Educational philosophy, revised

Educational Philosophy

My educational philosophy can be organized around five values: (1) Independence/Interdependence; (2) Anchoring knowledge; (3) Modeling; (4) Social Diversity; and (5) Economic Mobility.

1. Independence/Interdependence

The teacher’s goal ought to be to promote independence from the teacher and interdependence among the students. A teacher must fight the desire to be the center of attention and be prepared to cede the spotlight in order to develop students’ abilities. This does not mean a hands-off approach, especially in the early stages of learning. But it does mean structuring the class to provide greater opportunities for teamwork.

2. Anchoring knowledge

New knowledge must be anchored in old knowledge, both within and across subjects. My subject expertise, broad educational background and varied personal interests mean that I can generally come up with surprisingly apt analogies and tie-ins that relate to their world. By putting in the work to anchor new knowledge in their existing world, I’ve reduced the sense of inaccessibility and irrelevance that plagues science classes. Once that initial bridge has been made, anchoring occurs within a subject; this is especially true for physics, which depends upon earlier mastery in order to understand and solve progressively more complicated problems.

3. Modeling

Although the ultimate goal is to have students become collaborators and the teacher to become less a provider of information and more a facilitator of learning, students need to have a clear idea of what is expected and the accepted, proper ways of organizing and communicating information and understanding. Experience means nothing in itself; it is only experience paired with proper technique, process, and understanding that will take students toward greater mastery, both in the classroom and outside it. To this end, I make certain students see, clearly, how to solve problems in an organized, clear way, and emphasize process more than product (the right answer). The right answer matters, of course, but getting the right answer (or even asking the right questions) can happen reliably only if students have a solid, reliable understanding of the process of scientific inquiry and scientific problem-solving.

4. Social Diversity

Better decisions are reached when there is a combination of a diversity of backgrounds and a common purpose. An educational setting, and especially a community college, is built for that confluence of factors. I have had the challenge and opportunity to work across a wide variety of cultural, economics, and social lines, and one thing has become clear: there is an opportunity for every person to develop their awareness and appreciation of the universe. In so doing, the language of physics and astronomy provides a common experience and set of knowledge, one tied to the historical traditions of many cultures and peoples. Everyone, regardless of gender, sexual orientation, country of origin, ethnicity, economic means, or disability, can and should participate in the learning and doing of science.

5. Economic mobility

Scientific education isn't just about nurturing the soul. It’s about nourishing the body. I know that many of my students will be immigrants and refugees. Most will be near or in poverty. Historically, STEM training has been, and continue to be, a critical step in entering into higher-skilled, higher-wage jobs. I don’t expect, or even wish, that my students pursue academic research science careers. Instead, what they need, and what I hope to provide, are the fundamentals of problem-solving, technical knowledge, abstract reasoning, and deductive logic that will be fundamental to their success, whether they pursue a STEM career or not. To that end, I will make certain that, in every aspect, my course accommodates students of limited means without limiting their ambitions for a better life.

Why Mr. Weaver was both the best and the worst physics teacher ever

Given all the physics education stories I've told, I'm surprised I haven't written about this before. Granted, it was a while ago (pre-9/11), so my memories (and associated emotions) aren't nearly as strong. But it's worth writing, so non-Rosemead High School students get a taste of where I was coming from upon entering Harvey Mudd College.

In 1999, Mr. Weaver was a late-50s man who, by his own admission, was a burned-out mechanical engineer that had somehow ended up in teaching. I don't know if he started teaching right after college, or if he had been a practicing engineer until the aerospace layoffs of the 1970s or 1980s.

He bore a disturbing resemblance to Hannibal Lecter. Appearance-wise, not so much; he had a full head of hair, always parted to the left, and was less physically imposing than Lecter. But he did have these blueish-gray eyes, at once piercing and vacant. More than his appearance, his soft, vague, enervated tones made one think of Dr. Lecter.

It's not just me and my unhealthy fascination with serial killers, real or fictional. The Lecter-esque quality has been confirmed by multiple classmates. His language was not nearly as eloquent or energetic as Lecter, but was filled with these vague, straight-faced quirks of speech.

Anyway, this was how AP Physics B shook out. The first day, Mr. Weaver stood in front of class and gave a brief lecture about general things about this class. I honestly don't remember what he talked about, but some of it went over my head. I think he sprinkled a bit of statistics in there, and I would not take that [excellent] class until next year with -- and I'm not joking -- Mrs. Flaws.

During the next 180 or so days of instruction, there wasn't a single lecture. Not a one. He'd assign homework by writing it on the board. But, if memory serves, he wouldn't address the class as a whole again (barring, say, a fire drill).

So all of my introduction to physics was self-taught. I was helped along by the competitive pressures (some would say harassment) of a precocious Vietnamese student who was probably two or three years ahead of the rest of us in both math and science. (Contrary to expectations, he didn't major in physics -- he went the med school route, which I believe has been more financially and personally profitable than a physics trajectory, anyway. Huy, if you're reading this, you're welcome.)

We did have labs, and to his credit, Mr. Weaver did show us how to use the air track and other equipment. But only if we cared enough to ask.

