<<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
Goals for Mathematics Education
The story of the race in which an overconfident hare is
beaten to the finish line by a slower but non-stop plodding tortoise
gives a lesson on determination for all students, slow or gifted.
See For Greater Clarity
in Mathematics Course Design after the three
goals.
1. From Logic to Precision Reading and Writing
For the first goal, and for the greatest help from site pages to
your work and studies, start with and see if you enjoy logic chapters 2 to 5 in the
online book Three Skills
for Algebra
Logic chapters
2 to 5 aim to ease or avoid difficulties in school and at
work by making you aware of the need for care and precision in
reading. Once you can read exactly or precisely, you will
have a better chance of seeing the mistakes in your writing - the
difference between what you meant to say and what you said. Precision
writing allows one to state the maximum possible without becoming
inexact or wrong. Parents: If your teen has difficulties
in learning, once a year, try to guide your teen, through these
logic chapters, as many
as possible, or ask your teen's school to test and check logic
skills.
For literate math-phobics, reading online logic chapters 2 to 5 may
postpone mathematics studies for a few hours while building skills
and confidence for work and studies in general and for mathematics
education too
Mathematics studies may be further postponed by
reading Pattern Based
Reason, an online book, the source of the logic chapters. The
question of what rules or patterns are reliable and which ones apply,
if any, leaves room for thought and worry.
2. From missing words or links, to unifying themes
For the second goal,
learning how to describe numbers and equations with words to make
algebra mastery stronger and easier to obtain
site lessons will change how mathematics is understood and
explained.
There has been a silence, a lack of words, in
mathematics because arithmetic and algebraic expressions are difficult
to read aloud in a way that listeners understand or follow the order of
operations. So expressions of many kinds are better read in
silence and understood nonverbally in a glance. There-in lies the
silence.
With the passage of time, the following program for reducing the
silence, or adding words to lessen its effect, may be improved. But
here is the first draft, likely to be immediately effective in college
and high school instruction.
Online algebra chapters
8 to 14 in site volume Three Skills for Algebra
show (i) how to describe numbers with words apart from and
besides symbols, that is the first skill; and show (ii) how to
use describe formula usage as direct or indirect, and describe
indirect use as a numerical or algebraic solution. Item (ii) in
retrospect points to a fourth skill for algebra in the volume.
The second skill is given by our ability to describe calculations
with words and formulas. The third skills is given by ability to change
the description of how numbers and quantities are computed via
replacement or substitution operations. Learning the four skills
or emphasizing as unifying themes in high school and college will
change mathematics instruction. The online essay What is a Variable
provides a word-based and word-only explanation, and so illustrates the
first skills for algebra, our ability to describe numbers, amounts and
quantities with words alone, before any formalism of modern
mathematics, indeed before the use of any symbol.
Recognizing our ability to describe numbers, amounts and quantities, and
to describe operations on formulas or equations provides themes to unify
mathematics education from algebra to calculus. I kid you not.
To lessen the silence, we use names, descriptive phrases and more words
in mathematics by naming equations and formulas (old hat), by learning
how to describe numbers, amounts and quantities before or besides the
use of letters to stand for them (a first site contribution to
mathematics education), and by learning short descriptive phrases for
common operations on equations and formulas - direct and indirect use,
numerical or algebraic solution. There-in lies another site
contribution to mathematics education.
3. From Geometric Quantities to Algebraic Sense
For the third goal, we observe people have greater comfort and less
panic in algebra when letters are used in geometric formulas to stand for
lengths, areas and volumes which can be drawn or sketched. Those
geometric formulas can be evaluated when the geometric quantities in them
are known, given or measured. But the formulas are understandable,
they provide arithmetic recipes to follow, even when the geometric
quantities or numbers appear in drawings, but with values not yet given
or measured. There in lies a start for algebra - several steps to
provide an geometric start for algebraic skills and concepts.
-
The site three column method for solving linear equations with stick
diagrams to go from use of letters denoting a length in
formulas for area and perimeters to acceptance we hope of a
letter denoting a number in equation.
-
Geometric
(area) views of the distributive law leads to methods for
multiplying and adding polynomials and also the decimals
representation of whole numbers.
-
This complex number starter
lesson represents the last of several efforts in site pages, the
first appears in Volume 3, Why Slopes and More Math, to provide a
concrete coordinate-based development of complex numbers. Easy
consequence imply trig formulas for dot- and cross-product, and
algebraic proof of the cosine law.
-
Geometric & algebraic
previews of calculus to explain why slopes & factored
polynomials appear, and doing so provide a gradual, rather than
sudden start to the full strength use of algebra in calculus.
Thinking Part of Mathematics
and Logic: There are three kinds of rule-based intelligence in
mathematics, logic and most pattern-based subjects. The first
kind met in primary school arithmetic consists of skills with
repeatable, reproducible and therefore verifiable results - results
that are then considered right or wrong. The second kind also met in
primary school consists of pattern or rule recognition. The development
or exploitation of the ability to recognize or suggest simply patterns
in order to predict the next element in a sequence. If the prediction
fails, another pattern is required. The third kind appears after
inductive mastery of logic, that is mastery of implication rules If A
then B and their use. The second kind follows the use of implication
rules and definitions and assumptions, one at a time and one after
another, to arrive at logical conclusion in a repeatable, reproducible
and therefore verifiable manner.
Pure modern mathematics depends on or assumes the ability of students to
think logically and algebraically. Pure modern mathematics does not
require geometric diagrams nor physical assumptions for its thought-based
development from axioms (assumed patterns, algebraically described) about
sets and numbers.
To develop student ability to think logically and algebraically, and to
introduce the geometric concepts in trig and calculus, concrete geometric
diagrams if not physical assumptions are required. And the modern
mathematics high school and college mathematics of the last half of the
20th Century while inspired by the axiomatic structure of pure
mathematics followed in practice, impure mathematics programs of study,
with some contortions or routes more complicated than need-be in course
design. The latter was due to the conflict, not then fully understood,
between the demands of pure mathematics and the demands of mixed
mathematics, say trigonometry, calculus and pre-coordinate synthetic,
Euclidean geometry. That was the situation outside of the United
Kingdom in other parts of Europe and also in Canada and the rest of
North America.
Greater clarity in course design from primary school to college calculus
level could follow by using manipulatives, physical objects and geometric
suggestions and assumptions to develop an accessible, mixed mathematics
curricula which leads to results in a repeatable, reproducible and
therefore verifiable manner, in an empirical, if not pure manner.
|
|