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Stephen Downes post on Semantics at Half an Hour: Having Reasons is devoted largely to the issue of how to establish the well-foundedness aspect of knowledge as well-founded true belief.

A large part of the discussion was devoted to the question of how confidently it can be asserted that “interesting learning” occurred in #CritLit2010. (With “interesting learning” apparently referring to the acquisition of new knowledge rather than just unfounded beliefs or behavioural responses - which are also of course examples of learning but not so interesting)

In the course of that discussion Stephen referred to http://www.downes.ca/presentation/251 slide 23, where the inference of learning appears to be derived from observation of behaviour in a social network. But it requires quite sophisticated observation to confirm that the behaviour is based on beliefs that are founded on good reasons as discussed in this posting. (For example it might involve observation of exchanges between members of the network when solving problems together and evaluation of the explanations given to one another in that process.) Also, there would have to be a change of behaviour (beyond that attributable to increasing familiarity with that specific network) in order to infer that the demonstated knowledge was newly acquired and so evidence of learning.  Given the looseness and scale of the network involved it would be a huge task to sift through all of the exchanges to identify signs of increased knowledge in even just a few of the participants.  So I must say I agree with those who are skeptical of Stephen and co’s ability to provide convincing evidence that “interesting learning” has occurred (other than perhaps by direct testimony of the participants).

In the follow-up to #CritLit2010, Ulop OTaat (whoever he may be) has expressed a theory about what the educational theory called “Connectivism” is (or is not).

Basically he appears skeptical of its value as more than just a new lingo without practical consequences:

But that is all that connectivism does, add a different perspective couched in different terminology, jargon if you will.  It does not solve any issues, on its own. It is not the end game of learning theory.

I must admit that I have had similar concerns as to whether it had any direct paedagogical implications. But the jargon has some appeal to me because of its relation to the neural network model of how learning may occur at the physical level, and so I remain open minded as to the possibility that thinking in “connectivist” terms may in fact lead to techniques which might not otherwise have been discovered. (Perhaps it might help in dealing with the chronic problem of failure of “transference” of a concept or technique from one domain of application to another - but I have nothing specific in mind at this point.)
Also of course, any reasonably valid understanding of how learning works may help with the Critical Literacy of avoiding the learning of things that are not true.

Where I have even more difficulty is with imagining possible uses of the analogy between neural and social networks (which seems to get a lot of play in the “Connectivist” worldview). Yes, the development of patterns of connection in a social network may be “reinforced” by interaction with some external stimulus, but I guess I am not sufficiently social to have any idea of a non-trivial pattern as a goal - let alone how to design an appropriate stimulus to achieve it. Perhaps some complex business or political relationships mught be enhanced by forcing a social network to “learn” its way into a particular pattern though - so maybe  Hari Seldon will one day be quoting Stephen Downes as he takes the first steps towards the Foundation. And maybe that makes an understanding of how to train social networks a Critical Literacy for the future.

#CritLit2010 is now over.

I enrolled in this largely out of curiosity about what it would entail and in the knowledge that my travel plans for subsequent weeks would make it difficult to devote much time to it.  I was interested enough to go through most of the readings and to make some discussion entries and blog postings, but if I had had to pay for it then I think I would have been a bit disappointed.
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Both Heli Nurmi and MCMorgan have commented on the CritLit2010 week 4 reading from Clay Shirky Shirky: Ontology is Overrated — Categories, Links, and Tags.

I can’t help feeling that the idea that search based on content and tags will replace heirarchical categories is in one sense overstated, but in another sense doesn’t go far enough. …more »

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.

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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hardproblemsmovie.com is the website of a documentary made about the US team in the 2006 International Math Olympiad.

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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

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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)?

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).

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.

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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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.

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).

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 “All Old Teachers, Happily Out Camping, Have Amnesia Sometimes ” or “Odd Aged Teachers Are Happy Campers On Hot Sundays”?)

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.

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.)

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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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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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 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.

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.