Copasetic Flow

 I maintain that as an unschooling parent, I don’t teach, I facilitate. I try really hard to live by those words. One of the reason is the fringe benefits I reap by not ‘teaching’. Let me stop here  for a moment to summarize ahead of time. The point I’d like to make is that you don’t have to know the things to help someone else learn the things. Even better, frequently I find myself learning very cool new things I didn’t know before. Unschooling works, and it benefits everyone! Everything else below is rambling about math. Here we go! The kids and I have been talking about number bases for a few years now. Starting off in base two arithmetic—binary. It was an easy way to look at concepts without worrying about memorization. The addition and multiplication tables for that base have only four entries a piece. There’s not too much you have to memorize when the only numbers you have to work with are one and zero. A few years into this odyssey, the kids and I started looking into r...

9 y.o. Daize and I stumbled onto another math thing (I mentioned it briefly a few days back), and I’ve been aching to write about it so here goes. Daize and I started talking about binary numbers years ago.  At the time Daize wanted to build her own phone, so we talked about how computers work—in binary.  From there, things just kind of kept rolling becasue everything is easier in binary for us.  When you want to talk about addition, of multi-digit numbers for example, and how to carry and all that good stuff, it’s simple to talk about binary where there are only two numbers you need to know: zero and one.  With memorization out of the way, Daize and I could talk about carrying and adding to our hearts’ delight.  Same deal for multiplication which then led us to raising numbers to powers, and here we are! Raising numbers to powers—-for those who don’t mess with this stuff every day—iis usually illustrated by multiplying numbers by themselves.  So, 2...

"What!? Every teacher should be tested on the things they teach!" said the nine year-old unschooling kid this morning. We were talking about our word game that had turned into a math conversation and back into a word game, but not before I figured out why any number to the zero power is 1.  It turns out the answer isn't "Because I said so," as my 8th grade math teach would have had me believe.  There's a more intuitive answer that comes out right away if you talk about raising numbers to powers as shifting rather than multiplication or teacher-inspired mysticism.  It's simply that for the zero power of any number written in its own base, you just don't shift.  In other words you shift zero times.  The kid and I arrived on this purely by accident this morning because we had the time to play with numbers while we were talking about how many different words you could get out of a 26 letter alphabet for each size of word, (one characater, two chara...

Yesterday when I wrote about the 8 yo kid here learning algebra, I left out one caveat.  Math is kinda one of our things around the house.  We’re all immersed in it all the time.  My partner studied physics with a minor in math.  I studied engineering.  We both enjoy math, and consequently, we talk about math often, whether the kids are nearby or not.  So, math might be easier to pick up at our house just by virtue of being there. Here’s the thing though, when everyone frets “yes, but how will the kids learn math?”  Beyond the fact that if their interested in it they can find resources to learn it, beyond that fact, maybe it just doesn’t matter. Every family is into something, lots of things really.  If the family's cooking along without anyone knowing trigonometry, perhaps that’s because the things they’re passionate about just don’t need trigonometry.  And guess what?  The kids in that family will be immersed in those passions....

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 ...

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it. “Give me a math problem!” No. 1 asked Mom-person. “OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.” And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss. One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?” “I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?” One looked at me hopefully heading back into her mental math. I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?” I got a grin, and anoth...

Another conversation No. 1, our 6 year-old, and I had about number bases.  I'm not sure where I'm headed with all this.  No. 1 and I tend to talk as we ride through San Francisco on its various buses, trains, and cable cars... a lot.  It would be more concise to explain what we're doing math-wise by writing down a short description of the concepts.  It's not what we're actually doing though, so I'm not sure how much help that would be.  I'll just say for now, that I've discovered more about the math No. 1 and talk about by talking than I did by 'learning' it in school, so for now, I'll carry on. OK, so No. 1 and I had covered the basics of number bases .  You choose your base, you get that many numbers to place in a digit, and you have to include zero as a number.  Choose base 10, and you get our finger-counting system with ten different numbers represented by a single digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Choose base 3, and you ge...

Number 1 is learning division.  We're working on two different techniques.  I'm not sure which is working better, here they are. Method the first: When presented with the problem 12/4, we tell her to think about having twelve things she has to divide evenly between herself, her sibs, (Number 2, and Number 3), and a friend.  The downside of this version, is she has to guess.  The whole thing becomes experimental, (which has value in and of itself).  Number 1 draws twelve balloons, (she invented the technique), and then tries different groupings of the balloons until she finds one that's fair to all the sibs and their friend.  The upside is that there's a reason to want to divide in the first place; there's an application. Method the second: When presented with the same problem, we ask her how many groups of four she can make out of twelve things.  One of the upsides of this method is that it's mechanical.  No. 1 once again starts with...

