It’s a long time since we’ve had a MitPY. But, the plague goes on (including the plague of right-wing Creightons).

This one comes from frequent commenter Red Five, and we apologise for the huge delay in posting. It is targeted at those familiar with and, more likely, struggling with Victoria’s VCE rituals:

*VCAA uses some pretty strange words in exam questions, and the more exam papers I read, especially for Specialist Mathematics 34, the more I can’t get a firm idea of how they distinguish between the meanings of “*

**show that**“, “**verify that**” and “**prove that**“.*“*

**Verify**” seems to mean “by substitution”, “**show that**” seems to mean “given these very specific parameters” and “**prove that**” seems to be more general, but is it really this simple?
I remember asking this in specialist maths, the teacher couldn’t really give a straight answer. Think he said something like Verify means you can use a result in the question, and when they say Show it means you have to do less work than when they say Prove. And I think at the end of Verify (or was it Show?) you have to write “as required” or something.

The phrase “prove that” seems to only appear in questions where vectors are used to prove some geometric fact. Which makes sense, because that is the only part of the course where the language of proof is explicitly used. However, I don’t think VCAA really understand it to mean anything different from “show that”, in the sense that an exam question like “prove ABCD is a rhombus” could just as well be asked by “show that ABCD is a rhombus”, and student responses would be marked in exactly the same way.

My understanding of the way VCAA would like students to answer “show that” questions is to read it as “Find …” but make sure their answer matches the one given in the question. Whereas, in a “verify that” question, students can assume the truth of the information in the question.

Perhaps the clearest contrast is with questions involving roots of complex polynomials. “Verify that w is a root of P(z)” – students can write out P(w) and show that it equals 0. “Show that w is a root of P(z)” – students have to start with P(z) = 0 and find the roots, one of which must be w – making the serendipitous guess that w is a root and checking it by substituting in to P(z) is not considered a legitimate method.

An example illustrating SRK’s point is Q4a on Specialist Exam 2, 2017, which asks students to “Show that the roots of are and . The question is worth 1 mark and the Examination Report then comments:

“Correct solutions were obtained by using the quadratic formula or completing the square. Some students did not correctly follow the ‘show that’ instruction either by not showing key steps in their solution or by solely verifying the solutions given by substitution.”

Of course this is all insane. But the purpose of this MitPY is not to point out the obvious insanity, but to help people cope with the insanity.

I find that comment a bit unnerving. I wonder if the students wrote something like: “it is a quadratic and therefore has at most two distinct roots” AND verified the solutions by substitution, would they still be marked wrong?

Yes.

I’ll have to remember that. Giving students the answer kind of makes it hard for them to show they know how to get it, doesn’t it?

Yes. The technical term for this is “Really Fucking Stupid”.

No Marty, I disagree. It’s not stupid. It’s smart. It’s done to *give* the student a starting point to ensure s/he can progress through the rest of the question (and to reduce the total-pain-in-the-ass tracking of consequential marks). ‘Show’ questions are ‘giving’ questions.

Your technical term is applicable only when this is done *poorly*. For example, by using words such as ‘Show’, ‘Verify’, ‘Confirm’ etc. that have not been given precise definitions or meanings.

The following is an example (I think, anyway!) of doing it well:

(a) Prove the identity .

(b) Hence find .

Yes, you can simply avoid this kind of question altogether. But that can become very limiting, particularly in an extended response question, and can severely disadvantage weaker kids.

I like ‘Prove’ questions involving identities. Proving identities is part of the course, and can give a good hint for the next part. Another technique I often use to avoid ‘Show’ but also to ‘Give’ is the following:

(a) Find the trajectory of the ball when …

In fact, [blah blah] and so the trajectory of the ball is given by …

(b) Find …

I think the above two examples *ahem* show that the only word you need is ‘Prove’. It has a precise and well known meaning and the result is a lot more certainty for teachers and students (but less wriggle room for VCAA, but who cares).

And of course, if proof by induction was on the course, life could be even easier (https://mathematicalcrap.com/2020/05/28/inducting-mathematics-into-vce/) …

Like the NBN, Ultranet etc. the idea itself is not “fucking stupid”, it’s the execution of the idea.

This might be a stupid question, but if by “weaker kids”, you mean those who can’t do maths as well, is it really a bad thing if the exam disadvantages them? Isn’t that sort of the point of the exam? To distinguish between people who can solve maths problems and those who can’t?

