Last one. This is the final in our sequence of WitCHes on the Logic and Proof chapter of VicMaths, Nelson’s Specialist Mathematics Year 12 text; the previous WitCHes are here and here and here and here (and a PoSSW here). This WitCH is on the final section, Proof by mathematical induction. The worked examples are all similar in form to that given below. The exercises seem ok except for one, which is almost almost good, but which definitely isn’t good (and for which no solution is provided).

Yes, the multiplication is quite wrong. It is correct if a subtraction is used. Then reducing everything modulo 3 gives 2^(n+2) – 2^n = 3(2^n) = 0 (mod 3).

That’s a better resolution and now the typo makes more sense – they simply kept going with the superscript! And again, reducing modulo 3: 2^n + 2(2^n) = 3(2^n) = 0 (mod 3). Nice pick up! Why are these things not rigorously checked, and checked again?

Thanks, Alasdair. Without for a minute excusing the writers and vettors and publisher, the answer to your question also involves VCAA, who must take a good share of the blame. See my comment here.

What I wonder is how much experience the people writing the VCE textbooks, VCE resource material, commercial trial exams and VCE exams have with this material. Have they ever formally studied it and/or taught it? What are their credentials for posing as ‘subject matter experts’ with this material?

None of the content I’ve seen appears to be written by anyone with genuine and authentic experience and understanding behind them.

I saw this again and again when statistics was first added to Specialist Maths – people posing as ‘experts’ who did not understand the distinction between the random variables and where and are independent copies of X. (In fact, I still occasionally see it in places that should know better).

RF, I’m certain you’re correct. The +2 had delusions of grandeur in wanting to be a superscript.

A typesetting error that won’t necessarily have been picked up by the proof readers (and the editor would presumably be generic and therefore lack the skills to detect it). How hilarious that such a small error can lead to such a calamity.

It doesn’t excuse the overall poor setting out of the proof, but for the error I think the writer of the question can be let off the hook, and possibly the proof reader (depending on what stage of the process the error crept in).

Obviously the writers didn’t really believe a product of 5s and 11s was divisible by 3, and normally I’d let a typo like that go through to the keeper. But we were here anyway, and the question as written was so absurd. And, you have to ask: how could even a very rushed proofreading have not caught such a typo?

Ouch
I have spent too much of my life correcting students who, when verifying some formula, begin by assuming it and then work on both sides. When they end up obtaining say 4 = 4 they claim the result is proved. Now it is happening in a text book!

All the fun is gone now (I think a smart Yr 9 student would have picked up the error. It’s a shame that solutions aren’t available, I’d have loved to see them) so I’ll simply mention a bugbear of mine:

“Let .” Grrrrrr.

I would have started with

Let P(n) be the conjecture .

Then at ‘Step 2’ I would have said
Assume P(k) is true for some .

The at ‘Step 3’ I would have said

Show that P(k) true implies P(k+1) true.

Then I would have signed off with P(1) is true and P(k) true implies P(k+1) true therefore it follows from the axiom of mathematical induction that P(n) is true for .

Wow, a lot of action between when I started typing and when I submitted.

PS – Nice title. I wonder how many people have heard of induction cookers …
(I once made a still that uses an electromagnetic heating coil. I called it proof by induction).

It may not have been mentioned by commenters for it being so obvious, but just for the record, BiB corrects a blatant error in the mathematical expression on the left side in the line he starts with “Let P(n) be the conjecture …” (and which the original starts with “Assume”). It seems that whoever wrote this mixed up the two sides of the assertion.

Hi Christian, I’m not exactly sure if this is the point you’re making. But I thought the text defining P(n) = THING(n) and then, in Step 3, going forth to prove P(k+1) = THING(k+1) was pretty damn funny.

I hadn’t looked at the explanation until just now, and it’s beyond appalling. In fact, it could be considered a text-book example (ha!) of how NOT to present mathematical induction. No wonder students (and their long-suffering teachers) are confused and demoralized.

Even given that proof is central to mathematics, seeing this rubbish leads me to wonder if proof shouldn’t be removed from school mathematics. Or at least, remove proof by induction.

Another problem is that students can easily get so bogged down in the algebra that they miss the whole point of the proof.

Thanks again, Alasdair. I’ve also commented on this aspect on one of the associated WitCHes.

I think there are three things happening, which have created a perfect storm of nonsense. First, VCAA bulldozed in the new curriculum, which meant that the publishers, who in the best of times are not that careful, had to do a really rush job. Secondly, the new topic of logic and proof is out of the comfort zone of most of the usual textbook writers (whose comfort zone, anyway, is about the size of Eddie Gaedel’s strike zone). Thirdly, the textbook writers/publishers *never* pay much attention to the actual text, the words prefacing the examples; this is because they assume the students do not read the text, and which becomes at least in part a self-fulfilling prophecy. In general, the publishers can kind of get away with the text as filler. But, for logic and proof, …

(i) “5 and 11 are prime, so no combination of powers can be divisible by 3.” Not sure what Red Five was thinking, but “5 + 121” shows something is missing.

(ii) I confess to being less easily pleased than Marty!!
There can be no excuse for confusing beginners (in a difficult subject) with such nonsense as
“Prove P(n) for n=1”; and then writing “P(n) = …”.
Everyone loses if we continue such nonsense.

One proves statements, not formulae.
So P(n) should always be a statement (one can then write
“Prove (the statement) P(1)” or “Prove (the statement) P(n) when n = 1”;
one cannot write
“P(n) = …).

So induction has to start by underlining the fact that
* we are used to proving single statements, one at a time.
This creates problems if we want to prove that a formula is true *for every value of n*.
We may be able to prove it when n=1; and we may then prove it for n=2; and then … . But we can only ever prove finitely many cases.

* Induction is about the miracle of proving infinitely many statements *at a single blow*.

Every induction proof has to start with
– a clear statement P(n), that has a uniform structure for each positive integer n.
(So we never write “P(n) = …”, because P(n) is not a number or an expression: it is a statement.)

Thanks for correcting me on this – I felt something was very obviously wrong with the conjecture when but then, as you have quite rightly pointed out – the statement could have been true for if the exponents were different.

As to the rest of the induction process… yes. I suspect that a significant portion of Specialist Mathematics teachers have seen some form of proof by induction, maybe many years ago. This doesn’t mean that they remember the subtleties correctly – it is more than half a page of expressions followed by a half page of expressions.

I only ever understood the ‘assume what you’re trying to prove’ thing when I learnt that the only way for a statement to be false is if it’s (true thing) => (false thing). It doesn’t matter whether P(k) is true or not, if P(k+1) is true then we’re all good. Is that right? That’s the only way I can wrap my head around it. Because the statement P(k) looks identical to P(n) to me.

