One of the all-time great literary wallops, by one of the all-time great writers, is Mark Twain’s Fenimore Cooper’s Literary Offenses:
Cooper’s art has some defects. In one place in “Deerslayer,” and in the restricted space of two-thirds of a page, Cooper has scored 114 offenses against literary art out of a possible 115. It breaks the record.
There are nineteen rules governing literary art in the domain of romantic fiction — some say twenty-two. In “Deerslayer”, Cooper violated eighteen of them.
Twain lists eleven of these “rules”, which are on the basic principles of storytelling, and then continues:
In addition to these large rules, there are some little ones. These require that the author shall:
12. Say what he is proposing to say, not merely come near it.
13. Use the right word, not its second cousin.
14. Eschew surplusage.
15. Not omit necessary details.
16. Avoid slovenliness of form.
17. Use good grammar.
18. Employ a simple and straightforward style.
Fenimore Cooper’s Literary Offenses is hilarious, and anyone of good sense will put this post aside and go directly to Twain. But, for those remaining or returning, onto VCAA.
VCAA’s mathematics exams have some defects …
There are a number of compelling reasons for these defects: a thin and disconnected curriculum; a dearth of mathematical expertise; a domination by CAS; the ennoblement of an imagined real-world; an indifference to error; a small-minded culture of nitpicking and trivia. Separately and combined, these facets guarantee that VCE exams will be bad. But VCE exams are also bad because they are simply, gratuitously, badly written.
The principles for writing good exams differ to those for writing good historical fiction, and Twain’s larger literary rules do not automatically apply. They are not as irrelevant as one might imagine, and we plan to write soon upon VCAA’s larger literary offenses, but there are subtleties and it will take work. Twain’s little rules, however, apply directly to the writing of exams. This post is on VCAA’s continual, predictable violation of these little rules.
In the manner of Twain, we will focus upon one “work”: the 2022 Mathematical Methods Exam 1 (which, over a month after the exam was held, is still not publicly available). We will be close to comprehensive. We shall work through all but one question, commenting upon whether the sentences convey in a clear and straightforward manner what was intended to be conveyed.
The exam is now up, here.
Let . Find .
This is good. There are just two words, which is all that is required. But then,
Find and simplify the rule of where .
Human beings do not write this way.
In terms of mathematical content, 1(b) is the exact same type of question as 1(a) excepting for the extra (and not entirely clear or purposeful) instruction to “simplify”. But the wording is now bad. There is no point to wrapping the question in function notation. There is no point to introducing a domain which need never be considered and can anyway simply be implied to be all of R. There is no point to asking for “the rule of” f'(x), rather than simply asking for f'(x).
(On this last point, note that any attempted defense of “rule of” as being part of established VCE culture would be absurd, simply a justification of current madness by appealing to a tradition of madness.)
The wording is bad, and it is worse than bad. Even if one desires the function garb it is absurd to pile it all on at the beginning, requiring the somewhat busy student to first throw the garb aside in order to get to the only part that properly matters: the definition of the function. Much better would be to write something like,
Consider the function where . Find and simplify .
This could be and should be shorter. But at least, unlike the exam question, it can be read without straining.
This question is again dressed in function garb, again with a Who Cares domain, although at least the function g is clearly defined at the very beginning. But then we have,
Find the rule for an antiderivative of g(x).
Again with the needless “rule of”, but that is not the main problem. The main problem is that the question is not asking what was intended.
The question intended to ask for the general antiderivative of the function g. One way of asking for this is,
Find the general antiderivative of the function g.
What the exam question is asking, however, is different. The function x3 + π, for example, is an antiderivative of the function 3x2.
Does this garbling of the articles matter? Of course. As Twain commands, the writer should say what he is proposing to say, not merely come near it; as a simple matter of professionalism, major exams should not contain even minor errors. But it matters in practice as well as in principle. The students are, as we noted above, somewhat busy, and even a few seconds spent on deciphering incorrect or poor wording matter. These seconds add up. The minor flusters add up.
This question is better, although, in the same manner as 1(b), it suffers from the QFAIL format: Question First And Information Later. The bracketing style is also poor: the outer brackets in the first integrand are not required, but if employed would be better chosen to be square, to help distinguish them from the inner brackets; conversely, there is little point to the brackets in the second integrand being square. Either way, be consistent.
This question, on the number of solutions of a system of equations, is better, but there is sloppiness and inelegance. The question requires determining the variable k in the first equation and, atypical of VCAA, which tends to fuss such points beyond all reason, this variable is never introduced. Thus, although the role of k is reasonably understood, the question should naturally begin with “Let k ∈ R”.
After stating the system, there is the question:
Determine the value of k for which the system of equations above has an infinite number of solutions.
The preposition “above” is not required and “the system of equations” is clumsily repetitive. Simpler and better is,
Determine the value of k for which the system has an infinite number of solutions.
The writing in this question is not very bad; it is just not very good.
And so we come to one of the major literary offenses. We have already posted on this appalling, nasty question, and we won’t add anything here. The question exhibits numerous minor offenses, but these are difficult to disentangle from the intrinsic awfulness of the question. It is essentially impossible to word such a bad question well, although commenter SRK has given it a decent shot, and is much closer to the mark than the exam.
