### Lonely days are gone, I'm a-coming home

I just got a letter from SUNY Brockport.

It is a very heavy letter.

It is a very heavy letter.

I just got a letter from SUNY Brockport.

It is a very heavy letter.

It is a very heavy letter.

I should start getting used to talking about me, especially since I'm doing some interesting stuff that I'll probably have to be talking about before long.

I've been spending the last handful of weeks as a volunteer math tutor for the Rochester Educational Opportunity Center, helping folk to get their GED and qualify for an LPN license and similar sorts of job training and college prep opportunities. The organization I am working for is a non-profit affiliated with one of our local state universities that does all this training tuition-free for qualified prospective students (and it seems like pretty liberal standards to establish the economic need) and it really seems to do a lot of good in helping people to chart out a better future. For me as well: it's not a paid position and doesn't even count as field work for a Masters in Education (even though the affiliated university is the one at which I'll be studying), but it'll be a leg up to have the experience in both the schmoozing sense and in being actually prepared for a room full of adolescents.

And so far it's been going really well. The staff is thrilled to get all the help they can, and they've been very helpful and welcoming. The students have been very generous with their praise as well, and even though there is selection bias in my hearing primarily from the sorts of student that I am helping, I can also objectively see that they are growing from one class to the next. That has more to do with my students being eager to be taught and willing to invest the time and brainwork than in the role that I'm playing, but again I am a critical catalyst in that reaction and if I don't give myself credit for my part then I can hardly expect people down the road to independently offer me future opportunities.

At the same time, it has been very challenging and I have yet to even comprehend the scope of the challenges. Math anxiety is a significant hurdle, and I don't have any specific strategies for dealing with it. (Nor do I have any personal experience with it at this level no matter how much I know about other sorts of anxiety.) I have also come to appreciate that people who talk about anti-racism math have their thumb on a very real and important problem. Again, I see it when I see it, but I'm still soaking in it and I've got to figure out how to confront that. 95% of my students are women, 90% are people of color, and I'll go ahead and guess that at least 95% of them aren't commuting from my upper middle-class neighborhood. In short, the only thing that comes easily to me in this class is the math.

But those last two paragraphs are both true and coexist in balance. My best isn't good enough, and my students deserve even better than my best. But that doesn't mean that I shouldn't do what I can do. The alternative to me teaching this class isn't Jaime Escalente taking over the class for me; the alternative is no class at all. I need to fail a little bit better every day, because a lot of people are still going to succeed in spite of me because of me.

I've been spending the last handful of weeks as a volunteer math tutor for the Rochester Educational Opportunity Center, helping folk to get their GED and qualify for an LPN license and similar sorts of job training and college prep opportunities. The organization I am working for is a non-profit affiliated with one of our local state universities that does all this training tuition-free for qualified prospective students (and it seems like pretty liberal standards to establish the economic need) and it really seems to do a lot of good in helping people to chart out a better future. For me as well: it's not a paid position and doesn't even count as field work for a Masters in Education (even though the affiliated university is the one at which I'll be studying), but it'll be a leg up to have the experience in both the schmoozing sense and in being actually prepared for a room full of adolescents.

And so far it's been going really well. The staff is thrilled to get all the help they can, and they've been very helpful and welcoming. The students have been very generous with their praise as well, and even though there is selection bias in my hearing primarily from the sorts of student that I am helping, I can also objectively see that they are growing from one class to the next. That has more to do with my students being eager to be taught and willing to invest the time and brainwork than in the role that I'm playing, but again I am a critical catalyst in that reaction and if I don't give myself credit for my part then I can hardly expect people down the road to independently offer me future opportunities.

At the same time, it has been very challenging and I have yet to even comprehend the scope of the challenges. Math anxiety is a significant hurdle, and I don't have any specific strategies for dealing with it. (Nor do I have any personal experience with it at this level no matter how much I know about other sorts of anxiety.) I have also come to appreciate that people who talk about anti-racism math have their thumb on a very real and important problem. Again, I see it when I see it, but I'm still soaking in it and I've got to figure out how to confront that. 95% of my students are women, 90% are people of color, and I'll go ahead and guess that at least 95% of them aren't commuting from my upper middle-class neighborhood. In short, the only thing that comes easily to me in this class is the math.

But those last two paragraphs are both true and coexist in balance. My best isn't good enough, and my students deserve even better than my best. But that doesn't mean that I shouldn't do what I can do. The alternative to me teaching this class isn't Jaime Escalente taking over the class for me; the alternative is no class at all. I need to fail a little bit better every day, because a lot of people are still going to succeed in spite of me because of me.

Entry tags:

I've been spending the past few days playing with CaRMetal, which is an awesome application for generating mathematical figures. Here's a problem that I just finished working on.

In this diagram, the three internal circles all have the same radius and share the same point Q. ABC is a triangle "circumscribing" the three circles in the sense that each side is the unique line tangent to two of the three circles without crossing the third circle. R is the center of the circle that passes through the three vertices of the triangle (partially shown), and P is the center of the circle that is tangent to the three sides of the triangle (which isn't shown). In the diagram shown, P, Q, and R are on the same line, or at least they appear to be. The homework problem that was associated with this diagram was showing that this was the case, and that it works out this way no matter how you choose three circles of the same radius that share a point.

(For the record, I barely have the first clue of how you would show this to be true. I had some nasty punitive homework in college.)

The thing that makes CaRMetal a home run is that this diagram wasn't drawn so much as constructed. That line AC isn't "eyeballed" to sort of look like the tangent to those two circles; there are a whole structure of hidden points and lines that are all designed as being the perpendicular of this line through that point or the intersection that that line with the other circle and so on, so I can have confidence that the diagram is drawn "to scale". The thing that Makes CaRMetal a grand slam is that inside the application I can click and drag those three circles around and all of the dependent lines and points will recalculate themselves, so I can swing the circles around and watch the line segment PR move around in real time but still always pass through Q. Whoa.

