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, 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)

Just a few (?) words about this competition, and this blog.

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.

The Methods Intelligence Test

Psychologist Daniel Kahneman dedicates his book Thinking Fast and Slow  to the memory of Amos Tversky, his long-time collaborator. Tversky was considered so brilliant by his colleagues that they came up with the Tversky Intelligence Test:

The faster you realise that Amos Tversky is smarter than you, the smarter you are.

It has occurred to us that there is a similar Methods Intelligence Test:

The slower you realise that Methods is stupider than you, the stupider you are.

DIY Teaching Degrees

Dan Tehan, the Federal minister for screwing up education, has announced a rescue package for Australia’s universities. This was clearly necessary, since the universities are no longer in a position to fleece international students. The package guarantees funding for the universities, and introduces a range of cheap six-month courses in “areas considered national priorities”.

The government’s package is “unashamedly focused on domestic students”. That was inevitable since:

a) the government, and Tehan in particular, doesn’t give a stuff about international students;*

b) Tehan is a born to rule asshole, entirely unfamiliar with the notion of shame.

And, what of these “priority” courses? According to the ABC,

The Government said prices would be slashed for six-month, remotely delivered diplomas and graduate certificates in nursing, teaching, health, IT and science provided by universities and private tertiary educators.

OK, so ignoring all the other nonsense, we have a few questions about those six-month online teaching diplomas:

  • Will such a diploma entitle the bearer to teach?
  • If not, then what is it good for?
  • If so, then what is a school to do with the mix of 6-month diploma-qualified applicants and the standard 24-month Masters-qualified applicants?
  • And, if so, what does that tell us of the intrinsic worth of those standard 24-month Masters?

To be clear, we have no doubt that six months is plenty sufficient for the initial training of a teacher, and indeed is at least five months too many. We also have no doubt that a diploma-trained teacher has the same chance to be a good teacher as someone who has suffered a Masters. They have a better chance, in fact, since there will have been less time to pervert natural instincts and feelings and techniques with poisonous edu-babble.

But, good or bad, who is going to give these diploma teachers a shot? Then, if the teachers should be and are given a shot, who is going to address the contradiction, the expensive and idiotic orthodoxy of demanding two year post-grad teaching degrees?


*) Or anyone, but international students are near the bottom.

MitPY 4: Motivating Vector Products

A question from frequent commenter, Steve R:

Hi, interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of \boldsymbol{R^2} and \boldsymbol{R^3} respectively. One attempt explaining the difference between them is given here. The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane, around quadrilaterals, parallelograms and their diagonals .

Regards, Steve R

Guest Post: Failing the Coronavirus Test

The following was written by my brother Dan, who has been watching the governing of the coronavirus pandemic very closely. The views are, of course, Dan’s.* 

In a real emergency situation, it tends to be less than useful to have everyone imagining that they are an expert whose opinion absolutely deserves to be heard. One of the least enjoyable aspects of the general response to this pandemic, apart from the virus itself, has been watching everyone on Twitter and on television starting from the unreflective assumption that whatever crap they happen to believe is automatically worthy of public pronouncement. In such a situation, and despite one’s better judgment, one almost begins to feel sorry for those in actual power who have to make actual decisions with actual consequences. Nevertheless, and even though I have absolutely no grounds for claiming any expertise in any area related to the current pandemic situation, I feel compelled to at least raise some questions about how governments and journalists are responding, in particular in Australia.

Let’s begin with an article published yesterday in The Guardian:

How many cases of coronavirus are there in Australia? Live Covid-19 numbers and statistics

What followed in that article was nothing more than a representation of the information that Australian state governments have deigned to provide about the numbers of confirmed cases. Not one word was written about the question of how many actual cases there may be. And yet: do not GPs and doctors have reporting requirements concerning serious infectious diseases? In other words, are they not obligated to report probable cases to the relevant medical authorities?

We have to distinguish three categories: confirmed cases for those who have had a Covid-19 test with a positive result, probable cases, where a doctor has assessed a patient as showing symptoms that mean they are likely to have it but have not had an actual test for the presence of the virus, and actual cases, that is, everyone in the community who does have or has had the virus. Currently we have no way of knowing the precise number of actual cases, but the best clue about how many actual cases there may be is to know the number of probable or suspected cases that have been detected. For journalists to treat data about confirmed cases as indicating anything meaningful about actual cases is obvious and needless distortion.

