The whole book is a witty and quick 250 page read. I would even go so far as to say that it is probably the best, most witty and readable articulation of fundamental economics principles that I have come across. And while I enjoyed Steven Levitt’s Freakonomics book, this one was much better at articulating and detailing fundamental principles. Now many folks might be put off by that description, but trust me, the whole work is an engaging read with stories and anecdotes that relate the principles in the process, quite unlike any boring textbook on the subjects.

Here is a short excerpt from one of my favorite sections to give you a flavor for the rest. (As an aside, I’ve always been a little perplexed by the “carbon neutral” proponents which have proliferated at several recent meetings I have attended, but Harford’s exposition is way more articulate than I could ever be.)

“How did you travel here today?”

“I’m sorry?”

I’m puzzled. Here I am, going to a panel discussion organized by an environmental charity, and a very eager young member of staff is grilling me before I even get past the door of the lecture hall.

“How did you travel here today? We need to know for our carbon offset program.”

“What’s a carbon offset program?”

“We want all of our meetings to be carbon neutral. We ask everyone who attends to let us know how far they came and on what mode of transportation, and then we work out how much carbon dioxide was emitted and plant trees to offset the emissions.”

The Undercover Economist is about to blow his cover.

“I see. In that case, I came here in an antrhacite powered steamer from Australia.”

“Sorry, how do you spell anthracite?”

“It’s just a kind of coal–very dirty, lots of sulfur. OW!”

The Undercover Economist’s wife gives him a sharp dig in the ribs.

“Ignore him. We both cycled here.”

“Oh.”

Apart from being a good example of how irritating an Undercover Economist can be, this true story should, I hope, provoke a few questions. Why would an environmental charity organize a carbon neutral meeting? The obvious answer is “so that it can engage in debate without contributing to climate change.” And that is true, but misleading.

The Undercover Economist in me was looking at things from the point of view of efficiency. If planting trees is a good way to deal with climate change, why not forget about the meetings and plant as many trees as possible? (In which case, everybody should say they came by steamship.) If awareness-raising debate is the important thing, why not forget about the trees and organize extra debates?

In other words, why be “carbon neutral” when you can be “carbon-optimal,” especially since the meeting was not benzene neutral, lead-neutral, particulate-neutral, ozone-neutral, sulfur-neutral, congestion-neutral, noise-neutral, or accident-neutral? Instead of working out whether to improve the environment directly (by planting trees) or indirectly (by promoting discussion), the charity was spending considerable energy keeping itself precisely “neutral”–and not even precisely neutral on all externalities, nor even a modest range of environmental toxins, but preserving its neutrality on a single high-profile pollutant: carbon dioxide. And it was doing so in a very public way.

A kind view would be that the charity was setting a “good example,” if acting nonsensically can ever be a good example. An unkind view would be that it was indulging in moral posturing.

There is plenty more of that sort of thing on a wide range of topics…

Even if you never had any interest in economics micro or macro, go buy the book right now. In fact, you can get it here on Amazon right this instant.

[Update: For the record, I am in favor of establishing a market for Carbon credits which can address emissions just as the Sulfur market addressed acid rain so efficiently. I just tend to question misplaced emphasis and what may very well be PR-motivated grandstanding that has little real impact in return for so much attention and focus.]

You just have to love a school project wherein 4 students send a balloon up to an altitude of 20 miles to capture NASA quality photos  on a budget under a few hundred bucks.  Incredible.  This should be a model for all school science programs!  More Images, video, and detailed blog links after the break.

From the Telegraph article on the project:

Taking atmospheric readings and photographs 20 miles above the ground, the Meteotek team of IES La Bisbal school in Catalonia completed their incredible experiment at the end of February this year.

Building the electronic sensor components from scratch, Gerard Marull Paretas, Sergi Saballs Vila, Marta­ Gasull Morcillo and Jaume Puigmiquel Casamort managed to send their heavy duty £43 latex balloon to the edge of space and take readings of its ascent.

