Cocktail Party Physics

all you need is the rocket experience

If you've been paying any attention here, you know what a huge supporter of space exploration I am. Manned missions to Mars, moon bases, shuttle flights, space tourism: I love it all. It's probably part of my love of travel; what could possibly be more different from home than another planet? Than outer space? My fascination also stems from an early love of science fiction and early indoctrination with Star Trek (I was 6 when the show first aired). But its simultaneity with the first moon landing (the original series ran from 1966-1969, in case you actually don't know this) was probably the clincher. Not only could we dream about going to other planets, but hey! We could do it! Nothing could be more inspirational to a kid than watching a dream come true, or in this case, watching people actually setting foot on an alien world, when all you've done is read about it or seen fictional stories about it. It gives one a "damn we're good!" feeling not to be rivaled or duplicated in any other way.

This is one of the reasons I feel shuttle flights are so important: not just for the scientific knowledge that we gain with each one of them, but for their immeasurable inspirational quality. Having lived most of my life with the fact of my fellow humans flinging ourselves recklessly into space from right here at home, I can't imagine that not happening, or imagine having to look to, say, China, for the next exploratory manned rocket launch. I do think space exploration should be an international effort. Nothing makes us realize that we're all part of one world than seeing the big blue marble from lunar orbit. Yeah, CGI is a marvelous thing, but it ain't real. But with shuttles due to be retired and NASA opting for unmanned Mars landings, how do you inspire young people at home to get the kind of rocket fever so many of my contemporaries had?

Cue Snoop Dogg and Buzz Aldrin:

P.S. I'm still processing my China trip and working my way through the essays my students wrote on Science in China, but there will be future posts on those topics. I just figured it was about time I got back to work here.

nascar driver vs. wiley coyote: a real-life seriously inelastic collision

I'd like to claim that no animals were injured in the making of this blog, but (insert Joe Flaherty from SCTV doing Kirk Douglas) I'd be lying.  I put the most graphic picture near the very end of the post so those of who, like me, have weak stomachs could avoid it.  It isn't pretty. Especially if you like coyotes.

Brad Coleman, a NASCAR driver for Joe Gibbs Racing, was testing (like doing homework for those of you unfamiliar with NASCAR) at Toyota Arizona Proving Grounds when he hit a coyote going about 190 mph.  (Clarification:  Brad's car was going 190 mph.  The coyote was just meandering around in the middle of the track.)  Natural selection works:  If you're dumb enough to wander onto a racetrack with 130 dB cars roaring past, we don't really want those genes propagating, do we?  This is a perfect example of a real-life collision.

Collisions come in two extremes.  In a perfectly elastic collision, two objects bounce off each other and lose no energy.  We like those problems because students must apply conservation of energy and conservation of momentum, and you can write the problem so that they have to use both. The other extreme is the perfectly inelastic collision, in which the two objects stick together after the collision.  We always use "two cars hit each other and their bumpers stick together".  You see that a lot in real life, don't you?  If you scroll down the page, you'll notice (perhaps with the exception of legs), the two objects really did stick together after the collision. 

An American coyote, according to the Phoenix Zoo, weighs between 50 and 75 lbs. (Lemma:  Coyote weight increases as you move North.  Mexican coyotes may weight as little as 25 lbs while those in the northernmost reaches of the US tend to be 60-75 lbs.)  For the sake of argument, let's assume the coyote is 50 lbs (22.7 kg mass).  A NASCAR Car of Tomorrow weighs 3450 lbs and this particular driver weighs about 185 lbs, for a total weight of 3635 lbs, or a mass of 1649 kg. 

Momentum is the product of mass times velocity and measures essentially how hard it is to stop a moving object.  The faster something moves, the harder it is to stop.  The more massive it is, the harder it is to stop. It is much easier to stop a speeding coyote than a speeding race car and much easier to stop a slow car than a fast car. Writing it as a formula (at left), momentum is symbolized by p because m was already being used for mass.  v is velocity, so p=m times v.  Yeah, I know it's technically a vector, but we'll assume that the car was coming down the straightaway and make it a one-dimensional problem. 

Conservation of momentum says that the amount of momentum the "system" (meaning the car and the coyote together) had prior to the collision is the same as the amount of momentum the car and coyote had after the collision. If you don't want to go through the math, go ahead and skip down to where the equations end and just look at the answer.  It's OK.  Really. 

For those of you still with us:

Here's where real life makes the problem easier.  The dumb ole coyote is sitting on the track.  Even if it was moving, it was moving really slowly compared to the 190-mph race car, so we'll approximate the coyote as having zero velocity before the collision -- which means the second term on the left-hand side of the equation becomes zero (yay!). 

After the collision, the coyote and the car are stuck together, which means that the coyote is going the same speed as the car because - well, look at the picture below.  It is basically glued to the front fender.  Our equation thus simplifies to:

Driver Brad Coleman - the only surviving eyewitness - said

"I see this thing, it must've been 100 feet in front of me, just jump out.  Right when I saw it come out from under the guardrail, I was like "that's a coyote"...  I didn't see anything in the mirror, so I was like 'I wonder where it went'..."

Don't you think you might know if you hit a coyote?  Let's look at how much hitting the coyote would slow down the race car.  As I constantly remind my students, solve for the unknown before you plug in any numbers.  We want to know how much the coyote slowed down the car, so we want to know how fast the car and coyote are going after the collision.

Using the fact that 190 mph is about 85 m/s, the car has a mass of 1649 kilograms and the coyote is about 50 lbs (corresponding to a mass of 23 kg), the speed of the car and the coyote would be:

 

which is about 187.5 mph, so the coyote only slowed him down by about 2.5 mph (1.3% of the initial speed).  If you hit the coyote doing 60 mph (26.8 m/s), your speed after would be 26.4 m/s, which is about 59 mph or 1.6% of the initial speed.  The slower you're going, the larger the percentage speed you lose.

How did Brad know he hit the coyote? 

"It (the car) just started smoking like crazy and it smelled terrible...I said, "Guys, I hit a coyote.  I'm going to come in because I think it screwed up the radiator.  I think it clogged up the grille a little bit."

You have to realize that NASCAR crews are really into practical jokes, so I'm sure the crew was sitting back in the garage wondering if this was a joke, or if Coleman had really hit an animal.  I bet they were surprised to see the car come in looking like this:

This is going just a little too far for a hood ornament.  I hope whomever had to clean up the car got some type of bonus, as they really deserved it.  Car-car and car-wall collisions are never quite this messy. 

Here's a great problem for all the kids starting physics this fall.  Give them the same information that's available on the Internet:  the car was going 190 mph and the coyotes was pretty much 100 ft ahead of the car and just sitting there.  Have them calculate:

1. How fast the car was going after it hit the coyote?
2. How fast do you think you could move your foot from the gas to the brake?
3. How long would it take the car to go 100 feet at 190 mph? and
4. What approximations did you make and how would your answer change if your approximations weren't valid?

The last question is probably the most important. The picture suggests that coyote mass was not strictly conserved, so the mass of the coyote plus car is probably a little smaller than I estimated and the car didn't even slow down that much.  Real life almost always demands approximations.  The important thing is knowing how much error your approximations might introduce if they are wrong.