Why does this matter? It matters because currently GPs are seeing patients who have the symptoms of Covid-19, and they are telling those patients to go home and stay home. Clearly, this is a directive coming from above. How widespread is this practice? We cannot know, because we are not receiving information about probable cases. Whether it is a good idea is one thing (which we will come back to), but the main point is that there is a co-ordinated approach and one that systematically distorts the informational value of the data about confirmed cases. Creating pretty charts and graphs showing what is happening with confirmed cases only contributes to the obfuscation of the situation. But all those GPs must be following their reporting requirements, and government must be collecting that information, yet no Australian journalist seems to have thought of ever asking for it, let alone demanding it. Why?

Government should want to give out that data, because it will make the situation seem more serious than the perpetual reporting of confirmed cases can ever do, and this seriousness seems to be what they are struggling to convey. The governments of both China and New Zealand decided at a certain point that it was better to use probable cases as the official data, rather than confirmed cases, but other countries including Australia do not. Perhaps the reason the Australian government is reluctant to do so is because it will have to admit publicly that it is telling GPs to send home probable cases (as long as they are mild), and not to make any further therapeutic intervention. Perhaps they believe that such an admission will cause consternation, disruption, or panic. Who knows? But the passivity of journalists about this means that Australians are unable to form any clear picture of how widely the epidemic has spread throughout the country.

It matters because there is one fundamental question animating every decision in relation to this pandemic: the calculation of the health benefit of highly restrictive strategies compared to the economic cost of those same strategies. What’s the connection? The connection is that the real issue is not “how tight should the lockdown be?” or “when can we begin to relax the restrictions?”. The real question is “what possibilities does a lockdown open up?” In other words, the lockdown is not an end in itself, but a means to another end: a way of setting the conditions for doing those things that actually need to be done. This is exactly what Tedros Adhanom Ghebreyesus, Director-General of WHO, said yesterday:

“We understand that countries are trying to assess when and how they will be able to ease these measures. The answer depends on what countries do while these population-wide measures are in place.”

Dr. Ghebreyesus’s point is that a lockdown itself is not the “solution”. It is only the basis on which any possible solution could be enacted. After that, the WHO Director-General itemises what must be done during the lockdown. First, expand and train the health care workforce. OK, no problem: presumably, hopefully, governments and authorities understand this point and are trying to do something about it (and should have been trying since Wuhan).

But then we come to the second point:

“implement a system to find every suspected case at the community level”.

 This, it seems to me, is fundamental and crucial – and it is being completely avoided by Western countries. Currently we hear a lot about “how much testing is going on” and “how wide is the testing”. By “the testing” is meant Covid-19 tests, those that lead to the “confirmed cases” being confirmed. Everyone who wants to seem sensible and rational says “we need lots and lots of testing”. Is this true?

This, it seems to me, is not what WHO is saying. They are saying “find every suspected case”. Sure, Covid-19 tests are one way of finding those cases “at the community level”. If you have enough tests. If you have the resources to implement them. If you have enough time to do those tests. But it may be that this is a highly inefficient way of “finding every suspected case at the community level”.

What would be more efficient? Well, firstly, 90% of cases start with fever, and temperature testing is much simpler and quicker than testing for the virus itself. Anyone can do it. Anyone can be employed to do it. Once someone has been found to have an elevated temperature, then the question becomes what procedure to follow after that. It was precisely this point that Donald McNeil of the New York Times explained with great lucidity two weeks ago in this interview, describing in some detail what they actually did in China that actually worked:


From McNeil, we can see that “testing” in China actually meant a sequence of steps, so that the Covid-19 test would only be necessary after having passed through the earlier steps. Notice that this is exactly what Dr. Ghebreyesus said in point 2.

