Who is the ultimate mathematics teacher?

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.