Krantz has written a book on problem solving, a book on mathematical writing, and another on teaching mathematics. All of them are worth reading. I read "How to teach mathematics - A personal perspective" a long time ago. Here are the notes that I wrote down while reading the book. I am publishing them in this blog to give a flavour of the book, and to indicate that hunting this book down might be worth the effort.
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You cannot learn to play piano or to ski by watching someone else do it.
And the fact that having sat in a classroom for most of your life does not mean that you know how to teach.
Teaching is important.
The good news is that it requires no more effort, no more preparation, and no more time to be a good teacher than to be a bad teacher.
As with any endeavour that is worth doing well, teaching is one that will improve if it is subjected to periodic re-examination.
There are some things that we do not learn by osmosis. How to lecture and how to teach are among these.
After all, teaching is a rather personal activity.
Picasso's revolutionary techniques in painting were based on solid classical foundation. By analogy, I think that before you consider new teaching techniques you should acquaint yourself with the traditional ones.
You cannot be a good teacher if you do not respect yourself.
a] Dress appropriately for the occasion
b] Make an effort to communicate with your audience
c] Respect the point of view of the audience
Have your materials completely mastered before you enter the classroom.
To me, preparation is the core of teaching.
Treat the questions with respect.
Students will rise to a challenge, provided the teacher starts with small challenges ad works up to big ones.
There is something of value, of an intangible nature, about passing knowledge along to other people.
I have long felt that those who cannot teach are those who do not care about teaching.
Prepare, be organized, be fair, be receptive to questions, meet your office hours, and so forth.
Do not introduce distractions into the classroom atmosphere.
The attitude in your class should be that you and the students are working together to conquer the material.
Be comfortable with your class.
Be willing to try new things.
Over preparation will make you lose your spontaneity.
You cannot learn to play the piano by accident.
Teaching is a yoga. Your mantra is "am I getting through to them?"
One of Mozart's most effective tools in his compositions was to repeat a particularly beautiful passage. We can benefit from his example.
A teacher foes not just lecture and answer questions. A good teacher helps students to discover the ideas. There are a few things more stimulating and rewarding than a class in which the students are anticipating the ideas because of seeds that you have planted. The way that you construct your lecture and your course is one device for planting those seeds. The way that you answer questions is another.
"There is no substitute to knowing what you are talking about"
--Jonathan RT Hughes, Professor of Economic History
Mathematics can be understood deductively from certain axioms but it is learnt inductively.
Go from the simple to the complex.
The easiest thing in the world for a mathematician to do is to state theorems and prove them. It requires more effort tot each.
"Only wimps do the general case. Real teachers tackle examples"
--Beresford Parlett
"A small inaccuracy can save hours of explanation"
-- Saki
It is my opinion that the very best students tend to teach themselves.
No methodology is perfect.
Write neatly.
Keep as much material as possible visible at all times.
Homework assignment should neither be long nor short. It should touch on all important aspects and should drill the students on the material that you want them to learn and the material on which you will be testing them. At least part of the homework should be graded. Exams should be based only on material given as homework.
Make sure that the questions you ask elicit the basic information that you seek.
You must personally work the test out completely before you give it to your class.
An exam that needs you 10 to 15 minutes is ideal for a 50 minutes test.
Make the student read your solution before you agree to talk about grading the problem.
A good math student must be self-motivated.
It is a strange facet of the human condition that most of us don't know consciously what we think about most of the things most of the time.
An extreme example of a teaching style that is virtually orthogonal to what we Americans know is the one that has been attributed to the celebrated Hungarian analyst F. Riesz. He would come to class accompanied by an Assistant Professor and an Associate Professor. The Associate Professor would read Riesz's famous text aloud to the class. The Assistant Professor would write the words on the blackboard. Riesz would stand front and center with his hands clasped behind his back and nod sagely.
The famous mathematics teacher RL Moore is said to have once brought a Colt-45 to an unruly math class, set it conspicuously on the table, and then proceeded into his lecture in a room so quiet that one could have heard hair grow.
---------------------------------------------------------------------------------------
You cannot learn to play piano or to ski by watching someone else do it.
