This topic is a bit off-topic for this blog, but it struck me as an area which should interest many of our readers: High-School Mathematics. There is no shortage of criticisms and complaints about the deteriorating state of mathematics preparation for US high-school students. Rodney Brooks has recently written an essay suggesting the need for a new mathematics "that is so revolutionary and elegantly simple that it will appear in high-school curricula." He claims that understanding biological systems "demands it." (Thanks to Terra Nova and Nate Combs where I first picked this up).

Our community here is primarily composed of business practitioners -- as opposed to academics -- so I thought we might approach this issue from a bit of a different angle. We are likely mostly concerned about the "lowering of the bar" that has been occurring vis-á-vis math thinking skills. Skills which are of critical importance to economics, finance, computer science and related professional success.

*New Math* may be a poorly chosen title for a new approach to teaching math skills. But, the question is valid, is the current path of math pedagogy sufficient for preparing future professionals who will inevitably have a need to work with, conceptualize, and manipulate extremely complex computing systems sufficient? Brooks' essay struck me as a bit reminiscent of Wolfram's "New Kind of Science" (NKS), which is essentially a descriptive approach to Cellular Autonoma (CA). We've discussed predictive applications of systems based upon this framework, but I've never had the feeling that any of us felt particularly unprepared for the intellectual rigors despite our schooling in a traditional mathematics curriculum.

But has it been a "traditional" curriculum? Some of us were surely caught up, at least for a while, in the first "new math" where we were taught set theory from a very early age. And, there have continued to be shifts in approach to teaching math, so the idea of adapting math teaching approaches is not as radical as it may seem. Perhaps new approaches to teaching math skills are necessary as we move into an age of techniques like genetic algorithms (GA), neural networks (NN), and Bayesian networks (BN).

For closure, I offer the running Internet (although somewhat pejorative) joke that probably everyone has read in some similar form:*Teaching math through the years*

*
Teaching Math in 1950: A logger sells a truckload of lumber for $100.
His cost of production is 4/5 of the price. What is his profit?*

*
Teaching Math in 1960: A logger sells a truckload of lumber for $100.
His cost of production is 4/5 of the price, or $80. What is his profit?*

*
Teaching Math in 1970: A logger exchanges a set "L" of lumber for a set
"M" of money. The cardinality of set "M" is 100. Each element is worth
one dollar. Make 100 dots representing the elements of the set "M". The
set "C", the cost of production, contains 20 fewer points than set "M."
Represent the set "C" as a subset of set "M" and answer the following
question: What is the cardinality of the set "P" for profits?*

*
Teaching Math in 1980: A logger sells a truckload of lumber for $100.
Her cost of production is $80 and her profit is $20. Your assignment:
Underline the number 20.*

*
Teaching Math in 1990: By cutting down beautiful forest trees, the
logger makes $20. What do you think of this way of making a living?
Topic for class participation after answering the question: How did the
forest birds and squirrels feel as the logger cut down the trees? There
are no wrong answers.*

*
Teaching Math in 1996: By laying off 40% of its loggers, a company
improves its stock price from $80 to $100. How much capital gain per
share does the CEO make by exercising his stock options at $80? Assume
capital gains are no longer taxed, because this encourages investment.*

*
Teaching Math in 1997: A company outsourced all of its loggers. The
firm saves on benefits, and when demand for its product is down, the
logging work force can easily be cut back. The average logger employed
by the company earned $50,000, had three weeks vacation, a nice
retirement plan and medical insurance. The contracted logger charges
$50 an hour. Was outsourcing a good move?*

*
Teaching Math in 1998: A laid-off logger with four kids at home and a
ridiculous alimony from his first failed marriage comes into the
logging-company corporate offices and goes postal, mowing down 16
executives and a couple of secretaries, and gets lucky when he nails a
politician on the premises collecting his kickback. Was outsourcing the
loggers a good move for the company?*

*
Teaching Math in 1999: A laid-off logger serving time in Folsom for
blowing away several people is being trained as a COBOL programmer in
order to work on Y2K projects. What is the probability that the jail
cell doors will open automatically at midnight on 01/01/2000?*

Wow, read the Rodney Brooks article and this new math sounds hard and obscure. Since these deal with specific systems and require some computer power, maybe it’s better if the kids learn it in college in science and advanced stat classes. Most people forget math beyond arithmetic and basic geometry and algebra once they leave school, I'm not sure adding more to the pile is an efficient teaching technique.

