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Algebre
Définition d'une variable
La
raison basée sur les règles et modelés
Appetizers and Lessons for Mathematics and
Reason
A T3 (Teach the Teacher) website
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Parents: the folder Helping
Your Child or Teen Learn covers Speaking Skills,
Reading & Writing,
Preparing for Science &
Having Patience. Parents and teachers need to say no for small things of little consequence to build and maintain
authority to say no for larger matters. Authority not exercised
atrophies
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Vol. 1, Elements of Reason,
introduces all site Volumes on understanding and explaining
mathematics and logic. Site
Volumes are onlinewith
postscripts. Paperback
versions (no postscripts) are available.
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Vol. 1A, Pattern
Based Reason, describes benefits,
origins and limits of rules & patterns in
daily life, science, business &
technology. Not all is certain.
Vol. 1B. Math
Curriculum Notes,
begins with inductive principles for progressive and observable
skill development in education. A sequel POMME is
coming. |
Vol 2, Three
Skills for Algebra, wordily gives starter lessons for logic and algebra
- stuff calculus and high school students
should know.
Vol.
3,
Why
Slopes & More Maths, offers starter lessons for
differential and integral calculus -stuff instructors should
know. Appendices
gives starter lessons for real analysis
- only for a few. |
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Teachers: Do you
need technical help in mathematics? Help provided
via WiZiQ (Sign Up and
Contact site author) - Free of Charge to the end of March. |
| Are you interested in
online instruction as a student or tutor? Free 45 minute
lesson (Sign
Up) of how to use online whiteboards with voice
and real-time writing, make online tutoring possible. Written work
scanned into a pdf files may be uploaded for
live correction. Real-time writing is not with a mouse, but
with a pen-tablet. For each operating system, real-time
writing depends on which pen-tablet employed. |
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Three Types of Site Lessons:
(i) Lessons trying to make mathematics and logic
clearer - see site reviews.. (ii) Lessons trying to explain
skills and concepts - readings for people who want to
understand; and (iii) Lessons exploring mathematics
education possibilities - readings for the curious few.
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Math
How-TOs etc
1. Arithmetic
2. Algebra
3. More
Algebra
4. Geometry
5. More
Geometry
6. Calculus
7.
provide densely written skill checklists.
More for Teachers: (a)
Outline of
a new K1-12 Mathematics Program, (b) LAMP an
earlier program, (c). Math Education
Essays
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More Site
Sections
1. Decimal
Arith. & Integer Webvideos (Flash)
2. Fractions
Fractions with Units, etc
3.. Solving
Linear Equations
4. Euclidean-Geometry
5. Analytic
Geometry - slopes, quadratics, polynomials, functions
6. Number
Theory.
7. Exponents,
Radicals & logs.
8 Calculus
9. Real
Analysis
10. Electric
Circuits Etc
11. Maps,
Plans, Similarity &Trig, (alternate views)
12. Complex
numbers - a visual approach
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Site
Reviews, Easy Reads for the
not too young, Etc.
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Magellan, the McKinley Internet Directory, 1996:
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. (Magellan is no longer online)
The
World-Wide Web Virtual Library Education , 2005. Why Slopes: Appetizers and
Lessons for Math and Reason. This online
classroom offers appetizers and lessons for math from
arithmetic to calculus or why slopes; for deductive reason
(logic) and critical thinking; and for learning in
general. Included here are opinions on the communication
of skills and mathematics instruction. The logic
appetizers are math free. Each appetizer is different. If
one is not to your liking try another. Most are from three
books on understanding and explaining math and
reason.
Easy Reads for
the not too young
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Logic
chapters 1 to 5 in Three
Skills for Algebra may help in senior high school mathematics
where rules and patterns are employed forwards and backwards,
and may also help in daily life where understanding and writing
agreements or contracts depends on understanding and writing rules and
agreements. Read Logic chapters 1 to 5
for self- defense.
In playing games, we need read it rules carefully. At work instruction
have to be carefully read. In learning, we need to read the
rules and patterns of an art or discipline carefully.
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Apart from mathematics, Volume 1A, Pattern
Based Reason, describes in general, the benefits, origins and
limitations of some rules and patterns in use everyday life and
in politics. There-in lies a request for thoughtful and considerate
leadership.
