Archive for the Philosophy of learning Category


Posted in Philosophy of learning, practical math on April 28, 2012 by Alex

One of my favorite mental exercises is something I like to call “ballparking.” The idea is to make a quick, rough estimate for some quantity using prior knowledge and some mental arithmetic. Here is an example I often use with my students:

Finding the radius of the Earth

It’s about 3000 miles from the west coast of the United States to the east coast.

There are 4 time zones in the continental U.S.

There are 24 time zones in the whole world, so the circumference of the Earth is about 18000 miles.

To get the radius, we just divide by 2\pi, or about 6, and we get about 3000 miles for the radius.

If you prefer the metric system, then there are about 2 km in a mile, so the radius of the Earth is about 6000 km. According to Wikipedia, the radius of the Earth is 6378 km, so my rough estimate was good to one significant figure.

You may have noticed I used the word “about” an awful lot in that calculation. This was by design. I do not know the exact distance from the west coast to the east coast, or the exact fraction of the 4 time zones taken up by that distance. I did not take into account the fact that we are not located at the Equator, but a little more than 30 degrees above it. I divided by 6 instead of 2\pi, mostly because it required less effort, and I was a bit off in my conversion from miles to kilometers. Despite all these imprecisions, I was still able to make an estimate that was “in the ballpark” and then get on with my life. If, for whatever reason, I needed a more accurate value for the radius of the Earth, I could go back and correct all these estimates, and get it. I could also use the ballpark estimate as a reference in case I make a mistake in the more laborious calculations needed to find the exact value.

One physicist who was famous for this type of calculation was Enrico Fermi. He was able to estimate the blast power of the atomic bomb Trinity by eyeballing how far it blew some pieces of paper miles from the blast site. Another classic question answered using this technique was “how many piano tuners work in the city of Chicago?” He was so good at this type of estimate, that most people refer to this technique as a “Fermi Calculation.” It was a method that helped scientists determine what a reasonable value for something would be before anyone had figured out what kind of math to use in order to determine the exact value.

Things for my readers to do:

What are some good problems for ballparking? I would like to use some of your suggestions for a class I am teaching this summer.


Integral of the Day

Posted in business ventures, Philosophy of learning on November 10, 2011 by Alex

A while back I had this idea of creating an integral of the day calendar.  This would be one of those calendars where you tear a page off every day and there’s a new integral waiting to greet you every morning.  The idea came from experiences I have had with some of my students.  Only a few months after taking a class on integral calculus, many of them simply forgot how to do all but the most basic integrals.  This makes things difficult when you actually have to do an integral to solve a physics or engineering problem.  The goal of the calendar was to give people a little practice doing integrals, so they would stay fresh and remember.

I have now come to the conclusion that the page-a-day calendar is a bad idea.  For one, it is a passive educational tool.  Anyone motivated enough to flip through the pages and do every one of the integrals would also have the motivation for remembering the integrals without flipping through the pages.  The product would not get to its target audience (people who actually need practice with integration) this way.  Two, I know of very few people who have ever made it through an entire year with these types of calendars.  The month of January goes by just fine, but somewhere around the middle of February the pages stop getting torn away, and the calendar just sits there collecting dust.  In fact, I am currently staring at a Dilbert page-a-day calendar stuck on February 19, 2008. 

I do have a solution to both of these problems.  Clearly, people in need of integration practice need to be forced into it.  It is their mathematical medicine to be taken with a grimace because it is good for them.  The modern student also needs a technological solution, and here it is:


Here’s how it works.  Every day the program calls up an integral from an online library and displays it somewhere on the phone’s screen.  If you touch the place where the integral is located, it calls up a box and the user can enter the answer.  If the answer is correct, the integral disappears for the rest of the day, and a new one appears the next day.  But what if the integral doesn’t get solved during the day, you ask?   The screen can now display today’s integral and yesterday’s unsolved integral at the same time.  All unsolved integrals since the download date will scroll across the bottom of the screen like a news ticker.

Another version can be released for students who really, really have trouble remembering how to integrate.  This version would be tied into the alarm clock on the phone.  When the alarm goes off, the integral to be solved appears on the screen along with a box to enter the answer.  The alarm keeps sounding until the correct answer is entered.  If an incorrect answer is entered, then the alarm either gets louder or switches to a more abrasive tone. 

I figure I can contract this product out to universities, wishing to help students currently on academic probation.  This product is a guaranteed money maker, so all you VC’s reading this blog, take note!

The Ultimate Math Text

Posted in Philosophy of learning, unfinishable projects on May 28, 2011 by Alex

I have a new ambitious project for myself.  I am going to create a math text that covers every subject of math from high school algebra, geometry, trigonometry, to calculus, boolean algebra, probability and statistics, linear algebra, differential equations, complex analysis, Fourier analysis, and whatever else I can think of.  Oh, and I want to make it available online.  For free.

I have some ideas about the structure of this text.

Derivations:  I will have derivations of useful theorems, formulas, and known problem solving strategies as the main part of the text.  Each will be self contained, and will have links to the prerequisite material found in other parts of the text.

