The Examiners’ Report indicates that 81%, 48% and 45% of students received full marks for parts (a), (b) and (c), respectively.
The Examiners’ Report indicates that about half of the students gave the intended answer of D, with about a third giving the incorrect answer B. The Report notes:
Option B did not account for common factors and its last term is not irreducible, so should not have Dx in the numerator.
The worst kind of exam question is one that rewards mindless button-pushing and actively punishes intelligent consideration. The above question is of the worst kind. It is also pointless, nasty and self-trippingly overcute.
As John points out in the comments, the question can simply be done by pressing CAS buttons. But, alternatively, the question also just appears to require, and to invite, a simple understanding of partial fraction form. Which brings us to the nastiness: the expected partial fraction form is not a listed option.
So, what to make of it? Not surprisingly, many students opted for B, the superficially most plausible answer. A silly mistake, you silly, silly student! You shoulda just listened to your teacher and pushed the fucking buttons.
The trick, of course, is that the numerator factorises, cancelling with the denominator and leading to the intended answer, D. The problem with the trick is that it is antimathematical and wrong:
But of course all that is way, way too much to think about in a speed-test exam. Much better to just listen to your teacher and push the fucking buttons.
Due mostly to the hard work of Damo, all of the outstanding WitCHes have been resolved, with the exception of WitCH 8. That one will take time: it’s a jungle of half-maths. Our new WitCHes are not so tricky, although there is perhaps more to be said than indicated at first glance.
The first of our new batch of WitCHes is from the VCE 2018 Specialist Exam 1:
The Examiners’ Report gives the answer as . The Report also indicates that the average score on this question was 1.3/5, with 98% of students scoring 3 or lower, and over a third of students scoring 0.
A while ago we had cause to meet with a school principal. The principal happened to have a PhD in mathematics education, and it was on that basis that they began the conversation: “As a fellow mathematician …”. It will come as no great surprise that our association with the principal ended soon after.
The principal was doubly wrong: no, of course they are not remotely a mathematician; but, neither are we. Once upon a time, yes, but not now. We are no longer seriously engaged in mathematical research, in trying to discover the facts and the nature of mathematical truth.
But, to the principal and the principle. Of course it doesn’t matter whatsoever if a principal is not a mathematician. What matters a great deal, however, is if a principal falsely imagines that they are. If a school principal does not understand what it means to be a mathematician then they cannot possibly understand what a mathematician might offer to their school, or to education in general.
Such a lack of understanding, an ignorance of what it means to think deeply about mathematics, is now endemic in Australian mathematics education. The consequence is that mathematicians are treated as inferior teachers and education academics, merely as weirdos with relevant training a proper subset of that of the education pros. The consequence is that clear and informed and deep mathematical thought is marginalised to the point of non-existence. The consequence is a pointless mathematics curriculum taught using painfully bad textbooks by poorly trained teachers and administered by organisations with no respect for or understanding of the nature of mathematical thought.
Mathematicians can be arrogant and annoying, and wrong. But mathematics education without the deep and continued involvement of good and serious mathematicians is pure insanity.
Yesterday, I received an email from Stacey, a teacher and good friend and former student. Stacey was asking for my opinion of “order of operations”, having been encouraged to contact me by Dave, also a teacher and good friend and former student. Apparently, Dave had suggested that I had “strong opinions” on the matter. I dashed off a response which, in slightly tidied and toned form, follows.
1) The general principle is that if mathematicians don’t worry about something then there is good reason to doubt that students or teachers should. It’s not an axiom, but it’s a very good principle.
a) No mathematician would ever, ever write that.
b) I don’t know what the Hell the expression means. Honestly.
c) If I don’t know what it means, why should I expect anybody else to know?
The fact that schools don’t instruct this first and foremost, that demonstrates that BODMAS or whatever has almost nothing to do with learning or understanding. It is overwhelmingly a meaningless ritual to see which students best follow mindless rules and instruction. It is not in any sense mathematics. In fact, I think this all suggests a very worthwhile and catchy reform: don’t teach BODMAS, teach USBB.
[Note: the original acronym, which is to be preferred, was USFB]
4) It is a little more complicated than that, because mathematicians also write arguably ambiguous expressions, such ab + c and ab2 and a/bc. BUT, the concatenation/proximity and fractioning is much, much less ambiguous in practice. (a/bc is not great, and I would always look to write that with a horizontal fraction line or as a/(bc).)