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The algebra
chapters 8 to 18 in Three
Skills for Algebra use words to introduce or rationalize the
algebraic shorthand role of letters and symbols. . Chapter 14
introduces the backward use of a formula in the arithmetic and
algebraic solutions to problems. The forward and backward use of rules
and patterns is everywhere in mathematics, chemistry, engineering,
physics and business. Volume
2 covers topics that students starting calculus should have mastered..
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Ease or avoid
calculus fears & difficulties.
These arithmetic
exercises with hints of algebra for students starting calculus
may test and strengthen skills, or warn of weaknesses.
This geometric
preview and chapters 2
to 6 in Volume 3, Why Slopes and More
Math, give a context for the senior high school
level study of slopes and of factored polynomials. The same material
may be employed at the start of calculus to make it easier.
Calculus asks students to calculate derivatives (slopes for straight
lines) and to do sign analysis, that is, to say identify
interval where derivatives or slopes are zero, positive or negative.
While calculus upto the calculation of derivatives is algebraically
challenging, the sign analysis and interpretation as introduced
in geometric preview
and chapters 2 to 6 is very
simple. Moreover, it helps develop algebraic skills in a way that
makes the calculation of derivatives and before that limits, much
easier. Chapters
11 and 13 provide a geometric introduction to the concepts of
second derivatives in the context of graphing slope versus position
and speed versus time. Chapter
12 talks about units for slopes and rates of change.
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Here is an alternative
presentation - one with still more advice and directions.
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Fraction
Operations (comparison, multiplication & division included).
developed by raising terms to transform into easy
cases.
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Format for formula
evaluation ( a small idea useful for evaluation of arithmetic
& algebraic expressions in general) shows how to do and
present work more clearly
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Solving Linear Equations
via fractional operations on sticks (line
segments) makes use of letters and stick-free solution
understandable.
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Easily Solved Systems of equations in essentially one unknown
gives a model for formulating word problems, as a way to
ease or avoid mental gymnastics with algebra.
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Integers
introduced
Geometrically give a model for the geometric introduction
of most operations on signed numbers. See
Arithmetic
How-TOs for more. The full treatment may come later.
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Early Geometry in the form of Plan
and Map usage provide a practical, self-contained, early math (trig-free)
approach to solving similarity problems by drawing diagrams to scale
and then measuring on the diagram. Here is a self-standing prequel
to trig.
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Wordy Introductions to
Logic may develop precision reading and writing needed in maths,
all further studies, home life and work for better performance or
self-protection - Romeo and Juliet make mathematical
induction easily to understand anad explain - Chains
of reason provide a model for reason in Euclidean
Geometry outside mathematics
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Three Skills for Algebra
(Talking about Numbers, Describing Calculations, Describing when
calculations are equal, what is a variable) may ease or avoid fears &
difficulties and clarify concepts that obvious to some, but not ALL.
The algebraic way of writing and reasoning needs to be
introduced with words - rationalized.
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Essay What
is a variable puts words before and beside symbols at
a level the calculus or precalculus student will
understand. Mathematicians: Arguments against past
verbal descriptions of variables do not apply here.
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The Forward and Backward use
of formulas (also rules and patterns) is a unifying theme for
senior high school and college mathematics and science. Beyond
that, note in the mathematics
of proportionality this theme appears with
backward use (finding the proportionality constant) put
first.
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Complex numbers & properties
introduced geometrically & rigorously before the development of
periodic trig functions will simplify simplify the high
school level 2D geometric development of trig
and vectors. The simple geometric proof here of the
distributive law is the key. The advantages of using complex
numbers in the exposition of trig was well-known in the 1940s or
earlier. Why complex numbers were not geometrically
developed before trig in the course designs of the 1950s or 60s
is a bit of mystery. Some inquiry or research may explain
why. Since 1976, this site author looked for a simple
proof, found or re-invented several, only to learn in February
2010 that giving a geometric proof was an exercise in Secondary
Mathematics, A Functional Approach for Teachers, H. F. Fehr,
D. C Heath and Company Boston 1951.
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How to quickly introduce multiplication,
addition, subtraction and long division of polynomials
geometrically. Justification is another matter.