Needless to say, without management, classroom management fell apart. The seniors were the first casualty -- a lot of them stopped doing homework. Seniors and juniors would use the class (after lunch) as a second lunch hour, sitting cross-legged on the tables in circles to eat. After telling one of my students the story of this class, he asked, "Wasn't he worried about getting caught by the principal?" The answer had to be no, which I suppose demonstrated the systemic nature of the problem. My personal experience indicates that there was more attention paid to the slipping of the word "necrophilia" into the school newspaper, or illicit trips to the In-n-Out burger during classroom hours, than to physics instruction.

So yeah, almost no one cared. I remember doing a lab in which I was doing error analysis while the rest of the group was watching American Pie. I do remember generating some messed up system of error analysis; this is also the class where I taught myself Excel.

He did grade the homework and labs submitted, and did give tests.

Around second semester, some of the seniors started to realize that they were failing this course, and that an F in this (and other) courses could jeopardize their admission to various colleges. "Ruh roh!" (I think that's a direct quote from Mr. Weaver.) I don't know if he pity-passed anyone, but it was mildly amusing to see someone try to muscle through E&M and optics, having paid zero attention to any of the preceding physics.

Also, at some point, he dyed his hair brown. He then disappeared for a couple weeks. When he came back, we learned he had married a Japanese woman. Weird.

Yes, he was the worst physics teacher I'd ever known. He wasn't hostile; he wasn't ignorant. He was simply a non-factor. He demonstrated all the fucks he didn't give before the meme existed.

I don't know if people in those classes hated physics. It could be argued that they are actually more positive about physics than average precisely because it was less instructional, and more food-centered.

So my preparation for physics going into Harvey Mudd College was, well, less than adequate. And it probably did contribute to the disconnect between what I thought physics was and what it actually was.

But maybe he was a secret genius, and a master teacher. Maybe he knew that no one could get through a physics degree without a great deal of self-motivation. And I, being tested by the crucible of a nearly worthless teacher, learned to learn on my own, and passed this life test of self-learning.

Or maybe he was a useless piece of crap protected by seniority, union rules, a relatively inactive parent pool, and the fact that he didn't commit any actual crimes while teaching.

So to all of you who had legendary teachers that set you on the path to learning, I applaud you and celebrate your good fortune or the blessings of a good zip code. But for the rest of you, please don't use a bad teacher as an excuse for poor knowledge or hatred of a subject. Our wisdom and understanding are shaped by our experiences. But we do have agency of our own, and sometimes discover different and important things about ourselves when forced onto more lonely, less familiar paths.

Why is physics scarier than other math/science classes?

After some discussion with my student (geometry/chemistry) tonight, I think I have an answer to a question that plagued me for a while: why is physics scary?

It's scary because it's the first (and last) exposure to multi-step, multi-tool problem solving in math and science.
I had a high school student who found precalculus easy (or at least manageable, even that nasty stuff toward the end) but struggled mightily with physics problem solving. I couldn't figure it out for a while.

At first I thought it was that I found it easier to teach precalculus; the book is set up nicely, and I don't have to worry about abstract concepts (except insofar that I tried to tie it to useful stuff).

Then I thought maybe it had to do with the student having a weak science background. But by her own admission, she had both a weak math and science background.

It wasn't work ethic; this student worked a lot, both in class and outside. (I was dorm RA, so I knew studying was happening even outside of school hours.)

And I couldn't just chalk it up to "being Chinese", and the stereotype that Asian education consists of rote memorization and drills, leading to mathematical fluency but deficient creativity. (I secretly suspect this is an excuse perpetuated by Westerners to ignore the severe gap in educational readiness vis a vis other nations.) "Western" students in the same classes exhibited this pattern, too.

Eventually, it came down to this multi-step, multi-tool problem. Both she and I determined this independently.

***

In the typical high school math class, you solve problems by using a single trick or tool.

In biology, you mostly memorize a bunch of concepts and vocabulary, which are evidently important skills in the first year of med school.

Chemistry presents perhaps the strongest challenge to this thesis. It is possible to generate a multi-step, multi-tool problem in chemistry. There's a reaction, and you have to figure out it's yield and reaction type. First, you might have to do some stoichiometry to balance the equation. Then, you might have to figure out Lewis structures to determine the number and type of bonds, and then calculate the binding dissociation energies. You can then figure out if the equation is endothermic or exothermic. Maybe you adjust the calculation using phase transitions. Then you can determine molar masses, and compare the expected mass to the measured mass, or some other contrived number that allows you to calculate the yield.

So maybe chemistry is the first opportunity. But my experience indicates that plenty of students muddle through chemistry and hit a solid brick wall when they take physics. So there's something different about it.

I think chemistry, in principle, can be taught with emphasis on using multi-step, multi-tool problems to solve chemistry problems. But in practice, it looks like that's not done. I don't have a good explanation why that is; I've never taught a chemistry course. (Those who tutor, teach or study chemistry: your input is definitely welcome.)

