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6. Let them learn to guess: guessing can be a pathway to proof (in the case of mathematics) and discovery (in the case of sciences). The key is to make more educated guesses. It helps if an easier approach can be taken. In algorithm design, we are also used to first trying out easier methods and then resort to more subtle approaches.

7. Let them learn proving: proving is indeed the central aim in mathematics. However the logical thinking involved in proofs can be a useful lesson in many fields.

8. Look for useful features of the problem: while tackling any mathematical problem, we have always been taught to identify any features of the problem which may show the way to the solution. It is almost like a detective utilizing clues to understand what happened.

9. Do not give away the whole secret at once: a teacher who kills the thinking process of the student is hardly doing a good job. It is a challenge to keeps things interactive in a class every time. However it is a no-brainer that spoon-feeding does not help.

10.Suggest but do not force: keeping a student interested is better than putting too much pressure on him or her. The power of suggestion is also relevant when a student makes a mistake. It is helpful to nudge the student in the right direction rather than drive him there. This is also better for the student’s self-esteem.

The advanced reader who skips parts that appear too elementary may miss more than the reader who skips parts that appear too complex – G. Polya

I was going through some books when I re-discovered a classic work by George Polya. Polya, who is better known for his best selling work entitled ‘How to solve it‘, wrote extensively on methods of learning, teaching and problem solving. The book which I re-discovered is more in depth and is called ‘Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving’. In one of the essays in the book, Polya proposes ten perennial rules for teaching. I’ll summarise the first five with my personal thoughts on them:

1. Be interested in your subject: This is so true! If you are bored of the subject, you are definitely not doing justice to your students or your career. Admittedly, sometimes, there are pressures on researchers and teaching the same the course repeatedly might not be as exciting as it once was. However, it is a given that showing your boredom will automatically put students asleep.
2. Know your subject: More than any other education method, it is the in-depth understanding of the subject that is most important before entering the lecture room. Polya points out that interest should come before knowledge because true knowledge only comes with genuine interest. This makes sense.
3. Know the ways of learning: A basic grounding of learning methods and psychology is considered essential for any kind of teaching or mentoring. Moreover, Polya makes an excellent observation that self-discovery of a concept is the best way to learn and then teach it. It is easy to see that working out a proof from scratch makes one aware of the critical arguments instead of just reading a proof.
4. Read faces and empathise: If one is observant enough, it is easy to notice which students are following you and which ones are nodding off! If the whole class is nodding off like that, it might be a good idea to explain things again. Being able to put oneself in the place of the students is critical for communication. As a teaching assistant during my undergraduate years, it was easy for me to explain matters to students because I was in their place only a couple of years before.
5. Give not only info but ‘know-how’: With easy access to information online, good guides and textbooks, the teacher’s responsibility is increasingly toward directing the student than giving information. As a seminar tutor, I try to go over general methods to tackle problems instead of going over each case study or problem.

I’ll be back with the final five golden rules.

(The photograph is taken from http://download.srv.cs.cmu.edu/~pausch/)

Just when one starts to take degrees and careers too seriously one realizes that there are more important things: life itself, for instance! Lance Fortnow has an entry about Randy Pausch who passed away due to pancreatic cancer on 25th July 2008. Randy was a distinguished computer science professor at CMU with contributions in human-computer interaction. Randy had been commenting on his last days on this webpage.

His last lecture video puts many things in life in perspective. Randy gives us important lessons on courage, living to the fullest, pursuing one’s dreams and giving back to one’s family and community. Randy made an insightful comment that ‘We cannot change the cards we are dealt, just how we play the hand.’ His other quotations are:

The brick walls are there for a reason – they are not there to keep us out, they let us prove how badly we want things. Brick walls are there to stop people who didn’t want it badly enough; they are there to stop the other people.

Wait long enough and people will surprise and impress you. When you are pissed off at somebody and are angry at them, you just haven’t given them enough time. Just give them a little more time and they will come to impress you.

It’s not how you achieve your dreams; it’s how you live your life. If you live your life the right way, the karma will take care of itself and your dreams will come to you.

Randy’s last lecture was attended by Jeffrey Zaslow, a columnist for The Wall Street Journal who wrote a column about the lecture. Zaslow was so much inspired by the lecture that he has written a book about it and how Randy showed dignity and courage in his last months. I definitely plan to buy the book.