I. Three Goals to Set for students.
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A. Master the rules, methods or patterns in arithmetic, algebra, trig
and calculus so that in your hands, they lead to the same results as
others - repeatable, reproducible and hence correct results. If
you belong to a group of students whose results differ after using
the same method to arrive at them, you or the group have problem.
More drill and practice will be required alone or with help.
First Sign of Intelligence: The patience and ability to figure
well, to follow multistep method in arithmetic carefully,
precisely, to a right answer, points to and develops the
ability to read and write clearly, with precision and not with
confusion in many arts and disciplines.
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B. Watch for the use or combination of rules and patterns, one at a
time and one after another, in sequence, which gives or suggest new
rules and patterns.
Second Sign of Intelligence: If you see how
rules and patterns can be combined to get results or more rules and
patterns, you have found the key to the thought-based development of
more skills and concepts, those to come and if you like, those you
have seen.
Third Sign of Intelligence: If you develop
ability and interest to see and know the limitations of rule and
pattern based reasoning in theory and practice at home, at
school, at work and in society in general, you are becoming a
critical thinker. Good luck. This third sign of intelligence is
not always appreciated.
In arithmetic and beyond, students need to learn to apply
rule and patterns one at a time and then in combination, one after
another, in repeatable, reproducible and hence verifiable manner. In days
gone by, precision figuring skills were taken as a sign of intelligence
or potential to follow, if not bend, rules and methods, with precision to
meet the needs at hand. Rules and patterns with repeatable,
reproducible and therefore verifiable results literally provide a base
for society to function, but there is a caution. Rules and patterns
once found or given need not be fair, nor sustainable in the
long-term. Their assumption always involves some risk. The
knowledge of how to use rule and patterns in a precise, repeatable
and reproducible way, and the knowledge of the benefits, origin and
limitations are both needed for critical thinking and
intelligent problem solving at many levels.
II. Supporting Aims A and B
The thought-based development of mathematics is both inductive and
deductive. Key patterns (algebraically described?) can be suggested and
illustrated say by drawings and by numerical examples, that is the
inductive part, and then assumed for use in further chain of reason. That
is the deductive part. A balance is needed. Too much deductive detail
will alienate students. Ease of exposition or comprehension should be the
guide. Where derivation is too long, assumptions (axioms or assumed
patterns) should be explicitly stated for the sake of transparency.
The aim is an operational command of mathematics with a knowledge that an
inductive-deductive account of the discipline is possible. References
for fuller thought-based development should available for students
who want to go farther.
After arithmetic, an operational command of quantitative skills
sufficient for mathematics to pre-calculus level may follow from
lessening algebraic difficulties as indicated the site entrance, from mastering logic
and from the assumption and geometric interpretation of the
properties of real and complex numbers, from the easier consequences of
those properties, and from the assumption of that all real numbers have
decimal expansions, finite or infinite. The geometric introduction of
complex numbers only requires and involves the junior high school level
familiarity of coordinates, translations, reflections and rotations in
the plane, and may be use to develop that familiarity.
See site areas: 2. Linear Equations
& Fraction Skills 3. Fractions Ratios Rates Proportions
Units 5. Analytic
Geometry 8. Complex Numbers 10.
Secondary IV(?)
math; the online volumes 2. Three Skills for
Algebra and 3. Why Slopes
(A Calculus Intro) & More Math and the site area 7.
More Calculus.
The material here can be presented as rules and patterns to use and
combine with confidence in results coming from their repeatable,
reproducible, and therefore verifiable nature. The coverage of logic
here aims to develop precision reading and writing, and an
understanding of how implication rules B IF A can be used and combined
directly, one at a time and one after another.
The support of aims A and B advocated here may be simple, short and
effective enough to allow more students to start calculus while also
serving the needs of students heading for business and technical trades
(surveyor, plumber, carpenter or electrician) for which calculus is not
normally required.
In starting the development of trigonometry after the introduction of
complex numbers AND the assumption of the
field properties of complex numbers, turns the development into an
algebraic exercise and so make trigonometry and the properties of
vectors in the plane easier and more accessible. . Here students are
meeting and using rules and patterns that are easily understood or
assumed, and easily combined to arrive at further rules and
patterns. Ease of exposition is the guide. Harder routes in
which less is assumed can be left for students of mathematics
and/or the hard sciences.
III. Supporting Aims B and C
Axioms
and postulates in
mathematics are labels for rules or patterns that have been assumed in
order to secure a base for deduction. Further rules and patterns
are then tested in mathematics by looking for direct or indirect chains
of reason (arguments) that imply them. That provides a proof. Rules and
patterns thus proven may then being used in further tests or proofs.
