Learning Jigsaws

"Onus of making lectures more interesting is on the teacher. What about the onus of learning?"

I am an assistant professor in the Department of Electrical Engineering at IIT Madras. I joined in 2016 and have taught multiple courses to different class strengths varying from a low 20, all the way to 150+. I am very passionate about teaching and that should do for now. I decided to embark on writing this article to share my thoughts on teaching and learning. After all, of what use is teaching if learning cannot happen at the other end?

Teaching:

The end goal of teaching is to ensure that students learn and that is precisely why pedagogy experts  emphasize on the the role of "learning" objectives and not "teaching" objectives. Standing and simply delivering an hour long lecture will not ensure that students learn the material. Feedback plays an important role in this process and hence teachers are advised to partition their material into smaller chunks of 10-12 mins each, pause to ask a question and proceed based on the feedback. Obviously, throwing open a question in a classroom will result in a select few answering and the rest falling behind sooner or later. Teachers have thus adopted the idea of posing the question in the form of a quiz and students responding by way of buzzers or through mobile apps. Teachers also employ the idea of minute papers where students list out the concepts that they found hard to follow in that particular lecture. There are numerous other methods that teachers all over the world adopt to ensure better learning and it would suffice to say that teachers have adapted to changing times. There is a system in place to share such pedagogical knowledge with teachers and centers like the Teaching Learning Center (TLC) at IIT Madras does exactly this for their own faculty and many other faculty from the country. All this aside, at the heart of good teaching is technical depth that cannot be substituted by any amount of good teaching practices or other attempts of making the lecture interesting. 

That brings me to an interesting question. We live in a world where there is a deluge of information on the world wide web. What role does a teacher play in this Information Age? Of course, I do not have to stress on the fact that information and knowledge are vastly different and should not be confused with each other. Like Feynman says, you could know the name of the bird in ten different languages and yet know nothing about the the bird. The role of a teacher is to tell you about the bird! 

Is that enough? Certainly not.

One of the questions I had to answer, to myself, before I taught my first course at IIT was why a student should even sit down and listen to my lectures? For the moment let us forget about traditional classroom teaching and instead assume that I decide to record a lecture series and put it up on the internet for others to consume. There is no grading or exams or any form of evaluation that would force a student to sit through my lectures. There are plenty of courses available online from various universities and from the best of faculty. Both the student and I have access to the exact same information. The same video content, the same textbooks and the same lecture notes that other professors make publicly available. What would prompt a student to pick my lectures over other material available online? I did not have the answer to this question in the beginning of my preparation but nevertheless I decided to go ahead, hoping to find it on my way. A month or two into the preparation phase, the answer dawned on me and provided immense clarity that helped boost my preparation and deliver my lectures more effectively. I will answer this a little later but before that, we need to switch ends. 

Let us examine the other end of the channel:- Learning.

Learning:

In order to understand the process of learning we need to understand how one assimilates a new piece of information. I will restrict my examples to XII standard mathematics and physics so that I can reach out to a wider audience. Let us assume that you are sitting in the first lecture of integral calculus and the teacher says integration is the anti-derivative and that you are taught the solution to integral x^2 dx is x^3/3 (+C). This is new piece of information that has to enter your brain. What could prevent this from happening? At this point you have already been taught differential calculus and you know that the derivative of x^3 is 3x^2. How were you taught that piece of information, though? Well, you were taught polynomial expansion in earlier classes and you know how to derive the result from "first principles". You were also taught how to take limits and you put all of this together to get the answer 3x^2. Now you know why you have to scale the integral by a factor of 1/3, in order to be left with x^2. The topic is immaterial. The idea is pretty much the same! When a new piece of information is presented, a complete hierarchy of existing knowledge kicks in and checks to see if the new bit of information is consistent with what already exists and only then can the brain accept it, satisfactorily. All the information and knowledge in your brain is sitting in a carefully arranged hierarchical jigsaw and the new piece must find a place to fit exactly into this giant jigsaw. 

Can everything be broken down into its first principles like I explained above? Definitely not! Many times, the piece may fit the jigsaw almost perfectly but for a small kink somewhere. You might not have the maturity to sort that kink out as yet. Or you may have to learn many other ideas before that piece can fit perfectly. Nevertheless, there are a set of axioms that you accept and new pieces of information need to be built hierarchically from these axioms. The deeper the axiom sits in the hierarchy, the lesser you will have to memorize. For example, a charged particle gives rise to an Electric field governed by Coulomb's law while a moving charge gives rise to a current which in turn results in a magnetic field. The magnetic field is governed by the Biot Savar't law. These two laws are taught as independent pieces of information, and were historically discovered independently too. Therefore you build two different axioms one based on a static charge and the other based on a current. A deep question could arise at this point. What happens to the electric field when the charge is moving? Honestly, this question did not occur to me ever. A friend of mine posed this question around 2010 and it wasn't until four years later that my colleague explained it to me and pointed me to Feynman's lectures for the answer. It was a moment of joy for me. Feynman explains how the relativistic contraction of length leads to a small charge imbalance in the current carrying conductor and it is precisely this electric field which manifests as Magnetic field that we knew all along as Biot Savart's law. 

