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Topic: Khan Academy: Math instruction goes viral
Replies: 69   Last Post: Apr 6, 2012 9:58 PM

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Alain Schremmer

Posts: 876
Registered: 10/10/05
Re: Khan Academy: Math instruction goes viral > backlash
Posted: Sep 19, 2011 4:01 PM
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On Sep 17, 2011, at 3:18 AM, Clyde Greeno @ MALEI wrote:

> "... how haphazard the learning process is."
>
> A very pertinent truth ... one that is worth pondering ... perhaps
> by musing through a metaphor:



> (See below for the full text.)


Once more, a very nice summary of a complicated issue. Here are a few
thoughts it triggered.

(1) Allow me to give a personal example of what I meant by haphazard:

I am old enough that my education was in pre-Bourbaki mathematics and
then in mechanics (In France). But, round about 1960, I took a
graduate "pure", Bourbaki, analysis course starting with logic, set
theory, etc. And then came the definition of equivalence relation,
equivalence classes and quotient set which I absolutely and totally
did not understand. My wife, who was taking the same course, tried to
explain. Nothing worked. I just did not get it. She couldn't believe it.

I have no recollection as to how and when I came to understand but it
was quite a while later and, ever since, I have been wondering, but
still have no idea of, what it was that I did not understand.

(2) I would also mention (again?) a metaphor which, I believe, I got
from Dienes: The first task for the Educator is to use a "garden",
that is a safe area for the child to play in and explore and learn
from. By safe, he meant simplified enough that the child would not
come against issues that s/he could not handle at that point. For
instance, the diversity should be strictly bounded for the child to
be able to cope with. (Hence the attribute blocks and the multi-base
arithmetic blocks as "pre-abstracted" real world.) The second task
for the Educator is occasionally to bail the child out of any hole s/
he got him/herself into but, mostly, occasionally to ask "dumb"
questions. The third task for the Educator is, as soon as the child
is getting close to the boundaries of the garden, to push them back
some.

In other words---those of Einstein---"I never teach my pupils; I only
attempt to provide the conditions in which they can learn."

(3) We should also keep in mind that adults do not learn the same way
as children do if only for two reasons:

---in general because adults have to reconcile what they are
learning with all these pre-conceived notions. See <http://
www.ted.com/talks/
hans_rosling_shows_the_best_stats_you_ve_ever_seen.html> at the
beginning of which he mentions the very problem,

---in the case of developmental students, because of the well-known
resistance of adults to any learning that appears to them to nullify
the previous competence and experience they thought they had. The
competency from the ``old'' learning drops precipitously before the
competency from the ``new'' learning even starts growing, resulting
in anger and yearning for the ``old way''. See Atherton, J. S. (1999)
Resistance to Learning.

(4) As to how I would respond to your "Just how does one invoke that
awareness when designing instructional materials?", I design in a
model-theoretic mode, that is by setting as our goal encoding the
real-world. See for instance my <http://www.freemathtexts.org/
ChapterThirteen.php> But there are, of course, other design outlooks.
And, moreover, there is no unique Hamiltonian path in mathematics.

(5) Re "designing instructional materials ... as several of us now
are doing", the reason we can't make any difference against
commercial texts is that, as opposed to what is happening in other
fields (programming, <http://www.vanceintegral.com/>, <http://
pages.uoregon.edu/koch/texshop/> etc), we each work in our little
corner which we guard jealously.

Regards
--schremmer

On Sep 17, 2011, at 3:18 AM, Clyde Greeno @ MALEI wrote:

