Patterns of Change - Calculus as a Critical Literacy

Tuesday, June 8th, 2010

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.

OnLine Educational Resources

Saturday, January 9th, 2010

Scott Leslie may be on the right track with another 1/4-baked idea - OER “virtual reference librarian” at EdTechPost, but I suspect that it may be less with the idea itself than with the doubt he expresses as follows:

<<Is “discoverability” even actually the problem with resources getting reused, or is it possible that the whole model is so flawed, so disconnected from how educators construct course materials, that it wouldn’t make any difference..?>>

Commenter Mike Caulfield followed up on this with

<<the really interesting thing is how many people said they wanted that, and how few people contact us for help>>

Some educators want a complete package provided by a publisher while others want to develop their own way of engaging students with the material.

In the space between those two extremes it would seem that there was ample room for a style of preparation which involved searching for and combining the best of what is available, and many of us think that is where we belong - but when push comes to shove we bifurcate and either go with a complete package or "roll our own" completely.

As variously a creator, organizer, and user of OERs I think I may have gained some insight into why this is the case.
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Hard Problems

Friday, November 20th, 2009 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)?

Why Math?

Friday, November 6th, 2009

A couple of recent additions to the arsenal of reasons for promoting mathematics education are this recent article by Ian Stewart in the UK Telegraph and the collection of 'Math Matters - Apply It' posters developed by SIAM (the Society for Industrial and Applied Mathematics).

Stanford Study of Writing

Friday, November 6th, 2009

The Stanford Study of Writing provides a welcome counterpoint to some of the nonsense that has been put about regarding impact of the internet on literacy.

Crossing the Finish Line: SATs and GradRates

Friday, November 6th, 2009

Stephen Downes points to Chad Adelman posting on Crossing the Finish Line - a recent book about university graduation rates by William G. Bowen, Matthew M. Chingos & Michael S. McPherson.

I haven't read the book but am suspicious of any attempt to draw conclusions about social policy from statistical analysis - especially in reviews and commentaries that isolate particular statements about how variables are correlated (and even more so if they include references to "predictive power" going "below zero", since statistical power is defined as a probability and  a correlation of minus one has a very strong predictive power in any reasonable sense of the term).

A common "paradox" pointed out to students in an introductory statistics course is that it is possible to have a variable S (for, say, SAT score) that is positively correlated with some measure, say G, of success (eg graduation) in each of several subsets making up the whole of a population - while being negatively correlated in the population as a whole.

One way this might happen, for example, would be if there was a characteristic I (for, say, Inspiration) which was very highly correlated with G, and such that among the high I part of the population S was only weakly correlated with G but in the low I population S was very strongly correlated with G.

If among the population as a whole (in this case university entrants) low I was correlated with high S, then entrants with high S would be more likely to be in the low I group and so less likely to graduate and so S might be negatively correlated with G - even though in each of the low and high I groups separately, higher S does contribute to increasing G.

Of course, many readers of this  (if in fact there were any) might then say "but if I is the best predictor of G, let's just use it and forget about S".

And maybe they are right.  At least if the goal is soley to maximize the G rate we should just ignore the low I group and concentrate all of our efforts on those with the magic I factor.

If only we could identify it we could do away with all that "high stakes testing" and give our attention to those who deserve it.

Well the good news is that I have found I.

The bad news is that it is not Inspiration.
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A Widening Gap

Sunday, November 1st, 2009

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.

FAQ for Universities Interested in WPMu

Wednesday, October 14th, 2009

Since my institution has started using WPMu for faculty and departmental websites I had better have a look at this from bavatuesdays (which came to me  via Stephen Downes).

How to remember trig ratios

Monday, October 5th, 2009

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 "?)

Teaching math using interactive white boards

Monday, September 21st, 2009

This interview with a recent convert to teaching math using interactive white boards includes a lot of good ideas for using the computer display but  leaves me wondering what the IWB adds over what could be done with a tablet PC and projector.

One weakness of the WB is that it forces the presenter to face away from the audience for writing - something we are all used to and try to mitigate, but which could be avoided with the old style projector.

A possibly distinctive use of the IWB might be to have students come up and interact with it themselves, but the interviewee actually seemed to be saying that she tried that but found the benefits outweighed by the distraction of having people moving about so much.

Born on a Blue Day

Wednesday, April 15th, 2009

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.)

Lecturing - stupidest profession?

Saturday, April 4th, 2009

Donald Clark Plan B: Lecturing - stupidest profession?

An uninterruptable lecture is almost as much an abomination as a TV broadcast, but even a replayable recording is no better than a bad lecture - where the only response to a question is to repeat verbatim whatever was not understood.

The point of a lecture is to allow spontaneous rephrasing or elaboration to correct for misinterpretations. In the absence of that you might as well use a book or printed notes, since written information is both much easier to navigate and much quicker to take in than that which is spoken.
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Homework Help = Cheating ?

Tuesday, January 27th, 2009

In checking out some of the people mentioned in this posting by Seb Schmoller (which I learned of via Stephen Downes), I was led to consider where is the borderline between helping a student to learn and facilitating cheating.
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What is 0^0 equal to? - squareCircleZ

Sunday, January 25th, 2009

This post at squareCircleZ (a very nice enrichment and support website for students and teachers of mathematics) raises the conundrum of how to define 0^0 if all positive x give x^0=1 and 0^x=0.
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Open Culture

Saturday, January 17th, 2009

Open Culture is a website developed by Dan Colman (who moonlights as the director of Stanford's Continuing Studies program). It focuses on educational video offerings such as the Leonard Susskind
Physics Lectures, and includes
a page of links to other academic YouTube video collections.

By The Numbers

Thursday, January 15th, 2009

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.

Trigonometry tips @ squareCircleZ

Thursday, December 4th, 2008

Murray Bourne's IntMath Newsletter this week includes a nice preamble to the study of Trigonometry. I'd like to be able to link to that item specifically when introducing the topic, so maybe I'll ask him to isolate it if he has the time.

BCcampus OER site - Free Learning at EdTechPost

Wednesday, December 3rd, 2008

Scott Leslie writes about the BCcampus Open Educational Resources site with some new ideas for using social networking sites like

Computer Algebra Systems

Saturday, November 1st, 2008

(This posting was prompted in part by a brief mention of the issue in  squareCircleZ )

It is often claimed that Computer Algebra Systems will (or already do) eliminate the need for much of what is taught in high school and college math classes but I seriously doubt that that is the case.

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Goodbye College Diplomas ?

Monday, October 27th, 2008

Stephen Downes links to Tom Haskins saying Goodbye College Diplomas

Thank god that the time will soon arrive when a prospective employer will not be denied the pleasure of reading all my undergrad essays but will instead be able to compare that proof of my learning in detail with those of all of the other 126 applicants for his one available position rather than having to rely on brief summary documents "of a few words and transcript numbers".

And that in my old age there will be no shortage of doctors qualified to treat my ailments by having met their own self-determined educational outcomes.

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