So, much crap, so little time.

OK, after a long period of dealing with other stuff (shovelled on by the evil Mathologer), we’re back. There’s a big backlog, and in particular we’re working hard to find an ounce of sense in Gonski, Version N. But, first, there’s a competition to finalise, and an associated educational authority to whack.

It appears that no one pays any attention to Western Australian maths education. This, as we’ll see, is a good thing. (Alternatively, no one gave a stuff about the prize, in which case, fair enough.) So, congratulations to Number 8, who wins by default. We’ll be in touch.

A reminder, the competition was to point out the nonsense in Part 1 and Part 2 of the 2017 West Australian Mathematics Applications Exam. As with our previous challenge, this competition was inspired by one specifically awful question. The particular Applications question, however, should not distract from the Exam’s very general clunkiness. The entire Exam is amateurish, as one rabble rouser expressed it, plagued by clumsy mathematics and ambiguous phrasing.

The heavy lifting in the critique below is due to the semi-anonymous Charlie. So, a very big thanks to Charlie, specifically for his detailed remarks on the Exam, and more generally for not being willing to accept that a third rate exam is simply par for WA’s course. (Hello, Victorians? Anyone there? Hello?)

We’ll get to the singularly awful question, and the singularly awful formal response, below. First, however, we’ll provide a sample of some of the examiners’ lesser crimes. None of these other crimes are hanging offences, though some slapping wouldn’t go astray, and a couple questions probably warrant a whipping. We won’t go into much detail; clarification can be gained by referring to the Exam papers. We also don’t address the Exam as a whole in terms of the adequacy of its coverage of the Applications curriculum, though there are apparently significant issues in this regard.

**Question 1**, the phrasing is confusing in parts, as was noted by Number 8. It would have been worthwhile for the examiners to explicitly state that the first term *T _{n}* corresponds to

*n*= 1. Also, when asking for the first term ( i.e. the first

*T*) less than 500, it would have helped to have specifically asked for the corresponding index

_{n}*n*(which is naturally obtained as a first step), and then for

*T*.

_{n}**Question 2(b)(ii)**, it is a little slack to claim that “an allocation of delivery drivers cannot me made yet”.

**Question 5** deals with a survey, a table of answers to a yes-or-no question. It grates to have the responses to the *question* recorded as “agree” or “disagree”. In part (b), students are asked to identify the explanatory variable; the answer, however, depends upon what one is seeking to explain.

**Question 6(a) **is utterly ridiculous. The choice for the student is either to embark upon a laborious and calculator-free and who-gives-a-damn process of guess-and-check-and-cross-your-fingers, or to solve the travelling salesman problem.

**Question 8(b)** is clumsily and critically ambiguous, since it is not stated whether the payments are to be made at the beginning or the end of each quarter.

**Question 10** involves some pretty clunky modelling. In particular, starting with 400 bacteria in a dish is out by an order of magnitude, or six.

**Question 11(d)** is worded appallingly. We are told that one of two projects will require an extra three hours to compete. Then we have to *choose* which project “for the completion time to be at a minimum”. Yes, one can make sense of the question, but it requires a monster of an effort.

**Question 14** is fundamentally ambiguous, in the same manner as Question 8(b); it is not indicated whether the repayments are to be made at the beginning or end of each period.

That was good fun, especially the slapping. But now it’s time for the main event:

# QUESTION 3

Question 3(a) concerns a planar graph with five faces and five vertices, *A*, *B*, *C*, *D* and *E*:

- Determine the number of edges of the graph.
- Draw the planar graph.
- Determine a Hamiltonian cycle for the graph
- Determine whether the graph is Eulerian, semi-Eulerian or neither.

What is wrong with this question? As evinced by the graphs pictured above, pretty much everything.

As pointed out by Number 8, Part (i) can only be answered (by Euler’s formula) if the graph is assumed to be connected. In Part (ii), it is weird and it turns out to be seriously misleading to refer to “the” planar graph. Next, the Hamiltonian cycle requested in Part (iii) is only guaranteed to exist if the graph is assumed to be both connected and simple. Finally, in Part (iv) *any* answer is possible, and the answer is not uniquely determined even if we restrict to simple connected graphs.

It is evident that the entire question is a mess. Most of the question, though not Part (iv), is rescued by assuming that any graph should be connected and simple. There is also no reason, however, why students should feel free or obliged to make that assumption. Moreover, any such reading of 3(a) would implicitly conflict with 3(b), which explicitly refers to a “simple connected graph” three times.

So, how has WA’s Schools Curriculum and Standards Authority subsequently addressed their mess? This is where things get ridiculous, and seriously annoying. The only publicly available document discussing the Exam is the summary report, which is an accomplished exercise in saying nothing. Specifically, this report makes no mention of the many issues with the Exam. More generally, the summary report says little of substance or of interest to anyone, amounting to little more than admin box-ticking.

The first document that addresses Question 3 in detail is the non-public graders’ Marking Key. The Key begins with the declaration that it is “an *explicit* statement about [sic] what the examining panel expect of candidates when they respond to particular examination items.” [emphasis added].

What, then, are the explicit expectations in the Marking Key for Question 3(a)? In Part (i) Euler’s formula is applied without comment. For Part (ii) a sample graph is drawn, which happens to be simple, connected and semi-Eulerian; no indication is given that other, fundamentally different graphs are also possible. For Part (iii), a Hamiltonian cycle is indicated for the sample graph, with no indication that non-Hamiltonian graphs are also possible. In Part (iv), it is declared that “the” graph is semi-Eulerian, with no indication that the graph may non-Eulerian (even if simple and connected) or Eulerian.

In summary, the Marking Key makes not a single mention of graphs being simple or connected, nor what can happen if they are not. If the writers of the Key were properly aware of these issues they have given no such indication. The Key merely confirms and compounds the errors in the Exam.

Question 3 is also addressed, absurdly, in the non-public Examination Report. The Report notes that Question 3(a) failed to explicitly state “the” graph was assumed to be connected, but that “candidates made this assumption [but not the assumption of simplicity?]; particularly as they were required to determine a Hamiltonian cycle for the graph in part (iii)”. That’s it.

Well, yes, it’s obviously the students’ responsibility to look ahead at later parts of a question to determine what they should assume in earlier parts. Moreover, if they do so, they may, unlike the examiners, make proper and sufficient assumptions. Moreover, they may observe that no such assumptions are sufficient for the final part of the question.

Of course what almost certainly happened is that the students constructed the simplest graph they could, which in the vast majority of cases would have been simple and connected and Hamiltonian. But we simply cannot tell how many students were puzzled, or for how long, or whether they had to start from scratch after drawing a “wrong” graph.

In any case, the presumed fact that most (but not all) students were unaffected does not alter the other facts: that the examiners bollocksed the question; that they then bollocksed the Marking Key; that they then bollocksed the explanation of both. And, that SCSA‘s disingenuous and incompetent ass-covering is conveniently hidden from public view.

The SCSA is not the most dishonest or inept educational authority in Australia, and their Applications Exam is not the worst of 2017. But one has to hand it to them, they’ve given it the old college try.