A few thoughts about teaching the exact sciences at an advanced stage (perhaps even earlier? debatable). I find that in some disciplines (especially maths!) linear and step-by-step teaching is given extreme importance. Writing things down very very logically is considered the correct way when teaching in the class rooms. I think this is actually harmful. Students might get used to such presentations and then when it comes to thinking about a problem, they often try to analyse and think in the same logical and linear fashion. This is highly undesirable! Problems are not solved by thinking linearly! Analysis yes, but not problem solving. One often has to relate to several different concepts and draw intuition from other problems that people have solved. This means that you might be forced to jump from one idea to another, try out bold conjectures based on some intuition and so on. Many of these paths will fail but they will add to your intuition about the problem. And it is this process that one must try to teach students. This is not easy. It means that when the teacher lectures, he/she will not complete arguments, often will not provide all the details in one go and most importantly, may not give the complete picture immediately. The teacher will state the goal clearly and then wander around a bit, experiment with ideas and try to convey why some approach works the way it works. This will entail quite some work on the part of the student. It will also leave you confused till you see the whole picture. A good teacher will do this so that you vaguely see the direction but not so much that you will sit back and watch the teacher do all the thinking instead of you.
The benefits of this process are that albeit hard, it does teach you to think on your own with suitable guidance from the teacher. It also conveys the message that solving your own problems is not going to be anything like just learning a subject from a book.
This also is precisely why we need teachers. Books are good to learn subjects linearly. But they rarely convey the intuition behind the subject. A teacher can show us how to read between the lines in a book.
To conclude, linear presentations are over-rated.
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ReplyDeleteHarsh: I read your part and;
ReplyDeleteJyothi: I read your part as well, and now I have few things to say, though niether supporting to anyone nor offending anyone. The reason being that it is very understandable, and normal that we all like to see things the way WE want them to be or rather should be. Then be it Harsh, or the X person in the article which Jyothi is mentioning.
Harsh I think you are talking here in context of mathematics as you said at the start of the paragraph and then at the end of that paragraph itself you are trying to make it more generalized. Lets say, take an example of history, do you think the chronological sequences can be taught in non-linear way? Making students think linearly does not necessarily means that you are trying to give them a food for thought in a specific way, but you can also see it other way round, where you train them to approach any problem in a systematic way. Example: at an accident site you can not act in a non-linear way, they should be a linearity even when you want to take any step. I think we should not restrict ourselves the definition of LINEARITY to class rooms, as we all know that these things are carried way long in our life with us.
"This is highly undesirable! Problems are not solved by thinking linearly!" do you think this statement is not correct. Also dont you think that jumping from one option/idea to another, but maybe a most linear/sequential way can help, or do you think that non-linearity is helpfull here. Again same exapmle of accident, logical/linear way of thinking should tell one to call ambulance at first sight and not police van.
When we say about not defining the complete goal and just giving a transluscent idea about any topic and let the students find it for themselves what it could be, sounds a very good idea. But I think there are patterns in the class rooms where it is expected from the student to react in this way, say during exam in the form of Q&A, give reasons, etc. At the stage when you should be made understand things, if you are confdused can you imagine the situation and the base you will be creating for their future: then i am pretty sure the person will definitely call police van instead of ambulance.
Books are a good medium to teach us things linearly with the intuition that this is what is expected when you read this book. And this can be shown and/or taught by a teacher serving as an intermediate medium.
Finally at the end of the day it is important to know that schooling is considered as one of the crucial part as a base for your success in your life. And if at such stage you are taught things not in a linear/sequential/chronological way , but more like randomly, you can imagine what might be in your hand at the end of day?
I agree with Bhushan's view point. Schooling must not be nonlinear.
ReplyDeleteAlso, in most practical situations it is important that we think linearly and in a direct fashion. I would even go further and say that right up to Bachelor's level 80% of the teaching should be linear. It is only at masters or PhD level, where solutions to problems are not as clear as calling an ambulance, that one must shift to more and more nonlinear and original thinking.
Also, in the creative arts, like music, films, painting etc. linear thinking might not be the best way. I am inclined to think that the creative arts require even more nonlinear thinking than the sciences or mathematics. Poetry is by no means linear. Nor are ideas occurring to a writer writing a book or a play.
Another point is that it also depends on the person concerned. Some people have a flare for analysing problems carefully. This often asks for very systematic and linear thinking. On the other hand, some people are more creative than analytical. For these students, nonlinear thinking exercises will be pleasurable.
I think that most people are not born with linear systematic minds. Schooling trains us to organize our thoughts and think linearly. The tradeoff is that one has to spend a little less time on creative thinking. This tradeoff is not only good but even necessary. A very creative mind which cannot organize thoughts will find it difficult to get its ideas into practice. But one must not go to the extreme end of thinking only linearly; this will surely dampen creativity. Each person has to find a good tradeoff for him/her-self and find an appropriate vocation. Unfortunately, this is not always possible and one sometimes ends up in the situation that destiny decides the tradeoff for you. More often than not, situations in life call for linear thinking over creative thinking. However, contributions to society that are remembered over decades are almost always those that required radical thinking.
Very nice argument and good piece of writing.
ReplyDeleteWell I think so we both are thinking on the line that, linearity is an essential part during your initial days of progressive thinking. Am I right? I do agree with you about the Master's and PhD, but as far as I know most of the atleast Master's courses here are in the way you are expecting. Atleast that was my experience, so I will restrict it to myself.
