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.
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.
86 Replies to “WitCH 89: Proof Negative”
I haven’t gone through the video, but those examples of proof by contradiction in the slides are truly awful (the induction examples less so, but I wish textbooks would include examples beyond sum identities and inequalities, e.g. covering a board with one cell removed with L-trominoes).
Would it be too much to teach modular arithmetic in specialist? No one in their right mind would prove using bare-bones induction (even the binomial theorem would be better).
Apparently one can’t take for granted that Year 12 students can calculate 20 x 20, so I suppose expecting students to calculate 1 – 1 could be a bit iffy.
Some may find this a useful resource: https://web.maths.unsw.edu.au/~jim/proofs.html
Or may not.
I was under-impressed by this book, to be honest. The only proof by contradiction example is the irrationality of and there are better proofs of this easily accessible.
The induction chapter… I’m not convinced it is suitably clear on the reason why such proofs work.
As a quick reference for students who have never seen a proof before, maybe it passes. I’ve seen better though.
The proof of contradiction that loge(5) is irrational seems irrational to me. How do you know that e^a is not divisible by 5, without knowing that loge(5) is irrational?
It’s kind of like the “proofs” of pythagoras theorem that use the identity sin^2(x) + cos^2(x) = 1.
Indeed. That’s the most obvious defect I saw in the proof by contradiction material.
What’s irrational is that the contradiction actually lies in the fact that is transcendental (*). I doubt that this fact is within the scope (or even spirit) of the new Study Design. I’ve attached what I think is missing from the VCAA (Example 3) proof, as well as what I think is a more appropriate example (whose proof is available if there’s sufficient curiosity).
In general, I think any formal proof that uses the word “obvious” (the word used in the audio 7:34) is doomed. It’s clear (as if more clarity is needed) why the VCE exams consistently contain errors.
I’ll have more to add about induction later tonight.
* No trivial thing to prove! (The standard proof also uses proof by contradiction).
Corrections to VCAA Implementation Material – Proof by Contradiction
You don’t need to know that e is transcendental. Stick to the point, which is the flaws in the VCAA materials.
OK. But I think a subpoint might/should be to try and offer constructive suggestions on how to fix at least some of these flaws (*).
To this end, Re: transcendental, I’ve amended my earlier attachment for Example 3. The amendment is based on what Franz posted (**). I’m happy for it to be ignored. I’m happy for it to be criticised. I’m happy for it to be useful to anyone.
I can hardly wait for the VCAA Quality Assured Proof by Contraposition implementation advice.
As an aside, it’s been mandatory (according to the Study Design) to teach proof in Specialist Unit 1 since 2016 (***). So it would have been punctilious of VCAA to have paid some attention to proof much earlier than now. The current resources smell like a mad scramble to start ticking boxes.
* This in turn might make the flaws clearer to some readers. Mainly for teachers, but if the VCAA is following, it might also help it to amend these resources.
** @Franz: I don’t understand why you “absolutely LOVE this “proof ” [of Example 3]”. Were you being sarcastic, or is your love relative to horrible curriculum stuff in Germany?
(***) And VCAA made it explicitly mandatory to be teaching it in Unit 2 at the end of 2021.
Corrections to VCAA Implementation Material – Proof by Contradiction
John, fixing the proof of the irrationality of log5 is off the point. It’s an interesting side-point. But, since clearly the exercise will never appear as part of Specialist Maths, it’s definitely not the main point.
In terms of the side point, the most obvious thing to note is that unless you know something about e, you cannot know anything about .
Sorry if it looks like I’m flogging a dead horse (I don’t mean to) – The main bad thing for me about Example 3 is the (inappropriate) use of words and phrases such as “obvious”, “It’s clear at this point that” etc.
If nothing else (*), it sets a (unintended) precedent that such words and phrases can be used in proofs that teachers and students write. How is anyone to know what VCAA thinks is “obvious” (**)? Where’s the line (if any) in the sand? I can imagine students encouraged to use such language in a proof by induction when trying to show that P(k) => P(k+1) …
This will bite students on the bum come exam time.
