09/02/12

Yesterday was 20 February 2012, and my question is: had the best-before date passed yet?

In Europe, it is typical to write dates in the form “day/month/year,” so if this were Europe, then the best-before date would have passed, and I perhaps ought not to eat the bar. (Although the best-before date would not have passed by too much, and usually the time at which the food is spoiled is beyond the best-before date, so perhaps eating it would be OK.)

In North America, the year is usually last, but the day and month could be in either order. But there are six possible permutations of the six data, so although only two of the permutations are in common use (“year/month/day” is also used, but less commonly), there is still room for confusion. For something as important as best-before dates, shouldn’t we have an unambiguous usage convention? Clarity is courtesy, but when it comes to best-before dates, clarity could also save someone from illness.

But it is difficult to reach consensus on conventions, particularly in this case with products being produced around the world. So here is my proposal for ensuring that best-before dates will never be misinterpreted:

1. Always write all four digits of the year.
2. Always use a day that is at least 13.

Then it doesn’t matter which permutation is used. The dates 13/02/2012, 02/13/2012, 2012/13/02, etc., can only be interpreted in one way. And if a packaged food is good for a year or more, then altering the day by a few to ensure that it is at least 13 will cause no problems. In my situation, this would have saved me wondering whether the best-before date was 9 February or 2 September, or even some time in 2009 (2002 seemed unlikely).

If the food product is only good for a few days after packaging (which is the case for meats, bread, and some other foods), then it is unlikely that it will travel far, and the word for the month could be safely used.

The allegation of “muzzling” came up at a session of the AAAS meeting to discuss the impact of a media protocol introduced by the Conservative government shortly after it was elected in 2008.

The protocol requires that all interview requests for scientists employed by the government must first be cleared by officials. A decision as to whether to allow the interview can take several days, which can prevent government scientists commenting on breaking news stories.

Sources say that requests are often refused and when interviews are granted, government media relations officials can and do ask for written questions to be submitted in advance and elect to sit in on the interview.

‘Orwellian’ approach

Andrew Weaver, an environmental scientist at the University of Victoria in British Columbia, described the protocol as “Orwellian”.

The protocol states: “Just as we have one department we should have one voice. Interviews sometimes present surprises to ministers and senior management. Media relations will work with staff on how best to deal with the call (an interview request from a journalist). This should include asking the programme expert to respond with approved lines.”

Professor Weaver said that information is so tightly controlled that the public is “left in the dark”.

“The only information they are given is that which the government wants, which will then allow a supporting of a particular agenda,” he said.

Shame on the Canadian government, where ideology is regularly trumping reality. (And the mayor and city council of our largest city, Toronto, is evidently cut from the same autocratic cloth.) For those of us who have never experienced life under the thumb of a KGB-like apparatus, there is apparently no need to move to North Korea or Iran. All we have to do is wait here for the control freaks who run our country to stealthily grind us down into a parallel prison.

ps. Video of the AAAS symposium “Unmuzzling Government Scientists: How to Reopen the Discourse” (along with some recent actions along these lines) is available here.

However, what is also sorely needed is some sort of column that explains basic science to the lay public. I don’t mean a development of Newton’s laws of motion for beginners, but rather an understanding of how the enterprise of science works, and how to interpret what one reads in the popular media. When one reads shouting headlines that a recent study shows that Vitamin E is dangerous (without some perspective about the population of the study, the strength of the effect, the time over which the study took place, the control group, and so on), and then reads another shouting headline saying that coffee is good for you after all, really, one can’t help but have sympathy for the poor man in the street who doesn’t know what to think, and then just discounts it all because it is too difficult and troubling to think about (in the same way that the antics of some politicians cast them all into disrepute). And by the way, how did the Knicks fare last night against the Lakers? Linsanity!

This bewilderment leaves citizens particularly vulnerable to nefarious agents who actively foment confusion by loudly trumpeting complete nonsense, in the (unfortunately quite realistic) hope that doubt will be engendered in the minds of many people, which furthers their cause.

One case in point is the despicable Discovery Institute, which is completely anti-science, but tries to let on that its arguments for the existence of God are scientific. Their not-so-hidden agenda is to promote their religious views and do everything in their power (which is well-funded and quite considerable) to force their religious views to be taught in science classes. Amongst their many transgressions, their claim that evolution is “just a theory” confuses the every-day meaning of the term with its scientific meaning, a strategy that is intended to sow uncertainty in the minds of the uninformed. Citizens with a strong sense for what science is and how it works would be far less likely to be hoodwinked by such transparent fallacies.

