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,
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.
To be more precise, what does “digital technology” mean and, precisely as possible, how is Digital Technology X used in Year Y of schooling? If you confused, then why not find out more about this here.
It is now impossible, of course, to write a document on education without genuflecting to the God of Technology. The repetitious chanting of “technology”, like a wired Tibetan monk, is the way people with no sense of the past or the present indicate how hip they are with the future. But, what do they mean? What technology are they talking about? It is a serious question, of which we only vaguely know the answer. We want help.
Of course by “technology”, the Education Experts are never intending to refer to something like blackboards and chalk. They would not even recognise such primitive devices as products of technology, although of course they are. No, what the EE mean by “technology” is electronic devices, mostly computers and computer programs, and preferably devices that are internetted. So, calculators and electronic whiteboards and Mathletics and Reading Eggs and iPads, and so forth.
The question is, precisely how are these devices used in specific classrooms? For example, are calculators used in Year 5 to perform arithmetic calculations, or to check calculations that have been done by hand? Is Mathletics used in Year 7 to teach ideas or to test knowledge and/or skills?
The same question applies to all subjects. Are word processors used in Year 6 to check and/or teach spelling and grammar? Are iPads used in Year 8 to check the definitions of words?
We want to know as much as possible, and as specifically as possible, what electronic gizmos are being used, and with whom and how.
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.
Predictably, last week’s talk ran short of time, and we were forced to skip some slides. The most regrettable omission was a slide titled “How to Teach …”, the motivation for which was to talk about the man in the photograph above, and about the photograph.
Our approach to teaching is, shall we say, eccentric. We won’t comment on the effectiveness of our teaching but, if “method” is too strong a word, there is an underlying idea. This idea is best captured by Ralph Waldo Emerson, writing upon writing: “The way to write is to throw your body at the mark when your arrows are spent”. Even if it indicates one way to teach, however, Emerson’s quote is of course not a dictum on teaching. Teaching is communication, and every teacher has to determine for themselves how they can best communicate ideas to their students.
Which brings us, almost, to the man in the fuzzy photograph. For the twenty years we were involved in the popularisation of mathematics, including the giving of and arranging of presentations, we were privileged to witness a number of great teachers. The brilliant John Conway was a stand-out, of course, as was Art Benjamin. But there were also two Australian mathematicians that were truly and particularly memorable.
The first mathematician was Mike Deakin. We mentioned Mike in last week’s talk, as one of our go-to guys when we started LunchMaths at Monash, and he gave a number of beautiful talks. Before that, Mike was, for decades, an editor, proofreader, janitor and mega-contributor for Monash’s mathematics magazine, Function.
The other mathematician was, finally, the man in the fuzzy photograph above: that is E. R. Love, who was professor of mathematics at the University of Melbourne for about three hundred years. In 1992, when Professor Love was 80, Terry Mills encouraged us to invite Professor Love to give a talk to the mathematics department at LaTrobe, Bendigo. We did so and Professor Love accepted. Declining multiple offers to be driven, Professor Love took the train to Bendigo and gave an absolutely beautiful talk on Legendre functions. Afterwards, over lunch, Professor Love entertained all with stories of Cambridge in the 30s.
Why write about Mike Deakin and, especially, Professor Love? Well, why not, of course; Deakin and Love were great contributors to Australian mathematics and deserve to be remembered and honoured. There was a specific reason, however, why we thought they were relevant to our talk, and why we particularly regret not having included acknowledgment of Professor Love: they were great teachers in a manner ceasing to exist. They were great lecturers.
Mike Deakin, who was an undergraduate at the University of Melbourne and then a Masters student under Professor Love, reminisces here on Professor Love’s teaching:
Love, in particular, was a superb lecturer. It was said of him that he was a menace because he made his subject seem so straightforward and logical that one missed seeing its difficulties.
The point is not that Mike Deakin and Professor Love were popular lecturers; the point is that they lectured in a careful, scholarly manner that is being lost. Their lectures had no gimmicks, had none of the crazy showmanship of the Mathologer, or of the writer of this blog. They simply lectured, conveying carefully crafted ideas to an audience willing and keen to listen. And, the point is that almost no one now recognises this, or cares, or can even properly understand. Almost no one under the age of fifty can realise that what is being lost is an art form, and an extremely beautiful and valuable one.
