## WitCH 120: Unanalytic Continuation

Ten years ago, Numberphile came out with a now famous/infamous video featuring physicist Tony Padilla, in which Padilla “proves” that 1 + 2 + 3 + … = -1/12. It was quickly followed by a second, “what we really mean” video and some rebuttal notes by Padilla. At the time, Burkard and I (too) softly hammered Numberphile for this, in one of our final Age columns. A few years later, Burkard hammered Numberphile much harder, in an early Mathologer video.* Now, Numberphile has come out with two new videos: The return of -1/12, featuring Berkeley mathematician Tony Feng; and Does -1/12 protect us from infinity, with Padilla.

I have my views, but this a WitCH (thus summarising my views). So it’s your job. The two new videos are below. Go for it.

*) See also my post defending Burkard’s uncollegial hammering of Numberphile, an earlier WitCH on Numberphile (and Too Much Woo), a post on my general dislike of mathematics documentaries and on video as a medium for maths ed, and a video containing my proof that 1 + 2 + 3 (no dots) equals -1/12 (48:45 and 55:00).

## Can a Maths Ed Theory Give Back More Than It Takes?

A year or so ago, a decades-long friend and colleague reappeared. My friend also has a strong interest in maths ed, although she takes the “Let’s all be friends” approach. Readers of this blog know well that I’ve given up on that, but still my friend and I can argue amicably about this and that. In particular, she took some issue with my “all modern maths ed sucks” post. While conceding that most educational research is bad, she was unwilling to write off the discipline entirely and she suggested a few things for me to read. I gave them a semi-decent try, and my response was “meh”. While the stuff she suggested was mostly reasonable, or even good, I felt it was, at best, addressing third order issues. One article, however, has had me pondering a little, and I thought it may be worth sharing.

## A Maths Ed Lecture From Long Ago

I have a heavy post coming very soon, but it seems worthwhile first getting in this quick, light one.

Once upon a time, before going off the rails, I was a (semi)regular mathematician. I proved theorems and stuff like that. I was a committed lecturer and, with all due humility, a very good lecturer, but I had no specific interest in “mathematics education” and I knew nothing about school mathematics. That began to change around twenty years ago when, back in Melbourne, I somewhat randomly began talking to mathematics teachers. I soon realised that most Victorian mathematics teachers, even very dedicated ones, knew little mathematics and understood less. I began giving talks to teachers and then public talks, and I discovered the obvious about myself: I am significantly better at telling jokes than proving theorems. Burkard then appeared and it all took off, first with the popularisation, which Burkard has continued, and now with my gadflying. Some years earlier, however, before all this began, I bumped into my future occupation, and into a maths ed titan. This is the story of that bump. Continue reading “A Maths Ed Lecture From Long Ago”

## Do the Maths: Why Mathematics Education is Failing Our Kids

The following is our complete Arena article, which we announced a couple weeks ago. It includes footnotes and references and links that didn’t make it into the Arena version.

## Arena: Do the Maths

A few months ago, I asked blog readers for suggestions for what had gone wrong with mathematics education. Plenty of discussion ensued, and it was very enlightening.

The purpose of that blog post was to help with the writing of an article. Guy Rundle had suggested that I write something on mathematics education for the magazine Arena. The article has now appeared (and in print). The introduction to the article is below and the full article can be read at Arena. Thank you to Guy for the opportunity, thank you all for your suggestions, and thank you to my secret crack team of hypercritical editors. (I have no idea of the significance of Arena’s graphic, but that’s cool, and it’s cool.)

## An Educational Conjecture

OK, time for a competition (And, no, we haven’t forgotten our previous competition, which we shall revive at some stage.)

Here is a conjecture:

Every new idea in modern mathematics education is either trivial or false.

Can we prove this conjecture? Of course not. Not, at least, if we do not Learn More about it and without reading thousands of pages of educational gobbledegook. (Which. We. Will. Not. Do.) Is the conjecture true? We don’t know. But we are not aware of any counterexample.

The competition, then, is to attempt to prove the conjecture false. The winner is the commenter(s) who comes up with and argues most persuasively for a counterexample: the new idea in modern mathematics education that is true and the least trivial. The winner(s) will receive a signed copy of the number one best seller,* A Dingo Ate My Math Book.

Just a few notes on the parameters:

• By “idea”, we mean any claim about or approach to teaching or learning mathematics.
• By “new” we mean something other than dressing up a traditional idea in new clothing.
• By “modern”, we mean from the last fifty years or so, back to about 1970.
• By ”least trivial”, we mean something of genuine value, least trivial to mathematics education. So, deep ideas in neuroscience, for example, will score little if the subsequent application to mathematics education is trivial.
• By “true” we mean true.
• Suggestions, which can be made in the comments below, need not be long, specific or heavily documented. We will reply politely to any suggestion (and other are welcome to reply), querying and critiquing. Further argument and evidence can then be provided.
• Will we be fair? Probably not. But, we’ll honestly try.
• Multiple entries are permitted, and there may be multiple winners.

Go for it. We’re genuinely curious about what the responses may be.

*) In Polster and Ross households.

UPDATE (17/4)

The competition is, of course, a challenge: put up or shut up. If a reader cannot propose and defend one single idea of modern mathematics education then that reader should perhaps stop imagining that any such idea exists. And, if such an idea does not appear to exist, the reader should consider what that suggests about the mountains of Wow produced by the maths education industry, and what it suggests of the shovelers creating these mountains.