So I've been rather giddily making diagrams to supplement my old college math homework. It might not turn out to be the best tool to draw graphs, but at least I should be able to draw proper pentagons instead of hand-drawn misshapen messes. Shiny.

In this diagram, the three internal circles all have the same radius and share the same point Q. ABC is a triangle "circumscribing" the three circles in the sense that each side is the unique line tangent to two of the three circles without crossing the third circle. R is the center of the circle that passes through the three vertices of the triangle (partially shown), and P is the center of the circle that is tangent to the three sides of the triangle (which isn't shown). In the diagram shown, P, Q, and R are on the same line, or at least they appear to be. The homework problem that was associated with this diagram was showing that this was the case, and that it works out this way no matter how you choose three circles of the same radius that share a point.

(For the record, I barely have the first clue of how you would show this to be true. I had some nasty punitive homework in college.)

The thing that makes CaRMetal a home run is that this diagram wasn't drawn so much as constructed. That line AC isn't "eyeballed" to sort of look like the tangent to those two circles; there are a whole structure of hidden points and lines that are all designed as being the perpendicular of this line through that point or the intersection that that line with the other circle and so on, so I can have confidence that the diagram is drawn "to scale". The thing that Makes CaRMetal a grand slam is that inside the application I can click and drag those three circles around and all of the dependent lines and points will recalculate themselves, so I can swing the circles around and watch the line segment PR move around in real time but still always pass through Q. Whoa.

So I've been rather giddily making diagrams to supplement my old college math homework. It might not turn out to be the best tool to draw graphs, but at least I should be able to draw proper pentagons instead of hand-drawn misshapen messes. Shiny.

Entry tags:

- ← amusing links →,
- ← geekery →,
- ← mathematics →

Reading the xkcd forum has turned me on to the awesomesauce that is Manufactoria.

The concept is simple at the start. Your job is to build a machine to test robots to ensure that their "program" (expressed as a string of red and blue dots) meets certain characteristics. For instance, the layout above tests to see if a robot's program ends with two blue dots. The conveyor belts pushes robots (that start in the circle at the top) into a neighboring square in the direction indicated, and the branches eat the first character in the string and push the robot in the indicated direction (or in the direction of the gray arrow if the string is empty). In this case, you can see that if the input string ends with two blue dots, the robot will be pushed to the acceptance square in the bottom, otherwise it will fall on the floor and be destroyed.

So it starts off as a fun model for deterministic finite automata, and that's cool enough. But over the course of the 31 levels, the ability to print dots at the end of the input string and a greater range of colors is added, and then you've got an entire range of formal grammar problems available for challenges. And once you've solved a problem, you can go back and tinker with it to make it smaller or faster as you wish. Or you can keep going through some pretty dense challenges that get hard to fit on the factory floor, much less read. That above example was maybe level 10 or 11 of the set, and let's call this level 29:

Still have two more challenges to figure out myself, but it's a great time if this is the sort of thing that you're into.

The concept is simple at the start. Your job is to build a machine to test robots to ensure that their "program" (expressed as a string of red and blue dots) meets certain characteristics. For instance, the layout above tests to see if a robot's program ends with two blue dots. The conveyor belts pushes robots (that start in the circle at the top) into a neighboring square in the direction indicated, and the branches eat the first character in the string and push the robot in the indicated direction (or in the direction of the gray arrow if the string is empty). In this case, you can see that if the input string ends with two blue dots, the robot will be pushed to the acceptance square in the bottom, otherwise it will fall on the floor and be destroyed.

So it starts off as a fun model for deterministic finite automata, and that's cool enough. But over the course of the 31 levels, the ability to print dots at the end of the input string and a greater range of colors is added, and then you've got an entire range of formal grammar problems available for challenges. And once you've solved a problem, you can go back and tinker with it to make it smaller or faster as you wish. Or you can keep going through some pretty dense challenges that get hard to fit on the factory floor, much less read. That above example was maybe level 10 or 11 of the set, and let's call this level 29:

Still have two more challenges to figure out myself, but it's a great time if this is the sort of thing that you're into.

Entry tags:

NORMAN, Okla. — Prolific mathematics and science writer Martin Gardner, known for popularizing recreational mathematics and debunking paranormal claims, died Saturday. He was 95.

Martin Gardner was the most influential of my math teachers in junior high. I didn't get credit for his classes, but he taught me that math was far more interesting than what the official curriculum was revealing. He, along with Raymond Smullyan, turned me from a student who liked math into a nascent mathematician. I cannot imagine how my life's journey would have been different had I not been exposed to his work.

Rest in peace, Mr. Gardner.

Martin Gardner was the most influential of my math teachers in junior high. I didn't get credit for his classes, but he taught me that math was far more interesting than what the official curriculum was revealing. He, along with Raymond Smullyan, turned me from a student who liked math into a nascent mathematician. I cannot imagine how my life's journey would have been different had I not been exposed to his work.

Rest in peace, Mr. Gardner.

I think I may have decided what I want to be when I grow up. A schoolteacher. Which is mildly disturbing, since I was scared off of that when my graduating class deemed me most likely to return to my school as a teacher and that seemed non-complimentary. But I'm great at math and good with people as long as I'm not terrified of them and from talking to the teachers in my acquaintance they've been waiting for quite a while for me to want to jump into the square hole alongside them.