Created by the four students under the guidance of teacher Jordi Fanals Oriol, the budding scientists, all aged 18-19, followed the progress of their balloon using high tech sensors communicating with Google Earth.

Team leader Gerard Marull, 18, said: “We were overwhelmed at our results, especially the photographs, to send our handmade craft to the edge of space is incredible.”

Completing their landmark experiment on February, the Meteotek team had to account for a wide variety of variables and rely on a lot of luck.

“The balloon we chose was inflated with helium to just over two metres and weighed just 1500 grams,” said Gerard. “It was able to carry the sensor equipment and digital Nikon camera which weighed 1.5kg.

“However, when we launched at 9.10am on that morning the critical point for the experiment was to see if the balloon would make it past 10,000m, or 30,000ft, which is the altitude that commercial airliners fly at.”

Due to the changing atmospheric pressures, the helium weather balloon carrying the meteorological equipment was expected to inflate to a maximum of nine and a half metres as it traveled upwards at 270 metres-per-minute.

“We took readings as the balloon rose and mapped its progress using Google Earth and the onboard radio receiver,” said Gerard.

“At over 100,000ft the balloon lost its inflation and the equipment was returned to the earth.

“We traveled 10km to find the sensors and photographic card, which was still emitting its signal, even though it had been exposed to the most extreme conditions.”

Balloon flight path using Google Earth

Payload and camera chassis

An absolutely fantastic project that took great planning, electronic, thermal, imaging/sensing, control systems, computer science, and telecommunications expertise!  Three cheers to the team!

Don’t miss all the details at their blog (Thanks to Google for the English Translation.)

It will allow you to instantly crush your old-school opposition using the simple principles of impact and inertia.  It’s also instructive to consider some winning Jenga strategies using finger flicks even when you are forbidden the latest technologies!  Video after the break.

Here are some images and live video of the two engines running.

They’re little marvels of engineering.  You really can build anything with Legos.

These versions are actually up for sale for very reasonable prices (133 euros for the Inline 3, and 455 euroes for the V8) considering the myriad parts and assembly steps required.  They come pre-built and tested from Alex and Ivan’s new online store at LPE Power.  Or, if you want to source the parts and build and test your own engines in true Lego spirit, check out my earlier Lego Engines post with links to parts lists and step-by-step visual build instructions.

Last week’s post on the Globalization of Leadership ended with a clarion-call for change. Given that the entire US economy is built upon a foundation of energy and energy policy, it makes sense to start looking there to see where the biggest economic levers lie.  Here, I offer a somewhat more analytical approach than can be found in the general media.

Today’s Tilted Playing Field

The US’s problem with today’s global energy game is that the worldwide playing field is dramatically tilted.  The game of global macro-economics simply isn’t a fair game.  The odds are dramatically stacked through the distribution of resources, both natural (most notably fossil fuels) and human (in terms of population).  But the game is also biased based on which country has the technical and infrastructure wherewithal to exploit the resources wherever they might be distributed.  For half a century, America developed a significant global advantage as the direct result of national policy which strongly promoted technical innovation and large infrastructure investments based on those technologies.  When the whole world depended on US innovation to exploit their local natural resources, our country purposely stood in the position of gate-keeper to the world’s mineral and economic resources and it was a position we exploited handily to establish a long tenure of control, dominating petroleum and natural gas collection and refining.

But over the last twenty years, two disturbing trends have served to tilt the playing field in the opposite direction, away from the Yankees.  The first trend can be seen in the rise in traditional religious influence that has through financial and political support, largely coupled itself to the machinery of the Republican Party. In reaching to sustain traditional values and strongly supporting the historical practices which made America successful in the last century, the religiously influenced right has unfortunately set itself squarely against the fundamental principles of scientific and technical progress which underpinned the US emergence as a world power.    This national resurgence in Luddite attitudes is frightfully witnessed by the fact that we actually have a presidential candidate who does not know how to use a computer or even what the Internet really is much less what industry or infrastructure based on these might need to look like.  The paired Vice Presidential candidate believes that intelligent design should be taught in science classes despite the observable truth that the entire nation’s critical biotechnology industry, today’s most fundamental building block of health care advancement, rests on the repeatedly demonstrated facts of evolution.   Worse, these candidates are representative of giant swaths of the US that actually support them and their traditional view that last century’s outdated beliefs, technologies, and practices will suffice to lead tomorrow’s national policy.