Partial solution:  190 mph converts to about 279 feet every second.  Brad estimated that the coyote was 100 feet ahead of him, so he would have traveled the entire distance in (100 ft/279 ft =) 0.36 seconds.  Go see if you can shift your foot from full out on the throttle to the brake in a third of a second.  And there are a lot of other possible questions.  How fast would Brad have had to decelerate to miss the coyote.  Why didn't he swerve around to miss the coyote (remember reaction time!)  Why is hitting a deer on the highway with a passenger car so much worse than this was?  (Deer are a lot heavier!)  Deer also have a higher center of gravity (longer legs) - how would that affect the collision?

This reminds me of a situation in which I'd written a problem that asked, "Texas Motor Speedway sells enough hot dogs each year to circle the track six times.  How many hot dogs do they sell?"  and I got told by someone from a federal funding agency that I had made a mistake because I hadn't given the length of a hot dog and said whether it included the bun. (Your tax dollars at work, folks...)

You get on the web and look up that Texas Motor Speedway is 1.5 miles long and estimate that a hot dog is about six inches long (with or without bun, doesn't change much).  This isn't rocket surgery, folks!  I don't know about you, but my real-life job rarely presents me with precisely the information I need to solve a problem. 

I'm about to undergo a lot of changes in my life.  Until I moved out of my office last week, I had a stuffed Wiley Coyote hanging on the wall.  He was there for a very important reason - Wiley is my totem animal. Have you ever noticed that Wiley never falls until he looks down?  Seriously - as long as he doesn't realize he's doing something he really shouldn't be able to do, he can do it.  Sometimes, we are our own worst impediments when we talk ourselves out of being able to do things.  My motto for the next few weeks is going to be "Don't look down". 

"Meep meep" indeed.

science, hollywood and the web

We're down to the three-week sprint on The Damn Book, so there is no time at all for blogging at the cocktail party, and even my co-bloggers are uncharacteristically silent. So here's a recent online video interview (a vlog, as the kids are calling it these days) with Robert Scoble for Building 43 about my work with the National Academy of Science's Science and Entertainment Exchange. When not working on The Damn Book, this is what I am doing. As I say in the interview, I'm very excited about the potential of things like DVD extras and Web-based interactive sites -- like the marketing gurus for Lost came up with to launch the Season 5 DVD for that series. Hopefully the radio silence at the cocktail party won't last too long. In the meantime, maybe this can tide everyone over. (By the way, the Exchange now has a fledgling blog, called The X-Change Files. Check it out!)

assume a spherical zombie

Physicists often use 'model systems', which are systems that share properties with more complex systems.  We try to gain fundamental insight into the more complicated system by studying the model system.  For example, an alloy with seven elements in it might share some subset of properties with an alloy made from only two elements.  You might be able to learn something about the seven-element alloy from studying the two-element alloy because there are simply fewer variables.  The information you learn from your model system can give you a stepping stool for trying to understand the more complex system.

Biologists also have their own types of model systems, called model organisms.  Model organisms serve the same purpose - they are easier to deal with and share some of the features of more complex organisms that we don't yet understand.  There is a field of mathematics called biometry, biostatistics or biometrics that combines math and biology to study everything from epidemiology to ecological forecasting.  When you asked your math teacher "what am I ever going to use math for", this is the field that provides some really interesting answers.  Biometrists usually focus on real systems, but a recent paper in Infection Disease Modelling Research Progress 2009 features a very underutilized model system.

Zombies. 

In particular, the "classic zombie" as typified by Night of the Living Dead, as opposed to the more modern zombie.  Modern zombies are a little smarter and move a little faster than classic zombies and thus are harder to model.  As an experimentalist, I question the choice of zombies for a model system, as zombies are notoriously ill-behaved.  They physically decay on very short time scales (often losing limbs without notice), they are mindless, make annoying noises and have as their sole purposes eating people or turning them into zombies. (Now that I think about it, I'm questioning whether a couple of my colleagues might fall in that category...)

This is more than just a cheap shot at getting your research on NPR.  Maybe we don't have a zombie problem in real life, but we do have AIDS, river blindness, West Nile, malaria and a host of other diseases that spread as individuals make contact with either infected individuals or carriers (such as mosquitoes).

The paper was written by three students (Philip Munz, Ioan Hudea and Joe Imad) and a professor from the Department of Mathematics and Faculty of Medicine at the University of Ottawa Robert J. Smith?. No, the question mark is not a typo.  It's actually part of the professor's name.  Being a statistics type, he knows how many "Robert Smith"s there are in the world and thought he might distinguish himself through creative punctuation.  He notes

"Not that the question mark actually solves that, but at least it differentiates me from that guy from The Cure. It's been twenty years now and sadly his career shows no sign of drying up."

One of the things I like about this paper is that is demonstrates how you start with a relatively simple model and gradually add elements to make the model more complex - and more capable of modeling reality.  It also reminded me that every field has its own particular notation that makes no sense to people not in the field.  I'm going to depart a little from the notation used in the paper because I personally found it confusing to represent the number of dead people by the symbol "R".  But that's just me.  Stat mech taught me that anything representing a number is N with a subscript.

The first model in the paper assumes three classes of beings:  Humans, Zombies and Dead.  I'm calling the number of humans N_H, the number of zombies N_Z, and the number of dead N_D. We're interested not so much in the actual numbers in each category, but in how these numbers change with time. If the number of zombies gets bigger faster than the number of humans being born, we've got problems.  So we don't care as much about N_Z as we do about the rate of change of N_Z with time.  Just as speed is how fast your position changes and acceleration is how fast your speed changes, the rate of change in zombies is the net increase or decrease during some period of time.

To calculate this number, you have to understand the zombie rules.  A human becomes a zombie when bitten or otherwise infected by a zombie.  A dead person can also be resurrected into a zombie.  Let's say we had 6 humans turned into zombies every hour and four dead people resurrected into humans every hour, so there would be 10 zombies created every hour.

Much to my surprise, you can kill zombies, so calling them the 'undead' is technically wrong.  You can kill a zombie by removing their head or destroying their brain.  So let's say 3 zombies per hour are killed every hour.  Then the rate of change in the number of zombies is:
 

For those of you scoring along at home, this is a translation of the middle equation on page 135 of the paper into English. 

All that was just to calculate the rate of zombie number change, so we also have to write equations for how the number of dead and the number of humans change.  To make the interconnectedness of the numbers clear, I've taken the liberty of reproducing the diagram on page 135 and putting a translation into English immediately below it.

So the rates in the problem are the birth rate and death rate of humans, how many humans become zombies, how many zombies become dead and how many dead become zombies.  Mathematicians call the resulting equations a coupled system of ordinary differential equations.  It's a system of equations because we have to write three equations to describe the system - one for how the number of humans changes, one for how the number of zombies change and one for how the number of dead change.  Remember from seventh grade algebra that in order to be able to solve a system of equations, you have to have one equation for each variable.  They're coupled because the same variables appear in multiple equations.  For example, the number of zombies that become dead will appear in the equation for the rate of change in the number of dead and the equation for the rate of change in the number of zombies.  They're differential because we're looking at rates, which are derivatives.  They're ordinary because we're looking at the rate of change with time for all three equations. This isn't a pejorative designation: F=ma is an ordinary differential equation, too. 

Using this model and assuming that the infection occurs quickly (meaning we can ignore the birth and death rates because they are small during the time over which the infection occurs), the paper authors show mathematically that zombie-human coexistence is impossible.  The infection will spread and zombies will infect everyone in pretty short order.  Bummer.

