Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
to switch.
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
fractions.
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct?
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.
Solve
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
equations algebraically.
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.
Using
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 1A Pattern Based Reason >> Postscript D Reflections-on-Law-of-the-Excluded-Middle Previous: [Postscript C Consistency-as-a-Tool-for-Reason.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
Appendix D. Reflections on
Law of the Excluded Middle
Origins, Benefits, Limitations
In writing a story or developing a theory, there may be some benefit in
be able to say for a possibility A, that
A or NOT A
holds - or is true. The assumption that A or NOT A holds provides the law
of the excluded middle in logic, one that may be used in mathematics to
arrive at conclusions.
In talking about this law of excluded middle, I will play the role of a
devil`s advocate.
A material implication rule A if B is vacuously true when the situation B
never occurs. In the formulation of a theory, consistency requires that
the situation
A and NOT A
never occur. That in turn implies
A or NOT A
in the exclusive OR sense. But the requirement for consistency does not
tell us which one of the two, A and NOT A occurs. If the elements of a
story, play or theory so far written are unconnected to the occurrence of
A or its negation NOT A, then whether or not A occurs (provided it is
internally consistent) does affect the consistency of the existing part
of the story, play or theory. With respect to the latter, the situation A
and its negation NOT A unreachable and informally at least, undecidable.
Thus the law of excluded middle A or NOT A holds in a
vacuous manner. On the other hand, if one of the two
statements A and NOT A is within the reach of (decidable, proveable)
within the existing story or play or theory then consistency requires
exactly one be implied, and the law of the excluded middle holds for A
provided the reach is consistent. Thus the law of excluded middle can be
assumed without loss of consistency.
Remark: The discussion of logic in mathematics is a
delicate manner. There is a risk here and in the following page of making
some naive mistakes.
Law of the Excluded Middle: A or Not A.
Let A be the statement that some situation occurs. Then a story or theory
that suggests a statement A is both true and false is inconsistent. So
for the sake of consistency in our present and further reason, we may
require and assume the statement
A AND Not A
to be false - NEVER TO OCCUR. So in our story or theory in its present
and further development, we require
A OR not A
to be true but not both at any instance (except during a brief transition
period).
So A requires not (not A) for consistency with A AND not A, and not (not
A) requires A at any instance (except during a brief transition period).
Remark: The discussion of transition time
suggests the law of excluded middle might be broken momentarily when
situations are time-dependent or place dependent. For example, in
counting people in a room that has a door, we cannot say a person is
all in or all out because of the middle possibility of a person being
part in and part out. So a person has three static states namely, in,
out and partly both, and two transition state namely, going from in to
out, and going from out to in. During these transitions, the middle
state of partly in and partly out occurs for a short or long period of
time.
Law of the Excluded Middle
essay Dec 1, 2008
(To do - keep, rewrite or cut, what is best here?)
Logic I of Partial Inclusion : Let B be a
region in a one occupant house. We say B is true when the
occupant is partially in B. Likewise we say Not B is true when
the occupant is partially in the rest of the house. Now for that
occupant, the statement
B or Not B
holds, but the two events B and Not B may occur
simultaneously. That is the assertion
B and Not B
may be true. They are not mutually exclusive.
Whence to say B holds is not to say Not B does not, and
vice-versa.
Logic II of Full Exclusion: Again, let B be
a region in a one occupant house. We say B is true when the
occupant is fully in B. Likewise we say Not B is true when the
occupant is fully in the rest of the house. Now for that
occupant, the statements B and Not B are mutually exclusive.
Thus
B and Not B never both hold.
However the statement
B or Not B
fails when the occupant is partially in
both.
A More Careful Logic III: Yet again, again,
let B be a region in a one occupant house. But this time, let A
be the statement that
the occupant is partially in B.
Then Not A would be the statement
the occupants is not partially in B - the
occupant is fully out of it.
Then statement A and the Not A are mutually
exclusive: The statement
A and NOT A
can never hold. Moreover, the statement
A or Not A
will be hold as well. So Not (Not A) implies A and
A implies Not (Not A). That be said, even though the latter
holds, we still may be a state of ignorance which one occurs and
when.
The Law of Excluded Middle. This law holds
when the statement when an assertion C and Not C are (a) mutually
exclusive and (b) at least one of the two statement C or Not C
occurs. The law of excluded middle fails for logic I and II, but
holds for logic III.
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Selby A, Volume 1A, Pattern Based Reason, 1996.
www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript D Reflections-on-Law-of-the-Excluded-Middle Previous: [Postscript C Consistency-as-a-Tool-for-Reason.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
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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?
Site Reviews
1996 - Magellan, the McKinley
Internet Directory:
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,
Number 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
to scale.
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
or space.
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
cross-products.
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
trigonometry.
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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