***

In general, in the sciences, the problem statements are longer and more involved, making it less practical to have students do a number of problems on a single concept to hammer it in (as it's done in math). As much as we'd like to think otherwise, repetition and drills really do help cement a concept.

Maybe physics would be better if we could better segment subject material, and have more practice problems limited to one topic. This is different than how most textbooks are set up. Most of the better textbooks I've seen have, at the end of the chapter, problems grouped by topic, and prefaced with qualitative questions. It's not like a math book, where the section/chapter problems are divided into clusters in which you are asked to basically do the same thing over and over.

Physics, as a multi-step, multi-tool discipline, requires that all the tools work, and all the steps are clear. Maybe in other classes, even chemistry, a student can half-ass Lewis structures and still get an A. But it's just not possible to half-ass, say, linear momentum and be able to learn the rest. (Chemists: feel free to quibble and argue that the analogy ain't fair; I'll argue that even precious PV=nRT can be botched without irreparable damage to the rest of chemistry learning and the final grade.)

I'm not arguing that physics is better, or necessarily more complicated. But it is different.

Anyway, I think I'm going to revise how I teach physics. I basically need to generate drills, in addition to the problem-solving organizational methods that I'd ask them to use to convey their knowledge in a solution.

teaching physics to high school students - a true story

I've told this story before. But it's good enough to immortalize.

I used to teach high school physics (in addition to six other subjects every day) at a private boarding school. It was a pretty rotten job for a number of reasons, though the students themselves were pretty funny. The vast majority were from mainland China, though we had a couple Koreans, a Russian, a Dane, and two Americans.

Even though nearly everyone was not from "here", certain aspects of humor seemed universal.

I was trying to teach some of my students about rotational motion. This is often cited as the most difficult part of mechanics, and, for some, the most difficult part of the entire course. It was an algebra-based course, so it didn't involve cool integrals of nonuniform shapes or utilize nonuniform densities. Everything was pretty boring -- a rectangular prism, a sphere, a rod, a disk, a hoop, a sphere, and, occasionally, a hollow sphere. (That reminds me - I should look into calculations for a right cone. That would probably blow my mind.)

I was explaining the different rates of speed achieved by different shapes. Because different objects have different moments of inertia (an expression for how the mass is distributed throughout the shape that affects rotation), they split their energy in different ways between translational motion (moving in linear direction) and rotational motion (spinning). So a disk, a sphere, and a hoop, all with the same radius and mass, will move down a ramp at different speeds because they have different moments of inertia.

I drew something like this:

Needless to say, as a first-year teacher, I was unprepared for the howls of laughter. I looked at this and saw the finishing order for a sphere (gray), a disk/cylinder/can, and a hoop.

They, of course, saw a cock and balls.

Atwood's machine was also troubling:

Anyway, I ended up putting the question involving rotational motion down a ramp for a sphere, a cylinder, and a hoop, all of mass M and radius R, on a quiz. One particularly lazy/uninspired student answered this question, and this one only. He answered it by drawing the picture above, eight times in the space provided for an answer.

What was I to do? He was technically right. So I gave him full credit. Besides, that was the only thing he had written on the quiz, and I didn't feel like handing out a zero that day. He still failed the quiz and the class.

The moral of the story: high school boys are very much the same across cultures.

Why I wanted to become an astronomer

I can't believe I haven't told this story yet.

This is the story of how I got interested in astronomy and managed to will myself somewhat far down the professional path.

There were three primary influences: a man, books, and television.

The man:

I think I was about six when I met Uncle Kimo, a cousin-in-law-in-law. (My aunt married a man who had a sister who married Uncle Kimo.)  At the time, Uncle Kimo worked at JPL as an instrumentation engineer. I really liked "Uncle Kimo". He seemed to know a lot of things! We spent time drawing maps of continents and he would tell me about space. Over the years, he sent me some of those nice high-gloss photos from Voyager and other space missions. Some of my personal favorites were a radio reconstruction of Venus' surface, Jupiter's volcanic moon Io, black and white images of Uranus' satellites, and that stunning deep blue shot of Neptune and the Great Dark Spot, accented by some bright white storms (the largest called "Scooter", if memory serves).





I think I loved Uncle Kimo not just because of the cool space stuff. He was maybe the first male family member I really felt comfortable around. My dad was crazy and unstable; my uncles either scared me because of their anger or were just not that interesting/good with kids. I loved my grandpa, but he was intimidating (especially to everyone older than me), and the language barrier made us not quite as close as we might have been.

Also worth emphasizing: Uncle Kimo was the first adult who really tried to teach me about the world around me, and do it in a way that didn't assume I was just a dumb kid.

I only saw Uncle Kimo a few times growing up, but I'd still place him as a tremendous influence on my life in general, and my interest in astronomy in particular.

Books:

I spent a lot of time alone growing up. My parents divorced when I was three. My mother worked, and so my grandparents played a large role in raising me. They were kind and loving, but didn't speak much English, so I ended up turning to books. My mom says I initially hated to read, and would slam my little hand down on any book she tried to open and read to me. Maybe I was creeped out by Shel Silverstein's artwork in Where the Sidewalk Ends. But, eventually, I did start reading on my own.


