Dorai’s LearnLog

March 4, 2008

Study: Impact of Improving Cognitive Skills

Filed under: Mathematics — dorai @ 9:13 am
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A Study Finds Sharp Math, Science Skills Help Expand Economy.

Increased years of education boost economic growth — but only if students’ cognitive skills, as measured by math and science tests, are improved as a result, a new study says.

The study, released in this spring’s issue of Education Next, an education-policy journal, concluded that if the U.S. performed on par with the world’s leaders in science and math, it would add about two-thirds of a percentage point to the gross domestic product, or the total value of goods and services produced in a nation, every year.

People think of Math as a subject to learn. But we may be missing the point. To me it looks more like a basic skill for people to learn. Here is Jagjit Singh on Great Ideas of Modern Mathematics, a book first published in 1959.

It is true that physical sciences, such as physics and astronomy, did use a good deal of mathematics, but even in these sciences one could get along and often make useful contributions without it.

Nowadays, even descriptive sciences, e.g. biology, zoology, genetics, psychology, neurology, medicine, economics, philogy, etc., have begun to employ elaborate mathematical techniques.

Math can be made easy to learn by great teachers. On a more personal note,  I had wonderful teachers from elementary school till the end of my undergraduate (engineering) course. The early teachers were such an inspiration. They were mostly responsible for my interest in Math and later in Sciences and Engineering.

November 27, 2007

Math: Developing a Mindset

Filed under: Ideas, Mathematics — dorai @ 5:35 pm
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You keep hearing about Mathematical thinking. What is it? How do you develop a Mindset for Math?

Khalid has a nice post on How to Develop the Math Mindset.

Math uses made-up rules to create models and derive relationships. When learning, I ask:

  • What relationship does this model represent?
  • What real-world items share this relationship?
  • Does that relationship make sense to me?

I will add a couple of more:

  • How does one develop a mind-set for thinking beyond mere numbers, formulas and low level concepts.
  • How can we take these insights that come out of that mindset and apply to real problems

One of the slides I used to have on my “Thinking About Thinking” talks was to ask the audience (mostly CS students) to do a few of the following multiplications, mentally.

19 X 21

25 X 15

Very few actually find the simple algebraic patterns, till you point them out.

That brings us to one more insight (not my own):

  • A lot of problems can be solved by looking for patterns and applying some existing knowledge

The more mental models we build, the easier it is to apply them.

July 15, 2007

Little Innovations: Math Lab in Your Cell Phone

Filed under: Creativity, Ideas, Innovation, Mathematics, Tools — dorai @ 9:59 am
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From ZDNet’s Emerging Technology Trends

Israeli scientists have decided to put a math lab in your pocket. They developed a library of math modules which can be installed on almost cell phones available today. So you’ll be able to see graphs or solve equations on your phone while on a train or a bus ride. You’ll also be able to send graphs or formulas by SMS to other students — and to send the results of your exercises to your teacher.

This may just be the beginning. With cameras, ability to play flash, SMS and GPRS, these may become the new devices for augmenting learning.

September 12, 2006

Seven Reasons I use Wikipedia

Filed under: Mathematics, Wikis — dorai @ 5:51 am

About a couple of months ago we were discussing about Wikipedia and Encyclopedia Britanica. My brother-in-law was wondering why people would use Wikipedia instead of the Encyclopedia Britanica (which is perceived to be more accurate). I can cite my reasons.

  1. Wikipedia is accessible (if you have access to internet)
  2. Wikipedia is more easily searchable (if something is in Wikipedia, it comes up fairly high up on popular search engines)
  3. It is easy to provide a pointer to a wiki page on the topic. My blog on Palindromic Numbers was based on following such a link from a discussion forum.
  4. Wikipedia is dynamic and growing rapidly (it is constantly being updated). I track some of my favorite pages using InfoMinder
  5. It is free and the content is under GNU Free Documentation License.
  6. Wikipedia-ness (if there is such a thing) inspires lots of other similar useful projects:

    Wiktionary
    Dictionary and thesaurus

     

    Wikinews
    Free-content news

     

    Wikiquote
    Collection of quotations

     

    Wikibooks
    Free textbooks and manuals

     

    Wikispecies
    Directory of species

     

    Wikisource
    Free-content library

     

    Wikiversity
    Free learning materials and activities

     

    Commons
    Shared media repository

     

    Meta-Wiki
    Wikimedia project coordination
  7. The back links to a page some times provide a rich source of references. To find what refers to a page, try clicking on the link to the left titled “what links here”. For example, for the Math page here are the backlinks.

Feel free to pitch in and add comments on why you like or don’t like Wikipedia.

August 11, 2006

A Mathematician’s View of God and the Goal of Life

Filed under: Books, Inspiration, Mathematics — dorai @ 8:05 am
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Here is a fascinating piece I read in My Brain is Open. I made a simple substitution of one word to be politically correct.