The exciting news is that I went to the campus of our local state school that has an excellent reputation for education on Monday and talked to a whole bunch of different people who all had their own ideas of what it would take to get me to the literal head of the class. The not-so-exciting news is that this afternoon I talked to the person who actually seemed to know the answer to that and her answer was the most disappointing of the lot. No, I don't get to pursue an MS in mathematics, no I can't apply in the fall for an MSEd, no I can't actually teach for at least three years (maybe after two years in special circumstances). And, not being affiliated with the Math department, the prospect of a teaching fellowship has probably dropped to zero.

The thing about this that tweaks at me is that all I've ever heard is that there is a back door for people who want to teach math in under-served communities but have the knowledge without the credentials. I'm hoping that one of my friends can show me where that back door is, because the front door is mighty unappealing. I'm mildly tempted to talk to the Math department guy again and see if he thinks there is good placement for people with MS in Math, because I got such a buzz from looking over the course catalog at that program. As you might imagine, my advanced undergraduate studies at Carnegie Mellon make the first year of study for an advanced degree at a state school a cakewalk, plus the hope of a fellowship and the knowledge that admission for Fall 2010 is still a few months away. Oh well, I suppose that at least I might be able to clean up the leftovers part-time after I get my boring old MSEd.

I stopped by the state vocational rehab place with an update. When I told my assistant caseworker that my plan wouldn't start until January from what I can see and that I'd probably need some sort of work placement for the interim, she smiled and said that it would at least be better than what I had been doing. That wouldn't be hard. I continue to be glad at how much I'm able to get done and how wonderful and supportive my team is.

The exciting news is that I went to the campus of our local state school that has an excellent reputation for education on Monday and talked to a whole bunch of different people who all had their own ideas of what it would take to get me to the literal head of the class. The not-so-exciting news is that this afternoon I talked to the person who actually seemed to know the answer to that and her answer was the most disappointing of the lot. No, I don't get to pursue an MS in mathematics, no I can't apply in the fall for an MSEd, no I can't actually teach for at least three years (maybe after two years in special circumstances). And, not being affiliated with the Math department, the prospect of a teaching fellowship has probably dropped to zero.

The thing about this that tweaks at me is that all I've ever heard is that there is a back door for people who want to teach math in under-served communities but have the knowledge without the credentials. I'm hoping that one of my friends can show me where that back door is, because the front door is mighty unappealing. I'm mildly tempted to talk to the Math department guy again and see if he thinks there is good placement for people with MS in Math, because I got such a buzz from looking over the course catalog at that program. As you might imagine, my advanced undergraduate studies at Carnegie Mellon make the first year of study for an advanced degree at a state school a cakewalk, plus the hope of a fellowship and the knowledge that admission for Fall 2010 is still a few months away. Oh well, I suppose that at least I might be able to clean up the leftovers part-time after I get my boring old MSEd.

I stopped by the state vocational rehab place with an update. When I told my assistant caseworker that my plan wouldn't start until January from what I can see and that I'd probably need some sort of work placement for the interim, she smiled and said that it would at least be better than what I had been doing. That wouldn't be hard. I continue to be glad at how much I'm able to get done and how wonderful and supportive my team is.

"Some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way." - Tom Lehrer

I don't know so much about this question, but I got a really stark insight yesterday into how it has transformed me.

I was doing prep work for the enumeration of one of our local college dorms. In a nutshell, it is putting two hundred one-page census forms into an envelope. The form has to have a sticker and a control number written on it, and the envelope has to have its own sticker and the same control number, plus some extra information like details on where and when the respondent should return the form plus some god-awful fourteen digit "for official use only" code that I don't think any officials actually use. (Indeed, I think that I'm the official the code is intended for, and I'd make much better use of it if it had three or four digits.) And all this work has to be double-checked against two other forms to make sure that I'm assigning the proper control number to each student and that I write in the proper RA for each student's envelope. (Yeah, now you wish you had taken the Census test yourselves, amirite?)

And I'm doing this work, and it quickly becomes routine. And I suspect that an average person would put some music on and zone out and through passive consciousness would look up five hours later and see that the job was completed correctly. But my brain only does passive consciousness when driving long distances in nice weather. While doing grunt work, I get hyper-conscious and continually analyze whether it wouldn't be more efficient if this piece of paper was over THERE and whether I should do those two steps in the reverse order. And so I'm done in four hours but ready to publish a time-motion study on this process that, um, only gets done once every ten years. Curses.

And the punchline of the story is that average person made more money than I did because they could charge for their extra hour of work. That's me, always thinkin'.

I don't know so much about this question, but I got a really stark insight yesterday into how it has transformed me.

I was doing prep work for the enumeration of one of our local college dorms. In a nutshell, it is putting two hundred one-page census forms into an envelope. The form has to have a sticker and a control number written on it, and the envelope has to have its own sticker and the same control number, plus some extra information like details on where and when the respondent should return the form plus some god-awful fourteen digit "for official use only" code that I don't think any officials actually use. (Indeed, I think that I'm the official the code is intended for, and I'd make much better use of it if it had three or four digits.) And all this work has to be double-checked against two other forms to make sure that I'm assigning the proper control number to each student and that I write in the proper RA for each student's envelope. (Yeah, now you wish you had taken the Census test yourselves, amirite?)

And I'm doing this work, and it quickly becomes routine. And I suspect that an average person would put some music on and zone out and through passive consciousness would look up five hours later and see that the job was completed correctly. But my brain only does passive consciousness when driving long distances in nice weather. While doing grunt work, I get hyper-conscious and continually analyze whether it wouldn't be more efficient if this piece of paper was over THERE and whether I should do those two steps in the reverse order. And so I'm done in four hours but ready to publish a time-motion study on this process that, um, only gets done once every ten years. Curses.

And the punchline of the story is that average person made more money than I did because they could charge for their extra hour of work. That's me, always thinkin'.

