Date: Nov 17, 2013 7:22 AM
Author: GS Chandy
Subject: Re: Simplifying Algebraic Expressions with Subtracted  Expressions

Pam posted Nov 17, 2013 10:44 AM ( - GSC's remarks interspersed:
> >
> > On Nov 16, 2013, at 4:03 PM, Pam

> <>
> > wrote:
> >

> > > The biggest difference I see is that the algebra
> > student may be able to go through the motions
> > mechanically, using memorized procedures
> >
> > Is it possible Pam that you don't know what they

> are
> > doing in their head, and that it's anything but
> > mechanical?

> >
> > Bob Hansen
> >
> >

> You misunderstood. For those of us who did well in
> algebra, I would say that most of us had a lot going
> on in our heads that was anything but mechanical.

I'd go further: I suggest that ANYONE who has EVER learned ANYTHING has ALWAYS had a lot going on in his/ her/ their head(s) that was anything but mechanical. It is, in my opinion, not possible actually to learn anything '(purely) mechanically'. This is so very obvious to me that I find it astonishing that some people still do insist on 'by-rote learning' by students. The 'by-rote stuff' may well contribute to recall of specific facts, figures and stuff (which may contribute to making that available for application) - but not to real 'learning'.
> In my statement above, I said "may". Unfortunately,
> there are students who approach algebra mechanically,
> through memorized procedures, because they lack
> fundamental understanding.

This is (IMHO) the direct result of poor teaching.
> Also unfortunately, they
> think the memorized procedures are the only way to
> approach problems, that procedures *are* the algebra,
> so to speak. And, no, although they may be able to
> solve simple problems that follow exactly what they
> understand of the procedure, they are not generally
> successful. Prior to remediation, that is.

As I'm not a US-based teacher I don't know how good or effective 'remediation' is in the USA. In fact, I'm not a teacher at all - I'm just a person most keenly interested in the 'learning process' - which I claim is the only hope for humanity as a whole. Most of our planet's issues and problems derive from the fact that we (humanity) have not 'learned effectively' (though we've always had the opportunity to learn).
> As Wayne says, this won't happen with a solid
> grounding, but I think the relatively small
> percentage who get that solid grounding in the US had
> to do so mostly on their own. For the reasons that
> Liping Ma has found in her analyses of US math
> education, among others.
> Pam

I don't know about the 'solid grounding' provided in various disciplines/subject areas in various countries. I believe the solid grounding is lacking right through our educational system, in practically all fields, in practically all disciplines, in practically all countries - there are, of course, exceptional teachers everywhere who do manage to apply, effectively, the 'learning+teaching dyad' to their professional work as teachers. It is my claim that we'd have a great many more effective teachers if 'teaching' were, in practice, considered a part of the 'learning+teaching dyad' (and not a 'thing-in-itself' as it is too often). A little practical understanding of 'systems' by teachers would help significantly.