But the reality, as they go on to point out, is that most infections are latent:  that is, people don't turn into zombies immediately, but they walk around (for about 24 hours, according to The Zombie Survival Guide) looking normal even though they are doomed to become zombies.  This means we have to introduce another box in the diagrams above to represent people who will, but have not yet become zombies.  The researchers call this group 'Latents' which they represent by "I".  (I know, I shouldn't be smirking since my discipline represents momentum by "p" and angular momentum by "L".)  In addition to a fourth box, you'd have to include arrows from Human to Latent, Latent to Dead and Latent to Zombie, which gives you a system of four coupled ordinary differential equations to solve. I didn't bother drawing it because the authors show that, even with a latency period, we are doomed and zombies win in the end.
 
A null result is less interesting that a non-null result.  If there is no way for us to defeat the zombies, do you think NPR would care?  So the researchers asked themselves "what would happen if we were able to quarantine some of the Latents or Zombies?"  OK, now we've got a mess because we have five boxes, some Zombies or Latents would be killed trying to escape quarrantine and they also include the possibility that some dead people would become re-animated as 'free' zombies.  The results from this analysis are that stopping the zombies depends critically on getting infected people and zombies into quarantine.  If you don't get enough of them into quarantine fast enough, the zombies win.  Again.

Next, the researchers allow the possibility that a cure for zombie-ism is developed.  The cure transforms zombies into humans, but it doesn't protect them from becoming re-zombified.  If a zombie who was resurrected from the dead is cured, he becomes human again.  Got all that?  Well, if you don't, don't worry about it because we get only marginally better chances of survival:  Humans are not entirely eradicated, but not many survive.  At some point, the individual humans left are going to realize that curing the zombies is not statistically worth the chance.

Finally, the researchers consider a "brute force" method - what if we just killed as many zombies as we possibly could as fast as we possibly could?  When they solve the system of equations, they find that this model provides a much higher probability of human survival, but only if you kill zombies really aggressively.  The slower you kill the zombies, the higher the likelihood they win.

I love the tongue-in-cheek discussion section:  "The key difference between the models presented here and other models of infectious disease is that the dead can come back to life".  They do point out that there are plenty of real-world cases in which this model might apply, like modeling people's allegiance to political parties.  I've got to find some way that nanomagnetism might relate to zombies, because I'd love to see whether one could get something like that into Phys Rev.

I realize that these are students, but I do have to raise a quibble with their results. Given what we know about zombies, they really should have extended their final model (the one that gives us humans a chance of survival) to include a latency period.  I haven't worked through the math, but my gut-level instinct is that if you assume a 24-hour latency period, the zombies have a much better chance of winning unless we start aggressively killing off not only full-blown zombies, but people we suspect have been infected as well.  What a great plot for a movie...

IMPORTANT GUY AT CDC:  Well, Dr. Smith?, I don't care how right you think your model is, we're not taking action until it's been peer-reviewed in a major journal. 

GENERAL:  I don't pretend to understand anything having to do with math - never did get into that 'number thing', but if the numbers say we need to start killing millions of people to save the country, let's get going.

SMITH?: Um... did I explain the concept of a model system to you guys?

Really, this would be a great plot:  scientists develop a method for saving civilization and it hinges on whether the survival of the world is worth killing some number of people who aren't infected?  Uh, don't laugh.  Ethiopia ordered all pigs in the country to be killed as an approach to prevent the H1N1 virus from spreading.)  There's a good twist:  let's put the zombie invasion not in Los Angeles, but in Libya or Iran - somewhere a significant number of Americans might not feel so bad about invading wiping off the map. 

OK, that was a fun and fluffy piece of science - the paper got enough coverage that people are griping about why the Canadian government would fund such a thing. (For the record, it was a student project and the professor was acknowledging the agencies that fund him in general.)  But there is a serious issue here about science journalism that gets back to the 'science popularizer' vs. science journalist dichotomy.

You can't argue that the research results were being reported for the sake of science, so besides being cute, what was the message here?  I could come up with a couple slants on reporting on this that go beyond 'humans vs. zombies, ha ha'.  For example, this is a great case study of graduate students learning about communicating science beyond their own subdiscipline. For example, from canada.com

"If you look at it in a more realistic way, zombies are about the same as any other major infectious disease — they get out and we try to eliminate them," said Joe Imad, a University of Ottawa mathematics student and one of the paper's co-authors. "Modelling zombies would be the same as modelling swine flu, with some differences for sure, but it is much more interesting to read."

But the even bigger story -- which I haven't seen in any of the stories I've found is that Smith? and his students do work with important problems.  After I came up with the idea of setting my zombie vs. scientist movie in a less-developed country, I looked at some of Smith's other work.  One paper about to be published in the Journal of Health Care for the Poor and Underserved points out that there are a slew of neglected diseases, such as leishmanaisis and dracunculiasis that don't get as much attention (or funding) because they: affect less-developed countries, and aren't as fatal, but do have long-term impacts on large numbers of people not so much by killing them, but by socioeconomic effect and long-term suffering.  Diseases with high mortality rates attract a lot of attention (and funding), but chronic diseases can actually have a much greater impact on cost of care, lost days of work and social ostracism.  These diseases are much more like the zombie scenario - where the infected don't die, they live and can transmit the disease to others.

The graduate students who contributed to the paper are unlikely to specialize in zombie studies, but if they pick up their advisor's obvious passion for using math to make the world a better place one or all of them just might be the person(s) who make a difference in the lives of people who don't have the ability to lobby NIH or Congress. And if they pick up his sense of humor as well, they'll probably have an easier time getting in the door of those institutions as well.

NEW VOICES: the iceman cometh

NOTE: I would love to be blogging about zombies or the quantum Hamlet effect or The Time Traveler's Wife, or any number of cool science-y things, but instead I am STILL writing The Damn Book. Here's Alex Morgan again, saving the day with yet another nifty guest post.

I was at the Traverse City Film Festival a few weeks ago and saw No Impact Man, which opens in September. It documents a year in which Colin Beavan and his wife, who live in a small Manhattan apartment, give up everything they are doing that might harm the environment. They decide not to buy any new clothes or household items. They stop getting take-out and restaurant food (including Starbucks coffee!) and buy food only if it is produced within 250 miles of their home. They use bicycles for transport and take a train only for a few trips outside the city, still within the 250 mile limit. As the year progresses, they give up more and more. For example, they stop using toilet paper. After about six months they flip the circuit breakers and go without electricity, meaning candles for lighting and nothing for heat.

The sticking point in their experiment comes after giving up electricity. They have no working refrigerator. How are they going to cool food, for example, milk for their young daughter? A ceramic pot-within-a-pot device that is supposed to use evaporation for cooling doesn't work in their apartment. Finally, they end up "cheating" by borrowing ice from a neighbor's refrigerator. Their journey of no-impact living is documented further on Beavan's website. What interests me is the ice.

I was a kid in the 1950's, and at that time in Savannah the iceman would still make regular deliveries to people's houses. My mother's mother – we called her Nanny – had an icebox as well as a refrigerator. Nanny was born about 1880 and grew up without electricity in a small town in Tennessee. Her family used kerosene lamps for lighting and iceboxes for food. When electricity became available, she didn't trust it and never gave up her lamps or her icebox. She used both alongside of their electric descendants. The kerosene lamp on her night stand was never allowed to go out.

To my child's eye, the iceman was a scary giant. He would come to the house with a monster cube of ice on a leather pad on his shoulder. He used a hook to hold the ice in place and then swung it down into the icebox in one practiced sweep. Nanny's icebox was the smaller variety that came up to my eye level but to an adult's waist. It was white metal, like the refrigerator, not the wooden kind.