The weakness of this deductive (more rigorous) style of reason lies in
the choice of initial axioms and postulates. Chains of reason
provide stories to follow one at a time and one after another. But
these stories, rigorously or deductive put together through chains of
reason, become works of fiction when the initial axioms or postulate
are not true. On the other hand, if there are elements of truth in the
original axioms and postulates, these stories may be
non-fictional.
The mixed mathematics development of synthetic (coordinate-free)
4. Euclidean
Geometry in site pages inductively suggests and clearly identifies
geometric rules and patterns, those assumed for use in deductive
reasoning. There is motivation here for the indirect statement of the
parallel postulate as given by Euclid, namely the assumption that two
lines segments extended will meet on side of a transversal will if
the interior angles on that side of the transversal sum to less
than two right angles. This coverage of Euclidean geometry with the
selection of a unit length and assumptions about coordinates and
their decimal representation to imply the field properties of real and
complex numbers taken as assumptions in the support for aims A and B
above.
Most, if not all, of the deductive chains of reason offered here
will be direct. Ease of exposition, making the ideas more accessible, is
the objective here. That being said, in the development of
Euclidean and then Analytic Geometry here, there is focus on the possible
origins of assumptions - how they can be suggested by and extrapolated
from experience. Besides this, there is a focus on deductively deriving
the consequences.
IV. Why Rewrite the Curriculum
Preparation for calculus and other disciplines still requires to a
greater or lesser extent, a good command of arithmetic with
decimals and fractions, algebra alone and in further subjects, analytic
geometry and trigonometry and logic. For that good command, some drill
and practice is required along side a thought-based development of
skills and concepts. The support for aims A, B and C above points
to a leaner and more accessible development of the necessary skills and
concepts for calculus and for most technical subjects that require
algebra, trigonometry or complex numbers. This lean program in reaching
or developing key skills and concepts directly and systematically is
more likely to build skills and confidence necessary to retain students
in this program from start to end. Shorter and leaner provides a
remedy for longer, more awkward and less focused mathematics
programs that seem to continue year after year in a manner in a seeming
aimless, ad hoc, manner, an endless manner that may alienate students
and be difficult to motivate or justify in the classroom. Less
well-chosen would be better.
Modern mathematics with its set-based description of axioms for real
numbers given (or derived from assumptions about sets or natural numbers)
provides a geometry-free, model for understanding, describing and
developing the properties of real and complex numbers, and also
properties of functions which appear in calculus, all apart from the use
or mention of decimals. Geometry-free means there is no dependence
on diagrams or suggestive reason. But there is a great dependence
on direct and indirect deductive reason, and on the shorthand role of
letters and symbols to codify, record and develop concepts and results.
In North America, if not elsewhere, modern mathematics curricula from
birth in the late 1950s to abandonment in the late 1970s or 1980s with
their set-based presentation of axioms for real numbers aimed to prepare
students for the more rigorous treatment met in university programs of
study in pure mathematics. But the introduction of modern mathematics
curricula used geometry to introduce and illustrate the properties of
real numbers, avoided all mention of decimals in the representation and
properties of real numbers, and required arithmetic skills with decimals
in examples involving calculations or coordinates, and used geometric
diagrams to introduce functions. There are was also no systematic
development of the algebraic way of writing and reasoning. It was just
assumed. So the modern mathematics curricula while preparing
for pure, geometry-free and decimal-free, mathematics did so
inconsistently and awkwardly.
The modern mathematics curricula, its course content, has further
lingered in course design and delivery, preserved in an ad hoc manner as
the sway of pure mathematicians over mathematics course content has been
replaced by by the sway of others with a greater knowledge of pedagogy,
that is how a master teacher should cover material, and a lesser
knowledge of mathematics. Education reform since the 1990's has been
based on the assumption that a change of delivery style with less
content, and less drill and practice, would make mathematics
instruction more effective. The net result is an ad hoc dilution and
shift away from the modern mathematics curricula of the late 1950's.
All the foregoing with the inconsistency the modern mathematics curricula
from arithmetic to calculus with it nominal aim of preparing for
geometric-free development of ideas in pure mathematics provides the
freedom here to put aside the awkward elements of the modern mathematics
programs, and propose a gentler, mixed mathematics approach likely to
provide a good operational command of arithmetic, algebra, complex
numbers, trig and some calculus in a more accessible manner, and likely
to provide the algebraic, geometric and deductive maturity needed to
appreciate pure mathematics. There-in lies a context and a justification
for a change of substance and not just delivery style in mathematics
instruction. Indirect instruction, constructivism or guided
discovery say, when fully developed should be compared and
contrasted with direct instruction fully developed in accordance with
inductive principles for
intruction, and the following theory of knowledge for technology, the
hard science and mathematics.
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