induction comment on wiki (1 Viewer)

[buchanan]

On the new boredofstudies wiki (the biki), the following comment appeared today, regarding what to do at the end of an induction:

4) You've done all the hard work. Now you must make a summary paragraph to explain it. These vary from person to person, but something to the effect of "Thus, if S(n) is true for S(k), it is also true for S(k+1), but since it is true for n=1 (or whatever the starting value is), then it must also be true for n=2, 3, 4, 5... etc. Thus S(n) is true for all natural numbers, by the principle of mathematical induction."

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But look what else I found, from tywebb:

here's an email i got 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=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."

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Opinions on this issue are divided, and so I published the following comment at the end of my solution to the induction question in last year's HSC at
http://www4.tpgi.com.au/nanahcub/2005hscsol.pdf :

Now I'll be like Darth Vader and be the chosen one to bring balance to "the force", I'll placate mathematicians by ending the proof here and say the statement is therefore true for all integers n≥0 by induction. QED.

and to placate teachers I'll put the dreaded mantra here but say also that although I have for funny reasons decided to include it in this set of solutions it isn't part of the above proof:

"It is true for n=0 ∴ it is true for n=1, ∴ it is true for n=2, etc., i.e., by induction it is true for all integers n≥0."
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I have some questions for you on the matter.

• What's your opinion on the matter?
• How did your teacher teach it?
• Do you agree with your teacher?
• Do you agree with the HSC Exam Committee?
• Do you agree with the biki?

No abuse please. Only sensible comments thanks.

 

[Lazarus]

I hope you decide to make any relevant contributions to the wiki.

 

[buchanan]

I certainly have. I have already put a link to my solutions to the 2005 HSC exam in the biki (which contains the Darth Vader comment)

However, I have chosen NOT to edit Slide Rule's contribution, because it now leads to open discussion (hopefully, non-abusive).

 

[Pianpupodoel]

May I ask where one can access this wiki?

 

[pLuvia]

I usually end induction questions something like this:

"Hence by mathematical induction the statement is true for blah blah blah"

Also my tutor and school teacher told me to write the "By mathematical induction" or "by induction"

 

[Slidey]

However, I have chosen NOT to edit Slide Rule's contribution, because it now leads to open discussion (hopefully, non-abusive).

If you have a contribution to make to my article, or any article on the wiki, please don't hesitate to do so.

However, please note that:

1) I said "These vary from person to person, but something to the effect of"
2) I also said "by the principle of mathematical induction".
3) The 'mantra' as you call it, was placed there to help students understand why mathematical induction works and what it is doing. Often students have trouble grasping the concept of mathematical induction, and see it almost as cheating.

I feel you have a point about the 'mantra' being a waste of time in exams, however, so would you like to write an article for the wiki about exam time management tips?

Thanks.

 

[buchanan]

The contentious issue is whether or not to include the mantra in the proof itself. It's fine as an aid to explain the proof. But the mantra isn't actually part of the proof itself. It is this explicit distinction which was the substance of my Darth Vader comment above.

I am pleased that your subsequent examples in the biki did not include the mantra as part of the proofs.

Some teachers teach that the mantra should be included in every proof, as part of the proof, and it is this which the HSC Exam Committee had objected to.

when the hsc exam committee learned about the darth vader comment, they sent the following message:

"While I agree that the principle of Induction, which is an axiom not a theorem, is subtle, and indeed teachers may want to and perhaps should explain the reasonableness of the axiom using the ideas in the `mantra' (personally I use the idea of rows of dominoes, which is equivalent to the mantra anyway), it is quite another matter to insist students write it all out, especially when it is at best misleading. The average (or even above average) student is not likely to understand the true mathematical significance of the axiom, but this is not a good reason to make them write something which is nonsense. Any student who wrote such things in our discrete Maths course would LOSE marks, rather than gain them. We expect our students to end an induction with: Hence the statement is true for all n≥0 by induction.

The mathematical argument I put forward for its removal will probably gain little traction among teachers who perhaps don't properly understand the subtlety or perhaps don't care much about it anyway, but I am hoping that the fact the `mantra' receives NO marks in the HSC and indeed wastes valuable student time might prove more persuasive. Thus teachers who insist on their students writing it are in fact penalising their students and doing them a disservice in the HSC exam.

Spread the word!"

 

[Riviet]

May I ask where one can access this wiki?

Click here

Any contributions will be greatly appreciated.

On the topic of discussion, I would like to ask, after reading what has been said, which way of writing it is safer now, if at all safe? Because the short way is not widely accepted by all teachers, and I am perfectly fine with writing it the shot way, it makes logical sense to me. Also, I have seen an even shorter, but still mathematically correct way of writing the final statement, which goes like this:

.'. true ∀ n∈ℤ where ∀ means "for all" or "for any" or "for each".

Now how's that?

 

[buchanan]

the short way is not widely accepted by all teachers

That's right. Some teachers have stated their opposition to the HSC Exam Committee's stance.

I am more sympatetic to their cause. An explanation of a proof is not the same as a proof. They were likewise pleased with my Darth Vader comment.

 

[Riviet]

Buchanan, are you part of the hsc committee yourself?

