Whatever small good there is in me, it came from her.

## WitCH 44: Estimated Worth

This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, *Estimating the area under a graph*. (It is followed by 21B, *Finding the exact area: the definite integral*.)

We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.

## MAV’s Trials and Tribulations

Yeah, it’s the same joke, but it’s not our fault: if people keep screwing up trials, we’ll keep making “trial” jokes. In this case the trial is MAV‘s Trial Exam 1 for Mathematical Methods. The exam is, indeed, a trial.

Regular readers of this blog will be aware that we’re not exactly a fan of the MAV (and vice versa). The Association has, on occasion, been arrogant, inept, censorious, and demeaningly subservient to the VCAA. The MAV is also regularly extended red carpet invitations to VCAA committees and reviews, and they have somehow weaseled their way into being a member of AMSI. Acting thusly, and treated thusly, the MAV is a legitimate and important target. Nonetheless, we generally prefer to leave the MAV to their silly games and to focus upon the official screwer upperers. But, on occasion, someone throws some of MAV’s nonsense our way, and it is pretty much impossible to ignore; that is the situation here.

As we detail below, MAV’s Methods Trial Exam 1 is shoddy. Most of the questions are unimaginative, unmotivated and poorly written. The overwhelming emphasis is not on testing insight but, rather, on tedious computation towards a who-cares goal, with droning solutions to match. Still, we wouldn’t bother critiquing the exam, except for one question. This question simply must be slammed for the anti-mathematical crap that it is.

The final question, Question 10, of the trial exam concerns the function

on the domain . Part (a) asks students to find and its domain, and part (b) then asks,

* Find the coordinates of the point(s) of intersection of the graphs of and .*

Regular readers will know exactly the Hellhole to which this is heading. The solutions begin,

*Solve for ,*

which is suggested without a single accompanying comment, nor even a Magrittesque diagram. It is nonsense.

It was nonsense in 2010 when it appeared on the Methods exam and report, and it was nonsense again in 2011. It was nonsense in 2012 when we slammed it, and it was nonsense again when it reappeared in 2017 and we slammed it again. It is still nonsense, it will always be nonsense and, at this stage, the appearance of the nonsense is jaw-dropping and inexcusable.

It is simply not legitimate to swap the equation for , *unless* a specific argument is provided for the specific function. When valid, that can usually be done. Easily. We laid it all out, and if anybody in power gave a damn then this type of problem could be taught properly and tested properly. But, no.

What were the exam writers thinking? We can only see three possibilities:

a) The writers are too dumb or too ignorant to recognise the problem;

b) The writers recognise the problem but don’t give a damn;

c) The writers recognise the problem and give a damn, but presume that VCAA don’t give a damn.

We have no idea which it is, but we can see no fourth option. Whatever the reason, there is no longer any excuse for this crap. Even if one presumes or knows that VCAA will continue with the moronic, ritualistic testing of this type of problem, there is absolutely no excuse for not also including a clear and proper justification for the solution. None.

What of the rest of the MAV, what of the vetters and the reviewers? Did no one who checked the trial exam flag this nonsense? Or, were they simply overruled by others who were worse-informed but better-connected? What about the MAV Board? Is there anyone at all at the MAV who gives a damn?

*************************

Postscript: For the record, here, briefly, are other irritants from the exam:

Q2. There are infinitely many choices of integers and with equal to the indicated answer of .

Q3. This is not, or at least should not be, a Methods question. Integrals of the form with non-linear are not, or at least are not supposed to be, examinable.

Q4. The writers do not appear to know what “hence” means. There are, once again, infinitely many choices of and .

Q5. “Appropriate mathematical reasoning” is a pretty fancy title for the trivial application of a (stupid) definition. The choice of the subscripted is needlessly ugly and confusing. Part (c) is fundamentally independent of the boring nitpicking of parts (a) and (b). The writers still don’t appear to know what “hence” means.

Q6. An ugly question, guided by a poorly drawn graph. It is ridiculous to ask for “a rule” in part (a), since one can more directly ask for the coefficients , and .