This properly expanded notion of “testing” ties into Dr. Ghebreyesus’s third point, on the need to “ramp up production capacity and availability of testing”, as well as the fourth point, on the need to “identify, adapt and equip facilities you will use to treat patients”, and his fifth point, on the need to “develop a clear plan and process to quarantine contacts”. Of course, governments will say that they are doing precisely the things Dr. Ghebreyesus demands. But are they doing those things? Is this the reason that the Australian government prefers not to mention that GPs are quietly telling patients with mild symptoms to go home and stay home? Because doing so amounts to crossing their fingers that those with the virus will do as they are instructed, that they will know how not to infect their family members, that they will not take a trip to the supermarket, or to Dan Murphy’s, etcetera etcetera etcetera. And maybe crossing their fingers will work. It is also possible, however, that instead of finger-crossing, it just might actually be better to pull patients suspected of having Covid-19 out of wherever they are, send them to one place for further investigation, and then, if confirmed to be infected, send them to another place for treatment.

Of course, no one is keen to follow the Chinese example of welding people into their apartment buildings. But that’s not the real story of China, and we’ve known that for some time now. Pointing out what China might have done wrong is not very important, if they are not things we are likely to do ourselves. But pointing out what they did right might be very important, if anyone cares enough to think about it. The real story of China’s response is: having a systematic approach to finding every suspected case, having a multi-step and efficient way of finding out if they do have Covid-19, removing those people from general circulation before they have a chance to infect others, offering treatment in dedicated facilities, and then sending out teams to trace their contacts.

The questions are: What is the real reason that Western countries are not willing to do what WHO has told them needs to be done? Why are they not willing to do what China has shown them can work? Why are they not willing to hire masses of people (plenty of whom have recently had space freed up in their calendar) to stand in a mask and gloves at train stations and public building entrances to give each person a temperature test? Why have they not been willing to insist that suspected cases be isolated until they are confirmed or otherwise?

It seems to me that the answers to these questions are not just about a different assessment of what’s the best strategy, but instead have significant cultural and psychological aspects. That’s a discussion for another time. However that may be, the results of not asking these questions are clear in one Western country after another, as they take one step after another in what seems a completely reactive way, as the completely predictable progress of the pandemic unfolds.

The benefits of a systematic and carefully prepared approach are obvious: you know when to lock people down, in what way to do it, and what you hope to achieve. For example, consider public transport, which is obviously a place where infection can spread (see the photos and videos from the London tube). In the West, the question is always framed: should you stop it to prevent infection or keep it going because it fulfils an essential service? But this is the wrong question. No doubt it is dangerous, and no doubt it is essential, so the point is to create the conditions under which it is justifiable for it to continue. That may well mean significantly reducing the number of passengers. Why? Precisely so that it will be possible to give every single person who enters a train station a temperature test. Will that pick up every infected person? No, it won’t. But it will very significantly reduce the numbers exposed, while providing an opportunity to “find suspected cases at the community level”, and still allow some necessary movement.

The same thing applies to schools. Instead of “should we keep them open or shut them down?”, the question should be, “under what conditions would it make sense to keep them open?” Those conditions should include testing…not Covid-19 testing: temperature testing. Every single day. Why not? Why is that so impossible? Why is it never proposed and why do journalists never ask?

It is not that every country has to take exactly the same measures. Each country must evaluate its specific situation and make the most rational decisions possible in line with the points made by the Director-General of WHO yesterday. Some things that China did may be simply impossible for Western countries to do. So then, why not take a look at how Singapore was able to contain the virus so successfully? Was it by having a complete lockdown, close schools, and so on? No, schools remained open in Singapore and no total lockdown was necessary, yet it has been vastly more successful than Western countries. As Dale Fisher explained on 18 March, Singapore was already prepared by the end of January, had a vast number of hospitals (for a tiny island) specifically dedicated to handling patients, kept confirmed cases (even mild ones) in hospital and separated from friends and family, made sure that school children showing any illness whatsoever did not attend, and communicated clearly and explicitly with the public:

Why Singapore’s coronavirus response worked and what we can all learn

This kind of effective preparation simply did not occur in Australia, just as it did not take advantage of its specific situation as, like Singapore and Taiwan, an island nation.