Nanny kept mason jars of water and various other items in the icebox. I remember the butter sitting in a dish on top of the ice itself. The ice melted, but slowly, so that it lasted until the next day. The melt water flowed into a pan that had to be emptied regularly. The icebox was messy and smelly, but that didn't bother Nanny. She was used to it.

Where did the ice come from? In the 1950's, it was made at "the ice company," situated, I remember, in downtown Savannah near the post office. But where did it come from in 1880? I asked, and the answer surprised me. It came from lakes in New England.

Beginning in the early 1800's, "ice cultivation" became an economically significant industry in the United States, comparable to agriculture. Lakes in New England, and other parts of the country with hard winters and reasonable access to transportation, were designated for ice making. In the winter, after they were frozen to between one and two feet thick, they were cleared of snow. Then the ice was scraped clean, scored into squares, and sawed out as blocks, which were stored in warehouses. These "ice houses" needed effective insulation – commonly made from sawdust – and a system to drain the water, so that the ice would stay as dry as possible.

Ice farmers favored sawdust for insulation also when shipping their crop. Savannah, Charleston, and New Orleans were early ports of call in the ice trade, which eventually included Britain – American ice was considered higher quality than Norway ice – and even India. This cultivation and harvesting of ice diminished as artificial means of production became more efficient, but it lasted well into the Twentieth Century. The principles of ice manufacture had been known before the Nineteenth Century. William Cullen at the University of Glasgow in 1748 demonstrated that the evaporation of liquid ether generated cooling.

The International Exhibition of 1862 in London featured two ice-making machines – one based on the vaporization of ether, the other on ammonia – which thrilled the crowds in the summer heat. However, the journey from science to engineering to production can take many decades, where efficiency is often the issue after feasibility has been established. It takes energy to run the motor that operates the pump that compresses the refrigerant in a typical ice maker. Efficiency and reliability of refrigeration machinery improved to the point that seventy-five years after the London exhibition, harvesting "natural ice" was no longer economical. But now our society is reconsidering all its energy costs.

The no-impact man had to borrow refrigerator ice from his neighbor. But within 250 miles of Manhattan there are lakes that freeze in the winter, so the old technology of ice cultivation, harvesting, and transport could be resurrected to get no-impact ice. That is, most of the cost would be in labor and transportation, the same as for small-scale food farming. Such cultivated ice would be "more expensive" than machine-made ice, but perhaps only if the cost doesn't include global warming and other ecological disasters looming for our children and grandchildren.

My grandmother had no problem making do with less impact. Colin Beavan and his family were willing to be inconvenienced and to pay more for their food. Ice farming is perfectly feasible, perfectly aligned with the spirit of no-impact.

math class is tough

In Jr. High I failed basic algebra. I'll be the first to admit here that fourteen year olds have a very bad sense of what is the end of the world and what is not, so I try to keep my experience on this subject in perspective. But it seems clear from forums like this one and so many others that people's early experiences with mathematics resonate far down the line. And with math there's the feeling of absolutes. Either you are good at it or you're not, either you get it or you don't, either you are smart or you are not supposed to get it anyway. Because math is hard right?

At the start of the new school year I was embarrassed to show up on the first day and face the other kids who asked me why I wasn't in the "smart class." I wasn't by any means the highest achiever at school, but I still played on the team. Failing made me want to cling to my smart label even more, yet because of that one class I felt it all slipping away. My sense of identity was at a fragile state, resting timidly on band stickers I put on my book covers and the macrame in my locker.  I think failing math hit my sense of self confidence pretty hard.

I hated math. I hated it for making me feel stupid, and I hated it because there was nothing enjoyable about it. It was just there, like a big black wall I would run into every once in a while. It would stare me down, not letting me know why it was there or why I should care, just intimidating me and forcing me to walk around it. Like a good teenager, in the face of my frustration and my shortcomings, I rebelled.

I slunk into my new class with scowling eyes and a plan of attack: I would push my frustration back on the establishment. I would not conform to math! Math was pointless, suppressive and totalitarian. The whole thing stifled my creativity and individuality. I, forward thinking 8th grader that I was, would not be contained by such a bourgeoisie institution.

Like I said, it was 8th grade. Passions were high and political terminology was new.

I believed there was nothing this teacher could do to make math any less frustrating, or to make me feel better about being held back. I was ready to hate the crurel, toad-like miser who would waddle up to the chalk board and scratch out meaningless equations that he would force me to memorize. Oh, what a shock he was in for.

And then my plan immediately started to crumble when I found out that my teacher was young and female, and began the class with a hands-on demonstration about geometry and a short lesson in physics. This was the first real class she'd taught, and while I still harbored hopes that she would be mean or useless, she turned out to be neither. She was funny. She talked to us about what math meant, where we could find it in the world, how it was applied and why she loved it. Maybe that was more of a shocker than learning what math was good for. Finding someone who liked it just because. I didn't know on the first day that I would eventually come to trust this woman like a close friend, and confide in her the anger I felt at not understanding. And while I didn't realize it at the time, the fact that she was female was also having an affect on me.

That's not to say things turned around right away.

The first few weeks of class went something like the previous year - I'd sit up late at the kitchen table with my dad, trying to make headway on my assignments and frequently breaking down into tears of frustration. I felt like a failure, and I was anxious and afraid to go to school the next day with yet another incomplete assignment. Worst of all I felt helpless to change the situation. No text book could tell me what I needed to know, and this was before we were privy to any online homework sites. My dad did the best he could, but he wasn't a math whiz either. He tried to be patient and would sit and read through my text book, trying as hard as I did to understand it all. A few times I turned my frustration around on him and would storm up to my room, doors slamming. The big black wall was surrounding me. I couldn't find a way around it, I couldn't see where I was going and I was starting to feel like any confidence I had in my intelligence was all wrong.

The first Friday afternoon in class, my new teacher opened up a book called "Five Minute Mysteries." These short stories were all like episodes of Law and Order or Murder She Wrote, in which a tantalizing mystery needs to be solved. At the end of the story you are asked to solve the case, and the answer is listed at the end of the book. Personally, I thought of them more like episodes of The X-Files (in retrospect I still harbored the geek gene). I had read shorter versions of these stories as a kid and I loved them. I loved hearing them and thinking, thinking, thinking, and then getting the answer.

These stories were a strong teaching tool, because through them my teacher revealed the essence of mathematics. Each problem is a mystery waiting to be solved. If you learn to speak the language and acquire the tools, you can break down the mystery, interview the witnesses and balance the equation. Quite frequently, the path to the solution is not straight forward, but the result of a series of creative moves. This connection to something that I loved so much started to push back the big wall. It gave me some space to breath. I felt like maybe I could stick with a math problem if I thought the ending would be like solving a little mystery.

I can't recount all of the teaching techniques that my teacher used to help me understand things. I do remember afternoons in her classroom, still struggling with problems. She saw the tears a few times. I remember in class playing with demonstration toys and doing group exercises. I remember when the after school homework sessions started to change. I was still getting the answers wrong, but at some point I finally trusted my teacher, and her abilities, enough to start asking the "stupid" questions I'd kept hidden in my previous classes. I wasn't afraid to tell her, after fifteen minutes of struggling, that I still didn't get it. And I remember that she was not only patient but happy to be there, excited by math and glad to walk me through things as many times as I needed. It takes a lot of questions to get the answers in mathematics, and quite frankly, I think many teachers are not equipped or not willing to listen to them all.

