The reason I was seeking an education degree was because I had developed methods that would allow school to teach Calculus at a much earlier age. Bringing this method to fruition became a primary goal of my education career. I had hoped to turn it into a real career.
In my opinion, calculus should be taught immediately after or at in concurrence with algebra. Everything becomes easier when one knows calculus. Calculus is the basis of physics. Physics is the basis of chemistry and chemistry the basis of biology. In finance, you can't understand interest rates without calculus. Statistics leans heavily on calculus. Trigonometry is easier to understand once you know calculus. Calculus is primary and it should be taught earlier.
If we had a good method for teaching Calculus to your average high school student, then we would dramatically improve the quality of our math and science education and would dramatically improve the quality of life.
On a side note, my mother was interested in the history of logic. Logic was the center of education from the days of Aristotle through to the modern era. In this 2300 years history of teaching logic, there developed a plethora of approaches to the subject. Interestingly, the very first thing the left did when they gained hegemony in education was to rip the logic class out of the school. That is a different story.
… back to calculus ...
There are multiple ways to teach calculus. The traditional Cauchy method is a college level course. Proponents of new math favor a set theoretic approach which presents calculus as a construct of language. This method fits in well with the tradition of Hegel, Marx, Cantor, Russell and Chomsky. I find the approach paradox ridden. Again, I don't think it is suitable for high school. Marvin Kline proposed an Intuitive Approach to Calculus. His method fits in with the thinking of Montessori, however, it really depends on the skill of the teacher.
Not surprisingly, the method I wanted to develop fell squarely in the classical tradition of Aristotle, Descartes, Leibnitz, Newton, Gauss, Reimann, Einstein, etc.. It would work even better if schools taught logic.
I believed that there was merit to my method. Regardless of my personal thoughts, I would never force it on anyone without data to back up the claim. At the time, I figured I would need about $200,000 to develop and do alpha testing of the curriculum. If the data looked good, I would want to role out a controlled beta test. If the average high school student was able to master Calculus with my method, as I contended, then I would want it to compete with all of the other ideas on the market.
There are multiple ways of teaching subjects like logic and calculus. So, the question is: How do you create a mechanism that simultaneously allows for the development of multiple curriculums with the majority of students getting the best methods?
Unfortunately, in our current single payer system of education, there is market mechanism to allow for diversity of ideas.
In Utah, 96% of students go to public schools. The curriculum is set in Universities that have zero interest any method beyond New Math. Ideas, other than new math, don't even have the potential to survive.
With one extraordinarily corrupt group, the UEA, having absolute power over 96% of the education market, there is no longer any place for ideas such as teaching logic to primary students or teaching Calculus to all students in high school to exist. Survival for diverse ideas is not even possible. It is a stagnant system.
It is possible that new math really is the way to go, and that my method really should fail. That is entirely fine with me. Trying ideas and failing is part of life. For every profound scientific discovery, there are at least a thousand ideas thrown on the table that fail. For every blockbuster toy, there is a thousand flops. This method of developing and testing ideas through open inquiry is called science. The marketplace leads to prosperity. Our left dominated public school has destroyed the marketplace for ideas.
The fact that I never even got the opportunity to put any of my ideas on the table because a group of brownshirts in the education department weeded people out for political reasons is inexcusable and greatly devalues the value of our education.
In the past, I had rejected vouchers. I don't think it is as good as tax credits, neither is as good as a world where taxes are low and people have sufficient personal resources to send students to the school of their choice. In looking at the diversity of the charter and private schools that is starting to come to life in the wake of the State's legislature commitment to diversified education, I decided that the vouchers are a good interim step for breaking the monopoly in education.
Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Wednesday, October 10, 2007
Sunday, July 03, 2005
Trapped by Comtempt
I realize that my last post was very contemptuous. Temerity might be a better description of amateur daring to speak of what many consider to be the greatest triumph of modern thought...transfinite theory.
More than once, people have told me that I am a contemptuous little snit who does not recognize his betters. Most of the comments I have received about investigations into the subject point out that I am too small and petty of an individual to even utter the great words “diagonal method.”
The theory was not intended for the common man or for small minded rabble who drown their pettiness in pubs (people like me). It was meant for that small tiny transcendental elite who flutter in the upper heavens of thought and spirituality
My actions do wreak of contempt. When I speak of transfinite theory, I feel the part of an uncouth gentile soiling sacred temple grounds.