I just proved that multiplication of the natural numbers is commutative. It took a half a page (and I lost track of how many pieces of scratch paper to work out the details), and that includes assuming that addition is associative and commutative and cutting out as much formalistic dreck as I could. Just to prove that ab=ba for the positive integers only.

So, if you were concerned about that, you can relax.

So, if you were concerned about that, you can relax.

Entry tags:

The final NaTexWriMo tally for me is about 54000 words, 35 figures, and 91 big pages for my graph theory solution guide, plus a 4000 word appendix on the fundamentals on how to do mathematical writing that won't drive your papergrader up the wall. There are still about a third of the problems that are unsolved and there is more that I can say in the appendix, but I think that turnip is damned near dry. Still, wow, I'm glad I struggled on it, because I really managed to pick up a lot of knowledge about some deep and cool results. I think I want to track down a book on the Four Color Theorem now, because I have a much greater appreciation of the lead-up to it now. I'm vaguely reading through my algebra books now, but I don't think I'll do another one on a deadline for a while.

There was quite a bit of snow on the ground this morning. Enough to cover the ground but not high enough to blot out all of the grass, so it was like tiny green stars in a sea of white. It was kinda pretty, but also kinda cold. Now it's back to all being melted, but not much less cold.

There was quite a bit of snow on the ground this morning. Enough to cover the ground but not high enough to blot out all of the grass, so it was like tiny green stars in a sea of white. It was kinda pretty, but also kinda cold. Now it's back to all being melted, but not much less cold.

At the halfway point in the month, my writing project is up to 20,299 words and 16 diagrams. I've come to think that whoever said that a picture was worth a thousand words wasn't a mathematical writer, because my pictures would seem to be generally worth only about 250 words. It's laid out at 41 pages right now (and that's normal A4 paper, not tiny book paper), solving 135 problems out of my textbook.

I'm thinking that my output is probably due to drop like a rock for the rest of the month. Even though there are 192 problems left, I've nearly covered the entire syllabus of the undergraduate course that I took (and then graded for four semesters), so the sections that are left are some combination of difficult and esoteric. The solutions will probably be longer when they come, but I think that they probably will require much more thought per word, and 1000 words per day is probably too optimistic.

I'm thinking about writing a chapter of my own about the process of mathematical proof writing, as my papergrading experience lead me to the belief that a lot of people do not have the innate talent to write out a well-argued proof by induction or proof by contradiction even if they understand fundamental logic and the material they are trying to convey. Alternatively, I guess I could jump over to my abstract algebra, topology, or combinatorics textbooks to flesh out my word count for the rest of the month. Then again, I might keep on hitting my head against graph theory for a while and be pleased with the new revelations that come out of it even if I miss the "goal" of NaNoWriMo, which I've already perverted by writing non-fiction and having worked on the material ahead of time.

ETA: OMG, I just found out that I've been robbed all this time. How many words are there in the sentence "Let v be a vertex of G"? I think there are seven. But if I throw math tags around the two variable names to give them the italic look and spacings of math variables, LyX doesn't count them as words. I'm pretty shocked that a mathematical layout application would offer the feature of a word count and then undercount like that.

I'm thinking that my output is probably due to drop like a rock for the rest of the month. Even though there are 192 problems left, I've nearly covered the entire syllabus of the undergraduate course that I took (and then graded for four semesters), so the sections that are left are some combination of difficult and esoteric. The solutions will probably be longer when they come, but I think that they probably will require much more thought per word, and 1000 words per day is probably too optimistic.

I'm thinking about writing a chapter of my own about the process of mathematical proof writing, as my papergrading experience lead me to the belief that a lot of people do not have the innate talent to write out a well-argued proof by induction or proof by contradiction even if they understand fundamental logic and the material they are trying to convey. Alternatively, I guess I could jump over to my abstract algebra, topology, or combinatorics textbooks to flesh out my word count for the rest of the month. Then again, I might keep on hitting my head against graph theory for a while and be pleased with the new revelations that come out of it even if I miss the "goal" of NaNoWriMo, which I've already perverted by writing non-fiction and having worked on the material ahead of time.

ETA: OMG, I just found out that I've been robbed all this time. How many words are there in the sentence "Let v be a vertex of G"? I think there are seven. But if I throw math tags around the two variable names to give them the italic look and spacings of math variables, LyX doesn't count them as words. I'm pretty shocked that a mathematical layout application would offer the feature of a word count and then undercount like that.

Reading through my FOAF lists, it amuses me to point out that I've started LyX'ing up the solutions to all of the problems in Bondy and Murty's *Graph Theory with Applications*, my college graph theory textbook. Naturally, given the scope of the material, it might be unreasonable to expect that I could grind out 50K words in a month, since precision is more central to mathematical writing than to novelization and so editing is critical instead of counter-productive.

That being said, 8948 words so far, 13,948 if you use the time-honored conversion that a picture is worth a thousand words.

And how cool is it that authors are releasing PDF's of out-of-print college textbooks? This is not the only $50 book in my library that is now free for download.

That being said, 8948 words so far, 13,948 if you use the time-honored conversion that a picture is worth a thousand words.

And how cool is it that authors are releasing PDF's of out-of-print college textbooks? This is not the only $50 book in my library that is now free for download.

Entry tags:

I've been spending the weekend getting my math geek on. I have long thought that LaTeX deserved the Lisp Award for User Hostility in Otherwise Powerful Software. This has been largely muted by LyX, which is a GUI front end for LaTeX. They go out of their way to say that it isn't WYSIWYG, but what appears on the screen is really a great first-order approximation of the compiled output -- and not just PostScript anymore; you can also export to a PDF file and share it online with Acrobat. Here is a sample paragraph that that's pretty easy to create (at least once you've decided what it is that you want to say). Amazing!