Note I said “other parts of the text” and not “previous parts of the text.”  The text itself will not follow page numbers like ordinary books that are physically limited to them.  Certain areas of mathematics can be learned without knowing about certain other areas of mathematics.  For example, you can learn all about elementary linear algebra without knowing any calculus, but you do have to know how to do basic algebra.  Likewise, you can learn calculus without knowing any linear algebra.  The subjects are simply “independent.”  Now, if you want to learn how to solve systems of linear differential equations, then you have to know both calculus and linear algebra, and that will be reflected on the page for that.

Exercises and practice problems: These will consist of a few easy exercises that can serve as definition checks and concept checks.  They will be followed by more comprehensive problems designed to combine multiple topics together.  I would also like to include some open-ended long term type questions, but I’m not sure how to do that yet.  Since this will be available online (and for free!) I would like to have some interactive animations to help illustrate some of the topics.  I will include “historical problems of interest” to give the reader an idea of how these things came up in the first place, and some of the challenges that arose in the initial attempts to solve them.  Finally, I will include a list of open problems in the subject, to show the reader that there is always more to be done in a given field of study.

As I develop the text, I’m sure some of this will be modified greatly, and I would appreciate input from others on this project.  My ultimate goal is to provide a free resource for anybody who wants to learn any subject in math (and hopefully, eventually this format can be extended to any number of other subjects in the future.) Right now it’s just me doing this, although there are others who have created amazing online resources such as the stack exchange .  I don’t know how long this is going to take, either, but considering I spend a good chunk of my free time doing math problems for my own amusement anyway, I may as well amuse myself to someone else’s benefit.

My experience with the guitar

Posted in Philosophy of learning on March 26, 2009 by Alex

When I was nineteen years old I decided I wanted to learn to play the guitar. I went out and picked up an acoustic guitar, and started playing around with it. I had played a few musical instruments as a kid, the piano, the recorder (I could play Take me out to the ballgame with the best of them), the glockenspiel and the autoharp, but didn’t really have any formal music training. I didn’t take any guitar lessons, and I couldn’t read sheet music, but I could sit down and listen to songs that I liked over and over again and attempt to replicate the sounds on my CD’s with the guitar. I started with some easy songs, and gradually worked my way up to slightly more complex sounds. I learned how to read tablature, where to put my fingers to play all the various chords, learned a couple of scales (mostly just blues scales) and a few chord progressions. I never took any lessons for this; I just played around with my guitar in my spare time. At some point I got good enough to where I could pick up new songs relatively quickly, but it still requires a lot of effort for me to think about where I should put my fingers, which chord comes next, etc…

Today I am a reasonable facsimile of a guitarist. However, if you listened to me play, you would not mistake me for a professional musician, even though I can, in principle, do everything on the guitar that a professional guitarist can do. So what separates me from, say, Jimi Hendrix? Musical talent, you may say. There may be something to that, but I think that view is a bit of a cop-out. I would have a hard time learning any task if, in the back of my mind, I was always thinking “I’ll never be as good as so-and-so.” It would be counter-productive for any kind of learning. A more practical approach to take, in my view, would be to ask “how can I be better than I am right now?” I don’t a priori know my ceiling as a guitarist until I have actually reached that ceiling. The best I can do is to improve my skills in increments, and over time view the aggregate results.

Even though I wouldn’t compare my guitar playing talent to Jimi Hendrix’s (I don’t have any idea how to quantify such a thing), I can use him as a model for improving my guitar playing ability. He was a really good guitar player, how did he get so good? Here I have to speculate a bit, but I can imagine roughly how it went. To start off, he probably started a lot younger than me. That’s strike number one against my prospects, but not insurmountable. The second, most important piece of information would be to know how much he practiced as he learned to play. Since I can’t even picture the guy in my head without a guitar near by, I would imagine he practiced a lot. He probably practiced almost every free moment that he had, every time there was a break in a conversation he strummed out a little tune. Every time he had an hour to kill sitting at home he would try to find interesting combinations of notes. Maybe he even took the guitar into the bathroom with him, or played when he was driving around, or strummed an uplifting tune after he tasted how good his mashed potatoes were, I don’t know. I may be exaggerating just how much he probably practiced, but the point is he had to have spent a lot of time practicing in order to become as good as he was.

The lesson, I suppose, is that if I want to be a better guitar player, I’d better practice… a lot. If I am really serious about improving to the level of one of the greatest guitarists of all time, I had better spend every waking moment practicing. Even if I want to be a passable guitarist, like someone playing at open mike night at a coffee shop, I had better practice. So why don’t I? I have the usual assortment of excuses: I work all the time, so I don’t have time to practice, I don’t want to bother the neighbors with the horrible sounds the guitar will make while I am still an amateur, I’m too tired, my fingers are bleeding, and so on.

But I suppose the real reason I don’t practice that much is that I just don’t want to. Don’t get me wrong, I would like to be a great guitarist, but I guess don’t want to be one enough to put in all the hours of work. Every one of the usual excuses can be overcome. I could very easily make time in my seemingly busy schedule for five minutes of guitar here, ten minutes there, and get all the guitar playing motions into my muscle memory. I really shouldn’t care what my neighbors think of these sounds; in fact they should be honored to be listening to a future rock and roll hall of famer in his formative years. I should drink a cup of coffee, and I should want to play until my fingers are bleeding. Maybe I won’t become the best guitarist ever, but I could certainly become better than I am now, and nobody knows how good I can possibly be… with enough practice.