When asked what is the purpose of life, Erdos would reply: “To prove and conjecture and to keep the God’s score low”. He imagined that humans were constantly engaged in a deadly serious game with God in which, “if you do something bad the God gets at least two points. If you don’t do something good which you could have done, the God gets at least one point. And if you are okay nobody gets any point. Humanity cannot win this game, so that the goal of life is not victory. “The aim is to keep the God’s score low.”

It is simple enough. I especially like the part about “if you don’t do something good which you could have done”.

July 29, 2006

Book: Letters to a Young Mathematician

Filed under: Mathematics, Thinking — dorai @ 12:04 pm

I just finished it yesterday. I really enjoyed reading it. Here are some of my favorite chapters.

  • Why Do Math?
  • The Breadth of Mathematics
  • Surrounded by Math
  • How Mathematicians Think
  • How to Learn Math
  • Mathetmatical Story Telling
  • Pure or Applied?
  • Where Do You Get Those Crazy Ideas?
  • How to Teach Math
  • Is God a Mathematician?

The last chapter “Is God a Mathematician” is all about Symmetry. I really love this chapter. Here are a few quotes from it.

“God and mathematics both strike terror into the heart of the common humanity, but the connection must surely run deeper…. You needn’t subscribe to a personal deity to be awestruck by the astonishing patterns in the universe or to observe that they seem to be mathematical. Every spiral snail shell or circular ripple on a pond shouts that message at us.”

“What are the laws of nature? Are they deep truths about the world, or simplifications imposed on nature’s unutterable complexity by humanity’s limited brainpower?… Are mathematical patterns really present in nature, or do we invent them? Or, if real, are they merely a superficial aspect of nature that we fixate on because it’s what we comprehend?”

“Because we cannot experience the universe objectively, we sometimes see patterns that do not exist.”

“One of the simplest and most elegant sources of mathematical pattern in nature is symmetry. Symmetry is all around us. We ourselves are bilaterally symmetric…. There are symmetries in the structure of the atom and the swirl of galaxies.”

“Imagination is an activity of brains, which are made from the same kind of materials as the rest of the cosmos…”

“Symmetry is deep, elegant and general. It is also a geometric concept. So the geometer God is really a God of symmetry.”

This book, in my mind at least, raises more questions than it answers. But it provides  lots of hints on where to look, and what to look for.

July 23, 2006

Work and Satisfaction

Filed under: Books, Mathematics — dorai @ 7:34 am

Your satisfaction must come from the high you get when you suddenly, for the first time understand the problem you are working on and see your way to a solution. I use the word “high” advisedly. You need to be something like an addict for this feeling to provide sufficient recompense for all that work.

From Ian Stewart in “Letters to a Young Mathematician” in the chapter on “The Career Ladder”.

Also:

Definition of Work 

Conceptual Difficulties

Filed under: Books, Mathematics — dorai @ 7:24 am

Leaving conceptual difficulties unresolved s a bit like using new credit cards to pay off the debts on the old ones. You can keep going like that for sometime, but eventually the bills come due.

From Ian Stewart in “Letters to a Young Mathematician

July 15, 2006

Math Inside

Filed under: Books, Inspiration, Mathematics — dorai @ 7:22 am

From “Letters to a Young Mathematician” by Ian Stewart.

I sometimes think that the best way to change the public attitude to math would be to stick a red label on everything that uses mathematics. “Math inside.” There would a label on every computer, of course, and I suppose if we were to take the idea literally, we ought to slap one on every math teacher. But we should also place a red math sticker on every airline ticket, every telephone, every car, every airplane, every traffic light, every vegetable…

Vegetable?

Yes. The days when farmers simply planted what their fathers had planted, and their fathers before them, are long gone. Virtually any plant you can buy is the outcome of a long and complicated commercial breeding program. The whole topic of “experimental design” in the mathematical sense, was invented in the early 1900s to provide a systematic way to assess new breeds of plants, not to mention the new methods of genetic modification.

Wait, Isn’t this biology?

Biology, sure. But math, too. Genetics was the first parts of biology to go mathematical.

This was in the very first chapter “Why do Math?”, in the book. One of the most inspiring books I have laid my hands on. I don’t even recall, how I came to know about this book. Probably a blog. A big “Thank You” to the person who mentioned this. I am having the time of my life reading this book.

April 21, 2006

Mathematical Experience

Filed under: Mathematics — dorai @ 8:02 am

We often hear that Mathematics consists mainly in "proving theorems". Is a writer's job mainly in writing sentences?

A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

To master mathematics is to master an intangible view, it is to acquire the skill of a virtuoso who cannot pin his performance on criteria.

By Gian-Carlo Rota on "The Mathematical Experience" by Philip J.Davis and Reuben Hersh

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