Edit- hang on is that a statement or an implication? Don’t know anymore.

S&C, your confusion points out just how careful this material has to be worded, and so just how bad the above excerpt is.

I don’t think the T => F thing helps much with the intuition here. You wrote,

“It doesn’t matter whether P(k) is true or not, if P(k+1) is true then we’re all good.”

That’s almost it. It doesn’t matter whether P(k) is true, as long as ASSUMING P(k) true allows us to conclude that P(k+1) would also be true. (So, still, at that stage of our argument, neither may be true! But IF P(k) is true THEN P(k+1) is also true.) I think carefully read Tony’s comment above, and take a look at the chapter of his (free) book.

Since, many years ago (1988), I was a coauthor of a Grade 13 text that covered proof by induction, I thought I would check to see how we handled it. We began with a worked example: find the sum of the series 1(1!) + 2(2!) + … + n(n!). We worked out the first four cases (to get the sums 2! – 1, 3! – 1, 4! – 1, 5! – 1), saw a pattern and made a conjecture for n = 5, and checked it. We put in the sentence “Perhaps there is some way of using the information previously obtained to get the new equation.”
Then we worked how to get from the n = 5 to the n = 6 result. “Note that this procedure not only saves a fair bit of computation, but also provides the basis for providing new numerical patterns.” Then we did it for n = 6 to n = 7 and then stated the general conjecture. All of this was done on a single page of text.

On the second page, we made the analogy between a subroutine in a computer algorithm and the induction step and worked out going from n = k to n = k+1. Only at this point did we write a formal description of the induction involving a statement P(n). We concluded by comparing it to climbing a ladder rung by rung, starting at the bottom. All that took another page.

The point I want to make is that proof by induction is one of the hard ideas for students to grasp because so much depends on one’s perception of what is going on. It is easy to get hung up on “I’m assuming what I want to prove”. Actually, one of the things I can remember is how I first understood it. I was exposed to it in high school, but was as thick as a post at that time. Then we were assigned the problem of proving Leibniz Rule for the nth derivative of a product, and I struggled with it for a week. Then all of a sudden it hit me that what I was doing when going through the small values of n was a pattern of reasoning that could be applied to each step and formalized.

I think most students need to go through such an epiphany, so the textbook needs to help this to happen. It really does not need to use that much space. The presentation given above in the text is just plain gobbledygook given ex cathedra; the student will read it and her eyes will glaze over.

Ed, you alluded to a terrific point that I wish the textbook had made (maybe it does in the 1&2 textbook, which is why I held back on mentioning it):

We often see patterns in mathematics (like the one you mentioned), make a conjecture based on those patterns and then wonder if our conjecture is true. Proof by induction is a powerful technique in such cases.

Of course, the presentation above is plain awful. I quickly checked the introductions in two other Victorian texts (Jacaranda and Cambridge). They were pretty standard, appeared to be correct, and gave some honest effort to providing the intuition behind induction. Each, in different ways, had good aspects and bad aspects. So, all in all serviceable, which, for induction, is not really good enough. Neither text introduced the topic with a full blown example in the manner you described.

I am devoid of any experience in school education issues but I would nevertheless like to offer the following thought: to write P(n)=… is not only horrendous but also has the (pseudo-)advantage of nearly getting rid of one of the three bits that one considers at that junction of the proof, namely

(Statement, i.e. P(n)): A = B.

(Index A and B as A(n)=B(n) if you wish.) In what I see as related to this, a question: will the student be able to appreciate that if, say, the two sides of the equality to be proven were reversed (so that P(n) in the proper (!) form is unchanged), that it is still best to start with the sum expression, i.e. what is now the right-hand side, and prove it to be equal to the left? This is to illustrate that, from the mere point of logic, it is not necessary to mangle just the left side of any given P(n). I currently can’t think of a good example where working from “any” of the two sides yields equally valid paths in an inductive proof, but I think such examples do exist.

Yes, I would regard this as a ‘proof question’ (in particular, a direct proof).
(I’d re-word the question as “Let f(x) = …. Prove from first principles that f'(x) = …)

The reason “Prove” and “proof” is unfamiliar to students is that all the VCAA questions use “Show” instead of “Prove”. So in the classroom, monkey see, monkey do. Now that the VCAA has a veritable beehive in its bonnet about proof, it will be interesting to see how many “Show” questions appear on future exams.

A few (possibly many) years ago I watched a video of some MAV professional development presented by someone who I will refer to as ME. I can’t remember what the PD was about but I do remember there was a Q&A at the end. ME was asked why the VCAA uses the word ‘Show’ rather than ‘Prove’ in its exam questions. The answer given (perhaps half-jokingly) was that ‘Prove’ was considered too “scary” a word for students … I wonder why that ‘paradigm’ changed?

I thought it might have been because VCAA think current students are so much more resilient and courageous than past students. But yeah, your guess might be getting warmer.

I still don’t know what the real agenda is for including Logic and Proof. As commented previously, I’ve heard that it was to attract more female students to Specialist Maths. I can’t completely buy that. It just seems too much like someone’s thoughtless bubble in reaction to …?

When Logic was an optional module in Specialist Maths back in the 80’s, I believe less than 5% of the state chose to do it. (The overwhelming majority choice was mechanics).

We have been moaning for years about the lack of “proof” in the syllabuses. In the time BC students would meet proof in Geometry. There the proofs concerned lines and circles, things closer to the students’ real world experience. Steps in a proof corresponded to facts that a student could visualise in the diagram.
So how to fix this hole in the students’ experience? Let’s give them an abstract treatment of logic in Year 12! [Have these people ever taught real students?]

Do any problems at the VCE level “require”* strong induction?

I don’t agree with marty that strong induction shouldn’t be included at all (if that is what you were saying). Even if it’s not useful at the VCE level, it’s still something students should be taught, even if just tangentially. In fact, if I was first taught about “strong” induction when first learning it, I would have saved quite a few lines while writing proofs.

I don’t think it should be a central focus of the course, just maybe a dot point somewhere, accompanied with a proof that weak induction implies strong induction.

*Of course, no proofs actually “require” strong induction, instead of weak induction as strong induction is a trivial consequence of weak induction.

Hi, Joe. Yes, I was suggesting that not having strong induction in the curriculum is fine. (It’s not in the curriculum, and so no problems require it.) VCAA has made such a god awful mess introducing weak induction, I’d rather they fix that up first.

Am I right in assuming however, that you would see no problem with a teacher teaching it to their VCE class, even though it’s not in the curriculum? Even though I’d like for it to be taught, it’s probably for the best that it’s not in the curriculum knowing textbook writers and VCAA…

Well, I see a “problem” with anything in VCE that doesn’t contribute to the grade. It’s not the time for idealism. But obviously it’s a judgment call for the individual teachers. (That is, the 50 individual teachers who know what we’re discussing.) So, in that sense, yes, a teacher deciding SI is worth discussing is fine.