In truth, I do have a great deal of respect for the time and effort that scholars apply to their respective fields of study. I’ve always thought that it is more likely that I am wrong in my beliefs than it is likely that the great mathematicians of the modern era were misled. This assumption that I am wrong in my thoughts is a driving force in both my academic and professional life.
The idea that I am wrong drives my own personal investigations.
What I do in my research in start with the assumption that I am wrong. I assume that the great mathematicians who hold to the theory are right. Somehow, set theory really is the hidden forms that Plato saw when exited the cave of temporal illusion. I assume that I just somehow misunderstood the diagonal method.
The way I would approach the issue is to put the theory aside for several months or years. I would then pick up a few books on Set Theory. I would read through, doing exercises, etc.. I am usually impressed with the astute logic and structure of discrete set theory. Frege’s propositional and prepositional logics are a grand contribution in the history of thought.
Then I would get into the sections on transfinite theory. All of my old objections would flood back into mind...generally the objections would be added with new objections. I find myself wanting to take an even stronger stand against transfinite theory, transcendental logic and all of the silliness of the modern tradition.
Anyway, during the last iterations of this process. My mind was burning on the issue of how one could rid mathematics of this idea that mathematics is driven by a fundamental dichotomy.
A year ago, I lighted on the idea of creating something called “Rich Theory.”
Transfinite theory essentially says that there are only two layers of infinity. It starts with the claim that all denumerable sets are the same size. The diagonal method creates a set which is larger than this denumerable infinity. Transfinite theory holds the following. It holds that n = n+1 where n is infinity, 2*n = n, when n is infinity and n^2 = n when n is infinity. However, it states that 2^n is greater than n.
Rich Theory notes that n < n+1 < 2n < n^2 < 2^n when n is a large number. We can actually imaging each of these operations creating a different level of infinity. Essentially, infinity is a rich space.
For various reasons that will take a long time to explain, what we see in infinity is largely a reflection of what we see in the finite. When we look at complex issues in the finite world, we see that there’s a large number of different things going on (wheels within wheels). Transfinite Theory’s effort to sum up mathematics with a simple dichotomy between denumerable and nondenumerable was as off base as Marx’s attempt to sum up the economy in terms of a single conflict between the bourgeoisie and proletarian classes.
Transfinite theory does not open a paradise. It impoverishes a paradise. It wants to reduce the fantastically complex nature of infinity into two piles...denumerable and nondenumerable.
Rich Theory says that infinity is rich. We can find all sorts of wonderful things reflected in the study of infinite sets.
Calling this view of infinity “rich” is really a mean thing.
The theory was not satisfactory because it did not address the reason that attracted attention in the first place. Mathematicians want a way to address the important distinction between discrete and continuous mathematics.
So, my mind had been racing on how the issue of the distinction between discrete and continuous mathematics. The distinction of course, pretty much falls from definitions. Discrete mathematics is reduced to finite elements. Continuous mathematics allows of the use of infinite length entities.
My current thinking is that an approach that the best way to approach transfinite theory would be a discussion of binary strings. Such a course could present better and clearer models for discussing the infinite. Likewise it would fit in well with computer science.
Which, of course, brings me back to the problem that I am a contemptuous little snit.
There is a possibility that I have a good idea that could help students understand mathematics. The problem is that, even if I have a good idea, my being a contemptuous little snit immediately nullifies anything that I might say.
In classical mathematics, there would be a hope that serious mathematicians could look beyond my personality defects. If they found something worthwhile, they would run with it and discard the inane things I say.
Unfortunately, we live in the world of modern mathematics. Modern mathematics is driven by who a person is and not what they say. The Hegelian heart of modern mathematics has the world spirit realized in the writings of a few super intellectuals (philosopher kings).
The very act of speaking, researching or trying to understand mathematics is an act of contempt.
The modern world has created a culture of silence. In the days of classical mathematics, you saw people in all walks of life engaged in the discourse of mathematics.
The set theoretic model discards input from people judged as lesser men. They drove the classical mathematicians and scholastic scholars from the schools. Today all mathematics and logic is written in terse, arcane symbolic form. Commoners are simply to worship our transfinite betters and stay silent. To even speak or think is an act of contempt.
It is a trap that is extremely difficult to break.
More than once, people have told me that I am a contemptuous little snit who does not recognize his betters. Most of the comments I have received about investigations into the subject point out that I am too small and petty of an individual to even utter the great words “diagonal method.”