5) Extending that, brackets can also be overdone, if people jump to overinterpret every real or imagined ambiguousness. The notation sin(x), for example, is truly idiotic; in this case there is no ambiguity that requires clarification, and so the brackets do nothing but make the mathematics ugly and more difficult to read.
6) The issue is also more complicated because mathematicians seldom if ever use the signs ÷ or x. That’s partially because they’re dealing with algebra rather than arithmetic, and partially because “division” is eventually not its own thing, having been replaced by making the fraction directly, by dealing directly with the result of the division rather than the division.
So, this is a case where it is perfectly reasonable for schools to worry about something that mathematicians don’t. Arithmetic obviously requires a multiplication sign. And, primary students must learn what division means well before fractions, so of course it makes sense to have a sign for division. I doubt, however, that one needs a division sign in secondary school.
7) So, it’s not that the order of operations issues don’t exist. But they don’t exist nearly as much as way too many prissy teachers imagine. It’s not enough of a thing to be a tested thing.
Australian is going to the polls today, with that smirking, right wing clown attempting to be elected Prime Minister. And of course we’ll all be cheering for him to beat Scott Morrison.
The fact that the ignorant, science denying, happy clapping, coal-hugging thug pictured above even has a shot at winning indicates the appalling low level of political discourse. We shouldn’t be surprised, of course, but for some reason we are.
Back in 2014, the Maths Masters wrote a column on then Prime Minster Tony Abbott’s climate denialism, entitled How to be Liberal with the Truth. Our editor rejected the column as a “diatribe”, which was fair enough, and we took the rejection in our stride. Nonetheless, our editor passed the diatribe to the Age‘s op-ed desk which published the column as Tony Abbott is a liar: It’s a mathematical truth. Our diatribe was a hit.
The diatribe ended with a prediction:
But what of Tony? Will he be remembered as a liar? Probably, but probably he’ll be remembered for much more. Eventually, and more likely sooner rather than later, global warming will be undeniable. Truly undeniable.Which means Abbott should go down in history as the Australian Prime Minister, the last Australian Prime Minister, to deny physical reality.
We were wrong then. But, maybe now Australia will finally be done with anti-science idiots.
Geez, Australians are dumb. And Queenslanders are dumber.
Last week, AMSI released yet another paper on the issue of school mathematics being taught by “out of discipline” teachers. It will come as no surprise to readers of this blog that we have many issues with AMSI’s paper. Here, we’ll focus on just one aspect.
The Sydney Morning Herald’s report on AMSI’s paper begins:
Fewer than one in four Australian high school students have a qualified maths teacher …
That statement is, of course, utter nonsense. By any reasonable definition, a much higher percentage of secondary students are taught by formally “qualified” teachers. It is concerning that an “education reporter” would lead with such an implausible claim, but SMH was not alone. The news.com.au report was titled:
Only 1 in 4 high-schoolers are being taught maths by qualified teachers
The Australian’s barely comprehensible sentence, courtesy of another education reporter, appeared to suggest that matters are even worse:
Fewer than one in four students are taught by a qualified maths teacher — one with at least a university minor in the subject — at some stage between Years 7 to 10.
So, what is the source of all these inflated declarations of educational doom? It would appear to be on page 2 of AMSI’s paper. In the first of the paper’s eye-catching Key Points, the authors write:
The extent of the problem [with the supply of qualified teachers] is illustrated by the estimated amount of out‐of‐field teaching occurring with less than one in four students having a qualified mathematics teacher in each of Years 7 to 10.
That reminds us: we must buy AMSI a box of commas for Christmas.
The above sentence, which turned out to be the grabber of AMSI’s paper, is like an optical illusion: you think you’ve got the meaning, and then it slips around to mean something entirely different. It is no wonder if reporters misinterpreted.
What did the AMSI authors intend to convey, and on what basis? It is difficult to tell. A linked endnote in AMSI’s paper refers to a 2017 AMSI publication. The page reference to this second document is clearly incorrect, but it appears that the intention is to refer to page 4, which has its own list of key points, including:
At least 26% of Years 7–10 maths teachers are not fully qualified.
