Department of Education inTechnology and Science, Technion

Workshop onAdvanced Mathematical Thinking

October 16, 2007

15:00-16:00 Panel: Questions for future research in advanced mathematical thinking

Panelists: Prof. Avi Berman, TechnionProf. Ted Eisenberg, Ben Gurion

Univ.Prof. Orit Hazzan, TechnionDr. Boris Koichu, Technion

Prof. David Tall, University of Warwick

Moderator: Prof. Nitsa Movshovitz-Hadar

2

Teaching Mathematics – The Gulf between Semantics (meaning) and Syntax (form), Presented at ICTM3, Istanbul June 2006The problem: Why do students, irrespective of their level of talent or background, have trouble following an undergraduate mathematics lecture? Why are mathematical seminars frequently difficult to comprehend, even for experienced mathematicians?The goal: To begin to identify elementary but useful general principles which affect comprehension in the flow of information between teachers and students of mathematics.

David Easdown, School of Mathematics and Statistics, U. of Sydney, NSW, Australia

Oct. 16, 2007

3

On the surface lies the syntax, namely the basic shapes and forms of the language. Far below is the seabed which represents the semantics or meaning. This gives rise to the notion of depth of understanding: ‘superficial’ versus ‘deep’. The teacher needs strategies to enable the student to come down from the surface to reside comfortably along the seabed of understanding

David Easdown: One may imagine comprehension as an ‘ocean’

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

4

1. The Principle of Reflected BlindnessIf one is in a room with a light on, at night, with no curtains on the windows, it is impossible to see what is outside, and all one can see is the reflection of oneself and the contents of the room. To see outside, one must first turn the light off. As teachers of mathematics, unless we can ‘switch off’ the bright reflections caused by our own (profound!) knowledge, it may be difficult to perceive, or even empathize, with our students and their misunderstandings.

David Easdown:

Three Principles

5

In the lecturer’s mind the semantics is settled and s/he can say a complicated sentence with complete comprehension. Is it always the case that the listening student has the same comprehension???

: כרצוננו קטן חיובי אפסילון לכל א משנה זיכרונותשלכל כך טבעי אן -קיים - - גדול- מאן גדול קטן אן

... ש מתקייםלעומת

שלכל N > 0קיים ε > 0לכל , ...n>Nכך

an example

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

6

Choosing the quality and volume of detail are essential in effective mathematics teaching.

Certain details take the recipient deeper towards semantics. Other details can be a distraction towards syntactic oblivion.

As teachers we must constantly remind ourselves that information can be overwhelming, and our fluency in the subject is the result of many years of experience which our students simply don't have.

David Easdown:2. The Principle of Trivial Complexity

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

7

Halmos claimed that every good theory has accessible examples. “The heart of mathematics consists of concrete examples and concrete problems.” (P.R.Halmos, How to Write Mathematics, AMS, 1973)

The art of teaching mathematics is selecting appropriate examples.

David Easdown 3. The Halmos Principle

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

8

One of the most difficult learning thresholds for students of mathematics is the concept of proof. The difficulty manifests itself in several ways: (1) appreciating why proofs are important;(2) the tension between verification and understanding; (3) proof construction.

David Easdown: The role of proof in mathematics teaching and the Plateau Principle (Proceedings of the Uniserve Science Symposium 2007)

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

9

Easdown presents as a case study two specimen proofs of the Cayley-Hamilton Theorem, a deep theorem in linear algebra:

The first one is fallacious but convincing. This introduces briefly the idea of syntactic reasoning (symbol manipulation).

The second one is correct, but the routine verification does not in itself develop the student’s conceptual understanding. (recall G. Hanna’s ideas)

David Easdown: Proving the Cayley-Hamilton Theorem

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

10

In cases like that it is the author’s view that it is better to apply The Plateau Principle and engage the students in higher level consequences and applications rather than digging into the proof details.The Plateau Principle, is an unspoken credo for successful research in mathematics, which says simply: ‘look for and be prepared to use a variety of plateaus as starting points for a mathematical investigation.’

David Easdown: The Plateau Principle

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles

11

Please tell us how you see the agenda for future research in advanced mathematical thinking. I hope you can relate your vision to Easdown’s principles: 1. The Principle of Reflected Blindness2. The Principle of Trivial Complexity3. The Halmos Principle (examples)4. The Plateau PrincipleThen, we’ll open the floor for the audience reaction

To the panelists

Oct. 16, 2007N. Moivshovitz-Hadar Easdown's Principles