As the US in its recent growing obsession with “traditional values” (now recast as “small town” or rural values) and celebrity has retreated from supporting fundamental research, and national-scale technology and infrastructure investments, other countries, most notably China and India who each comprise the largest concentration of the world’s human resources, have turned in the opposite direction to invest ever more in technologies and infrastructure and technical education.  To their great advantage, our international competitors have managed to couple staggering national investments in science and technology education with staggering company and even national industry sponsorship to literally become the engines of manufacturing and production for the entire world.  They learned what made America great, and are doing more and more of it just as we are turning to do less and less.

The second troubling development is that while China and India develop and tune their human and infrastructure engines of progress, America has managed to teach those countries fortunate in their mineral wealth to become savvier in capturing and retaining the economic value of those natural resources. Each year more countries nationalize their utility and energy companies.  America is losing its position as the lone facilitator and gatekeeper (and toll-taker in the sense that by investing in, creating the technology and operating the infrastructure, US companies have made enormous profits on very high volumes of trade) for the world’s fuel and infrastructure technology and at the same time, abandoning the educational and government practices which could offer the hope of becoming the leader and gatekeeper for the world’s next set of critical resource.

We have been the technology leader of the world for almost a century, but that technological advantage has slipped.  Now our cars are slower and less efficient than almost every other industrialized nation that manufactures cars.  Our infrastructure, buildings and bridges, energy production and distribution, telecommunications and most recently our data network infrastructures led the world.  We now lag behind much of Europe and Asia in all of those areas including Internet usage and broadband penetration and performance where we now rank 17th among industrialized nations in a technology that we invented. We squandered a ten year head start.   At no time in the history of the United States has this decline been more apparent and measurable than over the last eight years.  We Americans now find ourselves paying ever more dearly to purchase the best cars and displays, wireless networking technologies, and countless other things from countries that have now surpassed the systems and technologies we invented and built.

But the single biggest tilt to the global playing field is the one cast by the regionally skewed distribution of fossil fuels.  Because we were the first and most ardently industrialized country, our rate of energy consumption is prodigious compared to the rest of the world.  As a result, despite being the third-largest producer of petroleum in the world, the US appetite is larger still.  Today we must purchase fuel from those countries that now control the local energy harvesting and refinement and extraction infrastructure, much of which we originally built for them.  As contracts expire, and countries nationalize their energy infrastructure, western corporations are rapidly losing their financial leverage in petroleum energy management.

This progressive disenfranchisement of the Exxons of the west is reflected in both the global price of oil and diminishing access to this critical resource as demand increases through the burgeoning growth of India, China, and other third-world countries.  We have become a middle-man that no longer produces or controls its vital resources and we face the classic case of a middle-man being cut out from the middle when the ends can manage without.  But this isn’t about a Midwest farming equipment reseller who can find another living in an economy rife with alternatives.  This is happening around today’s only viable fuel commodity on the planet, and on a global scale at that.

Financially then, we have to borrow more money every year to fuel an aging, increasingly obsolete, and increasingly expensive transportation and energy habit with decreasing economic leverage.  Every year, we pay ever more to other countries which supply our petroleum addiction, and every year, we buy more products and services from those countries that can now make and provide things more efficiently than we can.  The net flow of both leverage and money is away from “those who manage,” and more towards “those who have or produce” minerals or products to sell that they can generate more cost-effectively.

I think a lot of people get this general idea, but my sense is that there is an utter lack of appreciation for the massive SCALE of the problem in the energy sector, or of how much that imbalance really tilts the global macro-economic playing field. Otherwise, for example, the debate about Arctic drilling could have been settled with a single graph that made it clear that the marginal change in oil production due to newly tapped Arctic reserves would still be dwarfed by both the import volume AND the speculative price components due to future demand projections.  And that would be true even if the production could all come on line at once rather than the 5-15 years it would take to set up the infrastructure and start the actual flow of fuel.

