How does your teacher make you conclude inductions?
The hsc exam committee don't like typical textbook conclusions like since it is true for n=1 it is true for n=2 etc, therefore it is true for all positive integers:
The following is from the HSC exam committee:
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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=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3. The statement is true for n=3 and hence is true for n=4 and so on. Hence the statement is true for all integers n≥1 (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 positive integer n, it follows that it is true for all positive 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≥1 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 hsc exam committee don't like typical textbook conclusions like since it is true for n=1 it is true for n=2 etc, therefore it is true for all positive integers:
The following is from the HSC exam committee:
---------------------------
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=1 and hence is true for n=2. The statement is true for n=2 and hence is true for n=3. The statement is true for n=3 and hence is true for n=4 and so on. Hence the statement is true for all integers n≥1 (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 positive integer n, it follows that it is true for all positive 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≥1 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.
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