In her K/1 class (ages five to seven), the 'fuzz was doing "the weather" when an argument broke out between those who thought the clouds "move the Sun around" and those who thought the clouds "block out the Sun". She let the discussion proceed, encouraging contributions. In the end, the "move the Sun around" group got the consensus. What to do? You can correct them all, and then they learn that scientific truths are handed down by more knowledgeable authorities ("How do you know the Universe is expanding?" "I read it in a book."). Or you can let it lie, in which case they go home thinking they know something that in fact is wrong. Or (best, but extremely time-consuming), you can go through the process of having them turn their pseudo-scientific explanations into predictions about other phenomena, or have them extrapolate their model into other domains, and then see why or where it breaks. That's (to my mind) the only solution you could possibly call science, but it would require an absolutely radical replacement of the current curriculum and structure of school. The 'fuzz didn't have the right (it was someone else's classroom) to blow the schedule, and she didn't want to be a priestess, so she let it lie and moved on to the next activity.
The time derivative of velocity is acceleration, both vectors of course. But I was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.
I said in lecture that this issue was
deep but I didn't emphasize it enough. I feel like it is so big it almost needs its own week!
Macroscopically, air resistance is ram pressure (proportional to cross-sectional area times velocity squared). Microscopically, drag is Stokes-like (proportional to radius times velocity). Where does the cross-over happen? I didn't have the guts to put that on
problem set one of my course for pre-health students, but it will be in around problem set eight. It could be on problem set one, because the transition can be obtained purely by dimensional analysis.
In transport processes, there are often qualitatively different effects working at small scales than at large. Another good example is diffusion vs convection.
My big challenge in preparing my General Physics I syllabus is to figure out what to cut, when the majority of the students are pre-health. I cut thermodynamics, because we have learned that it is taught also in chemistry (and other places). I then wanted to add more material about fluids and elastic solids (pretty relevant to medicine, it seems), so what to cut? I ended up cutting most of rotation, spinning, and angular momentum. Why? To understand the body, you do need to know about torques (how does your arm work, static structures, and so on) but you don't really need to conserve angular momentum. Or do you? The centrifuge spins, but it doesn't have angular dynamics.
(I will be doing the centrifuge.)
The parallax distance to an object is the distance you get by moving your (single) eye's position by a transverse distance x, measuring the angular displacement θ of the object given that move, and dividing the transverse distance by the angle.
Problem-set problem: You are outside after the rain late in the day and you see a rainbow. What is the parallax distance to the rainbow?
(Thanks to Andrei Gruzinov at NYU.)
Cheryn (9 years old): What's the capital of Kansas?
Hogg (40 years old): I don't know; it is a useless piece of knowledge, because I can look it up in a few seconds on the internet.
Dustin (30 years old): In other words "I don't care, because I can just ask my iPhone." but then your teacher says "Ah, but what will you do if your iPhone isn't working?" answer: "I am in New York City; if the iPhones aren't working, I don't need to know the capital of Kansas, I need to know how to shoot, skin, and dress a squirrel!"
In my previous post I implied, inadvertently, that we should only teach useful math. That was not my point. My point was that we should not teach math if the effect of that teaching is to cause most students to hate it. Of course if we could teach it such that the effect was to cause most students to love it, I would be all for teaching it!
In a nice conversation about writing for education, Adam Gidwitz (the author of A Tale Dark and Grimm) pointed me to The Mathematician's Lament. The book makes (much more clearly than I) a point I have been making informally for years: If you want students to know and love math, you definitely should not teach it in school! (Same for literature.) The mathematics requirements in school empty the subject of its meaning and point, and are useless to boot. How many non-scientists use the quadratic formula, ever? Discovering the formula would be fun, using it is a drag (and exceedingly rare).
In the State of the Union address of 2011 January 25, Obama raised one of his standard themes, which is the great importance of college education:
If we take these steps—if we raise expectations for every child, and give them the best possible chance at an education, from the day they are born until the last job they take—we will reach the goal that I set two years ago: By the end of the decade, America will once again have the highest proportion of college graduates in the world.Though I don't disagree that we need to give all Americans great opportunities, I disagree that the best way to achieve that is through universal college attendance.
First of all, as pointed out here and many other places, though employers want to see college degrees, they do not really look for many of the things in a potential employee that college brings. That is, it is not clear that college is delivering useful things to employers. A good example is math skills; even high-tech employers do not look for math skills among potential employees.
Second, since most of the important skills for employers are communication skills, flexibility, problem-solving, and ability to learn (rather than specific content knowledge), employees ought to be well prepared by the end of high school. How can (more than) full-time work for twelve years not be enough to teach 18-year-olds the communication and common-sense skills they need to do most jobs? I realize that schools face huge challenges, but why not make helping the schools the top priority?
Third, and related to the second, the reason Obama and other politicians focus on sending students to college instead of improving K–12 is that college is private. Fixing schools takes public money; sending kids to college takes private money. Even state schools in America charge tens of thousands in tuition, and they don't have the capacity to take up Obama's slack. So requiring all kids to go to college is like levying an enormous quasi-voluntary tax on parents, relieving society of the responsibility of providing opportunity to the less fortunate. If a kid has trouble, in this vision, it is because his parents didn't do right by him.
Indeed, this ties it all together: Abandon the public schools, because improvements there would reduce disparity and improve the public sphere. Require (effectively) all potential employees to go to expensive colleges, because that is tax-neutral and increases disparity! What happened to the common good?