How to Read Mathematics

When I was a small kid, my grandfather used to teach me how to read English. I was born in India, so we did not get to reading English till we were 6 or 7. His technique was simple. He would mark portions of a sentence with a vertical mark (in pencil). He told us that we need to pause at the vertical bar, at comma and a pause a bit more at the period.

I wish I had seen this article on How to Read Mathematics, when I was young, since it was one of my favorite subjects. This is a profound statement, a quote in the article, made me take a quick read.

“Reading Mathematics … involves a return to the thinking that went into the writing” (ibid page 16).

Here are a list of things to do (in a nutshell).

• Don’t miss the big picture
• Don’t be a passive reader
• Don’t read too fast
• Make the idea your own
• Make sure that you are the intended audience

The Example of Mathematical writing towards the end, is a clear step by step dialog on applying these techniques.

For more math blogs try http://wordpress.com/tag/math/

A Tool Building Philosophy

I looked at WikiCalc a couple of months ago. A friend of mine mentioned it recently and I decided to take a look again. I read Dan Bricklin’s page on WikiCalc. I like his philosophy for developing software.

I do not have a business model that makes me want to force people through servers I control, so it gives me freedom in system architecture. I just try to make available a variety of architectures to choose from so you can use what best suits you, not just me.

There was a mention of ListGarden in the article, so I went to look for it in Dan’s home page and found this blog entry.

The basic idea is that the value of the Long Tail isn’t just that you make money selling an awful lot of unpopular things. The value comes from the fact that the long tail is the reason many people will choose one tool or system over another.

According to Dan, WikiCalc does the following (I reformatted the original paragraph into bullets for emphasis):

• The program can do many common spreadsheet operations while authoring, including numeric calculation and sums across ranges, formatting, and copy/paste/insert/fill (though it is not meant for heavy duty spreadsheeting or serious calculations).
• It handles freeform text in a wiki-like manner and works well with large blocks of text.
• It publishes through a template much like many blogging systems so you can easily create pages with a common graphic design and boilerplate.

You can see the combination of rich functionality, wikiness and blogness in his thinking. It is something to learn from.

Thinking About Thinking

Here is Edward De Bono on Thinking About Thinking.

It is not very often that anyone thinks about thinking. We do not think about walking or breathing. These things just happen.

Yet to watch our thinking in action can be very helpful and instructive. We can notice our usual habits. We can notice our weaknesses - and we can also notice our strengths.

Dealing with everyday matters is not a very good way to develop better thinking skills. The pressures are too great, and the search for a safe routine is simply too attractive.

Now I need to figure out how to “watch my thinking in action”.  I have made a few attempts but will be happy to hear about your own experience.

Curiosity Quotient

I was reading Jay Cross’s Insatiable Curiosity yesterday. It got me thinking about curiosity, in general. I went to Google for help. I tried to search for “curiosity quotient” but did not have much luck. So I broadened the search to “quotient”.

I came upon the following interesting ones.

IQ Page on Wikipedia. Following this, I found Flynn Factor which I did not know about. It was interesting to read about IQ Correlations.

To my surprise, Weired Quotient came up on top of the search. I am scared to take this test, afraid that my WQ may be a bit too high.

Social Quotient looked interesting. There is even a test from SELF magazine. Probably worth investigating.

Emotional Quotient (EQ) was the next interesting one. I had heard about it earlier. I think I even took a test once.

Charisma Quotient seemed interesting. But the link did not lead to anything interesting. So I let it go.

Nutritional Quotient is about your health and nutrition. Since I was munching on something not very healthy, I passed on that one.

By the time I got to Perseverance Quotient , I lost the perseverance to go on. I decided to get back to work.

After about 15 minutes and several links later, I decided that this topic deserves more time. I will come back to this later.

Mar 10, 2008: I came across Leadership Quotient.

How to Think like Designers?

Fast Company has fascinating articles. Their blog is a great resource. Here is an article on How To Think Like Designers.

Companies that want to win in the future have to understand how to make the most of design thinking. That means non-designers need to learn how to think like designers. Here’s how, from the six jurors who helped select our Masters of Design.

