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Elementary Mathematics from an Advanced Standpoint - Arithmetic - Algebra - Analysis, by Felix Klein
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FELIX KLEIN ELEMENTARY MATHEMATICS FROM AN ADVANCED STANDPOINT- ARITHMETIC- ALGEBRA -ANALYSIS. TRANSLATED FROM THE THIRD GERMAN EDlTION BY E. R. HEDRICK AND C, A. NOBLE PROFESSOR OF MATHEMATICS PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CALIFORNIA IN THE UNIVERSITY OF CALIFORNIA AT LOS ANGELES AT BERKELEY WITH 125 FIGURES MACMILLAN AND CO., LIMITED ST. MARTINS STREET, LONDON 1932 ALL RIGHTS RESERVED PRINTED IN GERMANY BY THE SPAMERSCHE BUCHDRUCKEREI LEIPZIG Preface to the First Edition. The new volume which I herewith offer to the mathematical public, and especially to the teachers of mathematics in our secondary schools, is to be looked upon as a first continuation of the lectures Uber den mathematischen Unterricht an den hoheren Schulen, in particular, of those on Die Organisation des mathematischen Unterrichts by Schimmack and me, which were published last year by Teubner. At that time our concern was with the different ways in which the problem of instruction can be presented to the mathematician. At present my concern is with deve lopments in the subject matter of instruction. I shall endeavor to put before the teacher, as well as the maturing student, from the view-point of modern science, but in a manner as simple, stimulating, and convincing as possible, both the content and the foundations of the topics of instruction, with due regard for the current methods of teaching. I shall not follow a systematically ordered presentation, as do, for example, Weber and Wellstein, but I shall allow myself free excursions as the changing stimulus of surroundings may lead me to do in the course of the actual lectures. The program thus indicated, which for the present is to be carried out only for the fields of Arithmetic, Algebra, and Analysis, was indicated in the preface to Klein-Schimmack April 1907. I had hoped then that Mr.. Schimmack, in spite of many obstacles, would still find the time to put my lectures into form suitable for printing. But I myself, in a way, prevented his doing this by continuously claiming his time for work in another direction upon pedagogical questions that interested us both. It soon became clear that the original plan could not be carried out, particularly if the work was to be finished in a short time, which seemed desirable if it was to have any real influence upon those problems of instruction which are just now in the foreground, As in previous years, then, I had recourse to the more convenient method of lithographing my lectures, especially since my present assistant, Dr. Ernst Hellinger, showed himself especially well qualified for this work. One should not underestimate the service which Dr. Hellinger rendered. For it is a far cry from the spoken word of the teacher, influenced as it is by accidental conditions, to the subsequently polished and readable record. On the teaching of mathematics in the secondary schools. The organization of mathematical instruction. IV In precision of statement and in uniformity of explanations, the lecturer stops short of what we are accustomed to consider necessary for a printed publication. I hesitate to commit myself to still further publications on the teaching of mathematics, at least for the field of geometry. I prefer to close with the wish that the present lithographed volume may prove useful by inducing many of the teachers of our higher schools to renewed use of independent thought in determining the best way of presenting the material of instruction. This book is designed solely as such a mental spur, not as a detailed handbook. The preparation of the latter I leave to those actively engaged in the schools. It is an error to assume, as some appear to have done, that my activity has ever had any other purpose...
- Sales Rank: #8007963 in Books
- Published on: 2008-11-04
- Original language: English
- Number of items: 1
- Dimensions: 8.50" h x .81" w x 5.51" l, 1.14 pounds
- Binding: Hardcover
- 292 pages
Most helpful customer reviews
0 of 0 people found the following review helpful.
Great resource for high school math teachers with a strong mathematical background
By Jeremy Gross
This book (and its second volume on geometry) saved my sanity when I taught high school mathematics. Felix Klein is one of the greatest mathematicians of the 19th century, the first person to put geometry on a group-theoretical footing, which led to the geometry of modern physics. This book comes out of a course that Klein taught to high school math teachers in Germany in 1906-7. At that time, his audience would have had Ph.D.s in mathematics, and so he taught them high school mathematics from the perspective of a mathematician at the Doctoral level. The result is absolutely fascinating, and gave me so many resources for teaching math courses that would be interesting to my students and also interesting to myself.
I'll give you an example. When I was 15, I asked my teacher how an expression like 2^(sqrt 2) could be well-defined. Exponentiation is well-defined for rational numbers, but how do we know that such a definition extends to all real numbers? My teacher was stumped, and I was very disappointed. It wasn't until I took a real analysis course in college and proved that the exponential function exp(x) is continuous at every real value of x (and indeed at every complex number of finite modulus as well) that I understood why 2^(sqrt 2) is well-defined.
