Mathematics Education Readings and References
Coffee Table Books
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Mathematics From the Birth of Numbers, by
Gullberg Norton Company, New York & London, 1997, ISBN
0-39304002-X, QA21.G78 1996, 1002 +xxiii pages, well-illustrated.
Very readable by masters of differential and integral calculus. A
copy of it should be in every school where calculus or preparation
for calculus is taught. If not, strongly suggest that one should be
ordered.
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The VNR Concise Encyclopedia of Mathematics by W.
Gellert, H. Küstner, M. Hellwich & H. Kästner, Van Nostrand
Reinhold Company, 1975 (or 1977). 450 West 33rd Street, New York,
N.Y. 10001 (circa 1977) 750+ pages. ISBN: 0-442-22646-2 (hard cover)
and ISBN:0-442-22647-0 (paperback).Applications of mathematics in
money computations, geometry, navigation, surveying and so on, are
found in this encyclopedia – one reference for subjects for further
inquiry. This is a beautiful work. It has many colored pages and many
diagrams. This work gives a broad overview of mathematical ideas from
advanced high school to specialized studies in college or university.
It contains many worked examples. Every high school math and science
teacher should own or have access to a copy of this encyclopedia. So
should every gifted student taking mathematics at the high school
level and above. A copy of it should be in every college and
community library. If not, strongly suggest that one should be
ordered.
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Mathematical Thought from Ancient to Modern
Times, by Morris Kline, as three volumes (1990,
published by Oxford University Press).It was first published as one
book in 1972 by the same press. This work gives an overview of the
discipline, the strands of reason and geometric thought that entered
into it in rigorous and not so rigorous fashion. This work describes
the changing nature of mathematics. Mathematics apart from geometry
was not a deductive exercise. In particular, the symbolic reasoning
of algebra, also called analysis from 1700 to 1900 was a tool with
useful results – faith in it would follow usage. There was
no rigorous and no precise thought-based foundation. The material
underlying algebraic or symbolic analysis treatment of calculation,
that is the concept of number (whole, fractional, negative,
imaginary, complex) was only clarified gradually. This work describes
mathematical knowledge before its deductive codification, that is,
its derivation in an axiomatic framework for sets and arithmetic.
This reference is more technical than the rest, and may need to be
sampled rather than read from end to end in the first instance. Its
eventual comprehension could be the target of a college student
specializing in mathematics.
Secondary Mathematic Education - technical base etc.
There is a difference between discussion of delivery style and content
matters. Delivery styles come and go quickly. Content matters change, but
does so more slowly. The 1960s, 1950s and even the 1940s set the stage for
the technical discussion and design of course content. That content lingers
on today in pre-university calculus oriented courses. In some of the texts
below we see discussion of the topics prior to the settling of conventions
regarding the extent, if any, of their inclusion in the curriculum.
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What is Mathematics, R. Courant & H. Robbins,
Oxford University Press, Fourth Edition.Classic Work. This may be
taken a prequel to the discussion in the 1950s of what should be
taught in pre-university mathematics. Very readable for undergraduate
students in mathematics.The geometric interpretation (or
representation) of complex numbers assumes the addition theorems
(angle sum formulas) for sine and cosines in order to show how to
multiply complex numbers using moduli and angles. Compare and
contrast that with the site development of complex
numbers.
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Secondary School Mathematics, J. J. Kinsella,
published by The Center for Applied Research in Education,
Inc., New York, 1965It describes mathematics instruction from
the early 1900s to the 1960s in North America. Many of its comments
are still valid.
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The Learning and Teaching of Mathematics, Its Theory and
Practice, The 21st year book of the National Council of
Teachers of Mathematics, Washington D. C. 1953,This work ends with
the following on pages 348-9. The phrases Learning Engineer
and Master Technician are noteworthy.
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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. My 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
... Opinion:
Those who say skills and
knowledge are not observable nor verifiable discount what is done
in carpentry, cooking, engineering, science and mathematics in an
observable and verifiable manner.
