VCAA’s new version of Specialist Mathematics contains “Proof” as a topic (which says everything one needs to know about VCE mathematics). A few commenters have alerted us to the fact that VCAA have now provided two videos: on induction (transcript and slides); and on proof by contradiction (transcript and slides). There are issues. (31/01/24 Link to the videos updated.)
This WitCH is continued here.
UPDATE: Induction (14/02/23) (Contradiction below.)
OK, this VCAA stuff is everywhere, as are our posts, but we have to update as best we can. So here goes: the induction webinar.
To begin, it is difficult to understand the purpose of these webinars. What can reasonably be done in a 10-ish minute slide-video that cannot be done better in a text? What makes VCAA thinks their proof webinars are better than the existing ten thousand videos? Of course, they are not.
Any New Topic video must be tight. Every word matters, especially when introducing proof concepts. But VCAA’s induction webinar is, at best, sloppy and pointless. And, as commenters have pointed, below and here, the webinar quickly descends into farce. And error. And there’s the sound. Why would VCAA think it’s acceptable to post something with such atrocious sound quality?
On to the content. The induction webinar begins with the familiar dominoes analogy. Fine, although the lack of dominoes is a trifle lazy. The slide then moves onto the natural numbers:
Prove for a starting value (often n = 1)
Prove what for a starting value? Then, the voiceover moves onto the natural numbers:
Now in the mathematical context, what we are doing is we are proving something is true for a particular number, a starting point, which we often label n-zero. We assume that the statement is true for n equals k and then we must use that assumption that the n equals k statement is true to prove that the statement for k plus one is then true.
There is no clear declaration that we are seeking to prove is a statement P(n) for a range of natural numbers n. The notation P(n) or similar is never employed in the webinar. The terms “base case” and “induction hypothesis” and “inductive step” are never employed. The language and presentation throughout is vague and sloppy, when absolute precision is required. The webinar is useless, and thus worse than useless.
The first example is a finite geometric series, the first of the Sample Exam 1 questions. In proving the unlabelled P(k+1),
… I like to write out the statement that I’m trying to prove
Fine. But for Christ’s sake label the statement as the statement to be proved. Don’t, ever, do this:
Remember, this is VCAA’s guide to the new topic of induction, the indication of what teachers and students should be thinking of when writing induction proofs. It is crazy.
It gets way crazier. The second question worked through in the webinar, Question 2 of the sample questions, is simply meaningless: see the discussion here, and in particular the excellent summary in edderiofer’s comment. The webinar, of course, simply proves (badly) what they imagine was asked.
The third and final question worked through, Question 3 of the sample questions, is a simple divisibility question, proving by induction that 9n – 5n is divisible by 4. The question was criticised below, since the binomial formula and/or modular arithmetic are more natural approaches. This criticism ignores the fact, however, that Specialist Mathematics is an algebra-free wasteland. The webinar presentation of the proof is bad, in the manner previously noted.
UPDATE: Contradiction (14/02/23)
This one makes the induction webinar look like a work of genius. It begins,
When doing proof by contradiction, the first thing we do is we assume that the given statement is false, so we usually then write a statement which contradicts that statement, the one which has to be proven.
The next step is to work through algebraically and prove that the contradictory statement or the assumption is false. When we prove that the contradictory statement is false, it follows that the original statement must be true.
This is appallingly worded, and it is false. Let’s suppose we want to prove the statement,
(O) There are no prime numbers between 10 and 20.
Now we want a statement “which contradicts that statement”. OK, how about
(M) Marty adores VCAA.
OK, we’re being cute, and we’ll stop being cute in a moment. But there is a point: words matter, especially when introducing a notion of proof. Two statements are contradictory if they cannot be simultaneously true. In particular if a statement is false then it contradicts every possible statement.
Let’s be less cute. Consider the statement,
(C) 12 is a prime number.
Does (C) contradict (O)? Clearly so, even without having a clue what “prime number” means. Can we prove (C) is false? Of course. So, “it follows that the original statement must be true”? Of course not.
What is said and written in the webinar is utter nonsense. As another example, where the statement to be proven is true, and interesting, but the contradictory statement is of no assistance, consider,
(O)’ There are infinitely many prime numbers.
(C)’ 7 is the largest prime number.
When doing proof by contradiction, we begin, precisely, with the negation of the statement to be proved:
(N) There is a prime number between 10 and 20.
(N)’ There are only finitely many prime numbers.
Great way to start the webinar. On to the first worked problem. Which sucks.
The first problem is to prove by contradiction that if n is odd then n3 + 1 is odd, and it is absurd. The problem has been hammered below in the comments, and in this WitCH. Predictably, the webinar makes it all worse:
… as with any proof by contradiction the first thing that we do is we assume that the opposite is true
The word is negation. Learn it. They then clutter and confuse an already bad proof by introducing an integer k to write n3 + 1 = 2k + 1. It serves no purpose whatsoever.
The second worked problem is better, although still not great, and it raises questions, as does the first problem, regarding what students can and cannot assume. See the discussion on this WitCH.
The third worked problem is so bad, and so wrong, it’s hilarious. The problem is to prove loge5 is irrational. To set up the proof by contradiction, they assume the “opposite”, that the thing is rational. They write loge5 = a/b, noting
where a can be any integer and b is a natural number, and the fraction is in simplest form.
Why “integer” and “natural number”? This is
to avoid a situation where there’s any possibility of getting a zero on the denominator …
Yeah, ok, but you’re still gonna allow the possibility that a is negative? And why declare the fraction to be in simplest form? We’re never told. (Hint: there’s no point whatsoever.)
This is all warm-up. Doing the standard log and powers stuff, they arrive at the equation,
ea = 5b.
And, now what?
It’s clear at this point that the right hand side must be divisible by five but the left hand side is not.
No, it is not clear. Nothing is clear about the left hand side. As it happens, proving anything about ea is pretty damn hard. If the left hand side is not clear, however, what is clear is that the authors of this nonsense have no proper sense of proof by contradiction, or of number systems.
This is a maddeningly bad, and wrong, presentation. And the sound really sucks.