## WitCH 43: Period Piece

This one comes courtesy of a smart VCE student, the issue having been flagged to them by a fellow student. It is a multiple choice question from the 2009 Mathematical Methods, Exam 2; the Examination Report indicates, without comment, that the correct answer is D.

## WitCH 42: Multiple Issues

We’re really, really trying to avoid new WitCHes right now, but people keep throwing them at us. This one comes courtesy of Simon the Likeable, who suggested we watch the following, and who will pay dearly for it.

UPDATE (11/09/20)

Commenter has SRK has flagged a related video (and will pay dearly for it):

## 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 . 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.

## 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.

## WitCH 39: Field of Dreams

This WitCH comes courtesy of frequent commenter, SRK. It comes from the 2016 Specialist Mathematics Exam 2. The examination report does not comment, except to indicate that about 2/3 of students gave the intended answer of B.

## WitCH 38: A Deep Hole

This one is due to commenter P.N., who raised it on another post, and the glaring issue has been discussed there. Still, for the record it should be WitCHed, and we’ve also decided to expand the WitCHiness slightly (and could have expanded it further).

The following questions appeared on 2019 Specialist Mathematics NHT, Exam 2 (CAS). The questions are followed by sample Mathematica solutions (screenshot corrected, to include final comment) provided by VCAA (presumably in the main for VCE students doing the Mathematica version of Methods). The examination report provides answers, identical to those in the Mathematica solutions, but indicates nothing further.

## UPDATE (05/07/20)

The obvious problem here, of course, is that the answer for Part (b), in both the examination report and VCAA’s Mathematica solutions, is flat out wrong: the function fk will also fail to have a stationary point if k = -2 or k = 0. Nearly as bad, and plenty bad, the method in VCAA’s Mathematica solutions to Part (c) is fundamentally incomplete: for a (twice-differentiable) function f to have an inflection point at some a, it is necessary but not sufficient to have f’’(a) = 0.

That’s all pretty awful, but we believe there is worse here. The question is, how did the VCAA get it wrong? Errors can always occur, but why specifically did the error in Part (b) occur, and why, for a year and counting, wasn’t it caught? Why was a half-method suggested for Part (c), and why was this half-method presumably considered reasonable strategy for the exam? Partly, the explanation can go down to this being a question from NHT, about which, as far as we can tell, no one really gives a stuff. This VCAA screw-up, however, points to a deeper, systemic and much more important issue.

The first thing to note is that Mathematica got it wrong: the Solve function did not return the solution to the equation fk‘ = 0. What does that imply for using Mathematica and other CAS software? It implies the user should be aware that the machine is not necessarily doing what the user might reasonably think it is doing. Which is a very, very stupid property of a black box: if Solve doesn’t mean “solve”, then what the hell does it mean? Now, as it happens, Mathematica’s/VCAA’s screw-up could have been avoided by using the function Reduce instead of Solve.* That would have saved VCAA’s solutions from being wrong, but not from being garbage.

Ask yourself, what is missing from VCAA’s solutions? Yes, yes, correct answers, but what else? This is it: there are no functions. There are no equations. There is nothing, nothing at all but an unreliable black box. Here we have a question about the derivatives of a function, but nowhere are those derivatives computed, displayed or contemplated in even the smallest sense.

For the NHT problem above, the massive elephant not in the room is an expression for the derivative function:

What do you see? Yep, if your algebraic sense hasn’t been totally destroyed by CAS, you see immediately that the values k = 0 and k = -2 are special, and that special behaviour is likely to occur. You’re aware of the function, alert to its properties, and you’re led back to the simplification of fk for these special values. Then, either way or both, you are much, much less likely to screw up in the way the VCAA did.

And that always happens. A mathematician always gets a sense of solutions not just from the solution values, but also from the structure of the equations being solved. And all of this is invisible, is impossible, all of it is obliterated by VCAA’s nuclear weapon approach.

And that is insane. To expect, to effectively demand that students “solve” equations without ever seeing those equations, without an iota of concern for what the equations look like, what the equations might tell us, is mathematical and pedagogical insanity.

*) Thanks to our ex-student and friend and colleague Sai for explaining some of Mathematica’s subtleties. Readers will be learning more about Sai in the very near future.

## WitCH 37: A Foolproof Argument

We’re amazed we didn’t know about this one, which was brought to our attention by commenter P.N.. It comes from the 2013 Specialist Mathematics Exam 2: The sole comment on this question in the Examination Report is:

“All students were awarded [the] mark for this question.”

Yep, the question is plain stuffed. We think, however, there is more here than the simple wrongness, which is why we’ve made it a WitCH rather than a PoSWW. Happy hunting.

