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Page Sections: [Quotes and Site Books] [Key
Appetizers and Lessons for Students/Teachers] [Mathematics Education
Revamped/Revisited] [Short Descriptions of Site
Books and Areas][Page Top]
Ideas for making the hard easier may be used in current
courses. They may also be used for reformulate course design and
delivery. The late Richard Feynmann in public lectures for a
general audience at McGill University in 1976 briefly implied his
subject, physics, was based on the addition and multiplication of
arrows (vectors) in the plane. The same can be said of.
Mathematics .Mathematics for general audiences and for secondary
students also can be based that addition and multiplication of arrows
in plane in ways that accelerate comprehension.
Three fall 1983 lessons
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two logic
puzzles - an attempt to point out the existence of logic in
mathematics and also to develop greater precision in reading and
writing, a must for work & study.
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three skills for
algebra - words before & besides symbols.
-
why
slopes, a geometric calculus preview
with inductive
principles for instruction should be sufficient for immediate
improvements in mathematics course design and delivery in secondary
mathematics and calculus. Inductive principles
demand all skills and concept be developed clearly, directly and
systematically.
Ends, Means and Values: Mathematics is an art and
discipline in which rules and patterns have to be met and carefully
used one at a time and one another, alone or in combination, to arrive
at good results. Drill, practice and correction are all required to
show and imply the importance of applying and combining steps and
methods carefully, in repeatable, reproducible and hence
verifiable ways. From arithmetic onwards, awareness that an error in
one step makes the rest wrong is a sign of intelligence appreciated and
present in all arts, trades and disciplines, an awareness very
much needed in their mastery. The ability and will to apply rules and
patterns, steps and methods, or customs and convention carefully
provide a value, an end and means for learning and teaching in
mathematics and in most arts, trades and disciplines.
The inductive
principles work best when there is motivation or clearly defined ends
and values for course design and digestion. Where teachers and students
say mastery of high school level mathematics is a natural talent,
there has been a failure in course design and delivery. When a problem is
recognized, remedies can be sought and investigated. We need effective
lessons and effective lessons plans, easily followed and repeated by
teachers, with technical and applied themes to guide and motivate skill
and concept development with verification, step by step.
Still More For Instructors and Tutors: At the
present time, site ideas and methods provide all the pieces of a
mathematics education jigsaw puzzle. The pieces are almost all
here in site areas lying about, waiting to be put-together. Hints
or directions for that appear in the current page section: Mathematics Education Revamped
and Revisited. Readers with less than an expert knowledge will see
that pieces are useful in many circumstances while experts in
mathematics will find in this page section and site pages a sufficient
number of hints and directions to put the pieces together to redesign
mathematics education from mastery of whole numbers and fractions to
calculus. For readers with less than an expert knowledge, the assembly
or putting-together of the pieces will come later - most likely before
fall 2008, time permitting. The long-term objective in site
development if not its terminal objective is the exposition of a
full inductive approach to mathematics education. Most unexpectedly,
the approach does not support and remedy shortcomings in the
modern mathematics curricula of the mid-1950s onward - the original
intent that persisted to say 2005. Instead the approach stems
from the physical and geometrically assumptions or empirically
practices needed to employ numbers as coordinates in 1, 2 and 3
dimensions. Whence the axioms for real assumed in the modern
mathematics curricula, and field properties of complex numbers too, are
seen as geometric necessities - there-in lies delights for
advocates of the use of manipulative in skill and concept development
from use of whole numbers and fractions to calculus. The proof is in
the details - pieces mostly online.
In writing Volume 1B, Math Curriculum
Notes, my aim was to make modern (pure) mathematics curricula
more accessible. Writing led to an identification of inconsistencies
in need of resolution, namely the necessity in the secondary
school level development or exposition of modern mathematics of
departing from pure context-free mathematics in the geometric
(physical?) introduction and application of trigonometry and
calculus. Those inconsistencies, andinconsistencies with common
needs, were overlooked in the 1950's sputnik-born rush (stampede) to
improve the curriculum. However on slow, very slow, further
reflection, there exists a mixed-mathematics scheme for
development of quantitative skills and concepts in which an operational
viewpoint systematically exploits geometric and physical assumptions
inherent in its operations to develop and derive the properties of real
and complex numbers, and whence to hasten & make easier the
development and application of trigonometry and complex numbers in
well-known ways.