One winter day, I distinctly recall sitting all the way through algebra, tapping my foot in anticipation of its end. As soon as the bell rang, I hopped up and ran to the front. I grabbed my teachers attention and began to spout off what I felt was a perfectly wonderful and exciting story of how I'd solved one of the homework problems. I'd told her how I'd thought it was funny that I couldn't do it at first and how I finally figured it out. After that day, I genuinely started to look forward to math class.

Slowly the dark, unreachable cliff that stretched up before me and all around me, fell away. The wind came in and the sun came out and I was sailing over a vast open sea. New skills appeared on the horizon like islands, and I navigated the rough waters with patience and confidence. It was fun to see how fast I could shake the answer out of a short problem, or take a tricky word problem and gently extract the equation. I was Moulder. I was Scully. I was good at math.

I still didn't think I'd become a physics major, even at the end of high school when I'd finished up two years of calculus. With two more great teachers along the way I'd fallen in love with that subject as well, although I was still never the very best at it. I just acquired the tools and continued to crack the cases. There are many people who can't focus on math unless they have the application close at hand, and while I like this, I think there are many people who would truly enjoy the satisfaction of solving math puzzles if they only had someone to help them overcome the big wall. It's a mind block for many people to learn the language of math, but it's not impossible. Like learning any language it's often stressful enough to try to string together the correct words in the correct order without considering implication and tone. But at some point the words just start to flow.

This topic seems to generate more discussion than many others on this site. There are probably many folks who feel the way I did, maybe without realizing it: that they want to understand math, and want to get around that wall, they just don't know how. I give so much credit to this teacher I had, and many others I had after her, and I continue to believe that good teaching has everything to do with getting kids to move past that wall early. Students should have the resources and support to approach their frustration with math as many times as it takes to move past it.

In my time at the American Physical Society I worked with members of the education department there. Their group explored how one of the biggest problems in our primary school education system so far as math and physics goes, comes down to a lack of communication between teaching departments and math and science departments at universities. Quite often, even if undergrads want to become math and science teachers, they are separated from taking real math and science classes. This lack of communication also means that those departments do not assist in building curriculum that the future math and science teachers take. Just like any other subject, teaching mathematics is not the same as teaching English or history, and the curriculum needs to reflect that.

So it's true that math class is tough. But I so titled this post in a swing of irony. When the phrase "Math class is tough," was uttered by Teen Talk Barbie at her debut in 1992, Mattel was met with some understandable outrage. The comment, coming from a female cultural icon, was not sympathetic or followed up by a "But we'll get through it!" It implied what too many people have already pointed out: giving girls the notion that they will do worse in mathematics than boys, or are less apt to do well, is a self fulfilling prophecy. Talking G.I. Joe never uttered such a phrase. I won't ever be able to measure how much of a difference it made that my eighth grade algebra teacher was a woman, but I know it did. The struggle that I had with math was definitely not unique to my gender, but in a male dominated field, it did change how I overcame it. I'm not sure where I'd be if I hadn't accepted the idea early on that not only could I do math, but I would not be anomaly or rarity if did. Women do math, just like men. And some women will fight to study what they love no matter what; but why should they have to? 

Talking Teen Barbie is only one example of the many gender biased sentiments expressed by kid's toys these days. Comedian Jared Logan remarks that commercials for girls' toys during Saturday morning cartoons are alarming in their disregard for eliminating stereotypical gender roles ("It's like Susan B. Anthony caught an assassin's bullet and suffrage never happened.") He goes on a rant about a fictional "Shop 'till You Cook" game for girls. On that note, I'll leave you with a segment of an NY Times article that some of you may remember from 1993, when a group of people took this issue into their own hands: 

Your son tears the wrapping paper off his fierce new "Talking Duke" G. I. Joe doll and eagerly presses the talk button. Out comes a painfully chirpy voice that sounds astonishingly like Barbie's saying, "Let's go shopping!"

Does your son:

A) Furiously vaporize the doll with his own phaser rifle?

B) Go shopping with Joe?

C) Say: "Mom, I suspect we're the lucky victims of an elaborate nationwide publicity stunt designed to ridicule sexual stereotyping in children's toys. This barbaric little action figure you gave me may turn out to be a valuable collector's item."

If the answer is C, your son may be a collector's item himself, for he has correctly divined the latest socially conscious news media prank to hit the nation's toy stores. Painstaking Alterations

For the last several months, a group of performance artists based in the East Village of Manhattan has been buying Talking Dukes and "Teen Talk" Barbies, which cost $40 to $50 each, painstakingly swapping their voice boxes and then, with the aid of cohorts, replacing dolls on the shelves of toy stores in at least two states.

The group, which asserts it has surgically altered 300 dolls, says its aim is to startle the public into thinking about the Stone Age-world view that the dolls reflect.

The result is a mutant colony of Barbies-on-steroids who roar things like "Attack!" "Vengeance is mine!" and "Eat lead, Cobra!" The emasculated G. I. Joe's, meanwhile, twitter, "Will we ever have enough clothes?" and "Let's plan our dream wedding!"

NEW VOICES: sense and sensor-ability

NOTE: Still crunch time on The Damn Book, but guest blogger Alex Morgan is back with a quick overview of the state of robotic cars -- and showing a bit of love for classic sci-fi B movie The Forbidden Planet in the process.

I loved robots when I was a kid. I used my erector set to built one, which rolled on wheels connected to an electric motor. It couldn't go far, since the motor had to be plugged into an electric outlet, but still … it was wonderful. I was inspired by Isaac Asimov's robot stories and the robots in the movies. Robby the Robot, from the 1956 movie Forbidden Planet, was a multitalented home-butler-type robot, the kind we were all supposed to own in the twenty-first century. He was as smart as a human, and he walked and talked like one, too.

Alas, such robots are still a long way off. But one kind of robot is coming along nicely, the robot car. I love the idea that one day I might be able to tell my car where I want to go and it would take me there, while I relax and use my cellphone or laptop without endangering anybody. Or, I could send the car to school to pick up my kids and cart them to soccer practice, while I do something productive, like relax. The U.S. military in the form of DARPA (Defense Advanced Research Project Agency) is now providing major support to bring robot vehicles into reality as quickly as possible, motivated in part by roadside bombs, suicide bombers, and other hazards of war.

Over the last few years, DARPA has been sponsoring robot car competitions. In 2007, the contest was called the "Urban Challenge," since it took place in a city-traffic environment. Roboticists at Carnegie Mellon University and General Motors (with support from many other companies) built the winning car, called "Boss," and it illustrates the state-of-the art in "autonomous vehicles." (I'm slighting the other contestants merely for simplicity. Other entrants also were impressive.) Boss is not ready to be unleashed on Anytown, U.S.A, but its descendants will be, probably over the next ten or twenty years. And parts of the technology will be used in production cars sooner.

If you have an ordinary car, you need three things to turn it into a robot car: first, a compact, rugged, powerful computer; second, some way of letting the computer operate the steering, brakes, gearshift, and engine; and third, sensors and a brain to enable the robot to "see" and "think." Good-enough computers are finally available. The steering, brakes, gearshift, and engine are relatively easy to rig for computer control with electric motors and electronics.

The problem of building a practical robot car comes down to the seeing and thinking, that is, sensors and software. A robot car has to maintain in its "mind" a picture of itself in its environment. The car has maps and a GPS, but it needs more. It needs to be able to "locate" itself, along with the roads and fixed obstacles it sees (for example, the curb, roadside trees) and also the positions and speeds of nearby moving objects, such as other cars. It needs to avoid collisions, not get stuck, and navigate itself to its destination.