But I digress.

I was playing around with transcribing my college notes into an electronic format when I came across a Math Studies Problem Seminar problem that I hadn't solved. (I had a lot on my plate that semester.) The problem was to find one or more interesting facts about antiprimes (often called "highly composite numbers", defined as numbers that have more factors than any number less than it) and to compute all antiprimes less than 100 without electronic devices. So I worked out a pretty routine algorithm that would do the latter part that needed two lemmas to justify it, and thought that those would be the interesting facts that I would prove. The first was routine (and is the first two observations about the prime decomposition of an antiprime given on the MathWorld link above if you were curious), but the second was harder. I wanted to show that if n is antiprime and greater than 1, then n has a prime factor p such that n/p is also antiprime. It seemed obvious from poring over the first few dozen antiprimes, but there wasn't an easy proof to be found. So I banged my head against that for a good long while, and then gave up and posted to xkcd's math forum to see if they could make anything of it. (Huh, in retrospect, Usenet is not only dead but the mourning period is over, because this is the first moment this weekend I've considered sci.math.) And after some other good people banged their heads for a while, damned if another of the good people there didn't find a counterexample.

362279431624673937974303738230488502933082643722886373107941760000

I know, why didn't I notice that before wasting the time of nice forum people, right? That is likely the biggest number that has ever been generated on my behalf. I suspect the hypothesis is more interesting than some boring old true one would be.

But I digress.

I was playing around with transcribing my college notes into an electronic format when I came across a Math Studies Problem Seminar problem that I hadn't solved. (I had a lot on my plate that semester.) The problem was to find one or more interesting facts about antiprimes (often called "highly composite numbers", defined as numbers that have more factors than any number less than it) and to compute all antiprimes less than 100 without electronic devices. So I worked out a pretty routine algorithm that would do the latter part that needed two lemmas to justify it, and thought that those would be the interesting facts that I would prove. The first was routine (and is the first two observations about the prime decomposition of an antiprime given on the MathWorld link above if you were curious), but the second was harder. I wanted to show that if n is antiprime and greater than 1, then n has a prime factor p such that n/p is also antiprime. It seemed obvious from poring over the first few dozen antiprimes, but there wasn't an easy proof to be found. So I banged my head against that for a good long while, and then gave up and posted to xkcd's math forum to see if they could make anything of it. (Huh, in retrospect, Usenet is not only dead but the mourning period is over, because this is the first moment this weekend I've considered sci.math.) And after some other good people banged their heads for a while, damned if another of the good people there didn't find a counterexample.

3622794316246739379743037382304885029330

I know, why didn't I notice that before wasting the time of nice forum people, right? That is likely the biggest number that has ever been generated on my behalf. I suspect the hypothesis is more interesting than some boring old true one would be.

I like living in the future. I was reading a blog a few weeks ago that reminded me of *Heaven & Earth*, the most excellent Scott Kim puzzle game that never got the publicity it deserved, so I dug out my copy and replayed it. Then, hitting the end, I had to Google to verify that the victory screen was actually the end and not an Easter Egg for a further challenge, which led me to a forum where someone was giving love to Bug Brain.

Wow. I've never had the opportunity to play much with practical logic circuit design. Electronics books get esoteric in a hurry (even when they're written for kids) and math books stay abstract. I've always wanted a platform that combined the relevance of robotics with the puzzle-oriented worldview of games like*The Incredible Machine*. Evidently, someone wrote this game for me in 2000 but didn't go to the trouble of telling me about it.

This is an example of an intermediate-level puzzle, designing an artificial brain for a worm that is supposed to crawl forward until it hits a door, and then back up for a few seconds (to allow space for the door to swing out), and then return to the forward crawl. The brain contains input nodes (the red circle, which fires if the head is currently against an obstacle) and output nodes (the four blue circles, which direct the motor skills for raising the middle segment, lowering it, grabbing the ground with the head segment, and grabbing the ground with the tail segment when they are charged). The green circles represent neurons, which do the very elementary computation of measuring whether the accumulated charge of all of the incoming synapses (which can be individually weighted and even negatively weighted to inhibit charge) and send a charge down all of its outgoing axons if the total is greater than a threshold you define. You can use that mechanism to create the common logical gates: the two neurons in the middle of the brain are an AND and OR gate, and combined with the two supplemental yellow nodes they form the logic of an XOR gate. In addition, you can set synapses to slowly lose their charge over time, which let you form natural constructs like the feedback loop in the upper left that allows the brain to briefly "remember" that it bumped its nose a few moments ago and the two neurons with mutual decaying inhibitors in the lower left that form the cadence that it steps to. As the worm "chapter" progresses, you get control of more input and output controls and have to design more complex brains that ultimately have you negotiating a 2-D landscape filled with obstacles to find mushrooms that you can sniff out.

The game is far from perfect. The learning curve can be very steep for some problems, and while there are hints and even full solutions available you just wind up building the author's model instead of conceptualizing your own. It would also be nice if you could "chunk" common components like this XOR gate into a single "bundle" of neurons because it can get tricky to read the brains as they grow. The game contains a final module that hit on the real apparent strength of neural networks, which is their capacity of adaptive self-programming, but the examples they give are either simple or don't solve their problems dependably. But the biggest flaw is that it's such a delight that it's over too quickly.