Re: “clearly the exercise will never appear as part of Specialist Maths” …. Bet your house on that?
* In fact, there’s plenty else.
** There are two things you should try to prove – the impossible and the obvious.
No, the main point about example 3 is that it should not exist.
I absolutely LOVE this “proof”. Over here, textbooks have replaced the old proof of the irrationality of by one-liners mentioning the parity of the powers of in the prime factorization. Of course prime factorization is not proved, so there’s no proof at all. What these guys were doing was extending this argument to real numbers: the powers of are not divisible by because, well, you know, prime factorization stuff. Actually they could similarly have reached the contradiction by saying that is not divisible by . Tempora mutantur . . . .
It’d be kinda nice if commenters here, and I’m talking about a number of people, exhibited a little discipline and common sense.
Come on you guys. I’m tired. This is a WitCH, which means you’re supposed to be doing the work, not me. You’re just making fun of the low hanging stupid fruit.
Are you ok with the introduction to induction? Why or why not?
Are you ok with the first induction proof? Why or why not?
Are you ok with the introduction to contradiction? Why or why not?
Are you ok with the first contradiction proof? Why or why not?
Re: Induction. The introduction and examples are poorly set out. In brief:
i) There’s no explicit use of the standard terminology ‘base case’ (step 1) or ‘inductive hypothesis’ (step 2) in the introduction.
ii) (I think) Every proof by induction should start with: Let P(n) be the conjecture ….
iii) The start of ‘Step 3’ is sloppy: “Prove true for n = k+1”. NO! You’re not proving anything true. You’re showing that if P(k) is true then P(k+1) follows.
iv) None of the examples have a conclusion.
To illustrate, I’ve attached how I’d set out an induction proof for Example 1. It can be read or ignored.
Corrections to VCAA Implementation Material – Proof by Induction
A clear concise summary suitable for secondary students IMO
without the added complication of logic/set theory given by Peano’s Axioms using the successor function
Wikipedia also does a reasonable job explaining the concepts so I wonder why the videos are actually needed?
Wikipedia isn’t remotely appropriate for what we’re discussing.
Maybe …but Wikipaedia or other crowd vetted sites could save the people who make the videos etc from reinventing the wheel when trying to explain induction and other mathematical proof concepts in simple terms IMO.
I’m a fan of Wikipedia, and there’s no question that VCAA’s reinvented wheel was unnecessary (and non-circular). But “simple terms” is not such an objective term.
1. The domino analogy is clunky and in my opinion unnecessary.
The best way to introduce induction in my opinion is as follows:
If P(0) is true and P(k) –> P(k+1), P(1) is true. Therefore P(2) is true, and P(3) is true, and P(4) is true… and eventually it shows that P(k) is true for all natural numbers. That explanation should be enough to show induction; unnecessarily including “analogies” to physical phenomena is just clouding the issue.
2. The first induction problem they give is unnecessarily complicated for a first introduction to induction. I don’t see what’s wrong with this simple problem as an introduction:
Prove that the sum of the natural numbers from 1 to n is n(n+1)/2.
This is a simple problem that can be nicely proven using induction.
The way they set out the proof for the problem they gave out is unclear. They should do the following: (a) explicitly define a proposition P; (b) prove that it’s true for n = 0, then (c) prove that P(k) –> P(k + 1), and finally (d) have a conclusion stating that because P(0) is true, and P(k) implies P(k+1), by induction, P(n) is true for all integers.
As an aside, a nice second problem a teacher could demonstrate would be proving DeMoivre’s theorem—it’s a nice theorem easily proven by induction; it showcases “backwards induction” for the negative integers, and it’s a theorem that’s not necessarily true for non-integers, showing a concrete example of how induction only applies to natural numbers. Plus, it’s used later in the course.
Their second teaching example is utter rubbish, as in order to properly answer part a), you need to prove part b), which seems to have gone over their head.
