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Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

January 2006 Mathematics Notebook
Introduction      
Goals
   
     

An Example of a "Learning Process" Journal
Tuesday January 31, 2006
Learning Log Number 5
8:00 am Ballina NSW Australia

The intent is to make some notes for Coincidences, Chaos, and All That Math Jazz.

I will begin with a mind map to sketch out the overall structure.



I like this. At a glance I can see that there are four main topics.

Once again, I am immediately faced with a decision as to how to organize my comments with the content. A related decision is how to handle the passages that I have yellow-highlighted in the book. I am going to begin with this type of inserted brown box for my comments. The actual mathematical content will appear in the light green background.

  • "... the cryptic equations, formulas, and graphs ... are not the only language of mathematics and it does not reside at the center of the subject. ... Ideas ... are truly at the heart of mathematics." [p. viii]

  • "... mathematics is an artistic pursuit." [p. viii]

Since ideas are at the heart of mathematics, it is important that I focus on clarifying these essential ideas for each topic or example that I encounter. In fact I need to create a special box to hold these ideas and I need to embed them into my mind maps. My original mind map at the top of this web page does this rather well: the four key ideas for this book are coincidences, numbers, chaos and infinity.

Not only are ideas important, but reflection about these ideas is critical. One must embed the ideas within one's own cognitive structure.

The above chart gives me a slightly better image of the relationships among coincidences, chaos and statistics. But these are the fundamental ideas: Coincidence, chaos and statistics.
  • "Coincidences surprise us because our intuition about the likelihood of an event is often wildly inaccurate." [p. 5]

  • "Coincidences ... arise when we ask so many questions that the vast numbers of opportunities make the chances for coincidences overwhelming." [p. 7]
    • Lincoln - Kennedy similarities
    • twin studies
    • monkeys at a typewriter
    • good karma
    • stock market experts
    • astrologers
    • throwing darts
    • lotteries
    • birthday parties

  • "Coincidences in twin studies may be significant or not. But what is clear is that interpreting such studies requires far more information than just the list of coincidences." [p. 9]

  • "Every particular poker hand we are dealt is exactly as unlikely as any other hand." [p. 10]

  • "... those monkeys would also type out every other book ever written, every book to be written, and every variation on them all." [p. 10]

  • "... the point is that if enough attempts are made, even very unlikely events will happen." [p. 16]

  • "In reality, each individual's life is actually a long sequence of incredibly unlikely events all strung together." [p. 19]

A nice first chapter. Many interesting examples and no equations.

The one difficult example is that of estimating the chance that there will be two people in a room of 40 that have the same birthday. The probability of one person having a match is small: 40/365, roughly 1/9. The probability of one person having no match is 1 - 1/9 or 8/9. The probability of 40 people having no match is 8/9 raised to the power 40. Any fraction less than one when multiplied by itself a large number of times gives a small number. Thus the probability of in a group of 40 people there is no match is very small. Said differently, the probability that there is at least one match is very high.

However to understand the above requires a basic understanding of probabilities, which few people have. Most people intuitively understand relative frequencies, but knowing when to add and when to multiply can be another example of rote memorization.

Making these notes has definitely helped bring the overall structure of the book as well as the idea of coincidences into a much clearer focus. 9:20 am

Total time for this session: 1 hour 20 minutes.