At last December’s World Economic Forum in Davos, I happened to end up sitting next to the Foreign Minister of Dubai at one of the Energy workshops.  His great lament was that they had over $250 Billion dollars of technical infrastructure they wanted to build in Dubai LAST YEAR with the money literally burning a hole in their government pockets. But sadly, they couldn’t find anywhere near enough technically trained personnel to actually build it.  Yes.  $250 Billion in one year.  The then CEO of Fleur was almost despaired thinking about the out-of-reach opportunity even having started several schools in India specifically to train thousands of technical construction workers per year specifically for Dubai’s infrastructure plans.  And keep in mind, Dubai is the capital of the UAE, a country about the size and population of Massachusetts.  Further, Dubai is one of the smaller oil and natural gas producers in the Middle East.  No wonder they can afford to have someone build the world’s largest building there.

Russia’s resurgence over the last decade from a difficult transition to a partial market economy has been almost entirely funded through their energy production as the second largest fossil fuel supplier in the world after Saudi Arabia.  The Gasprom execs in Davos were claiming they made a slight profit back when Oil was $40 a barrel (Even this was disingenuous as we know Saudi production costs run about $2 per barrel leaving LOTS of margin.)   Last January, they were all smiling very widely indeed, chauffeured in Bentleys and attended by strings of super-model type female staffers they had sent to US Ivy League schools.  They have almost completely resurrected the old Soviet military machinery and are sitting pretty with oil prices of over $100 per barrel and a very health national production-to-consumption ratio.  Take a look at this chart and note in particular the massive scale of production volume (in millions of barrels per day), ratios of consumption to production and the resulting foreign trade balances of petroleum, the world’s highest-value commodity.

Top global petroleum consumers (red) and Producers (blue) in barrels per day.

Now look at how each individual nation’s Oil consumption versus production imbalance skews its overall economic position on the international macro-economic playing field.  At today’s price of $100 per barrel, the tilt in the playing field amounts to a cash flow of ~$1.5 billion dollars PER DAY out of the US and into those with a positive net oil trade balance.  (Here I simply deducted our local US oil production from our consumption needs and multiplied the net consumption by the price per barrel.)  Remember that number, but also keep in mind that the $1.5B per day number is not really a complete measure because we haven’t even begun to discuss either natural gas, or the necessary ancillary costs in addition direct fuel purchases including having to pay another $720 million a day for a war in a region that would otherwise be as meaningless to the US economy as Ethiopia if it wasn’t replete with oil and natural gas, not to mention the ongoing cost of regional turbulence surrounding Israel and the Palestinians.  This simple estimate also fails to include the staggering costs that are accruing due to environmental damage and global warming from our fossil fuel programs as well, which has been estimated as a potential detriment in the multiple trillion dollar range.  To top it all off, there is another cost that nobody seems to even talk about, the opportunity cost of failing to otherwise invest that astronomical cash flow where it could do more economic good for the nation rather than fueling international competitors.

So where are these US dollars going?  Who is on the uphill side of the field of this $1.5+ billion dollar per day slope?  That’s right, Saudi Arabia, Russia, Norway, Venezuela, Iran, Nigeria, UAE, Kuwait, and Iraq.

US petroleum imports by origin for June 2008.

It’s tempting to take heart from the fact that around 40% of this month’s US oil imports come from Canada and Mexico.  But that ignores the truth of the open global petroleum market.  Simply by consuming at our prodigious rate, we leave all the other countries in the world no choice but to buy fuel from the remaining producers, the other OPEC nations. We set the price in our rapacious demand since the US alone comprises the majority of global consumption.  So whether we buy directly from Iran or not, they benefit economically from the balance of broad international demand while we pay Canada, Saudi Arabia, and Mexico.  Who we buy our oil from doesn’t really make any difference.  Money flows out of the international consumers and into the producers’ coffers.

