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# ðŸ“™ History in Mathematics Education by John Fauvel, J. A. Van Maanen â€” download pdf

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The importance of the subject matter of this book is reasserted again and again throughout, but never with the force and eloquence of Beltrami's statement of 1873:
"Students should learn to study at an early stage the great works of the great masters instead of making their minds sterile through the everlasting exercises of college, which are of no use whatever, except to produce a new Arcadia where indolence is veiled under the form of useless activity." (Beltrami, quoted on p. 36).
Teachers who think that sterility of student minds is innate rather than their doing had better consider that when a student calls mathematics teaching stupid he is merely echoing the opinion of the greatest mathematicians who ever lived. When the teacher blames his student for being too unmathematical to grasp his teaching, the truth is rather that the student is too mathematical to accept the anti-mathematical junk that is being taught.
Let us concretise this in the case of complex numbers. Here the teacher tries to trick the student into believing that complex numbers are useful because they enable us to "solve" otherwise unsolvable equations such as x^2+1=0. What a load of rubbish. The supposed "solutions" are nothing but fictitious combinations of symbols which serve absolutely no purpose whatsoever except that if you write them down on exams then the teachers tells you that you are a good student. A mathematically inclined student is not one who plays along with the charade but rather one who calls the bluff.
If we look at the history of complex numbers we find first of all that the nonsense about "solving" equations with no real roots is nowhere to be found. Secondly, we find that complex numbers were first conceived as computational shorthands to produce *real* solutions of higher-degree equations from certain formulas. But the inventor of this technique, Cardano, immediately condemned it as "as refined as it is useless," noting "the mental tortures involved" (Cardano, quoted on p. 305). Cardano's condemnation was not reactionary but perfectly sound and justified, for blind manipulation of symbols leads to paradoxes such as -2 = Sqrt(-2)Sqrt(-2) = Sqrt((-2)(-2)) = Sqrt(4) = 2. (This example is from Euler, quoted on p. 307.) These paradoxes dissolve with a proper geometric understanding of complex numbers. Only after such an understanding had been reached in the 19th century did the mathematical community take complex numbers to their heart (cf. pp. 304-305).
From this outline of history we learn not only that students are right to call their teachers charlatans and corrupters of honest knowledge, but also that students are in fact much more receptive to and enthusiastic about mathematics than mathematicians themselves. This is made clear in an interesting experiment conducted by Bagni (pp. 264-265). High school students who did not know complex numbers were interviewed. First they were shown complex numbers in the bogus context of examples such as x^2+1=0; then they were shown Cardano-style examples of complex numbers acting as computational aids in obtaining real solutions to cubic equations. In the first case "only 2% accepted the solution"; in the second 54%. But if the examples were given in the reverse order then 18% accepted complex numbers as solutions to x^2+1=0. In other words, students echoed the judgement of the masters of the past, except that they were more enthusiastic, being somewhat impressed by an idea referred to by its inventor as useless mental torture. Teachers should recognise what privilege it is to work with such admirably critical yet receptive students. The teacher should nourish this clarity of judgement and independent thought "instead of making their minds sterile."

## About book:

## About file:

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- Author:
**John Fauvel, J. A. Van Maanen** - Year:
**2002** - Publisher:
**Kluwer Academic Publishers** - Language:
**English** - ISBN:
**9780306472206,9780792363996,9781402009426,079236399X,1402009429**

- File size:
**4 895 513** - Format:
**pdf**

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This book explicates some of the fundamental philosophical tenets underpinning key theoretical frameworks, and demonstrates how these tenets inform particular kinds of research practice in mathematics education research. We believe that a deep understandi...

Educational equity and quality are not only research issues which cut across different disciplines but are major determinants of socio-economic and human development in both industrial and developing countries. The status and role of mathematics, a subjec...

In spite of the fact that APOS Theory has been used extensively in numerous scholarly publications, in the design of textbooks, and in teaching practice, there is no single references that contains all the relevant information about its components, and pr...

This book shares insights into the achievement gap in mathematics between East Asian and Western countries, and the ways to improve studentsâ€™ mathematics achievements. Especially, it highlights the importance to integrate case studies with large-scale int...

This volume is dedicated to the career of Jill Adler and the role she has played in growing mathematics education research in South Africa, Africa and beyond. Her work epitomises what is referred to as the â€˜engaged scholarâ€™: i.e. doing rigorous and theore...

This volume discusses semiotics in mathematics education as an activity with a formal sign system, in which each sign represents something else. Theories presented by Saussure, Peirce, Vygotsky and other writers on semiotics are summarized in their releva...

This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importanc...

With the ninth edition of the four-yearly review of mathematics education research in Australasia, the Mathematics Education Research Group of Australasia (MERGA) discusses the Australasian research in mathematics education in the four years from 2012-201...

This book is suitable for mathematics and vocational educators, researchers, and research students. Historical, sociological, and practical elements of mathematics within vocational education are set against the emerging impact of technology. Differenc...

A guide to how diverse sensors can be integrated to create enhanced controllers and estimators in a variety of situations....

Eternal People tells the story of Joseph Abrams, a Ukrainian Jew who finds his way to America at the end of the nineteenth-century. During a break from his studies in Russia, he returns to his shtetl in the Ukraine to find it is the target of a Cossack r...

The Joint FAO/IAEA Division of Nuclear Techniques in Food and Agriculture has a long history of coordinating isotope aided research projects for improving animal productivity in developing countries. These projects have focused on enhancing the efficiency...

This CCPS Concept book shows designers and operators of chemical facilities how to realistically estimate the flammable mass in a cloud of accidentally released material that is capable of igniting. It provides information on industry experience with flam...

Estuaries are among the most biologically productive ecosystems on the planet--critical to the life cycles of fish, other aquatic animals, and the creatures which feed on them. Estuarine Ecology, Second Edition, covers the physical and chemical aspects of...

About The ProductPublished by the American Geophysical Union as part of the Coastal and Estuarine Studies Series. The pollution of rivers, estuaries and the sea, the associated impact on these ecosystems and the effect on organisms, food-chains, water-sup...