3. “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.”
That is what’s written in the transcript, which is false. 1=2 contradicts every statement, but that’s useless when proving by contradiction. To prove a statement P is true using proof by contradiction, you have to assume P is false, then show that a contradiction arises. You can’t just use any statement that contradicts P, it has to be the direct negation of P.
“When we prove that the contradictory statement is false, it follows that the original statement must be true.”
Same issue as above.
4. Their first example doesn’t seem to bad, but their proof seems unclear for a proof to be presented to a class.
Thanks, anon. Point 3 is what really really really really annoyed me. Like really annoyed me. With a capital Really.
I don’t like their first example at all. I once refereed a manuscript in which the authors used proofs by contradiction for everything in sight. In essence, the structure was this: Claim: . Assume that . Then : contradiction. Therefore . This is exactly what they are doing in problem 1: the prove that the L.S. is even and reach a contradiction by assuming that the R.S. is odd.
At least they got the negation correct! Franz, just to be clear:
Are you saying that proof by contradiction is an inappropriate method to use for example 1? I totally agree, of course. A ‘direct proof’ would be the most appropriate choice.
Or are you saying that contradiction could be used, but not in the way they have done? Even then, it’s still in essence a direct proof.
I think a ‘meta-issue’ (suggested by example 1) is that the VCAA resources avoid the question of which method of proof is the most appropriate for a given problem (*). There’s ‘direct proof’ (which students have been doing for years), ‘proof by cases’ (method of exhaustion), proof by (i) contradiction, (ii) induction, (iii) contraposition. Franz, your anecdote about the paper you refereed maybe touches on this …?
Example 1 does no-one any favours in this respect.
* It’s like teaching factorising. For a given problem, there’s some appropriate methods and inappropriate methods. The first difficulty a student has is deciding/knowing which method to use. Although I suppose the VCAA exams will tell the student which method of proof to use. Nevertheless …
To be fair, I think it’s difficult to come up with appropriate problems at the VCE level that are best solved by proofs by contradiction. Induction problems are a dime a dozen. I don’t agree with tom means when he says “It’s difficult with induction to find examples that are not trivial but are easy enough for this level.” I could probably make a script that creates random induction problems appropriate for the VCE level. Creating a problem that you prove using contradiction that can’t be solved directly is much harder.
It may be hard, but I’m not sure it’s *that* hard.
These videos (and transcripts) had better not be an indication of how this stuff will be examined…
We can be comforted by knowing that they went through a “Quality Assurance process”.
I’m not convinced it went through a Sound Quality Assurance process.
I think Sound Quality was conducted – by the roadies for Disaster Area ( https://www.nme.com/blogs/nme-blogs/celebrate-towel-day-disaster-area-loudest-band-galaxy-2078072#:~:text=%E2%80%9CThe%20Hitch%20Hiker's%20Guide%20to,of%20any%20kind%20at%20all. )
I’ve just looked at the slides for induction.
It’s difficult with induction to find examples that are not trivial but are easy enough for this level. So I was pleasantly surprised with the three examples used.
But looking at the proof in example 2, I read:
“Show that, for the inequality statement to be true, “.
In fact the statement in question is also true for infinitely many other values; the induction could start at any integer greater than 4. There is no stated requirement that is the least such. Let’s not nit-pick the style of explanation. The bigger problem is the usual suspect, a logical fallacy.
“Let’s not nit-pick the style of explanation.”
Why the hell not!? The VCAA are a bunch of whiny little assholes when it comes to student presentation. And here we have a new topic, on proof for God’s sake, and this presentation is supposed to be a guide for teachers.
Everybody has every right to nitpick the living hell out of this tripe.
I don’t think it’s too difficult to find induction examples that are “not trivial but are easy enough for this level”. I have previously posted one such example
The sort of which I’d expect to see mainly in Exam1.
The NSW and SA exams also have good induction questions that I expect will be plagiarised. (They would have made good examples for the implementation material).