Some striking instances of the effectiveness of this strategy are to be found in this unintentionally funny clip, showing beauty pageant contestants answering the question, “Should evolution be taught in schools?” (This spawned the hilarious parody, “Should mathematics be taught in schools?” Sometimes humour is the best teacher. More scary, for me at least, is this “news report” from The Daily Show on 26 October 2011, entitled “Science — What’s it up to?” (Here is the link for those based in the U.S.; I haven’t been able to find a link that works in Canada (not sure about the rest of the world; try the U.S. link), but a partial transcript is here.) Interviewer Aasif Mandvi gets political strategist Noelle Nikpour to admit the most ridiculous things about science, including, “Scientists are scamming the American people right and left for their own financial gain.”

Nikpour: “It’s very confusing for a child to be only taught evolution, to go home to a household where their parents say, “Well, wait a minute, God created the Earth.”
Mandvi: “What is the point of teaching children facts if it’s just going to confuse them?”
Nikpour: “It confuses the children when they go home. We as Americans, we are paying tax dollars for our children to be educated. We need to offer them every theory that’s out there. It’s all about choice. It’s all about freedom.”
Mandvi: “I mean it should be up to the American people to decide what’s true.”
Nikpour: “Absolutely! Doesn’t it make common sense?”

Another case in point is climate change denialism, which has been supported by very rich and powerful interests. A news story yesterday reports that climate denialists want to get their propaganda into schools (see also here). Nobody knows what the climate is going to be like in the next decade, the next century, or the next millennium. The system is just too complex, and nonlinear, for any kind of prediction to be meaningful. But this is exactly what is alarming; it seems clear that the system is out of equilibrium, and like a car swerving out of control, who knows where it will end up? Maybe it will end up upright and the passengers will breathe a sigh of relief at a close brush with death, but maybe the car will roll over several times, end up in a ditch and all the passengers will die. Maybe we will turn the planet into a desert, maybe we will freeze. Who knows? So the conservative action would be to plan for a car crash, and to take steps to mitigate potential catastrophes that might arise. But for the giant oil companies, and related industries and hangers-on, any sense that our actions might be causing climate change must be crushed, because mitigating action would be very bad for business. And that’s the bottom line; we must protect profits at all costs, and business must proceed as usual.

What does a scientifically educated person think about all this? Well there are many possible reasonable opinions, but categorical denial that human actions could possibly have any effect on climate change is not one of them. Weighing the evidence and taking prudent action, including funding projects that monitor weather and research historical climate changes, is essential. Just because we don’t know exactly what will happen in future does not excuse us from taking wise prophylactic action now. Doing nothing is radical and dangerous. Advocating that nothing be done using self-serving fallacious arguments, and fraudulently casting doubt on solid scientific research, is reprehensible and dangerous.

What does a person who is ignorant of the ways of science think about all this? Oh, there is a controversy, who knows who is right, nobody knows, so let’s just hope for the best and not bother any further with the issue. Those elitist scientists are confusing, and they always make me feel dumb. Pass me the chips, and turn on the flat-screen, I want to watch my Knicks. (Full disclosure: I would never think like this; I am fanatical about the Raptors.)

ps. On a brighter note, Scientists In School is a group that is taking positive action to help interest young people in science (via hands-on activities) in my locale. Undoubtedly there is some group in your area that is similarly interested in promoting public understanding of science that is worth supporting.

After understanding that students were not responsible for setting up this dysfunctional system, it would be inappropriate to blame them for their unhealthy focus on grades. (Every teacher who ever lived can recall hearing some other teacher complain about “students nowadays, who only care about marks, not about the spirit of learning.”) Students are under enormous pressure to get high grades, because we have set up a system in which their entrance to university undergraduate programs, entrance to graduate and professional schools, scholarships — in short, their entire career success — depends on grades. It seems absolutely silly to base everything on a single number, but there you have it.

It’s worth noting that professors are subject to similar pressure for certification. Forgetting about the same competition for high grades while they were students, professors are under intense pressure to achieve high “grades” for their career success in academia. They need high “grades” to achieve tenure, to be promoted to full rank, and to receive their yearly merit increases in salary. In this case, their “grades” are composed of factors that account for their performance as teachers, their service to their department, faculty, and university community, but by far the most important factor is their research. (Despite protestations to the contrary from university administrators about how much universities value good teaching (and I’m sure they do), note that it is quite possible to become a full professor while being a consistently below-average teacher, while being a stellar teacher but a below-average researcher will never get you to the full professor rank.)

