On Friday, October 14, 2016 at 4:49:38 AM UTC, tkosan wrote:
> 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): 
> http://patternmatics.org/temp/scott_gray.html 
> Here are some passages from this blog post: 

"I eventually found out that all of the math exams in Russia are oral 
Nope, this is not true, and has never been so. Although it is correct that 
there were oral exams and written exams, too.

> "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." 

that's what you get if you teach to the test (and for good student 

> "...students were simply emulating calculation techniques, without 
> understanding where those techniques came from, or how to create them 
> themselves." 
> "...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 
> difficulty. 
> I'm not sure if ability to manipulate complicated formulae quickly has 
much to do with mathematics
and is so important as a skill.


> 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. 
> 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. 
> > 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 
> capabilities. 
> Ted 

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