A lot of teachers could only be described as amateurs and are not good mathematicians. Unfortunately, these amateurs have influence on the syllabus development process. The result is mediocrity, not good mathematics.
Several books have been written on the subject of dumbing down, my favourites being:
Donnelly, K. (2004), Why our schools are failing.
Furedi, F. (2004), Where have all the intellectuals gone? - Confronting 21st century philistinism.
Gare, S. (2006), The triumph of the airheads and the retreat from commonsense.
Smith, N. (ed.) (2004), Education and the ideal.
There will be a new one out on Feb 1, 2007, called
Donnelly, K. (2007), Dumbing down: outcomes-based and politically correct - the impact of the culture wars on our schools
I got the following message from a professional mathematician on the hsc exam committee regarding induction which shows how stupid and unmathematical some amateur teachers are when they insist on a mantra:
HSC exam committee said:
I would like to comment on the induction part of the question.
It has come to my attention that many teachers are training their students to write some form of the following mantra at the end of induction problems.
The statement is true for n=0 and hence is true for n=1. The statement is true for n=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3 and so on. Hence the statement is true for all integers n≥0 (by induction).
In many cases the words 'by induction' are omitted.
It needs to be pointed out that
(a) No marks are awarded for this mantra in the marking guidelines for the HSC.
(b) Much time is wasted writing it
(c) Most importantly, the above mantra, especially if the word induction is left out, is at best misleading.
There is a logical (and subtle) difficulty in trying to argue that because the statement is true for any (finite) integer n, it follows that it is true for all non-negative integers n. The axiom of induction is needed to fix this difficulty.
It would be better both mathematically, and for the students themselves, if they ended induction proofs with the simple statement
Hence the statement is true for all n≥0 by induction.
I might add that students who persist in writing this mantra actually LOSE marks in our discrete Mathematics courses at University, so teachers are not doing their students any service, either in the short term (HSC marks) or in the long term. I (and others) have been complaining about this for a long time but without success.
The current syllabus does
not insist on a mantra, and if induction remains in the syllabus I only hope the amateurs won't put one in it.
If the amateurs put a mantra in the syllabus the better teachers would instruct their students to burn it.