Worse yet, our fuel trade position is rapidly deteriorating as western petroleum reserves are taped and overall production in the region progressively declines. The tailing off in western production is unfortunately compounded by the rapidly rising US, and international demand particularly through the industrialization of China and India.  The countries with the largest remaining reserves that look to supply the next century’s fuel demands are Saudi Arabia, Russia, Iran, and other less savory international partners.  The OPEC nations would benefit enormously from a continued US focus on drilling for more oil because they know that even with the taps wide open, the US reserves (that petroleum yet waiting to be harvested from underground) are nowhere near large enough to meet even today’s demand even much less tomorrow’s with China and India in the mix. That focus would keep us dependent on them even as their economic leverage continues to increase.

Neither the US economy, nor the individual consumer would benefit from more drilling in the US because even the most optimistic flow rate projections wouldn’t ever amount to more than 2% of global production.  Relative to growing demand in India and China, that effect on pricing is completely negligible. Just consider that 15 years of effort to ramp up US production using the controversial new Arctic oil fields would be overtaken in less than half a year by growth in demand from China alone.  Another way of appreciating the insignificance of the drilling proposal is to realize that from one day to the next, OPEC decided last week to reduce production volume by a greater amount than the Alaska project would produce in the next three years. Just to keep prices high. We have no pricing leverage when we do not control the majority of production volume whether we drill in Alaska or not.

As a slight digression, it is worth asking, “if it won’t change the overall market dynamic, or production or consumption volume by any meaningful amount, and it won’t change the price of oil, which is still basically controlled by OPEC as a consortium, and it won’t do anything to reduce US foreign dependence given the negligible global contribution, why is ANYONE even discussing the idea? Why is it a political issue at all?” The answer is actually easily discovered by examining who would benefit from more drilling in Alaska; just follow the dollars.  It’s the local oil companies and worker’s organizations in Alaska that are spending huge amounts lobbying to induce the balance of cash flow from US production to shift towards their local efforts. Is it any surprise that the governor of Alaska is working to benefit her home state? I don’t think so. But that local benefit for the hometown is being purchased with a continued national addiction to a trade imbalance that is crushing the US economy as a whole. In this case, what is good and appropriate in a Governor’s role is actually in direct conflict with what is good and appropriate for the role of the Vice President. It is the very definition of partisan in its most negative sense that politicians fail to rise above local and personal interest to place the good of the entire country first. End of digression.

So in summary, we find ourselves in a situation where the very industrial success which made us a superpower is driving our need for a commodity which will bankrupt our country if we keep trying to milk the same economic structure long term.  This is an economic structure we have spent the last century building, and it was purposely built upon the cheapest energy source that would fuel our industrialization. The open and free market has led us naturally over the decades to incrementally invest staggering finance in an infrastructure that we are losing control of, both operationally and financially. We happen to be sitting on the wrong plot of land that doesn’t contain the Oil and natural gas commensurate with our industrialized needs and there is no incremental path that will change that fact.  The only good news here is that China and India are eventually headed for similar problems as they industrialize beyond the support of their local resources.  But given their trove of natural and human resources respectively across their vast geographies as yet untapped, the United States may no longer be competitive or even solvent by the time their rapidly growing fuel needs drive them to the same scale of foreign energy trade debt and economic impasse.

As long as the US transportation infrastructure is primarily focused on petroleum which our national reserves alone cannot supply, we are playing a game we cannot win on a field that is too tilted against us.  It’s not simply a matter of figuring out how to play the game better, though any broad efficiency and conservation efforts would certainly help.   We need to change the game completely and find a level playing field that doesn’t depend on raw materials we don’t have. On a level field, American ingenuity and innovation will tell, as long as we don’t abandon those values in the meantime.

More on fostering innovation and changing the global energy game in future posts.

“We threw the conventional microscope out the window and began again,” says John Sedat, a professor of biochemistry and biophysics at the University of California, San Francisco.