I’m currently teaching Extension 2 Maths for the first time and proof by contradiction is in the course.
I’ll see if I have half a clue and critique the contradiction slides (also, I’m happy to be whacked so I can improve).
1) They are not clear with their terminology. They talk about “false statement”, “opposite statement” but don’t clearly define what it is. They need to talk about negating a statement. When I taught how to negate a statement, some students had the misconception to negated both parts of a statement (e.g “if n is odd, then n^2 is odd” negated to “if n is even, then n^2 is even”).
2) The examples don’t illuminate much:
The first contradiction example is more of a direct proof of the statement. I also don’t think the RHS and LHS stuff is necessary.
The second one is a stupid example to use. Pretty much trivial stuff that doesn’t show much.
Finally, the last example is problematic with the N not being clear whether it includes zero as a natural number. If zero is included then e^a does equal 5^b when a=b=0. If they don’t include zero in the natural numbers that it not obvious if e^a = 5^b and how to set up a contradiction (at least for an Extension 2 student). Probably should have proved log2(3) is irrational instead. At least you get 2^a = 3^b and LHS being even contradicts RHS being odd.
Hi Potii. 1) is a really important teaching point. Well done on detecting this misconception among your students.
Which is why the VCAA thoroughly screwing this up is so damn infuriating.
I assume you go through truth tables and propositional logic ( a very nice topic by the way) along/before
proofs? I would not be petty about the misconception you mentioned, I had students laughing when they heard that disjunction is true when both propositions are true (so their common language interpretation of or is xor).
If you test your maths/science school colleague how many would know how to negate an implication?
It is hard. I think.
Not long till witch 100
Just like a Centurion to notice that. I’ll try to be ready with something special.
It’s worth noting that VCE algorithmics has contradiction and induction in the study design, and there are questions on past exams about it.
The only math-related one I could find is 2018Q8, which other than being rather trivial (still, the mean mark was 1.7/4.0) is a quite nice problem. It’s interesting to compare the proof in the examination report to the “proofs” given in the slides for specialist mathematics.
Thank, Anon. Yes, the comparison between Examination Report and VCAA implementation material is fascinating. What I also found interesting is how the question is asked:
“Prove via induction that the sum of the first n odd numbers is the same as , for all n ≥ 1.”
“Prove via induction that for .”
The latter being much more likely in Specialist, I think. Maybe 1 of the 4 marks in the Algo question was for converting the ‘words’ into a formula …? Interesting that 33 percent of the state got zero (I’d argue that students doing Algo aren’t exactly dummies – the scaling is higher than for Methods). It seems that the question might have been unexpected in the context of the questions on previous (and subsequent) exams … Which would not be the case in Specialist.
I hope that something like Algo 2019 Question 10 will NOT get asked in Specialist … (The mean mark was equally low)
I remember doing Algo that year and being excited when I saw the question. The precedent for proofs in Algo is unusual, because they usually look at a proof of correctness for an algorithm, i.e proving that Djikstra’s algorithm produces the correct output for specific graphs, using loop invariants and so on. The 2019 question is a cute question, it’s one of the first few things you prove in a graph theory course. There are many, many lovely and challenging graph theoretic proofs, but I wonder if HS students would have a framework to approach some of these questions…
Well, what I wonder is how fair would such a question be on a Specialist Maths exam (*)? And how well would it be testing (I assume) the student’s understanding of mathematical induction (**)?
Can we assume from the VCAA implementation material that proof questions drawing upon knowledge of Graph Theory will not be asked (***)?
* Very unfair, in my opinion.
** Very unwell, in my opinion.
*** I sincerely hope so!
Thanks very much, Anon. For others, here are the links: 2018 exam (Q8) and report; 2019 exam (Q10) and report.
The wording of the questions and solutions has aspects that annoy me, but they are obviously much closer to the mark than the Specialist nonsense.
