## BODMAS v USBB

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.

Strong opinions? Me? No, just gentle suggestions. I assume they’re the same as Dave’s, but this is it:

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.

2) Specifically, if I see something like
3 x 5 + 2 x -3
my response is

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?

3) The goal in writing mathematics is not to follow God-given rules, but to be clear. Of course clarity can require rules, but it also requires common sense. And in this case common sense dictates

# BRACKETS

For Christ’s sake, why is this so hard for people to understand? Just write (3 x 5) + 2 or 3 x (5 + 2), or whatever. It is almost always trivial to deambiguousize something, so do so.

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.

## It’s not the Heat, it’s the Stupidity

Australian is going to the polls today, with that smirkingright 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.

## Update

Geez, Australians are dumb. And Queenslanders are dumber.

## AMSI’s Brain Teaser

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.

## The Story of an Unqualified Teacher

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 of 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.

UPDATE (10/02/21)

I assume that pretty much anybody who will read this post has already done so. Thus, they are already aware that the story is about my mother, who died last year, at the age of 374 (approx).

The story below referred to a student of my mother’s coming to our house, triggering my mother’s fight with the VUSEB. I didn’t remember the student’s visit but I was pretty sure I knew who the student was, and it turned out I was correct. It was a guy called Ron Adams, who became a family friend (and who, when I was about 7, cheated me out of a million dollars.)

After our mother’s death, my brother Jeff contacted Ron, who read the story below and added his own pretty wild memories. Ron has generously allowed me to include his thoughts below, which I’ve done as interspersed footnotes.

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

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.1 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.2 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.

1. Mr Canty. Yes, he was a fatherly figure. Like your mother, I was interviewed by him to get into WHS in 1965. I’d completed to form 4 (Year 10) at Watsonia Junior Technical School, but couldn’t continue there as I’d failed Maths 2 and Science 2 in the form 4 exams. The Menzies government introduced a scholarship scheme in 1964 for senior students. You had to sit an external exam which mainly tested for aptitude, and included writing a creative story. I wrote one about answering an advert in a paper. Won a scholarship (about 2 pounds per year for years 11 and 12) and turned up with the scholarship testamur, which mightily impressed Mr Canty, who didn’t bother to ask to see my form 4 report showing all the subjects I’d failed. So, like your mum, I was a new addition to the school in 1965.

2. Which is why we transferred en masse to Macleod High for Matric in 1966 – the year Jeff did Year 7 there.

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.3 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.

3. And your mother made us all feel so special. She called us ‘lovely teenagers’ – we’d have called her a lovely teacher. A breath of fresh air, not stuck in a rut and exploring the syllabus with us. She went out of her way to work with every student. We had a double period prac class Wednesday morning after morning recess until lunch time. That’s when we got to dissect rats that had been soaked in formalin. It was impossible to wash away the smell and we’d have to eat our sandwiches which we’d pick up in paper.

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.4 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.

4. It never did wear off for the students. There was something tremendously exciting and exotic having a teacher with such a pronounced American accent.

Teaching for a Living

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.

5. I don’t remember this. But we did catch and bring frogs that your mother would let us cut up. (‘Dissect’ would inflate what we actually did.)

Staff Interactions

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

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.9 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.

6. This of course is me. Being able to visit a teacher was an unprecedented privilege, especially for me. Teachers were out of my league.

7. She didn’t actually cry, but the shock was visceral. I recall her shaking.

8. [Jeff and Ron believe this man was Lindsay Thompson, one of the rare humans in the loathsome Bolte government. Thompson, however, only became Minister for Education in 1967.] Your mother’s account is probably more accurate and I might be imagining what happened on the basis of the qualities of the two people:
1 Your mother had a bee in her bonnet on this issue (an issue of justice) and if provided with the opportunity would, I am sure, have bounced up to the minister.
2 Lindsay Thompson was an honourable person who took seriously what it meant to ‘minister’ to the stakeholders in his portfolio: the students and teachers, and he would have been open to giving your mother a fair hearing.

So while my recollection might be apocryphal rather than accurate, it does convey a truth.