I wonder also about the kind of behaviour questions encourage. For example, if it was known that there are questions where you need to get the first part right in order for the rest of question to make sense, this encourages students to thoroughly check their work, which is good behaviour to encourage (I think).

If there are questions where you have to understand opaque rules about VCAA-specific meanings of words that they won’t tell you, then that encourages people to worry about that sort of thing, and read past exam papers like tea leaves looking for clues. It communicates the idea that mathematics is about understanding other people’s absurd rules. I feel like this is a bad message to send.

An exam already ‘disadvantages’ the weak kids, in the same way that a foot race ‘disadvantages’ the slow runners.

You need to ask yourself a couple of questions:

How much *additional* disadvantage do you want for that student,

How much of what that student knows do you genuinely want to examine, and

How much time are you willing to devote to tracking the consequential marks (if the kid makes a mistake in answering part (a) and it was an essential result for answering parts (b), (c) and (d)).

A simple example (relevant to a max/min problem):

(a) Show that V(x) = ….

(b) Find the maximum value of V(x).

(c) Find the domain and range of V(x).

(d) Draw a graph of V = V(x) on the set of axes below.

Now change part (a) to “Find V(x).”

So what do you want to have happen to the kid who for whatever reason can’t find V(x) but knows what to do once s/he has it …? And what are you going to say to that kid afterwards … “Tough break chum, but you couldn’t find V(x) so you get 0.”

I can see that. But I think sometimes a flat zero can be very instructive (for example, if the most important and essential skill is tested in the first part) and also mitigated by having other questions on the exam.

I’ve graded questions like that simple example, but at university, which I guess is maybe different. And I think I would prefer the “Find V(x)” version, with consequential marks according to how reasonable the subsequent answers are.

It was sometimes a lot of work to follow through an unexpected approach that a student took (maybe not on a question like this, but if it was a proof or something) and I understand that high school teachers may have less time. I suppose I’m yet to find that out.

In that question (in the Specialist Mathematics exam), I worry that students would have to focus so much on following instructions that they wouldn’t have much time to engage with the mathematics.

What you *must* do as a teacher is be confident in what you tell students. If you give them clear and confident definitions of ‘Show’, ‘Verify’ ‘Prove’ etc. illustrated with examples and then give tests where they get practice writing solutions to questions using these words, students will not worry.

It doesn’t matter that in reality there’s ambiguity and bullshit – what matters is *confidently* giving the students your very best educated guess.

This shit is on the exams, so there’s no avoiding it. You’re doing students a grave disservice if you do try to avoid it. You have to run towards the fire (not away from it).

It’s really important that students don’t see teachers worrying and getting stressed and anxious about this stuff. It’s OK for them to see you being angry and fired up over it, but never ever let them see that you’re worried and uncertain. Because then they will worry at an exponential rate, and worry turns to panic. They will lose confidence. Confidence is very hard to instil, and very easy to lose.

It’s absolutely essential that students see their teachers confident and unworried, particularly in a year such as this. Students have to believe that you’re in control, you have a plan and that everything is going to be OK.

Whether you are confident or you aren’t, it is essential that students *see* that you are. Otherwise panic sets in and spreads like a wildfire.

Re: The flat zero. You have to pick your student very carefully. Some can take it and become better as a consequence, many others can’t and will give up. It is vital that you maintain student confidence. Very very rarely have I ever seen a flat zero of the type you describe being instructive.

s-t, I won’t get in the way, but I’m on your side here. The “make sure they can all go on to the next part” thing is infantilising, and it also screws up the evaluation of everyone. It means that no decently involved question is ever permitted.

It didn’t used to be like this.

John, yes and no. You give an acceptable and very good example of an “a) Prove … b) Hence …” question.

The difference is, your a) is not completely trivial, either in what to do or the mechanics of doing it. But having a prove/show/verify/whatever for a 1-point application of the quadratic formula is Really Fucking Stupid. And, the overwhelming majority of prove/show/verify/whatever questions in VCE are RFS.

OK Marty, for your illustrative example I take it all back.

Yes, it is RFS. And that’s being kind. How does the solution proposed by S-T NOT show it. Or alternatively and possible faster, constructing the linear factors from the roots and expanding to get the quadratic.