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Program for College Preparatory Mathematics,
Report of the Commission on Mathematics, College Entrance
Examination Board, New York 1959, 63 + x pages.This basic outline of
mathematics in grade 9 to 12 still echoes in US and Canadian courses.
This booklet focus on education for "university-capable" students
with a few remarks on mathematics in the general education for those
(most) not going. Appendices provide more details. The Preparation
for college here means preparation for calculus and analytic
geometry.There is a strong, college preparation orientation not just
for engineering and the physical sciences, but also for mathematics
itself. In that, the mathematical orientation (the striving for a
logical rigour) may be too much. The rigour present in the
diagram-free, algebraic-deductive axiomization of modern mathematics
is lost in the classroom with the use of diagrams in the development
and application of trigonometry and beyond calculus for the
exposition of ideas. In order to avoid details that are too technical
(overwhelming) for students and teachers, the classroom approach to
mathematics has to introduce mathematical practices and tools likely
to be of service in other disciplines in manner that prepares for but
does not provide the rigour of advanced studies. I object to the
criticism of earlier curricula on the basis of standards for rigour
that in retrospect, the new curricula in formation cannot meet. That
is the pot calling the kettle black.
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Program for College Preparatory Mathematics,
Report of the Commission on Mathematics, APPENDICES, College
Entrance Examination Board, New York 1959, 223 pages.In these
appendices, there is a strong, college preparation orientation not
just for engineering and the physical sciences, but also for
mathematics itself.
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Algebra
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Geometry
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Trigonometry
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1. An introduction to
Algebra2. Set, Relations and Functions 3. Classroom Approach to
Irrational Numbers. 4. Linear Function and Quadratics 5.
Complex Numbers 6. Limits 7. Permutations, selections and the
Binomial theorem 8. Mathematical Induction 9. Sets - How to
specify, Operations on Sets
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10. Reasoning for Modifying
the treatment of Geometry11. Deductive Reasoning12. Indirect
Proofs13. The first Theorems14. Coordinate Geometry Intro 15.
Theorems having easy analytic proofs 16. Solid and Spherical
Geometry 17. Transformations 18. Order Relations in Plane
Geometry.
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19. Vectors, Intro20.
Coordinate Trigonometry and vectors21. Trigonometric
Formulas22. Circular Functions
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Remarks:
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The names of some
chapters have been modified - abridge or
extended.
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With a pre-university
orientation, these appendices cover the essential elements of
algebra, geometry, trigonometry and vectors in grades 9
to 12 in 22 very detailed, lesson-plan oriented
chapters. The approach is authoritative. Rules and Patterns,
even axioms, are given for students to accept without
any attempt to rationalize them at the secondary level.
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The introduction of a
variable in the introduction of algebra,
topic 1, is a little too
formal. Site pages include a more intuitive, pre-algebraic approach
that could serve as a prequel, and separate the notion of variable
from the use of symbols.
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The coverage of complex
numbers is done in a formal, here is how
to calculate with a + bi
manner, with no geometric illustration,
except that implicit in
the use of order pairs (a, b) to represent a + bi. That being said,
the geometric representation of complex numbers as
vectors is introduced in coordinate trigonometry and
vectors.
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The directions on how to
specify sets is very clear - worth repeating.
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Deductive Reasoning and
Indirect Proofs are given in a how to do
it manner, with no
attempt at any rationalization. Site pages
point to
alternatives.
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The description of 30
theorem having easy analytic proofs is
neat.
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The coverage of solid and
spherical geometry is informal - neatly based on diagram to
demonstrate ideas. It is not axiomatic.
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The introduction of
vectors is pattern based. Here are some
practices to
follow.
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The coverage of
trigonometry while analytically is strongly based
on diagrams.
