Revised web site
Here is an updated URL:
Here is an updated URL:
I have been playing with a new web site since my last post. Here is the URL:
Here are two quotes that I have noted in the last few weeks. The first was from the Michael Spivak book on calculus: "A great deal of this book is devoted to elucidating the concept of numbers, and by the end of the book we will have become quite well acquainted with them. ... It is therefore reasonable to admit frankly that we do not yet thoroughly understand numbers. ... we will have to consider a little more carefully what we mean by 'numbers'. " [p. 12 - 13]
Then yesterday, while collecting my books, I came across a textbook from my last year of undergraduate study, "Principles of Mathematical Analysis" by Walter Rudin (1964). Here is the first sentence of the book: "A satisfactory discussion of the main concepts of analysis (e.g., convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept." [p. 1]
Thus I now feel that I should embark on a two-prong approach to Learning calculus. In addition to focusing on calculus, I now want to spend a similar amount of time on studying numbers themselves. If nothing else my approach to Learning remains flexible. In passing I notice that the undergraduate curriculum where one takes a few courses at the same time, but where the courses have some overlap, is consistent with what I am now endeavoring to undertake. The real issues are time, and intellectual background. Since almost all of the actual mathematics that I once studied has not been used in over 40 years it seems fair to assume that the actual detail is forgotten. Fortunately the positive aura surrounding the detail is still intact, which should be a valuable support to my future Learning.
The blog I am currently using does not have the power to capture my notes so I will begin a new web site that chronicles my efforts. I will post the link on this blog when I have it ready.
I have accumulated a substantial number of books on number. It is time I spent some time with them. The first task is to collect them into one area on my book shelves.
There are moments when I feel that I am collecting butterflies. There are many kinds of numbers: the natural numbers, the integers, prime numbers, odd numbers, even numbers, rational numbers, irrational numbers, imaginary numbers, complex numbers, transcendental numbers, cardinal numbers, ordinal numbers, congruent numbers, ... I have also seen a quote ascribed to the German mathematician Kronecker in a number of my books, "God created the natural numbers, the rest is the work of man."
I find it fascinating to look at the books I have on number. There are a total of 25 that have a major emphasis on numbers. I have about a dozen on calculus. All of these books describe some aspect of the topic with some having a fairly well-defined sequaence and structure while others are more like a collection of interesting facts and problems.
However none of the books describe the actual process of Learning the material, not do any of them attempt to describe the actual mathematical knowledge that the author possesses about the topic. In fairness some of the books do attempt to convey a sense of the excitement of playing with the topic, but I have not seen a book that tries to set out how the author conceptualizes the topic and how she explores ideas within the topic. This blog has been a preliminary attempt to do this, and my future web site will be a more detailed description of my personal journey over the next few years.
However the real goal is not this blog, nor my web site. The real goal is the modification of my mind to better incorporate a number of mathematical ideas and to improve my understanding of mathematics and hence of science and philosophy. The blog and web site are only cloud chambers of an underlying mental transformation.
The following comments apply to both my Learning of calculus as well as to other areas of Learning that I am interested in.
I have just finished reading and making a mindmap for the book "Tomorrow's People" (2003) by the neuroscientist Susan Greenfield. In my first read through I made a small ink mark in the margin when I noticed a point that I found particularly striking or important.
On the second reading, I created a mindmap using the software MindGenius. This really helped me identify the important points and see where they fitted in the overall description of her views about the future.
As I have indicated on many posts to this blog, I am keeping my notes as a website (currently located on my hard drive) using the web authoring package Dreamweaver. As with any website, I have the capability to make links to other web pages (both my own as well as to external web resources) and perhaps more importantly, to files created by other software programs. Links to these files will work, and as long as the user has the same software that originally created the file, the file will open in that environment where one has dynamic control of the file.
In the case of this mindmap, this is preferable to seeing a static portion of the map on the screen. I can scroll through the complete map or collapse and expand nodes to suit my current area of interest. The difference is dramatic.
Similarly, when I am Learning calculus, I can use sophisticated software such as Mathematica to explore my ideas and these files can be saved and accessed from within the website that contains my narrative notes. Thus my electronic notes are much more than just a static textual presentation. They permit me to quickly move among various software packages with just a couple of mouse clicks and give me the added power that each package provides.
There is no doubt in my mind that these web-based notes are superior to anything I might try to create using more conventional materials. Each reread of the book, each paragraph that is created as part of note-making, each activity that uses a software package to play with the ideas provides numerous opportunities for review and enhancement. The cumulative effect of all these activities is to modify my neural patterns (a point made repeatedly by Greenfield) which is then manifested by my being able to engage in other activities such as a discussion (perhaps on a blog) that exhibits aspects of this new Learning experience.
I began reading a new novel last evening, "the curious incident of the dog in the night-time" by Mark Haddon. I love it!
On page 12 I encountered the following: "Prime numbers are what is left when you have taken all the patterns away."
I bought two books last year that I have glanced at, but have yet to read slowly. The first is called "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics" by John Derbyshire. The second is "Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers" by Dan Rockmore.
If one needs extrinsic motivation, there is a prize of one million dollars for the individual who is able to prove that the Riemann Hypothesis is true (or false).
Number theory and the subset that deals with prime numbers (i.e. numbers that have no divisors other than 1 and the number itself) is a branch of mathematics that is fascinating in its own right. Surprisingly it is also relevant to the idea of developing codes that are unbreakable, a topic of current interest with the surge of interest in the Internet and the World Wide Web.
In addition to my interest in calculus, I am interested in learning more about number theory. However these topics all have a way of interlocking. The two fundamental ideas of calculus are those of function and limit. The same two ideas are at the heart of the Riemann Hypothesis.
The Mark Haddon novel continues, "I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your time thinking about them."
Mathematics is a discipline with a specialized notation. In the days when the pencil or pen was the tool of choice this meant coming to an agreement about how to "properly" write an expression to avoid ambiguity and miscommunication.
However with the advent of typewriters and computers the conventions previously arrived at have turned out to be very difficult to transcribe onto paper or the screen.
I am making (almost) all of my notes as web pages. In order to capture the notational complexities I use a combination of two different software packages. I begin by using Dreamweaver to create my basic web page. However when I encounter a mathematical expression that is difficult to express in Dreamweaver I switch to Microsoft Word which has been extended with the software package MathType 5. It is relatively easy to compose the mathematical expression using MathType, then save the result as an image (i.e. a gif file) and then paste this image into the Dreamweaver file. This actually sounds more complicated that it is in practice. Creating the expression is straightforward (once one becomes familiar with MathType - which is easy) and the saving and retrieving of the image is just a few mouse clicks).
Here is an example:
Although my primary goal for the next year or two is to become reasonably proficient in calculus, and perhaps also in number theory, I am equally interested in the psychology of Learning. There are obvious extensions to education where my focus is on the use of technological tools to facilitate and enhance Learning.
The above paragraph is my personal road map to the future. It is also a map of the present and without much additional thought, a map of the past. The map is created by myself and is self-referential in that it is continually being referred to and modified as time progresses.
While I have stated my preference for a constructivist approach to Learning, and have used that as a guidebook for the last 30 years, I continue to enjoy reading, studying, and thinking about the topic. My three favorite authors are Jean Piaget, Lev Vygotsky and Jerome Bruner. My favorite sources for information specifically on constructivism are Ernst Glasersfeld, Paul Cobb and Seymour Papert. Cobb has written extensively on constructivism and mathematics learning whereas Papert has emphasized the role that technology can play in learning to think mathematically.
I recently purchased the book "Constructivism: Theory, Perspectives, and Practice. Second edition" (2005) edited by Catherine Twomey Fosnot. The second chapter provides an introduction and overview of constructivism. I began reading this chapter about an hour ago and quickly realized that I needed a map to envisage the overall structure of the chapter. I immediately quit reading and opened up my MindGenius software.
It only took about 15 minutes to create a chart based on the headings and subheadings of the chapter.
As a result of creating the image, I was aware that the chapter contained a fairly traditional and reasonable structure consisting of a brief review of theories of learning, and of biological approaches to learning, followed by a major section on constructivism and then a very brief section on education and a concluding paragraph.
The short concluding paragraph contains only 4 sentences, the third one being "It is a theory based on complexity models of evolution and development." For me, this is an important sentence. I am familiar with complexity models and the important idea of emergence but this is the first time that I have seen these models explicitly utilized as the framework for understanding constructivism. I suspect that I have been falling behind the current literature on constructivism - this is a bit of a wake-up call. I have yet to read the chapter, but I now have a good sense of what to expect. Reading for me is an example of a constructivist activity where I first try to imagine/create a structure that captures the overall framework and then I begin to actually "read" the words and yellow-highlight important passages. Yes, I mark up my books. The action of highlighting forces me to slow down and reread a section while also allowing me to review the book later more efficiently. If I decide it is appropriate I can also use this highlighting to efficiently extend my cognitive map that I have created with the MindGenius software. Then I can refer to this map at a later time when for some reason I wish to review this material.
My next step was to read the short section on education. Once again, a sentence jumped off the page at me: "Constructivism is a theory about learning, not a description of teaching." That is my interest and the central theme of this blog. I am interested in how I, the learner, can learn. And in my case this learning is totally self-directed. I am learning about learning and about calculus simply because I find it enjoyable. It engages my mind.
Now to sit back with a fresh cup of coffee, my mental map of the chapter, a yellow highlighter and the book and see what flesh I can attach to the bones.