UPDATE (11/05) Steve C’s comment below has inspired an addition:

## Update (20/05/20)

The third greatest issue with the exam question is that it is wrong: none of the available answers is correct. The second greatest issue is that the wrongness is obvious: if lies in a sector then the natural guess is that will lie in one of three equally spaced sectors of a third the width, so God knows why the alarm bells weren’t ringing. The greatest issue is that VCAA didn’t have the guts or the basic integrity to fess up: not a single word of responsibility or remorse. Assholes.

Those are the elephants stomping through the room but, as commenters as have noted, there is plenty more awfulness in this question:

• “Letting” is sloppy, confusing and pointless;
• The term “quadrant” is undefined;
• The use of “principal” is unnecessary;
• “argument” is better thought as the measure of an angle not the angle itself;
• Given is a single complex number, “the complete set of values for Arg()” will consist of a single number.
• The grammar isn’t.

## WitCH 36: Sub Standard

This WitCH is a companion to our previous, MitPY post, and is a little different from most of our WitCHes. Typically in a WitCH the sin is unarguable, and it is only the egregiousness of the sin that is up for debate. In this case, however, there is room for disagreement, along with some blatant sinning. It comes, predictably, from Cambridge’s Specialist Mathematics 3 & 4 (2020).

## WitCH 35: Overly Resolute

This WitCH (arguably a PoSWW) comes courtesy of Damien, an occasional commenter and an ex-student of ours from the nineteenth century. It is from the 2019 Specialist Mathematics Exam 2. We’ll confess, we completely overlooked the issue when going through the MAV solutions.

Update (16/02/20)

What a mess. Thanks to Damo for pointing out the problem, and thanks to the commenters for figuring out the nonsense.

In general form, the (intended) scenario of the exam question is

The vector resolute of in the direction of is ,

which can be pictured as follows: For the exam question, we have , and .

Of course, given and it is standard to find . After a bit of trig and unit vectors, we have (in must useful form)

The exam question, however, is different: the question is, given and , how to find .

The problem with that is, unless the vectors and are appropriately related, the scenario simply cannot occur, meaning  cannot exist. Most obviously, the length of must be no greater than the length of . This requirement is clear from the triangle pictured, and can also be proved algebraically (with the dot product formula or the Cauchy-Schwarz inequality).

This implies, of course, that the exam question is ridiculous: for the vectors in the exam we have , and that’s the end of that. In fact, the situation is more delicate; given the pictured vectors form a right-angled triangle, we require that be perpendicular to . Which implies, once again, that the exam question is ridiculous.

Next, suppose we lucked out and began with perpendicular to . (Of course it is very easy to check whether we’ve lucked out.) How, then, do we find ? The answer is, as is made clear by the picture, “Well, duh”. The possible vectors are simply the (non-zero) scalar multiples of , and we’re done. Which shows that the mess in the intended solution, Answer A, is ridiculous.

There is a final question, however: the exam question is clearly ridiculous, but is the question also stuffed? The equations in answer A come from the equation for above and working backwards. And, these equations correctly return no solutions. Moreover, if the relationship between and had been such that there were solutions, then the A equations would have found them. So, completely ridiculous but still ok?

Nope.

The question is framed from start to end around definite, existing objects: we have THE vector resolute, resulting in THE values of m, n and p. If the VCAA had worded the question to find possible values, on the basis of a possible direction for the resolution, then, at least technically, the question would be consistent, with A a valid answer. Still an utterly ridiculous question, but consistent. But the VCAA didn’t do that and so the question isn’t that. The question is stuffed.

Further Update (26/06/20)

As commenters have noted, the Examination Report has finally appeared. And, as predicted, answer A was deemed correct, with the Report noting

Option A gives the set of equations that can be used to obtain the values of m, n and p. Explicit solution would result in a null set as it is not possible for a result of a vector to be of greater magnitude than the vector itself.

Well, it’s something. Presumably “result of a vector” was intended to be “resolute of a vector”, and the set framing is weirdly New Mathy. But, it’s something. Seriously. As John Friend notes, it is at least a small step along the way to indicating the question is not all hunky-dory.

That step, however, is way too small. We’ll close with two comments, reiterating the points made above.

1. The question is wrong

Read the question again, and read the first sentence of the Report’s comment. The question and report justification are fundamentally stuffed by the definite articles, by the language of existence. All answers should have been marked correct.

2. The question is worse than wrong

Even if the vectors and had been chosen appropriately, the question is utterly devoid of mathematical sense. It suggests a long and difficult method to solve a problem that, if indeed is solvable, is trivial.

## WitCH 34: Jumanji, the Numeracy Level

This adventure game comes courtesy of Victoria’s Department of Education and Training. Click on the graphic, or go here. (Don’t try to do it all. You will die.) If you can be bothered, you can also complete a survey here.