Johan wrote:

> As William stated, I think any functionality improving SageMath's appeal
> for, say, educating high school students would be very welcome. My main
> concern is how valuable what you propose with PRESS-like printing is in
> this respect.
> You gave a printout of your current PRESS implementation earlier, for
> solving a simple linear equation in one variable. That printout is
> extremely verbose and very micro-syntactically oriented. My guess would
> be that few high school teachers would wish to teach the simple isolation of a
> variable technique that way.

Unfortunately, what most mathematics teachers are teaching is not
mathematics. This observation is described well by Scott Gray in the
following blog post excerpt (Scott was a university mathematics
professor, and he also designed and orchestrated the building of the
O'Reilly School of Technology):

Here are some passages from this blog post:

"None of my students knew what they were talking about. Even students
who got perfect scores on my written exams didn't really understand
what it was that they were doing."

"...students were simply emulating calculation techniques, without
understanding where those techniques came from, or how to create them

"...students would ask me to solve every type of problem they could
find in the textbook. Even though I'd have them try the problems
before showing them the solution, they were really preparing a
decision matrix for a matching game. If the problem was like this,
then they would do this; if it was like that then they'd do that, and
so on."

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. In the
future, K-12 mathematics teachers will be replaced by AI mathematics
tutors. The software I am writing is being specifically designed for
this purpose.

> And students struggling with the technique
> would mostly just become more confused ("undefine the binary '-' operator",
> "change the association of +", etc.), and would fail building up
> mathematical intuition but rather focus on the syntax.

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

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

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

> In Denmark - and my impression is that the same goes for other countries
> as well - high school curriculum has largely moved away from solving of
> such equations using identities and "tricks". The cos(x+y) identities
> etc. and similar are not even taught anymore. In algebra, students
> mostly solve linear equations with 1 or 2 unknowns and second-degree
> equations. Simple isolation involving logs and exponentials, and there's
> also some simplification and/or factoring of e.g. polynomial fractions
> involved in function analysis.

Eventually I want to give my solver all of the abilities that PRESS
has. However, this is mostly from a personal desire to understand
everything there is to understand about how PRESS works. For most
students, I think the best use of PRESS-based software will probably
be for teaching them the fundamentals of how mathematics actually
works. After they are comfortable with this knowledge, and proficient
at using it, I think they should be taught how to program a
conventional CAS instead of being taught PRESS's more advanced


You received this message because you are subscribed to the Google Groups 
"sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
To post to this group, send email to
Visit this group at
For more options, visit

Reply via email to