Wow. I've never had the opportunity to play much with practical logic circuit design. Electronics books get esoteric in a hurry (even when they're written for kids) and math books stay abstract. I've always wanted a platform that combined the relevance of robotics with the puzzle-oriented worldview of games like

This is an example of an intermediate-level puzzle, designing an artificial brain for a worm that is supposed to crawl forward until it hits a door, and then back up for a few seconds (to allow space for the door to swing out), and then return to the forward crawl. The brain contains input nodes (the red circle, which fires if the head is currently against an obstacle) and output nodes (the four blue circles, which direct the motor skills for raising the middle segment, lowering it, grabbing the ground with the head segment, and grabbing the ground with the tail segment when they are charged). The green circles represent neurons, which do the very elementary computation of measuring whether the accumulated charge of all of the incoming synapses (which can be individually weighted and even negatively weighted to inhibit charge) and send a charge down all of its outgoing axons if the total is greater than a threshold you define. You can use that mechanism to create the common logical gates: the two neurons in the middle of the brain are an AND and OR gate, and combined with the two supplemental yellow nodes they form the logic of an XOR gate. In addition, you can set synapses to slowly lose their charge over time, which let you form natural constructs like the feedback loop in the upper left that allows the brain to briefly "remember" that it bumped its nose a few moments ago and the two neurons with mutual decaying inhibitors in the lower left that form the cadence that it steps to. As the worm "chapter" progresses, you get control of more input and output controls and have to design more complex brains that ultimately have you negotiating a 2-D landscape filled with obstacles to find mushrooms that you can sniff out.

The game is far from perfect. The learning curve can be very steep for some problems, and while there are hints and even full solutions available you just wind up building the author's model instead of conceptualizing your own. It would also be nice if you could "chunk" common components like this XOR gate into a single "bundle" of neurons because it can get tricky to read the brains as they grow. The game contains a final module that hit on the real apparent strength of neural networks, which is their capacity of adaptive self-programming, but the examples they give are either simple or don't solve their problems dependably. But the biggest flaw is that it's such a delight that it's over too quickly.

Entry tags:

Huh. I just saw a puzzle that I've never seen before. And it's one of those gems that makes you say "THIS is why combinatorics is so cool; learn how to count around the thing that is hard to calculate." Actually, while writing this I just thought of an even cooler proof than my original proof, which was pretty cool in its own right.

Morpheus and Neo each have a pile of fair coins. Morpheus has x coins, Neo has x+1. They flip all of their coins. What is the probability that Neo has more coins heads up than Morpheus?

Morpheus and Neo each have a pile of fair coins. Morpheus has x coins, Neo has x+1. They flip all of their coins. What is the probability that Neo has more coins heads up than Morpheus?

Entry tags:

So I went and laid out my argument a few days ago that high school mathematics education in the United States is not what any rational person would choose to implement if she were solely responsible for rebuilding it from scratch. I hope it didn't come across as an indictment that the asylum is being run by the inmates, but rather that it is a case of collective action where a community of rational individuals cannot be fairly expected to seem rational when considered as a single entity. It requires a more delicate analysis to untangle where we're headed in such a hurry and how we got into the handbasket. I hope that I illuminated a few pieces of that puzzle in my previous post.

Still, I am reminded of my eighth grade social studies teacher. When you made an argument in his class, he would pepper you like a four year-old with a mantra of "So what?" until you either reached a conclusion worthy of your data or (more often) broke down in tears. So, let's move on in that spirit. High school math is FUBAR; so what? Who cares? What do we do about it? I say: not much. Let's slap a disclaimer on it, supplement the ways in which it does not meet our individual needs, and spend our lives focused on more pleasant things.

I mean, truly, what branch of the high school curriculum isn't FUBAR? Science promotes the same dead recitation of the experiments of the eighteenth and nineteenth centuries without teaching the pleasures of research (plus it's not like anyone is claiming that the Bible contradicts the Quadratic Formula). English class is about studying only six stories a year, all by dead white men two of whom are always Shakespeare and Dickens. History is a cobbling of complex stories into oversimplified narratives that overlooks anything that gets in the way of "the moral of the story". The treatment of math is starting to look more comfortable. The only reason pure mathematicians have a stick up our collective asses is that the Death March to Calculus is only a part of the beautiful story of the "true math". It would be like if you were a Jordanian who adored the courageous outreach of Queen Rania and had to endure a larger world in which people thought that she was pretty hot but no Carla Bruni-Sarkozy.

Also, it's not new, and it's not dire. If this were the death knell for math, math would have died thirty years ago. I'm not actually even convinced of the unstated assumption that it is the role of formal education to teach students to love a field of study. (If it were, would the students fail if they didn't care for the topic?) In point of fact, there are currently many more vectors for the math bug than there were fifty years ago, and more than there were when it bit me twenty-five years ago. Here are some of them:

- Books. Martin Gardner wrote an amazing survey of recreational mathematics for*Scientific American* from 1956-81, and that has been collected into quite a few books that I greedily devoured in my formative years. Raymond Smullyan wrote fictional adventures that involved accessible but deep forays into non-elementary topics like combinatorial algebra and decidability. Kids today have access to these books (perhaps second-hand, admittedly) and a greater access to the layman-friendly works of John Conway and the fiction of Dennis Shasta.

- Programming. Computers are computational devices, and very usually one is brought to learn coding techniques with mathematical studies. I remember in high school that we were tasked with writing programs to estimate pi by a series of different algorithms and come to decisions about which were the fastest and most accurate. Project Euler is a much larger catalog of problems requiring a combination of mathematical investigation and algorithm design skills.

- Puzzles. Abstract logic puzzles of the sort that Nikoli produces on a commercial level and folks like the World Puzzle Championships craft at the OMGWTFBBQ difficulty level, and they existed to a lesser but still regular extent in my teenage days. Some of the puzzles are actually number-based to various degrees, but they're all mathematical. You're investigating an abstract environment that is unfamiliar to you and developing and refining heuristics to address the problems that you face there with increasing productivity. The specific tactics that you discover for each individual puzzle don't have much utility on their own, but the strategies for researching and formally codifying logical structures is a necessary skill for survival in our real world.

