This is a brief post, intended as a place for discussion of the 2021 NHT Specialist Mathematics exams, here and here. See also the accompanying 2021 NHT Methods exams post, here.
We’ve made a few remarks, below, on specific questions, and see this WitCH, but we haven’t gone carefully through the exams. We’ll update with further remarks if and when commenters raise substantive issues with the questions, or either exam as a whole.
The exam reports are out, here and here. We haven’t looked and don’t intend to look, except for Q10 on Exam 1. The suggested solution is, well, rooted. See here.
A few months ago, frequent commenter Red Five noted a pretty much forgotten book, The Saber-Tooth Curriculum. Written in 1939 by the mythical J. Abner Peddiwell – the creation of education professor H. R. W. Benjamin – the book is a series of drunken lectures on the nature of education during the paleolithic era. That education supposedly included lessons such as saber-tooth-tiger-scaring-with-fire, long after saber-tooth tigers had disappeared, and so on.
The book is crazily satirical, happily takes shots at everybody, and it holds up well. Maybe not well enough to bother hunting out – it is difficult to sustain such a parody for 100+ pages – but The Saber-Tooth Curriculum is clever and pretty funny. Surprisingly so since humour, particularly topical humour, tends to date quickly.
Below is our favourite passage from the book, concerned with the establishment of university courses for teachers, and the introduction of professors of paleolithic education.
The crude, naive work of the education professors was regarded with contempt by the subject-matter specialists. It was inevitable that a man who who had devoted a lifetime of productive scholarship or systematic speculation to such a problem as The Mystical Element in Sputtering Firebrands as Applied to Tiger-Whiskers or Variations in Thumb-Holds for Grabbing Fish Headed Outward from the Grabber at an Angle of Forty-Five Degrees Plus or Minus Three should be contemptuous of pseudo scholars who were merely trying to show students how to teach.
The academic contempt for pedagogy had a good effect on the education professors. Stung by justified references to their low cultural status, they resolved to make their discipline respectable. With a magnificent display of energy and self-denial, they achieved this goal. First, they organized their subject systematically, breaking it down into respectably small units, erecting barriers to keep professors conventionally isolated from ideas outside their restricted areas, and demanding specialization and more specialization in order to achieve the narrow knowledge and broad ignorance which the paleolithic university demanded of its most truly distinguished faculty members.
Second, they required all members of their group to engage in scientific research in education by counting and measuring quantitatively everything related to education which could be counted and measured. It was here that the professors of education showed the greatest courage and ingenuity. They confronted almost insuperable obstacles in the fact that education dealt with the changing of human minds, a most complex phenomenon. The task of measuring a learning situation involving an unknown number of factors continually modifying each other at unknown rates of speed and with unknown effects was a tremendous one, but the professors did not hesitate to attack it.
Finally, the professors of education worked for academic respectability by making their subject hard to learn. This, too, was a difficult task, but they succeeded admirably by imitating the procedures of their academic colleagues. They organized their subject logically. This necessarily resulted in their giving the abstract and philosophical courses in education first, delaying all practical work in the subject until the student was thoroughly familiar with the accustomed verbalizations of the craft and, thereby, immunized against infection from new ideas. They adopted the lecture method almost exclusively and labored with success to make it an even duller instrument of instruction than it was in the fields of ichthyology, equinology and defense engineering. They developed a special terminology for their lectures until they were as difficult to understand as any in the strictly cultural fields.
Thus the subject of education became respectable. It had as great a variety of specialists as any field. Some of its professors tried to cover the whole area of the psychology of learning, it is true, but most of them confined their efforts to some more manageable topic like the psychology of learning the preliminary water approach in fish-grabbing. Its research workers were so completely scientific that they could take a large error in the measurement of what they thought maybe was learning in a particular situation and refine it statistically until it seemed to be almost smaller and certainly more respectable than before. Its professors could lecture on modern activity methods of instruction with a scholarly dullness unequalled even by professors of equicephalic anatomy. Their cultured colleagues who had once treated them with contempt were now forced to regard them with suspicious but respectful envy. They had arrived academically.
Porter is also a sadistically destructive asshole. As Minister for Social Services, he brought in the Turnbull Government’s psychopathic Robodebt scheme. It was evil, and obviously evil, from the start, bringing needless misery to thousands of the most vulnerable and powerless, and prompting God knows how many suicides. And it took four years of the fucking obvious, four years of senate reviews and legal challenges to get rid of the fucking thing. Without an apology. What a cunt leprous schlong.*
No one knows now – except Porter – and we’re extremely unlikely to find out. What would a review, any review, accomplish? What would it tell us other than, perhaps, something questionable happened 30+ years ago, when the guy was a teenager, merely a trainee sadist. Which we already know.