As a society, are we still ok with this?


One of the first books I got was The Golden Book of Stars and Planets.

I read it probably a couple hundred times. I loved the artist's depictions (all images of the planets were hand-drawn).

I still remember it mentioning NGC 5128 (depicted on the cover, right side), thinking "That's a weird name for a galaxy. Why wasn't it called something like the Milky Way? Or Snickers?" (It's a radio source, and got special mention, though I thought it made weird sounds because the artist's depiction made it look like it was surrounded by hair.)

I also went through a weird phase as a kid in which questions in a book would freak me out. They scared me! Because of that, I'd have to skip over the last part of the Mars section? "Was there ever water on Mars? Could there have been life?" It was that and an earthquake preparedness pamphlet that creeped me out with those "?", which I must have mentally read in some sort of spooky voice. (Did anyone else have this, or was this a leading indicator for profound mental unsoundness? -- again, with scary questions!)

I loved that book so much.

More books. Remember Scholastic catalogs? Or Arrow? Or the other one? Classroom teachers would give us these catalogs filled with books that we could buy at (what I thought were) reasonable prices. Now, I didn't know how money worked, even though I loved Scrooge McDuck in Ducktales and tried to swim in a pile of dimes in my grandparent's living room. (Maybe another flag that this boy ain't right.) I remember that in first grade everyone wrote a letter to President George H. W. Bush. I wrote that the process of making change (money, not policy) seemed unfair -- why does someone get to keep more of the money? My teacher helped me write it, but I'm a bit annoyed she didn't try to sit down and explain it to me at the time. I got a photo back, but I don't think the letter had an answer.

Also, in retrospect, this was one place where economic differences started to show. A lot of my classmates probably couldn't afford any books. I always got to pick a few, as well as get a subscription to Highlights! magazine.



Goofus generally gets what he wants, even if he is an asshole. Gallant is a spineless appeaser and a fake, pretentious prick. Guess who I grew up to be?

One of the books I got this way was Planets: A Golden Guide to the Solar System.

This was not my favorite book. I don't know why -- I carried it around everywhere. It just seemed not as exciting, or less accessible because of more data. There were tables of numbers, I think, and maybe fewer dramatic, page-filling images.

I got a more important book that had both dramatic pictures and tons of numbers (though not derivations) around first or second grade. My Sunday School teacher gave me his old college astronomy textbook, a paperback that cost $34.75 at the Aschula's student store. (I still remember the sticker on the cover.) The book was Essentials of the Dynamic Universe: An Introduction to Astronomy, Second Edition by Theodore P. Snow. Sadly, I can't track down the cover image, but it must have been a saturated image of a star, or an AGN, or something like that, with some rainbow accents suggesting spectra.

I think I read that book cover to cover several times. This was the source of nearly all of my knowledge of  introductory astronomy. Memory being what it is, those early memories were retained much more easily than ones in college, making it sometimes challenging to rewrite my knowledge of certain important constants. (The distance light traveled in a second was, in my mind, 186,282 miles, and not 299,792 kilometers. Mercury orbited the Sun at a distance of 38 million miles -- and unfortunately all my distance scales inside the Solar System remain imperial.)

Same church: our rather conservative junior minister occasionally called on me, possibly for comedic effect, to quote some astronomical fact -- closest star, distance to the Sun, etc. -- that had some tie-in to his sermon.

Years later, after being accepted at CU Boulder's astronomy grad program, I happened to meet with Theodore P. Snow. (I think he went by Ted at the time.) Usually, meetings were about prospective research, but I spent the entire time as a fanboy gushing about how incredibly important his textbook was to my life. I have no idea whether it was flattering or scary for him, but I was thrilled to put a name to a face. (It also helped that he seemed nice -- read: Not A Professor Asshole.)

This book and interactions with that pastor explain why I never, ever thought of a conflict between science and religion until I started paying attention to politics/went to college. I still think it's misguided/overrated.

Books were a good substitute for technology. I was lucky enough to get a small telescope for Christmas, but quite frankly, it was a piece of crap Celestron. I probably should have read the manual more carefully, but none of us really took the time to figure out how to use the RA and DEC wheels, or how to use coordinates to find things in the sky. Oh, and perhaps most importantly, I couldn't see jack shit because I was too close to Los Angeles. Saturn's rings and the Galilean satellites were nice to see, but a bit anticlimactic after spending years staring at NASA images.

Television

As a kid, I watched a ton of TV, unsupervised. I remember being confused as to why "Orchie Bunkur" was so angry all the time. (This is a reference to All in the Family.) I even wrote a letter to that effect, to no one in particular. (I liked writing letters as a kid.) This letter was proudly placed in my grandpa's scrapbook without further comment, a testament to both his love and the complete absence of analytic evaluation of child behavior in my family.

I watched a lot of Star Trek: TNG. Remember that it ran from 1987-1994. From the age of four to eleven, I saw brand-new TNG episodes. I saw the very first airing of "The Best of Both Worlds". Eat your hearts out, young nerds.

I would record a lot of these episodes and re-watch them endlessly (VHS, in case the young people are curious). I didn't have particularly good taste -- I recorded as much as I could, and ended up with a skewed impression of the overall series, with Data being held up by Samuel Clemens with a .45 revolver playing a more important role than, say, "The Inner Light" or "Darmok". More fortunately, I also recorded and watched "Chain of Command", though maybe even scrubbed and tidied torture scenes weren't the best thing for a young child to process.

I also watched Babylon 5 and Star Trek: Deep Space Nine. Again, I had no taste -- I thought Babylon 5 was an incredibly well-acted show and ST: DS9 weak by comparison (I would reverse those judgments years later.)

But in addition to sci-fi, I watched some sci-fact. NOVA specials were great, and what I know of cosmology comes from them.

The most personally important program, however, was a National Geographic documentary titled Asteroids: Deadly Impact. It primarily focused on Gene Shoemaker and his study of asteroid and comet impacts. Of course, no one man or team can claim sole ownership of such a broad field, but it made for great watching.

Only the intro is available for free online: Asteroids Deadly Impact

Again, I recorded this, and watched in at least 50 times. I returned to it a few times in high school, especially when I had an abysmally poor physics teacher, to sort of remind me why I thought astronomy was worth studying.

Epilogue:

These influences were critical. I enjoyed reading books. I had a person who cared about me, and encouraged my interest. Even my religious authority figures fed this interest in astronomy.

I didn't really know what astronomers do until later, and in retrospect, I probably should have looked into it a bit more before embarking on a professional path.

But in some ways I was more successful than I should have been. How many of us dreamed of being a paleontologist as a kid? Or a marine biologist? At some point, most of us revise those dreams -- ideally because we discover other interests, but often because the impracticality of our dreams are beat out of us by parents, teachers, or others.

When I was about four, I said I wanted to be a pediatrician, probably because my pediatrician, Dr. Nakashima, was a hilarious and awesome guy who claimed to be a ninja turtle, and maintained that despite my challenges. But my mom's asshole friend said that that wasn't a good idea, and I never, ever considered being a medical doctor, even though, in retrospect, I probably would have been a good one.

I was lucky, in some sense. I was lucky to have enough resources and opportunities to follow my dreams without reality intruding. Sure, it was unfocused and overly idealistic. Sure, I hit points when the contradiction between what I felt were my skills and interests diverged from what I appeared to be doing. And yes, it ended pretty badly, and I'm still recovering from poor choices I made.

But I got away with it for a hell of a long time, partly because I was good enough at math and science to do it, but partly because I had just wanted it so badly and didn't know "better" not to irrationally pursue what should have been a more deliberate, cautious, and considered course of action.

Maybe I shouldn't think that I failed spectacularly. Maybe I should instead be grateful that I got away with it for so long.

A math problem and a general life lesson

A former student highlighted a problem that I found too intriguing to resist. It was challenging, but I managed to solve it.

Let me be clear - the point of this post isn't to shore up my ego by demonstrating my ability to do algebra better than someone else. That is not the point at all. I think the student showed just the right judgment in struggling with it for a couple hours, then giving up and searching for an answer. That's not intellectually dishonest -- it's quite smart. I probably would've given up and not ask anyone -- which I did do, to a distressing level, in college.

The point is to show how something intimidating and new can be connected with prior knowledge. College, especially in technical disciplines, is largely about extending this prior knowledge in ways that are new, but connected with what you already know. Don't worry, you'll get practice, training, and experience, which will refine your judgment and improve your ability to solve problems of increasing complexity.

Here's the question:

Here's my handwritten solution. Sorry if the photo isn't great - my scanner is broken.

It took about an hour. Most of that time was spent on wrongheaded paths. Once I realized what would work, it took about ten minutes. By the way, that's probably an optimistic estimate of the amount of time solving a problem/knowing what to do versus the amount of time wandering like blind moles in research, in my own terrible experience.

 Some of my friends might find that ridiculously slow. I suppose I could say I'm rusty, but, to be honest, I never really put as much energy and thoughtfulness into mathematical derivation in school as, say, understanding psychology or international relations. It just never interested me -- or it frightened me.

Remember, the point of this post is to illustrate a point about math, and life. Sometimes, when confronted by something that seems impossible, it can help to look at hints for how to derive something.

Repeating, in case it's not clear from the scan.

In this case, there were two key insights:

1. The solution form looks like solving for tan (x/2) was the penultimate step. This suggested a substitution of a dummy variable equal to x/2.

2. The radical on one side of the equation in the penultimate step suggested a quadratic solution, with tan(x/2) as "x" in the quadratic form of the equation. From that, I was able to figure out what A, B, and C are. (Note that it doesn't matter whether A is positive or negative, but the relative sign of A, B and C are important.)

Based on that, I was able, after some time, to figure out the right approach and the right substitutions to make, and solved it quasi-formally.

Note:
In this case, looking at half-angle and double-angle formulas was briefly counterproductive. I saw a radical in the half-angle formulas, and a radical in the solution, so I thought that was the proper approach. Big mistake - I ended up wasting time just because of a superficial connection.

