1. Who is the ultimate mathematics teacher? Johnny W. Lott, President National Council of Teachers of Mathematics

2. Is it a family member?  Parent  Grandparent  Sibling

3. Who really taught you to count?  Family member  Teacher

4. Who taught you geometry?  Measurement  Data Analysis  Probability  Algebra

5. What do you look for in an excellent teacher?  Heather Bridges, a student at the University of Georgia, suggested the following on her website.

6. Elements of a Master Teacher  Experience  Instructional Techniques  Assessment Techniques  Teacher/Student Environment  Professional Development

7. A different approach: Cases  Who is the better teacher?  Farkas Wolfgang Bolyai  Karl Friedrich Gauss

8. Consider Janos Bolyai.  Wrote his father:  Out of nothing I have created a strange new universe.  His chief interest was in “the absolute science of space” by which he meant those theorems that were independent of the parallel postulate.

9. Farkas (Woflgang) Bolyai  Urged the proposed paper be published as an appendix to his own two-volume work on elementary mathematics; suggested this in 1823  Eventually published a 26-page paper as an appendix to Volume I of his work in 1832.  Note that the 26 page paper was NINE years late.

10. Carl Friedrich Gauss  F. Bolyai sent Janos’ work to Gauss who replied.  I can hardly praise Janos’ work because is is basically like what I developed (but never published) over the last 30-35 years.

11. Who was the better teacher?

12. Consider Sophie Germain.  Worked in number theory and won prizes for mathematical physics.

13. Sophie Germain’s Parents  Viewed “brain work” as a dangerous strain on the minds of young women.  Did everything possible to discourage her  Took away heat and light so she couldn’t work at night  Hid her clothes so that she couldn’t sit up and study at night

14. Joseph Louis Lagrange  When Sophie was not allowed to register at Ecole Polytechnic, she obtained lecture notes of Lagrange.  Under the pen name, M. LeBlanc, Germain submitted a paper on analysis to Lagange  Lagrange  Was impressed by her originality  Became a help to her and introduced her to scientists

15. Who was the better teacher?

16. Consider Sonya Kaovalevskaya.  When looking at sheets of calculus notes wallpapering her room, Sonya  Spent hours trying to decipher even a single phrase, and to discover the order in which the sheets ought to follow each other.

17. The Tutor  Used “modern” techniques of punishment--not corporal  Wrote misbehaviors on a sign and pinned the sign to Sonya’s back for her to wear for all to see.

18. Karl Weierstrass  Could not have Sonya in class but gave her a set of very difficult problems to do alone convinced that “she would not succeed, and gave the matter no more thought.”  Met her privately  Shared his lecture notes and his time with her

19. Who was the better teacher?

20. Lipman Bers (Latvian mathematician 1914-)  Asked his mathematics teacher about a one-to-one correspondence between two segments of different lengths.  “You know, I have taught mathematics for fifteen year and have never heard such a silly question.”

21. Paul Cohen (New York 1934-)  A lot of teachers are very threatened when they find a child is studying advanced things. And I was reluctant at that time to talk to other children because I felt they found my interest in math somewhat strange.

22. George B. Dantzig (Maryland 1914)  On the importance of homework  “My father taught me by giving me problems to solve. He gave me thousands of geometry problems while I was still in high school.”

23. At Berkeley, Dantzig copied down problems from the board assuming that they were homework from Jerzy Neyman.  Turning them a few days later, they were thrown on Neyman’s desk.  About six weeks later, Dantzig was awakened by Neyman beating on the door early in the morning, “I’ve just written an introduction to one of your papers. Read it so I can send it out right away for publication.”  The homework problems became the doctoral dissertation.

24. Andrew M. Gleason (California 1921-)  I often frighten students by the way I answer questions. I answer very quickly, in purely mathematical terms, when I should be more concerned with what the questioner’s problem is. It’s very hard to understand what another person is having a problem with when you’ve never had a problem with it yourself. And, of course, even if you did, you’ve forgotten how it was.

25. Mina Rees (Ohio 1902-)  Did you have any special teacher?  No. The teachers were women who had been educated at good colleges, knew what they had learned originally, and continued to teach.

26. Elements of a Master Teacher  Experience  Instructional Techniques  Assessment Techniques  Teacher/Student Environment  Professional Development

27. Experience  Knowledgeable to start  Interested in content

28. Instructional Techniques  LISTEN  Ask provocative questions  Assign provocative homework  Make sure students have lecture notes

29. Assessment Techniques  Ask provocative questions.  Focus on good points as well as bad points.  Use to learn how to improve teaching.

30. Professional Development  Know that it never ends.  Look for new and different methods in order to teach well.  Look for new and different content in order to teach well.

31. What do you want in an ultimate teacher?  Look in a mirror!  Tell me.  jlott@nctm.org

32. Teacher/Student Environment  Caring  Listening  Challenging  Caring  Don’t hide clothes and remove light and heat.