What is Mathematical Understanding?

Unfortunately, I didn't prepare my ideas at all for AIED. So I mostly improvised on the topic of how logic & argumentation games can be used in science education, and the use of formalized theories.

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What is Mathematical Understanding?

My dream that you should be able to transfer mathematical knowledge from a formalized mathematics proof (as in Coq, Mizar, etc.) to a student was based on the incorrect assumption that formal proofs contain enough information.

The question that needs to be answered is
"What is mathematical understanding and how can we model/represent it?"

I think mathematical understanding of a problem/domain requires somewhat more than formal proofs. It requires:
* mapping formalisms to intuitions (which may be formalized as models) and witnessing that the homomorphism/analogy holds up: checking (weaker versions of) "soundness" and "completeness". While proving the relevant facts (completeness), the user fails to prove absurdities with the formalism (soundness).
* evaluating counterfactuals (proof critics: "would this proof go through without this assumption?")
* etc.


Btw, while some kinds of intuition can evaluate statements and can be relied on to guide formalizers (e.g. I *know* that squares are special cases of rectangles, and any formalism must respect that),
other kinds of intuitions are just intuitions, which tend to be acquired empirically through experience with examples one has already seen or imagined. (see Lakatos's(?) notion of intuition by experience) Alison Pease - A Multi-agent Approach to Modelling Interaction in Human Mathematical Reasoning (2001) )


from Bundy's "A Proof, Conjecture and Theory Editor":

Most work on automatic reasoning assumes a given theory and conjecture and develops a proof in the context of these. However, in practice, especially in common-sense reasoning, both the conjecture and the theory are adaptable. For instance, the failure to prove a theorem might result in a change to the conjecture or the theory in order to make proof possible.

Also, Paul Thompson - The Nature and Role of Intuition in Mathematical Epistemology
seems interesting.


Also, Mobius Stripper quotes reader "Susan":

It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.

We need to define and expect something similar as far as being numerate.

In my experience, this can be done by:
* engaging the student in argumentation
* make him/her imagine a slightly different situation, ask questions. Ask the student to compare: "why does this procedure work here but not there?"


I'll have to organize these ideas later.

is "Kolmogorov information distance" + "case-base reasoning" an original idea?

Gustavo - "Kolmogorov Complexity"! "information distance"! "case-based reasoning"!
Google - no major websites or papers connecting the two


Isn't the connection obvious??
Doesn't CBR require a similarity measure? Isn't information distance the most general similarity measure?

The Economics of Flames in Technical Forums

We need to find a balance between welcoming beginners and not annoying the hell out of the experts. Can't someone come up with mechanisms for this?

Beginners can sometimes be annoying, but they could easily be filtered out by those who don't like them. It seems most people keep a reputation system, about how much they respect each person.

The ignored person should ideally know who is ignoring them (filtering them out), and it should also be possible to label your messages in such a way as to indicate the probability of annoyance (statements, in the case of chat rooms). As usual, what I'm asking for is just collaborative filtering.



http://groups-beta.google.com/group/comp.lang.lisp/msg/7060ad1dc1defc9c

My _actually_ expressed motive, not the Coby Beck rewrite, is to drive
_stupidity_ out and make it so painful to act stupidly that people
stop being stupid.
...
I want a technically stimulating forum. When push comes to shove,
that is what most other people want here, too. However, USENET is a
virtual _magnet_ for idiots and opinionated air-heads,
...
Most people accept stupidity and incompetence in every form they come
across because they would rather be seen as easy-going and friendly
than to get what they pay for and want.

guitar workshop / instrument training systems

Today, Nelson gave a guitar workshop. I got to go for free, as a perk for being his housemate.

It turns out my accompaniment skill sucks (in large part because I get bored with repetitive lines: when you hear of people that play better in front of an audience, this is why: motivation), and I found it awkward and unnatural to do the upward thumb-strum that they assigned me. They told me to loosen my wrist more, but the whole effort of trying to get several things right at the same time made my arm a bit tense.

Later, I practiced a 12312312 fingering pattern, which works very nicely for that song (Feira de Mangaio). But I couldn't do it fast enough to play with the others.

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I'm looking forward to instrument training with biofeedback (or better: bio+visualfeedback). AFA I understand, all the learning that takes place when learning an instrument's technique is learning to coordinate muscle movements (i.e. create muscle memory for certain sequences/patterns). A biofeedback system could indicate which muscles you are using, and tell you to practice sequences appropriate to your level, and help you get rid of bad habits (whether static habits about holding the instrument or patterns that you are stuck doing one way, such as always changing bow direction in a certain position on the measure) ...and even tell you to relax if you're too tense or if you're not breathing in the right places; or help you if your mind hasn't gotten into the rhythm yet ("does this rhythm make you tick?").

In fact, with a display (e.g. goggles), one could probably practice the correct muscle movements even while sitting in the train without the instrument, despite the lack of haptic feedback.

About the visual input, to get around the vision problem, the student could practice in a dark room with light points all over his body and the instrument (I'm sure I've seen an animation like this before).

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Anyway, after the workshop, I went to Vondelpark, and found myself trapped under the bridge due to the rain. Joining me were two buskers (one of whom was a Goan fiddler who played with a hairless bow), a picnic, and a conference of Alcoholics Non-Anonymous (unlike AIED, this was a very local conference, so they had bottles instead of badges).

I think I look sexier when I'm carrying a guitar by the looks I get from the females, but it could just be my impression. Although I think I've heard of research where males pictured with a guitar were rated more sexy (though I can't find it).