And then there is the question of mandating the wearing of masks in public situations, and governments supplying them to those who need them… (Edit 27/03: links added)

It is only by asking questions such as these that it will be possible to adopt strategies that will allow a gradual transition from highly restrictive “lockdowns” to situations where there is greater freedom of movement and activity. This is what Singapore anticipated from the beginning, and it is clearly what needs to be thought about here and everywhere, if the medical and economic “balance” is going to be approached in anything like a rational way, and if there is in fact a way of preventing a global economic catastrophe. This shift to a less restricted approach is what China is beginning to do right now, but the problem is that, in a globalized world, each country is connected to all the others, and so such strategies must include very strong border control until such a time as everyone starts to behave like grown-ups. Currently, most of the Western countries are still behaving like scared or defiant children, to one degree or another.

In the wash-up of this pandemic, it will be found that governments, media and populations each had their true strengths and weaknesses exposed. Those strengths and weaknesses may not be the ones we thought they were beforehand. Based on current evidence, Western governments, media and populations seem, in general, very far from being top of the class.


*) Dan and I agree on my most things. Except, Dan believes that I have too much faith in the wisdom and goodness of human beings. 

Scott Morrison is Fucking Up

From the opening remarks of the March 25 WHO press briefing:

Asking people to stay at home and shutting down population movement is buying time and reducing the pressure on health systems. But on their own, these measures will not extinguish epidemics.

The point of these actions is to enable the more precise and targeted measures that are needed to stop transmission and save lives. We call on all countries who have introduced so-called “lockdown” measures to use this time to attack the virus. You have created a second window of opportunity. The question is, how will you use it?

There are six key actions that we recommend.

First, expand, train and deploy your health care and public health workforce;

Second, implement a system to find every suspected case at community level;

Third, ramp up the production, capacity and availability of testing;

Fourth, identify, adapt and equip facilities you will use to treat and isolate patients;

Fifth, develop a clear plan and process to quarantine contacts;

And sixth, refocus the whole of government on suppressing and controlling COVID-19.

These measures are the best way to suppress and stop transmission, so that when restrictions are lifted, the virus doesn’t resurge.

Dr. Tedros Adhanom Ghebreyesus, Director-General of WHO

UPDATE (26/03/20)

Donald McNeil Jr talks about what worked in China (and further discussion here)

The Funny Coronavirus Post

Well, yes, the humour is pretty black. But there is still humour to be had. In this post we’ll link to things that genuinely made us laugh (even if we were crying/yelling two minutes later). Commenters are of course welcome to suggest others. (Our standards are high.)

1) Michael Spicer: Trump and the Coronavirus

2) Marina Hyde: When Johnson says … it’s just another line for the side of a bus

3) First Dog on the Moon: Coronavirus 1 and 2 and 3 and 4 (27/3) and 5 (7/4) and 6 (16/4) and 7 and 8 (21/4) and 9 and 10 and 11 (1/5) and 12 and 13 (11/5) and 14 (14/5) and 15 (18/5) and 16 (23/5)

4) I confess, I’m a closet ScoMo fan:

5) Amanda Hyde: Who are the designated survivors? (7/4)

6) Amanda Hyde: Footballers are to blame (7/4)

7) Curb your going outside (7/4):

8) Michael Spicer: Trump and the Coronavirus II (7/4):

9) Marina Hyde: Our leaders should be facing tougher questions (16/4)

10) Marina Hyde: Eamonn Holmes, daytime TV’s fiercest sleuth (16/4)

11) Michael Spicer: Symptoms of Dominic Cummings (16/4):

12 Marina Hyde: At Least We’re Not the US (21/4)

13) Michael Spicer: Trump and the Coronavirus III (21/4):

14) Michael Spicer: Trump and the Coronavirus IV (1/5):

15) Marina Hyde: Richard Branson and bleach and Johnsonian linguistics (01/05):

16) Darryl Charles’ algorithm for responding to Trump (1/5)

17) Marina Hyde: Thank you for the opportunity (11/5)

18) Michael Spicer: Trump and the Coronavirus V (11/5):

17) Marina Hyde: Lessons on civic duty from Boris Johnson? (14/5)

18) Marina Hyde: Johnson’s cheerleaders (18/5)

19) Michael Spicer: Matt Hancock and the Coronavirus (18/5)

20) Michael Spicer: Trump and the Coronavirus VI (23/5):