This is an admirably clear statement and, if true, one may (or may not) regard it as a relatively major problem. The statement, however, is not remotely supportive of the educational catastrophe that AMSI’s garbled 2019 statement led gullible reporters to declare.
Also puzzling, it is not clear how AMSI’s 2017 statement, or any other AMSI declaration that we could find, leads reasonably to any natural interpretation of AMSI’s 2019 statement. This is the case even if one ignores that “not fully qualified” does not clearly equate to “not qualified”, and that 26% of teachers does not equate to 26% of classes, nor to 26% of students. Even with the most liberal assumptions and generous interpretations, we still cannot determine the basis, any basis, for the 2019 statement. The reader is invited to give it a go.
There are plenty more serious issues with AMSI’s paper which, though raising some very important issues and suggestions, also connects some distant and very disputable dots. It probably doesn’t matter, however. We worked hard to read AMSI’s clumsily written paper. It seems unlikely that many others will do likewise.
This final sabbatical post is a story from 1965. It is the story of Marian, who found herself as a single mother with two young children, in Australia with no extended family, and in need a job. It is quite a remarkable story, not least for Marian’s skirmish with the Victorian Universities and Schools Examinations Board, the pre-pre-precursor to the VCAA. The story is taken from Marian’s memoirs.
A New Profession
Near the end of the school year in 1964, I began seriously looking for a job starting in the New Year. In late November, I asked Eileen about the possibility my getting a job at the local Macleod High School where she was teaching. After making inquires, she told me that there would be no vacancies in the Science department. So I went out a bit further into the suburbs and applied to the principal at Watsonia High School. I was interviewed by the headmaster who was a very kind and fatherly type of man, not at all threatening. He asked me lots of questions about my education and work experiences. I played up my studies of the sciences and my teaching experiences within hospital settings. He and I both understood that I was not a qualified teacher and I had the feeling that he was reluctant to hire me. However part of his job, as a principal employed by the Victorian Education Department, was to cover all of the subjects and classes in his school and it seemed to me that they didn’t care how he did it. I soon learned that there were many schools in the same predicament. They also hired unqualified staff just to keep their schools functioning, at least at some basic level. Watsonia High was a new school and had been open for only a few years. 1965 was the first year that they would be offering year eleven subjects. It would be several more years before students could complete year twelve at that school. When the headmaster said I could have the job teaching year eleven Biology, as well as junior science and mathematics, I swear I heard him say a prayer. I know I did.
He handed me a copy of the Biology syllabus which was set by the Education Department of the State of Victoria. That was my complete introduction to the Victorian Education system. I knew absolutely nothing about how the local school related to the Education Department. What I would be teaching, other than Biology, remained a mystery until the following February when the school year actually started. As Biology had not previously been taught at the school a copy of the student text book was not yet available and materials and equipment for teaching the subject were extremely limited or completely absent. It would have been a huge job for a qualified teacher to set up a new department in the school and here was I, an absolute novice, being thrown in at the deep end. I felt I could not complain because I had known that I was applying for a job for which I was not qualified. It was because I was not qualified, that I did not even know what kind of help to ask for.
My Life as a New Teacher
My life as a schoolteacher began in February 1965. My first actual teaching experience on the first day of school was fronting a mathematics class of forty seventh grade boys and girls. I really had no idea of what to do but that didn’t stop me. I plowed in and opened the maths book at page one which just happened to be on operations with fractions. I wasn’t even aware that working with fractions was still in my memory bank. Discovering that I still remembered much from my own education came as a surprise and gave me some much needed confidence. Those were the days in the state of Victoria where temporary teachers made up a fair section of all teaching staffs. As well, the class sizes were ridiculous. I ended up with one grade eight general science classes which was composed of fifty students. The worst part of trying to teach science to such a large group was in supervising the lab work and finding enough equipment for all of the students. What I considered my real work and my real interest was teaching Biology to fifteen lovely teenagers. I really enjoyed those boys and girls and I so wanted to do the very best I could for them. No one on the staff, neither the principal or other science teachers, assisted me in any way whatsoever. I had to learn routine school procedures plus the requirements of my own discipline, including the laboratory procedures. I did know that these older students would sit for an external exam at the end of the year. That really did not worry me. I knew I could well and truly handle the subject matter and I knew that I would do everything I could to help those students gain a pass, so that they could go on to year twelve. In addition to my teaching duties, by virtue of my nursing credentials, I was appointed the job of ‘first aid’ to the entire student body of approximately three hundred plus students. It was a natural enough appointment considering my background but I often wondered what schools did where there was not a trained nurse on the staff. I did not get paid extra for that job. In a way, it made me feel better about my teaching appointment, as it was a part of the job that I really was qualified to do.
