Wednesday, February 27, 2008

Needing "school math" without using it

I was thinking about my previous posts about the UW College of Education's (CoE's) recent political polemic about so-called reform math. One of their major points is that engineers don't use what they call "school math": they just use computers. Please allow me to outline my own work, which is highly compute intensive and rarely involves what they would call school math, but which nevertheless I could not do without a healthy dose of school math -- not just in my education, but also in my work.

My research is in the area of computational neuroscience, in which I build mathematical models of individual nerve cells (neuron)s and groups of neurons, develop simulation software for single computers and clusters of computers, and analyze data from simulations and also from experiments on actual living tissue. This sort of work is very much like that done by anyone simulating physical systems, be they biological, chemical, mechanical, electrical, etc.

Like the engineers described in the CoE's publication, my work is heavily computational, as it isn't feasible to do this work with pencil and paper, as in "school math". The basics of the mathematical models involve a number of differential equations: equations that describe how some part of the system changes in response to other parts of the system. Now, it turns out that differential equations is covered by a pretty much standard college sophomore mathematics course. So, why isn't the stuff I do "school math"? It's simple:

  1. We only cover the mast basic type of differential equations in that class, linear equations. These are actually quite good for describing simple systems: electrical circuits made up of resistors and capacitors, mechanical systems with springs, etc. The advantage of these equations is that we can solve them on paper and they're easy to learn. The disadvantage is that they aren't very good descriptions of complicated systems like neurons (and many other, nonlinear systems). Once we move to nonlinear systems, we almost certainly need to use computers to do numerical simulations.
  2. We mostly just solve single equations in that class (there are other classes where we learn to solve groups of differential equations, later on in the curriculum). The systems I'm interested in can have hundreds or thousands of differential equations, and so I have no choice but use computer simulation.

If you were to watch me work, you would see the following (between the long periods of time in meetings, in class, preparing for said things): I decide on a question I'd like to answer, such as how the behavior of a network of neurons changes as some parameter (think: "tuning knob") is changed. I set up the parameters for a simulation or maybe bunch of simulations and, anytime from a few minutes to a few days later, I have some results. I load those results into MATLAB (numerical mathematics software) and plot the results. I then either exclaim, "Wow!", and hurriedly start writing a summary and thinking of what else I need to do to finish telling the story for a publication (rare), or I say, "Nuts!" and think again why the system either displays uninteresting behavior (Who knows; maybe its lack of interest is in itself something noteworthy? Or is that just wishful thinking?) or doesn't behave like the living nervous system. So, the observer sees that I don't "do" "school math". End of story?

Well, not quite. Because the observer doesn't see what's going on "behind the scenes" (i.e., in my mind). First of all, I would have no hope of even being able to start understanding what simulations I need to run without a very firm and extensive "school math" background. For instance, I work with a number of bright undergraduate students in my research. Some of them have math backgrounds that include differential equations and beyond and some don't. This has nothing to do with how smart they are; math beyond calculus isn't required for computer science and so only those students who come to us via a "nonstandard" pathway (e.g., changed major, previous degree/career) will have the more advanced math. Though all of these students can help out in my research, only those with more advanced "school math" are able to understand the underlying mathematical model well enough to mess with that aspect of the project (unless I teach a student some of the required "school math"). After all, unless you want to resort to randomly poking something just to see what it does, you really have to understand what's going on inside it; that's the only way you can intelligently select what kind of "poking" is likely to tell you something interesting. In fact, it's the only way you can begin to ask questions about the system, let alone start formulating experiments to answer those questions.

Even after the simulations are over, I still need to interpret the results, and this requires yet more "school math" running around in my head. What kind of result did I get? What relationship does it have to previous results I've gotten, or for that matter, results others may have gotten? What does this result mean in the overall context of the system in question and the thinks I'd like to know about it/do with it? And so on.

In other words, it is emphatically not the case that the computer has relieved me of the need to know math. All the computer has done is take over the grunt work: it has become an additional tool in my mathematical arsenal. But the computer can't think, and that thinking is where all the "school math" is. It just isn't apparent to the observer because I know it well enough that it happens in my brain automatically. This is no different than the automaticity with grammar that we use in everyday life. Just because we don't carefully label each of our utterances with "subject", "verb", "object" doesn't mean that grammar isn't necessary.

Finally, does this apply to "everyday" engineers, or just people doing research? Of course it applies to engineers (at least those who haven't "moved up" to management)! That's why businesses hire engineers: they need people who can think about solutions to problems and have the depth of background to understand the interrelationships among parts of solutions from "first principles" on up to final product. Some tasks may become routine and thus almost automatic or thoughtless, but its important to have someone who can look at a problem (or a solution proposed by some software) and say, "Wait a minute; something's fishy here." And, in the final analysis, that's the most important contribution of "school math": it is the language of creativity.

