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

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Is it a family member? Parent Grandparent Sibling

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Who really taught you to count? Family member Teacher

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Who taught you geometry? Measurement Data Analysis Probability Algebra

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

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

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

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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.

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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.

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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.

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Who was the better teacher?

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

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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

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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

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Who was the better teacher?

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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.

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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.

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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

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Who was the better teacher?

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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.”

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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.

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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.”

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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.

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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.

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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.

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

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Experience Knowledgeable to start Interested in content

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

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

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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.

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

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

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