There were a couple of men on the staff who deeply resented not only us temporary teachers but the fact that women actually held any teaching positions at all. One man teacher, fortyish, blatantly stated that women should not be allowed in any work force but rather should be at home full time. Misogyny was alive and well in Australia. He was quite serious and this, in spite of the fact that, when I started teaching in 1965, women did not receive equal pay even though they did the same jobs as the men. He didn’t want us to be paid at all. A couple of years later Australia joined the real world and women were awarded equal pay. A couple of the teachers tended to look down their noses at us temporary teachers but I was amazed at the thickness that my skin developed over the next years. I was not doing the job because I wanted it but because I needed to work to support myself and my sons. I was determined I was going to do whatever I had to do to manage that. After all I did not create the system that employed me and if they were willing to pay me for teaching in their school then I would do my very best.
I Hate this Job
After the initial pleasantries of meeting the students and the novelty of my Yankee accent wore off I found the large classes of boys and girls, aged twelve and thirteen, hellish. I often found myself at my wits end trying to keep classes of forty to fifty students interested and attentive. It is a fact that in that first year at Watsonia High School I said to myself (and sometimes aloud) every single day, “I hate this job, I hate this job’ and I absolutely meant it. However, I stuck it out, taking one day at a time, knowing that I had to work at something and no other job would allow me the luxury of being at home when my sons were home from school. At that time, as far as I was concerned, that was the only good thing about the teaching profession. I did however get a lot of enjoyment from the company of most of the other staff members. It was nice being out in the world again, meeting and interacting with intelligent adults. I got along very well with everyone who wanted to be friendly. It was not only the days before equal pay it was also the days before anyone ever talked about or recognized any such thing as single mothers. Out of a staff of fifty teachers only two of us fit that category and I generally escaped overt ridicule because I was an American and every one agreed they were weird.
Teaching for a Living
I worked extremely hard over the next couple of years but, by far, that first year was the hardest and steadiest of grinds. I tried to learn every thing I could about the noble art of teaching. I went to every seminar and tutorial which became available. I bought many books on subjects which I thought might enable me to become somewhat proficient as a teacher. I tried to learn class discipline from reading books and articles because, god knows, the senior teachers at the school, although they were quite friendly, seemed to be quite ready to see me sink or swim in my own good time. The prescribed laboratory work for the Biology students was very minimal to what it became in later years. However, since this was the first year the subject had been taught in the school, I had to start from scratch in setting up the materials for the course. So, although the actual course work was quite basic, I spent endless amounts of time trying to scrape together materials and equipment and, where these were not available in the Science Department, I had to chase around to find out where I could get the necessities. One good example of how I struggled was just one demonstration (when a word from a senior science teacher might have saved me much anxiety). I was required to dissect, as a class demonstration, the reproductive system of a female mammal. In order to complete this required part of the syllabus, one of the students brought in a freshly killed rabbit and the students and I suffered through the dreadful smell of dissecting it. Much later I learned that many such preserved specimens were available to be purchased from certain universities or supply houses. I probably could have ask for a lot more help but it was not in my nature to impose on other people. Besides I had no wish to flaunt my inadequacies.
My first year of teaching coincided with the advent of the “New Mathematics”. I was on a par with other junior school mathematics teachers in learning something which was new to them as well. We were all equal when we attended the classes teaching us the basics of this new approach to mathematics.
