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 2 Three Skills For Algebra >> Chapter 31 Direct-and-Indirect-Reason Next: [Appendix A. Reading Guide For Next Appendices.] Previous: [Chapter 30 Truth Tables.] [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] [31] [32] [33] [34] [35][36] [37] [38] [39] [40] [41] [42]
Chapter 24, Direct and
Indirect Reason
To prove a statement is to show that it must hold. To refute a statement
is to show that it cannot hold. In between proof and refutation1 lies
uncertainty or We don`t know.
PS. The concept of proof comes perversely from the idea of
testing. The first question can we test if a statement is false? If a
consequence of a statement or assertion fails to be true, then the
statements or assertion is false or refuted.
1 Proof and Refutation
Methods
Proof and Refutation was the title of a work by
the philosopher Latakos, Cambridge University Press, 1976. More recent
editions are available.
In a given situation:
- what evidence or proof do we need to say a rule if A then B is
never disobeyed?
- what evidence or proof do we need to say the rule is false?
False here means sometimes disobeyed. The proof or disproof techniques
described next are usually employed in a context where some implication
rules are assumed to be never disobeyed. The first question (1) then
becomes what further implications rules are never disobeyed. The
techniques described next provide answers in some but not all
circumstances.
Refutation: Proving Falseness
To show that a rule if A then B is false is simple, see if we can
find (or show there exists or must be) a situation in which A
occurs and B does not. Then the implication rule A implies
B is false. In other words, it is not always obeyed or it is
sometimes disobeyed.
A Direct Approach: Applying Implication Rules
PS: A proof in mathematics consists of a chain of reason, direct or
indirect, but carefully done, which implies that a statement or
assertion is a consequence of an assumption or set of assumptions. Then
if the assumptions and chains of reason are not false, the statement or
assertion must hold (we hope).
You may show that the rule If A then B is always true by showing
that when situation A occurs so must B. Such a proof could
employ a chain of reason (deduction) using trusted implication rules:
rules that are not disobeyed in the situation at hand. Such a proof shows
that when A occurs, so must B. This says and shows that the
situation A and NOT B never occurs. See the chapter Chains of
Reason for an illustration of this approach.
A Somewhat Indirect Approach: Proving The Contrapositive Method )
Alternatively we can show the (contrapositive) rule If NOT B then NOT
A is always true by showing that when situation NOT B occurs
then so must NOT A. Such a proof again shows the situation A
and not B never occurs. That is what we need. (See the chapter
Chapter 22, The Contrapositive, or see Chapter 4, Implication
Rules.
A Fully Indirect Approach: The Contradiction Method
The aim here is to find a situation C with the following
properties:
- the situation C does not occur (is obviously false) and
-
(A and NOT B) implies situation C.
These properties tell us that the situation (A and NOT B) cannot
occur - why? So that the implication rule A implies B is never
disobeyed. Next are a few words to explain why:
The contrapositive form of the implication rule if (A and NOT B) then
C is what we need. It says if NOT C then NOT (A and NOT B).
But the situation NOT C, by chance or discovery, occurs. Thus the
situation (A and not B) can never occur. And that is what we mean
when we say that the rule A implies B always holds.
Remark (a-looking we will go). This proof by
contradiction method can be applied without knowing in advance what
the situation C will be. We search for it. That is, for the sake
of finding such a situation C, we assume the situation A and
not B occurs. After this we follow whatever chains of reason we can
to reach a conclusion that an absurd (or obviously false) situation
C occurs.
Selby A, Volume 1A, Pattern Based Reason, 1996.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 31 Direct-and-Indirect-Reason Next: [Appendix A. Reading Guide For Next Appendices.] Previous: [Chapter 30 Truth Tables.] [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] [31] [32] [33] [34] [35][36] [37] [38] [39] [40] [41] [42]
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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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