DIY Math Post-Mortem
Joe Corneli
I feel like this course is effectively over (and maybe was
even "over before it began", though I think none of us saw
that). Rather than trying to squeeze more out of the
course than it has to offer, I'd like to share my
reflections on "what went wrong".
This mailing list, of course, continues to be a place to
discuss "DIY Math" in general, and perhaps the course will
rise again sometime. I'd like that, and I am not
discouraged by what I've learned so far. Further, I wish
to salute all of you for your efforts to be self-directed
learners. But I think we need a different approach.
The following is my own post-mortem analysis (cf. [0]) of
the course. If you would like to add to it please do. If
you feel like the course is not actually over, now would
be a good time to speak up about that, too.
* * *
My assessment of the course was that participants knew
*more or less* what they wanted to focus on, but they
didn't exactly know "how" to start, and they didn't
particularly want to discuss questions about "how to do
things".
Unfortunately, "how questions" were supposed to be the
central focus of this *discussion-based* "DIY" course.
At this point, I can see several profound shortcomings in
the way I designed the course, not the least being that
the implied tension between "doing it yourself" and
"discussing it with others" was not given enough attention
up front. I had hoped this would be a productive tension
or "dialectic", but it doesn't seem to have turned out
that way. There is a certain degree of commitment
required to work with(in) such a dialectic, and no one who
signed up for the course seemed particularly interested in
that. Rather, applicants for the most part "just wanted
to learn math". I did not do a very good job explaining
why I think working with the diy/p2p dialectic is *how* to
learn. Moreover, my personal philosophies aside, I simply
didn't do enough to get people talking in the course,
partly because I rather blindly assumed that the desire to
discuss would be shared by all participants. Or, more
precisely, I assumed that those who wanted to discuss the
most would carry the conversation forward.
Aside from these issues, which I take responsibility for,
some of the shortcomings of the course itself point to
design considerations that would best apply at a higher
level; something like "P2PU best practices". One could
say that course-level and P2PU-level issues have some
degree of "trade-off" between them.
As one very simple example: If P2PU was taking an ongoing
survey of the "wished for" course topics (to be tagged and
voted up or down by site visitors), I would presumably
have been able to create a mathematics course tailored to
the interests of pre-self-selected participants, instead
of fashioning a course that was supposed to serve all
mathematical interests simultaneously.
As it turned out, I learned at the beginning of the DIY
Math course that the interests of the applicants were
approximately equally distributed between (1) General
interest, e.g. learning how to learn mathematics; (2)
Technical interest, e.g mathematics for computer
programming; (3) Mainstream mathematics, e.g. precalculus
and calculus; and (4) Advanced mathematics,
e.g. probability and statistics.
One obvious thing to do "next time" would be to simply run
a "standard" mathematics course for each of these four
segments. This would be a lot of work to set up, but by
no means impossible: perhaps I could find some other
people who would want to co-facilitate these courses. My
guess is that with significant work and a design that
draws on P2PU's collective experience of how to teach (and
how *not* to teach) "academic subjects" in a peer-to-peer
setting, these courses would all be reasonably successful.
I'll also note that it would be possible to focus on "how
questions" in any of these courses (perhaps especially
obvious in the case of the first group, since one could
copy an existing course on mathematical foundations).
Nevertheless, I think the course would most likely need *a
lot* more structure in one form or another. There are
undoubtedly several different ways to go about structuring
something like this [1], but my guess is that *either*
we'd need a more step-by-step curriculum crafted to ease
people into thinking about "how they learn", *or else*
we'd need to know in advance that participants were really
committed to asking and answering "how" questions.
Regarding this issue of "commitment" -- which in fact
seems quite important independent of the course's
"peertagogic" style -- in the latest P2PU community call,
Alison Jean Cole again reminded me about the idea of
setting up a *social contract* for the course.
I said to her that this seemed like another issue that
could potentially benefit from support coming from a level
above individual courses, e.g. by involving an "course
mentor" who would help facilitate the creation of a social
contract for the course. This could be someone who had
previously run a "successful" course.
Such an idea might go along with re-thinking and nurturing
the "social contracts" implicit to the P2PU structure
itself (i.e. we would ask ourselves what we share as a
broader learning community). Some other grassroots
organizing styles are very strong on this aspect, both
those that are "radicalized" and others that are more
politically moderate. [2]
To sum up: I think a course built around "how to learn
mathematics" is a great idea. Trying to involve
mathematics learners at all levels in discussing this
question is also probably a good idea, but a course may
not be the best way to have such a discussion. Certainly
people need to know what they are getting into in any
case.
