A philosophy talk about teaching

1. Authority and Sensibility:
Lakatos and Pólya on teaching
Brendan Larvor
University of Hertfordshire
11 October 2007

2. 2
Mathematics presented in the Euclidean
way appears as a systematic, deductive
science; but mathematics in the making
appears as an experimental, inductive
science.
Preface to How to solve it (1945)

3. 3
Heuristic Presentation
The content of the
theorem and the moves
made in the course of the
proof motivated by the
experimental/ inductive/
heuristic/ dialectical
process by which they
were arrived at.
Euclidian Presentation
The content of the
theorem and the moves
made in the course of the
proof seem to appear by
magic.
Deus ex machina (Pólya)
Authoritarian mysticism
(Lakatos)

4. 4
If the terms of the sequence a1, a2, a3… are
non-negative real numbers, not all equal to
0, then:
How did anyone guess this?
What is e doing there?
ae<)a...aaa( nn321
1
1/n
1


Theorem:

5. 5
    n
nn
n
n
n
cacacaca
aaa
1
1
332211
1
1
21
1
...
... 



 




1
2211
1
...
nn
cacaca nn
  



 

1 1
1
k kn
kk
nn
ca
 












1 1
11
k kn
kk
nn
ca
 
kk
k
a
k
k
k
k
11
1
1





ae< n
1


Define the numbers c1, c2, c3,…cn… by:  n
n ncccc 1...321 
GM ≤ AM
 k
kk 1 Approaches e
monotonically from
below
Proof: