Foreword (an intro to all
site books)
Elements
of
Reason
understanding and explaining
reason and math
Volume 1
by
Alan M. Selby
Ph. D.
Printed in Canada
ISBN 0-9697564-1-0
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Mathematics teachers hould see the second
part - the first part provides a context for the second part and further
site volumes. Bon Appetit.
The first part Pattern
Based Reason of this volume Elements of Reason describes
rule and pattern based thought and processes in daily life, society, science and
technology. Reliable rules and patterns can be followed one at a time or one
after another to obtain conclusions or results. Not solved is the problem of
identifying reliable rules and patterns to employ. Instead, the empirical method
of coping with this problem is discussed.
Rule and pattern based thought and processes touch many arts and disciplines.
Awareness of the difference between one- and two-way implication rules will
improve reading, writing and argumentation skills. Students of critical
thinking, persuasion, philosophy, mathematics, science and technology may find
this first part worth reading.
In both arithmetic and logic, rules and patterns if followed carefully lead
to results which are repeatable and reproducible, and thus verifiable and
objective: two individuals following the same rules and patterns with the same
data or in similar circumstances should obtain the same or similar
results. Arithmetic and deductive reason are but examples of verifiable
rule and pattern based thought or processes.
Verifiability, repeatability and reproducibility form a basis for the
appreciation of, if not reliance on, rule and pattern based thought and
processes. This appreciation should not be too firm. The identification of
reliable rules and patterns, or reliable data to use with them is
not certain. Further, where rules and patterns do not apply
mechanically, there is room for thought. Still, verifiability,
repeatability and reproducibility may provide a basis for the common knowledge
and informal mastery of a subject.
Mathematics
Curriculum
Notes
understanding and explaining
Reason and Math
Volume 1B
by
Alan M. Selby
Ph. D.
Printed in Canada
ISBN 0-9697564-6-1
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The second part Mathematics
Curriculum Notes is for teachers and advanced students of mathematics
or a quantitative college discipline. This part describes simply yet
precisely, the role of rule-based reason, that is logic, in providing a
thought-based framework and codification for mathematical thought. This
second part further describes how an inductive educational philosophy provides
a context for math and logic instruction from primary school to college.
Ideas which are easily repeated and understood may provide a common
knowledge of mathematics and the rule-based reason sufficient for a more
formal and rigorous comprehension.
This two-part work and its the companion volumes Three
Skills for Algebra and Why
Slopes and More Math stem from a project to write a
single book, namely Ideas that Might Count for
Education, Reason and Mathematics (1994). That single book (no longer available) was written and
distributed. It covered a vast number of topics. Some of interest to one
audience but not to another. With further writing and rewriting, this first
endeavor was divided into three volumes, the first of which, the one before you,
was divided into two parts. Writing for some is an iterative affair.
The initial aim was to report some unique idea, innovations, for math and
logic instruction. These ideas or lessons had worked well with college students,
shy or curious about one or both disciplines. But in writing and rewriting,
the aim became wider. The possibility of a consistent and coherent scheme
for math and logic instruction from primary school to college was seen and
explored. The scheme is comprehensive save for the treatment of
geometry. How to fit or emphasize Euclidean geometry in the curriculum is
not covered.
Formal mathematics can be difficult to follow for students who fail to
grasp deductive thought and the symbol-based algebraic way of writing
and reasoning. The latter like arithmetic is better seen and written
than spoken aloud. Symbols like pictures can be worth a thousand words.
Words have been missing to explain the role of symbols in providing the
shorthand notation of mathematics or its algebraic way of writing and
reasoning. The latter consists of recording and developing thoughts on paper
at least for those among us afflicted with a short or too forgetful memory.
The absence of a verbal culture to introduce and explain the algebraic way of
writing and thinking leaves its mastery to immersion and osmosis.
Comprehension depends on one's aptitude for learning some basic ideas by
immersion. I am in the radical position of suggesting that a certain
change is possible and desirable. This work and its companions suggest
how. They have yet to be formally peer reviewed and so should be read with
caution. The discussion of math and logic instruction and the discussion
of reason and persuasion are both fraught with controversy. Scrutiny or critical
examination of this work may lead to its refinement.
Alan Selby
Montreal 1996.
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Volume 1, Elements of
Reason, its first part, entertains and informs apart from
mathematics, as it describes logic, critical thinking and problem solving
skills for many arts and disciplines: Learn about the benefits,
origins, limits and risks of rule- and pattern-based activities
and explanations. Develop a critical command and understanding
of science and technology before defending or attacking any part.
Learn how patterns are suspected or recognized, and learn what patterns can
be tested before jumping to conclusions or alternatives.
-
Volume 2, Three
Skills for Algebra, offers gifted & adult students a
clearer way to review logic and algebra Logic
is put first as difficulty in it leads to faults in reading and writing in
work and studies. Putting words before and besides symbols makes algebra
easier to learn and teach.
-
Volume 3, Why
Slopes and More Math, gives calculus students the
clearest way (?) to preview or introduce the subject and ease or avoid
initial algebra shocks. With a
few suggestive diagrams, the easiest part of calculus can be mastered
with only a knowledge of slopes and polynomials. That gives a calculus
preview or review plus a first context for calculus.
-
Volume 1B, Mathematics
Curriculum Notes, the second part of Volume 1, Elements
of Reason,, is also online in pdf
form. First
chapters give a 1996 view of old flaws in past course
design, expositions on which site material stands, a view that
mistakenly assumed no trouble with the mastery of arithmetic. Inductive
principles for education like those in mathematics
requires (i) all steps be well-put and well-defined, (ii) the first
steps be accessible, and (iii) further steps also be accessible from
earlier ones. The inaccessibility of some parts of mathematics points
to hidden curriculum - concepts or skills not explicitly described.
The triplet 2, 3 and 1A could enrich the knowledge of gifted
students or prepare for calculus; the pair 1A and 2 could build skill and
confidence for algebra beginners and literate mathphobics, while 1A alone is for
avid readers in school or out -
Canadian Cataloguing in
Publication Data
Selby, Alan M,
Understanding and Explaining reason and math
Contents: v. 1. Elements of Reason - v. 2. Three Skills
for algebra - v.3. Why Slopes and more math.
ISBN 0-9697564-4-5 (set) -
ISBN 0-9697564-1-0 (v. 1) -
ISBN 0-9697564-2-9 (v. 2) -
ISBN 0-9697564-3-7 (v. 3) -
1. Mathematics--Philosophy. 2. Reason.
3. Algebra. 4. Calculus. I. Title. II. Title: Elements of
reason. III. Three Skills for algebra. IV. Title: Why Slopes and more math.
QA8.4.S44
1995
510'.1 C95-900945-0
Reprinting may lead to new ISBN numbers.
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Lesson Plans - Sec I
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Vol 1. Elements of Reason
Maths Harder than need-be
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What may be learnt and when depends on how skills and
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YOU are
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Take
greater charge of your work or studies: Read like
a lawyer
for better work & study skills, but do not take
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particular, two
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Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
-new Arithmetic
Folder
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course
Design & Delivery
- Fractions
- alone
- by rote
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algebra
- videos
- Functions - introduction
hindsight
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substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in
Logic -
Occurrence
& Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition
- Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes
& (?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for
dot & cross - cosine
law
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