As with many things, looking for and identifying superficial connections between things can lead us astray.

For this problem, I needed to know how to recognize the solution of a quadratic equation, how to use a quadratic equation to solve for a function (versus a variable like x), how to substitute a variable for another one to take advantage of double-angle formulas, and a sense of which double-angle formulas I should use. (sin 2x is straightforward, while there are three choices for cos 2x; even though they are equivalent, not all are equally useful in a given situation. The handwritten work shows I momentarily chose the wrong one.)

So stuff from Algebra 2 could be put together to solve an algebraically intensive problem from first-year college physics.

Now I think this is generally true, even at the cutting edge of science, and perhaps other human endeavors. We build off of our prior knowledge. We extend our knowledge and tools to dazzling heights -- but, barring something really weird and advanced beyond my ken, every tool and every solution builds by analogy or by logical extension from previous ones.

So the next time you see something like this on a test -- don't panic! Take a deep breath. See if the form of the solution looks familiar. It might pay to make a list of the standard tricks -- in this particular case, substituting to take advantage of double-angle formulas, and solving for a function instead of a variable using the quadratic equation, with particular attention to coefficients.

Don't be afraid to work backwards -- sometimes it makes a lot more sense to do that. For the formal proof, you need to start from the beginning, but there's no rule saying you can't understand a derivation by reverse engineering it. Just practice, and try to see how anything that is "new" relates to things you know already. If you don't know those other, more fundamental things, make sure you do before you tackle the complicated things.

(Aside: I wish I thought this clearly 11 years ago. I would've probably done a bit better in college and gotten more out of it.)

Finally, to my former students -- sorry if sometimes it seemed like you were being thrown into the deep end of the pool when I was teaching. It was tough-- there were students at all different levels, and I think as an amateur teacher I didn't do as good a job as connecting the dots as I should've. Consider this a down payment on that deficit.

Now, left as an exercise to the reader: extending this analysis beyond math into daily life.

PBS and the battle over America's soul

This election has been about many things. It's been about jobs, tax policy, public debt, a sprinkling of foreign policy, and, occasionally, social issues like gay marriage, vaginal ultrasounds, and contraception.

But it's becoming clear that it is about principles in addition to policy. It's subtle, but it comes down to two different visions about the nature of America, its government, and its future.

We must decide, not once and for all, but as often as is necessary, as often as we are in danger of forgetting, what America really represents.

The battle for America's soul is cast as a false choice between equity and individualism. It really comes down to whether we care, and how much we care, about our fellow citizens, even if they are unknown to us.

I think the PBS debate has highlighted this.

***

First, facts about the funding:

The Corporation for Public Broadcasting receives $445 million from the federal government. The 2012 federal budget is $2.469 trillion. This means the CPB represents 0.018 percent of the overall federal budget.

This is important, because, as we've seen from surveys, people often overestimate the share of the budget that goes to things like NASA. One survey suggests Americans think NASA takes up 24 percent of the budget, when it in fact receives around 1 percent.

Obviously, zeroing the federal funding for public television will not, in itself, do much for the deficit. Neil deGrasse Tyson puts it particularly well:

2. Federal money represents about 15 percent of the overall public television budget. The rest is made up by sponsorship from foundations and, as they famously say, "viewers like you". On the face of it, that doesn't sound like a lot. But it's unclear what restrictions are placed on both the public and private funds. Contributions and local sponsorship of specific stations might vary a lot -- KPCC in the greater Los Angeles area probably does better than a number of rural networks. Some organizations might target specific programmatic development, and not fund operations -- or vice versa. Some organizations insist on matching funds. So there is the potential for a 15 percent reduction to affect far more than 15 percent of the operations of PBS.

But the reason this battle is important is not the specific numbers involved with the funding. It has to do with what PBS represents. I'm going to focus just on the children's programming, though a spirited, solid defense can be made for the rest of it -- including the News Hour, which the debate moderator, Jim Lehrer, helped create so many years ago.

***

Younger people might not know much about Mr. Rogers. He was the host of a long-running children's show on PBS called Mr. Rogers' Neighborhood. It has been satirized, sometimes acutely, but always from a place of love and respect for the impact it's had on generations of Americans (and Canadians).

The funding issue reminds me of Mister (Fred) Rogers' testimony before the US Senate Subcommittee on Communications, in which he successfully argued against a proposed PBS budget cut by Nixon. He did it by explaining, calmly, clearly, what he did, and why he did it. In short, he created a safe, educational environment for children, explaining calmly and carefully things about the world. He did this because he was worried about commercial programming for children, and what it could do to the emotional development of children.

(For those of you who don't know, Mr. Rogers had a background in child development and music composition, and was an ordained Presbyterian minister.)

Mister Rogers, more than Big Bird, embodied the spirit of PBS, especially its children's programming. It was only later, much later, that I realized he was talking to adults too.