Part of me (the large part, but the balance is unfortunately shifting slowly in the wrong direction) still believes that decent mathematics can be taught at Specialist 1&2 level. By Units 3&4, sure, the exam (and SACs) need to be the focus because the study score is really all that matters at that point. Drilling CAS practice is a bitter pill that simply has to be swallowed.

But maybe (just maybe) at Units 1&2 level, where the grade is simply S/N and all assessment is internal…

(I will also admit not being one of “the 50” before reading the attachment in the previous comment – thanks btw for the distinction. Very interesting.)

I reckon Methods is lost in the majority of schools. There is just too much damned content to get through. Specialist 1&2 is (from what I can see) holding on for now, and it may well survive the “Mathematical Investigation” revolution that has recently been thrust upon all Unit 1&2 studies out of nowhere (well… I suppose when the VCE first introduced CATs there were also BATs at Year 11 briefly…)

Good stuff can and will be done in Specialist Units 1&2 only for as long as a certain group of teachers avoid the lure of retirement.

So, until that day comes… May QED continue to adorn many a student workbook as a badge of honor!

I think Specialist Units 1&2 has well and truly taken the baton from Methods when it comes to “just too much damned content to get through.”

(In fact, it’s a poisoned baton because VCAA has said that Specialist 3&4 exams can include 1&2 content – as well as Methods content. So you have to grab that poisoned baton and run like a bat out of hell).

Wasn’t it always the case that, at least officially, the Year 12 exams could cover Year 11 material? I understand teachers believe that this more of a threat now, but I’m not sure I understand why.

The core material was examinable, which was fine because it consisted predominantly of topics that were also part of Specialist 3&4.
But as of 2023 it’s not impossible that there will be a question on Graph Theory, or Logic (Boolean Algebra) or Sequences and Series (which was part of the core but never a serious threat to be examined). The new Study Design and no more core is a game changer (for the worse).

Who would have thought the (NHT) Exams would ever have binomial random variables? And yet it happened and I wonder how well those questions were answered by students (admittedly, all 5 of them) who did Methods 3&4 the previous year.

Trust me, the potential for the Specialist Exams to really screw students (*) over has significantly increased.

* Particularly students who did Methods 3&4 the previous year and might therefore be among the stronger Specialist students.

In reply to BiB:

So, you seem to be saying two things:

a) It was always possible for non-core SM11 to be examined, but it was understood that it wouldn’t be.

b) Now that there is no core, then either all of SM11 will be considered fair game, or none will (excepting topics that get subsumed by SM12 topics).

What is the reason to believe that, in effect, all of SM11 will be fair game?

It’s silly to bring up NHT exams, which have always been erratic and can’t be taken as an indication of policy.

a) Non-core couldn’t be examined because it was chosen from a number of optional modules. Not everyone chose the same options so not every student had the same background. You can’t examine Graph Theory or Logic at the 3/4 level if it was optional at the 1/2 level because not every student will have done it. Many/most teachers chose options that aligned with the 3/4 syllabus (such as dynamics and statics) – in order to get a ‘headstart’ on that material.

b) VCAA has stated that all of 1&2 Specialist and all of 1&2&3&4 Methods can be examined in the 3/4 exams.

As for the NHT exams, whether they are ‘non-canon’ or not is debatable. I haven’t seen any statement by VCAA that those exams can’t set a precedent. A VCAA exam is a VCAA exam and we all know that the exams and reports provide a lens through which a Study Design lacking in clarity has to be interpreted.
An example is the requirement that students memorise critical values of z in statistics for Exam 1. But there’s an inconsistency – must students memorise and use z = 2 or z = 1.96 for a 95% confidence interval? One can reasonably argue that the 68-95-99.7 rule in Maths Methods means that z = 2 is the one to memorise. But I have seen questions where z = 1.96 was required (due to the required accuracy of the answer). This grey area could easily have been removed by including the critical values on the Formula Sheet but VCAA chose not to do this.

On (b), which is the important point, I’m not convinced. I understand that VCAA has indicated it’s all examinable, but why were no such SM11 questions on the sample exams and so on? Do you really think they’re going to have a Boolean question in November? I’m not suggesting what teachers or students should do about this, but it’s not obvious to me that assuming all is examinable is optimal.

On NHT, you’re playing games. You know as well as I do that whatever NHT is purported to be, in the past it has been a clunkily put together mess indicative of nothing but thoughtlessness. I hate this kind of wilfully ignorant arguing. It’s frustrating and confuses people. Just stop it.

Re: “I understand that VCAA has indicated it’s all examinable, but why were no such SM11 questions on the sample exams and so on? Do you really think they’re going to have a Boolean question in November?”

Probably not this November (2023) but I think it’s only a matter of time (2024, 2025?) before an exam writer thinks (or is instructed to think) that a question on Boolean or Graph Theory is a good idea.

People often get on VCAA exam writing panels because their CV shows experience writing trial exams. I think it’s fair to say that looking at many trial exams from many organisations, I’m not filled me with confidence that something like the above won’t happen at some stage.

Re: “it’s not obvious to me that assuming all is examinable is optimal.”

Of course it’s not optimal. I know teachers that are being strategic in what they teach, taking an education gamble on the roll of the dice.

Neither am I. Obviously, the entire situation is a disaster waiting to happen. VCAA screwed up.

But I’m still curious of the practical issues, of how teachers and students should approach this year’s exams. Which SM11 topics should they consider, and how?

“how teachers and students should approach this year’s exams. Which SM11 topics should they consider, and how?”

and then coming up with ways to punish the students of those teachers. I can see it now:
VCAA: “Let’s learn those teachers that they have to toe the line or their students will suffer the consequences.”

I’m sure the exams for this year are almost ready for printing. But next year (and the year after) is a whole new ball game.
I believe that man walked on the moon. I believe in anthropogenic climate change.
I don’t believe that Elvis is still alive. I don’t believe that the VCAA wouldn’t try and teach teachers a lesson if it thought that teachers were being strategic in their teaching of SM Units 1&2 and their approach to Units 3&4 exams.

I’m comfortable with my concern being called “way too conspiratorial” (I’d say more but this blog might be bugged. I’ve already said too much).

Your rhetoric is very confused, and pointless. (And it’s “try to”, not “try and”). But again, I don’t care. If others indicate they want the post then I’ll post; otherwise I’ll leave it.

Seek out the “advice to teachers” that was put out last year for Units 1&2. It was meant to give advice to teachers of 1&2 about how to prepare students for the new 3&4 units that had as yet not been published…

Made for some interesting reading! (At the time).

Only time will tell if it was worth the time it took to read.