The trouble is:

1) The writer has a particular solution in mind to the exclusion of all other reasonable solutions. A narrow-minded moron.

2) It’s really stupid to ask this question on a bloody CAS- exam!

3) The writer is a moron.

I get that this is trying to polish a turd, but I’m baffled why VCAA don’t just write the question as something like “Use an algebraic method to find the roots of …”

There should be no concern over students not getting the answer and being stuck on the latter parts of the question, since it’s Exam 2 and students can find the correct solutions using CAS.

There should be no concern anyway. It is that concern which leads to micro-grading, which amounts to ritual, and the show-prove-verify lunacy. Just ask real fucking questions.

That won’t necessarily work for my above examples. And it won’t work in Exam 1 where there’s no CAS technology but where you will still find ‘Show’ and ‘Verify’ questions etc.

Once upon a time, VCAA did use phrases like “Use calculus to …”. I will use that phrase on a SAC, along with the phrase “Use algebra to …” but NOT for the reason you mention …

You’re not allowed to have a separate ‘tech-free’ section for your SAC, the whole bloody thing has to be CAS-Enabled. So what do you do if you, quite reasonably, want the kids to demonstrate ‘by-hand’ skills …? You have to use dumb phrases like the above.

Something else that this blog has yet to raise – the number of marks in a CAS-Enabled assessment is also VCAAspeak for the intended way of answering it. A simple example:

Find in polar form all cube roots of -i. (3 marks)

There are 3 roots and you might think 1 mark for each so I’ll get them from a CAS and write them down. Ta da! I’ve found them.

Not today, Zurg! 3 marks means NO CAS. 3 marks means you have to show appropriate working that clearly leads to the 3 roots. No working, no marks.

Or perhaps just use the wording “Show, substitution, that …”

But then does the following solution use substitution or would it be considered an algebraic method:

.

Therefore ( ) and ( ) are factors therefore blah blah are roots.

Did students who used this method get penalised? (I’d argue that it requires more skill than just substituting stuff into the quadratic formula.)

So maybe the question should have been worded “Show by explicitly using the quadratic and explicitly simplifying that ….”

This is the bullshit that a stupid question written by a narrow-minded moron who has a particular solution in mind to the exclusion of all other reasonable solutions creates.

Actually I’m only kidding with the above suggested re-wording. It was just a dumb question to put on a CAS- exam and no amount of re-wording can change that simple fact.

By the way, here’s a question:

How could anyone “follow the ‘show that’ instruction” when, as RF correctly points out, this command term is not defined or exemplified by VCAA??

And this, dear readers, was the very point of my asking the question – I cannot find anywhere VCAA’s definitions of these command terms in a manner that distinguishes “show that” from “verify that”.

“Prove that” is in a totally different category altogether.

As far as I can tell, a student is allowed to use substitution to “verify” but not to “show”.

Does anyone have any evidence to the contrary?

It is long overdue for VCAA to be asked point blank what the definitions are or at least to provide an example of how each word is used.

I would agree with you except that I have sat in “meet the examiners” sessions where I (and others) have asked the examiners what is meant by “Show that”.

Different answer each time.

Although, I now know a lot of ways to NOT “show that” in a Methods Paper 1, for example…

I just wish I had the sharpness of wit to quote Poirot and say, “stop telling me what it isn’t and try telling me what it IS.”

The answer is probably different each time because there are no well-defined meanings – the meanings change to suit the circumstances. (Which is also probably why there is no definitive statement of definitions with examples from VCAA).

It seems that the intended meaning of ‘verify’ is very much the same as ‘confirm’.

It’s a disgrace that a student can get zero for a question simply because s/he doesn’t understand the current VCAAspeak. If VCAA just stuck with two words, prove and verify, and gave precise meanings and examples of each, students could get on with doing the maths and not have to worry about semantics. Is it too much to ask for?

As an aside, would anyone like to prove/show/verify that the Bernoulli *Distribution* is in the Study Design? Apparently a “natural connection” to stuff that *is* in the Study Design means that it is (even though it isn’t).

Thanks, John. You are probably correct, in which case the best approach is along the lines the SRK suggests: if the examiners want to insist that students do or do not use some technique then just bloody well say it.

Please hold you Bernoulli fire. I’ll posting on that in the next day or so.

“Prove”:

Proposition: Bernoulli random variable is a special case or Binomial random variable, for example: X~Bi(n=1,p),

Observation: each year Binomial distribution is assessed.

Conclusion: since A sequence of independent Bernoulli random variables can constitute Binomial distribution, we therefore conclude that Bernoulli distribution is in the study design.

“Show that”,

By inspecting the study design, on page 73, area of study 4 probability, we quote that “ bernoulli trials and the binomial distribution, Bi(n, p), as an example of a probability distribution for a

discrete random variable”

“Verify that”:

Let S = “Bernoulli trials will be assessed in 2020”, S’=Bernoulli trials will not be assessed in 2020“

We know it can either be assessed or not assessed.

Hence this event itself is a Bernoulli Trial.

As exam content must be based on the study design, we confirmed that it is in the study design.

P.N., a post is coming.

Sorry P.N. but you’re starting from the wrong premise and therefore proving the wrong thing.

The correct proposition is:

“*Bernoulli Distribution* is stated in the Study Design”.

The proof that this proposition is false is trivial.

Hi SRK. I have formulated the exact same understanding that you have – what you’ve said is exactly what I tell my students.

As for the VCAA using the word ‘Prove’, I honestly think that VCAA use that word when it just sounds too dumb to use the word “Show”:

“Show the identity ….” versus “Prove the identity …” I don’t see the point of the word ‘Show’. It’s pretty much ‘Prove Lite’.

Also, I suspect (although I cannot prove) that VCAA has a very limited definition of proof. VCAA just seem to invent their own meanings for words – VCAAspeak. Wankaster would undoubtedly have a lot of nothing to say about it all.

As the old sensei said – “Don’t show, PROOOVE!”

However, having said all this, students are also entitled to ask the difference between theorem, lemma, and corollary. The difference here is that these are well-defined technical words that have well-known (and precise) meanings.

If VCAA cannot define key terms, can we conclude they have a DiLLemma, perhaps?

*Droll* (not drool)

Here is my take on this very interesting question.

I was always told that “verify” means “check the truth of”. To verify your identity, someone might check your driving license.

To “prove” a statement in mathematics, means to create an argument that starts from some reasonable starting point and ends up with the statement of interest.

As for “show”, I don’t know. VCE teachers with more experience would be able to give their take on this. Judging by Marty’s report from examiners, it seems that “show” means “prove”.

Of course, in some cases, the verification might be a proof e.g. Verify/prove that x=2 is a solution of x+1=3.

Again, an excellent, but unsettling, question.

Terry, how many synonyms can you fit on the head of a pin?

For a normal, sane mathematician, plugging in the value of z would suffice to prove = verify = show that this z is a solution. It’s ridiculous to suggest otherwise.

VCAA can use words however they wish, but it’d be preferable if this usage didn’t conflict with standard mathematical usage. It’d also be preferable, as others have noted, if such usage was stated explicitly somewhere.

In my work at some schools, I have heard teachers discussing tests and SACs and they seem to know what “show” means. As I was not involved in the tests, I was not in the discussion. But it is interesting that they seem to know what these terms mean.

They *seem* to ‘know’. But do they *really* ‘know’? Or are they just guessing like everyone else? What is their evidence for ‘knowing’? Are they joining the dots from the pontificating bullshit in the Examination Reports and getting a horse when it’s actually a donkey? Are they privy to VCAA marking discussions?

If you over-heard SRK discussing it in the hallway in the context of an assessment (and btw I 100% agree with what s/he says), you’d think s/he knows … But it’s a guess, a very informed guess but still just a guess.

How can anyone ‘know’ when VCAA doesn’t know and just makes it up each year to suit its own opaque purposes. This is one of numerous problems ultimately caused by a lack of clarity, detail and transparency from VCAA, leading to uncertainty, confusion and guess-work by teachers.

I said “seem” deliberately.

Seem to know, or seem to act like they know?

Hi,

An interesting discussion.

I like the example given here suggesting that ” verify” is weaker than “prove” when testing

The Goldbach Conjecture for a couple of million cases say

https://math.stackexchange.com/questions/2636274/verify-vs-prove

Steve R

Yes, but it’s still just convention. You can just as well say Goldbach has been proved for these cases.

