[RUME] Mathematics Education How-TOs

[Alan Selby]

From: Alan Selby, Ph. D. Montreal, Quebec

Appetizers and Lessons for Math and Reason
(1000 webpages and 80 webvideos)
http://www.whyslopes.com

If anyone thinks these how-TOs would be the basis for work in their
department as a  Professor of Mathematics Education, let me know.
I have not had an college level post since 1989.


Direct Mathematics Education How-TOs

At www.whyslopes.com, I have posted on-line what may be the first  
mathematics education, teaching how-TOs
to support and reform calculus and the preparation for calculus in high  
school and remedial
college studies. The how-TOs are developed in accordance with a quest to  
support
inductive principles for instruction met in 1981. The proof is in the  
details.

     Inductive principles for instruction fail in the same or similar way to
     principles for mathematical induction

The how-TOs in particular introduce smaller and more steps to ease or  
avoid difficulties,
and to enrich comprehension in algebra and at the start of calculus. The  
Algebra & Calculus
starter how-TOs support and refine existing practices. In contrast, the  
full and very simple
geometrical development of complex numbers in the Euclidean Geometry area  
of my site
points to a reshaping of the high school math curriculum in geometry and  
trig.

The latter development of complex numbers stems from a 1979 lecture series  
of the late Richard Feynman
at McGill University. I saw Richard Feynman in say 20 minutes of a three  
evening of public
lectures on his discipline physic describe his subject as a simple  
addition and multiplication
of arrows in the plane. He did so without mentioning complex numbers as  
not to alarm audience
members.  Since then I have been wondering how to use his idea in a manner  
that would
speed the high school or early college development of trig on the unit  
circle.
The Euclidean Geometry area of my site finally solves the problem in a  
manner which is simple
and clear enough  to change the high school mathematics and college  
curriculum.

The how-TOs are logically developed in accordance with the standards of  
applied
mathematics or Euclidean Geometry appropiate for mathematics education  
prior
to specialization in pure mathematics. But only some are tried and tested  
in the classroom
due to my intermittent career in and out of education in suboptimal  
circumstances..
The how-TOs are further light on set notation (an issue perhaps).

When writing began in the last days of 1990, I set myself the task of  
supporting
and reporting inductive principles for instruction.  Just after the task  
began, I looked at the
education literature to see if my idea of providing clear and direct  
how-TOs for
instruction was moot. What I saw was a discussion of delivery styles and  
no concern
for how-TOs that teachers pushed into mathematics education without a  
quantitative
background might find easy to understand and follow in class. The posting  
on-line of how-TOs
to support, refine and modify instructional practices furthermore points  
to documentation standard
that advocates of education reform may follow. The how-TOs now present a  
lower bound for
instruction in a manner that no doubt can be improved by fellow  
mathematicians via the refinement
of technical detail and the additional of historical notes, notes beyond  
my current knowledge
of the historical development of mathematics.

I was fortunate at McGill to see Gian-Carlo Rota once and
L. Nirenberg several times give guest lectures in which their exposition  
made elementary to advance
topics clearer to understand and explain.  Their expositional delights  
implied room for
change in the high school and college curriculum. With the majority of  
high school mathematics
instruction provided by teachers without a quantitative background pushed  
into their duties
and a greater majority unaware of that calculus demands a full strength  
mastery of geometry,
algebra and exact arithmetic with whole numbers and fractions, mathematics  
education reform
needs to be based on clearly put how-TOs for direct or indirect  
instruction, how-TOs easily
understood and followed by all teachers to set a lower bound, if not a  
standard, for
instruction.