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Chapter 5
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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). |
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 school and community library. If not, strongly suggest that one should be ordered [1].
[1] This work is now out of print. The Thompson publishing company, telephone 1-800-865-5840 in North America, now has the rights to the work, or its successor with a different ISBN number. This author hopes that the work will be reprinted, hardcover, with multi-coloured pages as before.
VNR also produced the James and James, VNR Mathematics Dictionary, third edition of 1968. This may also be of interest. The first edition appeared in 1942. (VNR has or had a remarkable collection of works in science and mathematics. Their issuance was a public service.)
Providing information about the origin of terms and methods is one way to nurture a knowledge of mathematics and its origins. An effort in this direction is provided by the book
| Historical Topics for the Mathematics Classroom, by J. K. Baumgart et al, published by the National Council of Teachers of Mathematics, 1969, second edition 1989, 1906 Association Drive, Reston, Virginia, USA 22091. |
In this reference, two sections or articles provide background information which support, I think, the perspectives on algebra and the development of mathematics given in this work and its companions.
1. The elementary section: The History of Algebra, an Overview, by J. K. Baumgart. This section briefly mentions the transition of algebraic thought from words only to symbolic.
2. The less elementary section: Development of Modern Mathematics, an Overview, by R. L. Wilder. This section briefly indicates that from 1930 to 1950, the set-theory perspective went from a curious part to an essential part of mathematics.The bibliographies A and B in this book, one more recent than the other, provide further references for the study of mathematics or its instruction.
The previously mentioned work, the 1965 book Secondary School Mathematics by J. J. Kinsella, published by The Center for Applied Research in Education, Inc., New York, is another reference. It describes mathematics instruction from the early 1900s to the 1960s in North America. Many of its comments are still valid.
Morris Kline’s work Mathematical Thought from Ancient to Modern Times, today appears 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.
Next: Chapter 6, Rule
and Pattern Based Reason in Mathematics
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Volume 1B, Mathematics Curriculum Notes,Foreword + Chapters 1 to 10 + 12
Area Map Inductive Principles 1 Introduction 2 For & Against Math 3 Algebraic Thought 4 Why Slopes & SQRT of -1 5 Books & Articles to Read 6 Unruly Origins of Algebra 6. Axiomatic Civilization 7 Geometry, 2 Ways 8 Modern Instruction 9 The Two Ends 10 The Transition 10 Explaining Logic 10 Explaining Algebra 10 Why Sets in Math. 12 Four Phases Links
Chapter 11: Primary School Mathematics
11 Primary Math 11 Cue Cards 11 Counting 11 Decimals - Addition 11 Decimals -Times 11 Decimals & Subtraction 11 Fractions and Division 11 Notational Conflict 11 Reciprocals Etc 11 Decimals - Ratios 11 Size Comparison 11 Numbers, +ve or -ve 11 Rename < Sign 11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen LearnMost students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus. Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context for the very optional study of pure mathematis.
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