A few days a ago, an occasional commenter told us about the teacher shortage at their school. They suggested the shortage was going to “play havoc” with their teaching load. We’re not quite sure how that works, since we thought there were strong and weird restrictions on what could be demanded of teachers, but we’re not doubting the reality on the ground. Our teacher correspondent also offered tentative reasons for the shortage: boomer teachers retiring, both naturally and motivated by covid; little incentive for people become new teachers; new teachers not lasting.
Category: teaching
Marty Talk – Not Quite The Prime Number Theorem
Having foolishly ventured out to Monash Uni last Tuesday to see the Evil Mathologre give a Lunchmaths talk, I found myself roped in to giving the next one. So, anyone who is around Monash next Tuesday and has nothing better to do is welcome to attend. Details below, and here. Continue reading “Marty Talk – Not Quite The Prime Number Theorem”
MAV’s Valuing of Mathematics
The MAV have started the engines for their 2022 Annual Conference. The organisers have already lined up an impressive list of speakers and, as always, the MAV have put out the call for anyone and everyone to present, at a cost of only $500 or so for the two-day extravaganza.
Of course, if planning to attend or present, one should keep in mind the theme and sub-themes of the conference: Continue reading “MAV’s Valuing of Mathematics”
Mr. McRae’s Triple Gift
This is a story from long, long ago. It is about Mr. McRae, who was our grade 4 teacher, at Macleod State School. We have written about Macleod before, and we have written, briefly, about Mr. McRae before, in regard to the moon landing:
I still have vivid-grainy memories of watching Armstrong’s first steps. A random few students from each class in Macleod State School were selected to go to the library to watch the event on the school’s one TV. I was not one of the lucky few. But Mr. Macrae, our wonderful Grade 4 teacher, just declared “Bugger it!”, determined which student in our class lived closest to the school, and sent out a posse to haul back the kid’s 2-ton TV. We then all watched the moon landing, enthralled and eternally grateful to Mr. Macrae.
He was that kind of guy. No-nonsense and intelligent and cultured.
The year he taught us, Mr. McRae was new to Macleod. He had just appeared on the playground before the first class of the year, tall and commanding. Rumour had it that he had played Under 19s for the Richmond Football Club, making Mr. McRae just shy of a Greek god. (The actual Greek god was, of course, Carl Ditterich.) He was a standard and excellent teacher. Firm, disciplined and disciplining, but kind, and with a calm and intelligent air of bemusement. He was the boss, but a thoughtful and unpredictable boss. Hence, our class getting to watch the moon landing. And, how else to explain the boxing match?
One day, Mr. McRae inadvertently started a harmless play-scuffle between two students. He then decided the dispute should be settled by a proper boxing match in front of the class. Once, of course, a kid had been sent home to fetch a couple pairs of boxing gloves. We can’t remember whether we lost, although we remember we didn’t win. In any case, neither of us had a clue how to box, and so the match was followed by Mr. McRae giving the class an impromptu lesson on technique. This was, to explain it a little, the era of Lionel Rose and Johnny Famechon and TV Ringside.
That’s all by way of background. The story we want to tell is of a mathematics lesson.
One Friday afternoon, Mr. McRae introduced his grade 4 class to Pythagoras’s theorem. Or, at least, to Pythagorean triples; we can’t specifically remember the triangles, or anything, but undoubtedly 
made an appearance. Why he showed us this, God only knows, but Mr. McRae ended the class with a challenge: find more triples. Our memory is that the specific challenge was to find a certain number of triples, maybe three, maybe five.
We have no idea what Mr. McRae hoped to achieve with this challenge, but we remember pondering, aimlessly, hoping to find triples. Eventually, by smart persistence and dumb luck, we stumbled upon the trick: doubling a triple gives a new triple. So, 
, and so on. With this kid-Eureka insight, we then happily spent the week-end doubling away.
Come Monday morning, Mr. McRae asked for the class’s triples. We proudly went to the blackboard and wrote up our largest creation. By memory, it was something in the millions. So,
![\[\boldsymbol{1572864^2 + 2096152^2 = 2621440^2}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-e71dd4bbb3c3495f57d48a7da8bd0770_l3.svg "Rendered by QuickLaTeX.com")
or thereabouts. And then Mr. McRae uttered the fateful words:
“Let’s check it!”
There were the inevitable groans from the class, and the little Archimedes hero of the story was more popular than ever. But, Mr. McRae was the boss, and so we all set down to multiplying, including Mr. McRae himself. And, ten or so minutes later, the class collectively started to conclude … the equation was wrong. Yep, Little Archimedes had stuffed up. Which led to more fateful words:
“Let’s find the mistake!”
More groans, more multiplying, and eventually the error was found. By memory, after quite a few doubles, somewhere in the mid thousands. And, satisfied, Mr. McRae led the class on to whatever he had been planned for that day.
What is the moral? We have a reason for telling the story, beyond a simple tribute to a great, memorable teacher. We think there are morals there. We’ll leave it for the reader to ponder.
