Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
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For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
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www.whyslopes.com >> Volume 1 Elements of Reason
Volume 1, Elements of Reason
The first part Pattern Based Reason (Volume 1A) of this work
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.
understanding and explaining
reason and math
Alan M. Selby
Printed in Canada
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.
The second part Mathematics Curriculum Notes (Volume 1B) 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 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
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.
December 2011 Postscript
Site chapters and steps now stands at
the sharp edge of mathematics education reform. Site material stems
from olde and continuing gaps and inconsistencies in ends and methods - there was no pleasing all.
The essay which way to
go "lightly" introduces a more detailed, five phase framework and nearly plain-language remedy.
Phases 1 to 3 focus on skills of value for adult or daily life - precision in reading-writing-figuring included.
Phases 4 & 5 focus on calculus and preparation for it. Many university programs demand calculus. Preparing for
it has value in senior high school science too, and some value for trades-professions not taught in university.
The framework addresses and remedies all the difficulties identified above, and implement most of the ideas
in the subvolume 1B, Mathematics Curriculum Notes. The framework being done sets the stage for
yet another consolidation of site material.
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.
www.whyslopes.com >> Volume 1 Elements of Reason
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
1996 - Magellan, the McKinley
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why factor polynomials - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
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