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YOU are better than YOU think. Show
yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful,
Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens
eyes. Leads to greater precision.
in reading and writing
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice is
approximately correct, for some circumstances, not all. Site How-TOs
are logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site
area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
For online automated help in senior
high school maths & calculus, visit quickmath.com
For Automatic Calculus and Algebra Help with derivatives, integrals,
graphs, linear equations, matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different
range of services, some free, some not, all based on webmathematica.
Good luck.
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Explore collaborative whiteboards from groupboard,
twiddla or
scriblink.
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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
-
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.
-
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, and
inconsistencies 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.
| 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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www.whyslopes.com
Mathematics Education Essays etc
[ Up ] [ Next ]
Area Intro
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
In for a Penny
Constructivism and Cognitive Theory
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course
Design & Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in
Logic -
Occurrence
& Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition
- Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes
& (?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for
dot & cross - cosine
law
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