Boss has eighteen sensors. Sixteen of them are "active," meaning they send out signals whose reflections are interpreted to determine what is in front of them. For example, it has radar (which sends out radio signals) and LIDAR (which sends out visible light, usually via lasers). It also has two cameras, the GPS, and an inertial navigation system (keeping track of how far it has gone in what directions).

The primary sensor is a LIDAR unit mounted on the top of the vehicle to provide a general sense of what's around. Three laser range finders are used to provide a very short-range "virtual bumper." Two radars are mounted in the front and one in the rear to detect and track moving vehicles at long ranges. In addition, two pairs of LIDAR and radar sensors are mounted together on pointable bases, enabling Boss to focus its attention down cross streets and merging roads (and conjuring up creepy images of the Terminator turning its head.

The fact that Boss needs all these sensors makes a point about artificial intelligence (AI) and robotics. Our human brains and sensors (eyes, ears) are so powerful and subtle, and we take them so for granted, it can be surprising how difficult they are to simulate in a robot. Current technology just doesn't offer anything as multipurpose as a human eye connected to a human brain. In one part of the 2007 Urban Challenge, Boss was confused by a cloud of dust, thought an obstacle was blocking the road, and stopped. As the car went through an error-recovery procedure, it eventually realized the obstacle wasn't there, and it was able to continue.

One of the key "lessons learned" from the competition is that better sensor technology is needed for robot cars, but some of the difficulty is in the brain, that is, the software that interprets the sensor data. Perception is only one of the components of Boss's brain. Another is mission planning, finding the fastest route to the next destination. Two others handle trajectory planning and behavioral execution. Trajectory planning is the most basic level of the brain, deciding exactly where and how the vehicle will move.

The behavioral-execution module determines which stock situations are relevant to the current phase of the mission plan. For example, this module "understands" intersections and, when an intersection is encountered, passes a strategy for dealing with it to the trajectory planner. This behavioral executive contains some of the more conceptual "thinking and reasoning" parts of Boss's brain. Consider this description of an intersection scenario from the Summer 2009 issue of AI Magazine:

In cases where Boss stops at an intersection and none of the other vehicles move, Boss will wait 10 seconds before breaking the deadlock. In this case [Boss] assumes that the other traffic at the intersection is not taking precedence and asserts that [it] has precedence. In these situations [Boss's] maximum speed [is limited] to 5 mph, causing it to proceed cautiously through the intersection. Once [Boss] has begun moving through the intersection, there is no facility for aborting the maneuver. [Boss] instead relies on the … collision-avoidance algorithms and, ultimately, the [error] recovery algorithms, to guarantee safe, forward progress should another [vehicle] enter the intersection at the same time."

This intersection-strategy description illustrates elements of the way Boss's brain works: First, it thinks in detail about what it will do in very specific expected situations (in this case, "intersections"). Second, it favors conservative driving strategies; the human drivers I've encountered at intersections do not limit themselves to 5 mph. Third, it is programmed to have some "mental flexibility" to avoid its two major challenges: not having an accident and not getting stuck. These are addressed by "problem solving" portions of its programming for collision avoidance and error recovery, respectively.

Eleven robot cars were qualified to enter the final phase of the 2007 Urban Challenge. They had to deal with merging into traffic, passing, parking, and negotiating intersections. According to the DARPA Urban-Challenge website, "This event was … the first time autonomous vehicles have interacted with both manned and unmanned vehicle traffic in an urban environment." The human drivers were in some jeopardy. The cars driven by people " … were retrofitted with safety cages, race seats, fire systems, radios and tracking systems, and were driven by professional drivers. In all, over 50 vehicles, both manned and unmanned, were navigating the city streets simultaneously during the final event."

Safety may be the issue that brings robot car technology into civilian use. The problem of "distracted drivers" is a growing concern. In "Driven to Distraction: Drivers and Legislators Dismiss Cellphone Risks" (New York Times, 7/18/09), Matt Richtel documents the dangers of driving and doing something else: cell phoning, texting (!), eating, operating GPS devices. Such multitasking is as dangerous as driving drunk, according to his article. Maureen Dowd picked up the same theme on 7/21/09 in her New York Times column "Whirling Dervish Drivers." AI tools can make cars more "accident proof."

Of course, some drivers will be upset about losing the freedom to drive "their way." My father ranted about seat belts, claiming nobody had the right to force him to wear them. But most of us will be relieved to drive cars that refuse to crash. After all, over 40,000 people a year are killed in car crashes in the U.S., more than the total number of deaths of army troops in the 16 years of American involvement in the Vietnam war.

Over time, and after the legal and liability hurdles have been overcome, cars will be allowed to drive themselves. I'm looking forward to the day I can send my car on autonomous errands. Robby the Robot is still too advanced to be perfected in the next few decades, but -– good news – that's also true of the Terminator.

game changer

Oh blogosphere, I miss you so! There are so many cool science-y items and fascinating debates to tempt me away from the task at hand, and I hate being cut off from it all for so long, but I must stay the course and finish The Damn Book, as it has come to be known. (Yes, I have reached the dark night of the writer's soul, but we shall overcome.) I'm nearly done, the manuscript is due September 18, and after that, I will return to blogging (and life in general) in full force, rather than the intermittent appearances I've been making for the last few months. In the meantime, we've been debating two options for The Damn Book's eventual title, and I figure, why not let my readers weigh in?

1. The Calculus Diaries. This was my original title, way back in the book proposal stage. It's direct, high-concept, and captures the nature of the book pretty well -- i.e., it's about me learning calculus by seeking it out in the real world. The question is, does putting "calculus" right there, front and center, turn off the potential general reader (as opposed to those who are already mathematically inclined)?

2. Dangerous Curves: How I Learned to Stop Worrying and Love the Calculus. This is the current working title. It's clever, playing on the "area under the curve"/"face of the function" aspects of calculus; and it's more subtle in its appeal to the general reader. However, it might be a little TOO clever, in that it's not as immediately apparent what the book is about. You kinda have to know a wee bit about calculus already to get the title.

I could go either way at this point, so feel free to cast your vote in the comments, with a brief explanation as to why you'd make that choice.

I have also reached that point of the writing process where I must sum up my conclusions/lessons learned from the long journey into calculus, and condense it into a compelling, readable epilogue. Much soul-searching has taken place during the last couple of years. I have talked to dozens of people (scientists, educators, my fellow "mathogynists") about their attitudes towards math (calculus in particular), how it is taught, how they learned it (assuming they did), etc., and compared it to my own experience. Here are some of my random musings thus far (much of which probably will not end up in the book, although it all feeds into the final product).

First, where does this knee-jerk dislike of math come from? Two years later, I can only say, who the hell knows? There is no one single factor, as far as I can tell. For many people, their struggles with math set in with high school algebra. Co-blogger Allyson performed so well in her other high school subjects that she was placed in advanced math classes. Alas, she was ill-prepared for that level, and the placement set her up for failure. “I distinctly recall the humiliation of my eyes welling in frustration at algebra,” she said. By the time her friends were taking calculus, Allyson had been demoted to what one might charitably call “math for dummies,” learning how to calculate compound interest and how to do her taxes – useful skills, no doubt, but she remains haunted by the memory of her failure.