An excerpt from a fantastic article that deserves to be read in full:

He is losing, of course. The revolution he started--a half hour a day, five days a week--it wasn't enough, it didn't spread, and so, forced to fight his battles alone, Mister Rogers is losing, as we all are losing. He is losing to it, to our twenty-four-hour-a-day pie fight, to the dizzying cut and the disorienting edit, to the message of fragmentation, to the flicker and pulse and shudder and strobe, to the constant, hivey drone of the electroculture … and yet still he fights, deathly afraid that the medium he chose is consuming the very things he tried to protect: childhood and silence. Yes, at seventy years old and 143 pounds, Mister Rogers still fights, and indeed, early this year, when television handed him its highest honor, he responded by telling television--gently, of course--to just shut up for once, and television listened. He had already won his third Daytime Emmy, and now he went onstage to accept Emmy's Lifetime Achievement Award, and there, in front of all the soap-opera stars and talk-show sinceratrons, in front of all the jutting man-tanned jaws and jutting saltwater bosoms, he made his small bow and said into the microphone, "All of us have special ones who have loved us into being. Would you just take, along with me, ten seconds to think of the people who have helped you become who you are ... Ten seconds of silence." And then he lifted his wrist, and looked at the audience, and looked at his watch, and said softly, "I'll watch the time," and there was, at first, a small whoop from the crowd, a giddy, strangled hiccup of laughter, as people realized that he wasn't kidding, that Mister Rogers was not some convenient eunuch but rather a man, an authority figure who actually expected them to do what he asked … and so they did. One second, two seconds, three seconds … and now the jaws clenched, and the bosoms heaved, and the mascara ran, and the tears fell upon the beglittered gathering like rain leaking down a crystal chandelier, and Mister Rogers finally looked up from his watch and said, "May God be with you" to all his vanquished children.

Seriously, read the whole thing.

***

PBS in general, and its children's programming in particular, serves the needs of educational and emotional development of children without being bombarded by commercial messages.

It is especially important for kids who don't have a fully developed social support network.

I came from a loving family. But I spent a lot of time with grandparents who didn't speak English while my mother was at work. I watched a lot of TV. In retrospect, watching All in the Family when I was three years old was probably not good for me. (I remember writing, "Why is Orchie [sic] Bunker so mad?" on a letter to my grandpa.) But I did watch Mr. Rogers, and Sesame Street, and it helped me with my language development.

So PBS isn't for the kids privileged with wealth and attention and top-notch early childhood education. It's for the rest of us. It's for those of us whose parents, bless their hearts, needed the TV as a babysitter occasionally, but were concerned about the epileptic-inducing nature of Power Rangers and similar shows. It was for those of us who grew up around non-English speakers. It was for those of us who didn't have anyone to talk to us about divorce, or Desert Storm, or about anger -- all of which Mr. Rogers did.

PBS is for the rest of us, for all of us. It's not enough--no television program could replace parenting, in-person education, and hugs. But it helped, especially when it brought kids and adults into the same room, the same emotional space, and helped adults to actually talk with their children.

***

There's a difference between asking for shared sacrifice and cutting a budget completely. If it were about the deficit, then I think Mitt Romney would have used PBS as an example of something we all value, but also something that will need to be cut somewhat in order to ensure a firm financial footing for the children in question.

He didn't do that. He made a joke, a joke to the face of a man employed by that network for decades, a joke at a member of an iconic show, Sesame Street, with twenty different versions around the world. In particular, a version for Palestinian children, Sharaa Simsim, has already been the victim of politics. In early 2012, it was defunded after the US Congress suspended Palestinian aid after Palestine's appeal to the UN for statehood. (The funding was later reinstated by Congress, with restrictions, but the restrictions were overridden by the President.)

It was a joke because he doesn't get how important it is to families not like his -- flawed families, with flawed parents, where TV is on more than it should be, but where it can be a source of learning and even healing. That's what Mr. Rogers saw, and Mr. Romney does not. It's about families that may or may not be like ours, children who may not be ours, but belong to us -- and we to them -- all the same.

Why should I try to say it in words, when Mr. Rogers did it for me?

We live in a world in which we need to share responsibility. It's easy to say 'It's not my child, not my community, not my world, not my problem.' Then there are those who see the need and respond. I consider those people my heroes."

Sesame Street, you're my hero. Fred Rogers, you're my hero. You have helped me develop into a person who, however imperfectly, believes that liberty and responsibility are partners, not antagonists, in the building of a more perfect union. And it is that partnership, with all its tensions, that is the soul of America.

Calculus memories from Harvey Mudd College

I had a multivariable calculus professor in college who was first-generation Chinese. She's a world expert on differential geometry, and a bit quirky. How much of it is her being Chinese, and how much of it is her quirkiness, I don't know.

Evidently we weren't the brightest class in recent memory, because she decided to review some single-variable calc during a review session. Here's how she helped us remember the derivatives of the exponential and natural logarithmic functions.

"e^x is like a strong child. You hit it [with a derivative] and nothing happens."

"ln x is like a weak child. You hit it and it dies."

I think there was audible consternation. But we remembered.

Sample question from a student, and my reply

Looking for feedback on whether I'm providing good help to my students via email. A sample (in fact, to this point, the only) email exchange for precalc.