Tuesday, February 26, 2008

A brilliant strategic move

Microsoft is making their software development tools free downloads for all students.

"It's a brilliant strategic move on the part of Microsoft," said Chris Swenson, a software industry analyst with NPD Group. "This is one of the core audiences you have to hit if you really want to make a difference in the rich Internet application market going forward."
Hmm. Aren't there many other software development tools available for free to anyone? Like Apple's XCode and Eclipse? And didn't Microsoft already arrange nearly-free licensing agreements with many schools? What are the qualifications one needs to become a software industry analyst? Enough free time to read a range of trade publications?

Thursday, February 21, 2008

More on "modern" math

I'm continuing my reading of the UW College of Education's little treatise on mathematics education. The author(s) are writing about multiple-choice testing on page 25, such as many standardized tests:

Taylor’s research shows that, although males tend to perform well on multiple-choice questions, females do not. The test questions that are most effective for non-Asian minorities and females are conceptual math problems... show how they arrived at their solutions.

These “performance-based” questions offer partial credit for partial understanding...

“The kind of algorithmic math traditionally taught in middle and high school might make sense — with no further explanation — to future theoretical mathematicians, but it seems a fairly elitist thing to push algorithmic math as mathematics instruction for all students.”

“Most kids are not going to become mathematicians, but they are still going to need to use mathematical ideas. What happens is that the largely abstract mathematics instruction becomes a turn-off for many students so they drop out of mathematics. We are one of the very few nations in the world where it’s acceptable to say, ‘I don’t do math.’”

Allow me to deconstruct:
  • Again, another false dichotomy. The type of tests that one uses is mostly independent of the way one teaches, and one could use either "show your work" or "choose the correct answer" testing (or even other testing approaches) with either "traditional" or "modern" math.
  • Just another straw-man argument meant to equate "traditional" math with what the author(s) consider poor math practices, whether or not such a connection actually exists.
  • I'm not sure what to make about the statement that women and minorities don't do as well on multiple-choice tests. Even assuming this is true, this is not necessarily an argument for modifying testing (assuming there are good reasons for such tests). Instead, it would seem to me to be motivation for getting at the underlying reasons and addressing those. And what are these students supposed to do when they hit a point in their education that requires them to take a standardized, multiple-choice test? Bitch and moan about how unfair things are? Well, yes, the world is unfair. Complaining rarely helps. This is a recipe for setting these children up for failure later in life.
  • Algorithmic thinking is just for theoretical mathematicians?!? Algorithmic math is elitist?!? What about chefs; they write and follow algorithms all the time. Are they elite mathematicians? The process of computation (which is what we're talking about at the lower grades) is algorithmic; there is no other kind of computation. This is simply incomprehensible, and smacks of someone who doesn't understand what an algorithm is (yes, this is quite a nerve to hit for me as a computer scientist).
  • Very few students will become mathematicians? Strictly speaking, this is true; allow me to neglect to discuss what fraction may actually need math beyond those who become mathematicians. The problem is, which ones will go on to need the math? Not so easy to answer. What if the alternative approach to instruction rules out mathematically intensive careers for a good chunk of students? I submit that that's what has been happening: the absolutely brightest students, with parents who have the resources to help them go beyond "modern" math instruction, will do OK, students who will actually never need math may not be harmed one way or another, and a big group in the middle who would struggle under "traditional" math but gain sufficient mastery to continue onward in their studies will be shut out of a wide range of careers.
  • The overall tone of the quote implies a mind-set (like much of this document) that "math is hard", "math is irrelevant to everyday life", "math is for mutant theoretical mathematicians", "math is elitist". A wise colleague of mine once said, "Mathematics is a social justice issue." I certainly agree. We need to stop treating math, including algorithms, like something so complex that only Star Trek-like disembodied brains can understand and start treating it as a common human birthright and the only truly international and intercultural language.
  • I especially like the last sentence in the quote. After giving a bunch of reasons why algorithms aren't central to mathematics, why certain groups of children need alternative approaches to testing, and why most children won't need "abstract" or "algorithmic" mathematics ("school math" earlier in the brochure), Taylor bemoans the acceptability of people saying that they don't do math! Wait a minute... OK, I've banged my head against the wall, and that still seems like a contradiction of the thesis of the earlier material ("certain types of math are too hard for most children").
I hope that this little brochure isn't indicative of the overall level of scholarship at the UW College of Education...