Some excerpts:

• Observe at the intersection of things
• A diverse read on culture is essential
• Defer judgment
• Focus on front-end opportunities — the end user — before any ideation begins
• Design thinking is iterative. It’s okay to be approximate in the beginning and then narrow and narrow
• Creative leaps of faith are part of how innovation happens
• It’s about discovery - prototypes provide a common language
• Business thinking is about crisp, binary conditions. Design thinking assumes that there are alternative options
• The word “design” comes originally from Latin “designare,” which means to draw the essence of. Design is a process of abstraction.

Mathematics and Programming

In this wonderful blog post, Ted Dziuba talks about learning programming like a mathematician. I am glad that the emphasis of mathetic thinking is coming back.

This article is a little more abstract, talking about the similarities between learning mathematics and learning a programming language. The goal of this article is to give programmers a framework by which they can effectively learn a new programming language, much in the way that a mathematician learns a new area of mathematics to the point where he or she can be effective.

In conclusion, Ted summarizes the essential similarity between math and programming.

1. Find the fundamental theorem, the sine qua non, of the language.
2. Understand how this fundamental theorem influences the structures and design decisions of the language, and how it is used to establish relationship between different parts of the language.
3. Practice and read documentation to the point where you can mentally picture how to fit the language structures together in the most efficient possible way to solve a problem.

Peter Norvig in his Teach Yourself Programming in 10 years says:

Learn at least a half dozen programming languages. Include one language that supports class abstractions (like Java or C++), one that supports functional abstraction (like Lisp or ML), one that supports syntactic abstraction (like Lisp), one that supports declarative specifications (like Prolog or C++ templates), one that supports coroutines (like Icon or Scheme), and one that supports parallelism (like Sisal).

I think the essense of programming is to build powerful mental models of the problem and the ability to deal with abstractions. A training in Math definitely helps in developing this skill.

Doing Algebra Mentally

A few days ago, I posted an entry It’s a magnificient time to know math. So I decided to keep an eye out for any resources that teach math or articles that show how to teach math. Today I came across this link in del.icio.us on Inner Algebra. It is the text of a book and looks very interesting.

Whenever I talk to a small group about Thinking and Problem Solving, I always try a few mental math questions. For example, what is 29*31. You can instantly answer this as 899 if you know a simple algebraic equation. Here are the mental steps:

Step-1: 29 * 31 = (30-1)* (30+1)

Step-2: (30-1)*(30+1) = 30*30 - 1*1

This based on the algebraic equation (a+b)*(a-b) = a^2-b^2

Step-3: 900 -1 = 899

You may not encounter these type of problems in every day life. But if you develop a mind set to solve problems mentally, life will be a lot more fun.

When I was in India in August, one of my favoire nephews (he is 11 years old) asked us to solve a simple equation. a^b+b^a= 100. What are the values of a and b? ( “a” to the power of “b” + “b” to the power of “a”).

Thinking - A Mind Map

I use this as one of the illustrations in a talk on “Thinking About Thinking”. Please click on the thumnail image to see a full graphic at Flickr. Thanks to Jay for pointing this out.

Discovery of Phosphorous or Why No Idea is Really Bad

A while ago, I blogged about Keeping an Idea Log. Just list down your ideas in a book or a blog or a document and come back and re-visit it. I tell people that even if the idea is not great, it can lead to other ideas.

I bought the book more than a year ago. I even read some portions of it. Since it was too bulky, I could not read it on my treadmill, which is where most of my reading is done nowadays. Then I located the audio verison -  A short history of nearly everything By Bill Bryson.

A oddball story caught my attention about the discovery of phosphorous. I could not believe it. So I came back home and tried it in my new favorite search engine Lexxe. Go ahead. Try it. Click on the link and type “How Phosphorous was discovered?” and see what you get.

Just proves my point. Even the most ridiculous idea some times leads you somewhere. Not every one does. But how do you know, unless you try it?

P.S: While I was searching for the story of discovery, I found this useful link from The Age.

Getting Started as a Thought Leader

Here is what Brian Fling has to say about Getting Started as a Thought Leader.

Start a Blog

Start Small

Create a Publishing Calendar

Know Your Audience

Turn on Comments

Write Guidelines

Stand By Your Principles

I can definitely see how getting started and starting small can help. The next steps have to be iterative. In the initial stages of blogging I just write to share. I am just thinking aloud.