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Mathematics Education References, Etc:
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,
1965
It 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.
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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The Growth of Mathematical Ideas, Grades K -
12, The 24th year book of the National Council of Teachers of
Mathematics, Washington D. C. 1959.
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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 Algebra
2. 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
Geometry
11. Deductive Reasoning
12. Indirect Proofs
13. The first Theorems
14. 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, Intro
20. Coordinate Trigonometry and vectors
21. Trigonometric Formulas
22. Circular Functions
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Remarks:
- The names of some chapters have been modified - abridge or
extended.
- 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.
- 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.
- 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.
- The directions on how to specify sets is very clear - worth
repeating.
- 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.
- The description of 30 theorem having easy analytic proofs
is neat.
- The coverage of solid and spherical geometry is informal -
neatly based on diagram to demonstrate ideas. It is not
axiomatic.
- The introduction of vectors is pattern based. Here are some
practices to follow.
- 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 pp254-296 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
pychologist 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 03-065240-5
A 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.
- 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.
- 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
- 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.
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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:
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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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