College Mathematics Professors: The plan for site
area Maps, Plans &
Drawings indicates a key part of the propose mixed curricula
in which the target is an operational (hand-waving, even manipulative
based) command of arithmetic, logic, algebra, geometry, trig, complex
numbers and calculus sufficient for the needs of TCPITS,
sufficient for the needs of arts, trades and professions outside of
mathematics, and sufficient to set the stage and to provide a
context for the optional, further study of modern mathematics. An
Euclidean geometry, style assumption that the addition and
multiplication of points in the plane can be defined geometrically
before and independently of any coordinates systems implies through use
of coordinate system, definitions and field properties for the addition
and multiplication of real numbers and also complex numbers. Simple add
the assumption that signed decimal expansion - finite, periodic and
non-periodic - can be used as coordinates to provide a pre-modern base
for instruction of mathematics to the level of advanced calculus. With
that include set notions where those notion ease skill and concept
development as in the discussion of functions and as in the function
& set description of combinatorics & probability
theory.
The scheme is online in the form of a mathematics education jigsaw puzzle
with its pieces indicated and present in site webpages on Maps, Plans &
Drawings (description in full), on Complex
Numbers (some details), and Number
Theory (more details) in a form accessible to applied mathematicians,
electrical engineers and physics students/teachers.
Potential Headlines: Physics
Magazines: Galilean Relativity implies definition and properties of
addition and multiplication of real and complex numbers. Consequences
of special relativity being investigated. Mathematics
Journals: Sequel to Modern Mathematics Curricula Uses Extrinsic
Viewpoint of Euclidean Plane to derive properties of signed numbers and
complex numbers from properties of vectors and coordinate
systems. Education Magazines: Consistent, Handwaving
Mathematics Curricula Adopted. Retreat in rigour makes
secondary school and college mathematics simpler for teachers to
understand and explain in a repeatable and reproducible fashion.
PostMortem Comments: (i) Newton, I prefer Geometry to Symbols in
proofs. (ii) Poincare, I did not see the need for set theory.
(iii) Hilbert - we will have to compare this to my geometric theory of
numbers.
The late physicist Richard Feynman in a brief 20 minutes or so of three
evening, guest lectures to a general audience at McGill University in
1976 (and most likely in lectures elsewhere) entertainingly described his
subject as the addition and multiplication of arrows in the plane.
That brief account sowed the seed for the mixed-mathematics scheme
described indicated above. The scheme is the shortest of several attempts
in site pages to develop complex numbers and their properties using
geometry with and without coordinates.
Technical Hints: counting principles
provide the properties of non-negative numbers, as in the site
area on (1) Number
Theory, while a lean treatment of (2) Euclidean Geometry sufficient
to imply parallelogram law shows head to tail arrow addition is
commutative, and the operational assumption that arrows (vector)
addition in the plane is associative and independent of choice of
coordinate systems in which represented (or done) leads to
a geometric representation and properties of (3) complex and real numbers - and with that,
imply yet another proof of the Pythagorean. (4) Easy consequences then hasten the
development of unit circle trigonometry, and trigonometric
expressions for dot- and cross-products in the plane. If the
derivation in (3) is not clear enough, see (1). The development
of real numbers and their properties in (1) Number Theory, includes besides zero, both
positive & negative numbers. But there is no need to define
addition and multiplication with negative numbers before
introducing the addition and multiplication of arrows and points in the
plane.