You may be interested but also disappointed to see “show” in hsc examinations (just scientific calculator allowed though) . Clearly for the purpose of denying a student marks for further sections of a problem. Still micro marking, but I read that they allocate groups of markers to different questions, I guess it is improving accuracy and consistency.

Hi, Banacek. You mean current NWS exams?

I’ll step back somewhat. I think John Friend is correct and there’s nothing intrinsically wrong with “show” questions. And, the word “show” doesn’t bother me as long as it is absolutely clear what is permissible, and it’s not too egregious an abuse of language.

The problem is when the “show” is absolutely routine, essentially one-step. Then, the majority of students spend no time demonstrating mathematical knowledge or skill or ingenuity. It’s all just guessing what “show” means for that triviality, and then guessing what the examiner regards as evidence of the step. The result is that *plenty* of students score 0/1 or 1/2, when they know exactly what’s going on. That kind of nickel-and-diming is entirely routine in VCE.

To ask students to “show” the roots of a quadratic is batshit insane. To penalise a student for “showing” this by plugging in is batshit insaner.

I think my “favourite” question along these lines was 9b from 2017 Maths Methods Exam 1, where not writing down the two fractions with the same denominator BEFORE grouping them together was enough to be denied the mark.

https://www.vcaa.vic.edu.au/assessment/vce-assessment/past-examinations/Pages/Mathematical-Methods.aspx

Yeah, classic muppets. Clearly it isn’t enough for a student to get the correct derivative (arguably worth 1 mark in itself), they also had to show how to get it over a common denominator. I’d have loved to be a fly on the wall at that Marx Brothers Meeting.

Let’s screw students over in part (b) but hope everyone cuts us some slack for our fuck-up in part (d).

SRK, your favourite example is perfect for demonstrating:

1) the problem with VCAA’s idiotic 1 mark ‘Show’ question, and

2) VCAA’s brobdingnagian hypocrisy and hubris (we’re allowed to fuck up but students better write down every trivial detail or we’ll punish them).

This is why the advice I give all my students is to treat ‘Show’ questions as (in this case)

1) “Find the derivative of …. and express your answer as a single term.”

and then

2) Pretend you have to write a solution such that even a clueless moron can understand where the answer comes from.

Jesus. That’s obscene, even before you consider the fact that the question is screwed up. I loathe these little Hitler fuckwits.

I’ll add a suitable blast to my post on the question.

Speaking a while ago to some “oldie” I realised that the questions in 60s? 70s? were close to one liners, say find the maximum area of … inscribed in … . Well, may be bit longer. Anyway, what does it say about the treatment os students these days. Clearly most of them are considered halfwits, need a problem cut into pieces, pushed in the right direction and even given some intermediate answers because they would fail in calculations.

Extended response questions are collections of

eclectic pieces leading to I have no idea where to and designed to check skills which are hard to connect in one question describing “real life problem” (as with sacs). It is a disaster and picking up the pieces will take at least a generatipn.

Thanks, Banacek. It is a hugely important point. I made this point, with an example, in my recent talk (at 35:45 and 46:20).

On the other hand, by breaking a question down into small pieces, the students are given more direction about what is expected (thereby improving the validity of the student’s score), and the assessors can allocate marks with greater precision (thereby improving the reliability of the student’s score).

http://www.mrao.cam.ac.uk/~steve/astrophysics/webpages/barometer_story.htm

Big Macs are made with great precision.

Re: More direction about what is expected.

A max/min problem (for example) where the dimensions of, say the box with the largest volume, is required surely does not require the level of direction about what’s expected that breaking it up into a series of ‘Show’ question parts gives it …?

See Maths Methods 2010 Exam 2 Q3. Are all those ‘Show’ parts (including the part worth 1 mark – and who knows how many lines of work must be shown to earn that 1 mark*) really required to give students direction on what is expected?

This highlights the stupidity of 1 mark ‘Show’ questions (also highlighted elsewhere).

* Hint: More than 3 lines of work.

Here is a question (worth 11 marks) from the 1974 Pure Maths Exam Section B:

A rectangular chicken-run is to be built on flat ground from a 16 metre length of chicken wire will will form three of the sides , the fourth side being part of a straight wooden fence. What is the largest area of ground which the chicken-run can cover? If no side can be longer than 6 metres, what is the largest area of ground which the chicken-run could then cover?