Why bother? Why not focus on Porter’s much more recent and much more provable awfulness? We know what Christian Porter is, and his disgusting colleagues know what he is. It has been, and still is, up to Morrison and his fellow thugs whether Porter should be in charge of the country’s law. And it’s now up to them if they want a show trial or a cover up. But no one else needs it. No one with an ounce of humanity can believe that Porter should be responsible for a dog, let alone a country.
And what if Morrison and his thugs don’t want an inquiry? Then so be it. Porter can stay, as a dead, festering lizard hanging around the neck of Morrison’s revolting, thoroughly evil government. And if that’s not enough to hang the whole fucking lot of them, then we’re fucked anyway.
Leave the loathsome cunt leprous schlong alone.*
*) We do our best to keep sanctimonious twats happy.
The 2019 TIMSS results are just about to be released, and the question is should we care? The answer is “Hell yes”.
TIMSS is an international maths and science test, given at the end of year 4 and year 8 (in October in the Southern Hemisphere). Unlike PISA, which, as we have noted, is a Pisa crap, TIMSS tests mathematics. TIMSS has some wordy scenario problems, but TIMSS also tests straight arithmetic and algebra, in a manner that PISA smugly and idiotically rejects.
The best guide to what TIMSS is testing, and to what Australian students don’t know and can’t do, are the released 2011 test items and country-by-country results, here and here. We’ll leave it for now for others to explore and to comment. Later, we’ll update the post with sample items, and once the 2019 results have appeared.
The report is out, with the ACER summary here, and the full report can be downloaded from here. The suggestion is that Australia’s year 8 (but not year 4) maths results have improved significantly from the (appalling) results of 2015 and earlier. If so, that is good, and very surprising.
For now, we’ll take the results at face value. We’ll update if (an attempt at) reading the report sheds any light.
FURTHER UPDATE (08/12/20)
OK, it starts to become clear. Table 9.5 on page 19 of the Australian Highlights indicates that year 8 maths in NSW improved dramatically from 2015, while the rest of the country stood still. This is consistent with our view of NSW as an educational Switzerland, to which everyone should flee. We’re not sure why NSW improved, and there’s plenty to try to figure out, but the mystery of “Australia’s” dramatic improvement in year 8 maths appears to be solved.
OK, no one is biting on the questions, so we’ll add a couple teasers. Here are the first two released mathematics questions from the 2011 year 8 TIMSS test:
1. Ann and Jenny divide 560 zeds between them. If Jenny gets 3/8 of the money, how many zeds will Ann get?
(The second question is multiple choice, with options 0.043, 0.1043, 0.403 and 0.43.)
To see the percentage of finishing year 8 students from each country who got these questions correct, you’ll have to go the document (pp 1-3).
What follows is the article Mathematica and the Potential Gaming of VCE, by Sai kumar Murali krishnan, which has just appeared in Vinculum and which we have written about here. Sai’s article is reproduced here with Sai’s permission. Sai can be contacted by email here.
Last year I completed VCE, including Mathematical Methods (CBE) and Specialist Mathematics. At my school these subjects employed the computer system Mathematica in place of handheld CAS calculators. The CBE (Computer-Based Examination) version of Methods also entailed the direct submission of SACs and the second (tech-active) exam on the Mathematica platform.1
Mathematica is extraordinarily powerful and, as it happens, I consider myself a decent programmer. During VCE, I entertained myself by creating custom functions to automate tedious computations, which I then shared with my fellow students. We were able to store these functions in a paclet (package), ready for use on the SACs and the exam. While handheld CAS calculators can also store (less complex) custom-made functions, Mathematica’s vast in-built library and ease of use moves it into a different class. Mathematica enables the creation of exam-ready functions to perform any computation a student might require.
I have witnessed, and experienced, many problems with the implementation of Mathematica, but in this article I will focus upon the two most glaring and most important issues. First and foremost, Mathematica is so powerful that it can trivialise the testing of the mathematics for which it is purported to be a tool. It enables any student who can program in Mathematica or, more perversely, who has a friend, teacher or tutor who can program in Mathematica, to perform well in VCE mathematics. Secondly, and as an inevitable consequence of this trivialisation, the current partial implementation of Mathematica could create a grossly unfair competition, an unfairness enhanced in Methods CBE by effectively permitting Mathematica code to be submitted as an answer. The students equipped with handheld CAS calculators are the victims. Armed with toys sporting 70s Nintendo displays, they are being outgunned by students deploying full-screen guided missiles.
In this article I will illustrate how Mathematica can trivialise exam questions in Mathematical Methods. In Part 2, I provide an example of the use of Mathematica’s in-built functions. In Part 3, I consider the application of custom-built functions. In Part 4, I summarise, and I indicate why I believe the problems with the implementation of Mathematica are only likely to worsen.