 

[pLuvia]

He sounds like a teacher himself

by the way, Riviet I can't read your maths codes it's just boxes to me

 

[Riviet]

He sounds like a teacher himself

by the way, Riviet I can't read your maths codes it's just boxes to me

I'm pretty sure buchanan teaches maths, and probably has close relations to the committee alhough probably not in the committee.

Hmm that's strange, I can see my symbols perfectly fine.

 

[A l]

Here is how I was taught how to write the conclusion on a simple mathematical induction equality proof where it is true for all positive integers:
"Since the statement is true for n = 1, it is also true for n = 1 + 1 = 2, n = 2 + 1 = 3 and so on. Hence, by the principle of mathematical induction: (............) = (............) for all positive integers" (or n ≥ 1)

On a book consisting of HSC answers, the conclusion written for a simple mathematical induction equality proof is:
"Since S(n) is true for n = 1, it is true for n = 2,
.: it is true for n = 3, and so on
.: S(n) is true for all positive integers"

I would also like to note that in the Notes From The Marking Centre of the 2004 Mathematics Extension 2 exam, that in one of the comments on mathematical induction states:
After checking an initial case and showing that the result for n = k +1 follows from the assumed result for n = k, it is sufficient to conclude the proof with something like‘So by mathematical induction, the statement is true for every n’. A lengthy paragraph justifying induction as a method is not necessary.

Unfortunately, many teachers do not approve of such short conclusions, so writing the short form may not lead to reduction in marks in the external HSC exam, but it may lead to deduction in marks in internal assessment tasks.

 

[buchanan]

Thanks for that A l.

Those mantras are precisely what the HSC Exam committee are objecting to.

They would prefer it to be concluded simply with

"Hence the statement is true for all positive integers n≥1, by induction".

Some teachers however, insist that it be done the way you showed, with the mantras included.

I should point out that the HSC Exam Committee don't just write your HSC exams, but they also write the marking guidelines for the HSC markers, including the comment published in the 2004 Notes From The Marking Centre.

So how do you think the conclusion should be done?

BTW, which book are you referring to exactly? I know for example MANSW and Coroneos are repeat offenders. Or is it something else?

Unfortunately, many teachers do not approve of such short conclusions, so writing the short form may not lead to reduction in marks in the external HSC exam, but it may lead to deduction in marks in internal assessment tasks.

Such reduction in marks in such internal assessments would constitute an indictment of the teachers' lack of mathematical knowledge, and not be in any way reflective of the students' mathematical ability - thereby rendering such internal assessments unreliable and invalid.

An explanation of a proof is not the same as a proof.

And this applies to other forms of proof.

One of the most spectacular proofs of the 20th century was that of Fermat's Last Theorem. Wiles proved it. But the job of explaining it has largely been left to others.

This is a slight misrepresentation of the historical events however and I will qualify that comment by saying that Wiles himself did however give a course at Princeton in the 1990's explaining the proof. But the crucial point is that that explanation was separate to the proof itself.

On a slightly humbler level, do you have to explain Pythagoras' Theorem every time you use it?

 

[A l]

At least the one I was taught wasn't too long; it's just adding an extra sentence to the main conclusion. I've seen other students who were taught to write like 5+ lines for it.

Such reduction in marks in such internal assessments would constitute an indictment of the teachers' lack of mathematical knowledge, and not be in any way reflective of the students' mathematical ability - thereby rendering such internal assessments unreliable and invalid

Unfortunately that is the case of the majority of schools. However, the scale of this "problem" is hardly significant in the HSC since mathematical induction questions usually appear only once or twice in every major assessment task and this concern is with a minor section of a proof. I'd imagine that very few would lose marks with that "mantra" in the HSC anyway because that particular section of the proof is often not in the marking guidelines so whether the student writes the shorthand or writes the "mantra", they should gain the same marks. The only main difference is that the students writing the "mantra" would lose some seconds of time.
Also, I think that HSC answers book with the particular conclusion I posted earlier was MANSW. I have yet to see the published answers from other authors. I wonder, particularly, what the Success One Mathematics Extension 1 book has published because that one is fairly popular.

 

[tywebb]

the cssa 2006 paper has an induction question and is at

http://www.angelfire.com/ab7/fourunit/cssa2006.pdf

 

[A l]

the mansw executive has been informed of the contents of this thread.

Well, I did say "I think" meaning that I'm not too sure it was MANSW. I was actually referring to a sheet (possibly out of date?) I received at school with answers to HSC mathematical induction questions and I thought it might have come from that book, so don't quote me on that.....

 

[Slidey]

On the topic of discussion, I would like to ask, after reading what has been said, which way of writing it is safer now, if at all safe? Because the short way is not widely accepted by all teachers, and I am perfectly fine with writing it the shot way, it makes logical sense to me. Also, I have seen an even shorter, but still mathematically correct way of writing the final statement, which goes like this:

.'. true ∀ n∈ℤ where ∀ means "for all" or "for any" or "for each".

Now how's that?

You forgot ", by mathematical induction.".

EDIT: Haha. Grammar.

 

[Riviet]

You forgot ", by the mathematical induction.".

Ah thanks for reminding me, totally forgot.

 

[buchanan]

I noticed last night Timbk2 changed the name of the BOS wiki to Biki.

That's a funny one!

lol.