Ted Kosan writes:
> Unfortunately, what most mathematics teachers are teaching is not
> mathematics. This observation is described well by Scott Gray ...

Indeed, I'm sure there's many examples of such teaching going on, and it
is important to try to improve this.

> High school mathematics teachers are not qualified to decide how
> mathematics should be taught because they don't know enough about how
> mathematics actually works to be entrusted with this decision.

Possibly true for some math teachers. However, I'm generally in favour
of putting more trust in the teachers and let them decide how to teach.
More control, canon and centrally-chosen systems for teaching, more
tests and more admin for teachers take time and passion away from what
they should be doing.

> In the future, K-12 mathematics teachers will be replaced by AI
> mathematics tutors.

That's a bold claim.

> As Scott Gray's blog post indicates, the kind of mathematical
> intuition that most students are being taught to build up is the wrong
> kind of intuition.

The kind of mathematical intuition that Scott Gray taught was wrong. As
Dima pointed out, teaching for the test is wrong.

> The right kind of mathematical intuition consists of the techniques
> that PRESS uses.

Wow, another incredibly bold claim. And it hardly follows from Scott
Gray's experience.

> These techniques are very syntax-oriented, but they
> are so simple and straightforward that even young children should be
> able to learn them without too much difficulty.

I'll strongly contest that. The ability to crank a handle on a musical
box doesn't make you a musician.

Almost no mathematicians think in axioms and syntax. I doubt that good
mathematical insight can be grown this way -- just see the pedagogical
failure of Bourbaki and New Math in the 60s.

> Before I started testing my solver with students, I was also concerned
> that the rules it used might be too detailed for them. However, what I
> discovered is that many students found the high level of detail to be
> illuminating because for the first time in their lives they were
> seeing how mathematics actually worked with nothing being hidden from
> them.

OK. Following Scott Gray, did you test your students on their
understanding? The students' own feedback is close to worthless here, as
Scott Gray pointed out.

> Having said that, the set of rules that are shown in the demo video
> are just the set I decided to experiment with first. The software can
> be easily configured to work with other sets of rules, and it
> shouldn't take too much effort to replace the rules shown in the demo
> with fewer more abstract rules. It is going to be interesting to
> discover which sets of rules are best suited for various educational
> goals.

I really believe that a major challenge lies in striking this balance.
To me, the level of detail in the earlier demo does not seem useful for
teaching. Automated solving of certain types of equations in "the right
amount of detail", and with more explanation, I could see as useful as
*one element* of teaching *one part* of mathematics.

But it's very difficult to let students sit with an automated tool which
spews out endless similar-looking equations and *not* actually be
encouraging them to do the pattern-matching-crank-the-handle

I'm not shooting down your project here. But my humble guess is, that if
the current version of your PRESS gets included into Sage, it will rot
there and noone will use it. It has to be more polished, and present a
compelling use case in a realistic, contemporary teaching environment.


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