I would like to comment though on the ambulance example, where your views are "solutions for problems are not as easy as calling ambulance, well said Harsh". Now lets take an example from mathematics itself. Suppose you are appraoching a very tedious problem, one of its own kind. Now according to your way, you do here trial and error method in a non-linear way. So you want to anyhow tackle that problem with non-linearity, do you think if you will get it done? Don't you have to be more practical and realistic here to get to the solution. There might be another case where you might end up in going around one thing again and again whereas the solution will be somewhere else. But maybe, I am saying maybe a linear/systematic way of thinking could make you solve this problem at ease. And then maybe you will get another challenging problem to solve. So instead of just going for non-linear way of thinking and coming to solution of solving one problem on longer time scale. It would be even nicer to solve more things and give more input in your field of interest to the community of your field.
I also think even if you take examples of all the great people who did some out of the box things, they were not non-linear thinker. Yes, it will be right to say that they thought out of the box than what the present other big people at that time were thinking, but that never means that these people thought non-linearly. A person from your own field of mathematics John Nash, do you really think he thought everything about Game theory in a non-lienar way, I doubt though I am not sure as well.
Ineteresting. On reading all this I asked my self "what do I mean by nonlinearity?". I must make that clearer. What I mean by nonlinear thinking is really thinking that relies on intuition drawn from various experiences. For instance, consider a problem in mathematics. One usually starts out with a hunch. On trying it out, it is common to find that it is not entirely true. Then one spends time finding out what exactly is going wrong with the hunch. Based on this intuition one tries to come up with examples which show why the hunch does not work. Fiddling around with examples yields quite some intuition about the problem at hand. Then based on the experience with the examples/counter-examples, one comes up with a modified hunch and checks if this hunch can be proved. This goes on for a while before one comes up with the perfect hunch which on proving becomes a theorem. It is this process that I am calling non-linear.
ReplyDeleteThere is no 'systematic' way of coming up with hunches. The whole process I've described above is an attempt to try and be systematic. But most good problems do not yield results purely logical and linear thinking. In particular, coming up with counter examples is something that is guided by logic but is impossible by using purely logical/linear thinking.
Now suppose a teacher is teaching this solved problem as a theorem in the class. Then s/he will usually just state the theorem and give a proof of it; completely losing all the intuition that comes from trying out different approaches (even ones that fail). It is this teaching methodology that I am referring to as 'linear'. Clearly, if the teacher wants to train students for research, then this 'linear' methodology is not the best.
I don't know about John Nash, but Sriram Abhyankar (who by the way knew John Nash personally), a big shot in algebraic geometry, once gave a lecture at IIT Bombay and I remeber him saying that his papers are linear and systematic but that his thinking was far from organized and linear. In fact, he often takes up to six months to actually write down the solution to the problem in a linear and systematic manner.
Linear teaching does not encourage 'out of the box' thinking. Just as it is extremely important to have linear teaching initially, it is equally important to encounter non-linear teaching at some point in the training towards research.
Every system has its limitations, flaws and even more importantly, merits. We should also not forget the context of their use.
ReplyDeleteI agree with both Bhushan and Harsh on the statement that linear teaching does not encourage 'out of the box' thinking. I agree with you that this methodology is disadvantageous when it come to training research personnel. [Quoting Harsh, "Now suppose a teacher is teaching this solved problem as a theorem in the class. Then s/he will usually just state the theorem and give a proof of it; completely losing all the intuition that comes from trying out different approaches (even ones that fail). It is this teaching methodology that I am referring to as 'linear'."]
But, like I said this is its limitation. Most importantly, we should not forget the context in which linear teaching is done, namely schools and maybe to a certain degree in bachelors .The initial days of schooling is spent gathering facts. Solving small problems. IMO, up until Bachelors, the aim is to create well rounded individuals armed with basic knowledge of the world around them.
As one specializes further, a new way of teaching is required, which encourages you to solve problems that have multiple solutions and multiple paths (experimental design and interdisciplinary approaches) to these answers. Importantly, nobody knows what the answers will be. You will be required to draw in from you pool of knowledge, collect information, ask questions, design your own experiments and become creative.
The linear teaching serves a different purpose altogether, which may be summed up as a short-cut transfer of basic knowledge that humanity has gained over the period of time it has existed.Literally and figuratively, reinventing the wheel at schools seems impractical to me. It does not mean that I am in favour of a schooling period that is merely fact cramming sessions. There is a balance and its for educational experts to find this.
Nice topic Harsh. I agree with Harsh that studennt should be ready to understand and precisely learn to understand.
ReplyDeleteI believe that two ways are to make you understand (1) you understand complex (2) I teach simple.
My take on this debate is it depend on our brain and capability. Human brain understand simple things quickly and complex things it may take time. Its not that brain functioning is slow, no its fast only but i think some time interpretation takes time and you often experience that you understand after 2-3 days of some lecture. Why? Because your brain is processing and trying to make logical interpretation and this also depend on other factors like interest, how much you understand etc etc.
Now coming to school, are kids really interested to understand? mostly not, but teacher has to make them understand so simplistic way is better otherwise only matured student will understand and not the normal one.
My conclusion is any thing taken by brain depend on interest and some other factors, therefore try to make yo story interesting and to do so one need logic and simplest way possible.