Studying Math
In the middle of the lockdown last year, I felt a strong pull towards mathematics. I majored in math during my undergrad and though I was not a natural in it, I loved it when it revealed its beauty to me. I couldn't resist this pull which came now, and so I wondered how I should get started. Initially I thought I will dip into advanced level abstract math, which is filled with a lot of incredible beauty, but later I decided to start with the basics. Basics in mathematics at the later high school and college level is algebra and trigonometry, which are together called Pre-Calculus these days (or Pre-Calc for short). I was familiar with most of the material here and so I thought it would be a breeze and it would help me build my confidence, before I ventured into the tougher parts of the subject. I had an Algebra text book with me which was popular when I was a student, but it was a 19th century textbook (Yes, we used to use 19th century texts once upon a time. I don't know what's happening now.), and I thought that as we are living in the 21st century now, it would be more appropriate to get something which was newer. After doing some research, I discovered that Cynthia Young's 'College Algebra' was written more recently and I zeroed in on it.
Buying a math textbook is like firing a shot in the dark. You never know what you are going to get. Online reviews almost never help, and so I don't bother with them. I use a homemade thumb-rule for choosing a math textbook. I look at the titles and authors of different books on the subject (Russian, German and British authors are harder, American authors are more accessible), find out when a book was first published (older textbooks are harder, newer textbooks are easier), look at who is the publisher (a Springer Verlag book will be notoriously difficult, a McGraw Hill or a Wiley book will be more accessible), and then check which book has the most number of pages (All things being equal, a book with more pages was better than a book with less pages, because the book with more pages will introduce the subject more slowly, more nicely). Cynthia Young's book passed all these tests and so I got it.
Of course, after doing all this, there was no guarantee that the textbook was going to be good and accessible. It could still end up being intimidating and I might give up on it after studying a few pages. But I needn't have worried. Cynthia Young's book was beautiful and accessible. It started from the very basics and took the reader forward step-by-step. It was like my mom or my favourite aunt took me on a walk in the garden and showed me the beauty of the different flowers and the trees and the butterflies and the birds. Of course, if things were too easy, we might wonder whether the book was too simple for us. But Cynthia Young didn't shy away from complexity. While she was leading the reader through the garden, the garden transformed and became a forest. Of course, the reader being immersed in the subject doesn't realize it. At some point of time, we discover complex zeroes and the fundamental theorem of algebra and linear programming and nonlinear equations and probability and things are no longer simple. But by that time our confidence has gone up because of the gentle way the book nurtures us, that we don't see the complexity as intimidating, but we embrace it with pleasure.
Women professors are rare in math. There are many women professors in other physical sciences like biology and chemistry, but it is hard to find women professors in the physical sciences which involve a lot of math, like mathematics and physics. This is reflected in the textbooks which are out there. Most math and physics textbooks are written by men. Cynthia Young seems to be an honourable exception here. I don’t know how she got into math and what kind of challenges she faced while finishing her doctorate and how difficult it was for her to get a faculty job. But I am glad she is there, and I am happy she wrote this book. It is an incredibly beautiful book and I would highly recommend it, if you are crazy like me and would like to dip your toes into the mathematical ocean again, for fun. Here is an interview with Cynthia Young in which she talks about how it is like to be one of the few female math professors out there, how she got into the field and the kind of work she does, and her life outside work and her family. It is very inspiring.
I have finished just three textbooks, completely from the beginning to the end, in my whole life before. I read all these three when I did my masters degree. I was hoping that Cynthia Young's book would be my fourth. It looked easy and it had ten chapters and when I started I thought I'll study a chapter everyday and finish ten chapters in ten days. Simple. So every afternoon, after I had finished my lunch, and cleared the dishes and washed them and put them away and cleared the kitchen, and I was done with domestic chores for the afternoon, somewhere between 3 and 4pm, I went to the front room of the house which was bright and airy and started studying the book. I studied mostly without any interruptions till around 3 or 4am in the morning. I took a short break for having a cup of tea in the evening and I took a longer break when I had to have dinner and clear the dishes and the kitchen for the night. I watched TV only while having dinner, and I avoided my smartphone and I avoided reading other books which I normally loved reading. It didn't matter to me whether it was Monday or Saturday or Sunday. I just followed my pleasurable routine like a metronome. I had never studied like this even during my student days. But things didn't go according to plan.