Another co-blogger, Lee, had a similar experience, with more dire consequences: her inability to grasp algebra – despite top grades in all her other classes -- kept her from becoming a marine biologist, and she has a visceral hatred of mathematics to this day. “It wrecked my self-confidence in a way nothing else ever did, and still knots my stomach,” she told me. “I’m not totally innumerate, but anything that looks like an equation makes me break out into a cold sweat and run screaming in the other direction.”(My bloggy buddy Brian over at Laelaps has written extensively about his struggles with math, which keep getting in the way of his desire to be a scientist.)

On the surface, at least, I have no good reason for my own negative reaction to mathematical symbols. I did very well in my high school geometry and algebra classes, yet somehow I never self-identified as someone with a penchant for math. The truth is that fear and loathing in math class does not necessarily arise from a lack of aptitude, but from a belief in a lack of aptitude. Where did I acquire that belief?

No doubt part of it stems from gender bias. There is a well-documented prejudice against women in math and science dating back thousands of years, although history gives us the rare exceptions, such as the plucky Sophie Germain. The century before, a French noblewoman named Emilie du Chatelet translated Newton's Principia and became the lover of Voltaire before dying in childbirth in her early 40s. Victorian England had Mary Somerville, another self-taught female mathematician who defied the cultural stereotypes of her age. Her passion for math was so strong, she was revising a paper the day before she died at 92.

Such women often have been dismissed as mere statistical anomalies, but evidence is mounting that there is, in fact, no innate difference in the mathematical ability of girls and boys. Any gap in performance is due primarily to sociological factors. This is a controversial statement, as evidenced by the heated comment threads that ensue whenever someone in the blogosphere dares to bring up the touchy subject of women in math (and science). We would prefer to believe that the overt sexism experienced by Sophie Germain et al are a thing of the past, and simply not an issue in this enlightened age, but the reality is that these attitudes persist. Women have come forth with innumerable horror stories ranging from mere discouragement to overt sexual harassment.

A geometry teacher tells the entire class that the girls would probably do the worst in his course because they lacked spatial reasoning ability. A guidance counselor shunts female students into “practical math” classes where they learn how many ham slices each guest would need at a wedding. A physics professor insists on checking his female students’ work before they can leave the lab, yet doesn’t feel the need to check the work of his male students. A computer science professor dismisses any questions from female students as “lazy little girl whining.” And a calculus teacher thinks it’s perfectly appropriate to measure his female students’ bodies and use those measurements as part of his volume calculations in class. One woman on a comment thread over at Tiny Cat Pants last year told of her high school math teacher who made the three female students sit in the front row, “because girls have a harder time with math than boys do.” It was really a flimsy excuse to ogle their cleavage and brush his crotch up against them suggestively during exams. Quoth the commenter: “Guess which three people in that class were not about to be stuck in a basement computer lab with that dude?”

While I have no doubt those things happened (and still do), I never experienced anything so horrific; my math teachers were kind and, if not openly encouraging, they certainly were not discouraging or hostile, nor was I ever sexually harassed. My parents were supportive of my intellectual pursuits, if a bit bemused by my headier inclinations. Nobody ever told me explicitly that girls weren’t as good as boys at math, yet somehow I absorbed that message anyway. Carol Tavris, a cognitive psychologist and author of several popular books (The Mismeasure of Woman should be mandatory reading for young women), explained to me that there are subtle, situational social cues that seep into our consciousness, like osmosis, even if we never encounter overt negative messaging about gender.

The phenomenon is known in psychological circles as stereotype threat, and it has been confirmed in more than 100 scientific articles. For example, a 2007 study in Psychological Science found that female math majors who viewed a video of a conference with more men than women reported feeling less desire to participate in the conference, and less of a sense of belonging, than female math majors who viewed a gender-balanced version of the video. The male math majors were immune to those subtle situational cues. 

That’s stereotype threat in a nutshell. The 2004 film Mean Girls perfectly captures the subtle influence of cultural factors. Home-schooled in Africa for most of her early life by her anthropologist parents, Cady (played by Lindsay Lohan) didn’t absorb the subliminal message that women can’t do math, or that liking the subject is uncool – in fact, she appreciates the universality of math, because “it’s the same in every country.” Then she begins attending a regular high school and the inevitable peer pressure kicks in. She is urged by her peers not to join the high school math team (“It’s social suicide!”), and pretends to be bad at math to win over the cute boy in her calculus class. This being a movie, she gets over it, and ends up copping to her love of math and snagging the cutest boy in school. Alas, peer pressure doesn’t excuse me, either.  I was a painfully shy, socially awkward, brainy sort in high school, and pretending suddenly to be bad at math would not have transformed me magically into the homecoming queen.

Tavris also cited our fascination in the US with the notion of innate ability. We are born with certain built-in talents, this reasoning goes; you either have a gift for math, or you don’t, and no amount of hard work can make up for that lack of innate ability. The reality is much more complicated. My friend Deborah self-identified as being good at math early on in her education. Her fourth-grade teacher held multiplication table competitions in class. Deborah was highly competitive, so she worked very hard on memorizing her multiplication tables and practicing at home. As a result, she excelled in these competitions and became known as being “good at math.” This had a significant impact on her later on: whenever she struggled with an especially tough problem, she pushed through, thinking, “I should be able to do this because I’m good at math.” Yet her belief in her innate ability, and success at math, were actually the product of a lot of hard work.

Another part of the problem has to be the way the subject matter is presented and/or taught. Guest blogger Alex Morgan offered a clue when it wrote about his own daughter's ambivalence towards math, despite having a certain aptitude for the subject. She just doesn't like it! And after perusing her math homework, Alex found he didn't much blame her. The material was dry, uninspiring, and completely divorced from any real-world experience. (Hence my use of real-world environments in which I seek out possible calculus problems in The Damn Book.)

Students need to feel inspired, particularly when it comes to a difficult subject. While I was at the Kavli Institute for Theoretical Physics last year as journalist in residence, I got to know UC-Santa Barbara mathematician Bisi Agboola, who generously shared his own story with me. Bisi was educated in the UK and failed most of his math classes through their equivalent of high school. “I found it dull, confusing and difficult.” As a child, he was determined to find a career where he wouldn’t need any math, finally announcing to his skeptical parents that he would be a woodcutter. He was crushed when they pointed out that he would need to measure the wood.

But one summer he encountered a Time-Life book on mathematics –- Mathematics by David Bergamini -– that offered “an account of the history of some of the main ideas of mathematics, from the Babylonians up until the 1960s, and it captured my imagination and made the subject come alive to me for the very first time.” It changed his mind about this seemingly dry subject. He realized there was beauty in it. He wound up teaching himself calculus, and told me he is convinced most physicists also do this. Today he is a PhD mathematician specializing in number theory, and exotic multidimensional topologies. Ironically, he still doesn’t much like basic arithmetic: “I find it boring.”

Some students respond well to how calculus (and physics) is traditionally taught, others don't. The Spousal Unit sent me a link to a fairly new blog called Gravity and Levity, written by a physicist with a way with words:

For those of us who immediately liked physics class in high school, physics was a game.  It was like a little logic puzzle where the rules of the game were given to you (usually on a formula sheet) and you were asked to use them cleverly to come up with a solution.  A friend of mine once put it succinctly: “Physics is all about finding out which variables you know and which variable you want, and then searching through your formula sheet for an equation that has all of those letters in it.”  That, more or less, was the physics game.  You rearrange some symbols on a paper and you come up with an answer.  Instant gratification.