I don’t see why not. Testing fraction skills in Year 10 is totally fair game, surely and so any pre-requisite should be examinable whether explicitly or implicitly. It should not be a large part of the exam, but I have no issue with a mark here or there that looks at a skill explicitly covered in Units 1&2. Methods exams have been doing this for a while, although Methods 3&4 has, historically, been a lot more structurally similar to 1&2 than was ever the case with Specialist.

General 1&2 is not a pre-requisite for 3&4 and a lot of Methods 1&2 students will choose General 3&4 over Methods 3&4 so I can see why the argument need not apply in that case.

Whether it will happen or not, we only have to wait a couple more months to find out…

I think the “why not” is encapsulated by Boolean algebra. Are they really gonna test that? I’m not arguing what is legal according to their documents. I’m arguing what is likely or unlikely to happen, and how students and teachers should plan accordingly.

I’m not claiming in any way to have the answers. I’m claiming this is a massive unanswered question. I also don’t see that anybody can answer this question except VCAA, and they do not appear to be of a mind to do so.

Boolean was not explicitly mentioned, but they use the phrase “may include…” so they are not ruling it out either.

I would anticipate no Boolean in the exam and I think there are many, more likely topics that need revising (Sequences and Series being top of the list).

Time is a rather limited ingredient in this recipe.

Ugh! I *know* VCAA are not ruling it out. For the hundredth time, I’m not asking about the legal restraints: I’m asking about the reality on the ground. And yes this is about teachers “anticipating” things, because they have no damn choice. But it is also about (a) the fact they have no damn choice; (b) the reasonable and unreasonable ways in which teachers can anticipate.

You, all teachers, have to be clear-headed and insistent about this, and I see no evidence that you are.

“What can one fairly include in formal assessments?” is a question which affects all education systems. But I am not aware of any explicit principle concerning material from earlier phases or school years.

Since mathematics is cumulative, it seems reasonable for assessment to also be cumulative. This becomes slightly silly where a topic or method was a “one-off” – something that was covered and then dropped, with no subsequent maths topics depending on it: an example in the UK might be “Roman Numerals”, which are covered in upper Primary (perhaps for links to history, general education, and as an opportunity to highlight the miraculous simplicity of the usual base 10 representation of integers and associated algorithms).
However, where material forms the bedrock of subsequent mathematics (e.g. anything to do with fractions), there are good reasons for allowing (even encouraging) examiners to test old ideas or
methods modestly within a question about more recent techniques.

Thanks, Tony. Of course I agree, to the extent that the later subject is built on top of or deepens the earlier subject, then the earlier subject should be considered fair game. That’s the case with Methods12 and Methods34: to be honest, I’m never sure what is in one subject and what is in the other. But the relationship between Specialist12 and Specialist34 hasn’t been like that, and it’s not exactly clear now what the relationship is, or can reasonably be presumed to be.

Again, I think a telling example is Boolean algebra, which is a sizeable topic in SM12, and kinda sorta leads into the logic and proof of SM34. But should kids be studying BA in preparation for the SM34 exams? It seems to me odd if they should, but it seems to me that formally BA may be fair game.

I’m not trying to nitpick or stir up trouble here. It seems to me there’s a serious issue here. And, maybe there is not. But no one has responded in a way to convince me there is not.

I’ll add graph theory to the list of
“should kids be studying […] in preparation for the SM34 exams?”

Boolean algebra and graph theory are the two topics in SM12 that do not “[build] on top of or [deepen] the earlier subject.”

Specialist is now extremely content dense and I do not think it is fair that either of those two topics would be explicitly examined even in the context of proof. For example, a question similar to 2022 Algorithmics Exam Section A question 3 would be very unfair in my opinion. But according to VCAA it could be asked in a SM 34 exam.

For what it’s worth, I have told my students to guess any MCQ involving pseudocode (also Methods students), graph theory or Boolean algebra and move on. The 1 or 2 marks they would be guessing are negligible in the bigger picture of finishing the exam. They can always go back to those questions if they have time. Most students struggle to finish Exam 2 anyway so it’s a matter of being strategic.

When I am King and also Chief Examiner of Everything, I will set Question 1 in the Year 12 Maths exams to be a collection of very short and simple questions examining a fairly comprehensive swathe of the Year 11 and 12 topics. There will be a published list of the topics; candidates will be assured that each topic in the list will be examined. Will I need more than 2 three hour exams? (Calculators and formula sheets will not be needed.)

That question 10 is a joke, right?

5 and 11 are prime, so no combination of powers can be divisible by 3.

Did the question put the +2 in the wrong spot? Maybe?

Or did it mean to say Plus, not Times? (Edit: nope, that doesn’t work either)

(Really just guessing now what the author meant)

It’s a good puzzle, to figure out what might have been intended. And of course it’s hilarious as written.

Well, it works if you just do the “induction” part but forget the “base step” part.

Yes, that’s what I liked about .

Yes, the multiplication is quite wrong. It is correct if a subtraction is used. Then reducing everything modulo 3 gives 2^(n+2) – 2^n = 3(2^n) = 0 (mod 3).

Ah, I didn’t try that. I tried addition, which turned out to be interesting (at least for me).

I also tried

Seems to work.

Ah. Maybe that was what was intended.

I first tried , which was interesting.

That’s a better resolution and now the typo makes more sense – they simply kept going with the superscript! And again, reducing modulo 3: 2^n + 2(2^n) = 3(2^n) = 0 (mod 3). Nice pick up! Why are these things not rigorously checked, and checked again?

Thanks, Alasdair. Without for a minute excusing the writers and vettors and publisher, the answer to your question also involves VCAA, who must take a good share of the blame. See my comment here.

What I wonder is how much experience the people writing the VCE textbooks, VCE resource material, commercial trial exams and VCE exams have with this material. Have they ever formally studied it and/or taught it? What are their credentials for posing as ‘subject matter experts’ with this material?

None of the content I’ve seen appears to be written by anyone with genuine and authentic experience and understanding behind them.

I saw this again and again when statistics was first added to Specialist Maths – people posing as ‘experts’ who did not understand the distinction between the random variables and where and are independent copies of X. (In fact, I still occasionally see it in places that should know better).

Thanks, BiB. This is clearly a big part of the problem. (I fixed the LaTeX in your comment: maybe you didn’t include the semicolons?)

A recent commercial trial exam had this very error.

The point stands of course, but short of actually hiring a mathematician, I’m not sure how these publishers can rectify the situation.

The publishers don’t care, or at least don’t care enough. They have no reason to.

RF, I’m certain you’re correct. The +2 had delusions of grandeur in wanting to be a superscript.

A typesetting error that won’t necessarily have been picked up by the proof readers (and the editor would presumably be generic and therefore lack the skills to detect it). How hilarious that such a small error can lead to such a calamity.