How does Klein approach the problem? He asks the student to draw the hyperbola xy = 1 on the blackboard. Then he asks the student to draw the line x = 1 on the same axes. Then he suggests using a yardstick as a slider to slide forward along the x-axis, or backwards towards x = 0, all the while noticing where the yardstick meets the graph. Then he asks the student to notice that the yardstick sweeps out area under the curve xy = 1, positive area to the right of x = 1, and negative area to the left of x = 1, Klein invites the student to consider that sweep of area to be a continuous function, zero at x = 1, negative between x = 0 and x = 1, and positive when x > 1. Klein notices that the value of that function at ab is equal to its value at a plus its value at b, and that its value at a squared is double its value at a.
Klein then says that this function is invertible since it is monotonic, and that its inverse has an interesting property, that its value at ab is equal to its value at a multiplied by its value at b. He then asks the student to name the original function the natural logarithm of x, and its inverse exp(x).
This demonstrates that both functions are continuous. From here, you can show that 2^x is equal to exp ((ln 2) x), and that 2^(sqrt 2) = exp ((ln 2) (sqrt 2)), and you have shown that 2^x is continuous at sqrt 2. This can obviously be made more rigorous, but it gives an easy introduction to natural logarithms and the exponential function, and I've used his demonstration in my classroom whenever I've introduced the subject since I read the book.
His understanding of mathematics is so beautiful and fluid and expressive. It is a great privilege to see how he views mathematics, and share it with my students.
20 of 21 people found the following review helpful.
Still the best book for Math education today !
By WuBing
I read Felix Klein's [1849-1925] 3-volume Math book [publised in German, 1924] in Chinese translation bought recently in Beijing bookshop. The chinese translators are themselves mathematicians, so they corrected some translation mistakes made by the 1939 English translators from whom they translate into Chinese.
Klein was a great mathematician, being the Father of Modern Geometry based on Group Theory [Erlangen Program]. He was also a great Math educationist, who built the Gottingen University into the World Center of Mathematics before WW II, attracted the Germany's and the World's best mathematicians to teach and learn there. eg. David Hilbert, Noether, Laskar, etc.
This 3-volume Math books were compiled by his assistants from the lectures he gave over 20+ years in breaching the gap of Secondary School Elementary Math and University Modern Math, hence the title "Elementary Math from an advanced standpoint". Although written 80 years ago, his concerns and teaching are still valid today. Except for France where they teach modern math concepts in Lycee (secondary school from 3eme to Terminale), other countries' secondary schools don't introduce these modern concepts (Group, Ring, Field...), some with little Set theory at best. For Singapore and the English commonwealth countries, secondary and high school math are still using the same syllabus 100 years ago: 100% computational-bias (calculus, trigo, classical algebra).
Klein spotted this gap 80 years ago, as indicated in the volume 1, which he labelled as Syllabus A, Syllabus B and Syllabus C.
In Syllabus A (which is still used worldwide): all math subjects (algebra, geometry, trigo...) are taught as separate independent modules. The secondary school students don't see them as inter-connected.
In Syllabus B, Klein proposed to base whole math teaching on Functions, which is intuitive as functions can be visualised in Graphs, and link to all branches of math.
eg. Function exponential e: e^x = 1 + x + 1/2! x^2 + 1/3! x^3 +....
also e^ix = Cos x +i.Sin x
where the function exponential has linkage to trigo (cosine and sine), and algebra's complex number i. Same can be applied to calculus: integration and differentiation link to functions like Log, exponential, trigo, etc.
Although he didn't elaborate on Syllabus C (out of his life / book context), but I would presume he had predicted the Bourbaki school of Math based on Set Theory as math structures. This approach of teaching was very prominent in French Mathematics Education. It has proved with little success to students because of being too abstract, distancing the Math from real-life (pure math vs applied math). Klein, although a pure math par excellence, didn't encourage this approach which was advocated after his death by the French Bourbaki mathematicians.
I admire Klein's foresight in Syllabus B intuitive pedagogy which would introduce secondary school students gradually to the University modern math concepts.
After 80 years, his dream is still not fully appreciated by math educationists. It is hard to change the mindset in the math education bureaucrats worldwide.
58 of 59 people found the following review helpful.
Still a great resource
By Viktor Blasjo
About the set: Klein's Elementarmathematik lectures was intended as a survey of mathematics for those who already knew most of the technical detail, especially future teachers, but who perhaps lacked a good understanding of mathematics as a whole. The lack of a broad perspective is probably at least as big a problem today as it was then, so Klein's text is still valuable. Klein also frequently discusses historical and pedagogical aspects, and the tone is quite informal throughout.
About this volume: The first part discusses basic arithmetic, with great emphasis on how it should be taught. Being such an enthusiastic lecturer, Klein cannot resist also including little sections on things like Fermat's last theorem and quaternions. The second part suddenly becomes much more sophisticated, as Klein sketches a rather non-standard approach to understanding algebraic equations, in particular incorporating a bit of Klein's trademarke icosahedron material. To keep up with this presentation one should probably consult his icosahedron book as well. For the third part, on analysis, we are back on familiar ground again, which makes it easier to appreciate this masterful overview, covering many of the great ideas of analysis and putting them in context with interesting discussions that are often partly historical.
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