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Secondary Mathematics, A Functional Approach for
Teachers, H. F. Fehr, D. C Heath and Company Boston 1951.The
book is interesting for its exploration of possibilities, it rigour,
and it frequent mention of physical applications. I wonder if modern
calls for cross-curricula development of mathematics and other
disciplines recognize as such the possible interplay between high
school mathematics and physics and/or the mathematics of finance,
growth and decay.This work is written by a Professor of Mathematic
Education who has great expertise in mathematics. The book explores
possible routes for for the development of geometry, linear and
quadratic functions; numbers, constant, variable, function, equation
and graphs; elementary curve tracing; loci and the conic sections (a
must read for me), etc. etc. Professor Fehr use of the word
functional may refer to the common use of the word functional in
response to the question: does it work? Alternatively, it may refer
to the books emphasis of the role of functions in mathematics.The
chapter (pages 254-296) on complex number systems and trigonometry
gives as a exercise for students (!) the task of giving a geometric
proof of the distributive law for complex numbers when multiplication
is defined by multiplying moduli and adding angles. Professor Fehr
must have had a few proofs in mind. .Perhaps they will found the end
of chapter references on page 296. The site development of complex
numbers was updated December 2009 gives proof. It give the most
recent and simplest site proof of the distributive law, the simple
proof I have looking for since seeing Feynman in 1976 describe
physics in terms of adding and multiplying vectors in the plane. That
being said the site
development gives a simple proof of the distributive law for
complex numbers, independent of trigonometry. Whence complex number
methods may be employed to develop circular, periodic function,
trigonometry. That implies a simplification of the high school
development of trigonometry which I have seen in high schools and
colleges since the mid-1960s. Thus the site development gives
methods, fresh and re-invented, the exact division is not clear to
me, for making complex numbers, trig and vectors easier to learn and
teach.
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New Thinking in School Mathematics, Organization for
European Economic Cooperation, Office for Scientific and Technical
Personnel, May 1961.The text discusses what should be in or out in
mathematics skill development. The selection of topics appears to be
college oriented. That being said, given the experience of the last
five decades, I suggest common or likely needs of student in daily
life, immediate or long-term, should be the first focus of
quantitative skill development in say K-8 or 9, so mathematics
instruction is concrete for teachers, parents and these students.
That being said, we should weave advance level ideas into this early
instruction only where that inclusion makes skill and concept
development clearer, since the inclusion may be seen as needless
overhead by teachers - those not familiar with the long-term value of
that inclusion.. Given the choice between two routes for skill
development, both being of equal service for common or likely needs,
the route which serves advanced mathematics most should be chosen.
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Synopses for Modern Secondary School Mathematics,
Organization for European Economic Cooperation, Office for Scientific
and Technical Personnel, 1961.This cover secondary school education,
1961, European style for cycle I (ages 11 to 15) and cycle II (ages
15 to 18). Arithmetic, algebra, geometry, analysis are all cover from
an advanced level, with preparation for university studies very much
in evidence.
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L' Enseignement des mathematiques: J. Paiget, Beth,
J. Dieudonne, A. Lichnerowicz, G. Choquet, C. Gattegno, published by
Delachaux & Niestle, Nechatel (Switzerland).Of interest here is
the fact that this is a joint work of the pyschologist Paiget and
first rate mathematicians with positions in France and the USA. This
work connects Paiget with the very abstract Bourbaki school of
mathematics in a way that implies an alliance but not opposition.
That should be food for thought for present day interpreters of
Paiget work, constructivists included.
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1985 Curriculum Guidelines, Mathematics Intermediate
and Senior Divisions, Grades 7 & 8, Grades 9 & 10 Advanced
Level, Grades 11 & 12 Advanced Level, Ontario Academic Courses,
Minister of Education, Ontario.The description in detail of skills
and concepts is worth noting. It indicates a progression. This
curriculum guides names or describes in detail the skill and concepts
to be covered, but does not specify the teaching technique for each.
The curriculum clearly represent preparation for university or
college level studies in mathematics science or business. That being
said, I think students in grade 7 and 8 would benefit from a focus on
the quantitative skills and concepts likely to be needed in daily
life, sooner or later. That focus most likely occurs outside the
advance level versions of grades 9 to 12. But the underlying subject
matter would be of great benefit to pre-university students, and
would give common ground between them and others not heading for
university.