Instead of focusing a small spot of light onto cells, the new microscope, which has a resolution of about 100 nanometers, illuminates cells with stripes of light called an interference pattern. When a fine cellular structure, such as a single cluster of proteins embedded in a cell nucleus, reflects this light, it changes the pattern slightly. The microscope collects this light; software is used to interpret changes in its pattern and create an image.”

Check out some of the amazing images:

Two adjoining cells prepare for division by condensing their DNA into chromosomes (red). The membranes around the cell nuclei are stained blue. The green filaments are protein structures called microtubules, which divide the cell’s genome into two equal parts and pull each part into the resulting daughter cells.

Shown here is another mouse nucleus, with the cell’s DNA stained red. The DNA is condensing into chromosomes. The envelope surrounding the nucleus, stained green, is beginning to puncture and distort in preparation for cell division.

Read the entire article over at Tech Review.

It may not look all that interesting in the static photo above, but check out this video of the model in motion!

The natural motion of the model is derived from a neat construct of linkages driven by a single small motor.  Here’s a small animated GIF of the CAD model that abstracts the linkages and the drive motor.

James’ site has some great images of his initial brainstorming that led to a few art pieces and the eventual paper kit product.

(Note that it all starts with a little brainstorming on the chalk board or notebook!)

The folks over at Make magazine have a nice sequence of images outlining their step-by-step assembly of the ~$15 kit which only took them a few hours.

For any of you participating in the FIRST or other similar robotics competitions, this kit is a great intro to customizing complex motions through multi-arm linkage design.  For you artist-types, don’t let the enginering mumbo-jumbo put you off, as it’s only a few hours of cutting, folding and pasting!

For those of you REALLY into this type of design/art, you can find some nice curriculum materials with real-life applications of kinematic design and hands-on crafty engineering excellence over at the fantastic Cornell University KMODDLE site.  They are perfect for high school math, physics, and robotics classes, and some would even inspire the precocious middle schooler.  If there is any interest I can post some follow-on materials and pointers to those lessons.

In the meantime, go forth and design.  And don’t forget to post your comments, pictures and results here if you build one of these or anything inspired herefrom!

I just ordered the pneumatic cylinder and switch parts to start fiddling around with these designs over the summer and will report on progress as I go.  In the meantime, start fiddling yourself!

Here are the assembly instructions for the inline 4, and the parts list

Check out this video of the inline 4 engine running at 1200 RPMs.

Here are the assembly instructions for the V8, and the parts list

Check out this video of the V8 engine running at 1700+ RPMs.

But in defense of the common man, I am here to tell you that math is simply misunderstood.  Worse yet, even the broader institution of K-12 Math Education (note the capital letters to indicate the authorities that comprise “Big Math”) as a whole fails rather drastically to understand what math is.  There are even a number of Universities that don’t really get it (see below). So it’s not really all that surprising that our schools and even our culture at large fail to grasp the significance, the beauty, the elegance, or the joy of practicing mathematics.

When you say “math” at a party, most people remember those painful moments in high school of rote formula memorization and the mechanical repetition of plugging in numbers, the otherwise meaningless manipulation of abstract symbols to achieve higher test scores.

And while yes, the manipulation of abstract symbols can be a useful tool in math, the mechanics and minutia of symbolic manipulation are all too often mistaken for what math really is.  Exercises in notation are the obvious and visible attendants when people are really doing math.  But to anyone unversed in the practice, it is not obvious at all that anything else is going on.  Is it any surprise that the medium is mistaken for the message? Math must be what most people can see, just as written paragraphs are what literature is all about.  So what is the message, you ask?  What is math?  What are all the assiduous formula manipulators missing?

There is logical structure to our universe.  How things are, and can be arranged, how they move, what colors and shapes they comprise, how things sound, how they feel, and what they will do in just a few moments; all of these things can be organized and understood by realizing underlying patterns and regularities.  Then we can use these patterns and concepts to understand and predict other things.  We have developed written languages and notation to illuminate these patterns, the symbols and operations of written mathematics, and better yet, we have developed the symbolic language in such a way that their very structure and regularity don’t just describe like verbal languages but also mimic the very structure and pattern of the reality that they describe.