The theory was not intended for the common man or for small minded rabble who drown their pettiness in pubs (people like me). It was meant for that small tiny transcendental elite who flutter in the upper heavens of thought and spirituality
My actions do wreak of contempt. When I speak of transfinite theory, I feel the part of an uncouth gentile soiling sacred temple grounds.
In truth, I do have a great deal of respect for the time and effort that scholars apply to their respective fields of study. I’ve always thought that it is more likely that I am wrong in my beliefs than it is likely that the great mathematicians of the modern era were misled. This assumption that I am wrong in my thoughts is a driving force in both my academic and professional life.
The idea that I am wrong drives my own personal investigations.
What I do in my research in start with the assumption that I am wrong. I assume that the great mathematicians who hold to the theory are right. Somehow, set theory really is the hidden forms that Plato saw when exited the cave of temporal illusion. I assume that I just somehow misunderstood the diagonal method.
The way I would approach the issue is to put the theory aside for several months or years. I would then pick up a few books on Set Theory. I would read through, doing exercises, etc.. I am usually impressed with the astute logic and structure of discrete set theory. Frege’s propositional and prepositional logics are a grand contribution in the history of thought.
Then I would get into the sections on transfinite theory. All of my old objections would flood back into mind...generally the objections would be added with new objections. I find myself wanting to take an even stronger stand against transfinite theory, transcendental logic and all of the silliness of the modern tradition.
Anyway, during the last iterations of this process. My mind was burning on the issue of how one could rid mathematics of this idea that mathematics is driven by a fundamental dichotomy.
A year ago, I lighted on the idea of creating something called “Rich Theory.”
Transfinite theory essentially says that there are only two layers of infinity. It starts with the claim that all denumerable sets are the same size. The diagonal method creates a set which is larger than this denumerable infinity. Transfinite theory holds the following. It holds that n = n+1 where n is infinity, 2*n = n, when n is infinity and n^2 = n when n is infinity. However, it states that 2^n is greater than n.
Rich Theory notes that n < n+1 < 2n < n^2 < 2^n when n is a large number. We can actually imaging each of these operations creating a different level of infinity. Essentially, infinity is a rich space.
For various reasons that will take a long time to explain, what we see in infinity is largely a reflection of what we see in the finite. When we look at complex issues in the finite world, we see that there’s a large number of different things going on (wheels within wheels). Transfinite Theory’s effort to sum up mathematics with a simple dichotomy between denumerable and nondenumerable was as off base as Marx’s attempt to sum up the economy in terms of a single conflict between the bourgeoisie and proletarian classes.
Transfinite theory does not open a paradise. It impoverishes a paradise. It wants to reduce the fantastically complex nature of infinity into two piles...denumerable and nondenumerable.
Rich Theory says that infinity is rich. We can find all sorts of wonderful things reflected in the study of infinite sets.
Calling this view of infinity “rich” is really a mean thing.
The theory was not satisfactory because it did not address the reason that attracted attention in the first place. Mathematicians want a way to address the important distinction between discrete and continuous mathematics.
So, my mind had been racing on how the issue of the distinction between discrete and continuous mathematics. The distinction of course, pretty much falls from definitions. Discrete mathematics is reduced to finite elements. Continuous mathematics allows of the use of infinite length entities.
My current thinking is that an approach that the best way to approach transfinite theory would be a discussion of binary strings. Such a course could present better and clearer models for discussing the infinite. Likewise it would fit in well with computer science.
Which, of course, brings me back to the problem that I am a contemptuous little snit.
There is a possibility that I have a good idea that could help students understand mathematics. The problem is that, even if I have a good idea, my being a contemptuous little snit immediately nullifies anything that I might say.
In classical mathematics, there would be a hope that serious mathematicians could look beyond my personality defects. If they found something worthwhile, they would run with it and discard the inane things I say.
Unfortunately, we live in the world of modern mathematics. Modern mathematics is driven by who a person is and not what they say. The Hegelian heart of modern mathematics has the world spirit realized in the writings of a few super intellectuals (philosopher kings).
The very act of speaking, researching or trying to understand mathematics is an act of contempt.
The modern world has created a culture of silence. In the days of classical mathematics, you saw people in all walks of life engaged in the discourse of mathematics.
The set theoretic model discards input from people judged as lesser men. They drove the classical mathematicians and scholastic scholars from the schools. Today all mathematics and logic is written in terse, arcane symbolic form. Commoners are simply to worship our transfinite betters and stay silent. To even speak or think is an act of contempt.
It is a trap that is extremely difficult to break.
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