Solve for .
VCAA were concerned that students might solve the equation for 13?
Solve the equation .
Find the maximal domain of , where .
There is no need to specify the logarithm, and the sentence can be shortened to negate the QFAIL:
Find the maximal domain of the function .
A sustained major literary offense, and numerous minor literary offenses; here, we document the latter.
The graph of , where is shown below.
The collar of my dog, who is a corgi, is outside.
Consider the function where . The graph of is shown below.
On the axes above, draw the graph of , where is the reflection of f(x) in the horizontal axis.
Over there, please feed Tom, who is Nancy’s brother.
Let the function be the reflection of across the -axis. Draw the graph of on the axes above.
Find all values of such that and .
Why introduce k?
Solve the equation .
Or, if one wants to remind the students of the domain of f, which is reasonable, do it conventionally and cleanly:
Solve the equation for .
Let , where has the same rule as with a different domain.
The graph of is translated units in the positive horizontal direction and units in the positive vertical direction so that it is mapped onto the graph of , where .
Who writes this and thinks “OK”? Who vets this and thinks “OK”? Who are these people?
It’s not easy, but here’s a go at writing this nonsense in a clear and monotonic manner:
Let , the same as , but with a different domain, D.
With , the graph of is then translated units in the -direction and units in the -direction. Suppose the transformed graph of then coincides with the graph of .
Find the value for .
It’s “of”, not “for”, and it is unnecessary:
Find the smallest positive value for .
VCAA has already declared that a is positive, although admittedly not directly, and it took a while. Moreover, a is a single unknown number, not a set of numbers:
Determine the smallest possible value of .
Hence, or otherwise, state the domain, , of .
It’s difficult to not give those commas a proper whack, but there is a much greater problem: as with Q1(a), the question is not asking what was intended. Finding the smallest possible value of a in (c)(ii) is not the same as declaring a to have that value.
Given that has the value found in (ii), determine the domain, , of .
Another major literary fail, and appalling question. As with Q4, it is difficult to isolate the minor offenses from the general mess, but we’ll make a few comments and suggestions.
For Type A, the colours on the tiles are divided using the rule , where .
The colours on Type A tiles are separated by the function for some constant .
The corners of each tile have the coordinates (0,0), (20,0), (20,20), (0,20), as shown below.
All the tiles have those coordinates? They’re piled up on top of one another? And only now VCAA tells us where the tile is, after having already given us the separating function? And it’s small in comparison, but the coordinates ordering is weird: it’s capturing a region, not walking around a polygon.
Each tile can be considered to have the coordinates (0,0), (0,20), (20,0), (20,20), as pictured below.
Find the area of the front surface of each tile.
Sigh. VCAA has already spent an entire page talking only about the front surface of these absurd tiles, but OK.
Find the value of a so that a type A tile meets Condition 1.
Apart from “satisfies” being preferable to “meets”, the sentence is avoidably clumsy. It is clumsy because there is a parameter “a” and an article “a” and a type “A”. The parameter need not have been a, and the article “a” could have been avoided.
Type B tiles, an example of which is shown below, are divided using the rule .
The colours on Type B tiles are separated by the function , as shown below.
Show that a Type B tile meets Condition 1.
Show that Type B tiles satisfy Condition 1.
Determine the endpoints of and on each tile.
It’s “respective tile types”, not “each tile”, it’s entirely unnecessary and makes no proper sense:
Determine the endpoints of the functions and .
Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2.
The use of “Hence” is clumsy, and “continuous pattern” is not a mathematical term. (How does repetition of y = |x| make a more “continuous pattern” than repetition of y = x/|x|? Are the regions of colour not “continuous”?)
Hence, show that any arrangement of Type A and Type B tiles in a row produces a (ugh!) continuous pattern, as required by Condition 2.
A boring, pointless, idiotic question throughout, almost impossible to word accurately, and it has not been.
A better question, storywise, and wordwise, but far from good.
Part of the graph of y = f(x) is shown below.
Why “Part” of the graph? There is no need for this qualification.
The graph of the function f(x) is shown below.
The rule gives the area bounded by the graph of f, the horizontal axis and the line x = k.
The area bounded by the graph of f, the x-axis and the line x = k is given by the function .
State the value of .
Consider the average value of the function f over the interval , where .
Find the value of k that results in the maximum average value.
It’s a QFAIL, there is no point in including the range of x, and it is confusing to do so. Of course the range of k should be included, but the range should also be correct: as discussed here, k = 0 should have been excluded.
For , consider the average of the function f over the interval .
Find the value of k that maximises this average.
And we’re done. Thank God.
Some final comments. This very long post is just on the bad writing of the exam. The broader issue of the intrinsic badness of the exam would make for an essay. The even broader issue of the intrinsic badness of VCE mathematics would make for a tome. The bad writing should be the least of one’s concerns with VCE mathematics.
Nonetheless, the bad exam writing is a very serious problem, and it is the easiest problem for VCAA to address. Even on VCAA’s own terms, even given the appalling curriculum, even given the trivial and non-sensical questions, the writing can and should be much, much better than it is.
Simply for the writing, the 2022 Methods Exam 1 is not close to acceptable. No competent and attentive mathematician would have signed off on this exam. The conclusions are obvious.