So ultimately, math is about patterns, logical and inter-related processes, puzzles, models, consistencies, inconsistencies, causality, correlations, and all sorts of profoundly interesting ideas.  Practicing real math is nothing less than a creative exercise in discovery, realization, insight, and deduction.  But sadly, most school curricula mistake facility with the tools of notation to be the sole component of math, leaving out the heart and the joy of it all.  Justifiably bored and disinterested students flee as a result.  I take heart from the fact that what they are fleeing isn’t really math, but an empty simulacrum thereof. But someone needs to tell people that what they fear is not math itself, the wonders of which they might actually enjoy.  I hereby offer two points in particular that will help illuminate the difference between math itself, and its written expression most often practiced in schools.

The Real World and Mathematical Abstraction
The first issue is the prevalent confusion between the overall subject of math; the ideas, the patterns, the observation of the real world and physical objects, and the ‘mathematical’ abstractions thereof, as written in equations.  Both are important and fundamentally intertwined, but rarely clearly related in practice.  Most schools tend to focus on the mechanical practice of the latter, i.e. “…do every odd problem (for which the answers are in the appendix) in chapter 7.”  The operations on the symbols seem arbitrary things to memorize and repeat without any apparent connection to reality or future life other than through the example problem template at the start of the chapter.  Many curricula try to offer “real-world” type problems, but often fail to clearly link the real-world components and steps in solving life’s actual problems with the power of written abstraction that we can use to reason about the real world without having to actually touch it.

Other schools, such as those using Montessori programs, focus on the real world with physical manipulatives, beads, blocks and so on, but then fail to clearly relate what the kids can touch and arrange with their abstractions that can be manipulated mentally and with paper absent the blocks.  Students in these types of programs learn to manipulate and reason physically, but not how to reason abstractly without the manipulatives. Nor can either set in isolation appreciate the intimate relationship between the real and the abstract or the power of understanding various and complementary approaches and representations of fundamental truths.

Neither type of program successfully relates the abstract to the real in such a way that both are clear expressions of the other, and that math is at the same time both grounded in reality and still open to intellectual creativity and insight in the abstract.  This fundamental beauty of mathematics is that the language we have invented to talk about math has as its very essence, a logical structure and process which mimics the real world, but in its abstraction, distills a purified essence of the real world that can be approached intellectually without requiring a descent into the messy and noisy real world.  We can use the abstraction to think about and learn about the real.  We can use the real to refine the abstraction, which in turn further illuminates the real in a continuing cycle of learning.  The real world and its abstraction are simply two sides of the same coin, but it is a very rare program indeed that clearly articulates this realization, and an even rarer one that specifically trains students to use and relate both sides of the coin.

Interestingly enough, even university researchers often miss this point.  Not long ago, a friend pointed me to a New York Times article a few months ago about an Ohio State study claiming that manipulatives and real-life examples did not seem to help in learning math.  You can read the full research article here with a subscription to Science.  Let’s just say that the conclusion of the study surprised me enough to pony up a few dollars to read the full article on the Science site.  Much to my relief, I found the study deeply flawed in its very structure of asking whether students learned math better by studying the abstraction or observing the real world.  Just by asking the divisive question, they missed the fundamental truth of mathematics as an abstraction of the real world with a deep understanding of both and their interrelationship as necessary for true enlightenment.

Math, Art, and Music:  Which of these is least like the others?
This is a trick question, because to many people’s surprise, they are all very similar.  My second point is that Math, as most commonly practiced in repetitive school calculation exercises, bears little, if any, relation to the creative and intellectual exploration of math outside of school.