And there's more than this, of course, but it's certainly more than enough to pique the interest of someone who is earnestly searching for it. And there is so much injustice in the world that I simply can't get myself worked up over the issue that these avenues of research have to be found instead of spoon-fed from a licensed teacher and you don't get "credit" for doing it. It'd be nice if there were more respect and support for this sort of learning, but for all I know it's even more beautiful because it is done out of personal curiosity rather than that it was a homework assignment.

Still, I am reminded of my eighth grade social studies teacher. When you made an argument in his class, he would pepper you like a four year-old with a mantra of "So what?" until you either reached a conclusion worthy of your data or (more often) broke down in tears. So, let's move on in that spirit. High school math is FUBAR; so what? Who cares? What do we do about it? I say: not much. Let's slap a disclaimer on it, supplement the ways in which it does not meet our individual needs, and spend our lives focused on more pleasant things.

I mean, truly, what branch of the high school curriculum isn't FUBAR? Science promotes the same dead recitation of the experiments of the eighteenth and nineteenth centuries without teaching the pleasures of research (plus it's not like anyone is claiming that the Bible contradicts the Quadratic Formula). English class is about studying only six stories a year, all by dead white men two of whom are always Shakespeare and Dickens. History is a cobbling of complex stories into oversimplified narratives that overlooks anything that gets in the way of "the moral of the story". The treatment of math is starting to look more comfortable. The only reason pure mathematicians have a stick up our collective asses is that the Death March to Calculus is only a part of the beautiful story of the "true math". It would be like if you were a Jordanian who adored the courageous outreach of Queen Rania and had to endure a larger world in which people thought that she was pretty hot but no Carla Bruni-Sarkozy.

Also, it's not new, and it's not dire. If this were the death knell for math, math would have died thirty years ago. I'm not actually even convinced of the unstated assumption that it is the role of formal education to teach students to love a field of study. (If it were, would the students fail if they didn't care for the topic?) In point of fact, there are currently many more vectors for the math bug than there were fifty years ago, and more than there were when it bit me twenty-five years ago. Here are some of them:

- Books. Martin Gardner wrote an amazing survey of recreational mathematics for

- Programming. Computers are computational devices, and very usually one is brought to learn coding techniques with mathematical studies. I remember in high school that we were tasked with writing programs to estimate pi by a series of different algorithms and come to decisions about which were the fastest and most accurate. Project Euler is a much larger catalog of problems requiring a combination of mathematical investigation and algorithm design skills.

- Puzzles. Abstract logic puzzles of the sort that Nikoli produces on a commercial level and folks like the World Puzzle Championships craft at the OMGWTFBBQ difficulty level, and they existed to a lesser but still regular extent in my teenage days. Some of the puzzles are actually number-based to various degrees, but they're all mathematical. You're investigating an abstract environment that is unfamiliar to you and developing and refining heuristics to address the problems that you face there with increasing productivity. The specific tactics that you discover for each individual puzzle don't have much utility on their own, but the strategies for researching and formally codifying logical structures is a necessary skill for survival in our real world.

And there's more than this, of course, but it's certainly more than enough to pique the interest of someone who is earnestly searching for it. And there is so much injustice in the world that I simply can't get myself worked up over the issue that these avenues of research have to be found instead of spoon-fed from a licensed teacher and you don't get "credit" for doing it. It'd be nice if there were more respect and support for this sort of learning, but for all I know it's even more beautiful because it is done out of personal curiosity rather than that it was a homework assignment.

Alas, **rivka** doesn't crosspost her livejournal stuff to dw and I don't want to get another lj account just to respond to her posts (as thoughtful and marvelous as they are), but I was struck by her article here referencing this thesis (in PDF form) by Paul Lockhart (evidently a private high school teacher with the cache to teach an elective math course) that the high school mathematics curriculum in the United States has no redeeming values at all. Of course, his conclusion is nearly entirely correct, but his dualistic over-reaching and insipid straw-man dialogs are far more amusing than persuasive. His argument glorifying his research-focused methods and lambasting the soul-killing of everyone else seems to say "My students are able to see so far (due to *my* training, natch) in spite of the fact that giants are standing on their shoulders." Meh, it lacks that intuitive ring, which really strikes at the heart of whether Lockhart is the sort of mathematical leader we want to lead us into the next generation.

It's hard to have this conversation without an agreement on what mathematics IS, and what mathematical skills we need from the general population. This won't come without a struggle, and the way we do things now reflects our lack of consensus. Math is traditionally the language of both accountants and engineers, who each use their own fields with their own language. And you would work your way up the tower until you hit the limits of either your talent or leisure time and that would determine whether you were qualified to be a laborer, a manager, or an expert in some field like surveying or astronomy or what have you. This has served us for centuries all the way up through the time that Generation X (including me) was in high school, with the prize that people who have mastered calculus could train to study science and engineering in college.

There has been a rebellion against that model of education over the past ten or fifteen years, and quite a bit of it was well-deserved. The main problem (as I see it) was that we have been holding people in high school for the same length of time regardless of their ambition. When high school lasts for eight periods a day for four years whether you're training for a prestigious diploma or a lesser one, one might well ask what the sense is of anyone signing up for the lesser one besides the obvious conclusion that educators can't be bothered to challenge everyone. Plus, of course, mid-level bureaucrats had far too much power to limit the potential of women and minorities through the self-fulfilling argument that they didn't seem like the sorts of folk who could become engineers.