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Function
Theory for Senior High School and Calculus Students -
Multiple Viewpoints explained and reconciled.
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Geometric and
Algebraic (Chapters
2 to 6 in Volume 3) Calculus Previews: These offer an end earlier
studies or a start for calculus in a manner that strengthens algebra
skills and eases or postpones calculus difficulties.
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Welcome. If you are a high school or college
level student, explore site pages to develop or check skills
you have just seen in class or explore site material if you would
like to become a tutor or a teacher, or to decide whether or not to
inform your school or college math department about this site.
For starters, if you have time, read
Logic Chapters 1 to 5 and Algebra
Chapters 8 to 14 in online Volume 2, Three
Skills for Algebra. Logic mastery will help skills
and performance at work, in study and
YES, in playing games. In each area, rules and instruction
understood exactly may give an advantage. Chapter 4 gives a Romeo and Juliet
view of mathematical induction. There is a unifying theme for
algebra in Chapters 14.
Most of your instructors have
not seen the skill development methods, fresh and recycled, in site
pages. Invite them to explore this site. Mention site
reviews. Say what you like or dislike.
Instructors: First,
this T3 (Teach the Teacher) offers skill development methods from arithmetic to the start of
calculus. With
regrets, the mathematics curricula put forth by English educational authorities in North America and
the UK, those I have seen, are not satisfactory. Technical gaps persist, are accepted or go unnoticed. See
below. Such gaps compound difficulties.
Moreover the
underlying premise in old course design (a premise inherited in presented
course designs) of preparing for calculus
and advance mathematics does not serve common needs. We could put first the common and likely needs of daily life in the streets
before preparation for advanced mathematics (calculus)
begins in earnest. In that, skill development should be observable and
verifiable. Third, teachers need to
observe and discuss each others methods - what works, what
does not, and when. Schools and colleges need the
foregoing discussion for improvement of and continuity in course
design and delivery. Calling
this a T3 (Teach the Teacher) site is not a claim to perfection.
Much can be improved. It is
a invitation to take site methods as a lower bound for
instruction, methods to exceed or equal.
Help Elsewhere for students - Do not put all
your eggs in one basket: Three
text-based sites
mathsisfun,
purplemath and themathpage
are well-done. The BBC also provides help (examples) in: mathematics
and many other subjects
for students. The Bright storm
Flash Video Site: (membership required) for secondary
mathematics US style and some calculus lessons with an emphasis
on the mechanics (the how, not the why), Brightstorm
flash videos are neat and usually well-done except for notational lapses - doing
calculations in place instead of doing one step per line, one step after
another.
Three Gaps - Why Mathematics is harder and less liked than
need-be.
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The olde Algebra Gap:
The shorthand
roles of letters and symbols are not fully explained or rationalized
from solving equations to the very challenging use of algebra in
advanced mathematics (calculus). Solving linear equations
starting with fractional
operations on stick diagrams gives an entry level, geometric
introduction to algebra with letters referring to visible lengths. Chapters
8 to 12 in Volume 2 and the essay What
is a Variable put more words into the explanation
and comprehension of algebra. Chapter
14 in the same Volume 2 with its detailed discussion of the direct and indirect
use a formulas identifies a unifying theme for algebra and logic - all
rules and patterns may and will be used forward and backwards in mathematics, science,
technology and logic or reason. The very challenging use of algebra in calculus
is made easier by (i) this why slopes, geometric
preview of calculus, by (ii) this factored polynomial, algebraic preview in Chapters
2 to 6 in Volume 3, and by (iii) the further discussion of
slopes, limits, derivatives and integration in Chapters
11 to 18 of Volume 3. Mathematical Fact:
Calculus requires earlier high school mathematics and logic at full strength:
(i)
This long complex
numbers lesson on shows how to simplify the development of
periodic trig functions,
the derivation of their properties, and the derivation of trig identities and formulas in the plane for vectors
dot and cross-products. For
further algebra skill development, see the site coverage of fraction
with units, proportionality,
polynomials, quadratics
functions
and straight
line slopes and equations. And for logic mastery, start
with the math-free chapters
1 to 5 in Volume 2 as early as possible for the sake
of precision or greater precision in reading, writing, reason.