The title of this blog post is a play on Neil Postman‘s book titled Building a Bridge to the Eighteenth Century, which was in turn a play on a Clintonism. Postman’s excellent, and final, book was written in 1999. It was concerned with society’s inability to understand and to cope with technology, and the consequent loss of tradition and authority, of wisdom and plain meaning. Subtitled How the Past can Improve our Future, Postman’s book argued that we should look back to the 18th century, to the Enlightenment, for guidance into the future.
And now, twenty years later? The idea of building a bridge to the eighteenth century seems utterly fantastic, and perhaps always was. Twenty years on, and there is scarcely a memory of the twentieth century. The photo above was the best, the only photo we could find of Professor Love.
Mike Deakin and E. R. Love are dead, and they are being forgotten. The scholarly tradition they represented, the gift they gave, is being lost. And no one cares.
Gareth Ainsworth has contacted us, noting that Scotch College had an obituary for E. R. Love, which included a short biography and a photograph.
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.
This is way unimportant in comparison to the current idiocy of the neoliberal nutjobs. But, as they say in the theatre, the shitshow must go on.*
We had thought of taking this further whack at Bambi a while back, but had decided against it. Over the week-end, however we were discussing related mathematics with Simon the Likeable, and that has made us reconsider:
Get to work.
*) Mostly Andrew Lloyd Webber productions.
We were working on an update to polish off this one, when Simon the Likeable pointed out to us the video below. It could easily be its own WitCH, but it fits in naturally here (and also with this WitCH).
We’ll give people a chance to digest (?) this second video, and then we’ll round things off.
OK, time to round this one off, although our rounding off may inspire objection and further comment. We will comment on four aspects of the videos, the third of which we regard the most important, and the fourth of which is mostly likely to cause objection.
The first thing to say about these videos is that, as examples of teaching, they are appalling; they are slow and boring and confusing, simultaneously vague and muddyingly detailed. In particular, the “repeated addition” nonsense is excruciating, and entirely unnecessary. You want us to think of division as “how many”, then fine, but don’t deliver a kindergarten-level speech on it.
Eddie Woo’s video has the added charm that at times no one seems to give a damn what anyone else is saying; particularly notable is the 6:00 mark, where the girl suggests “Therefore it’s [i.e. 1/0 is] undefined?”, the very point Eddie wants to make, and Eddie pointedly ignores her so he can get on with his self-aggrandizing I’m-So-Wonderful performance. Dick.
The second thing to say is that the Numberphile video is littered with errors and non sequiturs, the highlights being their dismissing infinity as an “idea” (as if 3 isn’t), and their insane graph of . We’ll go through this in detail when we update this WitCH (scheduled for sometime in 2023).
The third thing to say is that the videos’ discussion of the impossibility of defining 1/0 gives a fundamentally flawed view of mathematical thought. The entire history of mathematics is of mathematicians breaking the rules, of doing the impossible. (John Stillwell has written a beautiful book, in fact two beautiful books, on the history of mathematics from this perspective.) As such, one should be very careful in declaring mathematical ideas to be impossible. So, 1/0 may generally not be defined (at school), but is it, as Eddie declares, “undefinable”?
Of course taken literally, Eddie’s claim is silly; as we suggested in the comments, we can define 1/0 to be 37. The real question is, can one define 1/0 in a meaningful manner? There are reasonable arguments that the answer is “no”, but these arguments should be laid out with significantly more care than was done in the videos.
The first argument for the (practical) undefinability of 1/0 is that we’ll end up with 1/0 = 2/0, leading to 1 = 2. What is really being claimed here? Why is 1/0 = 2/0, and why should it lead to 1 = 2?
The heart of this approach is asking whether 0 can have a multiplicative inverse. That is, is there a number, let’s call it V, with 0 x V = 1? Of course V couldn’t be an everyday real number (not that real numbers are remotely everyday), but that’s neither here nor there. It took a hugely long time, for example, for mathematicians to leave the safety of the world of everyday (?) integers and to discover/create an inverse for 3.
Well, what goes wrong? If we have such a number V then 1/0 stands for 1 x V. Similarly 2/0 stands for 2 x V. So, does it follow that 1 x V = 2 x V? No, it does not. V only has the properties we declare it to have, and all we have declared so far is that V x 0 = 1.
Of course this is cheating a little. After all, we want V to be an infinityish thing, so let’s concede that 1 x V and 2 x V will be equal. Then, if we assume that the normal (field) rules of algebra apply to V, it is not hard to prove that 1 = 2. That assumption is not necessarily unreasonable but it is, nonetheless, an assumption, the consequences of that assumption require proof, and all of this should be clearly spelled out. The videos do bugger all.
