Copasetic Flow

Another question I see come up all the time in the context of homeschooling and especially unschooling is “How do you teach  math?”  There are lots of different ways.  I know people that use curricula, I know kids that attend math circles where they work out math problems with other kids, I know kids that learn math as they run into a need for it in the real world. When the real world example pops up, people tend to ask, “Yes, but how will they learn complex math like algebra and trigonometry?" To which I respond, “The kids here learn those things mostly by talking.” And that’s how we do it.  Talking.  Usually in  tiny snippets at a time .  My partner and I started working with the kids on math as we hung around in coffee shops .  We'd ask the kids—now 8 y.o. No. One, 6 y.o. No. Two, and 4 y.o. No. Three—questions about adding or subtracting.  They'd generally work them out on their fingers.  This worked great all the way through mul...

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went. I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back. "I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :) After 2 had worked ...

It was Christmas time when the kid and I started talking about number bases.  The air outside was more than normally chill for San Francisco.  In the winter here, thanks to the fog, the air’s still damp, so cold feels really cold, but the chill was compensated for by town being even prettier than usual; sporting all it’s holiday lights.  People on public transit were more tired than in other seasons; the holiday rush, and December’s early sunsets combined to make a sleepy, almost lethargic atmosphere.  The season also seemed to have made our generally friendly fellow bus riders even a little more affable.  Smiles swept across their faces a little more quickly.  People scooched and shuffled to help each other get into the crowded buses. The kid and I were on one of these buses, returning to the house from who knows where when, mostly just to liven up the ride, and with only the slightest hint of an ulterior motive, I asked her, “How many numbers can you ma...

It's that time of year again when physics students everywhere are deriving the spherical and cylindrical del, nabla, gradient, or Laplacian operators.  Every derivation I saw prior to this week involved lots of algebra and the chain rule... even mine .  Fortunately for me, a comment on my derivation, and a homework assignment from Rutgers  [pdf] led me to a far simpler and more intuitive way of doing things.  You just start from the differential displacement in a given coordinate system and go from there. The differential displacement in spherical coordinates is: The element in the r direction is easy to understand.  A small displacement along the r direction is represented as dr.  The theta and phi displacements might not be as obvious.  The graphic to the left illustrates what's going on.  With small displacement along the theta direction you're moving along a circle with radius r.  The distance you've moved is equal to the len...

On page 123 of “Gropus and Their Graphs” by Grossman and Magnus I struggled understanding Theorem 6 and the test it provides. The theorem read more clearly for me with the following substitution: “the set K that contains all elements of G such that...” becomes “the set K containing every element of the group G such that...” The following paragraph that contained the test was easier for me to understand with the following substitution: “If f maps all elements onto I...” becomes “If f maps all elements of G onto I...” Hope this helps! Any comments or questions?

Just recording my notes on learning group theory into a series of short highlight videos.

We’ve previously looked at how to derive divergence for cylindrical coordinates . If you’re like me though, knowing the rather lengthy derivation won’t help you understand or memorize the resulting formula. So, let’s take a look at why the result makes sense. The formula for the gradient of a function in cylindrical coordinates is: Why is the factor of 1/r in the phi term? Remember what question the divergence is asking. We want to find out the amount the function changes vs. a small change in distance along each coordinate’s direction. For coordinates that actually correspond to distances, like x, y, and z of Cartesian coordinates, or r and z of cylindrical coordinates, this is straightforward. The change in the coordinate corresponds to the change in distance along the coordinate. For coordinates that correspond to angles in the cylindrical coordinate system, there’s an extra twist. The direction of phi always points tangent to a circle centered on the z axis. The small chang...

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain ...