Further Mathematics is not our thing, but it seemed possible that some frequent visitors may want to discuss today’s exam. (We haven’t seen the exam, and don’t plan to seek it out unless something interesting is flagged.)

Have fun.

**UDPATE (31/10/22): **Exam 2 discussion is here.

The lack of a single comment (the deafening silence) on this post is about as good a proof as any that Further Maths is not mathematics at all.

The astounding size of the Further Maths cohort year after year is testament to the fact that the populace is getting dumber by the day, and it’s so painful to witness.

Hi, marty (from the official marty).

I don’t know. I agree entirely with the conclusion, but I don’t know that the silence on this post is evidence for the conclusion. The blog appears to be read by a decent number of people, but I don’t much of who, or why.

Marty (and Marty) – with Methods and Specialist next week, I’m prioritising answering student emails over working through a pile of multiple-choice-crap that, so far, I have found pretty vanilla, mostly button-pushing, the occasional counting of some dots or squares.

If and when I find anything worth raising… I will do so.

Paper 2 on Monday is more fertile ground for crap (pun intended).

Thanks, RF. I’m not at all fussed at the lack of FM posts. Previously, I ignored FM exams and then later issues arose. I figured this time to have the posts up and ready for whatever eventuates.

Maybe few of your correspondents (including me) do not teach Further Mathematics and hence are disinterested (as well as uninterested).

I taught it this year, a copy turned up on my desk, but didn’t look at it. Who wants to look at a Further Maths exam on a Friday afternoon?

I’ve heard that the very first question was dodgy – apparently students had to choose whether a distribution was best described as either negatively skewed or approximately symmetric (amongst other clearly incorrect options), but to the eye it was a borderline case (and students aren’t expected to calculate skew).

I’ll probably take a look tomorrow, since I won’t be teaching.

Thanks, SRK. Let us know if it’s worth a look.

That question is baffling. You can assume that the graph is representative of the precision of the data, and make a calculation (really not valid, and not something that Further students are taught anything about)… or you can do what’s expected, and make an arbitrary choice based on the picture. Which is an utterly absurd way to assess people.

Although, option C, I know some people took the rigorous approach of a true mathematician and put numbers into a spreadsheet in their CAS to generate a box plot. But I believe VCAA expect the Q to be done by visual inspection.

The distribution is skewed, the question is screwed and we can all have a happy November…

Seriously though… for a Q1 it is not what I would call “nice”.

Although, the outlier is moderately obvious in this instance (by VCAA standards)

Thanks, SRK, and everyone. I’ve added both this question and the nominal rates question to the error list.

Turns out I’m actually so uninterested it didn’t occur to me to even seek out a copy. I should probably make an effort next week…

As an aside, it appears VCAA amended the 2017 Exam 2 report a few days ago (24/10/22).

Apparently VCAA have (five YEARS after the fact) admitted to the mistake they made on Matrices Q1cii.

” Through an exhaustive process, the error in the introduction was found to have been an issue for only a very small number of students.

The overwhelming majority answered the question in the manner intended without a problem.”

That is very funny. What is even funnier is, and notwithstanding VCAA’s huffy defense, the question is so bad it is still technically wrong. There are infinitely many different matrices L that will work. I talk about the question here and here, and will update those posts.

Thanks, DM. Are you just guessing that 1(c)(ii) was amended on 24/10, or do you know that the error was still unacknowledged before then?

This question from the Further exam really is awful by VCAA, with such ambiguity with definitions that itll be easy for VCAA to spin it their way. Take Question 21, where a theory based question is asked. That being:

Consider the following four statements regarding nominal and effective intrest rates as they apply to compound interest investments and loans

(1)An effective interest rate is the same as a nominal interest rate if intrests compounds annually

(2)Effective intrest rates increase as the number of compounding periods per year increases

(3)A nominal intrest rate of 12 percent per annum is equivalent to a nominal intrest rate of 1 percent per month

(4)An effective intrest rate can be lower than a nominal interest rate.

How many of these statements are true?

Well if we consider adding intrest once every few years (ie once every 5 years) is it technically possible to have an effective intrest rate that is lower than the nominal?. According to the CAS, if we sub in an intrest of 12 percent per annum into reff calculator, and get the value for effective intrest is less than 12 percent per annum meaning that statment 4 can be true in a theoretical sense.

