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Mathematics Education References, Etc:
Coffee Table Books:
- 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.
- 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.
- 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.
- 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.
- 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.
- The Growth of Mathematics Ideas, Grades K-12, Twenty-Fourth
Yearbook, The National Council of Teachers of Mathematics,
Washington. D. C. 1959.
This work falls in my to be read category.
- The Learning of Mathematics, Its theory and practice, Twenty-First
Year Book, The National Council of Teachers of Mathematics,
Washington. D. C. 1953.
This work falls in my to be read category.
- 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.
- 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.
| Algebra |
Geometry |
Trigonometry |
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 |
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. |
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 highschools 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.
- 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.
- 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.
- 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.
- 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.
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.
-
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.
-
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.
-
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.
-
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.
-
The Growth of Mathematical
Ideas, Grades K - 12, The 24th year book
of the National Council of Teachers of Mathematics, Washington D. C.
1959.
College Level Mathematics Education
- 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.
- 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
- 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.
- 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
- 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
- A History of Mathematics, 1968 C. B. Boyer, Princeton Paperbacks,
Princeton University Press 1985, ISBN 0-691-02391-3
- Makers of Mathematics, S. Hollingdale, 1989 & 1991, Penguin
Books ISBN 0-14-01922-8
- The Nature and Growth of Modern Mathematics, 1970 E. E.
Kramer, Princeton Paperbacks, Princeton University Press 1982. ISBN
0-691-02372-7
- 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
- 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
- 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.
- Foundations and Fundamental Concepts of Mathematics 1958, H.
Eves, Dover Publications 1997, ISBN -0486-69609-X
- Logic for Mathematicians, A. G. Hamilton, Cambridge
University Press, ISBN 0-521-36865.
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Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Site Areas
and SubAreas
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations - making alg easier
4. Formulas
forwards & Backwards - a theme
5. Proportionality,
Back- & For-wards
6. Euclidean-Geometry
(lean intro)
7.
Logic - Math Free, good for work & studies
8. Slopes
and Lines
9. Why
Study Slopes - Advanced Motivation
10. Quadratics
and Polynomials
11. Application
of Factored Polynomials
12 Functions
- Forwards & Backwards
13 Number
Theory, Richly
14. Exponents,
Radicals & logs.
15 Calculus
- Examples & Blah, Blah, Blah
16. Real
Analysis
17.
Electric
Circuits Etc (So So)
18. Maps,
Plans, Similarity & Trig, (alt view)
19. Complex
numbers - a visual approach
20.
Logic with Symbols (and truth tables)
21.
Logic
& Consistent Story Telling
22. Even
More Logic
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