And just as rich businessmen are leading the attack on public school teachers (to the benefit of rich publishing companies), and are thereby influencing in an unhealthy way the environment and conditions of our childrens’ education, rich publishing companies are in control of the certification system for professors’ research, to the benefit of themselves, at great costs to universities, and therefore to the detriment of society.

For professors, their research contribution is graded based on the frequency and quality of their published papers, and on the number and value of their research grants. (The latter also depends on the former. Also, I know that there are other factors, such as participation in conferences, but published research is the most important factor.)

However, how on earth is a university administrator supposed to judge the value of research in a field in which he or she is not an expert? (Which is to say, practically all fields of research.)

This is where the academic journal publishers step in. They provide a service by publishing a large number of journals, with varying levels of prestige, and so a professor’s research can be judged by a non-expert by counting the number of papers published, weighting them according to the prestige of the journal, accounting for impact factors, and so on.

In days gone by, journal publishers provided services that could not be easily provided by individual professors. For example, typesetting mathematical texts used to be a very expensive task that only highly specialized typesetters could accomplish. Mathematical manuscripts were often hand-written, and turning them into professional-quality publications was a very useful step in disseminating and archiving a paper. Publishers hired copy-editors to make manuscripts more readable. The tasks of editing academic journals and refereeing papers was (and is) done on a volunteer basis by professors.

Nowadays, however, with , , and all of its friends freely available to the vast majority of mathematics and science researchers, the need for publishers is questionable. If professors are doing the work of typesetting papers, refereeing them, and editing journals anyway, what value do journal publishers add? Journal publishers have cut back drastically on copy-editing, their function as an archiver is unnecessary now that electronic means of archiving are widely available, so the only thing that they really provide is prestige. But prestige is a chimera, and can be easily accounted for in other ways.

What exacerbates the situation is that journal publishers make absolutely enormous profits, which ultimately are paid for by our taxpayers. Publishers often place the published papers behind paywalls so that the same taxpayers who paid for the research in the first place have to pay again to read the results.

Still worse is the way that publishers of academic journals shake down university libraries. Taxpayers pay for the research done at universities (yes, private universities receive some private funding, but they are still financially supported by taxpayers), and a powerful argument for open access is that taxpayers should not have to pay a second time to have access to published research. This argument gains in strength when one realizes the enormous profits made by publishers of academic journals, on the backs of volunteer labour (the same academics who referee papers submitted to journals, edit the journals, etc.).

Reed Elsevier has been judged the worst offender , and thanks to an initiative spearheaded by Tim Gowers, a boycott of Elsevier has been organized. To learn more about the boycott, which has been in the news and all over the internet for the past few weeks, see the links in the following paragraph.

Authoritative arguments in favour of open access (see here and here), and against large publishers of academic journals, are made by Tim Gowers (here and here; the latter includes an incisive open letter outlining the motivation for the boycott, signed by 34 prominent mathematicians), and Terence Tao (here and here). Other valuable resources are John Baez’s page on journal publishing reform (also see here and here, where Baez makes recommendations for introducing new systems, including Math 2.0 (a discussion forum started by Andrew Stacey and Scott Morrison), and here), and Michael Nielsen’s page on journal publishing reform (also see here for a concise summary of the Elsevier issue). Prominent bloggers who have commented on the issue include Scott Aaronson, Nassif Ghoussoub, Cathy O’Neil (here and here), and Peter Krautzberger (here and here), and Sean Carroll. The issue has even hit the mainstream, at The New York Times and The Boston Globe.

If you would like to add your signature to the protest site, The Cost of Knowledge, it is here. I just checked the site and currently 6251 mathematicians and scientists from around the world have signed to publicize their decision to boycott (in various degrees) Reed Elsevier.

This whole issue deserves serious discussion within the scientific community, and the wider public needs to become aware of it. Most universities are continually crying about how they are strapped for resources, and their libraries are blackmailed by large publishers of academic journals. Reed Elsevier is the worst offender, but not the only one.  The issue has been discussed for a long time (see some of David Mermin’s contributions, for example here, which is reprinted on pages 57–62 here; other references are listed in the open letter signed by Gowers et al linked here; a recent example is an article by Peter Olver in the September 2011 issue of the Notices of the AMS here) and it’s good to see some serious action.

It’s heart-warming to see academics working to wrest control of their research publications away from large, powerful, greedy academic publishers such as Reed Elsevier. One hopes that along with this initiative there will be movement towards a more fair and humane system for evaluating the contributions of professors. And maybe we can figure out a way to evaluate students in a more humane way while we’re at it.

Update: Cosma Shalizi makes the point a lot better and more concisely than I do here. (Thanks to Sam Alexander for the link.)