The second argument for the undefinability of 1/0, at least as an infinity thing, is the limit argument, that since tiny numbers may be either positive or negative, we end up with 1/0 being both and , which seems a strange and undesirable thing for infinity to do. But, can we avoid this problem and/or is there some value, in a school setting, of considering the two infinities and having them equal? The videos do not even consider the possibilities.
The fourth and final thing to note is that, as we will now argue, we can indeed make sense of 1/0 as an infinity thing. Moreover, we believe this sense is relevant and valuable in the school context. Now, to be clear, even if teachers can introduce infinity and 1/0, that doesn’t imply they necessarily should. Perhaps they should, but it would require further argument; just because something is relevant and useful does not imply it’s wise to give kids access to it. If you’re collecting wood, for example, chain saws are very handy, however …
First, let’s leave 1/0 alone and head straight to infinity. As most readers will know, and as has been raised in the comments, mathematicians make sense of infinity in various ways: there is the notion of cardinality (and ordinality), of countable and uncountable sets; there is the Riemann sphere, adding a point at infinity to the complex plane; there is the real projective line, effectively the set of slopes of lines. Cardinality is not relevant here, but the Riemann sphere and projective line definitely are; they are both capturing 1/0 as an infinity thing, in contexts very close to standard school mathematics. And, in both cases there is a single infinity, without plusses or minuses or whatever. Is this sufficient to argue for introducing these infinities into the classroom? Perhaps not, but not obviously not; infinite slopes for vertical lines, for example, and with no need for a plus or minus, is very natural.
What about the two-pronged infinity, the version that kids naturally try to imagine, with a monster thing at the plus end and another monster thing at the minus end? Can we make sense of that?
Yes, we can. This world is called the Extended Real Line. You can watch a significantly younger, and significantly hairier, Marty discussing the notions here.
The Extended Real Line may be less well known but it is very natural. What is in this world? Take a guess. Or, ? It all works just how one wishes.
But what about when it doesn’t work? You want to throw or or at us? No problem: we simply don’t take the bait, and any such “indeterminate form” we leave undefined. In particular, we make no attempt to have be the multiplicative inverse of 0. And, then, modulo these no-go zones, the algebra of the Extended Real Line works exactly as one would wish.
Can these ideas be introduced in school, and for some purpose? No question. Again, whether one should is a trickier question. But as soon as the teacher, perhaps in hushed and secretive tones, is suggesting or , then maybe they should also think about this in a less Commandments From God manner, and let come properly out of the closet.
Finally, what about 1/0 in the Extended Real Line? Well, the positive or negative thing is definitely an issue. Unless it isn’t.
There are many contexts where we naturally restrict our attention to the nonnegative real numbers. And, in any such context 1/0 is not at all conflicted or ambiguous, and we can happily declare . The exact trig values from 0 to 90 is just such a context: in this context we think it is correct and distinctly helpful to write , rather than resorting to a what-the-hell-does-that-mean “undefined”.
That’s it. That’s a glimpse of the huge world of possibilities for thinking about infinity that Numberphile and Woo dismiss with an arrogant, too-clever-by-half hand. Their videos are not just bad, they are poisonously misleading for their millions of adoring, gullible fans.
This is one of those WitCHes we’re going to regret. Ideally, we’d just write a straight post but we just have no time at the moment, and so we’ll WitCH it, hoping some loyal commenters will do some of the hard work. But, in the end, the thing will still be there and we’ll still have to come back to polish it off.
This WitCH, which fits perfectly with the discussion on this post, is an article (paywalled – Update: draft here) in the Journal of Mathematical Behaviour, titled
Elementary teachers’ beliefs on the role of struggle in the mathematics classroom
The article is by (mostly) Monash University academics, and a relevant disclosure: we’ve previously had significant run-ins with two of the paper’s authors. The article appeared in March and was promoted by Monash University a couple weeks ago, after which it received the knee-jerk positive treatment from education reporters stenographers.