Ugh! I’ll let Further teachers comment, but that doesn’t look great.

Agreed Raschool, just looked at the exam this morning, and that is a poor question.

Far less pertinently for Further Maths (but perhaps not, since some Specialist students also do it), but I also wondered if we could consider negative nominal interest rates, in which case I *think* statement 2 is not always true (even if we ignore non-real effective interest rates).

Ok, thank you both. I don’t think it’s worth a post, but I’ll add it to the Further error list once the exams somehow float onto my desk, and the dust settles.

The idea of also considering negative interest rates is very funny, and not crazy. I assume it’s fair enough to not consider this as part of the question, but I don’t know. In a purportedly real-world subject like Further, would it be a totally unusual idea for a teacher to raise?

I’d say perhaps not for most students at least a few students considered the effective interest being lower than the nominal interest rate to be true for when the compounding periods is greater than one year, however haven’t heard any students or teachers consider the possibility of a negative nominal interest rate. Considering the exam was written like how a typical methods exam was written its only fair for VCAA to accept theoretically possible answer (although i doubt they will).

Thanks, Raschool. This is not my fight, but if you (and/or SRK) think there are two properly legitimate answers to a question, and you think it worth the effort, I would be raising it with VCAA now and strongly, while they are grading. VCAA may fob people off, but I know of at least one occasion (involving cheese) when (after some effort) they did not.

As someone who last taught Further in 2004… (forgive my lack of up-to-date knowledge, especially CAS):

Statement 1 is true.

Statement 2 is true if you assume a positive interest rate which seems fair and reasonable in a FM exam.

Statement 3 is how students are taught to determine nominal monthly rates, so reasonable to say it is true.

If statements 1 and 2 are assumed to be true, a fair conclusion is that statement 4 is false.

Hence 3 statements are true, answer D.

HOWEVER… if either of statements 1 or 2 are false (which is easily done by changing the assumptions made) then statement 4’s truth value is affected as well.

There is also the elephant in the room idea that n>1 where n is the number of compounding periods per year. Again, FM students in my experience are really only shown examples of n=1, n=4, n=12, n=52, n=365 so… perhaps a fair assumption?

On balance, I would expect a good FM student to answer D and not question their assumptions.

Then again… negative interest rates have been seen around the world in the last decade, so…

Thanks, RF and everyone. It doesn’t sound great, but VCAA setting a not great question is news like Peter Dutton being a thug is news.

For my purposes (apart from general interest), the issue is whether the not greatness in these questions is sufficient to categorise them as errors. In that regard, they would appear borderline.

I’ll ask some FM teachers next week, but my guess is that none of them (n=5) will see an issue with the question.

Whether or not that makes the question OK is very, very debatable!

A correlation of about 0.1.

Positive r value? That too is debatable.

Hi Raschool, and everyone. I’m just taking a look at Q21 now. I don’t know the conventions/definitions here. So I’m sure of commenters’ view, here is my understanding (and please correct me if I garbled something):

*) The first statement is true.

*) The second statement is true as long as we stick to positive rates, which commenters (with the possible exception of SRK) consider a reasonable assumption for Further.

*) The third statement commenters accept as true (but see below).

*) The fourth statement is false if we allow the nominal rate to for a period greater than a year (or if we allow negative rates), and commenters regard this as actively and badly ambiguous for Further students.

I have trouble understanding the third statement, but I think that may be me straining too hard to make “nominal interest rate” more meaningful than it is.

I WAS just guessing that 1(c)(ii) was the subject of the amendment made on 24/10.

After reading your reply, though, I checked the wayback machine. The most recent snap shot of the further maths page was 11/10/22. That page indicates the 2017 FM examination report 2 had not been amended at that date.

Also, after reading the amended report a couple of things occur to me:

(1) How on earth do VCAA support the notion that “[t]he overwhelming majority answered the question in the manner intended without a problem” when 72% of candidates scored zero on the question?

(2) If VCAA carried out “an exhaustive process” to determine “the error in the introduction was found to have been an issue for only a very small number of students”, why was the error ony acknowledged FIVE YEARS later?