Wednesday, February 20, 2008

Even engineers apparently don't need math

From the UW College of Education comes the very attractive publication linked from the title above. It addresses the "math wars" between "traditionalists" and advocates of "modern math education". Of course, like many political debates, they use titles such as those to pre-dispose their readers to see things their way. This is a sign right off the bat that this is not a scholarly work, but a political argument. Other signs are straw-man examples that are supposed to show how "traditional" math is misguided, such as this (p. 9):

A toy is hidden in one of two cakes. One cake is a circle, cut into fourths. The other is a rectangle, cut into sixths. Students must choose the cake that gives them the best chance of finding the toy.

Some choose the rectangle. Why? Because “most toys come in square boxes.”

Of course this is a poor question for students who would answer that way, because it is varying two things at a time (cake shape -- and therefore slice shape -- and fractions of total area). Presumably, this question is trying to get at more abstract thinking; that the shape of the cake and its slices doesn't matter, all that matters is fraction of total area. But the publication doesn't say anything about this, all it does is use this as a straw man to set up the argument that we shouldn't tell students how to do things (like optimal methods for mathematical calculations). All methods are equally valid:
One student may add 28 + 34 with traditional column carryover. Another adds 2 to 28 and subtracts 2 from 34 before adding the two results. A third student adds 8 and 4 to make 12, then 12 and 30 to make 42, and 20 more to make 62. In an effective classroom, all those solutions are studied, the links between them established, and the connection made to larger mathematical concepts (such as place value, the properties of addition, and developing generalized strategies).
This sounds very nice until you consider, "How did these students all arrive at different methods for addition?" The answer is that they weren't taught how to add; they were expected to "discover" it themselves. Go read a history of mathematics book sometime and consider how long humanity has worked to discover what we know about mathematics; how many geniuses have been involved. Does it make sense to systematically (not as an occasional teaching device) expect children to re-create any fraction of this? And is the only way to teach about place value, etc. to compare multiple methods?

Oh, and the implication is that "traditionalists" teach by giving out problems and just marking them right or wrong and "modernists" look at student mistakes and seek to understand why they make them. Nice false dichotomy.

I find this anecdote on page 13 especially interesting. One of the UW Education faculty has spent time observing engineers, scientists, and architects working, and here are his conclusions:

The architects, he discovered, worked problems out with visuals, not textbook algorithms. Engineers use mathematics, but much of that is embedded in their computational tools, and they too use forms of quantitative reasoning that looked very different from the activities of school math. It turned out that school math was a fairly rare species of activity outside of school.

“If you spend a month with architects, you’ll never once see them write an equation,” says Stevens.

The story was the same when he studied roadway engineers. “All the calculations were done on the computer,” says Stevens.

As the brochure continues, the distinction is between "school math" and the math that people actually use in the real world. Well, except for mathematicians, who are like poets, viewing the world in a different way than most people. Apparently, engineers don't need to know math; it's already in the computers (how it got there is unanswered). It's unfortunate that engineering schools and the accreditation folks require math through differential equations, multivariate calculus, etc. They must not know what engineers do as well as UW Education folks.

Tuesday, February 19, 2008

Get by with a little help from your (non-Windows-using) friends

Seems that a lot of digital picture frames are shipping with a Windows virus pre-installed. This is very convenient; it eliminates all of the hassle of connecting to the net and waiting for infection.

Seriously, and as John Gruber pointed out in Daring Fireball, here is one general approach to avoiding Windows viruses (quoted from the article linked above):

Deborah Hale at SANS suggested that PC users find friends with Macintosh or Linux machines and have them check for malware before plugging any device into a PC.

Wednesday, February 13, 2008

Tech companies to educators, commuters, the poor: screw you

Washington state is currently ranked 37 in college degree production, though it's in the top 10 in terms of employment of high tech workers. The Seattle area is ranked 3 in traffic congestion. Technology companies in this state complain loudly about the impact of congestion and lack of local college graduates on their companies. Bill Gates started a large foundation dedicated to solving other problems: in public health. But, when you get down to it (as you'll read by following the link above), when push comes to shove, public dollars in these companies pockets is more important to them than anything else. In this case, they're arranging things so they won't have to pay state sales tax on their equipment purchases for their server farms. To put this in perspective, the University of Washington -- a state institution -- pays sales tax on its purchases, even when using money allocated to it from state taxes. It's nice to know for whom this country is really run.

Monday, February 11, 2008

Single point of failure?

Do people really rely on a device that can fail, nationwide, if someone screws up a software upgrade at a single location in the country (apparently the reason for the April 2007 Blackberry outage)? A quick perusal of Google indicates two outages last year and one in 2007. And this is supposed to be a device better suited to corporate use than an iPhone?