Here is the abstract of the article:
Reform-oriented approaches to mathematics instruction view struggle as critical to learning; however, research suggests many teachers resist providing opportunities for students to struggle. Ninety-three early-years Australian elementary teachers completed a questionnaire about their understanding of the role of struggle in the mathematics classroom. Thematic analysis of data revealed that most teachers (75 %) held positive beliefs about struggle, with four overlapping themes emerging: building resilience, central to learning mathematics, developing problem solving skills and facilitating peer-to-peer learning. Many of the remaining teachers (16 %) held what constituted conditionally positive beliefs about struggle, emphasising that the level of challenge provided needed to be suitable for a given student and adequately scaffolded. The overwhelmingly positive characterisation of student struggle was surprising given prior research but consistent with our contention that an emphasis on growth mindsets in educational contexts over the last decade has seen a shift in teachers’ willingness to embrace struggle.
And, here is the first part of the introduction:
Productive struggle has been framed as a meta-cognitive ability connected to student perseverance (Pasquale, 2016). It involves students expending effort “in order to make sense of mathematics, to figure out something that is not immediately apparent” (Hiebert & Grouws, 2007, p. 387). Productive struggle is one of several broadly analogous terms that have emerged from the research literature in the past three decades. Others include: “productive failure” (Kapur, 2008, p. 379), “controlled floundering” (Pogrow, 1988, p. 83), and the “zone of confusion” (Clarke, Cheeseman, Roche, & van der Schans, 2014, p. 58). All these terms describe a similar phenomenon involving the intersection of particular learner and learning environment characteristics in a mathematics classroom context. On the one hand, productive struggle suggests that students are cultivating a persistent disposition underpinned by a growth mindset when confronted with a problem they cannot immediately solve. On the other hand, it implies that the teacher is helping to orchestrate a challenging, student-centred, learning environment characterised by a supportive classroom culture. Important factors contributing to the creation of such a learning environment include the choice of task, and the structure of lessons. Specifically, it is frequently suggested that teachers need to incorporate more cognitively demanding mathematical tasks into their lessons and employ problem-based approaches to learning where students are afforded opportunities to explore concepts prior to any teacher instruction (Kapur, 2014; Stein, Engle, Smith, & Hughes, 2008; Sullivan, Borcek, Walker, & Rennie, 2016). This emphasis on challenging tasks, student-centred pedagogies, and learning through problem solving is analogous to what has been described as reform-oriented mathematics instruction (Sherin, 2002).
Stein et al. (2008) suggest that reform-oriented lessons offer a particular vision of mathematics instruction whereby “students are presented with more realistic and complex mathematical problems, use each other as resources for working through those problems, and then share their strategies and solutions in whole-class discussions that are orchestrated by the teacher” (p. 315). An extensive body of research links teachers’ willingness to adopt reform-oriented practices with their beliefs about teaching and learning mathematics (e.g., Stipek, Givvin, Salmon, & MacGyvers, 2001; Wilkins, 2008). Exploring teacher beliefs that are related to reform-oriented approaches is essential if we are to better understand how to change their classroom practices to ways that might promote students’ learning of mathematics.
Although teacher beliefs about, and attitudes towards, reform-oriented pedagogies have been a focus of previous research (e.g., Anderson, White, & Sullivan, 2005; Leikin, Levav-Waynberg, Gurevich, & Mednikov, 2006), teacher beliefs about the specific role of student struggle has only been considered tangentially. This is despite the fact that allowing students time to struggle with tasks appears to be a central aspect to learning mathematics with understanding (Hiebert & Grouws, 2007), and that teaching mathematics for understanding is fundamental to mathematics reform (Stein et al., 2008). The purpose of the current study, therefore, was to examine teacher beliefs about the role of student struggle in the mathematics classroom.
The full article is available here, but is paywalled (Update: draft here). (If you really want it …)
It is not appropriate this time to suggest readers have fun. We’ll go with “Good luck”.
Jerry in the comments has located a draft version of the article, available here. We haven’t compared the draft to the published version.
Yesterday, Bach had an op-ed in the official organ of the Liberal Party (paywalled, thank God). Titled We must raise our grades on teacher quality, Bach’s piece was the predictable mix of obvious truth and poisonous nonsense, promoting the testing of “numeracy” and so forth. One line, however, stood out as a beacon of Bachism:
“But, as in any profession, a small number of teachers is not up to the mark.”
This MitPY is a request from frequent commenter, Red Five:
I’d like to ask what others think of teaching (mostly linear) Diophantine equations in early secondary school. They are nowhere in the curriculum but seem to be everywhere in competitions, including the AMC junior papers on occasion. I don’t see any reason to not teach them (even as an extension idea) but others may have some insights into why it won’t work.