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
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Caution: some programs are rewarding. Others lead
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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 C Consistency-as-a-Tool-for-Reason Next: [Postscript D Reflections-on-Law-of-the-Excluded-Middle.] Previous: [Postscript B More-on-Story-Telling-and-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 3, Consistency
as a Tool for Reason
While we may not know that a theory or set of assumptions is consistent,
or free of contradiction, we may use the requirement for consistency as
part of the reasoning process without loss of generality or harm we hope.
That is a gamble. Logician may have more say.
Example One: A detective in solving a crime may have a suspect.
Then he may found the suspect has an alibi which directly or indirectly
implies she did not committed the crime. So the alibi and suspicion are
inconsistent - that is incompatible. The detective may drop the
suspicion or challenge the alibi. Lawyers for the prosecution and
defense may erect competing chains of reason, and leave it to a jury or
judge to decide which one, if any, appears to be true. A conclusion may
follow or not.
Example Two: In developing a story or theory, we may require its
elements to be consistent. We writing or developing a story, we note
that a situation A is inconsistent with what has so far been written or
determined, we will not add situation A to the plot or theory. We will
instead accept the situation did not occur and may employ that in the
further composition of the plot, story or theory. Likewise, if that the
non-occurrence of a situation A is inconsistent with what has so far
been written or determined, we will be forced to add the occurrence of
situation A to the plot or theory. Finally, if the occurrence of A or
its non-occurrence has is independent of the plot, story or theory, we
extend the story or theory to include A or not as we like, if we are
writing fiction or describing what may be possible. On the other hand,
if we are trying to write non-fiction then the addition of statements
of patterns A to plot depends on their correspondence with reality.
In telling a story or developing a theory, we may look at the
consequences of our assumptions - the situations we tend to assume as
holding or being true. If a chain of reason implies that a situation C
occurs and does not occur, then the story or theory is inconsistent -
becomes absurd. For the sake of consistency, the story or theory needs to
be revised or abandoned.
That being said in developing a theory of how matters work, we hope that
the theory will be logically consistent. That we hope that there will be
no contradictions as a consequence of our assumptions. The consistency of
a theory is hard to prove or test, and impossible in a mathematical
theory large enough to include counting with whole numbers 1, 2, 3, 4,
...
In telling a story or developing a theory, there is an inconsistency if a
situation A and its negation Not A both occur. While we are developing a
theory from assumptions, we cannot be certain that an inconsistency A
and Not A will not occur. However, the assuming the law of excluded
middle in the development of a theory or is equivalent to the statement
that the situation A and Not A does not happen. It equivalent to
the assumption that the theory under development is consistent. That be
said, when we are developing a theory from logic and underlying
assumptions, we may not be able to prove that the theory is consistent.
If the theory is consistent, the law of excluded middle holds and so we
may used in our logical development of theory without loss of
consistency. But if the theory under development is inconsistent,
assuming or using the law of excluded middle in its development may lead
to the discovery of the inconsistency, sooner rather than later, if at
all. Assuming the law of excluded middle, simply adds another
inconsistency. .
Example Three: Assume any infinite decimal expansion locates a
point or distance on a real number line. Assume further that each ratio
of two whole numbers can be expressed a ratio of two whole numbers with
no common divisors? The Pythagorean theorem then suggests in an isosceles
right triangle, the ratio of the hypotenuse to each of the others sides,
the legs, by length given by the square root of 2. Is that square root
equal to a rational number? The suspicion or assumption that YES, the
square root of 2 equals a rational number implies an inconsistency.
Namely, that in any ratio or fraction that represents the square root of
2, the denominator and numerator will both be multiples of 2. So the
square root of two cannot be rational.
The Pythagoreans in finding the inconsistency in example 2 had a problem.
They assume lined segments in the plane represented numbers and they
assumed all such lengths were rational multiples of each other. When
these assumptions or their consequences clashed, reconciliation was not
obvious. Their view of numbers collapsed and a replacement was not
available. That was a serious problem for the Pythagorean school in their
theory of knowledge was based on and assumed rational numbers and only
rational numbers.
Today, however, we have an advantage or two. One advantage is a our
assumption that infinite decimals expansions represent rational and
irrational numbers. Physically, if you imagine ruler with a unit length
and its division into tenths, hundredths, thousandths and so on, then you
can count the maximum number of units, tenths, hundredths, thousandths
and that may fit in a line segment. The result is a sequence of numbers
which provide a better and better approximation to the line segment
length. You can further imagine that sequence of approximations given by
two, three, four, five and more decimal places in a decimal expansion
locate the end of a line segment. The decimal representation of numbers
does not depend on nor require the "numbers" to be rational.
Aside: The number 1.0000 represent a single unit
of length. It also represents the limit of the sequence 0.9, 0.99,
0.999, 0.9999 and so on. The sequence of decimal approximations 0.99999
(9 recurring) is shorthand for a sequence of lengths L(j) =1 -
10-j < 1 where the lengths are increasing, and where the
differences d(j) = 1 - L(j) = 10-j is getting smaller and
smaller as j increases. So the sequence approaches 1. Because the
difference tends to zero, the number 1 has several decimal
expansions
-
1, 1.0, 1.00 (0 recurring finitely many times)
-
1.000 (0 repeating indefinitely)
-
0.99999 (9 recurring)
where the last one represents 1 as the limit of a
sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ...
.
Selby A, Volume 1A, Pattern Based Reason, 1996.
www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript C Consistency-as-a-Tool-for-Reason Next: [Postscript D Reflections-on-Law-of-the-Excluded-Middle.] Previous: [Postscript B More-on-Story-Telling-and-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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