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More for Teachers and Tutors: Site content
stems from a long dissatisfaction with the secondary and college
level introduction of the algebraic shorthand role of letters and
symbols, and a more recent dissatisfaction with the incomplete
development of arithmetic skills with whole numbers and
fractions. In the introduction of algebra, words
have been missing from the first use of arithmetic and formulas in
primary school to the full-strength use of algebraic ways of writing
and reasoning in senior high school and college mathematics courses
on calculus. Chapters in Volume 2 and 3, and lessons on solving
linear equations, provide the missing words, and simultaneously
add a geometric and even numerical viewpoints to ease or avoid
difficulties in learning and teaching algebra. The site area on
fractions illustrate and develop arithmetic in the context of
fractional operations on line segments. Site lessons
solving linear equations also begin fractional operations on line
segments (sticks) visually, geometrically & simultaneously
develop, introduce, re-enforce and connect algebra and fraction
skills. The lesson continue with solving systems of equations in
essentially one unknown (an exercise which leads to an operational if
not explicit command of associative and distributive laws) and
triangular systems of equations before introducing general
systems. Three skills for algebra, and fourth namely the
backward use of formulas, numerically and algebraic
(literal) introduce words and themes to employ and emphasize in
secondary and remedial college mathematics instruction. Whence site
ideas, values and methods for instruction a greater vulgarization of
mathematics can be employed to support existing course designs from
re-enforcement of whole number and fraction skills to the
introduction of calculus.
The aim in writing volumes 2, 3 and 1B of
understanding and compensating for shortcoming in the modern math
curricula dating from the mid-1950's, a curricula that lingers
today in course design in a diluted, ritualistic manner, has
fallen aside. Instead, as of say fall 2007, this site
proposes and even details an alternative geometry- and
manipulative-based curriculum for numbers and algebra etc in
which the properties of both real and complex numbers are developed
from the very assumptions needed for the use of signed coordinates
in maps and plans to describe location and vectorial movements or
displacements, independent of the choice of unit length and unit
vector or vectors. The net result is an applied mathematics
curricula which supports education in general while providing
a solid algebraic, deductive base for further pure and impure
quantitative studies. The site objective (implicit and not fully
explicit in site material, fall 2007 onward) is to describe
mathematics education and its nuances - how it is possible and how
it may proceed from the introduction of writing , the similar shape
recognition and drawing of characters and decimal digits, at home
or the first years of schooling to the introduction of advanced
calculus, with less confusion and greater clarity. Details or
clues are mostly online.
A new dimensions in learning and teaching
algebra: In 1966 as a student, I met the quadratic
formula. As a matter of principle, I was not going to use the
formula until I understood its explanation. So I spent
three evenings trying to understand its justification..
Finally, I did. But I did not have the words to fully
introduce and explain that understanding to others.
Thereafter , in every mathematics textbook I met as a student and
later as a teacher , I looked for but did not see a clear
introduction to algebra, or the shorthand way of writing and
reasoning with letters and symbols. Finally in fall 1983, I
invented a review and starter lesson "Three Skills for
Algebra" That lesson lead to chapters 8 to 14 in Volume 2 ,
Three Skills for Algebra. In reading about the skills, you will see
that words have been missing in understanding, explaining and
applying algebra. That may be since arithmetic and algebraic
expressions, formulas included, are often too hard or awkward
to read aloud, precisely, term by term, symbol by symbol. So
arithmetic and algebraic expressions and formulas are better read,
written and even understood in silence as non-verbal code.
That silence, the non-verbal aspect, a missing dimension in the
comprehension of algebra and beyond, has been a source of
confusion or mystery. . Volume 2 breaks the silence. Yet
Volume 2 is misnamed as in Chapter 14, there is a fourth skill for
algebra which can be described in words, as the forward
and backward use of equations and formulas. The backward use
has two forms: arithmetical (or numerical) and algebraic. Chapter
15 describes arithmetic and numerical solution of linear equations
- read it after the larger site area Solving Linear Equations,
first with and then without stick diagrams. Good Luck
How to Avoid or lessen algebra shock in
calculus: In fall 1983, I also invented a lesson "Why Slopes"
to show students how their knowledge of slopes was one key to
calculus and to extend their algebraic thinking skills slowly and
thus avoid algebra shock in calculus. Calculus is the subject of
slope related calculations, their reversal and interpretation. It
is the reason for skill and concept development and perfection in
arithmetic, algebra, geometry and trig in high school mathematics
before algebra. To learn more, see the geometric and
algebraic previews of calculus in chapters 1 to 18 in Volume 3,
Why Slopes and More Math. Skip chapters 7 to 10 on the role of
units in calculations. Good luck.
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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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