So now everyone is on the pre-college math track, which is probably great except that we didn't actually change the curriculum when we made that decision. The train is still making all of the local stops even though everyone is going to the end of the line, resulting in a fair amount of busywork that makes little sense in the broader context. For example, one spends quite a bit of time learning strategies for factoring polynomials in "Algebra II" that never get applied outside that cocoon because virtually all polynomials in "the real world" are irreducible. If we were to take side trips for the sake of showing off the breadth of mathematics, would that really be a part of anyone's plan?

And, needless to say, we haven't talked about the elephant in the room which is whether the four years of high school math should be obligated to shoot its entire wad in the name of satisfying the prereq for college freshman physics. Some of my best friends are engineers, but there are other things to be too. There are some mathematicians (including me and evidently Lockhart) who would argue that mathematics is bigger than the science of creating abstract models of physical phenomena for the sake of making better and simpler scientific predictions of real-world behavior, and should be broadened to consider the entire range of intellectual strategies to solve problems. Ordinary folk in my experience can get by without the Fundamental Theorem of Calculus but would be well served with some discrete topics like logic and graph theory. But it would take a larger mathematical revolution than I've ever experienced or even read about to knock calculus off the top of the mountain.

It's hard to have this conversation without an agreement on what mathematics IS, and what mathematical skills we need from the general population. This won't come without a struggle, and the way we do things now reflects our lack of consensus. Math is traditionally the language of both accountants and engineers, who each use their own fields with their own language. And you would work your way up the tower until you hit the limits of either your talent or leisure time and that would determine whether you were qualified to be a laborer, a manager, or an expert in some field like surveying or astronomy or what have you. This has served us for centuries all the way up through the time that Generation X (including me) was in high school, with the prize that people who have mastered calculus could train to study science and engineering in college.

There has been a rebellion against that model of education over the past ten or fifteen years, and quite a bit of it was well-deserved. The main problem (as I see it) was that we have been holding people in high school for the same length of time regardless of their ambition. When high school lasts for eight periods a day for four years whether you're training for a prestigious diploma or a lesser one, one might well ask what the sense is of anyone signing up for the lesser one besides the obvious conclusion that educators can't be bothered to challenge everyone. Plus, of course, mid-level bureaucrats had far too much power to limit the potential of women and minorities through the self-fulfilling argument that they didn't seem like the sorts of folk who could become engineers.

So now everyone is on the pre-college math track, which is probably great except that we didn't actually change the curriculum when we made that decision. The train is still making all of the local stops even though everyone is going to the end of the line, resulting in a fair amount of busywork that makes little sense in the broader context. For example, one spends quite a bit of time learning strategies for factoring polynomials in "Algebra II" that never get applied outside that cocoon because virtually all polynomials in "the real world" are irreducible. If we were to take side trips for the sake of showing off the breadth of mathematics, would that really be a part of anyone's plan?

And, needless to say, we haven't talked about the elephant in the room which is whether the four years of high school math should be obligated to shoot its entire wad in the name of satisfying the prereq for college freshman physics. Some of my best friends are engineers, but there are other things to be too. There are some mathematicians (including me and evidently Lockhart) who would argue that mathematics is bigger than the science of creating abstract models of physical phenomena for the sake of making better and simpler scientific predictions of real-world behavior, and should be broadened to consider the entire range of intellectual strategies to solve problems. Ordinary folk in my experience can get by without the Fundamental Theorem of Calculus but would be well served with some discrete topics like logic and graph theory. But it would take a larger mathematical revolution than I've ever experienced or even read about to knock calculus off the top of the mountain.

Entry tags:

- ← amusing links →,
- ← geekery →,
- ← mathematics →

I'm probably not the first person to say this, but I haven't heard it said yet: Wolfram|Alpha rocks in stereo.

My first exposure to it was this evening, when my brother told me that he wants to program an object to move around "in a figure 8 pattern (well, a figure 8 on it's side)" but doesn't know the math behind it. My copy of Schaum's Mathematical Handbook seems to have wandered off, so I turned to Google, which offered no response to "lemenscate". No huge surprise, since I pulled that word out of the back corner of my mind and for all I know I made it up. So I eventually hit on a search for "infinity symbol" and Wiki tells me that the correct word is actually "lemniscate". A properly-spelled Google search works out, but this time a short Wiki trail from the top result fails me and I actually need MathWorld's site (and paging two screens down) to get the parametric equation I was looking for. Total time elapsed to get the answer I wanted: maybe five minutes. Bad Google-Fu, no biscuit!

With WA, http://www44.wolframalpha.com/input/?i=lemenscate . Not only does it instantly strike gold with the misspelled word, but it turns that gold into ... um ... some sort of valuable gold thing. I suppose it's no surprise that Wolfram is going to have math subjects covered well, but it's still quite impressive. Once their engines start being able to parse searches related to the softer sciences, it's sure to be quite a tool.

(And, just in case some wouldn't know how Google-Fu would lead me to the subject line, please for you to clap hands and cheering for The Ministry of Unknown Science.)

My first exposure to it was this evening, when my brother told me that he wants to program an object to move around "in a figure 8 pattern (well, a figure 8 on it's side)" but doesn't know the math behind it. My copy of Schaum's Mathematical Handbook seems to have wandered off, so I turned to Google, which offered no response to "lemenscate". No huge surprise, since I pulled that word out of the back corner of my mind and for all I know I made it up. So I eventually hit on a search for "infinity symbol" and Wiki tells me that the correct word is actually "lemniscate". A properly-spelled Google search works out, but this time a short Wiki trail from the top result fails me and I actually need MathWorld's site (and paging two screens down) to get the parametric equation I was looking for. Total time elapsed to get the answer I wanted: maybe five minutes. Bad Google-Fu, no biscuit!

With WA, http://www44.wolframalpha.com/input/

(And, just in case some wouldn't know how Google-Fu would lead me to the subject line, please for you to clap hands and cheering for The Ministry of Unknown Science.)