Incidentally, that question caused me significant consternation over several years. Until I read your blog, it simply never occured to me that VCAA was simply wrong without acknowledging thier error. So thank-you for keeping me (somewhat) sane!

DB, you’re most welcome. I write this blog to keep

mesane. If it also works for someone else, then great.Your further two points are both very good. Thank you.

On (1), yes the VCAA quote is completely insane. If 72% scored 0/1 then clearly the large majority didn’t “answer in the manner intended”. I will add this observation to the other posts.

Now, why did students screw up? I suspect that it was not because of the food-weeks screw up, but because the question is intrinsically nonsense: you just don’t think to code specific arithmetic (as opposed to algebraic set-up) in that manner. But in any case, 72% getting 0/1 means 72% got 0/1. Shouldn’t happen, not there.

2) I have no doubt that VCAA were aware of the error in 2017, within half an hour of the exam beginning. It is unclear why VCAA decided to acknowledge the error now. It is clear that they were not pleased to do so, and they never are. VCAA are under some pressure and scrutiny now, and there is a new curriculum manager. I don’t know what combination of positive or negative forces may have pushed them to finally acknowledge the bleeding obvious.

Re: If VCAA carried out “an exhaustive process” to determine “the error in the introduction was found to have been an issue for only a very small number of students”, why was the error ony [sic] acknowledged FIVE YEARS later?

Answer: I suppose because it was a exhaustive process.

Re: ” How on earth do VCAA support the notion that “[t]he overwhelming majority answered the question in the manner intended without a problem” when 72% of candidates scored zero on the question?”

Excellent question. Particularly since the Report says “ were correct but students gave a 3 × 1 matrix” (Friend emphasis).

Because I would have thought that, to stretch a point, the intended manner would have been to give a 1 × 3 matrix. We can only wonder what the “intended manner” was. Or maybe VCAA thinks that 28% is an overwhelming majority …?

Or maybe the assertion “overwhelming majority answered the question in the manner intended without a problem” is simply trying to make students who have a problem (potentially 72% of students!!) guiltily feel like they’re a stupid minority that had a problem and keep it to themselves … (The old “blame the victim of the screw-up” trick).

May I be pedantic? The question is “How many of these statements are true?” It is a fact that one of the statements true.

Raymond Smulllyan strikes again. (But you have a point.)

So, because the answers did not say “at least one” or “exactly one”…

…yep, you have a point.

And I don’t think this is overly pedantic at all. Whether VCAA can (they won’t) defend themselves by saying the instructions are to “select the best answer” is another matter.

If 3 statements are true then B, C and D are all “correct” answers, but which is “the best”?

Yeah, even I think that is clutching at clouds (beats yelling at them).

The instruction to “select the best answer” is simply VCAA code for ‘We take no responsibility when we screw up and there’s either no correct answer or more than one correct answer’.

Exam 1 has really highlighted a issue that I have had with this subject for some time now. VCAA’s unwillingness to give questions that genuinely test Further students in tandem with the harsh marking (in no small part due to Methods and even Specialist students screwing over the upper end of the scale in pursuit of the ‘easy 50’!) has led to a system where the difference between a ‘brilliant’ and a ‘mediocre’ student is the ability to decipher the puzzle of what VCAA thinks the right answer should be for one or two nonsense questions.

And of course, CAS is a disaster for testing genuine mathematical skill. For example, I was looking at the last question on graphs and relations, which seemed to be a decent enough test of a students understanding of objective functions and linear programming. This was before I snapped back to reality and remembered you could just bang out Maximize and Minimize in Mathematica until it gave you the right answer, irrespective of understanding.

Thanks, Thomas. No question the Further And Methods group screws it up for the Further Only group, and VCAA just Pontius Pilates the whole issue. Can you indicate precisely the graphs and relations question you mentioned? Does handheld CAS provide a similar nuclear weapon solution?

I’ve attached the question.* Apologies for the poor quality, I’ve only been able to get my hands on the one copy.

As for the query about CAS, I’ve only worked with students who use Mathematica, so I’m not aware of a similar capability.

* I wasn’t aware how images worked on this site, looks like its been compressed beyond reading. In any case, students were presented with an objective function of the form Z = ax + by, where x and y are integers, and then asked to determine which one of a set of statements on a/b ratios was true.

Thanks very much, Thomas. That’ll do for now.