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 one could be a WitCH or a PoSWW, except we’re withholding some information. So, consider it an analysis puzzle.
The following was posted to an Australia-NZ statistics list. The email was from an American statistician, referring to and quoting from this article(Update: link fixed), on a Texas hospital adopting critical care guidelines. See if you can identify the problem.
A weird hint: we are not playing fair. (And, boy are people gonna be pissed.)
As we wrote in reply to a comment “There is the post, and the links to which the post refers. You can consider what ever issues you see.” So maybe list quickly, without elaboration, anything you see as an issue.
OK, time to end this, and so a final hint. As indicated in the comments, the “problem” is with the poster’s line “Statistics is a bitch”. We’re very, very pleased that no one has hit upon the “problem” with this line.
FINAL UPDATE (01/08)
Well, wasn’t that fun? Thanks to everyone for playing along.
As indicated above, the “problem” is with the line “Statistics is a bitch”. And what’s wrong with that line? Not a whole lot. It’s not a great line, since it can be read as treating the “statistics” as a from-nowhere reality, rather than the disastrous consequences of Republican screw-up. But no big deal.
So, why post on this as a problem? Because the stats email list to which the comment was posted thought it was a problem. A number of commenters took very serious issue with the poster’s use of the word “bitch”.
This began with an off-post email to the poster, indicating “the language used is not at all appropriate for [such] an email list” and a request: “I’d appreciate it if you could apologise for this choice of words.” (To whom?) The poster replied to the email list, with a long and unhelpful, but fundamentally reasonable, non-apology apology. In brief, the poster, who is in Texas, suggested that they had much, much bigger things to worry about. And then the bashing kicked in.
There were calls for the poster to “grow up”, to “stop using hurtful, offensive language”, suggestions that “the problem is the use of a term that is all too often directed at women in a derogatory way” (ignoring that this was not the case here), whining about “gendered name-calling”, and all manner of nitpicky and gratuitous complaint.
It was crazy and it was revolting, and all of it coming from proud and proper academics. Eventually there was some tepid defence of the poster, but way too little and way too late. No one stood up properly to these ridiculous, self-important language nazis.
Which is why we posted about it here. Of course this is the type of blog where those offended by strong language are unlikely to hang around. And, maybe some shyer types here agree with the poster’s critics. But, it was still very, very pleasing that no one who engaged here had a clue what we could have been on about.
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.
This is an open offer to review Methods and Specialist SACs. Here are the conditions:
0) The review is free. (You can consider donating to Tenderfeet.)
1) You may email me any Methods or Specialist SAC, by anyone.
2) You should indicate whether or not you are the writer of the SAC.
3) If you are the writer of the SAC, I will be diplomatic.*
4) It’s on your head, in particular for future SACs, if you’re breaking confidentiality rules or conventions. This is not my concern.
5) I will keep all SACs confidential, except to the extent there is explicit agreement otherwise. (See 12-14, below.)
6) Future SACs should, at minimum, be close to a final draft.
7) All SACs should include solutions and a grading scheme.
8) I may decline to review a SAC for being too old, or for other reasons.
9) I will review only for mathematical sense and mathematical correctness.
10) In particular, I will not check for, and do not give a stuff about, VCAA compliance.
11) I will not check all arithmetic and a review should not be taken as a guarantee that the SAC is error-free.
12) Each time I review a SAC I will record so below, with brief and, modulo points 13 and 14, anonymity-preserving comments.
13) I will identify commercial SACs as such, possibly indicating the commercial entity.
14) If you are the author of the SAC and you agree, I will consider making a separate post, to review the SAC in detail and to allow for comment.
I will be interested to see who is brave enough to enter (and who is tossed into) the lion’s den.
*) Yes, I am capable of diplomacy. I just prefer to do without.
We have our first taker: a brave soul has entered the den. I’ll look at the proffered SAC asap. I was also asked what I am after, in making this offer, which is a fair question. The answer is two-fold:
a) (Jekyll) I’m making a genuine offer to provide a critique of a SAC from a mathematical perspective, for any writer who wants it. I’m hoping that by providing such a critique, the writer will become more attuned to any mathematical shortcomings in their (and all) SACs, and in VCE generally. Hopefully then, to the limited extent that VCAA’s idiot curriculum permits it, this will help the writer produce more mathematically coherent and rich SACs in the future.
b) (Hyde) I’m looking to see as much as I can of the nonsense the SAC system is producing. This will allow me to confirm for any teacher or student who has been served swill that they have indeed been served swill. It will also allow me to write upon such SACs, even if in very oblique terms.
OK, this post is being steered away from what I intended, but I’m happy to let others steer.
First, a clarification. By “SAC”, I mean any school-based Year 12 assessment that counts towards the final VCE grade. I don’t care if the assessment takes five minutes or five days.
Now, the question is what to do with SACs offered to me by authors? I have two currently. I can either
a) Make the SACs into posts on this blog. The SACs would then be a basis for discussion, and a model for future SACs, but the SACs themselves would presumably not be usable. (Again, I don’t give a stuff about protocol, but obviously teachers must.)
b) Keep the SACs off the site, except for brief comments below, and set up a Free SACs to Good Home post. Teachers can then contact me to obtain copies.
Readers can suggest to me what they prefer. They can also suggest how (b) might work in practice.