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 and 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 .
If you were teaching a (very) mixed ability Year 7 class in their first term of secondary school and had COMPLETE control over the curriculum, what would you start with as the first topic/lesson sequence?
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:
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:
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.
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
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.)
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) and 17 (15/6)
We had plans a week ago (seems like a year ago) for a PoSSW coronavirus post. But, God, there is so much stupid right now. Who could possibly keep up?
Here is a dedicated post for coronavirus-related stupidity. Commenters are welcome to point to and give links to specific idiocies, or simply to vent. We’ll update the post with the hyper-stupids. (So, standard Morrison/Trump/Johnson incompetence doesn’t cut it, but unloading a whole fucking boat of sick people most definitely does.)
PoSWW 2 (26/2) Courtesy of occasional commenter Franz, cardiopraxis.de gives us a graph showing what “uncontrolled exponential spreading of the infection” would look like:
PoSWW 3 (27/2) The Guardian royally fucks up their 96pt bold headline:
This post is an offshoot of our offer of free maths help to anyone and everyone. Frequent commenter RF started the ball rolling, asking how one might teach (= understand) the change of base for logarithms:
I’ll leave it as open as possible for teacher-commenters to discuss. I honestly don’t know how I’d teach it, and I have difficulty understanding it myself. But here are some preliminary thoughts.
Logarithms are intrinsically difficult because they are inverse things, implying that untangling any logarithmic statement also requires untangling the inverses, to get to the exponentials. This suggests any reasonable approach to the above identity must be grounded in some uninverted, exponential fact(s).
Pondering it quickly, I can see three ways that, together or separately, may lead to some understanding of the above identity:
First, multiply through to get rid of the fractions. (Never a bad bet.)
Second, think of a special case(s) that are easier to think of exponentially and to understand.
Third, give things names: if you want to understand, for example, what means then write . That then gives you the bones to be able to play around with the underlying exponential meaning.
I’ll leave it there, until others have commented.
Thanks to everyone who has commented so far. I’ll look more carefully at the discussion later on, when I can breathe. But, as long as people are happily discussing things, I’ll take a back seat. After the conversation has about run its course, I’ll try to summarise up top the smartness in the various approaches.
Just a couple quick points:
I definitely should have included an experiment/special-cases dot point along the lines suggested by Storyteller in the comments.
For those who want to experiment with what LaTeX does and doesn’t work in comments, you can experiment here.
I can also edit comments (a power I plan to use only for niceness, rather than evil). So, I’ll fix up some of the TeX glitches in comments later today.
Thanks to all the commenters below. Here’s an, um, “summary” of the discussion. (It’s intended to be slow and gentle, and could be further gentled for students/classes by the inclusion of numerical examples.)
Part 1 (Where log rules come from)
We want to make sense of the change of base formula
That’s not obvious, since the inverseness of logarithms obscures everything. So, let’s forget about chasing the weird formula, and first get back to thinking about powers. Remember the fundamental meaning of logarithms:
That is, any logarithm equation is just an exponential/power equation, thought of from a different direction: what power of gives us (answer ), rather than what does to the power of give us (answer )?
Ultimately, as Terry emphasised, any log rule must come from a corresponding power rule via this equivalence. For example, note that
That simple power manipulation gives us the log rule
(In words, the power needed to give us is times the power to give us .)
Part 2 (Experimentation)
We’ll give a more direct approach to the change of base rule in Part 3. First, we can experiment in the manner suggested by RF, and explored at length by Storyteller (aka Proust).
Let’s think about powers of 3. We have or, in log form, . But then, as powers of , we have or . We can summarise this in log form as
Notice that at its heart this calculation is just the blue power rule we proved above. And, critically, it is no coincidence that the 2 appears twice: on the left as a power and on the right as the denominator.
We now wave the Mathologer magic wand (perhaps after more experimentation). We replace the 3 by , the 2 by and (with much more trepidation) the 729 by . With fingers crossed, that gives
This is a change of base rule for logs we can actually understand: it tells us that if is the base then we need th the power than if were the base. And, again, this is just the blue power rule, but written in log form.
Lastly, let’s write . Then , and our blue log rule now takes the form
This is exactly the magenta log rule we’re after, and we’ve kind of semi-proved it. The gaps are justifying that:
(i) Any can be written in terms of the bases and ;
(ii) Any can be written in terms of the base ;
(iii) The blue power rule holds in this more general context.
That is, the proper justification of the change of base rule requires a deeper exploration of the real numbers and is, therefore, pretty much outside the school world.
Part 3 (More direct “proof”)
This is essentially the proof given by Franz, Glen and Anonymous, but framed more like SRK’s argument.
The change of base rule for logarithms has to do with quantities written with different bases. So, let’s ask that question directly. Suppose we have something with base and we want to write it as something with base . That is, if we have
how can we rewrite
(The question can be made more concrete by specifying to common numerical bases. So, one can ask how to rewrite , or even , as or . Even more concretely, we can be back in the experimental world of Part 2.)
Well, what power of do we need to get ? The answer to that must be , by the very definition of logarithms. But then our blue power rule tells us
This magenta power rule tells us how to change from one base to another, so it is really all we need to know. In fact, we don’t even need to that much: in this case, it is better to remember the technique rather than the formula. But, let’s go one step further.
Write for the common quantity . Then and , and our magenta power rule becomes
This is exactly the magenta log rule we’re after, with the fraction multiplied out.
As in Part 2, the proof assumes that logs and the blue power rule work for general real numbers, not just for and the like.
Just a “quick” update in response to some log conversation on this post. There is the question of whether beginning with logarithms as the inverse of exponentials is the “right” place to start. The answer is both “Of course” and “Well, maybe not”. The “Of course” comes, of course, from wanting a language early on to deal with undoing exponential equations, and logarithms provide that language: is just the symbolic manner of saying “The number you raise 2 to to get 8”.
So, why the “Maybe not”? What’s the problem? The problem is that this convenient logarithm language tricks us into thinking we know things that we don’t. For example, we can blithely write , but what does this mean? Yes it’s “the number you raise 2 to to get 5”, but what is that number? How do we know our log rules work for such numbers, numbers that we can’t simply grab like 2 and 3 and 8?
This trickery actually arises earlier, with exponentials. We happily write, for instance, that , without concerning ourselves with the a and the b. So, if a happens to be the number giving , that’s just fine and we go ahead, logging away or whatever. At some point, however, we have to think about what exponentials really are. How do we know that is true, and what does it even mean? What does , for example, mean? Or, ? Or, ?
In summary, we want to know that exponentials and exponential rules make sense and are true for any real numbers, not just natural numbers or (more ambitiously) integers or (more ambitiously) fractions. Without that, we can’t make proper sense of logarithms and log rules, unless we’re explicitly or implicitly sticking to and the like.
So, what do we do? The first thing to do is to follow a strong and proud mathematical tradition: we cheat. We want exponentials and logarithms, and we think we have some sense of how they work? Then let’s just fake it and cross our fingers and carry on, hoping nothing bad happens. So, teachers draw the graph of as if it all makes perfect sense, even though there’s not a hope in Hell of justifying that graph to most school students, and the overwhelming majority of school teachers and more than a few university lecturers would be unable to do it.
Is this cheating ok? Yes, no and no. Yes, it is ok, because there is no choice. Such cheating is unavoidable, in all areas of school mathematics. But no, it is not ok, because teachers should be much more aware of and, when appropriate, much more upfront about the existence of and the nature of such cheating. And no, it is not ok because in the end, we want our mathematics to be as solid as possible, rather than faith-based.
So how, in the end, do we sort this stuff out? Well, we can’t really do anything until we get a proper sense of real numbers. That’s standard undergrad stuff, although many maths majors (and thus many, many teachers) avoid it or are fed a pointlessly token version of it. And then, with real numbers in hand, we have a choice. The first choice is to fix the standard school approach: make proper sense of general exponentials, and whatnot, prove the exponential rules, and then go on to logarithms as inverses. The second approach, which is deeper and weirder but ends up being easier, is to first define the natural logarithm via integration. Then is defined as the inverse of the natural logarithm. The exponential and log rules can be proved (mostly via calculus), and finally other exponentials and logarithms can be defined by change of basis calculations. It is a big project (which we can write about sometime, if people wish), but it is nice stuff.
In summary, to make real, proper sense of logarithms and log rules is a lot of work, and work that goes well beyond school mathematics. The moral? We do doodily do what we must muddily must.
I don’t really know if or how this’ll work, but I figure it’s worth a try. While you’re all locked at home in your individual countries/cities/houses/rooms, you may request help here on any maths problem, of any level: just ask your question in a comment on this post.
God knows what will happen, but I will do my best to give you some guidance in a reasonably prompt manner (within a day-ish).* Others are of course free to offer help, and if they do so then I will try to ensure any subsequent discussion progresses naturally and helpfully.
A couple quick points:
Do your best to ask the question briefly but clearly, and indicate why you’re asking it.
Hopefully LaTeX works in the comments (try $ latex [Your LaTeX code] $ ).
If the question is small and easily resolved then the discussion can stay on this page; for more involved questions, I’ll create a separate post for the discussion.
Please ask new (unrelated) questions in new comments, rather than replies to existing comments.
My approach to this kind of teaching is to be pretty Socratic, to try lead a student to the answer, rather than just providing the answer. So, don’t be surprised if you’re asked to go away and ponder some specific aspect of the question.
I don’t particularly care if the question comes from an assignment or whatever, though I prefer honesty on this point. (And, the more I suspect the question is somehow officially assigned work, the more Socratic I’m likely to be.)
No CAS garbage, in either the questions or the replies. This will be ruthlessly enforced.
*) The Riemann hypothesis may take a little longer.
UPDATE (25/03/20) Here is MitPY 2 (change of base for logarithms).
UPDATE (28/03/20)MitPY 2 is done and dusted. Any offerings for MitPY 3?
UPDATE (03/04/20) MitPY 3: What to teach at the beginning on Year 7.
I’ve been busy the last couple of days, and will be for the foreseeable future, since my girlfriend and I have taken our two young children out of school.
I have informed my parent friends of this decision, but I am not advocating that they, or anyone, follow our lead. My girlfriend and I are lucky in that we are financially secure (for now), and are currently freer of work than we might otherwise be.* It is easy for us to bring the kids home, and we could see no good argument against it. Other parents are much less fortunate, and may have a very difficult decision ahead, very soon. I really feel for them, and for everyone dealing with this mess.
This brings up a general and hugely important question: should schools stay open? Honestly, I have no idea. It is an aspect of Australian discussion that I have been trying, and failing, to get my head around. It seems that the main argument for keeping schools open is simply as a childminding service, so that the oldies don’t do the minding and the doctors and the nurses can get on with running themselves ragged. Is that a sufficient argument? I’m sceptical, but I don’t feel confident to say “no”.
Here are two links to articles discussing the matter (in Australia), neither of which I either vouch for or reject:
A number of prominent public Health Professionals have written an open letter to Australia’s health ministers and (the stunningly appropriately titled) gambling ministers. The letter is written in a predictably calm, professional and diplomatic manner, but we’ll translate: you people who signed off to keep pokies venues open are out of your fucking minds.