I have another question for VCAA (but anyone else is permitted to guess an answer):
Please help me to reconcile the following statements, which I believe to be true:
1. There are lots of resources available that demonstrate proof by induction and proof by contradiction really well, really clearly and at a VCE-level (actually, at an IB level, so… not VCE level, but I’m sure teachers would cope).
2. Any teacher with half a brain who does not remember the finer points of these proof techniques knows that a brief internet search can deliver access to many of the resources named in point (1) above.
3. Surely VCAA means the same as everyone else when they say “proof by induction” and “proof by contradiction”.
So… WHY THE VIDEOS?
The VCAA believes – correctly – that it has a responsibility to provide resources for new curriculum. So the question is not “Why the videos?” The question is “Why the APPALLING videos?”
Resources? Yes please. Some ideas include:
1. SPECIMEN EXAMS. Show us how you are going to actually assess this. Bonus points if more than 10% of these specimen exams are actually NEW MATERIAL and not a cut-and-paste from the last decade.
2. ANSWERS TO FAQs such as “what are students/teachers meant to do when the exam is seriously flawed”?
3. Copies of the actual exam papers online in a time frame similar to what NSW seems to manage without difficulty.
Stick to the topic. Don’t harp on other issues.
Some specific nit-picks:
Proof by induction – there are a few steps left out in the working which I do not consider obvious in the VCAA sense of the word and based on examiners’ reports commenting on “show that” questions, but perhaps I can let that go; it isn’t great but I can see where they are going with the argument.
Proof by contradiction – you do not (in my opinion) prove that the contradictory statement is false, you prove that the negation of the original statement leads to a logical contradiction. These (in my opinion) are VERY different.
General nitpicks – examples are poorly chosen. Proving something is odd/even does not need proof by contradiction (even induction is a better choice!). There are so many obvious candidates that could have been chosen (even log base 10 of 3 being irrational would be easier – since an even number cannot ever also be odd for finite, real numbers)
Induction video was better than contradiction but I still would not recommend to anyone.
I’d also like to see the guidelines distinguish between induction and strong induction
one third party attempt attached
I very much doubt VCE will cover strong induction.
Yep, the only induction VCE covers is weak.
(*Ahem* It had to be said).
An interesting read, thanks Steve.
I found the Graph Theory example particularly relevant.
(Still wondering if VCAA will ask induction proofs of any Graph Theory content – as far as I can tell the only topic that has been ruled out definitively is Boolean Algebra)
Well, VCAA is off the hook. Quoted from https://www.vcaa.vic.edu.au/Footer/Pages/On-demand-video-teacher-resources-terms-and-conditions.aspx :
“Our webinars and on-demand videos support the implementation of the VCE study designs. They include presentations by external educators and subject matter experts, who are highly experienced and deliver their considered, individual interpretations of the study designs in the context of the classroom. These interpretations should not be seen as fixed or mandated but, rather, as suggestions or possibilities. The VCE study designs remain the official documents for schools and VCE providers to use in preparing teaching, learning and assessment plans. If you have any questions, contact the curriculum manager of the relevant VCE study.:
The presentations obviously did not fall into the latter category of “subject matter expert”.
Or expert in any relevant manner whatsoever. There must be a million videos on induction and contradiction out there, but you’d be struggling to find any as bad as these.
The speaker mentions his name at the start, and it seems like he has had a long and successful career as a maths teacher. I worry about judging teachers because I think they are almost always over-worked and tired, which makes it hard to ever do their best on anything.
wst, you make an important point. I think this post and the comments are fair, but I appreciate it is debatable.
I work very hard on this blog to punch up, not down. On occasion, I tie myself in knots in order to avoid mentioning individuals by name, even i feel the individual is sufficiently “up” for it to be warranted to do so. Similarly, I have deleted a number of such comments over the years.
The simple reality is, the videos are very bad and VCAA bears the responsibility for this. I have absolutely no idea how these videos came about, or who might have contributed to them, or how much. I made no judgments of any individual, and I do not interpret any comment as having done so, with the exception of one, which I quickly deleted (and then told off the commenter). Indeed, I am tempted to delete your comment exactly because it is individualising, even while appreciating the good reasons for your comment.