9. I kept in touch with your mother throughout my undergraduate years, and then dropped out of touch. I caught up with Jeff later when he went to La Trobe and I was doing post-grad. I often thought of your mother and the way, once riled, she took on the system. I’ve been taking on the system in different ways to hold it to account. Like your mother with the education department episode, when faced with an injustice towards students I’ve thrown caution to the wind, and gone to the top to get decisions reversed. Like going straight to the Vice-Chancellor to bring an injustice to his or her attention. I can’t say that I’ve endeared myself to them, but it’s led to some very productive relationships with a string of VCs – and I think has positively impacted on university processes. I can’t say that I’ve modelled myself on your mum, but I can say that actions like her confronting the education department have influenced me and emboldened me not to be intimidated by systems.

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 1965.10

10. One of my enduring regrets is that I never visited her in Echuca after meeting up with Jeff a few years ago and finding out she’d retired there. The intention was still in the back of my mind when I heard she’d passed away. Yes, she certainly was principled, fearless and good-hearted, and there are not too many people who wear all those qualities.

## Signs of the Times

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.

## Feynman on Modernity

We plan to have more posts on VCAA’s ridiculous curriculum review. Unfortunately.

Now, however, we’ll take a semi-break with three related posts. The nonsensical nature of VCAA’s review stems largely from its cloaking of all discussion in a slavish devotion to “modernity”, from the self-fulfilling prediction of the inevitability of “technology”, and from the presumption that teachers will genuflect to black box authority. We’ll have a post on each of these corrupting influences.

Our first such post is on a quote by Richard Feynman. For another project, and as an antidote to VCAA poison, we’ve been reading The Character of Physical Law, Feynman’s brilliant public lectures on physical truth and its discovery. Videos of the lectures are easy to find, and the first lecture is embedded above. Feynman’s purpose in the lectures is to talk very generally about laws in physics, but in order to ground the discussion he devotes his first lecture to just one specific law. Feynman begins this lecture by discussing his possibly surprising choice:

Now I’ve chosen for my special example of physical law to tell you about the theory of gravitation, the phenomena of gravity. Why I chose gravity, I don’t know. Whatever I chose you would’ve asked the same question. Actually it was one of the first great laws to be discovered and it has an interesting history. You might say ‘Yes, but then it’s old hat. I would like to hear something about more modern science’. More recent perhaps, but not more modern. Modern science is exactly in the same tradition as the discoveries of the law of gravitation. It is only more recent discoveries that we would be talking about. And so I do not feel at all bad about telling you of the law of gravitation, because in describing its history and the methods, the character of its discovery and its quality, I am talking about modern science. Completely modern.

Newer does not mean more modern. Moreover, there can be compelling arguments for focussing upon the old rather than the new. Feynman was perfectly aware of those arguments, of course. Notwithstanding his humorous claim of ignorance, Feynman knew exactly why he chose the law of gravitation.

This could, but will not, lead us into a discussion of VCE physics. It suffices to point out the irony that the clumsy attempts to modernise this subject have shifted it towards the medieval. But the conflation of “recent” with “modern” is of course endemic in modern recent education. We shall just point out one specific effect of this disease on VCE mathematics.

Once upon a time, Victoria had a beautiful Year 12 subject called Applied Mathematics. One learned this subject from properly trained teachers and from a beautiful textbook, written by the legendary J. B. “Bernie” Fitzpatrick and the deserves-to-be-legendary Peter Galbraith. Perhaps we’ll devote some future posts on Applied and its Pure companion. It is enough to note that simply throwing out VCE’s Methods and Specialist in their entirety and replacing them with dusty old Pure and Applied would result in a vastly superior, and more modern, curriculum.

Here, we just want to mention one extended topic in that curriculum: dynamics. As it was once taught, dynamics was a deep and incredibly rich topic, a strong and natural reinforcement of calculus and trigonometry and vector algebra, and a stunning demonstration of their power. Such dynamics is “old”, however, and is thus a ready-made target for modernising zealots. And so, over the years this beautiful, coherent and cohering topic has been cut and carved and trivialised, so that in VCE’s Specialist all that remains are a few disconnected, meat-free bones.

But, whatever is bad the VCAA can strive to make worse. It is clear that, failing the unlikely event that the current curriculum structure is kept, VCAA’s review will result in dynamics disappearing from VCE mathematics entirely. Forever.

Welcome to the Dark Ages.