How are you today? I am a student in your pre-calculus class.I have some questions on my homework.Could you please give me some help?

1.How to find out real zeros from all possible rational zeros of a function?(I have read the samples in the texrbook ,but I can't understand.Please give me a specific sample like page 160 :11)I know how to find all possible rational zeros ,but real zeros ,I can't.

2.Page 160:In exrcise 25-28 ,I have no idea to sketch the graph of f so that some of the possible zeros in part (a) can be disregarded)

Thank you for your help!Have a nice weekend.

My reply:

Hope you're enjoying your weekend so far!


1.How to find out real zeros from all possible rational zeros of a function?(I have read the samples in the texrbook ,but I can't understand.Please give me a specific sample like page 160 :11)I know how to find all possible rational zeros ,but real zeros ,I can't.

So the first thing to do is to use the rational zero test.

Remember that the rational zero test can be used on functions of the form

f(x) = an*x^n + a(n-1)*x^(n-1)+... + a1*x + a0.

where an is the coefficient of the term of order n (x^n).

We find the rational zeroes by taking the factors of the a0 term (the constant, which is -6) divided by the coefficient on the highest order term. In this case, the function is of order 3. The coefficient for x^3, a3, is 1.

If a0 is -6, the factors of a0 are {-6, -3, -2, -1, +1, +2, +3, +6}.

and the factors of a3 are {-1, 1}.

So, dividing the factors of a0 by the factors of a3 give us the possible rational zeroes:

a0/a3 = {-6, -3, -2, -1, 1, 2, 3, 6}.

I picked one root to try, +1. I substitute x = 1 into the function.

f(1) = (1)^3 - 6(1)^2 + 11(1) -6
= 0.

I got lucky! So I know that x = 1 is a rational root of f(x).

Now, there are two things I can do at this point.

1. Keep trying the other rational roots.

If you do this, you try the other possible rational roots (factors of a0 / factors of a3) and see if any of them are a zero.


2. Divide by x-1 to get a quadratic equation, which I will solve either by guessing or by using the quadratic formula.

Remember that if x = 1 is a root of f(x), (x - 1) is a factor of f(x), and we can divide f(x) by the factor to find the other roots.

This is the approach I used, because I didn't want to waste time in case there are no other rational roots. If x = 1 is the only rational root, then the quadratic formula will give us the real (or complex) roots, guranteed!

So, using long division,

(x^3 - 6x^2 + 11x - 6)/(x-1) = x^2 - 5x + 6


Now, I can guess the factors - it looks like x = 2 and x = 3 will work. Sure enough,

x^2 - 5x + 6 = (x - 2)*(x - 3)

which means the real roots of f(x) are x = {1, 2, 3}.

But it's a good idea to solve it using the quadratic formula, in case we can't factor it easily.

The quadratic formula can be used for any equation of the form

ax^2 + bx + c = 0

where a, b, and c are real numbers.

The solutions are found using this formula

roots = (- b +/- sqrt(b^2 - 4ac)/(2a)

where sqrt("something") equals the square root of "something".

For

x^2 - 5x + 6

a = 1, b = -5, and c = 6.

Using the formula,

roots = (- (-5) +/- sqrt((-5)^2 - 4*1*6))/(2*1)

= (5 +/- sqrt(25-24))/2

= (5 +/- 1)/2

= {2, 3}.

So we see that the solutions are x = 2 and x = 3.

In summary, for a polynomial function, you do the following steps:

1. Use the rational root test to find possible roots.

2. Try the roots and see if any of them work.

3. For every root that works, divide the function by its factor. For example, if the rational root is x = 4, you divide the function by the factor (x - 4).

4. After dividing, see if the function is of order 2 (looks like ax^2 + bx +c). If so, you can use the quadratic formula. This is how you get real roots that are irrational, and also complex roots.


2.Page 160:In exrcise 25-28 ,I have no idea to sketch the graph of f so that some of the possible zeros in part (a) can be disregarded)

The easiest way to graph it is to use your calculator. But if you don't have that, then the easiest way to graph is to find out what f(x) equals for a number of x values.

For example, on problem 25, f(x) = x^3 + x^2 - 4x - 4

Using the rational zero test, the possible rational roots are

x = {-4, -2, -1, 1, 2, 4}.

So I need to graph it at least over the domain of x:[-4, 4] (everywhere from x = -4 to x = 4).

I would write two columns like this

x f(x)

-4

-3

-2

-1

0

1

2

3

4

And calculate what f(x) is for each x.


x f(x)

-4 -36

-3 -10

-2 0

-1 0

0 -4

1 -6

2 0

3 20

4 60


If you draw the points and connect them, they look something like the picture I've attached in this email.

Note that from this picture, it's easy to eliminate -4, 1, and 4 as possible roots, but -2, 1, and 2 look like rational roots of f(x).

This is an easy example, but sometimes you'll get a function that is more complicated. That's why it's a good idea to graph it, especially in cases where you have a lot of possible rational roots to test.

Good luck! And thanks for asking good questions!