College Level Mathematics Education
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Calculus, Lipman Bers, Holt, Rinehart and
Winston 1969, SBN 03-065240-5A leading mathematics favors the decimal
viewpoint of real numbers, at least for students not in mathematics.
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How to Teach Mathematics, second edition, S. Kranz,
American Mathematics Society, 1991. ISBN 0-8218-138-6 Here are
recommendations for college level instruction. I tried to follow them
at the high school level. But they did not apply. In particular, I
announced my marking scheme for the current term early on, only to
discover end of term that the school required a new one, made-up at
the last minute, by a school committee. Ouch.
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Committee on the Undergraduate Program in Mathematics: A
Compendium of CUPM Recommendations, Volume I , Mathematical
Association of America, circa 1972 Volume I offers recommendations
for Training of Teachers, Two Year Colleges and Basic Mathematics,
Pre-Graduate Training.
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Committee on the Undergraduate Program in Mathematics: A
Compendium of CUPM Recommendations, Volume II, Mathematical
Association of America, circa 1972. Volume II offers recommends for
college level programs in statistics, computing and applied
mathematics, circa 1972
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Mathematics as a Service Subject, ICMI Study Series,
Udine 1987, Cambridge University Press 1988, ISBN 0-521-35395-5
(Hardcover) and -9 Paperback.The title of the conference is what
catches the eye.
Mathematics - Foundations, History, Logic, Philosophy Etc.
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History and Philosophy of Modern Mathematics,
Editors W. Aspray & P. Kitcher, Minnesota Studies in the
Philosophy of Science, Volume XI, University of Minnesota Press,
Minneapolis USA ISBN 0-8166-1567-5
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A Short Account of the History of Mathematics, W. W.
Rouse Ball, 4th edition 1908, Dover Publication Inc, paperback 1960.
ISBN 0-486-20630-0
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A History of Mathematics, 1968 C. B. Boyer,
Princeton Paperbacks, Princeton University Press 1985, ISBN
0-691-02391-3
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Makers of Mathematics, S. Hollingdale, 1989 &
1991, Penguin Books ISBN 0-14-01922-8
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The Nature and Growth of Modern Mathematics, 1970 E.
E. Kramer, Princeton Paperbacks, Princeton University Press 1982.
ISBN 0-691-02372-7
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Number Theory and Its History, Oystein Ore 1948,
Dover Publications 1988, ISBN 0486-65620-9
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A Source Book in Mathematics, D. E. Smith, 1929, Dover Publications
1959. IBSN 0-486-64690-4
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A History of Algebra from al-Khwarizmi to Emmy
Noether, B. L. van der Waerden. Springer Verlag, ISBN
3-540-13610-X, 260+ pages.Page 178 says the following regarding
complex numbers: Euler ... did not give a satisfactory definition.
Clear, geometrical definitions ... were given by Caspar Wessel in
1997, by Jean Robert Argand in 1806, by John Warren in 1828, and by
Carl Fredrick Gauss in 1831. ...William Rowen Hamilton defined (1843)
the complex numbers as pairs of real numbers subject to ... rules of
addition and multiplication. Augustin Cauchy interpreted (1847) the
complex numbers as residue classes of polynomials,..., modolo
x2 +1
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Evolution of Mathematical Concepts, An Elementary
Study, R L. Wilder, John Wiley & Sons 1968.Wilder is a
former President of the American Mathematics Society. From the
Jacket: This book attempts to explain how mathematics came into
being from the types of numerals found in primitive cultures, and to
determine the cultural forces that have governed its development.The
realization that mathematical content evolves implies mathematics
education content may evolve. That is liberating.
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Foundations and Fundamental Concepts of Mathematics
1958, H. Eves, Dover Publications 1997, ISBN -0486-69609-X
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Logic for Mathematicians, A. G. Hamilton, Cambridge
University Press, ISBN 0-521-36865.
References added, Fall 2009
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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
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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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