LeftoverPi

Some thoughts on the leading analogies in Lockhart's Lament

2008-05-02T00:04:00Z

I would like to do a series of posts on "Lockhart's Lament". Keith Devlin reposted the Lament on his "Devlin's Angle" column in March 2008. (The direct link to the pdf was http://www.maa.org/devlin/LockhartsLament.pdf in case the column is less permanent.) Lockhart's Lament is a hefty (in blogger terms) 25-page screed about mathematics education. It covers a lot of ground and makes a lot of claims. It's way too much to respond to in a blog post, so I'm going to break it up and take it a piece at a time. The first thing that Lockhart does is to present an absolutely beautiful analogy describing what music education might look like if our educational system ever really got its hands on it.]]> http://fulcrum.org/features/mountain.html I'll also mention another thing I think I saw on the Math Forum on perhaps on the math-teach listserv. Someone said that it is very unlikely for anyone to be able to accurately claim that "they don't like Chinese food". Their point was that there are so many varieties of Chinese food, it's highly likely that the person would find something they liked if they looked around a bit. But our general exposure (in America, at least) to Chinese food is very limited and we have a completely incorrect idea of what "Chinese food" means. What we eat may very well be Chinese food, but it in no way represents all of what is native to that country. And the same can be said for mathematics. People say that they "don't like math", but in reality they have only been exposed to a tiny slice of the pie [one half of the double entendre of "leftover pi", btw]. It seems very likely that the vast mathematical landscape, if we allowed or encouraged people to explore it, would have something for pretty much anyone. Back on topic, I think Lockhart has done us all a great service just in these first two pages. As he notes himself, we're up against a huge problem, as we have decades of heavily reinforced and widely believed misconceptions about mathematics to knock down. I think that analogies like this are some of the best tools for taking the first chunks out of the wall.

How I became an enemy combatant in the Math Wars, Part 2: Wrong War

2008-04-18T14:08:49Z

[This is part 2 of a two part series, parts 1 and 1a may also be of interest.] So, one thing that kind of cooled off the passion about the Math Wars (see part 1 for an introduction the Math Wars and my brief participation therein) for me was the fact that I planned on home schooling. I thought, "Well, I know how I am going to present math to my kids, and that's the most important issue for me right now, anyway." And so I went to teacher supply stores and bought fun counter things and pattern blocks and base 10 fries and all sorts of stuff. I was giddy. What fun these things will be to use! And my kids will totally "get it"--no half-formed ideas or things they don't really understand because they just learned by rote, etc, etc, etc. Guess how that turned out?]]> is a lot more fun to use manipulatives than to learn by rote. But, like "waist deep", "more fun" is a relative measure. What manipulatives don't (in general) do is impart relevance to a subject which has no intrinsic meaning to a kid. Base 10 notation and operation were not things that my kids wanted to know about. They had not seen them, used them, and wondered how they worked. There were not curious about them. The manipulatives did not stimulate the formation of that curiosity (nor were they designed to--they were designed to help get a concept across, and, as I said, I think they fulfill that role remarkably). Some time around then I realized that the Math Wars were essentially an argument about how best to teach kids math that kids weren't interested in learning. It was at that point that I "took off my uniform" and started fighting a different war (or perhaps fighting both sides in the current war? All I'm really sure about is that there are very few people on my side, and we're not organized :). So, shortly after that realization, I left the education foundation I was working for, realizing that I was way, way too radical to have much hope of getting funding, and I've been doing what I can on the side ever since. I wrote a kind of vision statement (still available at http://fulcrum.org/projects/yampod/), and described some projects I thought would be worth doing, and I've pushed them slowly forward in my spare time ever since. So, that's how I got here. Fulcrum.org was the site I started right when this realization happened. I started leftoverpi on a lark when I realized that the day after pi day could be called "Leftover Pi Day", and back-fit the idea of exposing people to the rest of the "mathematical pie" that they don't see because school focuses on such a narrow slice of it. As should be clear to you now, I have no idea where I'm going from here :). There are a lot of things I would like to do, but I'm only able to do them where they fit in between supporting a family and being part of that family in a meaningful way. But I like where I am, and I intend to keep moving forward, even if time restrictions mean it's only one post at a time, one software feature or improvement per month, etc. Because this, in my opinion, is a fight worth fighting.

Props to my Noollabs!

