Tootering Your Own Horn

Eddie Woo is reportedly concerned about private tutoring. His warning comes courtesy of SMH‘s education editor, Jordan Baker, in an article entitled ‘Be very, very careful’: Experts raise warning on private tutoring. The article begins,

Maths teachers including high-profile mathematician Eddie Woo have sounded an alarm on private tutoring, warning that bad tutors could be “fatal” to students’ future in the subject.

Eddie said it, so it must be true. And, Baker quotes another expert, the chief executive of the Australian Tutoring Association, Mohan Dhall:

I am absolutely dismayed at the lack of creativity and lack of real-world applicability most tutors bring to maths …The main problem stems from this idea that they focus on the outcome – ‘this is what students need to know’, rather than ‘this is what kids need to learn to be interested and engage’.

Finally, Baker quotes expert Katherin Cartwright, a lecturer in mathematics education at The University of Sydney. Cartwright, according to Baker, is concerned that poor tutoring could lead to a lack of confidence:

If it becomes about skill and drill and speed, and it becomes an anxious, emotional issue for students, then they are not going to like it, and they will not want to take it further.

Yep, of course. The most important consideration when framing an education is to be sure to never make a student anxious or emotional. Poor, fragile little petals that they are.

Baker’s fear-mongering is nonsense. Almost every line of her article is contentious and a number contain flat out falsehoods. Beginning with the title. Woo and Dhall and Cartwright are “experts” on the issues of tutoring? According to whom? Based on what? Perhaps they are experts, but Baker provides no evidence.

OK, we could concede Baker’s point that Eddie is a mathematician. Except that he isn’t and we don’t. Not that it matters here, since most mathematicians are unlikely to know much about the role of tutoring in Australian education. But the false and pointless puffery exemplifies Baker’s unjustified appeals to authority.

What of the declared concerns of Baker’s “experts”? Cartwright is supposedly worried about “skill and drill and speed”. This in contrast to school, according to Baker:

Most schools no longer emphasise speed and rote learning when teaching maths, and now focus on students’ understanding of key concepts as part of a concerted effort to improve engagement in maths across the system.

This hilarious half-truth undercuts the whole thrust of Baker’s article. It is true that many schools, particularly primary schools, have drunk the educational Kool-Aid and have turned their maths lessons into constructivist swamplands. But that just means the main and massive job of competent Year 7 maths teachers is to undo the damage inflicted by snake-oilers, and to instil in their students, much too late, an appreciation of the importance of memory and skill and efficient technique. Such technique is critical for formal success in school mathematics and, which is sadly different, for the learning of mathematics. Baker seems entirely unaware, for example, that, for better or worse, Year 12 mathematics is first and foremost a speed test, a succession of sprints.

As for Dhall, does he really expects tutors to be more offering of “creativity” and “real-world applicability”? Dhall seems blissfully unaware that most “real-world” applications that students must suffer through are pedagogically worthless, and are either trivial or infinitely tedious. Dhall seems unaware that some subjects have warped “applicability” into a surrealist nightmare.

And Eddie? What worries Eddie? Not much, as it happens, but too much. Eddie’s quoted comments come from a NSW podcast, which appears to have been the genesis of Baker’s piece; stenographic fluffing is of course the standard for modern reportage, the cheap and easy alternative to proper investigation and considered reflection.

Eddie’s podcast is a happy public chat about teaching mathematics. Eddie is demonstrably a great teacher and he is very engaging. He says a number of smart things, the half-hour podcast only being offensive for its inoffensiveness; Eddie, or his interviewer, was seemingly too scared to venture into a deep public discussion of mathematics and the sense of it. The result is that, except for the occasional genuflection to “pattern”, Eddie may as well have been talking about turtle farming as teaching mathematics.

Eddie’s comments on tutoring are a very minor part of the podcast, a response in the final question time. This is Eddie’s response in full:

When I think about external tuition – again just like before this is a really complex question – there is tuition and then there is ‘tuition’. There is some which is enormously helpful to individual students to come in at a point of need and say “you have got gaps in your knowledge, I can identify that and then help you with those and then you can get back on the horse and off you go, fantastic”. There are other kinds of tuition which are frankly just pumping out an industrial model of education which parents who are very well intentioned and feel like they cannot do anything else, it is like “at least I can throw money at the problem and at least they are spending more time on maths hopefully that will help”. Maybe it does and maybe it is making your child hate maths because they are doing it until 9pm at night after a whole day? That to me is heartbreaking.

I think that students need to be very, very careful and parents need to be very, very careful about how they experience mathematics. Because yes the time is a worthwhile investment, it is a practical subject, but if you are just churning through, often tragically learning things which actually are just machine processes. I have students come to me and they say “I can differentiate, I am really good at that, I am only fifteen years old”. You don’t need to know what differentiation is, but they come to me with this ability to turn a handle on this algorithm this set of steps. Just like me; I don’t know how to bake, but I can follow a recipe. I have no idea what baking powder does or why 180 degrees Celsius is important but I can follow steps. That is okay for a cake because you can still eat it at the end, but that is fatal for mathematics because you don’t know why you are doing any of the things that you are doing. If that is what you are, you are not a mathematician, you are a machine and that is not what we want our children to become. We have to be careful.

Eddie says plenty right here, touching on various forms of and issues with tutoring, and school teaching. The issues do not get fleshed out, but that is the nature of Q & A.

Eddie also gets things smugly wrong. Sure, some tutoring might be characterised as “industrial”. But more so than schools? How can mass education not be industrial? This isn’t necessarily bad: mostly, it just is. Unless, of course, little Tarquin’s parents have the time and the money to arrange for individual or small-group lessons with an, um, tutor.

All the concerns Baker and her experts raise about tutoring apply as much or more so to school education and, as a matter of business necessity, are largely a reflection of school education. And, how do tutors and tutoring companies deal with this? Some well, some poorly. But mostly with industry, which is not a dirty word, and with good and honest intent.

Baker notes the underlying issue, seemingly without even realising it:

However, Australian students’ performance in maths has either stalled or declined on all major indicators over that period, and academics have raised concerns about students arriving at university without the maths skills they need.

Why do parents employ tutors? Having enjoyed and suffered forty years of tutoring, in pretty much all its forms, we can give the obvious answer: there’s a zillion different, individual reasons. Some, as Eddie suggests, are looking for a little damage control, the filling of gaps and a little polishing. Some, as Eddie suggests, think of mathematics, falsely, as a syntactic game, and are looking for lessons in playing that dangerously meaningless game. Some believe, correctly or otherwise, that their teacher/school is responsible for little Johnny’s struggling. Some are trying to get darling Diana into law school. Some are hothousing precious little Perry so he/she can get a scholarship into Polo Grammar or Mildred’s College for Christian Ladies.

But, underlying it all, there is one obvious, central reason why parents employ tutors: parents are unsatisfied with the education their child receives at school.

Why are parents unsatisfied? Are they right to be? Of course, it depends. But, whatever the individual analyses, the massive growth of the tutoring industry indicates a major disconnect, and either a major failing in schools’ performance or a major blindness in parents’ expectations, or both.

That would be a much more worthwhile issue for Baker, and everyone, to consider.

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

USE 

SOME 

BLOODY 

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.

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 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.

Staff Interactions

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.