Electric field from a static charge and Magnetic field from a current carrying conductor. 

What if this profound question had occurred to me when I learnt Biot Savart's law for the first time? Well, I might have done poorly in my XII grade exams :). But the point is that this deep concern can prevent you from accepting the new piece of information as presented by the teacher. At the other end of the profoundity spectrum, lies incorrect understanding of a basic concept that needs to be rectified in order to accept the new piece of information. In the end of the day, as a student, you are the only one who knows what this jigsaw puzzle in your brain is. The teacher is faced with a class of 100 students, each one with their own jigsaw and it is therefore impossible to fit it precisely into each one's brain. That is where asking questions help. A good teacher encourages doubts and questions in class. These questions give the teacher a sneak peek into your jigsaw and can help you understand it better. In some cases, you may benefit from questions that others ask and you are a mere spectator to the exchange. The bottom line is : onus of understanding is on you, the student!

I can teach but you have to understand.

Understanding:

There are numerous videos on YouTube that claim to demystify a certain scientific concept. Not all of them, even if well made, and well understood by your brain, appeal to you in the same way. A select video from that set makes you go "This is what I have been looking for all these days!" Ever wondered why? It probably means the content resonates with your jigsaw very well. The same is true with books but I will focus on videos because they are more widely accessible to Gen-Z and teaching is also moving in that direction. The pandemic has only accelerated this move significantly. Gone are the days where Brilliant tutorials would send you material on a weekly basis and assess you with the practice problems. We now live in an era where short videos are made to explain a concept, sometimes through interesting animations and illustrations. Educational start ups have business models built on this teaching and learning mode. These videos are almost always well made, require a non-trivial amount of effort and have been designed with the intent of better understanding. So what is wrong? Before you conclude that I have a problem with the money that one could make through educational startups, let me state that I have no problem with any business model as long as the customer is satisfied. Nothing else matters. However, somewhere along the line from the Brilliant tutorial model to this enterprising and entertaining video mode of teaching, the onus of understanding has become a little blur. Students expect the video to magically make them understand the concept. It is here that I would like to remind you, as a student, the onus of understanding is entirely on you. The video in isolation may be very accurate and very interesting but what matters is the jigsaw in your brain. For example, a video might explain the concept of a rigid body rolling down an arbitrary incline, animate the concept of moment of inertia, explain how friction causes it to roll (without slipping) and how one can apply conservation of energy at the two end points of the incline to solve the problem. While this may be interesting on its own and the video may be extremely well made, it may not answer the question of why no energy* is lost to friction durning rolling but a significant amount is lost to it while sliding! This might be the stumbling block in your jigsaw and no one but you will be able to sort this out, unless you explicitly ask this question. More importantly, a good teacher can only make the content interesting but at no cost should it be made trivial. Some concepts are hard. They need time to sink in. This time could be days, weeks, months or even years. Occasionally, some require a life time and lead to breakthroughs in scientific understanding. The sooner you accept this reality the better your learning experience will be. It is ironical that teaching has evolved significantly in the last couple of decades and teachers have adapted beautifully but the process of learning is still the same. 

A rigid body rolling down an arbitrary incline

Conclusion:

I will conclude the article by addressing the unanswered question, as promised. Why would someone choose to view my lectures among numerous other choices? As you would have rightly guessed, it may simply appeal to certain students because it helps fit the knowledge into their jigsaw more accurately than other lectures would. Is this a matter of chance? Not really! There is some method to this process. While preparing for the course I read books and watched many videos and prepared my own notes and found that various pieces did not fit my own jigsaw very well. I had to iron out the kinks and that required enormous amount of thinking and analysis. Typically, one hour of lecture would require five hours of preparation. Now this is only for my own jigsaw. How do I ensure it fits a students' jigsaw as well? That is not guaranteed but here is something one could try. In the process of all this thinking I identified the minimum set of axioms that are needed in order to hierarchically derive any concept taught in the course. In engineering, this is easier said than done. Numerous approximations need to be made to make the analysis tractable and not all of them will appear to be consistent with the other at first glance. I meticulously kept track of these aspects and explicitly spent time clarifying these seemingly inconsistent approximations during my lectures. This way a student only needs to ensure the axioms fit their jigsaw well and the rest will follow automatically. I recorded my lectures for NPTEL and they were made available online in Jan-2020. I have  since received a couple emails from unknown students about how they enjoyed the course and how they resonated with what was taught. While it is very gratifying for me as a teacher to read these emails from unknown students, let us not forget it was some kink in their jigsaw that set them off in pursuit of better understanding. I taught but I am happy they put in the effort to understand.

* Technically the ball needs to start sliding for rolling to kick in and a negligible amount of energy may be lost to friction in the process. However, once the rolling starts, friction does no work on it.