> Alain:
>
> "... how haphazard the learning process is."
>
> A very pertinent truth ... one that is worth pondering ... perhaps
> by musing through a metaphor:
>
> Human learning might be likened to a sailing ship at sea ... but
> ever in the vicinity of reefs and shoals that cause various degrees
> of damage to the learner's potentials for future learning ... with
> the learner-helmsman sometimes on duty, sometimes not ... sometimes
> allowing progress to be dictated by the environment ... sometimes
> steering in some general or specific direction, but often turning
> toward whatever sirens are being heeding at the time.
>
> When contextually viewed only in the more narrow arena of core-
> curricular mathematics, the learning naturally is of narrower
> kinds ... being much more "directed" by the prevailing proceedings
> of the core curriculum ... but nonetheless relatively haphazard
> within that arena ... reminiscent of schools of fishes being
> corralled for domestication.
>
> From that viewpoint, the only *intentional* function of core-
> curricular instruction is to induce the learner to "move" (i.e. to
> grow) in instructor-desired directions [which might or might not be
> healthy for the learners, themselves] ... by using its own learning-
> modes [which might or might not be healthy for the learners,
> themselves].
>
> Always, the learner is at the helm, and the learning can be wildely
> haphazard. The instructor is, at best, a navigator who can serve
> only to *prescribe* where the learner should go, next ... hopefully
> serving to influence the progress, but never able to actually
> control it ... but ever hoping that even the haphazard learning
> would be channeled so that the progress is in the right general
> direction.
>
> Like a navigator's charted course over the sea, and also like the
> vessel's actual track, instructionally prescribed curricular
> mathematics courses traditionally are linear ... partly because
> state-transition (while multi-dimensional) is a function of time.
> So are speaking and hearing, and the reading of English sentences.
> So, it is hardly surprising that scholastic math curricula long
> have been designed for teachers and students to "follow the book".
>
> Of course, the wise navigator ... and the wise teacher ... does not
> count on the vessel exactly following the charted course. Always,
> there is room for some deviation ... which is why highway lanes are
> wider than are cars. The charted course is best used as an epsilon-
> band of alternative acceptable courses. Within that band, a
> student's actual progress might be quite erratic, with the net
> movement still being in right direction.
>
> In a more global, encompassing arena, the overall process of
> concurrently learning in numerous diverse directions might be quite
> haphazard, while still being generally directed within various,
> more limited contextual sub-arenas ... perhaps even being more
> "disciplined" in some arenas. [So arises the question of what kind
> of math-education actually helps students learn how to learn and/or
> how to think.]
>
> By putting his *library* of math "lectures" into an electronic
> format, Kahn departs from the linear organization. He much better
> allows for jumping to any one of numerous topics ... helicopter-
> mode ... always at the risk of landing on a topic which the viewer
> is quite under-equipped to digest that topic into personal
> mathematical sense ... thereby "pipering" many unwary users toward
> unhealthy parroting of his own performances.
>
> A library is a bit like a satellite map that shows where the cities
> are, without also showing the roads that connect them. If Kahn's
> "academy" shares with its users a cognitive map of mathematical
> topics, I have yet to see it.
>
> Although any such map definitely is partially-ordered, but not
> linearly ordered ... any learner's progress along it is linearly
> ordered, by time. A cognitive map seriously delimits what
> developmental states one can next grow into, from one's present
> state of development. [ALEKS is supposedly designed on that basis.]
>
> That says that for cohesive learning of mathematics, the core-
> curriculum cannot be haphazardly jumpy. Rather, it must be
> developmentally continuous with respect to students' cognitive
> growth. Most textbooks are wrought with jump-discontinuities ...
> thereby producing only fragmented knowledge. [May the math
> curriculum developer beware of making the same error.]
>
> Necessarily, each lecture is linear over time. Without a cognitive
> map, a library of lectures cannot yet be regarded as being even a
> partially-ordered curriculum ... and certainly not an "academy".
> Like any library, it might be explored quite hap-hazardly.
> Unfortunately, his math lectures ... like most core-curriculum
> lectures ... serve far more to expose what he knows (or does not),
> than to develop the viewers' own functional mathematical intelligence.
>
> It would seem that the key to instructional success is NOT to try
> to tow the learner to follow in the footsteps of the instructor,
> but to *shepherd* the erratic learner to keep moving in some
> "right" direction ... however wander-ous the learner's progress-
> path might be. So much the easier if the learner can perceive and
> adopt the instructor's goals -- even better if the learner owns a
> map for how to move toward those goals ... still better if the
> learner owns and can use good tools for moving along that map ...
> albeit the inconsistent and often clumsy use of those tools might
> give the impression of being haphazard.
>
> It seems that learning, as a whole, is as haphazard as the
> environmental encounters that provoke it [which is the real cause
> for the escalation of home-schooling]. But trying to tightly
> harness even special-context learning to some pre-scribed line ...
> especially to a long and badly discontinuous line ... might be a
> lost cause.
>
> It would be interesting to compile descriptions of any workable
> alternatives.
>
> "... how haphazard the learning process is."
> Just how does one invoke that awareness when designing
> instructional materials ... as several of us now are doing?
>
> Thanks for the stimulus to contemplate it.
>
> Cordially,
> Clyde
>
>
>
>
> --------------------------------------------------
> From: "Alain Schremmer" <schremmer.alain@gmail.com>
> Sent: Friday, September 16, 2011 9:24 AM
> To: "=?UTF-8?Q??=" <"AMATYC_=E2=80=8E[mathedcc"@mathforum.org]
> =E2=80=8Emathedcc@mathforum.org>
> Subject: Re: Khan Academy: Math instruction goes viral > backlash
>

>> A very nice summary of the issue.
>>
>> I would add that there was also an esthetic dimension: it seemed
>> to us that the construction was so beautiful that, per force, it
>> could not but be optimal for learning purposes. We had forgotten
>> how haphazard the learning process is.
>>
>> Best regards
>> --schremmer
>>

>> >
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