RBNZ “Jawboning” or Intervening to Lower the Exchange Rate?
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Professor Fekete, thank you for agreeing to this interview. Professor Fekete is the author of a book Real Linear Algebra and half a dozen papers on mathematics. In preparation are his monographs Quotient Set Theory and Stepnumbers. And you thought mathematics was just a means to attract the opposite sex. But for now we will begin with linear algebra. Professor Fekete, can antal fekete bitcoin exchange rate please share with us how and why you became interested in mathematics?
I understand that Hungary has produced many superb mathematicians. Hungary was exceptional in the second half of the 19th and the first half of the 20th century in that, by a quirk of fate, there were lots of brilliant math teachers at all levels. Hundreds of schools in Hungary competed in graduating thousands of outstanding mathematicians.
The problem today, in Hungary as elsewhere, is that an educational bureaucracy has taken over from dedicated teachers. This bureaucracy has interest neither in searching for budding mathematical talent, nor in grooming talent if it sprouts spontaneously.
It has absolutely no interest in teaching young people to think for themselves, for which mathematics is so superbly qualified. It has been ordered to train technicians how to turn the crank.
And this includes engineers and the like! Is it the students? Is it the teachers? Is it mathematics itself? It is definitely the teachers. Their job has been reduced to that of a drill sergeant. They have neither inspiration nor love for the subject they are teaching. They hold their job to earn a paycheck while they enjoy extra long holidays and summer vacation. If there are a few exceptions, they are not held in high esteem for their devotion to their students and for loving the subject they teach.
One subject that virtually everyone in any field of science needs to have a command of is linear algebra. You chose to write the textbook Real Linear Algebra. Why linear algebra vs the infinite other possible subjects? This extension is real linear algebra.
Why is linear algebra antal fekete bitcoin exchange rate beyond the standard viewand more specifically, why should math teachers and students antal fekete bitcoin exchange rate interested in your text over the many others? Recall that the cross product has all kinds of weird properties: It even fails to be associative ; instead, it satisfies the Jacobi identity:.
Moreover, there is a distinguished projection also called by its technical name: Without this motivation linear algebra is an empty shell. The motivation for this study is the cross product of real linear algebra.
The geometric images that come along are compelling and lasting. It is a scandal that this connection is kept from sophomore students and, even more so, from graduate students. Another disaster area in teaching mathematics, next to the teaching of linear algebra, is advanced multidimensional calculus. Here we find the worst textbooks ever written.
There is only one exception: It is a superb textbook, by now completely forgotten. In that book the definite integral and multi-dimensional differentiation are put into proper context. You would have never guessed it: Professor, this is a truly remarkable insight, looking at multiple integrals as a bilinear scalar product! Defining integration and differentiation simultaneously in a single formula is a magnificent unifying idea.
I sure wish this. Indeed it is, as opposed to the fragmentation as they are treated in the run-of-the-mill accounts of the subject. I hurl criticism to teachers of advanced calculus: Perhaps the reason is that they are ignorant of these facts themselves. It is not their fault. Their teachers were just as ignorant. They are teaching a bewildering variety of differential operators: The whole edifice of advanced calculus is falling apart for lack of any logical cohesion.
As the author, are there any other thoughts you might like share about Real Linear Algebra? Sometimes I describe this process, tongue in cheek, as rushing to generalize the concept of antal fekete bitcoin exchange rate group to semigroups, to pseudo groups, to pseudo semigroups, to quasi-groups, to quasi-semigroups, to quasi-pseudo-semigroups, etc.
This is not a condemnation of studying semigroups. It is not appropriate as a first exposure of antal fekete bitcoin exchange rate to fundamental mathematical ideas.
My answer to the race to stratospheric generalizations in the antal fekete bitcoin exchange rate atmosphere of the abstraction of the abstract is: What made antal fekete bitcoin exchange rate special?
In Professor D. Spencer of Princeton University invited me [Antal Fekete] to spend a year at Princeton and he put at my disposal that part of the mathematical archives of his friend, the late Norman Earl Antal fekete bitcoin exchange rate, which had to do with sophomore mathematics courses. I found a veritable treasure trove in those archives. Although geometry pervades all mathematics and is present at every stage of development, too often we fail to point this out to our students.
We rely on analytic formulations since we realize that they are complete and we are in a hurry to get on. We do not take time to look at geometric formulations.
We are too greatly impressed by the rigor of analysis. We seem to feel that geometry is not rigorous, or at least that the background needed for rigor is not available. We feel that it is better not to do anything that is not rigorous.
When we do present geometry, it is often the instructor who does the geometry while the student is merely a passive antal fekete bitcoin exchange rate. We present geometry to him in order to explain the analysis, but then we require him to do only the analysis — no geometry. We tend to avoid geometric formulations of questions in examinations. Questions are hard to formulate geometrically. Almost every time you try such a question, you find that a large group of students misinterpret it.
Such questions are hard to grade because they are so varied. The absence of geometric questions on final exams tends to degrade the geometric content of the course, and leads to its neglect. What has bothered me through the years is the control the exam seems to have on the course. Somehow the tail wags the god. In the exam we are supposed to take a sample of what the student knows. This process of sampling has a feedback effect that is very serious. The most famous example of this is the College Board exam and its influence on the teaching of mathematics in secondary schools.
The examiners, in order to be fair to students in all parts of the country, tended to take the intersection of antal fekete bitcoin exchange rate topics taught in various schools. A greater number of students were taking the exams, and schools were rated by the results. If a particular high school had a poor rating, they did something about it: The examiners, on their part, observed the shrinkage and narrowed the range of questions accordingly. At one time it was projected that after forty years only one topic would survive this elimination process, and that would be the factoring of quadratics.
Some say this cannot happen in college because the instructor is in charge of his course. Well, he is not, because in many colleges there are freshman courses with large enrollments and many sections. To avoid troubles with young instructors giving wide varieties of grades we insist on uniform exams and uniform grading. I have seen the feedback effect time and time again while teaching a section of the freshman course. Along comes a bright fresh Ph. Knowing that the concept of limit is central to calculus, he settles down and does a good job of teaching limits for two months.
His students do very well on that one question, but not so well on the other four of a more routine nature. Having learned his lesson, he runs a statistical analysis on the final exams for the last five years, and starts teaching his students how to turn the crank.
By the end of the semester he normally brings up their average up to antal fekete bitcoin exchange rate it should be. I do not know how to defeat this, but I do have one suggestions to offer. Harness the feedback effect to upgrade geometry by putting more geometric questions into their final exams and then face the problem antal fekete bitcoin exchange rate grading them. If, in the earlier parts of the course, on the ten-minute quizzes and the homework, you have inflicted geometry on the students over and over again, then on the final exam you have some chance of getting a good reaction out antal fekete bitcoin exchange rate the geometric questions.
This is, of course, not the only, nor the main reason why to bother with geometry at all The main reason is that most problems are presented in geometric form in the first place.
Reformulation and solution in analytic terms is merely a second step. To complete the process, there is an indispensable third, namely the interpretation of the analytic solution in geometric terms. There is another reason, which is psychological. Two views of the same thing reinforce one another.