If this question was re-purposed for a contemporary Maths Methods exam, it would have at least five parts and at least two of those parts would be pissant ‘Show’ questions.

What all these parts and shows are *NOT* doing is testing a student’s ability to independently construct a mathematical solution consisting of a logical sequence of parts.

A genuine question I have is to what extent the current micro-marking insanity is an aspect of the dumbing down of education, and to what extent it is an aspect of the dumbing down of grading. I wonder if there is a lack of trust in the ability of graders to grade a 10-mark question. In which case the question is, since 10-point questions use to be commonplace, where did the trust go?

I think it is perhaps also related to VCAA’s concept of ‘fairness’? From around 2010, they started publishing an article each year in The Age like this one: https://www.theage.com.au/education/complicated-yes-but-a-fair-system-for-all-20101212-18tvx.html

Thanks, s-t. It’s a shame VCAA don’t give a stuff about “informative” or “motivating to learn”.

I don’t know the answer to Marty’s question, but I doubt it’s about fairness. If English, History, etc. can apparently fairly grade a student response worth 20 marks, then I’m not sure why Mathematics can’t.

SRK, I think the argument may still be about fairness. You’ve just pointed out that their argument is stupid.

According to Bourdieu, it is important that educational assessment is perceived to be fair, in order to legitimize the reproduction of social dominance.

So not just fairness, but the perception of fairness is important.

Thank God we have Bourdieu’s genius to promote the bleeding obvious. But, once again: McDonald’s. Everybody getting the exact same hamburger is demonstrably fair. And, everybody gets shit.

Can you guys after all serious business give me some advice for this situation, middle school pythagoras and trig test for a not very strong group of students. Questions to be different from routine ones provided with the textbook subscription. I try “Verfy that the triangle with sides (here: some triple, different from 3 4 5) is right, then find all its angles”. After reviewing, it comes back: “Verify by drawing that a triangle with sides…”

How do you respond if it comes from: a. HoD b. A teacher with more years at the school than me but equal in responsibilities in math department

c. fresh from uni, in 20s. Regards.

Hi HB. I’ll set up a separate MitPY.

Hi, HB. I’ve posted a new MitPY on your question here. I did a minor clean-up of your language.

Just for anybody’s interests, I post it here – a “nearly official explanation” to “what is the difference between proof that and show that”:

Go to 28:17 directly and listen to what the gentleman said there.

Wow! Un-be-lievable (almost). And straight from the mouth of a Brobdingnagian-name.

What he said doesn’t surprise me – it’s consistent with the homeopathic dilution of VCE mathematics. I wonder who decided this (or maybe it was a shared responsibility and no-one made the decision).

Sheesh. So is the claim then that plugging in suggested roots and getting 0 doesn’t “prove” that they are indeed roots? Is there really a lunatic who believes that?

I suppose it follows from the claim that the standard method used to solve all those ‘cooked’ polynomials of degree three and higher in Methods and Specialist is invalid. Getting 0 during the trial-and-error stage of the method is no proof that you have a root … So to solve a cubic equation VCAA requires that a student must first solve the cubic equation.

This is what happens when fruit-loops with egos the size of a planet make stupid decisions and double down on those decisions.

It is this sort of shit that should (but probably won’t) get fixed during the ‘Expert’ Review on mathematics.

Just to make things interesting, you should all take a look at the 2007 Specialist Maths Exam 1 Q2. The Examiners Report says that substitution acceptable …. So there’s batshit insanity AND batshit inconsistency. What hope does anyone have of knowing what to do when VCAA just makes it up each year to the point where it contradicts itself in Examination Reports?

That’s pretty funny. (Well, it’s funny if you’re not teaching or studying VCE maths.)

Notably on that question, it’s not just that substitution was considered acceptable: substitution was the only practical method available. As to the reason for VCAA’s inconsistency, I can think of three possible reasons, none of them sensible. First, since 2007 there may have been a tightening up (and screwing up) of usage of “show that”. Secondly, VCAA may be distinguishing between checking one root and determining all roots. Thirdly, VCAA may be distinguishing between quadratics and symmetric quartics, (where the quadratic formula etc can be used directly), and cooked up cubics (where guessing/checking is the only available option). Whatever the dumb reason, the current denial of substitution as a valid method is anti-mathematical idiocy.