Q7. A tedious question, which tests very little other than arithmetic. There are, once again, infinitely many forms of the answer.

Q8. The endpoints of the domain for are needlessly and confusingly excluded. The sole purpose of the question is to provide a painful, Magrittesque method of solving , which can be solved simply and directly.

Q9. A tedious question with little purpose. The factorisation of the cubic can easily be done without resorting to fractions.

Q10. Above. The waste of a precious opportunity to present and to teach mathematical thought.

**UPDATE (28/09/20)**

John (no) Friend has located an excellent paper by two Singaporean maths ed guys, Ng Wee Leng and Ho Foo Him. Their paper investigates (and justifies) various aspects of solving .

## Bernoulli Trials and Tribulations

This one feels relatively minor to us. It is, however, a clear own goal from the VCAA, and it is one that has annoyed many Mathematical Methods teachers. So, as a public service, we’re offering a place for teachers to bitch about it.*

One of the standard topics in Methods is the *binomial distribution*: the probabilities you get when repeatedly performing a hit-or-miss trial. Binomial probability was once a valuable and elegant VCE topic, before it was destroyed by CAS. That, however, is a story is for another time; here, we have smaller fish to fry.

The hits-or-misses of a Binomial distribution are sometimes called *Bernoulli trials*, and this is how they are referred to in VCE. That is just jargon, and it doesn’t strike us as particularly useful jargon, but it’s ok.** There is also what is referred to as the *Bernoulli distribution,* where the hit-or-miss is performed exactly once. That is, the Bernoulli distribution is just the n = 1 case of the binomial distribution. Again, just jargon, and close to useless jargon, but still sort of ok. Except it’s not ok.

Neither the VCE study design nor, we’re guessing, any of the VCE textbooks, makes any reference to the Bernoulli distribution. Which is why the special, Plague Year formula sheet listing the Bernoulli distribution has caused such confusion and annoyance:

Now, to be fair, the VCAA were trying to be helpful. It’s a crazy year, with big adjustments on the run, and the formula sheet*** was heavily adapted for the pruned syllabus. But still, why would one think to *add* a distribution, even a gratuitous one? What the Hell were they thinking?

Does it really matter? Well, yes. If “Bernoulli distribution” is a thing, then students must be prepared for that thing to appear in exam questions; they must be familiar with that jargon. But then, a few weeks after the Plague Year formula sheet appeared, schools were alerted and VCAA’s Plague Year FAQ sheet**** was updated:

This very wordy weaseling is VCAA-speak for “We stuffed up but, in line with long-standing VCAA policy, we refuse to acknowledge we stuffed up”. The story of the big-name teachers who failed to have this issue addressed, and of the little-name teacher who succeeded, is also very interesting. But, it is not our story to tell.

*) We extend our standard apology to all precious statisticians for our language.

**) Not close to ok is the studied and foot-shooting refusal of the VCAA and textbooks to use the standard and very useful notation q = 1 – p.

***) Why on Earth do the exams *have *a formula sheet?

****) The most frequently asked question is, “Why do you guys keep stuffing up?”, but VCAA haven’t gotten around to answering that one yet.

## One FEL Swoop: The Further Error List

This is the home for Further Mathematics exam errors. The guidelines are given on the Methods error post, and there is also a Specialist error post.

***********************

**2019, Exam 1 (Here, and report here)**

**QB6 (added 21/09/20)** The solution requires that a Markov process is involved, although this is not stated, either in the question or in the report.

**2018 NHT, Exam 1 (Here, and report here)**

**MCQ4 (added 23/09/20)** The question provides a histogram for a continuous distribution (bird beak sizes), and asks for the “closest” of five listed values to the interquartile range. As the examination report almost acknowledges (presumably in time for the grading), this cannot be determined from the histogram; three of the listed values may be closest, depending upon the precise distribution. The report suggests one of these values as the “best” estimate, but does not rely upon this suggestion. See the comments below.

