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

Seven Ways to Use Mindmaps

Not all these are my ideas. I have read about most of them.

1. As a note taking tool

While listening to a talk or a radio program (like NPR), write down concepts. Scatter them on the page. Start connecting them. Later, after the talk/program is done, name the associations. I use a blank white sheet with no lines.

2. As a thinking tool

Start with the central idea/theme in the middle of the page. For about 5-10 minutes just write down every other related idea that occurs to you. Draw the connections. Take a look the the map and if needed, re-organize it as another diagam.

3. As a problem solving tool

Write down the problem in the center. Write every associated problem that may occur if this problem is not solved on one half of the page. Write every possible solution you can think of (for the initial problem) on the other half of the page.

4. As a brainstorming tool

This is done with a group. Write down the idea/problem you want to brainstorm about. Write it in the middle of a whiteboard. Let people shout out whatever comes to their mind. Write those ideas. Connnect them to the central idea and to other related ones. A visual dispersion of topics has one advantage. It has no implied ordering which allows you to think more clearly. If the mindmap gets too dense (more than 7-9 nodes), repartition it.

5. As a To-do list

Draw a simple map first with three nodes – short term, medium term, long term. Take each one of your todo list item and connect to one of these. Explore other alternative ways of classifying your todo lists:

6. As a learning tool

After reading each chapter, write down a mind map of the chapter. Then write down a consolidated map of the whole book (or topic). Use it to refresh your memory once in a while. You can also use the nodes of the maps as pointers to important resources.

7. Improvement Map

Make a list of ideas for improving things about you and around you. Pick a few and transfer them to the todo list (not-urgent but important category).

Go to Mind Mapping Resource Center for more ideas

Web as a Personal Learning Center

In a presentation to eLearning forum, Stephen does an amazing job of talking about the convergence of various technologies and trends.

The Interactive Web according to Stephen is changing.

From medium to platform

From static to dynamic

From a communication medium to a network of interactions, resources

From centralized to distributed resource descriptions

In addition to demystifying web 2.0, Stephen describes the new challenges germinating from these shifts.

Google Page Creator

The pieces are falling in place one by one. An online database, a portal, a page creator, a blog hosting service. What will be next? A wiki? Backed by a set of APIs the elements required to build online applications, websites and portals are taking shape at Google.

I had some difficulties in creating a page with an image. Hopefully will be fixed soon. Here is my page (I spent a couple of minutes).
Search Engine Watch has an article on the product here. TechCrunch has a mini-review.

XML Everywhere

In this outspoken, InfoWorld interview, Tim Bray covers a wide variety of subjects related to XML. Tim Bray, is one of the co-authors of the original XML specification. One of his most notable contributions is The Annotated XML Reference. In this interview he talks about everything XML – webservices, RSS, xml for data interchange, xml document formats etc. Here are a few snippets from the interview.

We thought we were building something that would enable somewhat more efficient publishing of Web pages to multiple devices and so on, and the explosion of creativity and energy around XML has wildly exceeded anything we could have possibly dreamed.

XML and Interoperability:

XML is basically being used everywhere that you have heterogenous computers, that’s to say heterogeneous technology where you’ve got different operating systems or different databases or different anythings, and you need to interchange data, well, XML now seems to be pretty much the de facto way to do that.

XML’s visibility:

So XML is essentially everywhere, but at the same time, it’s more or less invisible because [of] all that’s happening behind the scenes where you don’t see it. I think that the recent interest in XML as an office document format, both in the standardized version of ODF (Open Document Format) and the Microsoft stuff is a harbinger.

XML and RSS:

The world’s most successful XML application right at the moment is RSS. In terms of the volume of data, the number of feeds, and that is a huge source of change, not just on the technology front, but also culturally.


We’re[he is referring to Sun Microsystems] not promising to interoperate with whatever the WS-* castle in the sky is, we’re promising to interoperate with what Microsoft shipped. And we will do that. And I think IBM will too. So the core big picture vendors will actually interoperate with each other.

Blogs Accuracy:

InfoWorld: Of course there’s blogs where you can’t really trust he accuracy of everything that’s out there.

Bray: You know, I hate to say it, but I have read gross inaccuracies in your publication.

My Job is to Live in the Future

My job is to live in the future 20 years from today

Mr. Moss, who is 56, expects that technology will change society more profoundly in the next 20 years than it has in the past 20, by easing the burden of aging and improving communication, health care and education. He is enticed, for example, by the concept of cellphones that silence themselves upon entering a theater, or phones that convey the urgency of a call from an elderly parent at an unusual time of day.

The emphasis on the type of applications of technology will definitely shift more rapidly in the next 20 years. For the cellphone stuff Mr.Moss mentions, however, we may not have to wait that long.

Three Billion Mobile Phones by 2008?

The two billionth mobile phone subscriber worldwide signed up at the end of September, 2005, says ABI Research. The market data firm expects the three billionth mobile phone subscriber to sign up sometime before the end of 2008.

From Mobile Phones: More than 2 billion served in 2005

Just think of the impact this will have on the opportunity for new class of applications! Definitely Mind Boggling. Here is a link to Windows For Devices.