It doesn’t excuse the overall poor setting out of the proof, but for the error I think the writer of the question can be let off the hook, and possibly the proof reader (depending on what stage of the process the error crept in).

Obviously the writers didn’t really believe a product of 5s and 11s was divisible by 3, and normally I’d let a typo like that go through to the keeper. But we were here anyway, and the question as written was so absurd. And, you have to ask: how could even a very rushed proofreading have not caught such a typo?

Ah, subtraction… I didn’t even consider that as a possible typo.

Well done.

(To you, not the author of the question, nor the proof-reader)

Ouch

I have spent too much of my life correcting students who, when verifying some formula, begin by assuming it and then work on both sides. When they end up obtaining say 4 = 4 they claim the result is proved. Now it is happening in a text book!

Yep. Incredible.

Looking at the body of the text, I’m not sure why the notation was introduced in the worked solution but not in the earlier section.

Surely, if the author(s) thought it relevant they would also use and in the introduction and DEFINE them clearly.

As with my other criticism on this chapter: there is so much GOOD, CORRECT stuff out there on this topic. Selling books full of errors is… not good.

The P(n) aspect is way, way worse than you are suggesting.

I just don’t think the authors understand the difference between as a proposition for and the polynomial evaluated when

Yes. And so what have they actually written?

The latter.

Yes, but what have done with it? What have they actually written about P(k)?

Oh. I see. That is bad. Very bad.

At least I think I’ve fixed the question by moving the (see above)

Yes. Like 4 = 4 bad.

All the fun is gone now (I think a smart Yr 9 student would have picked up the error. It’s a shame that solutions aren’t available, I’d have loved to see them) so I’ll simply mention a bugbear of mine:

“Let .” Grrrrrr.

I would have started with

Let P(n) be the conjecture .

Then at ‘Step 2’ I would have said

Assume P(k) is true for some .

The at ‘Step 3’ I would have said

Show that P(k) true implies P(k+1) true.

Then I would have signed off with P(1) is true and P(k) true implies P(k+1) true therefore it follows from the axiom of mathematical induction that P(n) is true for .

Yeah, it’s more of a POSWW. But as I replied to RF, it’s not just that P(n) is an expression: it’s how the writers employed that expression.

Wow, a lot of action between when I started typing and when I submitted.

PS – Nice title. I wonder how many people have heard of induction cookers …

(I once made a still that uses an electromagnetic heating coil. I called it proof by induction).

It may not have been mentioned by commenters for it being so obvious, but just for the record, BiB corrects a blatant error in the mathematical expression on the left side in the line he starts with “Let P(n) be the conjecture …” (and which the original starts with “Assume”). It seems that whoever wrote this mixed up the two sides of the assertion.

Hi Christian, I’m not exactly sure if this is the point you’re making. But I thought the text defining P(n) = THING(n) and then, in Step 3, going forth to prove P(k+1) = THING(k+1) was pretty damn funny.

Hi Marty, my point was not the one you made, but instead that

“Assume ” should have read “Assume “. But what you point out is also hilarious for sure.

Ah, that is also very funny. I hadn’t even noticed that one.

I hadn’t looked at the explanation until just now, and it’s beyond appalling. In fact, it could be considered a text-book example (ha!) of how NOT to present mathematical induction. No wonder students (and their long-suffering teachers) are confused and demoralized.

Even given that proof is central to mathematics, seeing this rubbish leads me to wonder if proof shouldn’t be removed from school mathematics. Or at least, remove proof by induction.

Another problem is that students can easily get so bogged down in the algebra that they miss the whole point of the proof.

Thanks again, Alasdair. I’ve also commented on this aspect on one of the associated WitCHes.

I think there are three things happening, which have created a perfect storm of nonsense. First, VCAA bulldozed in the new curriculum, which meant that the publishers, who in the best of times are not that careful, had to do a really rush job. Secondly, the new topic of logic and proof is out of the comfort zone of most of the usual textbook writers (whose comfort zone, anyway, is about the size of Eddie Gaedel’s strike zone). Thirdly, the textbook writers/publishers *never* pay much attention to the actual text, the words prefacing the examples; this is because they assume the students do not read the text, and which becomes at least in part a self-fulfilling prophecy. In general, the publishers can kind of get away with the text as filler. But, for logic and proof, …

Two comments:

(i) “5 and 11 are prime, so no combination of powers can be divisible by 3.” Not sure what Red Five was thinking, but “5 + 121” shows something is missing.

(ii) I confess to being less easily pleased than Marty!!

There can be no excuse for confusing beginners (in a difficult subject) with such nonsense as

“Prove P(n) for n=1”; and then writing “P(n) = …”.

Everyone loses if we continue such nonsense.

One proves statements, not formulae.

So P(n) should always be a statement (one can then write

“Prove (the statement) P(1)” or “Prove (the statement) P(n) when n = 1”;

one cannot write

“P(n) = …).

So induction has to start by underlining the fact that

* we are used to proving single statements, one at a time.

This creates problems if we want to prove that a formula is true *for every value of n*.

We may be able to prove it when n=1; and we may then prove it for n=2; and then … . But we can only ever prove finitely many cases.

* Induction is about the miracle of proving infinitely many statements *at a single blow*.

Every induction proof has to start with

– a clear statement P(n), that has a uniform structure for each positive integer n.

(So we never write “P(n) = …”, because P(n) is not a number or an expression: it is a statement.)

I refer to Chapter 6 in https://www.openbookpublishers.com/books/10.11647/obp.0168 for what all maths teachers need to understand (and then mediate to their students as they choose).

Thanks for correcting me on this – I felt something was very obviously wrong with the conjecture when but then, as you have quite rightly pointed out – the statement could have been true for if the exponents were different.

As to the rest of the induction process… yes. I suspect that a significant portion of Specialist Mathematics teachers have seen some form of proof by induction, maybe many years ago. This doesn’t mean that they remember the subtleties correctly – it is more than half a page of expressions followed by a half page of expressions.

I only ever understood the ‘assume what you’re trying to prove’ thing when I learnt that the only way for a statement to be false is if it’s (true thing) => (false thing). It doesn’t matter whether P(k) is true or not, if P(k+1) is true then we’re all good. Is that right? That’s the only way I can wrap my head around it. Because the statement P(k) looks identical to P(n) to me.

Edit- hang on is that a statement or an implication? Don’t know anymore.

S&C, your confusion points out just how careful this material has to be worded, and so just how bad the above excerpt is.

I don’t think the T => F thing helps much with the intuition here. You wrote,

“It doesn’t matter whether P(k) is true or not, if P(k+1) is true then we’re all good.”