**2017 Exam 2 (Here, and report here)**

**Q1(c)(ii) (added 13/11/20) – discussed here****.** The question is fundamentally nonsense, since there are infinitely many 1 x 3 matrices L that will solve the equation. As well, the 3 x 1 matrix given in the question does not represent the total value of the three products as indicated in Q(c)(i). The examination does not acknowledge either error, but does add irony to the error by whining about students incorrectly answering with a 3 x 1 matrix.

**2017 Exam 1 (Here, and report here)**

**MCQ11 (added 13/11/20) – discussed here.** None of the available answers is correct, since seasonal indices can be negative. The examination report does not acknowledge the error.

**2015 Exam 1 (Here, and report here)**

**MCQ9 Module 2 (added 30/09/20)** The question refers to cutting a wedge of cheese to make a “similar” wedge of cheese, but the new wedge is not (mathematically) similar. The exam report states that the word “similar” was intended “in its everyday sense” but noted the confusion, albeit in a weasely, “who woulda thought?” manner. A second answer was marked correct, although only after a fight over the issue.

## Hard SEL: The Specialist Error List

This is the home for Specialist Mathematics exam errors. The guidelines are given on the Methods error post, and there is also a Further error post.

******************************

**2017 EXAM 1 (Here, and report here)**

**Q10(c) (added 13/11/20) – discussed here. **The intended solution requires computing a doubly improper integral, which is beyond the scope of the subject. The examination report ducks the issue, by providing only an answer, with no accompanying solution.

**2017 NHT EXAM 2 (Here, and report here)**

**Q3(b) (added 13/11/20) – discussed here. **The wording of the question is fundamentally flawed, since the “maximum possible proportion” of the function does not exist here, and in any case need not be equal to the “limiting value” of the function. The examination “report” contains nothing but the intended answer.

**2006 EXAM 2 (Here, and report here)**

**MCQ20 (added 24/09/20) **The notation refers to the forces in the question being asked, and seemingly also in the diagram for the question, but to the magnitudes of these forces in the suggested answers. The examination report doesn’t acknowledge the error.

## MELting Pot: The Methods Error List

We’re not really ready to embark upon this post, but it seems best to get it underway ASAP, and have commenters begin making suggestions.

It seems worthwhile to have *all* the Mathematical Methods exam errors collected in one place: this is to be the place.*

Our plan is to update this post as commenters point out the exam errors, and so slowly (or quickly) we will compile a comprehensive list.

To be as clear as possible, by “error”, we mean a definite mistake, something more directly wrong than pointlessness or poor wording or stupid modelling. The mistake can be intrinsic to the question, or in the solution as indicated in the examination report; examples of the latter could include an insufficient or incomplete solution, or a solution that goes beyond the curriculum. Minor errors are still errors and will be listed.

With each error, we shall also indicate whether the error is (in our opinion) **major** or minor, and we’ll indicate whether the examination report acknowledges the error, updating as appropriate. Of course there will be judgment calls, and we’re the boss. But, we’ll happily argue the tosses in the comments.

Get to work!

*) Yes, there are also homes for Specialist Mathematics and Further Mathematics errors.

******************************

**2017 EXAM 1 (Here, and report here) **

**Q9(c), Section B (added 13/11/20) – discussed here. **The question contains a fundamentally misleading diagram, and the solution involves the derivative of a function at the endpoint of a closed interval, which is beyond the scope of the course. The examination report is silent on on both issues.

**2016 EXAM 2 (Here, and report here)**

**Q3(h), Section B (added 06/10/20) – discussed here. **This is the error that convinced us to start this blog. The question concerns a “probability density function”, but with integral unequal to 1. As a consequence, the requested “mean” (part (i)) and “median” (part (ii)) make no definite sense.

There are three natural approaches to defining the “median” for part (ii), leading to three different answers to the requested two decimal places. Initially, the examination report acknowledged the issue, while weasely avoiding direct admission of the fundamental screw-up; answers to the nearest integer were accepted. A subsequent amendment, made over two years later, made the report slightly more honest, although the term “screw-up” still does not appear.