The Past is a Foreign Country
We’ll fill one more gap from our presentation. Our previous gap-fill was on Professor E. R. Love and the disappearing art of lecturing. The past is a foreign country and, indeed, they do things differently there. This post is about the past and that foreign country. The country is called China.
The above photo is of a Nanjing school, the sister to our daughters’ school here in Melbourne. It is considered a good public school, but no more than that, and the photo is of a Year 5 class. What does one notice? What does one notice, that is, apart from the algebra and the general formulae, material that Australia typically covers, thinly and badly, in around Year 8?
There is no colour. The room is dressed in drab tiles and off-white walls. There are no posters. There’s just rows of students at their desks, and a teacher up front with nothing but an overhead projector and a blackboard. What a Hell.
It’s a Hell we would kill for.
The photo is of a class, of a teacher teaching, of students learning. The students are respectful and attentive. They are undistracted, in no small part because there is nothing to invite distraction. It may not be apparent from the photo but was obvious from our observations, the students also enjoyed and appreciated the class. They were happy and engaged, and the teacher was engaged with them. The students presented their work and asked questions, and the teacher responded and, when need be, corrected. She was kind, and she was firm. The class had a purpose and everyone clearly understood and appreciated that purpose.
The Nanjing school is not just a Hell we would kill for, it’s a Hell we know very well. The Nanjing class reminded us of nothing as much as our primary school from the 60s. Macleod State School was completely ordinary, just another cheap, flung-up middle class Melbourne school. It had grey walls and desks in rows, and hilariously bad heating. It also had bullies and authoritarian assholes and corporal punishment, and the worst teacher we ever experienced or ever witnessed.
Macleod State School also had classes where the teacher was the boss and was, properly, respected. There was a clear and meaningful curriculum. The teachers were expected to, and generally wished to, teach the curriculum. The students were expected to and generally wished to, learn the curriculum. The students also had very little say in the matter. The school had a purpose, a proper purpose, and in general everyone went about that purpose in a thoughtless and efficient manner.
The past is a foreign country.
Video: Mathematics in Hell
Below is the video of our recent LunchMaths talk. You can comment/correct below and/or at the YouTube link.
A big thanks to Lawrence and Emma-Jane for arranging the talk, and for making the zooming as painless as possible. A couple of aspects that I intended to talk about, and some probably valuable clarification, were only covered in the Q and A. I’ll leave it be except in reply to comments, except for one aspect that I really regret not getting to and which I’ll cover in a separate post ASAP.
Zooming into Friday: Mathematics in Hell
Because we’re so in love with technology, and because we’re so short of things to do and, mainly, because we’re so, so stupid, we’ve agreed to give a LunchMaths/MUMS talk via Zoom this Friday.
The details are below, and this link is supposed to work. Attempt to enter at your own risk.
UPDATE (24/08)
The video of the talk has been uploaded and can be viewed on YouTube and/or on this post.
A Question from a Teacher
A few days ago we received an email from Aaron, a primary school teacher in South Australia. Apparently motivated by some of our posts, and our recent thumping of PISA in particular, Aaron wrote on his confusion on what type of mathematics teaching was valuable and asked for our opinion. Though we are less familiar with primary teaching, of course we intend to respond to Aaron. (As readers of this blog should know by now, we’re happy to give our opinion on any topic, at any time, whether or not there has been a request to do so, and whether or not we have a clue about the topic. We’re generous that way.) It seemed to us, however, that some of the commenters on this blog may be better placed to respond, and also that any resulting discussion may be of general interest.
With Aaron’s permission, we have reprinted his email, below, and readers are invited to comment. Note that Aaron’s query is on primary school teaching, and commenters may wish to keep that in mind, but the issues are clearly broader and all relevant discussion is welcome.
Good afternoon, my name is Aaron and I am a primary teacher based in South Australia. I have both suffered at the hands of terrible maths teachers in my life and had to line manage awful maths teachers in the past. I have returned to the classroom and am now responsible for turning students who loathe maths and have big challenges with it, into stimulated, curious and adventure seeking mathematicians.
Upon commencing following your blog some time ago I have become increasingly concerned I may not know what it is students need to do in maths after all!
I am a believer that desperately seeking to make maths “contextual and relevant” is a waste, and that learning maths for the sake of advancing intellectual curiosity and a capacity to analyse and solve problems should be reason enough to do maths. I had not recognised the dumbing-down affect of renaming maths as numeracy, and its attendant repurposing of school as a job-skills training ground (similarly with STEM!) until I started reading your work. Your recent post on PISA crap highlighting how the questions were only testing low level mathematics but disguising that with lots of words was also really important in terms of helping me assess my readiness to teach. I have to admit I thought having students uncover the maths in word problems was important and have done a lot of work around that in the past.
I would like to know what practices you believe constitutes great practice for teaching in the primary classroom. I get the sense it involves not much word-problem work, but rather operating from the gradual release of responsibility (I do – we do – you do) explicit teaching model.
I would really value your thoughts around this.
Warm regards,
Aaron
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.