That’s almost it. It doesn’t matter whether P(k) is true, as long as ASSUMING P(k) true allows us to conclude that P(k+1) would also be true. (So, still, at that stage of our argument, neither may be true! But IF P(k) is true THEN P(k+1) is also true.) I think carefully read Tony’s comment above, and take a look at the chapter of his (free) book.

“I confess to being less easily pleased than Marty!!” That could be a t-shirt.

How many people could, honestly, wear such a slogan though…?

The point isn’t for it to be true. The point is to dream it.

And we all know how easily pleased Marty is!

Since, many years ago (1988), I was a coauthor of a Grade 13 text that covered proof by induction, I thought I would check to see how we handled it. We began with a worked example: find the sum of the series 1(1!) + 2(2!) + … + n(n!). We worked out the first four cases (to get the sums 2! – 1, 3! – 1, 4! – 1, 5! – 1), saw a pattern and made a conjecture for n = 5, and checked it. We put in the sentence “Perhaps there is some way of using the information previously obtained to get the new equation.”

Then we worked how to get from the n = 5 to the n = 6 result. “Note that this procedure not only saves a fair bit of computation, but also provides the basis for providing new numerical patterns.” Then we did it for n = 6 to n = 7 and then stated the general conjecture. All of this was done on a single page of text.

On the second page, we made the analogy between a subroutine in a computer algorithm and the induction step and worked out going from n = k to n = k+1. Only at this point did we write a formal description of the induction involving a statement P(n). We concluded by comparing it to climbing a ladder rung by rung, starting at the bottom. All that took another page.

The point I want to make is that proof by induction is one of the hard ideas for students to grasp because so much depends on one’s perception of what is going on. It is easy to get hung up on “I’m assuming what I want to prove”. Actually, one of the things I can remember is how I first understood it. I was exposed to it in high school, but was as thick as a post at that time. Then we were assigned the problem of proving Leibniz Rule for the nth derivative of a product, and I struggled with it for a week. Then all of a sudden it hit me that what I was doing when going through the small values of n was a pattern of reasoning that could be applied to each step and formalized.

I think most students need to go through such an epiphany, so the textbook needs to help this to happen. It really does not need to use that much space. The presentation given above in the text is just plain gobbledygook given ex cathedra; the student will read it and her eyes will glaze over.

Ed, you alluded to a terrific point that I wish the textbook had made (maybe it does in the 1&2 textbook, which is why I held back on mentioning it):

We often see patterns in mathematics (like the one you mentioned), make a conjecture based on those patterns and then wonder if our conjecture is true. Proof by induction is a powerful technique in such cases.

Thanks very much, Ed.

Of course, the presentation above is plain awful. I quickly checked the introductions in two other Victorian texts (Jacaranda and Cambridge). They were pretty standard, appeared to be correct, and gave some honest effort to providing the intuition behind induction. Each, in different ways, had good aspects and bad aspects. So, all in all serviceable, which, for induction, is not really good enough. Neither text introduced the topic with a full blown example in the manner you described.

I am devoid of any experience in school education issues but I would nevertheless like to offer the following thought: to write P(n)=… is not only horrendous but also has the (pseudo-)advantage of nearly getting rid of one of the three bits that one considers at that junction of the proof, namely

(Statement, i.e. P(n)): A = B.

(Index A and B as A(n)=B(n) if you wish.) In what I see as related to this, a question: will the student be able to appreciate that if, say, the two sides of the equality to be proven were reversed (so that P(n) in the proper (!) form is unchanged), that it is still best to start with the sum expression, i.e. what is now the right-hand side, and prove it to be equal to the left? This is to illustrate that, from the mere point of logic, it is not necessary to mangle just the left side of any given P(n). I currently can’t think of a good example where working from “any” of the two sides yields equally valid paths in an inductive proof, but I think such examples do exist.

Suppose that I ask students to solve a problem like this.

Let . Prove, from the definition of the derivative of a function, that .

Do readers regard this question as a question about proof?

I ask because this problem connects (perhaps) unfamiliar “proof” with familiar ideas from calculus.

Yes, I would regard this as a ‘proof question’ (in particular, a direct proof).

(I’d re-word the question as “Let f(x) = …. Prove from first principles that f'(x) = …)

The reason “Prove” and “proof” is unfamiliar to students is that all the VCAA questions use “Show” instead of “Prove”. So in the classroom, monkey see, monkey do. Now that the VCAA has a veritable beehive in its bonnet about proof, it will be interesting to see how many “Show” questions appear on future exams.

A few (possibly many) years ago I watched a video of some MAV professional development presented by someone who I will refer to as ME. I can’t remember what the PD was about but I do remember there was a Q&A at the end. ME was asked why the VCAA uses the word ‘Show’ rather than ‘Prove’ in its exam questions. The answer given (perhaps half-jokingly) was that ‘Prove’ was considered too “scary” a word for students … I wonder why that ‘paradigm’ changed?

Thanks; very helpful.

Because “proof” is now a topic heading in Specialist 3&4.

(I’m only guessing).

I thought it might have been because VCAA think current students are so much more resilient and courageous than past students. But yeah, your guess might be getting warmer.

I still don’t know what the real agenda is for including Logic and Proof. As commented previously, I’ve heard that it was to attract more female students to Specialist Maths. I can’t completely buy that. It just seems too much like someone’s thoughtless bubble in reaction to …?

When Logic was an optional module in Specialist Maths back in the 80’s, I believe less than 5% of the state chose to do it. (The overwhelming majority choice was mechanics).

Back when Physics and Specialist were actually pre-requisites for an engineering degree…

Now…

I think someone (or some group) might be trying to “purify” Specialist – as in more pure mathematics, less applied mathematics.

Which, historically, is not what Specialist was about.

But hey, Methods is screwed, why not screw Specialist while we’re at it…

We have been moaning for years about the lack of “proof” in the syllabuses. In the time BC students would meet proof in Geometry. There the proofs concerned lines and circles, things closer to the students’ real world experience. Steps in a proof corresponded to facts that a student could visualise in the diagram.

So how to fix this hole in the students’ experience? Let’s give them an abstract treatment of logic in Year 12! [Have these people ever taught real students?]

The people who wrote this curriculum were demonstrably out of their minds.

I suppose we could say this section has been inducted into the Hall of Infamy.

Hi,

I often wondered why the difference between strong and weak inductive proofs

Are not included in the curricula.?

One summary here

Click to access induction.pdf

Not only to refute fallacies such as

All sheep are white

A heap of sand is always a heap

…

SteveR

Jesus. Because no one can properly handle weak induction.

Do any problems at the VCE level “require”* strong induction?

I don’t agree with marty that strong induction shouldn’t be included at all (if that is what you were saying). Even if it’s not useful at the VCE level, it’s still something students should be taught, even if just tangentially. In fact, if I was first taught about “strong” induction when first learning it, I would have saved quite a few lines while writing proofs.