That seems fair. I guess the videos could be a collaborative effort even if they are presented by an individual. VCAA seems to be trying to avoid institutional responsibility with the disclaimer above, but we can still consider them responsible.
Of course VCAA is responsible.
One can reasonably argue individual responsibility, but not here. And it’s not the point. The point is the institution is demonstrably dysfunctional.
So, is the point not that the videos were made but that they were published under the “authorised by VCAA” disclaimer?
That is interesting, and it may well be the main point some commenters here are trying to make, but I’m not sure it is the main point.
To my mind, the main point is that the same organisation that produced/published some crappy videos with crappy examples is going to be judging students soon on their ability to write proofs.
And that worries me a lot.
Hopefully it all comes to nothing.
For those who like historical matters, Euclid Book 1, Proposition 6, employs proof by contradiction.
While the talk here is about the quality of even the simplest induction questions, for at least 5-6years HSC Ext 2
has it clarified that variations are extected to be mastered by students in order to answer exam questions.
a. proving for all natural > n0 via separate odd and even
b. via P(n0) and P(n0+1) verifies, assumed P(k) and P(k+1) implying P(k+2) is true
c. via (step 2): assume P(n0) all through P(k) are True*).
Plus some other variations I dont remember.
* Personal note. It waited for me since my teenage years.
I imagine that proof by contradiction has its roots in logic in ancient Greece. Given that, one could look for examples in logic. Smullyan’s knights-and-knaves problems lend themselves to such analysis.
I am not sure where this question belongs, but I figure that this is as good a place as any. Does anyone know whether VCAA has decided on standard notation for logical symbols such as “not”, “and”, “or”, “implies” and “is logically equivalent to”?
Excellent question. It would also fit on WitCH 84.
I saw the new Cambridge text book for SM today; it uses words such as “and”, “or”, “not” etc rather than symbols.
I did not like what I saw about induction – but I had only a quick look.
VCAA has not.
I have long advocated that the logical (pun intended) place for defining all notation (including logical symbols) is the Study Design. But logic is something that VCAA does not understand. It is a case of the blind leading the (mostly) blind. And the latter are meekly happy to be led.
@JF: Good idea anyway.
Algorithmics tends to use words on the examination: and, or, not; implication does not seem to arise
I don’t think it will matter as the study design for Units 3&4 specifically states that Boolean Logic will not be examined in the Proof and Logic topic.
If all the textbooks start using the exact same notation though, I will be suspicious that VCAA has made a decision but not told anyone who doesn’t make money from it.
(Sorry if this is off-topic…)
RF, where does it say that in the Study Design? (It’s very good news btw, maybe I just don’t expect to see good news in the Study Design and so don’t).
RF, it is certain (in my view, anyway) that textbook writers received informal advice from VCAA ahead of the rest of us schmucks (*). There is far too much money at stake for this not too have happened. But I can’t feel angry about it – I’d rather a half-decent textbook as a result of advice given to writers ahead of time, rather than a crap textbook (**) as a result of writers guessing along with the rest of us saps.
* Look at the names of some of the textbook contributors.
** OK, they may well still be crap. Let’s just say that the textbooks will be less crap as a result of secret advice given.
Maybe I was reading an earlier draft of the study design…
“Proofs will involve concepts from topics such as: divisibility, inequalities, graph theory, combinatorics, sequences and series including partial sums and partial products and related notations, complex numbers, matrices, vectors and calculus.” (Page 109)
I don’t see Boolean Algebra mentioned anywhere…
“from topics such as: ….” rather than “from the topics: ….”
So VCAA are leaving the back door of uncertainty unlocked if not wide open.
OK, fair point…
…and VCAA does have form here (especially in Methods)
But I am comfortable in the knowledge that the Unit 1&2 teacher has given students a good grounding in Boolean Algebra and won’t be revisiting it in Units 3&4.