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The New Arithmetic Gap: An exact and efficient mastery
of arithmetic with decimals and fractions is needed for proper, full
strength, high level study of mathematics alone and in science, technology and
business. In site material, webpages with html, and real
player and flash format webvidoes on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill this gap. The exact
and efficient command of arithmetic should be obtained in the last years of primary
school and the first years of secondary school, partly to serve these ends,
values & methods for work & study - learning
to avoid mistakes in multistep methods via the early
mastery of exact arithmetic; and partly to set the stage for an
exact and careful mastery of algebra. Good luck. Skill in
arithmetic to be credible has to be seen: observable &
verifiable.
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The Motivation Gap - Put Common Needs
First, Specialized Needs Second: The
motivation of secondary school students and their families for mathematics
studies would increase if primary and secondary school
explicitly put first the logic and quantitative skills and routines needed
or likely to be needed in the daily life of daily life before preparation
for calculus begins in earnest. These ends,
values & methods for work & study in
mathematics should give student skills to hold and value even they do not
go further due to lack of interest or opportunity. To
learn about the motivation for high school mathematics, see below
- you may not like what you read, but those are the circumstances
seen.
Help Elsewhere For Teachers: the BBC gives many
many resources for preschool to senior
high school subjects taught in the Wales, Scotland, England &
Northern Ireland. For mathematics students and
teachers, the Cambridge based website nrich.maths.org
offers dozens or hundreds of rich, not simple, tasks and in
them, for mathematics learning & teaching. Technical
Gem: The New
Zealand Math Curriculum Page focuses on content matters,
not delivery style!
Teachers:
The 21st year book
of the National Council of Teachers of Mathematics, Washington D. C.
1953, ends with the following on pages 348-9
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page 348. a teacher is a learning
engineer, a builder of minds that will solve problems. As
such, the must first know the total mathematics he will
teach, that is, the framework.
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page 348. The lack of correct
concepts in arithmetic may be one of the great reasons for the
difficulty algebra presents to so many of our students.
(Opinion: adds the algebra gaps above as a
further reasons. )
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page 349. .. in a sense the
teacher must be a master technician. He must know how to build
any known kind of learning. .. must weigh, balance, and
appraise the possible learning. ... know their
relative worth both for the individual and for society.
(Opinion: put the relative worth for the
individual first. That would serve best the needs of society.)
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page 248 There are some persons
who say one who knows cannot teach for he cannot fathom the
difficulties of his students. These persons say that as a
teacher work with his students through a problematic situation
which is new to both teacher and student, real learning takes
place and then only. We believe this assumption to
be entirely erroneous and assert that a teacher is a learning
engineer ...
Entrance Level Pages: [ Site Entrance ] [ Site Entrance Continued ] [ Permissions ] [ Privacy Policy ] [ Proof of Concept ] [ Live Math Help ] [ Dig Deep ] [ Vol 1, Elements of Reason ] [ Applied Maths Program K1-12 ] [ Site Reviews ] [ Site Map ] [ Paperback Copies ] [ Technical Site Map ] [ Math Ed. References ] [ Ends & Values ] [ How to Improve Marks ] [ Grades 7 to 12 Math Guide ] [ Raising Standards ] [ Sec I, Teaching Ideas ] [ Sec II , Teaching Ideas ] [ Sec III, Possibilities ] [ Preparation for Calculus ] [ Cafe Logos ] [ Cafe Math Tips ] [ 1. Arithmetic How-TOs ] [ 2. Algebra How-TOs ] [ 3. More Algebra How-TOs ] [ 4. Beginner Geometry ] [ 5. More Geometry ] [ 6. Calculus How-TOs ] [ 7. Logic How-TOs ] [ 8. Complex Numbers ]
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What would you give to
be an Algebra
Power User?
Would you give time and effort to read
(i)
the site folder on Solving
Linear Equations; (ii) algebra chapters 8 to 18 and the
essay what
is a variable. in the book, Three
Skills for Algebra. That is a good start.
Look for what is different is in the site intro to algebra.
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