This is how children learn at school. They open a chapter, say trigonometry. They learn the relationships between side lengths and angles of triangles; they memorise the trigonometric ratios; then they move on to solve some trigonometry problems.
If you think about it, children are effectively learning a couple of algorithms or procedures to solve certain kinds of problems in a certain way. Then they are doing exercises that involve those very same procedures. Essentially, what they are practising isn’t trigonometry, but how to apply certain procedures to get the desired results without making “silly” mistakes. They are basically learning how to become an efficient calculator.
This worked pretty well fifty years ago when the world was extremely enthused to pay good salaries for procedural tasks such as typing, calculating, and working on an assembly line. People were given specific instructions that they needed to follow blindly without getting very “creative”.
But the world has moved on. These days, jobs that pay well, the jobs that have the most impact on the world require employees to be able to solve unexpected problems. Problems that cannot be pointed somewhere in a textbook. Problems for which there aren’t any preset procedures. Problems that are wicked. Unfortunately, current school education isn’t equipped for that. The way children are taught at schools has a lot of catching up to do.
As a kid, I was taught both by my mother and my father. My mother is a teacher, so she has her own way of teaching. Apart from teaching the lessons, it also involves teaching certain tricks to get the correct answers so that I do well in exams. Naturally, I was a good in school.
My father on the other hand is a non-teacher, so he didn’t care much about these tropes. His idea was to make sure that I understood the fundamentals very well—naturally it involved lots of questions. What do you think of this? What if this happens? How about this? Why do you think this will be the case? And he had no respect for the bounds of a particular subject. While teaching science, he often jumped to maths, also history.
If I have to be honest, it was a mental struggle for me whenever it was his turn to teach. Even if it was a simple chapter from literature, what my teacher-mother would finish up in 35 mins, my non-teacher-father would take 70+ mins and we would still be nowhere near the finish line.
Naturally, my parents employed completely different teaching methods. My mother was a trained teacher, so she naturally employed an algorithmic, procedural, or a rule-based approach that involved lots of tips and tricks. My father focussed on holistic learning instead. He followed a method that involved a lot of questions, a good deal of thinking, problem solving, and connection building.
My mother’s methods surely got me the grades. But interestingly enough, my father’s methods, although frustrated in the short-term, led to long-term gains. It was difficult indeed, but it had a desired amount of difficulty that facilitated broader learning, and enabled deeper understanding of the material.
If you ask a bunch of random school students whether a/8 or a/12 is greater, about 50 percent of them would answer correctly. It’s as good as randomly guessing. Asked to explain their answers, they would most likely point to some algorithm—like focussing on the denominator, or trying to get a common denominator (without really knowing why).
Very few would start with a broad conceptual reasoning that if you divide something into eight parts, each piece would be larger than if you divide the same thing into twelve parts. I don’t blame them. They aren’t taught this way in schools. Instead of concepts, the mainly learn math procedures.
Just as it is in cricket, procedure practice is indeed important in math—especially to solve problems quickly. But when it comprises the entire math learning strategy, it’s a problem. Students do not view mathematics as a system anymore; they view it as just a set of procedures that they have to memorise in order to solve problems in exams.
When asked what’s the purpose of learning trigonometry, or geometry, or fractions, or even calculus, hardly any student can come up with a better answer than some version of “to correctly answer in exams.” They have no idea how maths is relevant in any way beyond the curriculum. Again, we cannot blame them. They are taught in this manner.
And given the humongous syllabus that a teacher has to cover in a year, I cannot blame them either for not making the classroom a place to engage in connection building, rather than memorising procedures.
Even if they focus on teaching concepts, still the most difficult job of a teacher is to make sure that a student understands them. Just asking questions doesn’t cut. Students figure out how to answer questions correctly. That doesn’t necessarily mean they have understood the fundamentals of a topic.
For example, a teacher asks, “The cost of one ice-cream is 3 bucks. What is the cost of N ice-creams?” “Three over N?” one student offers. The teacher does not respond immediately, so they try something else. “Three times N.” This time the response is immediate. Bingo! “Now what if I say: six less than a number?” “Six minus N.” Incorrect. “Sorry, N minus six.” Great!