Zen Valentine

Before Valentine’s Day: Ride subway, walk briskly, look grim.

On Valentine’s Day: Ride subway, walk briskly, carry flowers, look grim.

After Valentine’s Day: Ride subway, walk briskly, look grim.

1 is 3, 3 is 5, 5 is 4, and 4 is cosmic.

Why is 4 cosmic? What happens to the other numbers?

As I mentioned in the earlier post, I slept on this before solving it. I initially thought about the riddle “mathematically,” but that was no help.

A mathematical way to think about the phrases involving the word “is” in the statement of the problem is to consider them as mappings. That is, consider a function (or mapping) f that maps 1 to 3, 3 to 5, and so on: f(1) = 3, f(3) = 5, f(5) = 4. I guessed that “4 is cosmic” meant that 4 is a fixed point of this mapping; that is, 4 is mapped to itself so f(4) = 4.

So the question is, can we guess a formula for f? What came to me in a flash is that it is possible to express the action of f in words: f(number) = number of letters in the English word for the number.

Thus, f(1) = number of letters in “one” = 3; f(2) = number of letters in “two” = 3; and so on. And 4 is indeed a fixed point of this action because the word “four” has 4 letters.

Now, it could be that “cosmic” means that whichever number you begin with, by repeatedly applying the function f you will end up at 4. To check this, we would first have to determine if there are any other fixed points of this action. (I assume the domain of the function is the whole numbers.) There are no other fixed points, but I will let you check this.

The other thing that must be checked is that there are no “cycles” in this action. For example, perhaps 11 gets mapped to 8, which gets mapped to 7, which gets mapped to 11, and round and round we go indefinitely. This does not happen either, but again I’ll let you check this for yourself.

In checking each of these facts, we are helped by knowing that the “word-lengths” of numbers increase much more slowly than the numbers themselves. Numbers less than 1000, for instance, are mapped to numbers in the twenties or less.

Functions that are applied repeatedly amount to iterative procedures, and have many applications. For example, Picard’s theorem is an iterative procedure for solving certain ordinary differential equations. An important method for solving equations of many types is to cast the equation into the form f(x) = x; if the function f has a fixed point and satisfies other properties (which depend on the type of equation being considered), then a fixed-point theorem (that guarantees a solution exists) may be available. The solution process then involves an iterative procedure in which the solution is approximated more and more accurately, with the limit being the fixed point. A general overview of fixed-point theorems is here.

All kinds of behaviours are possible with maps that are applied iteratively. For example, fractals are generated in this way. Certain dynamical systems are defined via iterative maps; the logistic map is a famous example, and one that provides an interesting entry into this field, as it includes both deterministic and chaotic behaviour, and interesting jumps in behaviour from one type to another depending on the value of a parameter.

One of the most famous unsolved problems involving an iterated map is the Collatz conjecture (also known as the hailstone conjecture), which can be phrased in the form of a game. The game works like this: start with any natural number. If the number is even, then divide it by 2. If the number is odd, then multiply it by 3 and add 1. Repeat the process with the resulting number. The conjecture is that if you repeat the process continually, you will eventually end up with the number 1.

Nobody has yet been able to prove that the Collatz conjecture is true or false, but you might have fun exploring it.

1 is 3, 3 is 5, 5 is 4, and 4 is cosmic.

Why is 4 cosmic? What happens to the other numbers?

As is the case for most riddles, the answer is all over the internet, so give this one a good try if you wish to solve it for yourself. I slept on it for several days, spending a few moments of thought on it each day, until the answer came to me in a flash. The feeling of satisfaction when the flash of insight comes is well worth the wait, so if you’re interested I encourage you to just play with it and then sleep on it if you don’t get it right away.

The greatest mathematicians and scientists report similar flashes of insights that helped them solve great problems and create great advances. My own thesis supervisor, himself a great man, told me that when you work hard enough for long enough you begin to solve problems in your sleep.

As educators, we need to find a way to encourage/train students to be patient with themselves, and to persist in working on problems for however long is necessary to solve them. This would be more beneficial to their intellectual development than any content knowledge we stuff into them.

I’ll post an answer and a few comments on related mathematics in a few days.

memorize the minimum

As a teacher, I took it as part of my responsibility to help students identify the essential core of material that must be memorized, and then to help students see how they could cope with the rest of the enormous amount of material by relating it to the essential core.

My original motivation for this approach is not very noble: I was a very lazy student, and I hated memorizing things, so I preferred to practice “playing around” with the material so that I could learn tricks to avoid memorizing. I suppose that this also reflects my natural love of sports, games, and play in general … I would much rather be playing than working!