What happened was this. I started slowly and I didn't rush through things. I found a natural rhythm and learnt at a natural pace. Learning beautiful mathematical concepts and applying them to solve problems was very pleasurable and therapeutic and I was on a constant high. Each of the chapters in the book had many sections, and a section had around a 100 problems and each chapter had anywhere between 800 and 1000 problems. A math book is not worth that piece of paper it is printed on if you don't solve the problems. I was studying the book out of love for the subject and I wanted to solve every one of those problems. The problems were at varying levels of difficulty, but they were not really that hard. It was possible to solve most of them if one had learnt the material in the chapter well. There were some proof type problems too in every section, and some of them looked tricky, but I could solve them too. But the sheer weight of these 1000 problems pressed down on me. It was impossible to finish one chapter everyday. The book laughed at me when I tried. It was hard to even complete one section in a day. It took me typically two days to complete a section. I immersed myself in the mathematical universe and lost sense of time. When I came up for breath, I discovered that two months had passed, and I was only halfway through the book. Yes, that is correct – two months! I had thought that studying the book would be a piece of cake, and I would be able to bend it to my will and breeze through it, but after two months, the book was laughing at me with contempt. Not the contempt of an evil person, but the gentle contempt of a wise person.
I thought about this. This was a textbook which was around 850 pages long with a moderate level of complexity. After some dedicated studying, I was halfway through the textbook and it had taken me two months. The second half of the textbook was harder than the first half and it would take more time. Even if it did take the same time, the whole textbook would take atleast four months to complete. Four months! That was way off from the ten days that I had planned! I thought about all the semesters I did during my undergrad and the terms I did during my postgrad. During my student days, I was always a moderate student. I had the usual distractions that college students have – catching up with friends, participating in college events, following and playing sport, watching movies, reading non-academic books, weekend parties. Luckily I didn't have any romantic distractions. My academic grades were always moderate. I think I had above-average talent, but they were never reflected in the grades. I wondered how other students managed to crack the exams and the courses and top the class. I thought that they probably put in more hours of hardwork and maybe they were smarter than me. So I always thought that if I put in more hours and worked harder, I'd do better, my grades would improve. But my present learning experience proved that it was all an illusion and I was wrong. Let me try to explain with a back-of-an-envelope calculation. During my undergrad, we had a semester which lasted around five months. We studied six courses in each semester. The difficulty level of the courses in the first year was similar to that of the algebra book I was studying now. But from the second year onwards the difficulty level increased. Still, for the sake of argument, based on my current experience, if we assume that it takes four months of dedicated effort to study one textbook properly and complete it, in five months one can learn one subject properly and then a little bit of another subject. This, if there are no distractions, no cricket matches, no movies and other things. That is all that we can do. While the expectation is that one should complete studying six courses in five months, one can really complete only one course and a little bit more in that period of time, if one learnt properly. During my postgrad, things were worse. We had three terms in a year, and each term lasted eleven weeks – that is slightly less than three months. In each term we had six courses. So in a year, we had eighteen courses. So we had to study and complete 50% more than we did during our undergrad in the same period of time. During my postgrad, the distractions were ten times more. Luckily, I didn't have any romantic distractions, but many of my classmates did. Ignoring all of that, the reality is that, with proper dedicated study, I wouldn't have been able to complete studying even one textbook during a term of my postgrad.
