The Trouble With Algebra
People have difficulties in algebra as the shorthand role of letters and
symbols appears is required but steps for its full development are
too large or missing. Remedies are offered in site material.
The remedies (a) describe and illustrate three skills for algebra;
(b) describe and illustrate the direct and indirect use of
equations, proportionality equations included, and so vocalizes a
unifying and hitherto theme in secondary school
mathematics; and (c) re-introduces the concept of what is a
variable with words instead of symbols. The remedies are
continued in site pages on solving linear equations, arithmetic with
polynomials, complex numbers and calculus.
Before or besides the early use of formulas in primary school and
of the mastery via examples of methods for addition,
subtraction, multiplication and division of fractions, the shorthand
role of letters and symbols in describing the properties of arithmetic
operations with further numbers (real or complex) can be introduced or
hinted at by giving and illustrating algebraic descriptions of
these operations.
The unavoidable occurrence of arithmetic and algebraic expressions,
better seen and read silently, has been a huge barrier to the role of
words in understanding and describing algebraic skills and
concepts. Formulas and equations like pictures are worth a thousand
words. To lower the barrier but to circumvent it, site pages provide a
more vocal, a more audible paths. In retrospect, shortcomings in
the exposition or explanation of algebraic skills and concepts has
twisted or complicated learning and teaching from from first appearance
of algebra to their full-strength use of algebra in elementary and
advanced calculus. Finally, skill development and perfection requires
some drill and practice - not too much but enough. The algebraic way of
writing and reasoning before or during its development. requires mastery
of arithmetic with fractions without a calculator - that arithmetic
should be repeatable, reproducible, verifiable and automatic.
Course design and delivery may now follow the easier path of
developing skills and concepts in solving linear equations, of arithmetic
with polynomials, of complex numbers, from assumptions about arithmetic
with real and complex numbers.
Optional would be a mixed inductive & deductive development of
Euclidean Geometry and the properties of real and complex numbers where
decimal representation of real numbers assumed..
All the foregoing has the aim of providing (i) an operational command of
mixed mathematics and providing (ii) the algebraic-deductive maturity
needed in the further and optional study of modern
mathematics, axiom based.
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