Now, what could or should one expect the response to be to such a challenge on an aggressively antiestablishment blog such as this? This blog has a decently large (but not huge) readership, although we can only determine the nature of the readership from the minority who comment, which is obviously a very biased sample. Still, it is probably reasonable to place readers of this blog into three camps:

1. There are the fellow travellers: like thinkers and “Marty fans”.
2. There are puzzled and/or annoyed teachers, who smell that there is something wrong with their teaching world, while still maintaining some faith in the orthodoxy. They may appreciate some of the specific critiques on this blog, while not buying the overall message of contempt.
3. There are the Marty haters, people who are convinced that Marty is an asshole or a nutcase,** who loathe this “nasty” blog, but who visit occasionally in order to feel superior.

This competition is primarily directed at members of Group 3, those who create and promote and value modern maths education. Again, it is a challenge: put up or shut up. If such a person cannot defend such ideas outside of the comfort of their cult, then there is no reason for anyone else to take them seriously.

Do we expect responses from members of Group 3? No, of course not. They regard debating on a blog such as this as beneath them. But the challenge is there, and it will remain there.

What about Group 2? Here, we’re guessing there are some thoughts of possibly defensible ideas, but there is probably some nervousness in proposing them. Such ideas will of course be critiqued (that’s the whole point), and strongly. So, we totally understand any such trepidation, although it is misguided. This blog is scathing of bad ideas, but it is respectful to all good faith commenters, which has been pretty much everyone.

Group 1 can take care of itself, of course, although its members could be more actively critical of this blog …

**) Both are true.

UPDATE (12/05)

OK, it seems like a good time to begin rounding this off. So, who is the winner? We’re not convinced anybody “won” in the sense that anyone has suggested a significant counterexample to our conjecture. None of the suggestions compares, for example, to the elephant truth that mathematics teachers need to understand mathematics a hell of a lot better than they do. None of the suggestions deals with the fundamental flaws of modern education, with, in particular, the deification of technology and the demonisation of discipline. We’re convinced as much as ever that modern educational “research” is fundamentally useless, when it is not actively destructive.

Still, if the suggestions below are minor in effect, some are good and sensible. We have thoughts on a winner, but we thought to let readers have a shot at it first. So, if you have an opinion on the best response to our challenge, please indicate below. We’ll consider, and we’ll announce our winner later in the week.

## Update (29/07/21)

We’ve finally ended this. The winner is student-teacher (wst). See here for details.

## Numberphile and the Cult of Collegiality

Mathologer recently posted a long video addressing the “proof” by Numberphile of the “astounding result” that 1 + 2 + 3 + … = -1/12. As well as carefully explaining the underlying mathematical truth, Mathologer tore into Numberphile for their video. Mathologer’s video has been very popular (17K thumbs up), and very unpopular (1K thumbs down).

Many who objected to Mathologer’s video were Numberphile fans or semi-literate physicists who were incapable of contemplating the idea that Numberphile could have gotten it wrong. Many others, however, while begrudgingly accepting there were issues with the Numberphile video, strongly objected to the tone of Mathologer’s critique. And it’s true, Mathologer’s video might have been improved without the snarky jokes from that annoying cameraman. (Although, awarding Numberfile a score of -1/12 for their video is pretty funny.) But whining about Mathologer’s tone was mostly a cheap distraction from the main point. Fundamentally, the objections were to Mathologer’s engaging in strong and public criticism, to his lack of collegiality, and these objections were ridiculous. Mathologer had every right to hammer Numberphile hard.

Numberphile’s video is mathematical crap and it continues to do great damage. The video has been viewed over six million times, with the vast majority of viewers having absolutely no clue that they’ve been sold mathematical snake oil. Numberphile made a bad mistake in posting that video, and they’re making a much worse mistake in not admitting it, apologising for it and taking it down.

The underlying issue, a misguided concern for collegiality, extends far beyond one stupid video. There is so much godawful crap around and there are plenty of people who know it, but not nearly enough people willing to say it.

Which brings us to Australian mathematics education.

There is no shortage of people happy to acknowledge privately their frustration with or contempt for the Australian Curriculum, NAPLAN, VCE, AMSI, AAMT, MAV, teacher training, textbooks, and on and on. Rarely are these people willing to formally or publicly express any such opinions, even if they have a natural platform for doing so. Why?

Many feel that any objection is pointless, that there is no hope that they will be listened to. That may well be true, though it may also be self-fulfilling prophecy. If all those who were pissed off spoke up it would be pretty noisy and pretty difficult to ignore.

More than a few teachers have indicated to us that they are fearful of speaking out. They do not trust the VCAA, for example, to not be vindictive. To us, this seems far-fetched. The VCAA has always struck us as petty and inept and devoid of empathy and plain dumb, but not vengeful. The fear, however, is clearly genuine. Such fear is an argument, though not a clinching argument, for remaining silent.

It is also clear, however, that many teachers and academics believe that complaining, either formally or publicly, is simply not nice, not collegial. This is ridiculous. Collegiality is valuable, and it is obviously rude, pointless and damaging to nitpick over every minor disagreement. But collegiality should be a principle, not a fetish.

At a time when educational authorities and prominent “experts” are arrogantly and systemically screwing things up there is a professional obligation for those with a voice to use it. There is an obligation for professional organisations to encourage dissenting voices, and of course it is reprehensible for such organisations to attempt to diminish or outright censor such voices. (Yes, MAV, we’re talking about you, and not only you.)

If there is ever a time to be quietly respectful of educational authority, it is not now.