2008-04-06T06:00:27Z

Woohoo! Talk about your old "todo" items. For a looong time now I've wanted to add some kind of animation to the noollabs to make it clear that they are able to propel themselves through the air. Originally they were going to have jet propulsion, and the things on either side of their body were supposed to be the intake/exhaust for that. I tried to do an animation of one filling with air, and pushing that air out one side. It was...pathetically bad. I thought about having little puffs of steam coming out, and I guess that's still a viable option (viable meaning one that I can do with my limited artistic skills). However, for the time being, I've settled on a very poor animation of a propeller. You can see it in action in the noollabs game: http://www.leftoverpi.com/play/balloons. (I did not add it to the introductory part, which becomes more and more amusingly-out-of-date with each enhancement. You have to get to the game itself to see it working.) The impetus for this wasn't really checking the todo item off as much as it was getting a step closer to removing the placeholder introductory part entirely, and replacing it with gameplay that teaches you what that text was. But that's a topic for another day.

How I became an enemy combatant in the Math Wars, Part 1a, A Brief Digression on the Futility of Force

2008-02-28T04:39:13Z

So, in my last post I mentioned that I started seeing the plight of the anti-reformists in the Math Wars side, and in a recent conversation with my brother he asked me what I saw in their side. Below I have included excerpts from my conversation with him and some expansion on the ideas. At some point I realized that it was counterproductive to try to force a person who wants to do direct instruction to do constructivist approaches. ...they aren't going to do it well...the kids would learn better from them if the teacher taught the way the teacher was comfortable/experienced. That's what I meant when I said I saw the plight of the other side--what about those teachers being forced into teaching with a constructivist approach, when they might actually know in their hearts that they (personallly) would be able to do a better job via direct instruction?

I saw the [constructivist] curricula and said "yeah, I would love this, and the kids [who I was teaching] would love it. And most [or perhaps "many"] teachers would be lost with it."

So [eventually I came to the position] that they were fighting about an utterly ridiculous question, which is "how should all teachers teach all students?"

I just think there are teachers that do well with direct instruction, or relatively well, and they do not have the disposition to coax understanding from students. If you try to force them into that by mandating a constructivist curriculum, you're likely to have both aptitude and attitude problems that will more than erase what gains you were going to get from the curriculum. Not only that, it will look like it's the curriculum that failed, even if that's not really the problem. I should note that this was not the biggest problem I had with the framing of the debate, by the way--all I am doing with this post is expanding on the comment from earlier. But more on that in part 2 :).]]>

How I became an enemy combatant in the Math Wars, Part 1, Introduction to the Math Wars

2008-02-16T15:13:11Z

I don't actually know when the Math Wars started. You might say that the first battles were with the New Math back in the 60's when I was still an accident waiting to happen (just kidding, Mom!). I don't really know what the "New Math" was, but I have some sense that there was an attempt to get kids to understand the concepts of math rather than just learning rote algorithms. So, rather than saying to the kids "just follow this procedure, and you'll get the right answer", you talk to them about the base ten number system, explain the concept of regrouping, etc when you teach subtraction. Which sounds like a good idea--we want kids to understand, not just robotically replicate. The devil was in the details of course. New Math entry on Wikipedia for a better informed take on what the New Math actually entailed]. But it was way too theoretical, or way too advanced, or way too done-by-scientists-who-didn't-remember-being-kids, or way too ohmigosh-sputnik-make-our-kids-smarter, and it didn't last. And it was largely looked back at as a failed experiment. You can hear a hilarious Tom Lehrer song about it (that link is from the Wikipedia page, and claims to be "official"--there are various YouTube videos that act it out, which are reasonably fun to watch as well) that gives you an idea of the sentiment, or at least the kinds of jokes about it that people thought were funny. When the National Council of Teachers of Mathematics brought forth a document commonly referred to as "The NCTM Standards" (shortened to "the Standards" in most discussions), a lot of people capitalized on this earlier (reputation as a) failure by calling the Standards "the New New Math". I don't know when that particular salvo was fired, but I'll bet it was pretty early on. In any event, very shortly after their publication, the Math Wars were on. The "reformers" wanted to implement Standards-based curriculum everywhere. "Conservatives" or "anti-reformers" (I'm not sure what they call themselves--"sane", probably :) ) were pretty much horrified with the movement, and created their own organizations (see Mathematically Correct's website for what I think is the best-known and largest example) to work against what they saw as the destruction of mathematics education in our country. I started out pretty much reform-minded. Reformers were Constructivists, which is an educational philosophy I generally agree with, and a lot of the Standards-based stuff was focused on giving the kids "ownership" of the material. Like leading them to discover an algorithm for themselves, for example, or letting them pick their own strategy to solve a problem (e.g. draw a picture of five groups of three and count, count by fives verbally, just use the fact that you know 5x3 is 15, etc). So, for a while, I considered myself a reformer. I participated in NCTM-l, a listserv (email mailing list) that was formed to discuss the use of the Standards, and generally when I argued I argued on the reformer side. As conversations on the internet frequently do, the ones on that list got heated (and often personal). I remember mostly the amount of time it was taking me from work to try to read it and participate. :) I actually did try to listen to the "other side" and try to see where they were coming from. (In case you are interested, you can read some of my posts and the discussion around them then at my barely-breathing site, fulcrum.org.) Perhaps it was the fact that I started to see the plight of the "other side" that led me to drop out of the discussion. I am sure that time was a factor...hard to recall exactly. But I know that at some point I started wondering if reform of the current system, in the form people were talking about, had any hope of doing what I thought needed to be done. And I think then was when I realized that I didn't fit on a "side". Tune in later for another exciting episode about the Math Wars and how I got orthogonal to them!]]>

