This came up in a Math Ed discussion at LinkedIn.
Here A, B, and C are three finite sets.
If half of the As are Bs and half of the Bs are Cs and half of the Cs are As, then what are the maximum and minimum possible ratios of the size of A to the size of C?
This came up in a Math Ed discussion at LinkedIn.
I have nothing to add to this, just want to keep the link.
Stephen Downes' introductory blog posting for the second week of the Critical Literacies Online Course ( CritLit2010 ) deals mainly with how we describe change, and in fact it would (with some minor edits) be the basis of a good motivational piece for the introduction to a calculus course.
This prompts me to make a suggestion that may well lead to howls of protest. Namely that not only should calculus remain as the mathematical topic through which all mathematics students must pass, but that it should in fact be considered as a Critical Literacy for everyone - without which no person can be considered to be properly, or even minimally, educated.
Certainly there are even mathematicians who would disagree. Many feel that various kinds of discrete mathematics are more appropriate to a digital age, others favour geometry and the study of symmetry as motivation for group theory and abstract algebra, and so on. All of these do have value, and it might well be argued that a survey of all areas of mathematics is also something that everyone should have some exposure to. But actually I believe that none of them is critical, and that while a global appreciation of mathematics is as important to a well-rounded education as an appreciation of literature or art, none of these is in fact a fundamental component of basic functional literacy. Calculus, on the other hand, is crucial.
To what? To having any capacity for understanding the questions, let alone the answers, to any of the key problems facing our survival as a species. All of these key problems have to do with rates of change - whether it is economic, environmental, or political.
Many who have struggled with calculus may think that it was just a bunch of abstract formulas and procedures that couldn't possibly be useful, and in one part of this they are right. Memorizing the formulas and procedures is not useful. This has nothing to do with the fact that computers can now do that work for us, and in fact it has always been true. Anyone who understands how change works doesn't actually need the "Product Rule", and the same applies to almost everything else students think they need to memorize. Calculus is not these things and never has been. What it is is the language we need for describing the various kinds of change that Stephen is talking about - and for understanding the long term consequences of different kinds of change patterns.
Without a commonly understood language of change, political debate about things like energy supply and global warming is pointless. And that language is calculus.
Jason Green's response to the readings for Week 1 of the Downes&Kop Critical Literacies course concludes with the question “how does one think critically without it coming across as a baseline of distrust?”
I actually think that a “baseline of distrust” is appropriate, but that social harmony often demands that the level of distrust not be fully expressed. This phenomenon (of people taking offense at any hesitation to accept their assertions and beliefs uncritically) is presumably related to the importance of trust in building a social network but ironically we appear to have evolved a need to be trusted that exceeds our need to be trust-worthy.
One of the puzzles of life as a mathematician is how proud many people are of their incompetence in the discipline, but perhaps it really is a social advantage - after all the one who can claim to have "never been any good at math" is less likely to bring the threat of unequivocal exposure of error which all of us in the field have to live with (and suffer repeatedly), but which most others would rather, it seems, avoid by various kinds of waffling, ambiguity, and evasion.
Perhaps the same psychology motivates those who accuse the so-called "new atheists" of stridency and belligerence where I only see bluntness (and perhaps admittedly a bit of smugness as well).
In my comment to Jason's post, I suggest that "perhaps the key in such cases is not to deny the validity of a proposed conclusion, but just to provide enough feedback to allow the proponent to re-consider".
One comment in particular rang a bell for me.
Often the use of a mathematical model is considered as giving predictions greater credibility when all it really does is ensure that they are consistent with the assumptions of the model.
In areas of technology based on well established assumptions this may actually justify some faith in the predictions, but in scientific practice the role of the mathematics is more often not to establish the prediction but rather to pin down the blame for its failure, and so to discredit the faulty assumptions of an inadequate theory.
Perhaps more emphasis on this aspect in our instruction would help to diminish the mystified acceptance of arguments to the effect that "mathematics predicts that..."
hardproblemsmovie.com is the website of a documentary made about the US team in the 2006 International Math Olympiad.
Although American students on the whole rank well behind many countries in mathematics, American math Olympiad teams regularly finish among the top teams. While aiming to inspire and entertain, Hard Problems provides an insightful and thoughtful look at the process that produces successful teams, and ultimately, great mathematicians of the future
The first part of the above quote raises some interesting questions about how educational effort should be prioritized. Does effort directed to strong performance at the top levels compensate for, or compete with, that needed to maintain the basic levels of verbal and mathematical literacy that are needed for effective democratic decision making (as opposed to the woefully ill-informed nonsense that passes for debate about health care in the US for example)?
The Back Page article by Joseph Ganem in this month's APS News suggests that nominal content and student capability outcomes in US high school mathematics are moving in opposite directions - and attributes this largely to attempts to introduce abstract topics before the students are ready.
This year's Blog Action Day is devoted to the theme of Climate Change and an understanding of mathematics is certainly essential for anyone involved in making making decisions about how to respond to this issue (which in a democracy is presumably all of us).
The choice of Math and Climate as the theme of this year's Math Awareness Month emphasized this connection, and Murray Bourne at squareCircleZ today points to a number of articles in which he has used related topics as a source of examples for teaching mathematics.
A good source of background on the science of CO2 related climate change is this excellent history prepared by Spencer Weart at the American Institute of Physics, as is also the RealClimate site managed by a team of well-reputed climate scientists, and the question of how to compare the effectiveness of different policy choices is addressed in this on-line book by UK physicist David McKay (reviewed by theRegister).
The fact that no amount of restraint or conservation can counterbalance the harmful effects of increasing population is not often noted in the CO2 debate so I was a bit disappointed that Murray did not include his discussion of that topic in his list.
Inventing cute mnemonics is fun, and the process of inventing and checking them may help reinforce the definitions, but beyond that they really are useless - and I believe they do more harm than good when people actually try to use them.
It takes much more time (and mental effort) to correctly recall and decode one of these than anyone who needs to use the terms can afford. And there is a much better way.
Just think 'Sine is the Side' or 'Cos goes Across' (we don't need both)
This takes negligible time to decode, reinforces the concept directly, and is immune to the vagaries of failing memory. (Was that "Odd Aged Teachers Are Happy Campers On Hot Sundays" or "All Old Teachers, Happily Out Camping, Have Amnesia Sometimes "?)
Coincidentally I read 'Born on a Blue day' just yesterday - i.e. one day before zac at squareCircleZ posted his summary review - (having been led to the order the book after watching a video posted - also at SqCZ I think - a couple of months ago). My only difference with the review is that I would reverse what Zac says about the last quarter and the finale. (And anyone who reads any of my views about climate etc may rightly suspect that I couldn't help having reservations about the breeding practices of Daniel's parents - admirable though their parenting may have been.)
This article discusses the latest round of changes in the WNCP Math Curriculum. Somehow, after seeing perhaps half a dozen rounds of this game, the rhetoric of revolutionary change wears a bit thin.
The part I find most encouraging in Murray Bourne's discussion of the latest TIMSS 2007 report on mathematics performance around the world is the distribution of gender differences - in particular the fact that the relative performance of females is stronger most especially in a number of Islamic countries. My top-of-the-head explanation for this is that perhaps mathematics is seen as relatively value-free (and perhaps more artistic than useful) and so is something that females can study without attracting adverse attention, and also perhaps that religious education is seen as more of a male prerogative. If so, then perhaps that portends a wonderful change of culture over the next generation as those with the greater power of reason will eventually find a way to take more control of their society.
I don't hold back from challenging the way mathematics is taught in schools myself, but Keith Devlin's recent MAAonline column is off base and out of line.