Real math is not about mindlessly copying example problem steps on homework to nail the SAT test. It is about wonder, and exploration, and discovery, the untangling of interesting puzzles, a creative exercise at every step. I could go on in this vein for a while, but someone has already done a better job of it than I could myself.  If you are even vaguely curious about what real math is like, or fear its evil twin that currently dominates our schools, or just worry about the technical and economic strength of our nation, you MUST read “A Mathematician’s Lament,” by Paul Lockhart. There are some truly sublime sections on musical and artistic analogies with mathematics, and the most on-point critique of K-12 math that I have ever seen.  As a friend of mine recently mentioned, “At first I thought I would get tired of it, but ultimately, I found it to be perfectly on target.”  Amen brother.  If there were anything I would add to the piece it would be about math’s fundamental utility as a tool in science and physics and how each field and approach further illuminates the others.  Here are a few of my favorite excerpts, but seriously, follow the above link to read the whole thing:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.  The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
If your art teacher were to tell you that painting is all about filling in numbered regions, you would know that something was wrong. The culture informs you— there are museums and galleries, as well as the art in your own home. Painting is well understood by society as a medium of human expression. Likewise, if your science teacher tried to convince you that astronomy is about predicting a person’s future based on their date of birth, you would know she was crazy— science has seeped into the culture to such an extent that almost everyone knows about atoms and galaxies and laws of nature. But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight?

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The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.

But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them). A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?

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So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time
to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.

The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.

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There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.

About Paul Lockhart from the MMA web site:

Paul is a mathematics teacher at Saint Ann’s School in Brooklyn, New York. His article has been circulating through parts of the mathematics and math ed communities ever since, but he never published it. I came across it by accident a few months ago, and decided at once I wanted to give it wider exposure. I contacted Paul, and he agreed to have me publish his “lament” on MAA Online. It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen. Written by a first-class research mathematician who elected to devote his teaching career to K-12 education.

Paul became interested in mathematics when he was about 14 (outside of the school math class, he points out) and read voraciously, becoming especially interested in analytic number theory. He dropped out of college after one semester to devote himself to math, supporting himself by working as a computer programmer and as an elementary school teacher. Eventually he started working with Ernst Strauss at UCLA, and the two published a few papers together. Strauss introduced him to Paul Erdos, and they somehow arranged it so that he became a graduate student there. He ended up getting a Ph.D. from Columbia in 1990, and went on to be a fellow at MSRI and an assistant professor at Brown. He also taught at UC Santa Cruz. His main research interests were, and are, automorphic forms and Diophantine geometry.

After several years teaching university mathematics, Paul eventually tired of it and decided he wanted to get back to teaching children. He secured a position at Saint Ann’s School, where he says “I have happily been subversively teaching mathematics (the real thing) since 2000.”

He teaches all grade levels at Saint Ann’s (K-12), and says he is especially interested in bringing a mathematician’s point of view to very young children. “I want them to understand that there is a playground in their minds and that that is where mathematics happens. So far I have met with tremendous enthusiasm among the parents and kids, less so among the mid-level administrators.”

So Where Can You Find REAL Math Materials?

The single program I have ever seen that best captures and relates the importance and interrelationship of the real world and abstract expression in math is Henri Piccioto’s Algebra Lab materials. They offer a fantastic combination of cleverly-designed manipulatives together with sets of problems using the symbolic abstractions, all supplemented with open-ended creative problems that really get you thinking and understanding the BIG PICTURE and how everything is related.  The system explicitly helps students develop facility with approaching problems from either physical or abstract directions, and how one approach can inform and illuminate the other.  Students can directly observe how different approaches can be used to best advantage for different types of problems. The crowning component of the system is the great set of open creative questions that get students exploring patterns and symbolic reasoning on their own using the physical and abstract tools they have developed.

The Lab Gear that comes with the program is comprised of a series of carefully-sized blocks that help articulate what algebra is really about, arrangements and grouping, and what things like factorization really mean.  You fiddle with the blocks, and get an intuitive feel for how things really are, and then learn about the abstractions, the principles and equations.  You expose fundamental truths of the world and learn how to discuss them!

Here are some animated graphics that were created by George Collison of INTEC
(International Netcourse Teacher Enhancement Coalition) to demonstrate the Lab Gear materials in use to help illuminate polynomials and factoring.

Anyone else want to get together and write some materials on Trig, Calc, or Pre-Calc?  At the very least, leave your comments below as to where you have found strong materials that make the real-to-abstract connection!