Just went through a few of the new textbook’s page proofs to compare their logic notation. There is not too much variance (from what I could see). Most books use a different notation for logical statements and Boolean algebras
Cambridge : VICMaths / Nelson : Jacaranda
And: ∧ : ∧ & • : ∧ & ·
Or: ∨ : ∨ & + : ∨ & +
Neg/Not: ¬ & ‘ : ¬ & overbar : ¬ & ‘
Implies: ⇒ : → : →
Iff: ⇔ : ↔ : ↔
Equivalence: ≡ : ≡ : =
The pseudocode conventions (yes, Marty, I know you won’t care too much!) also vary in style and constructs used
Cambridge is lowercase eg if …. then; … ;end if
VICMaths is uppercase eg IF … THEN; …; ENDIF
Jacaranda is lowercase but they insist on parentheses eg, if (….) then; …; endif
Cambridge is the only text to talk about defining functions/procedures in pseudocode, but their syntax is inconsistent with their other blocks, in that it requires a : and has no end statement (very Pythonic).
Jacaranda goes overboard on parentheses, eg `elseif ((time ≥ 18) and (time < 22)) then` and strangely they don't use a monospace font for their code samples. The Jacaranda pseudocode is clearly C or JS inspired – eg their for loop. And their Case statement is very ugly – Cambridge and VICMath don't bother with a Case statement.
These things really should be defined in the study designs. The fact that what aspects of pseudocode needed are not clearly stated is a worry in terms of possible exam questions – the dot point says "sequencing, decision-making, repetition and representation" – really does not clarify what they want.
The negation is an interesting one.
Cambridge seems to use different symbols depending on whether it is a statement being negated or the negation of a variable – thus distinguishing “algebra” from “logic”.
Which is perhaps even more of a reason to not worry too much about how/if this will be examined.
To be honest, I’m a lot more concerned about “pseudocode” in Methods.
Apologies to all if this is off-topic.
Well, it’s kind of segued from the original post (maybe Marty will make a new post dedicated to a pseudo discussion – VCE Mathematics Has Become a Pseudo Echo). For what it’s worth, here’s the official VCAA clarification (*) on pseudocode:
Students need to identify and understand iterations and sequencing.
It provides a language for students to articulate the problem solving process.
Some reserve words.
* Clear as mud. It will be interesting to see what ‘implementation advice’ and sample questions are provided. The new Mathematics Manager has inherited a shambles. I’m sure someone somewhere – hopefully an iceberg – is laughing their pinhead off.
Re: “The fact that what aspects of pseudocode needed are not clearly stated is a worry in terms of possible exam questions – the dot point says “sequencing, decision-making, repetition and representation” – really does not clarify what they want.”
VCAA has not clarified because VCAA has no idea what to do with it. VCAA will make it up on the run. It’s going to be the Wild West. The exams will define what’s needed, even then I doubt there’ll be consistency.
“VCAA has not clarified because VCAA has no idea what to do with it” – sounds cromulent.
Have you planned or taught any algorithms and pseudocode to your methods/specialist students yet?
I’m waiting until we get more clarification – and preferably the sample exam. It is a lot of time/learning to build this skill for only a couple dot points – and it’s not well connected to the rest of the topics, so it’s not a natural thing to include.
There are some algorithms already in the course, eg Bisection & Newton’s method for finding zeros, Euler (forward difference), trapezoid rule for numerical integrals, LCM/HCF, … any others? But they can be discussed without the formality of pseudocode and tracetables/desk checks.
I’ve updated my VCAA SD post with a bunch of links to new VCAA materials for the new SD.
I’ve updated the post with comments on the induction webinar. Nothing you don’t know. On to contradiction …
I’ve updated the post with comments on the contradiction webinar. Nothing you don’t know. On to the vodka …
Looks like the (N) statement has a typo.
Thanks very much. I’ll leave your comment, to indicate that there was a typo now fixed. (Seemed better than cluttering (N) with strikethrough text.)