There was the most minimal supervision of my work that one could imagine. I am certain that the headmaster visited my classroom no more than twice in that first year and even then he had nothing to say to me concerning my teaching, either good or bad. Nothing. In one staff meeting he made a point of mentioning my name. He had asked for some written information from each of the teachers and he reported that I was the only member of staff to complete the task. At the time I felt embarrassed, fearful that I appeared sycophantic. I mention this incident only to demonstrate how keen I was to do whatever was required of me. The headmaster had what I considered a bad approach to the staff in that he often remarked, critically, that he had seen or heard a teacher do this or that. But he never said who he was talking about. At first, I used to think to myself, ‘is he talking about me?’ I would wrack my brain trying to recall recent incidences and I would worry about it. After he did this several times I quietly let myself off the hook. I decided if he had something to say to me he should say it to me personally. As for his generic remarks in future staff meetings, I just said to myself, ‘he doesn’t mean me’ and I immediately put whatever he had said out of my mind. I do believe I was being watched, or listened to, plenty of times when I was not aware of it. I base this on one incident which occurred in a staff meeting chaired by the assistant headmaster, the physics teacher. In speaking to the staff he chose to quote something I had said to my students when teaching them about Mendel’s experiments in genetics. It was clear he been listening to my lesson from the preparation room which separated the two science classrooms. I suppose I should have felt flattered.
During the school year all fifty teachers on the staff had to endure several days of visits by several men, ‘inspectors’, from the central office of the State Education Department. They were the closest things to gods that I had ever run into. Everyone, even senior teachers, would shake in their shoes. I suppose some teachers’ promotions depended on the assessments made but that certainly was no concern of mine. I was at the bottom of the totem pole with every prospect of staying there. They came to our school, usually three or four, in a bunch, all dressed in dark suits, and looking exceedingly furtive, rarely saying anything, and making everyone nervous as hell. Then they left, not to return for another year. They sat in on a couple of my classes and at least one Biology class. They ask to see a couple of the students’ practical books which recorded results of experiments we had completed. At no time did they say anything to me regarding my teaching, either constructive or critical. If they said anything to the headmaster, he never mentioned it to me. I never went crying to the headmaster or anyone else when things got tough. I worked things out by myself and I was determined not to show any anxiety or distress. By the end of the school year no one was happier than I to see the summer holidays roll around. I still hated the job but I had become very fond of my fifteen Biology students and I was even finding that the younger kids could be lots of fun, at times. However, I still had no real love for the job and I continued solely out of necessity.
Fighting the Good Fight
My Biology students had taken their final exam in November and, according to the system, as their regular teacher, I had nothing to do with setting the exam, supervising or correcting it. This was all done by specially chosen people within the Education Department. One of my Biology students had become very friendly with me and my sons and he visited us often during that holiday period. From him I learned that the exam was not too difficult and he felt he had passed it. Results were routinely sent to the students during the following January. One day this young man came to the house visibly upset because, when he got his exam results, he found he had not passed Biology. Not only that, he had checked with the other students and none of them had passed either. One of the parents had phoned the Department to question the results. He was told that his daughter had passed the exam but the practical work, as evidenced by the submitted ‘prac’ books, was not of a passable standard. I was shocked. I knew I had followed the syllabus to the letter. I had taught every required topic and completed every experiment as laid down in the syllabus. I had been much too new to the business to try anything tricky or try to cut corners. I knew it was essential that I teach the subject as prescribed. It took me about ten minutes to absorb the shock and then I realized I had to do something. But what? The school was still closed for the holidays and I did not know how to get in touch with the headmaster at his home. So I decided the only thing I could do was to go to the source of the problem. Since the exam was set and corrected by people working in the main offices of the Victorian Education Department in the city, I determined that was where I had to go. I did not bother to make a phone call or arrange for an appointment. I was angry as well as mystified and I was determined my students were not going to suffer if I could help it.