And the fact that having sat in a classroom for most of your life does not mean that you know how to teach.
Teaching is important.
The good news is that it requires no more effort, no more preparation, and no more time to be a good teacher than to be a bad teacher.
As with any endeavour that is worth doing well, teaching is one that will improve if it is subjected to periodic re-examination.
There are some things that we do not learn by osmosis. How to lecture and how to teach are among these.
After all, teaching is a rather personal activity.
Picasso's revolutionary techniques in painting were based on solid classical foundation. By analogy, I think that before you consider new teaching techniques you should acquaint yourself with the traditional ones.
You cannot be a good teacher if you do not respect yourself.
a] Dress appropriately for the occasion
b] Make an effort to communicate with your audience
c] Respect the point of view of the audience
Have your materials completely mastered before you enter the classroom.
To me, preparation is the core of teaching.
Treat the questions with respect.
Students will rise to a challenge, provided the teacher starts with small challenges ad works up to big ones.
There is something of value, of an intangible nature, about passing knowledge along to other people.
I have long felt that those who cannot teach are those who do not care about teaching.
Prepare, be organized, be fair, be receptive to questions, meet your office hours, and so forth.
Do not introduce distractions into the classroom atmosphere.
The attitude in your class should be that you and the students are working together to conquer the material.
Be comfortable with your class.
Be willing to try new things.
Over preparation will make you lose your spontaneity.
You cannot learn to play the piano by accident.
Teaching is a yoga. Your mantra is "am I getting through to them?"
One of Mozart's most effective tools in his compositions was to repeat a particularly beautiful passage. We can benefit from his example.
A teacher foes not just lecture and answer questions. A good teacher helps students to discover the ideas. There are a few things more stimulating and rewarding than a class in which the students are anticipating the ideas because of seeds that you have planted. The way that you construct your lecture and your course is one device for planting those seeds. The way that you answer questions is another.
"There is no substitute to knowing what you are talking about"
--Jonathan RT Hughes, Professor of Economic History
Mathematics can be understood deductively from certain axioms but it is learnt inductively.
Go from the simple to the complex.
The easiest thing in the world for a mathematician to do is to state theorems and prove them. It requires more effort tot each.
"Only wimps do the general case. Real teachers tackle examples"
--Beresford Parlett
"A small inaccuracy can save hours of explanation"
-- Saki
It is my opinion that the very best students tend to teach themselves.
No methodology is perfect.
Write neatly.
Keep as much material as possible visible at all times.
Homework assignment should neither be long nor short. It should touch on all important aspects and should drill the students on the material that you want them to learn and the material on which you will be testing them. At least part of the homework should be graded. Exams should be based only on material given as homework.
Make sure that the questions you ask elicit the basic information that you seek.
You must personally work the test out completely before you give it to your class.
An exam that needs you 10 to 15 minutes is ideal for a 50 minutes test.
Make the student read your solution before you agree to talk about grading the problem.
A good math student must be self-motivated.
It is a strange facet of the human condition that most of us don't know consciously what we think about most of the things most of the time.
An extreme example of a teaching style that is virtually orthogonal to what we Americans know is the one that has been attributed to the celebrated Hungarian analyst F. Riesz. He would come to class accompanied by an Assistant Professor and an Associate Professor. The Associate Professor would read Riesz's famous text aloud to the class. The Assistant Professor would write the words on the blackboard. Riesz would stand front and center with his hands clasped behind his back and nod sagely.
The famous mathematics teacher RL Moore is said to have once brought a Colt-45 to an unruly math class, set it conspicuously on the table, and then proceeded into his lecture in a room so quiet that one could have heard hair grow.
3 comments:
A very interesting post...
Here what I have to share about teaching...
I remeber one of my teacher opening up the first class (Field Theory) in this fashion (though not verbatim), "This course is like a hiking trip we all are going to climb this hill and I am going to be your guide. Some of you may go with me and some may go before me and some may lag behind me. But together let us make it in an interesting hiking experience. The course was indeed one of the memorable hiking experience for most of us in class.
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