If I was queen… A basic high school statistics and logic course would be nice. So would a basic economics course covering supply and demand, scarcity, game theory and time value of money. Maybe a couple classes each on mass psychology, selective memory, the decline and fall of the Roman empire and Thomas Hobbes.

Posted by: astrid | Wednesday, March 29, 2006 at 17:38

Here's a paste of my comment on Terra Nova. I used a quote from Fewlesh from an earlier thread on Genetic Algorithms. My gut reaction is similar to Astrid's: I am deeply concerned about teaching basics in high-school that prepare students to succeed (or at least feel confident they might succeed) in college as engineering, business/finance, economics or computer science majors. We're seeing the opposite trend today and I suspect it has as much to do with mathematics skills as it does sentiment about what jobs await after college.

I think Brooks is a bit ambitious in his essay. Perhaps such a curriculum would be appropriate for extra AP high-school math students, but given the amount of Calculus and perhaps Linear Algebra that gets left for college still, it would seem better to just cover that material and leave the emerging math for college 300+ level courses.

--

There do seem to be echos of Wolfram's proposals in the essay. And wasn't Wolfram also blasted pretty hard at the time for lifting his work without attributing Schmidhuber and others?

Regardless, I question whether the same goals can be reached by teaching theory and application of Bayesian Networks. This quote came out of a discussion we were having on my blog regarding Genetic Algorithms, Neural Networks and Bayesian Networks (for a different purpose than the subject here):

Way to low level. PNN's are likely good for modelling the low level dendrite-axon firing of neurons.

But what are the neurons doing? They are making predictions. What is the best statistical framework for making predictions? BN's are. It is very likely that collections neurons are implementing BN's (conveys a survival advantage).I'm not convinced teaching CA should be a high priority given some very serious deficiencies in first-principals mathematics and sciences (not to mention language-arts). I definitely resist the notion that higher maths can taught in any useful fashion without first mastering dependent or related basics. We should endeavor to create mathematical *thinking*, not just mechanised doing.

Posted by: randolfe | Wednesday, March 29, 2006 at 18:22

If anyone has had recent experience with the AP high school system for stats, econ or logic I'd be very interested in the current state of those disciplines. I know basic computer science is taught regularly, but isn't a lot of this more vocational than theoretical? Logical reasoning would be a very good place to start, and should be a prerequisite of most of the sciences and mathematics. It always struck me as odd that they teach the scientific method at a very young age but leave logic until college.

Posted by: randolfe | Thursday, March 30, 2006 at 16:04

Now that I think more about it, astrid brings up a good point about Economics and Statistics too. Perhaps, more fundamental theory of those subjects taught in high school (and not just at the AP level) would go a long way towards solving many of the problems which currently plague popular society. I personally think that every voting citizen *needs* to understand both statistics and economics at a basic level to have a fighting chance of interpreting the sea of information spewing forth from the media.

Posted by: randolfe | Thursday, March 30, 2006 at 16:09

Randy,

I finished HS in 1998. Economics was offered but not logic or stats. When I took my college intro classes in stats and economics, I kept thinking how powerfully explanatory the concepts were. I’m a terrible math person so if I can understand the concepts, I think most people have the ability to understand it. Same with calculus. I can’t finish a calculus problem for the life of me, but I found the theory incredibly powerful in explaining how stuff worked.

Funny enough, while I didn’t have exposure to well established economics and statistics knowledge in HS, my 11th and 12th grade English teachers were teaching us highfalutin literary theories (phallic imagery, death, desconstructionism) and my history teachers were teaching us about historiography. I guess that’s what one gets with a liberal arts education.

Posted by: astrid | Thursday, March 30, 2006 at 16:56