As noted in the comment and update to this post, the “mean” in part (i) is most naturally defined in a manner different to that suggested in the examination report, leading to a different answer. The examination report still fails to acknowledge any issue with part (i).

**Q4(c), Section B (added 25/09/20) **The solution in the examination report sets up (but doesn’t use) the equation dy/dx = stuff = 0, instead of the correct d/dx(stuff) = 0.

**2016 EXAM 1 (Here, and report here)**

**Q5(b)(i) (added 24/09/20) **The solution in the examination report gives the incorrect expression in the working, rather than the correct .

**2015 EXAM 2 (Here, report here)**

**Q5(c) (added 13/11/20) – discussed here. **The method suggested in the examination report is fundamentally invalid.

**2014 EXAM 2 (Here, report here)**

**MCQ4 (added 21/09/20) – discussed here. **The described function need not satisfy any of the suggested conditions, as discussed here. The underlying issue is the notion of “inflection point”, which was (and is) undefined in the syllabus material. The examination report ignores the issue.

**2011 EXAM 2 (Here, and report here)**

**Q4, Section 2 (added 23/09/20) **The vertex of the parabola is incorrectly labelled (-1,0), instead of (0,-1). The error is not acknowledged in the examination report.

**2011 EXAM 1 (Here, and report here)**

**Q7(b) (added 23/09/20) **The question asks students to “find *p*“, where is the probability that a biased coin comes up heads, and where it turns out that . The question is fatally ambiguous, since there is no definitive answer to whether is possible for a “biased coin”.

The examination report answer includes both values of , while also noting *“The cancelling out of p was rarely supported; many students incorrectly *[sic]* assumed that p could not be 0.” *The implication, but not the certainty, is that although 0 was intended as a correct answer, students who left out or excluded 0 could receive full marks IF they explicitly “supported” this exclusion.

This is an archetypal example of the examiners stuffing up, refusing to acknowledge their stuff up, and refusing to attempt any proper repair of their stuff up. Entirely unprofessional and utterly disgraceful.

**2010 EXAM 2 (Here, report here)**

**MCQ17 (added 28/09/20) – discussed here. **Same as in the 2014 Exam 2, above: the described function need not satisfy any of the suggested conditions, as discussed here.

**2007 EXAM 2 (Here, report here and discussed here)**

**MCQ12 (added 26/09/20) **Same as in the 2014 Exam 2, above: the described function need not satisfy any of the suggested conditions, as discussed here.

## MitPY 9: Team Games

This MitPY is from commenter HollyBolly, who asked on the previous MitPY for some advice on diplomacy.*

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

*How do you respond if that review has come from: *

*A. The HoD;*

*B. A teacher with more years at the school than me but equal in responsibilities in the maths department;*

*C. A teacher fresh from uni, in their20s. *

*Regards.*

*) Yeah, yeah. We’ll stay right out of the discussion on this one.

## MitPY 8: Verification Code

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?## Shuffling NAPLAN’s Deckchairs

We’re late to this, but it’s gotta be done.

Some State education ministers, unhappy with NAPLAN, commissioned a review, which appeared a couple weeks ago. The Review considers many contentious aspects of NAPLAN, but we’ll focus upon “numeracy”, NAPLAN’s homeopathic proxy for mathematics. We’ll leave others to debate “literacy” and the writing tests, and the timing and reporting and so forth.

So, what might the Review entail for the Son-of-NAPLAN testing of mathematics? Bugger all.

Which was always going to happen. For all the endless public and pundit whining about NAPLAN, which is what prompted this latest Review, none of the criticism has been aimed at the two elephants: the Australian Curriculum, which underpins NAPLAN, is a meatless mass of gristle and fat; and “numeracy” is not mathematics, is not arithmetic, and is barely anything. The inevitable consequence is that NAPLAN amounts to the aimless testing of untestable fuzz. As Gertrude Stein would have put it, there is no *there* there to test.

