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 Simple Message to Primary Schools About Multiplication Tables

Dear Primary Schools,

If your students are not learning their multiplication tables, up to 12, by heart, then you are fucking up.

If you think giving your students a grab-bag of tricks replaces multiplication tables, then you are fucking up.

If you think orchestrating play-based, student-centred theatricalities replaces multiplication tables, then you are fucking up.

If you think quoting the Australian Curriculum gives you license to not teach multiplication tables, then you are fucking up.

If you think quoting some education twat gives you license to not teach multiplication tables, then you are fucking up.

Thank you for your attention.

WitCH 41: Zero Understanding

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.

UPDATE (9/8)

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.

UPDATE (12/08/20)

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 \boldsymbol{x^x}. 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 \boldsymbol{+\infty} and \boldsymbol{-\infty}, 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 \boldsymbol{\infty + \infty} in this world? Take a guess. Or, \boldsymbol{\infty +3}? It all works just how one wishes.

But what about when it doesn’t work? You want to throw \boldsymbol{\infty  - \infty} or \boldsymbol{0\times  \infty} or \boldsymbol{\frac{\infty}{\infty}} 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 \boldsymbol{\infty} 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 \boldsymbol{\infty + \infty =\infty} or \boldsymbol{\frac1{\infty} =0}, then maybe they should also think about this in a less Commandments From God manner, and let \boldsymbol{\infty} 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 \boldsymbol{\frac10 =\infty}. 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 \boldsymbol{\tan(90) = \infty}, 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.

Critical Issues: Spot the Error

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.

UPDATE (30/7)

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.

UPDATE (31/7)

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.

WitCH 40: The Primary Struggle

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”.

UPDATE (28/7)

Jerry in the comments has located a draft version of the article, available here. We haven’t compared the draft to the published version.

An Offer: Checking For SACking Offences

Pauline Baynes

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.

UPDATE (26/7)

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.

UPDATE (27/7)

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.)

or

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.

A Request: Auditory Lapses

It’s SAC time and, as indicated below, I have a request for people to contact me.

A couple days ago I was talking to a Head of Maths, who suggested/requested/pleaded that I take a whack at the auditing of SACs. In principle, I’d love nothing more. (Well, I’d love a bottle of Laphroaig more, but you get the point.) Maths SACs are a soul-drowning swampland and consequently, and independently, the auditing of SACs is a Kafkaesque nightmare. That is currently amplified to 11, with the VCAA making its astonishingly stupid and ineptly delayed decision to maintain SACs during a plague year.

The difficulty with me writing on SAC audits is that, although I am generally aware of the brain-drilling arbitrariness in SAC auditing, I seldom see the specific idiocies with a duty drawback. And, this is a case where the idiot devil is most clearly evident in the idiot details. Hence my request:

If anyone has a SAC audit horror story, please DO NOT provide the details as a comment below, but please feel free to email me.

Then, if you wish, we can chat about your horror story. Of course, I will maintain all confidences, and I will not use any information, even in an anonymised manner, without clear and specific agreement.

Even with information in hand, natural concerns for confidentially make this a very difficult topic upon which to write. I have no idea what I might be able to do. But, the first step is to see what there is to see.

Again, please don’t include specifics in the comments below, although general bitching is appropriate, welcome and to be expected.

PoSWW 12: They is Bach

There’s much we could write about Matthew Bach, who recently gave up teaching and deputying to become a full-time Liberal clown. But, with great restraint, we’ll keep to ourselves the colourful opinions of Bach’s former school colleagues; we’ll ignore Bach’s sophomoric sense of class and his cartoon-American cry for “freedom”; we’ll just let sit there Bach’s memory of “the sense of optimism in Maggie Thatcher’s Britain”.

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.”

We is thinking that is very, very true.

How to Cancel Culture

Yes, of course we’ve read on this nonsense, as have probably most readers of this blog; it’s either that or reruns of Bachelor Chef Island. For those divorced from virtual reality, however, here is:

There are plenty of left-wing thugs pretending that “cancel culture” is a fraud, and there are plenty of right-wing thugs pretending that they and their thuggish cronies don’t play the same nasty sport. For us, the smartest takes are:

But, really, “cancel culture” is what we gotta solve right now? One would think that being inundated with plague, and having the World heating to the point of no return, and with the three superpowers being led by homicidal maniacs, that would be plenty enough on our to-do list. But, if taking hair-trigger offense and being a Blockleiter is your thing then, sure, go ahead and have fun.

UPDATE (21/07)

A new article:

Read it. Yeah, you’re busy but, trust me, read it.