Logic Island - Math-Free View of Euclidean Reason
The online chapters
offer a wordy and mostly mathematics-free introduction to logic. Math-free Logic mastery is one way to may ease or avoid difficulties at
home, at work and in studies as it gives greater precision and hence skill in reading, writing and talking. Chapter 2
covers two logic puzzles to show the difference between saying B IF A and saying B IF and ONLY IF A. Not seeing the difference between
condition and biconditional implication rules is
a source of confusion in learning and working. The chapter here calls them one- and two- implication rules after the concept of
one- and two-way streets in town. Seeing the difference will help people read and write with precision, and recognize
when the reason of others is unclear or incomplete. The question of when a one-way implication does not apply is covered.
Chapter 2 by itself may be studied alone. Greater precision and skill
in paying attention to details means less confusion. Alternatively,
greater precision and skill in understanding and explaining may help
people recognize when are missing or inconsistent.
Through examples, Chapter 3 shows how to employ implication rules IF A THEN B, one at a time, and one after another in sequences or chains of implication
based reason. The question of when an implication rule is isolated, cannot be used in a chain of reason, is illustrated in one example
where several implication rules hooked or connected together. This chapters provides a math-free hint of the kind of logic students
will being meeting in Euclidean Geometry where the latter is limited to the direct use of implication rules.
Chapter 5 adds further to the image of logic in mathematics and other fields by talking about islands and bodies
of implication based knowledge connected by one way chains of reason. Instructors may talk about connections using two-way
implications as well.
Chapter 4 introduces mathematical induction via a math-free, romantic, ladder climbing story, and questions about when
is it possible to reach the top, or when is it possible to start climbing. The story is a metaphor for mathematical induction,
one that may read now or in the study of mathematical induction later in this phase.
The question of when
to introduce site logic chapters or equivalent material and to what extent
is open. Early as possible is one answer, perhaps the best. Early
math-free logic mastery also as model for the use implications in making
arguments in and outside of mathematics. The direct use of logic or
implications in development of Euclidean Geometry in this phase provides
a neutral model for reason in general - that is a model apart from human
conflicts, except those appearing in mathematics course design
committees.
Arithmetic Island: Non-Terminating Decimals
With decimals, students may learn to represent and approximate real
numbers or their coordinates along a real number line. Through decimal
calculation by hand or with calculators, students may learn to add,
multiply, subtract and divide and find square roots of real numbers
exactly and approximately. Earlier instruction may have shown how to add,
compare, subtract, multiply and divide rational numbers with finite
decimal expansion. In that long division may imply quotient given by
non-terminating, periodic decimal expansions. That provides one context
for the discussion of error control and significant digits.
Students may be shown how
Pythagoras theorem implies the existence of lengths that are
irrational multiples of a given unit length. The multiples and the the
number $\pi$ have non-terminating decimal explansions. That raise the
question of how to do calculation with irrational numbers. The answer in
practice is approximately with the notion that more decimal used and
carried in calculations, here calculators are useful, gives more precise
results. In the limit, as more and more decimals are carried through the
calculations, more and more decimals in the result can be found - that is
the assumption.
Now or later, in this phase or the next, the foregoing provides a
context for a the discussion of limited and unlimited error control in
measurement and computation. Limited means decimal approximations is
possible only to finitely many decimal places. That is the case with
calculators. Unlimited error discussion in full would require
consideration of the same inequalities and analysis appearing the
dicussion of limits and continuity of sequences and function in
enriched courses on calculus.
The introduction of non-terminating decimals here is descriptive. It
includes some common assumptions about "convergence". A consistent set of
rules and patterns is given. Anything more would be too complicated for
this level of skill development. Questions about whether
the story of mathematics is fiction or not are not raised here.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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