I reached the Department in the center of Melbourne during the middle of the morning and the place was a hive of activity, absolutely teeming with energy. There were loads of people running around like a bunch of mice, in and out of offices, up and down hallways, occasionally one of them stopping for a quick word with someone, then quickly running off again. It reminded me of a scene from Alice in Wonderland. As I watched their hyperactivity I felt reassured because it looked to me like they didn’t know what they were doing. Their kinetic behavior suggested indecision and confusion. They wouldn’t have looked any different if they had been told that the world was coming to an end and they were trying to find a place to hide. Watching them gave me courage, if I needed it. I waited for a very long time before I was finally ushered into the office of a properly suited male. I told him my story and explained that I could not understand why all of my students had been denied a pass in my subject. He had some records in front of him and he said (with a straight face) the students were failed because the practical work was not corrected properly. WHAT!! ?? I could not believe my ears. Not a word about the work completed, nothing about the content of the prac books, and nothing about the results of the students’ three hour exam. Their only criticism was the lack of corrections by me. Assuming I had made mistakes, how the hell could they make my students pay for it by failing them? If I had neglected to correct a statement or allowed misspelling of words to go unchecked, so what? Now they were going to fail the entire Biology class because of something I did or did not do with their practical reports. I saw this as raw injustice and whenever I come into close contact with blatant unfairness, I see red. I started by giving this man some home truths, saying what I honestly felt. I reminded this man that I was hired by the Victorian Education Department as an untrained teacher just one year ago. The school itself had not been set up for teaching Biology. I was not only setting up the department but also learning the job as I went. That was no secret. No one at the school had instructed me on how precise the corrections of the practical work had to be. No one supervised my work or the work of my students. Now at the end of a very gruelling year, if the practical work had not been properly corrected, then sack me, boot me out of the job, but for gods sake don’t ruin a year of these young students’ lives because of something I had or had not done. Amazingly, the man did not even make a pretence of arguing with me. The logic of my argument was sound and he knew it and, although he could not give me a decision immediately, he said he would get in touch with the school. I left that office and that building feeling quite satisfied. They were not going to get away with this. I knew it and so did that man I had just spoken to. As for my future as a teacher, I could not have cared less. I was not doing the job for the love of it and if they fired me, so be it. I would survive without the stupid system that not only set me up for trouble but worse still treated these young people with such disdain.
As it turned out the matter was dealt with quite promptly. A few days after my complaint session my young student came to the house to tell me that he had received another letter from the Education Department stating that he had passed Biology after all. In fact, eight of the fifteen students had passed, based on the exam results. This better than 50% pass rate was considered a good result in a very new, barely established, state school where students were not screened out, as is the practice in most, if not all, elite private schools. On the first day of the new school year in February, I made a point of approaching the headmaster to tell him what I had done. He was quite satisfied and told me if I had not gone to the Department he would have done so himself.
So all was well that ended well and my eight successful students went on to do year twelve Biology at nearby Macleod High School and most went on to do University courses. As for me I was relieved of teaching Biology when the school hired an ambitious university trained male teacher. He had a very big head and was sure he was God’s gift to the teaching profession. This well qualified male was given a whole year in which to set up the Biology Department in preparation for expansion to teaching Year 12 the following year. No such consideration was ever even hinted at when I started the department the year before. It just proves some people have clout and some don’t. In those days it certainly helped to be a male. I spent nine years altogether teaching in that school and as the years went by I enjoyed it more and more. The year eight students who had given me so much grief in my first year eventually gave me endless pleasure and many laughs. I was always aware of being an untrained teacher but through the years I learned a lot and after nine years in the State system I spent another six years at a Catholic Girls’ School which was a wonderful experience as well. I always worked hard, being employed as a teacher, but I also had a lot of fun with the students and fellow teachers.
My mother, Marian Ross, is now 90, and is still going sort of strong. She is as principled, as fearless and as good-hearted as she was in 1965.
Our second sabbatical post concerns, well, the reader can decide what it concerns.
Last year, diagnostic quizzes were given to a large class of first year mathematics students at a Victorian tertiary institution. The majority of these students had completed Specialist Mathematics or an equivalent. On average, these would not have been the top Specialist students, nor would they have been the weakest. The results of these quizzes were, let’s say, interesting.
It was notable, for example, that around 2/5 of these students failed to simplify the likes of 81-3/4. And, around 2/3 of the students failed to solve an inequality such as 2 + 4x ≥ x2 + 5. And, around 3/5 of the students failed to correctly evaluate or similar. There were many such notable outcomes.
Most striking for us, however, were questions concerning lists of numbers, such as those displayed above. Students were asked to write the listed numbers in ascending order. And, though a majority of the students answered correctly, about 1/4 of the students did not.
What, then, does it tell us if a quarter of post-Specialist students cannot order a list of common numbers? Is this acceptable? If not, what or whom are we to blame? Will the outcome of the current VCAA review improve things, or will it make matters worse?
Tricky, tricky questions.