We begin by looking at Question 5, Section B of 2019 Exam 2, which concerns the cubic .
The question first prompts us to find the tangent at x = a, which we perform in one step with the function TangentLine.2 We then find the intersection points Q and P with two applications of the function Solve. Next, the area of the shaded region as a function of a is found by subtracting the area under the cubic from a triangular area: the former is found using the function Integrate, and the latter is found directly from the coordinates using the functions Polygon and Area. Finally, we are required to find the value of a that minimises the area, which is found in one step with the function ArgMin.
What follows is the complete Mathematica code to answer this five-part question:
This solution requires little mathematical understanding beyond being able to make sense of the questions. In particular, the standard CAS approach of setting up integrals and differentiating is entirely circumvented, as is the transcription. In Methods CBE, the above input and output would be considered sufficient answers.
We’ll now venture into the world of custom Mathematica functions, where programmers can really go to town. We’ll first look at the topic of functions and the features of their graphs. Mathematica does not have a built-in function to give all the desired features, so I created the function DetailPlot. To begin, I use a module to gather data about a function, including endpoints, axial intercepts, stationary points, inflection points and, if required, asymptotes. I then turn the module into an image to place over the graph.
Let’s fire this new weapon at Q2(c), Section B of 2016 Exam 2, which concerns the pictured quartic. We are given the equation of the graph and the point A, and we are told that the tangents at A and D are parallel. We are then required to find the point D and the length of AE.
And, here we are:
With very little input, DetailPlot has provided a rich graph, with every feature one might require within easy reach. The intersection points are ‘callouts’, which means that the points are labelled with their coordinates. In particular the coordinates of D and E have been revealed by DetailPlot, without any explicit calculation. We can then press forward and finish by finding the length of AE, a trivial calculation with the in-built function EuclideanDistance.
In the next example I demonstrate that a multi-stage question can still be trivialised by a single piece of pre-arranged code. In the multiple choice question MCQ10 from 2017 Exam 2, the function undergoes the transformation and we are required to identify the resulting function. For such questions I created the function Transform, and then the in-built function FullSimplify polishes off the question:
My last example is on functional equations, for which I created two functions, FTest and RFTest. I will illustrate the use of the latter function. For MCQ11 on 2016 Exam 2, the equation f(x) – f(y) = (y – x) f(xy) is given and it is required to determine which of the given functions satisfies the equation. Here is my entire solution:
It is impossible to have a proper sense of the power of Mathematica unless one is a programmer familiar with the package. This article presents just a few examples from the vast library of functions I created for Mathematical Methods and I found even more so for Specialist Mathematics. My libraries for both subjects barely scratch the surface of what is possible.
Creating such packages requires skill in both programming and mathematics, but the salient point is that any subsequent application of those programs by another student requires no comparable skill. The programs I have written may improve the performance of mathematically weaker students. Conversely, any student without access to such programs or, worse, is required to use handheld CAS instead of Mathematica, will be at a significant disadvantage.
This demonstrates the potential power of Mathematica to change the focus of VCE mathematics and, consequently, to debase its teaching and its assessment. True, the same issues had already arisen with the introduction of handheld CAS; clever teachers and cleverer students have always engaged in creating and sharing push-a-button CAS programs. Mathematica, however, has massively elevated the seriousness of these issues, all the more so since only a fraction of students have access to the platform.3
Technology, including Mathematica, calculators, spreadsheets and the many online programs, have tremendous potential to assist students with learning, understanding and applying mathematics. What is important for educators is to be careful that students are not using this technology to bypass learning and understanding mathematics.
1. All non-CBE students take the same tech-active exams and are considered in the same cohort for ATAR purposes. The Methods (CBE) exam appears to differ in only a superficial manner, and it appears that CBE students have not been considered a separate cohort since 2016.
2. The examination diagrams have been redrawn for greater clarity.
3. Although the Victorian Government offers Mathematica to all schools, to date many schools have not implemented it.
Most people are familiar with doctors’ “Hippocratic Oath”:
First, do no harm.
Yes, this aphorism does not appear in theHippocratic Oath, which is also probably not Hippocrates’. And, yes, the meaning of and fealty to this statement are not nearly so straight-forward. Still, the statement gives a beautifully clear and human principle, a guide on how to think about the difficult work of treating a person in one’s care.
Which brings us to mathematics. We feel that mathematics teachers need a similar guiding light, the Mathematic Oath:*
First, tell no lies.
As with the doctors’ oath, the implications of the Mathematic Oath are not obvious, respecting the oath is not always so simple. But it is a light, telling us the way. And it also tells us the wrong way: if you are a mathematics teacher and you are not telling your students the truth, then you are doing wrong.
Hi, interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of and respectively. One attempt explaining the difference between them is given here. The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane, around quadrilaterals, parallelograms and their diagonals .