioning laboratory is the school. In school children are made to engage with the books, textbooks in most cases, in the most artificial and dishonest matter. Another problem is the quality of textbooks themselves. Though the school has a “textbook culture”, not enough effort is put in by the writers and designers of the textbooks to make the best that they can offer. Instead cheap, copy-paste techniques, and a mix-and-match fashioned content is crammed and printed onto those pages glued together called as textbooks. No wonder, people when they grow up don’t like books or run away at the sight of them. Its just behaviorism at work with Pavlov portrait in the background.

Schooled and unschooled education

It is difficult now to challenge the school as a system because we are
so used to it. Our industrial categories tend to define results as
products of specialized institutions and instruments, Armies produce
defence for countries. Churches procure salvation in an
afterlife. Binet defined intelligence as that which his tests
test. Why not, then, conceive of education as the product of schools?
Once this tag has been accepted, unschooled education gives the
impression of something spurious, illegitimate and certainly
– Ivan Illich (Celebration of Awareness)

Einstein on his school experience

One had to cram all this stuff into one’s mind, whether one liked it or not. This coercion had such a deterring effect that, after I had passed the final examination, I found the consideration of any scientific problems distasteful to me for an entire year … is in fact nothing short of a miracle that the modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wreck and ruin without fail. It is a very grave mistake to think that the enjoyment of seeing and searching can be promoted by means of coercion and a sense of duty. To the contrary, I believe that it would be possible to rob even a healthy beast of prey of its voraciousness, if it were possible, with the aid of a whip, to force the beast to devour continuously, even when not hungry – especially if the food, handed out under such coercion, were to be selected accordingly.

Seeing that even almost a hundred years later it is almost unchanged gives one an idea of how little effort has gone into changing how we learn.

On respect in the classroom

If you are a teacher (of any sort) and teach young people, don’t be disheartened if the students in your class don’t respect you or listen to you or maintain discipline. Even great philosophers like Socrates and Aristotle has a tough time dealing with their students

Socrates grumbled that he don’t get no respect: his pupils “fail to rise when their elders enter the room. They chatter before company, gobble up dainties at the table, and tyrannize over their teachers.” Aristotle was similarly pissed off by his stu­dents’ attitude: “They regard themselves as omniscient and are positive in their assertions; this is, in fact, the reason for their carrying everything too far.”Their jokes left the philosopher unamused: “They are fond of laughter and conse­quently facetious, facetiousness being disciplined insolence.”
– Judith Harris The Nurture Assumption

That being said, the students are also very perceptive about the knowledge of the teachers, and know who is trying to be a cosmetic intellectual.