I don’t think it should be a central focus of the course, just maybe a dot point somewhere, accompanied with a proof that weak induction implies strong induction.

*Of course, no proofs actually “require” strong induction, instead of weak induction as strong induction is a trivial consequence of weak induction.

Hi, Joe. Yes, I was suggesting that not having strong induction in the curriculum is fine. (It’s not in the curriculum, and so no problems require it.) VCAA has made such a god awful mess introducing weak induction, I’d rather they fix that up first.

Am I right in assuming however, that you would see no problem with a teacher teaching it to their VCE class, even though it’s not in the curriculum? Even though I’d like for it to be taught, it’s probably for the best that it’s not in the curriculum knowing textbook writers and VCAA…

Well, I see a “problem” with anything in VCE that doesn’t contribute to the grade. It’s not the time for idealism. But obviously it’s a judgment call for the individual teachers. (That is, the 50 individual teachers who know what we’re discussing.) So, in that sense, yes, a teacher deciding SI is worth discussing is fine.

What about at Units 1&2 then?

Part of me (the large part, but the balance is unfortunately shifting slowly in the wrong direction) still believes that decent mathematics can be taught at Specialist 1&2 level. By Units 3&4, sure, the exam (and SACs) need to be the focus because the study score is really all that matters at that point. Drilling CAS practice is a bitter pill that simply has to be swallowed.

But maybe (just maybe) at Units 1&2 level, where the grade is simply S/N and all assessment is internal…

(I will also admit not being one of “the 50” before reading the attachment in the previous comment – thanks btw for the distinction. Very interesting.)

Good point, RF. SM12 is, or was, a place where it made sense to do good stuff. With the new curriculum, that may not be as true.

I reckon Methods is lost in the majority of schools. There is just too much damned content to get through. Specialist 1&2 is (from what I can see) holding on for now, and it may well survive the “Mathematical Investigation” revolution that has recently been thrust upon all Unit 1&2 studies out of nowhere (well… I suppose when the VCE first introduced CATs there were also BATs at Year 11 briefly…)

Good stuff can and will be done in Specialist Units 1&2 only for as long as a certain group of teachers avoid the lure of retirement.

So, until that day comes… May QED continue to adorn many a student workbook as a badge of honor!

I think Specialist Units 1&2 has well and truly taken the baton from Methods when it comes to “just too much damned content to get through.”

(In fact, it’s a poisoned baton because VCAA has said that Specialist 3&4 exams can include 1&2 content – as well as Methods content. So you have to grab that poisoned baton and run like a bat out of hell).

Quilting Engineering Divas …?

Wasn’t it always the case that, at least officially, the Year 12 exams could cover Year 11 material? I understand teachers believe that this more of a threat now, but I’m not sure I understand why.

The core material was examinable, which was fine because it consisted predominantly of topics that were also part of Specialist 3&4.

But as of 2023 it’s not impossible that there will be a question on Graph Theory, or Logic (Boolean Algebra) or Sequences and Series (which was part of the core but never a serious threat to be examined). The new Study Design and no more core is a game changer (for the worse).

Who would have thought the (NHT) Exams would ever have binomial random variables? And yet it happened and I wonder how well those questions were answered by students (admittedly, all 5 of them) who did Methods 3&4 the previous year.

Trust me, the potential for the Specialist Exams to really screw students (*) over has significantly increased.

* Particularly students who did Methods 3&4 the previous year and might therefore be among the stronger Specialist students.

In reply to BiB:

So, you seem to be saying two things:

a) It was always possible for non-core SM11 to be examined, but it was understood that it wouldn’t be.

b) Now that there is no core, then either all of SM11 will be considered fair game, or none will (excepting topics that get subsumed by SM12 topics).

What is the reason to believe that, in effect, all of SM11 will be fair game?

It’s silly to bring up NHT exams, which have always been erratic and can’t be taken as an indication of policy.

a) Non-core couldn’t be examined because it was chosen from a number of optional modules. Not everyone chose the same options so not every student had the same background. You can’t examine Graph Theory or Logic at the 3/4 level if it was optional at the 1/2 level because not every student will have done it. Many/most teachers chose options that aligned with the 3/4 syllabus (such as dynamics and statics) – in order to get a ‘headstart’ on that material.

b) VCAA has stated that all of 1&2 Specialist and all of 1&2&3&4 Methods can be examined in the 3/4 exams.

As for the NHT exams, whether they are ‘non-canon’ or not is debatable. I haven’t seen any statement by VCAA that those exams can’t set a precedent. A VCAA exam is a VCAA exam and we all know that the exams and reports provide a lens through which a Study Design lacking in clarity has to be interpreted.

An example is the requirement that students memorise critical values of z in statistics for Exam 1. But there’s an inconsistency – must students memorise and use z = 2 or z = 1.96 for a 95% confidence interval? One can reasonably argue that the 68-95-99.7 rule in Maths Methods means that z = 2 is the one to memorise. But I have seen questions where z = 1.96 was required (due to the required accuracy of the answer). This grey area could easily have been removed by including the critical values on the Formula Sheet but VCAA chose not to do this.

Thanks, BiB.

On (a), I get it.

On (b), which is the important point, I’m not convinced. I understand that VCAA has indicated it’s all examinable, but why were no such SM11 questions on the sample exams and so on? Do you really think they’re going to have a Boolean question in November? I’m not suggesting what teachers or students should do about this, but it’s not obvious to me that assuming all is examinable is optimal.

On NHT, you’re playing games. You know as well as I do that whatever NHT is purported to be, in the past it has been a clunkily put together mess indicative of nothing but thoughtlessness. I hate this kind of wilfully ignorant arguing. It’s frustrating and confuses people. Just stop it.

Re: “I understand that VCAA has indicated it’s all examinable, but why were no such SM11 questions on the sample exams and so on? Do you really think they’re going to have a Boolean question in November?”

Probably not this November (2023) but I think it’s only a matter of time (2024, 2025?) before an exam writer thinks (or is instructed to think) that a question on Boolean or Graph Theory is a good idea.

People often get on VCAA exam writing panels because their CV shows experience writing trial exams. I think it’s fair to say that looking at many trial exams from many organisations, I’m not filled me with confidence that something like the above won’t happen at some stage.

Re: “it’s not obvious to me that assuming all is examinable is optimal.”

Of course it’s not optimal. I know teachers that are being strategic in what they teach, taking an education gamble on the roll of the dice.

Neither am I. Obviously, the entire situation is a disaster waiting to happen. VCAA screwed up.

But I’m still curious of the practical issues, of how teachers and students should approach this year’s exams. Which SM11 topics should they consider, and how?

Is this worth a separate post?