This misdirection of the Review was locked in by the terms of reference. No mention is made of “mathematics” or “arithmetic”. The single reference in the Terms to “numeracy” is a deadpan call for “the most efficient and effective system for assessing key literacy and numeracy outcomes”, as if this were a clear and unproblematic and worthy goal. It is no surprise, therefore, that the Review gives almost no attention to arithmetic and mathematics, and the meaning(lessness) of numeracy, and indeed works actively to avoid it.

The Review includes a capsule summary of the Numeracy tests, a superficial comparison to PISA and TIMSS, and Australia’s relative performance over time on these tests (pp 34-42). There is no proper exposition, however, of the nature of the tests. There is nothing reflecting the hard fact that NAPLAN and PISA are pseudomathematical garbage. TIMSS, on the other hand, is decidedly not garbage, so what does the Review do with that? That is interesting.

In what could have been a beacon paragraph, the Review compares the Australian Curriculum with expectations on TIMSS:

*“… The Australian Curriculum emphasis on knowing and applying is similar to TIMSS but the Australian Curriculum does not appear to cover some of the complexity that is described in the TIMSS framework under reasoning. It seems likely, too, that a substantial number of TIMSS mathematics items are beyond Australian Curriculum expectations for achievement, especially at the Year 4 level.”*

In summary, the emphasis on “knowing and applying” mathematics in the Australian Curriculum is just like TIMSS, as long as you don’t really care how much students know, or how deeply they can apply it, or how successful you “expect” them to be at it. Yep, two peas in a pod.

What does the Review then do with this critical paragraph? Nothing. They just drone ahead. Here is the indication that their entire Review is doomed to idiot trivialities, but they can’t see it, or won’t admit it. They see the smoke, note the smoke, but it doesn’t occur to them, or they just can’t be bothered, or it wasn’t in their idiot Terms, to look for the damn gun.

Finally, what of the recommendations proposed by the Review? There are two that concern the testing of numeracy and/or mathematics. The first, Recommendation 2.2, is that authorities

*“Rename the numeracy test as mathematics …”*

Huh. And what would be the purpose of that? Well, supposedly it would “clarify that [the test] assesses the content and proficiency strands of the Australian Curriculum: Mathematics”. Except, of course, and as the Review itself acknowledges, the Numeracy test doesn’t do anything of the sort. And, even to the minimal extent that it does, it just points back to Elephant Number One, that the Australian Curriculum is not a properly sound basis for anything.

The isolated suggestion to rename a test is of course a distracting triviality. Alas, not all of the Review’s recommendations are so trivial. Recommendation 2.3 proposes a new test, for

*“… [the] assessment of critical and creative thinking in science, technology, engineering and mathematics (STEM) …”*

Ah, Yes. Let’s test whether ten-year-old Tommy is the new Einstein.

This is a monumentally stupid recommendation. Is Jenny the next Newton? Maybe. But can she manipulate numbers and expressions with sufficient speed and accuracy to hold, let alone mould, a substantial mathematical thought in her head? Just maybe you might want to test for that first? Is Carol the new Capote? Then perhaps first teach her the basics of grammar, first teach her how to construct a clear and correct sentence. *Then* you can think to tease out all the great works inside her. Is Fritz another Mozart? Gee, I dunno. How are his scales? And on and on.

This constant, idiot call for the teaching of and, worse, the testing of “higher order” thinking, this mindless genuflection to reasoning and creativity, is maddening. It ignores the stubborn fact that deeper thought and creativity in any discipline can only be built upon the craft, upon the basic knowledge and skills of that discipline. The Review’s call is even worse for that, since STEM isn’t a discipline, it’s just a foggy con job.

This Godzilla versus Mothra battle is never likely to end, nor likely to end well. On the one side are the numeracy nuts, who can’t see the value of skills independent of some ridiculous application. On the other side are the creativity clowns, who ludicrously denigrate “the basics”, and ludicrously paint NAPLAN as the basics they’re denigrating. Neither side exhibits any understanding of what the basics are, or their critical importance. Neither side has a clue. Which means, unless and until these two monsters somehow destroy each other, we’re all doomed.