Today’s math fun involves portions of yesterday's, but with a few more steps.  It may also—dare I say it?—involve partitioning!  I might be using the word partitioning in an incorrect way, and if so, then pardons please, (also, please let me know).  The question is, what are the fewest number of coins you need to make change for up to a dollar. Here's how I worked, using an iterative algorithm, (fancy words for: "I'm going to use the same trick over and over").  It was all about granularity of coins, and getting quickly from one amount to the next.  The quickest way, (where quick is defined by using the smallest number of coins), to get to a large amount of change is with large coins.  So, as we move around within 99 cents, the biggest step we can make is with a half dollar. Using yesterday’s method, we can fit one half dollar into 99 cents.  That leaves us with 49 cents left move around in.  The quickest way to make progress within that interva...

Working through the problems in Niven's book on combinatorics, I came across the following one that cleverly introduces least-common-multiples without saying any of those words.  The book asks the following question: How many numbers that are evenly divisible by 11 exist between 1 and 2000?  How many that aren't also evenly divisible by 3?  How many numbers that are evenly divisible by 6, but not by 4 exist between 1 and 2000? The hastily scribbled answer can be seen below, with each of the answers boxed in succession down the screen. By simply dividing 2000 by 11, we find out how many integers between 1 and 2000 are evenly divisible by 11.  In other words, we ask how many multiples of 11 can fit between 1 and 2000.  When we want to eliminate the multiples of 3, that's when the least common multiple comes in.  We already have the answer for all numbers divisible by 11, but how to eliminate those also divisible by 3?  By first asking what number...

Remember when "The New Math" had us all learning set theory in elementary school?  I hated set theory, and had no idea why we were learning it.  This week, years, (ahem, many), year later, I found out what it was all about: Sputnik!  While researching a book in the--as it turns out--aptly named New Mathematical Library, I found the historical link. The NML was a series of books commissioned by the School Math Study Group (SMSG).  A quick dive into Wikipedia turned up the interesting fact that the SMSG was formed after Sputnik flew over.  The US decided we were going to need far more scientists and engineers than we had available.  The solution to the problem back then made logical sense: increase the number of math literate students available.  Hence, the SMSG gave us the NML, a series of excellent books that: "make available to high school students short expository books on various topics not usually covered in the high school curriculum", and the...

I didn't make it into the lab today what with the holiday and all, but I did have time to read one of my favorite journals, American Mathematical Monthly from the  +Mathematical Association of America  .  The journal features a very interesting article[1] by Marshall Hampton[3] about cosine identities.  The article got me back to musing about solving for potentials with spherical symmetries and Legendre polynomials again[5].  I don't have time to work through this now, so I'm just recording my meandering thoughts here for future self, and anyone else that would like to take a look. Hampton writes down the generalization of the law of cosines for polyhedra rather than just the plane, (pun intended), old triangle.  Here it is $$0 = \sum_j\vec{n}_i\cdot\vec{n}_j\Delta_j = \Delta\left(i\right) - \sum_j c_{ij}\Delta_j$$ Where, $$c_{ij}$$ is the cosine between two faces of the polyhedra i and j, and $$\vec{n}_i$$ is a vector field normal to th...

Summary:  Having worked through the examples that looked the most difficult, today's notes contain examples that are pick-up work from the easy problems.  These are simple-ish tensor index identities, including the divergence of the position vector, the cross product of the position vector, the Laplacian of one over the displacement squared, and the curl of a gradient. $\nabla \cdot \vec{r} = 3$ $= \dfrac{\partial}{\partial x_i} r_i$ Keep in mind that $r_1 = x$, $r_2 = y$, and $r_3 = z$.  Using the rules of partial differentiation, when the partial operates on the variable it is with respect to it will return 1, and when it operates on any other variable, it will return 0.  The results sum to 3. $\vec{\nabla} \times \vec{r} = 0$ $=\epsilon_{ijk} \partial_j r_k$ $= 0$ For the $\epsilon{ijk}$ to evaluate to a non-zero result, $j$ and $k$ have to not be equal.  However, as discussed above, if $J \ne k$, then the partial derivative evaluates to zero. ...

OK, so that was quite the title.  I haven't done one of these in a while, but classes are about to start again and i figured I may as well get started deriving things again.  Plus, I had to do it for the can crusher magnet simulation code [1] for the experiment [3].  Here's what's really going on.  I have a Sage function that will give me the magnetic field in the z direction produced by a coil of wire that sits at z = 0 and a has a radius of 'rcoil'.  I'd like to know the magnetic field produced by the loop of wire along a circular path that is perpendicular to the plane of the current carrying coil.  A circular path perpendicular to the plane of a coil kind of begs for spherical coordinates, but the routine I have takes a z coordinate and a radius coordinate in the cylindrical coordinate system.  In the picture above, the circular path is shown, and the coil of wire is at the diameter of the circle and perpendicular to the page.  Note:  ...

Imagine living thousands of years ago in ancient Sumeria as a mathematician.  Your medium for storing infomration is cuneiform on clay tablets.  As you work, you stamp each equation into wet clay by making wedge shaped marks using the blunt end of a reed to make a finished document that looks like this (picture 1)[1] When your instructor tells you to investigate the properties of a table of let's say, a hundred numbers or so, you might sigh in resignation, and plan on having results by sometime next week. With the advent of paper and pencil, things become much easier.  There's still lots of work to be done, but the recording of the information so that it can be viewed and worked with is, comparatively speaking, a piece of cake. Finally, though, the computer comes along and getting a table of 100 numbers is more like playing.  With  +The SageMathCloud   the 100 number task suggested in the +Mathematical Association of America  video below...

The factor of kinetic energy over force in the focus equations got me thinking about work integrals. The work integral also led me to think about the vertex height as a projection back on the y axis. Why? Because work is only done in the direction tangent to the force, which in this case is in the y direction. The calculation of the projection angle was a bit messy, but wound up with a clean if rather obscure result. The following is neither here nor there, and I suspect will waste more time than it's worth at the moment, so I'm just including it as extra notes to go back to later. I already know there's a Gudermannian lurking in all of this. The function for the arc length of the parabola contains one. The projection angle consisted of a tanget half angle formula which also leads back to Gudermannians, (see https://en.wikipedia.org/wiki/Tangent_half-angle_formula#The_Gudermannian_function ).

I mentioned yesterday that the anti-Gudermannian had come up in several articles I’d read, but that the authors hadn’t recognized, or pointed out the anti-Gudermannians lurking in their formulas.  This is a brief set of instructions on how to recognize anti-Gudermannians in their natural state which can be pretty  messy looking.  The first thing you’ll need to know how to spot the multiplicative inverse of something known as the quotient function[5].  It looks like this,  but may be frequently disguised as The quotient function is very interesting in its own right and turns up all over in things like electrical transmission line formulas, quantum mechanical transmission and reflection coefficients, and in optically active material formulas.  For complete coverage of the quotient function, see Lindell in AJP[5], (sorry I couldn’t find an open access version, so you’ll have to head to your closest university library).  As cool as it is t...

I just spent the last several minute taking it easy at the end of my day reading a little bit about set theory.  It wasn't the dry and/or indecipherable stuff some of you might remember from first grade thanks to the 'new math'.  Nope, this article on set theory was on something called the Cookie Monster Number, the least number of moves that Cookie Monster can take to empty a set of jars of cookies if he can only take the same amount of cookies from every jar on each move.  Since I'm headed off to bed for the night, rather than rehash the subject here, I'll spend competitora little bit of time telling you about where you can read more on the subject. For a brief intro to the Cookie Monster Number, check out  +Richard Green 's mini-blog on G+.  Specifically, you'll want the post on Cookie Monster [1].  For those reading this on my blog as opposed to G+, the Facebook competitor from Google has actually become quite the mini-blogging site for scientists...

Suppose you have an integral that looks like the following seen during yesterday's quantum lecture:   Your professor turns, looks at you and says, "Who can do this integral?"  After a bit, no one answers and he grins and writes down that the answer is pi.  Then he waves it away as being easy as a contour integral.  Well yeah, but how?  Here's how... We already said we're going to a contour integral, so that mystery is solved.  We're going to move the integral into the complex plane, and choose a contour that skips the pole at u equal to zero.  Something like this The portion lableled B skips around the pole at u equals 0, the portion labeled A is on the real axis and stretches from negative infinity to infinity.  The slightly off-screen semi-circle labled C is the return path that in this case integrates out to zero at infinity, (Jordan's lemma and whatnot if you're into the details).  We kept the pole outside of the con...