After many years of teaching and learning, and reflecting on both, I realize that although my original motivation may not have been high-minded, it represents good practice. “Knowledge keeps no better than fish,” said Alfred North Whitehead, which I am fond of paraphrasing as follows:

The more you understand, the less you have to memorize.

In calculus class, I would often present two ways to solve a problem, the main method which depends on practicing an essential technique and requires minimal memorization of formulas, and an alternative method that requires simply memorizing and applying a formula. Once I presented the two methods, I would poll the class to see who preferred which method. Invariably, 60%-80% would prefer the method that required the formula.

I understand why students would feel this way: Most of them had little interest in mathematics, were under enormous pressure, and were simply trying to survive with minimal effort. Survival in a world of high-stakes exams often means placing stimulus-response items in short-term memory, to be pulled out on the exam, and permanently forgotten thereafter. They reckoned that this is less work than having to put in the hard work of thinking in order to understand the material. This is a symptom of our crazy education system, but that is a subject for another day. Suffice it to say that I never blamed my students, but I also recognize that for someone who truly desires to understand the subject, the short-term strategy is counterproductive.

Here are a couple of examples of what I mean. I remember the easiest formulas for the derivatives of exponential and logarithmic functions, which I consider part of the essential core:

What about the other exponential and logarithmic functions? I don’t happen to have these formulas memorized, as I rarely have the need to use them, and have not made a special effort to memorize them. I consider them peripheral, and I would much rather have students practice the following methods for deriving them when needed. In practicing the derivations, students naturally also reinforce the connections between logarithmic and exponential functions, and the means for transforming between them.

Suppose you wish to determine the derivative of the exponential function , where $b$ is some positive number. Strategy: Take the natural logarithm of both sides, simplify the right side using properties of logarithms, differentiate implicitly, and then solve for :

On the other hand, suppose you wish to differentiate the function . The strategy here is to switch to exponent form, take the natural logarithm of both sides, and continue as above:

With practice, one can perform these calculations in seconds. And in practicing them one deepens and demonstrates one’s understanding of many essential process skills.

If you seek deep understanding, don’t memorize the results of these two calculations! Practice the calculations instead.

* * *

Earlier posts in this series:

How Much Mathematics Should a Student Memorize? Part 5, The Multiplication Table

How Much Mathematics Should a Student Memorize? Part 4, Geometric Series

How Much Mathematics Should a Student Memorize? Part 3, The Graphs of Power Functions

How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

How Much Mathematics Should a Student Memorize? (Part 1, Trigonometric Identities)

Goldman was discussing mistakes in the context of medical practice, and he observed that the secrecy surrounding mistakes (doctors don’t tell each other about them, and if it becomes known that a doctor has made a serious mistake, then it is difficult for other doctors to look the “mistaken” doctor in the eye) inhibits discussions about mistakes and efforts to minimize them. He said that in places where doctors do admit mistakes to patients, malpractice lawsuits are actually less frequent!

You can read media reports about a 2010 study published in Annals of Internal Medicine in Business Week and The New York Times, for example. The latter does a good job of describing the strain that doctors suffer, particularly when they are ordered by their legal departments not to speak to grieving families of dead patients, and includes links to other studies.

A particularly memorable phrase from Goldman, which he attributes to one of his mentors, is:

“Good judgement comes from experience, and experience comes from bad judgement.”

Isn’t this a wonderful insight, which applies much more generally than medicine? The implications for parenting and education are clear to me: As parents and as teachers, we need to provide a safe, supportive environment where children and students are able to fail without penalty. Once students learn that failure is a normal part of the learning process, they will become more creative and dynamic learners.

If one fails correctly, then one actually learns, whereas if one is paralyzed and ceases trying because of fear of failure, then one does not learn. Consider current widespread grading and testing policies and the anxiety they induce. University professors out there, how many students do you have that are truly enthusiastic about learning? And how many are concerned primarily with what will be on the exam?

John A. Wheeler: “Make as many mistakes as you can, as fast as you can.”

Winston Churchill: “Success is the ability to go from one failure to the next with no loss of enthusiasm.”

In the basketball world, if a defender is not fouling at all, it’s usually a sign that he is not exerting enough effort. Similarly in life, if you are not failing at all, then you are not trying hard enough. This makes “Failure is not an option” a silly way to think, as pointed out by Seth Godin.

Related posts:

• Failing to Succeed
• Why Do Some People Learn Faster? (Jonah Lehrer in Wired)
• Using Failure As A Friendly Tool For Learning

(via Daniel Coyle’s blog, The Talent Code)