To conclude: Hopefully we can find the "right" group to
make progress on these issues in the future. This course
has brought up some of the core problems that apply to
peertagogy. We should cut ourselves some slack, because
this field is so new that I had to come up with a
neologism here and now to describe it! If that's to be a
lasting positive outcome of the course, I suppose we
should offer a more formal version, "paragogy" (not to be
too heavily confused with the etymologically related
"paragoge"). Whatever we decide to call it, the key point
of DIY Math as a genre is to do away with the artificial
distinction between pedagogues and learners. This is a
power struggle par excellance, easy enough to mock
perhaps, but one I think we should not ignore.
Thank you for your attention,
Joe
[0] M. Myllyaho, O. Salo, J. Kääriäinen, J. Koskela, A
review of small and large post-mortem analysis methods,
in: Proceedings of the ICSSEA, Paris, 2004.
(http://www.vtt.fi/merlin/pub/pma_full_1.00-icssea-layout.pdf)
The authors mention four key topics to consider:
* what did we do well that does not have to be further
discussed (the positive aspects)?
* what did we learn (the negative aspects)?
* what should we do differently the next time (the
negative aspects which require improvements)?
* what still puzzles us (share the things that are
confusing or do not make sense)?
[1] For example, I think it would be *possible* to design
a "Discovery Learning" course where the very *foundations*
of the course were to be discovered on the fly by
beginning students, but I think we should acknowledge that
this would be going far beyond the "standard" Discovery
Learning programme for mathematics education, which asks
students to prove certain results from certain axioms,
thus providing at least a basic *logical* scaffolding;
cf. http://legacyrlmoore.org/reference/mahavier1.html.
[2] Alison recommends, for building community, and
creating social contracts, "Building Community in Schools"
by Thomas J Sergiovanni, and, "Two Bits" by Christopher
Kelty.
jonsul
be babies sometimes. A lot of us want to be pushed in the direction
that we may be too scared to find out for ourselves. I'm starting to
realize that to be successful or master anything you need to be able
to push yourself. I believe this is a course based on that. Most
courses aren't built like this and have the teacher doing the pushing,
and students are used to this. I think it would have been a good idea
to keep some of the aspects of pushing students for those used to it.
Maybe you could have a topic of the week to show a direction for them
to go based on your conversations with the students.
In any case I'll be on here still, I'm sorry for my absence I've been
trying to create a legal organization based on some technology I'm
designing so it's been hectic the past week. I fear I may have
contributed to the downfall by being absent. So I'm sorry, I'll try to
be more present on here in the future^^
Joe Corneli
course. I'm glad you'll be around. And I hope this post-mortem
analysis wasn't too disturbing. I'm noticing that PMA could
also stand for Pre-Mature Autopsy, as in
"You have to beware of premature autopsies. A noble sound might not
lie still in the dark cave where the dragons have taken it."
-- http://www.mdcbowen.org/p1/cobb/premature_autopsies.htm
I think you're right that students are used to being pushed, and
I also think that you're right was that the course is about participants
learning to push themselves.
I could have been more clear about that up front, but I think I'm
seeing it much more clearly now... and I'll have to keep thinking
about how to facilitate this well. Thank you for continuing the
discussion.
Joe
Alan Cooper
I think that your emphasis here on the goal of this course being to
discuss *how* to learn is a good point which may have got lost a bit
while people were focussing on *what* to study and then getting started
in doing it. It is hard to imagine guiding many people through
different programmes of study at the same time, and there may not be
enough of us to provide a pool of mutual mentorship with regard to
content, but the goal of providing guidance about how to proceed is
still valid in its own right.
What has been happening so far with regard to establishing goals is
actually I think the right first step. Next comes finding suitable
materials and resources (including how to evaluate them and interpret
the evaluations provided by others)- and I think it would be useful to
approach this in terms of discussing general strategies that we can use
in future, rather than (or as well as) giving specific recommendations
to each of us.
Further topics might include what to do when stuck, how to build
our own examples (and counter-examples) for things that are given to us
abstractly, how to find (or make up) examples of the practical
application of what we are learning, how to test ourselves for
understanding (both on a line-by-line basis and cumulatively), and so on.
Perhaps, given the range of demands on all of our time and the
extra difficulty of working independently, it may be too ambitious to
expect to progress through our projects at the same rate as a standard
class, and the goal of this course should just be to get us started with
some good strategies and tools for going on on our own.
To that end I think that this discussion may well continue to be useful,
and (perhaps after some editing and/or consilidation) might serve as a
resource for others wanting to get started on similar projects later.
So keep up the good work.
cheers,
Alan
> [0] M. Myllyaho, O. Salo, J. K��ri�inen, J. Koskela, A
Joe Corneli
I like your list of "further topics" and I've started to sketch some ideas
around them here: http://www.p2pu.org/general/node/5571/document/10225
I remain optimistic about the mutual mentorship with regard to content
and thanks for your help with that :)
Joe
>> [0] M. Myllyaho, O. Salo, J. Kääriäinen, J. Koskela, A