it's...aliFe!--better explosion

2008-02-11T01:50:32Z

So, the game of life now has a better explosion. Took more time to do this than any of the other features, but my son's gleeful giggle upon seeing it made it worth it. First I just set it up to send all the currently "on" squares radially outward (actually I think it was/is a little more complex than that--they go out in such a way that everything gets to the outside at the same time). I had to struggle with it for quite a while until I figured out that Director was doing something weird when you called "mod" against floating point numbers. Anyway, when I finally got it all worked out, the explosion was disappointingly mundane. So I added a couple of rings around the center so there's always something to explode, and voila! giggles.

its_aliFe: nuclear option (plus some on new focus)

2008-02-07T01:39:13Z

Updated the Conway's Game of Life game to let you nuke all the current cells and start over. This was requested by my kids shortly after they started playing with it. For the forseeable future (1.5 weeks is usually about how long that lasts for me), I'm going to work exclusively on things my kids are interested in. That means I'm only going to be adding features they request or that their usage shows a strong need for. This "nuke button" is an example. The funny thing about the nuke button is that when my wife saw it, she said "you should have some kind of explosion show on the screen". I said something like "maybe if the kids ask for it". My new focus was already saving me work! But it was practically the first thing Jacob said when he saw the button. So it looks like I can't win.

It's...aliFe!

2008-02-06T16:15:12Z

Specifically, it's Conway's Game of Life. Even (or especially) if you don't know what that is, go to this url: http://www.leftoverpi.com/play/life/ draw a little picture by clicking on the white canvas. Click the play button and watch it go. It's amazingly complex--in fact, complex enough that you could build a computer out of it!--and equally mesmerizing. You can read more about it here: Conway's Game of Life [Wikipedia]. All for now, hopefully I'll get time to update this later.

Threevenness and other neologisms

2008-01-24T13:33:39Z

So, once upon a time I was thinking about the statement that 2 is the only even prime. People observe this from time to time and think "Wow, that's weird--of all the primes, only 2 is even". Well, it doesn't take too much to realize that it's not really weird at all. One way of describing "evenness" is to say "any multiple of 2". There can't be any multiple of 2 in the primes (except 2), because it would be 2x[something], and therefore not a prime. But why is it that people think it's weird? ]]> threeven. And 3 is--amazingly--the only threeven prime! It's prime, and it's threeven! Wow! All the other primes are throdd! I actually came up with the word threeven specifically to help people see that 2's loneliness as "the only even prime" is not that strange. 3 is the only threeven prime, and five is the only fiven prime. Every prime is the only prime that is a multiple of that prime. But an interesting unintended consequence was the word "throdd", which was the natural opposite of "threeven". Probably one of the reasons we don't have a word like threeven is the fact that throddness is much less of a tidy designation than oddness. Evenness and oddness split the natural numbers neatly into two groups, but threeven and throdd are of course lopsided. Use of the words threeven and throdd might be a fun way to introduce modular arithmetic--you could talk about how there are two kinds of throdd numbers, those one more than a multiple of three and those one less than a multiple of three, and go from there. Ah, now I'm rambling. What actually prompted my post was when I got into work and realized that there is no word for "out between" to mean the opposite of "in between". We were looking at storing hierarchical data in a SQL database and...well, never mind, that's a story for another day.

Nurture Capital (tm) :)

2008-01-23T12:55:49Z

So yesterday I was talking to a friend of mine about an alternative funding model I call "Nurture Capital". The basic idea is that some ideas that might normally take form as a non-profit organization might be able to be run at a profit. It would be different from a normal for-profit business in the sense that the profit wouldn't be the main motivation, but rather a means to keep the enterprise running. The main motivation would be whatever cause the business was designed to further.

Certification--Identifying companies that fit the Nurture Capital Criteria. Something along the lines of "they have identified a real need and sources of revenue that could come from filling that need, they understand the terms under which the money is being invested, they have a believable plan for coming into profitability in a given time frame".

