Pure Mathematics
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Algebraic Structures: Groups, Relations, Fields
Food for thought: The algebraic structures of modern mathematics
should only be introduced after the algebraic way of writing and
reasoning has been introduced step by step. The algebraic description
of (i) properties of real numbers, of (ii) how to efficiently do
arithmetic with fractions; of (iii) the properties of logs and
exponentials, of (iv) how to calculate radicals, roots and powers
with the latter (including equivalent ways to describe constant rate
growth and decay) may all provide algebraic structure of the tangible
kind. Item (iv) includes a systematic development of powers of numbers,
positive and negative, with fractional exponents.
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Logical Structure: The role of axioms and logics in pure
mathematics - what is wanted, what is presently possible. What further
practices are present in the employment of mathematics daily life,
science and engineering.
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An alternate base for common mathematics from counting to
calculus.
The Problem: In my high school days, axioms for
real numbers were given - hung in midair without further explanation -
to provide a base for arithmetic with signed numbers and algebra in
general and to provide the hope that all would be very carefully
(rigorously) explained and derived. Later in an advanced college
mathematics courses (taken only by a few), those axioms were derived
from assumptions about sets - the latter may be viewed counting
principles in disguise. But in the modern mathematic education
programs, rigour was not possible and incomplete. The common knowledge
of decimal arithmetic, employed when needed, was not sanctioned. The
drawing triangles and unit circles to define six trig functions and
develop their properties represent a departure from pure mathematics -
the latter does not sanction the use of drawings. But in advanced math
courses, taken by a few, a pure math development of trig functions was
given. And in retrospect, the employment of decimals in calculations
but their avoidance in theory complicated the theoretical description
of limits, continuity and convergence in calculus - the mathematics
subject key to practice and theory in accounting, engineering, science,
technology. The modern mathematics programs for secondary maths and
calculus in my school days pointed to a unified and coherent
development but did not deliver it. That was a disappointment for a
student who wanted to take the development of mathematics
literally.
The following remedy is for keen students who would like a
thought-based development of the arithmetic properties above. The
remedy provides a development at a level of difficulty and complexity
less than the full development in pure mathematics from say set
theory.
A Remedy: Knowledge of numbers begins with counting using
single digits and then place value in decimals (compound symbols).
Following that addition, subtraction, multiplication and even division
operations on whole numbers (counting numbers) represent counting
methods. Given a finite set of elements to count one at a time, one
after another, we expect the count or number to be independent of the
order of counting. If two different counts are different, recounts
follow to provide a correction. This counting may be done by grouping
the objects to be count into subsets and adding subcounts. So addition
becomes a tool for counting objects. The grouping of elements into
subsets, each with the same number of elements, leads to multiplies of
that number and counting through multiplication. Here again any two
different methods for counting the same set of objects are expect to
give the same result. That principle and the linear ordering of
elements implies addition is commutative: a + b = b +a for counting
numbers a and b. That principle also implies the associative law for
addition of counts and more generally, the sums of whole numbers as
counts may be added by subtotaling the addends or counts in an
arbitrary manner. Now the rectangular ordering of elements for
counting implies multiplication of whole numbers commute: a \times b =
b \times a. The associative law for multiplication a(bc) = (ab) c also
follows from the equality of two different ways to count a
rectangular groups of b \times c elements. More generally, the
calculation of products as the product of subproducts may be justified
by mathematical induction. Counting principles also imply the sum and
product of nonzero whole numbers are nonzero. All being said, the
foregoing properties of counting and arithmetic with whole numbers
imply the corresponding or like properties for unsigned fractions and
in the decimal limit, for all unsigned finite and infinite decimals.
Following that, the arithmetic properties of real and complex numbers
mentioned above are easily developed - inherited or implied - with the
aid of mathematical induction and in the case of complex numbers, some
geometric assumptions about the use of coordinates, distance
calculation in the plane and the interaction between rotation of
midpoints of line segments endpoints - here complex number addends.
This remedy includes the thought-based derivation of properties of
numbers - whole to complex - from counting principles, mathematical
induction and geometric assumptions. That derivation is sufficient for
students of mathematics, outside of advanced college mathematics
courses on the algebraic, analysis and/or the set theory development of
mathematics. The latter courses may support and refine the remedy by
replace geometry and the physical viewpoint of rectangular arrays by
methods of pure mathematics - mathematical induction and analysis in a
variation I saw in my college days as a student of pure and applied
mathematics.
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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
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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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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