Those of us who went on to study physics in college almost invariably did so because we liked the game.  I personally loved it, and I was good at it.  But as I went further into physics, it began to be more than a game.  Little by little, all the equations and “rules of the game” started coming together into a coherent perspective on the universe and how it works.  ...  Over years of study my interest in physics gradually but completely shifted from “the game is fun” to “I want to know how to think about what the universe is made of and how it works.”

I hated the game; I couldn't really see the point -- at least until I got interested in physics and realized that calculus was relevant to my world. I'm one of those people who really needs to understand the context, and the "why" of things. (Although I must admit to a fondness for word games. I killed at Boggle and Scrabble in college.)

The point is, different people learn in different ways, and that makes coming up with a standardized educational system particularly challenging. The best model I heard about was from a woman scientist I met at a party, whose daughter went to an elite private school on the East Coast. There were several teachers for each subject, and students could transfer out of a class if the teaching style didn't agree with them, and try another instructor. The daughter's attitude towards her history class did an about-face once she found an instructor who inspired her. Just like science communication, learning (and teaching) science, or any subject, is about making that critical connection. I just don't see how it would be possible to scale up that approach to a nation-wide level; there's a reason this system was implemented at a pricey private school.

So: stereotype threat, gender bias, peer pressure, bad teaching, poor subject presentation -- all of these play a role in discouraging people (especially those of the female persuasion) from taking/liking math. There are countless efforts worldwide to combat these sweeping socio-cultural factors, and we should continue to fight the good fight in that regard. That said, Sophie Germain and Mary Somerville didn't let socio-cultural factors keep them from pursuing their love of math. What made the difference? Personal mentorship helped at some point, but it started with inspiration and falling in love with the subject in the first place.

Ultimately, when I closely examine my dubious history with math, I can't really cast too much blame on those Big Picture factors for my prolonged ignorance/avoidance of math. They discouraged me because I let them discourage me -- because I wanted an excuse to avoid calculus. I can't say I was intellectually lazy, because I avidly pursued knowledge in any subject that caught my interest, and worked very hard in those areas. Like Alex's daughter, I just wasn't all that interested, and hence was more than happy to be pushed away from math and science. What is the cure for willful ignorance and very deliberate avoidance?

I'm not sure there was a single game-changing moment, since the process of thawing towards math was gradual. It started with going to work for The American Physical Society, then becoming a science writer specializing in physics. And one day, out of curiosity, I asked a physicist named Alan Chodos (associate executive officer of the APS) about why objects fall at the same rate regardless of mass -- it seemed really counter-intuitive to me, although I had no doubt it was true. He insisted that I didn't have to take the matter on faith, and walked me (kicking and screaming) through the basic algebraic equation. Suddenly the numbers had relevance to something I'd actually experienced. And my kneejerk defenses started lowering bit by bit. I can't say I love math and calculus, but I understand the basics now, and I no longer break into a cold sweat at the sight of an equation. Believe me, that's tremendous progress.

It just goes to show that real learning is personal and individual. A good teacher can change someone's life and undo years of willful ignorance. I owe Alan a great debt, I realized. The least I can do is say thank you in public.

a quick one while i'm away

Greetings all and I apologize for the long silence here. As always we ladies are busy with many things outside this blog that enrich us and make us more entertaining and enlightening to all of you. I'm so thrilled by the dialogue that goes on and charmed by the great community that lives here at the cocktail party. I wish I could bask in it more often.

This week my excuse, however, adds up to a little buffet of science stories from the American Institute of Physics Industrial Physics Forum, where I blogged the week away in Anaheim, California. This year's meeting collaborated with the annual meeting of the American Association of Medical Physicist, with a theme focused on quantitative medical imaging for cancer detection and treatment. It was inspiring and sobering listening to very talented people talk about science that has such high stakes, and also to hear about the amazing advances they're making. You can read more about my adventures at the AIP blog, but here are a few of my favorites:

Karl Deisseroth, an assistant professor of bioengineering and psychiatry at Stanford University was the closest thing I saw to a mad scientists at the meeting, although I believe his intentions are good. He and his team have implanted mouse brains with halo rhodopsin pumps that they found in algae. Algae don't have brains, and they rely on light cues to tell them when to move. It's these halo rhodopsin pumps that respond to light and send a chemical message to do something different. By injecting these into mouse brains, then inserting an optical light fiber into the mouse, the team can control the mouse's behavior by turning the light on and off. So far they can get the mice to run in a circle or wake from a deep sleep. They've found different pumps that could make different colors induce different behaviors.

You gotta have faith in basic research and the Industrial Physics Forum usually reminds me why. Scientists at CERN, in their hunt for more and more tiny tiny particles, pushed photon counter technology ahead by a factor of 100. Photon counters are just light detectors that count individual photons and measure their energy. So, if you only have one second to take a picture, you need a high photon count detector to gather as many photons as possible and provide information about them. In the optical range the energy tells you a photon's color, so you can see how knowing photon energies can provide useful information. New results at the meeting showed the first human trials of a CT scanner with one of these individual photon counters. Most CT scans don't need photon counters, but eventually they hope to use them to detect early cancer cells or other disease indicators.

The grand imaginations of particle physicists and accelerator scientists will continue to push for larger and larger accelerators in the world (like the planned 31 kilometer ILC); but to spare money and resources, ther eis also a push for smaller accelerators. This comes especially from medicine, where things like proton cancer therapy are burdened by the large size and high cost of accelerators. Plasma wakefield physics is the front runner amid theories on ways to accelerating particles to higher energies over shorter distances, but I also heard about another technique in the works that still uses plasma, but accelerates the particles slightly differently. The details are rough, and I'm just going to point you to the AIP blog post. But if this new technique pans out, the advantages will include, yes, smaller size as well as no need for magnets.The scientists working on it are gearing it specifically toward proton therapy, which is perhaps unique to plasma wakefield acceleration. 

Also in the range of accelerator science, Joseph Lykken of Fermilab gave a great overview talk on the challenges for next generation accelerators. He mentioned a type of accelerator technology that could drive nuclear reactors run on thorium, rather than uranium or plutonium. The advantages are that the waste product decays in about 60 years instead of millions. But to do this we'll need very powerful accelerators that don't exist yet. Apparently this technology has been on the table for a few years (see the articles in Wired and COSMOS, where I got this absolutely stunning image courtesy of Justin Randall). 

I also heard from the Chief Technology Officer of Pacific Biosciences, a company that is planning to release a commercial DNA sequencing tool that will be 20,000 times faster than current technologies. Using at totally new technique, they've found a way to peek in on DNA polymerase as it sequences DNA in real time. The technology will also turn around results in minutes as opposed to days. In four or five years they say new technology will become available that will advance their technique even further. In, say, ten years, you might be able to sequence your entire genome in 15 minutes.

On another front in cancer treatment, I learned about thermal therapy, or the simple notion that if nothing else seems to work, try killing a tumor by pointing a laser at it. You cook the cells, with the main risk of damaging nearby cells as well. So to focus the heat and increase the gradient, scientists from Wake Forest University deposited multi-walled carbon nanotubes into tumors and used them to conduct heat. The tumors in mice deposited with these MWCN's were not only obliterated but came back far less often. MWCN's aren't approved by the FDA yet so clinical human trials can't begin for a while.

With many major meetings held during the summer months, there's a smorgasbord of good science stories to read by the pool. If you've read good story, blog post or other internet tid bit that has whet your science palate in the hot summer sun, I'd love to hear about it.