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When I think about this, I realized that any amount of hardwork wouldn't have helped. The system was against people like me, who loved acquiring knowledge and building expertise the old-fashioned way, with long hours of dedicated study. There was no connection between the time available and the knowledge one was expected to acquire. The time available was scarce and the knowledge one was expected to acquire was vast. It was an impossible task. The only way one could get through this was by adopting shortcuts. By not worrying about textbooks but by focusing on the notes taken during classes, by trying to get question papers of previous years' exams and analysing which kind of questions got repeated and studying the solutions for them alone, by cramming stuff in the eleventh hour into the wee hours of the morning of the exam. This is the kind of behaviour that the system rewarded. Everyone probably knew about it. The students definitely did. The smart ones didn't bother with dedicated studying, but used shortcuts to jump through the hoops and get through. The dumb ones like me spent long hours with a textbook and became great in one subject but struggled in five others. The teachers probably knew all this, but they probably turned a blind eye.
This is not the situation in just my college or my country. I studied in a small provincial college during my undergrad, but I did my postgrad in one of the finest colleges in the country. But the higher one's college is rated, the more crazier things get, the more shortcuts people follow. I know that now. In whichever country you live in, in whichever college you study in, even if you are at Harvard or Cambridge, if you want to complete one textbook, and you are ready to put in dedicated hours everyday, it takes four months. You can complete three textbooks in a year. This is what it is. There is no way around this, if you are a dedicated learner. If you want to do more, you have to follow shortcuts. But if you follow shortcuts, your learning won't be comprehensive or long lasting and you can't build expertise in the subject, though in the short-term you can improve your grades.
This finding was a huge eye-opener for me. But this also led to a question. Students who complete their college and go on to become scientists and engineers in the real-world, these students invent new things or they design new things. We see this happening everyday. Science and technology keeps progressing everyday with new findings, inventions and discoveries. If all these students used shortcuts to get through their courses (there was no other way), their knowledge is at best half-baked. It might be good enough to fake their way through, but not good enough to invent something new. How are these students inventing new stuff? I don't know the answer to this question. But the fact remains that it is happening. It is pure magic.
I asked myself why things are the way they are, and why courses are impossible to complete if studied properly in the old-fashioned way, and why they are designed the way they are, even though educators probably know that students wouldn't be able to complete them without following shortcuts. After all, educators were students once, and they would have employed the same shortcuts themselves. I can think of only one thing. As a field advances across time, new stuff gets included in a course. As this continues happening, the amount of material that has to be mastered in a course increases to an impossible level. When our parents were studying the same subject in college, they were studying far less. We are probably studying ten times more. And our kids will probably be studying ten times more than us. But the time that each person has at their disposal is the same. In the same time that our parents had, we are expected to master ten times more material. This is like stuffing more and more stuff in a suitcase and trying to close it. At some point the suitcase won't close. Or even if we manage to sit on it and close it, the lock might explode and the suitcase might open again. The only way an infinite number of things can be stuffed into that suitcase is if it is a magical thing and its inside is bigger than its outside. But in the real world, you can't squeeze more learning into the same amount of time. The only way out is by using shortcuts.
I realize that all the analysis above is based on the study of a mathematical textbook. Things might be different in a softer physical science like chemistry or biology, or in a science like psychology, or in a social science like economics. Things might be very different in an arts field like literature or languages or history or philosophy. I need to repeat this studying experiment, when I have time, in these other areas and see what happens. Though I have a feeling that the basic argument will hold true in these fields too – that the time at our disposal is far less than we need and the knowledge that we are expected to master is so vast, that the only way we can expect to do this is by adopting shortcuts.
I don't know what the solution to this problem is and how the system can be modified to discourage shortcuts and encourage old-fashioned comprehensive learning. I personally don't see any way out. It is sad and depressing to contemplate on.
As far as my particular learning project was concerned, I took a small break after completing half of the book. I thought I'll get back to it soon, but distractions and real life got in the way and I've not been able to continue. I'm hoping that I'll be able to resume from where I left off, later this year, and get many hours of pleasurable learning from it.
#Math #Mathematics #StudyingMath #CynthiaYoung #WomenInMathematics #WomenInScience