It would be very worthwhile.

Do NOT do it.

The last thing we need is VCAA reading

“how teachers and students should approach this year’s exams. Which SM11 topics should they consider, and how?”

and then coming up with ways to punish the students of those teachers. I can see it now:

VCAA: “Let’s learn those teachers that they have to toe the line or their students will suffer the consequences.”

I think that is way too conspiratorial, and the exams must be well under production already.

But I’ll leave it others to decide. If people want such a post, I’ll do it.

I’m sure the exams for this year are almost ready for printing. But next year (and the year after) is a whole new ball game.

I believe that man walked on the moon. I believe in anthropogenic climate change.

I don’t believe that Elvis is still alive. I don’t believe that the VCAA wouldn’t try and teach teachers a lesson if it thought that teachers were being strategic in their teaching of SM Units 1&2 and their approach to Units 3&4 exams.

I’m comfortable with my concern being called “way too conspiratorial” (I’d say more but this blog might be bugged. I’ve already said too much).

Your rhetoric is very confused, and pointless. (And it’s “try to”, not “try and”). But again, I don’t care. If others indicate they want the post then I’ll post; otherwise I’ll leave it.

Seek out the “advice to teachers” that was put out last year for Units 1&2. It was meant to give advice to teachers of 1&2 about how to prepare students for the new 3&4 units that had as yet not been published…

Made for some interesting reading! (At the time).

Only time will tell if it was worth the time it took to read.

If you or someone finds the link, I’ll highlight it in a suitable manner.

The link seems to have gone with the publication of the new study design.

I’m trawling my VCAA authored Word files for a document to upload or email through. So far, no luck.

OK, thanks. It would be interesting, but don’t stress it.

Marty et al,

My reading of the new Specialist lines in the study design lead me to conclude the following:

1. Yes, SM1&2 content can and will be assessed in a 3&4 exam.

2. In accepting (1) above, it is likely to only be assessed under the umbrella of “proof” or “pseudocode”.

3. NHT papers seem to have different authors and I expect the NHT authors use the November papers for inspiration, rather than the other way around.

As with everything I write on this blog, I could well be very wrong…

Thanks, RF. Of course proof and (ugh!) pseudocode will be examined. The real question is whether SM12 is generally fair game.

I don’t see why not. Testing fraction skills in Year 10 is totally fair game, surely and so any pre-requisite should be examinable whether explicitly or implicitly. It should not be a large part of the exam, but I have no issue with a mark here or there that looks at a skill explicitly covered in Units 1&2. Methods exams have been doing this for a while, although Methods 3&4 has, historically, been a lot more structurally similar to 1&2 than was ever the case with Specialist.

General 1&2 is not a pre-requisite for 3&4 and a lot of Methods 1&2 students will choose General 3&4 over Methods 3&4 so I can see why the argument need not apply in that case.

Whether it will happen or not, we only have to wait a couple more months to find out…

I think the “why not” is encapsulated by Boolean algebra. Are they really gonna test that? I’m not arguing what is legal according to their documents. I’m arguing what is likely or unlikely to happen, and how students and teachers should plan accordingly.

I’m not claiming in any way to have the answers. I’m claiming this is a massive unanswered question. I also don’t see that anybody can answer this question except VCAA, and they do not appear to be of a mind to do so.

Boolean was not explicitly mentioned, but they use the phrase “may include…” so they are not ruling it out either.

I would anticipate no Boolean in the exam and I think there are many, more likely topics that need revising (Sequences and Series being top of the list).

Time is a rather limited ingredient in this recipe.

Ugh! I *know* VCAA are not ruling it out. For the hundredth time, I’m not asking about the legal restraints: I’m asking about the reality on the ground. And yes this is about teachers “anticipating” things, because they have no damn choice. But it is also about (a) the fact they have no damn choice; (b) the reasonable and unreasonable ways in which teachers can anticipate.

You, all teachers, have to be clear-headed and insistent about this, and I see no evidence that you are.

I actually had a reply from VCAA… was meaning to forward it your way. Will do so when I get a chance.

“What can one fairly include in formal assessments?” is a question which affects all education systems. But I am not aware of any explicit principle concerning material from earlier phases or school years.

Since mathematics is cumulative, it seems reasonable for assessment to also be cumulative. This becomes slightly silly where a topic or method was a “one-off” – something that was covered and then dropped, with no subsequent maths topics depending on it: an example in the UK might be “Roman Numerals”, which are covered in upper Primary (perhaps for links to history, general education, and as an opportunity to highlight the miraculous simplicity of the usual base 10 representation of integers and associated algorithms).

However, where material forms the bedrock of subsequent mathematics (e.g. anything to do with fractions), there are good reasons for allowing (even encouraging) examiners to test old ideas or

methods modestly within a question about more recent techniques.

Thanks, Tony. Of course I agree, to the extent that the later subject is built on top of or deepens the earlier subject, then the earlier subject should be considered fair game. That’s the case with Methods12 and Methods34: to be honest, I’m never sure what is in one subject and what is in the other. But the relationship between Specialist12 and Specialist34 hasn’t been like that, and it’s not exactly clear now what the relationship is, or can reasonably be presumed to be.

Again, I think a telling example is Boolean algebra, which is a sizeable topic in SM12, and kinda sorta leads into the logic and proof of SM34. But should kids be studying BA in preparation for the SM34 exams? It seems to me odd if they should, but it seems to me that formally BA may be fair game.

I’m not trying to nitpick or stir up trouble here. It seems to me there’s a serious issue here. And, maybe there is not. But no one has responded in a way to convince me there is not.

I’ll add graph theory to the list of

“should kids be studying […] in preparation for the SM34 exams?”

Boolean algebra and graph theory are the two topics in SM12 that do not “[build] on top of or [deepen] the earlier subject.”

Specialist is now extremely content dense and I do not think it is fair that either of those two topics would be explicitly examined even in the context of proof. For example, a question similar to 2022 Algorithmics Exam Section A question 3 would be very unfair in my opinion. But according to VCAA it could be asked in a SM 34 exam.

For what it’s worth, I have told my students to guess any MCQ involving pseudocode (also Methods students), graph theory or Boolean algebra and move on. The 1 or 2 marks they would be guessing are negligible in the bigger picture of finishing the exam. They can always go back to those questions if they have time. Most students struggle to finish Exam 2 anyway so it’s a matter of being strategic.

Any chance the punchline can be posted here?

When I am King and also Chief Examiner of Everything, I will set Question 1 in the Year 12 Maths exams to be a collection of very short and simple questions examining a fairly comprehensive swathe of the Year 11 and 12 topics. There will be a published list of the topics; candidates will be assured that each topic in the list will be examined. Will I need more than 2 three hour exams? (Calculators and formula sheets will not be needed.)