What just happened can give the impression that the student understands the concepts, but what was really happening is this: the student was looking for mathematical formulas that might fit in this context. In doing so, they were actually converting a conceptual problem they don’t understand into a procedural one they could just execute. On top of that, they were reading the facial cues of the teacher to go on guessing answers until they got it right, thereby turning it into a multiple choice question. It might look like they understood the concepts, but they were only figuring out the correct procedure through trial and error. I know this because I have done this a lot of times as a kid. Just getting the correct answer never means that a student gets the concept as well.
This hack can appear smart. But failing to learn concepts, and relying only upon procedures often leads to some drastic long-term effects, even though they get good marks in the short-term. A student who is asked to verify that 462 + 253 = 715, would subtract 253 from 715, and get 462. But when they are asked for another strategy, they may not be able to come up with subtracting 462 from 715 to see that it equals 253. Because the rule they learned was to subtract the number to the right of the plus sign to check the answer. 100 in procedure. 0 in concept.
Once in a while there would be a teacher who would ask counter questions to the student. “Explain how you came to the answer.” This naturally confuses the student, and they have to think hard before coming up with a plausible explanation. But rather than letting students grapple with some confusion, the teacher would often resort to hint-giving that again morphs a making-connections problem into a using-procedures one. Again, the teachers cannot be blamed. They have to cover the whole curriculum, and there’s very little time that can be wasted in a 35 mins class.
This takes me back to my father’s teaching method of adding “desirable difficulties” in the learning process by constantly asking questions, and forcing the student to think hard without giving any hint whatsoever. This process is slow and frustrating in the short term, but very effective in the long run. Excessive hint-giving doesn’t help much. In fact, it does the opposite. It boosts immediate performance, but undermines progress in the long run.
In her excellent book, A Mind For Numbers, Barbara Oakley talks about the “recall and retrieval” process. Real learning happens when you recall something that you have read or learnt a while back. The struggle to recall is what actually cements the knowledge. Looking it up or revising it doesn’t do that. It just gives you an illusion of productivity.
Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material. Psychologist Jeffrey Karpicke and his colleagues have shown that many students experience illusions of competence when they are studying. Most students, Karpicke found, “repeatedly read their notes or textbook (despite the limited benefits of this strategy), but relatively few engage in self-testing or retrieval practice while studying.” When you have the book (or Google!) open right in front of you, it provides the illusion that the material is also in your brain. But it’s not. Because it can be easier to look at the book instead of recalling, students persist in their illusion—studying in a far less productive way.
The process of recall and retrieval is a desirable difficulty. In fact, if you deliberately wait it out longer than usual to make it extra hard for you to recall, the better you learn. How counterintuitive!
Another process is to ask a student to generate an answer to a question without giving them any information in advance. For example, “What is the English word for: a product of laborious study?” You may not know the answer, but struggling to retrieve information despite that primes the brain for subsequent learning, even when the retrieval itself is unsuccessful. The struggle is real, and really useful. This is known as the “generation effect,” and a product of laborious study is called lucubration.
Interestingly, being forced to generate answers improves subsequent learning even if the generated answer is wrong. It can even help to be wildly wrong. In fact, the more confident a learner is of their wrong answer, the better the information sticks when they subsequently learn the right answer. This is called the “hypercorrection effect”. This essentially means that tolerating, even encouraging big mistakes during the learning process can create the best learning opportunities.
The internet is rife with content about shiny, new, and unsupported learning hacks that lead to rapid progress. But the right way to learn is taking it slow, doing away with hints, deliberately waiting longer to make recall and retrieval harder, and testing knowledge—sometimes even before learning the concepts.
The best learning process is slow and frustrating. Struggling now is essential for better performance later. It is so deeply counterintuitive that it fools the learners themselves—both about their own progress and their teachers’ skills. Therefore, like my father, the teachers who deliberately make the learning process hard and frustrating for us in the short term so that we can develop a deeper understanding in the long term are our best teachers.