Consulting--for startups that haven't quite figured out any of the above, the company would help them figure out ways to make their good works pay for themselves

Introductions--connecting the people with the money to the people with the ideas, making sure that both sides understand the risks and obligations, matching up interests, etc. For this, the company would take some small, perhaps percentage-based fee from the funding, and it would live off of those fees. Would it work? Well, I know that it is a problem for people with money who would like to give something back to find reliable entities through which to do that. Many nonprofits are unfortunately little more than a place to work for the principals, with huge overhead ratios and much of the money going (ironically) into fundraising. The requirement of profit is in some sense a harsh one--it forces the company to face reality as it exists today and make all the hard choices that go along with it. Furthermore, it would probably not work for every nonprofit-type venture, and it may only be viable for a small fraction. But for those few it could be a great way to match up givers with people who want to work to make a difference in the world. I think it's worth a try--and that's essentially what the whole concept of Nurture Capital is about. High risk, sure, but a reasonable possibility of becoming independent through profitability and all the while working to promote a good cause.]]>

New Stretcher Feature--Interfacius Dissapearicus

2007-09-29T13:11:55Z

A new feature has been added to Stretcher to give you more room to play with the fractals. You click the diagonal arrow to make the interface elements move out, and click it again to make them come back when you need them again. Perhaps I should make it default to being "out", and have a toolbox icon or something to let people know that there are more controls available. I would like to make the effect a little more smooth, with the stuff sliding out to the edge, and then perhaps having the mouse entering that area make them automatically slide back into place. All in good time, I suppose :). I should take a look at my todo list for that project and see what's missing. Anyway, enjoy!

Benezet fan on the web!

2007-09-26T05:39:25Z

This was exciting to me, because it's the first time I have seen Benezet's experiment mentioned on the web. I've been carrying around a copy of his paper (not because of my fanatical devotion, but because it's still in my laptop bag from when I put it there to read on the plane to OSCON) and referring to it every so often, but now I can actually point to it on the intarwebs! Apparently someone at this place in the UK that does really cool computer science research read about Benezet and loved it, and created the "Benezet Centre": http://www.inference.phy.cam.ac.uk/sanjoy/benezet/ One of the best things about this is that now I can link to the three articles instead of depending on my paper copy to always be handy. Oh, I guess you're wondering what Benezet did, eh? He quit teaching formal arithmetic to kids who were too young to care. Instead, he had teachers make up questions that might stimulate some mathematical thinking, develop concepts of scale and ratio, etc. By the end of sixth grade these kids were far, far beyond their traditionally-"educated" peers. I've read that he got fired for this, but I don't have the background on that, I'll check it out and update. Benezet is one of the people I cite for proof to my assertion in an earlier post that mathematics instruction in school is worse than a waste of time. It's nice to see him getting attention from other folks.

A cute little mini-app with some cute history

2007-09-13T05:20:22Z

I am pretty sure I referenced the "oodometer" in an earlier post. I had read someone talking about a little counting gadget (press the button and it adds one to the count, like a car's mechanical odometer registering another mile) and how they were surprised at how much interest it generated in their kids. I figured I could bang one of those out in software in pretty short order, so I did, and then presented it to my son (maybe five years old at the time? Not sure exactly) to see what his reaction was. hundred times the first time he played with it. Amazing. So, if you want to do a similar experiment, you can try the original version (or something very similar--I think the "return to zero" button is a later addition) here: http://leftoverpi.com/play/oodometer_simple/ Later I added

other powers of ten
subtraction
base "tree"

The "advanced" version is here: http://leftoverpi.com/play/oodometer We suspect it was his play with the "advanced" version that led the aforementioned tyke to spontaneously declare during a car ride one day that "two fousand is twenty hundred" It is, indeed. :) Let me know in the comments if you try it with your own kid, and how many times she clicks the first time. ]]>

New Stretcher screencast--multipoint

2007-09-12T13:05:39Z

I was doing some screen capture of Stretcher to show off its many cool features, and realized partway through editing that nobody would want to sit and watch all of it. So I started trying to pick out the cool stuff and release it separately. Here's the first from that:

Apropos Quotes from Simone de Beauvoir

2007-08-28T14:09:37Z

The writer of originality, unless dead, is always shocking, scandalous; novelty disturbs and repels. Simone de Beauvoir

Of course, that doesn't necessarily mean that shocking, scandalous, disturbing, or repellent ideas are all original :). Nor that the original is worthwhile. But it's a good thing to keep in mind. A lot of what I'm talking about is kind of hard to swallow, but I think when we look back at it later people will wonder why we ever saw it differently. And here's another from the same person--this really just says it all:

I wish that every human life might be pure transparent freedom.