During the Cold War, the usual way nations compete (direct warfare) was taken off the table by the invention of the hydrogen bomb. With the alternative being mutually-assured destruction, the two sides, led by the USA and the USSR, had to find other ways to compete. Some of those ways were harmful, such as proxy wars, as happened in Vietnam. Others, however, were helpful, such as the space race. The United States put men on the Moon in order to beat the Soviet Union there, as this iconic 1969 photograph makes evident (source: NASA).
We are all still reaping the benefits of the technological and scientific advances made during this period, and for this purpose. The most obvious example of such a benefit is the computer you are using to read this blog-post, for computer technology had to be advanced dramatically, on both sides, in order to escape the tremendously-challenging gravity-well of the Earth.
Wouldn’t it be wonderful if other conflicts in society took beneficial forms, as happened in this historical example?
This could happen in many ways, but the one that gave me the idea for this post is the conflict between teachers’ unions and school districts’ administrators, now taking place in school districts all over the place. I think it would be awesome if this previously-harmful competition changed, to take a helpful form: book drives, to help school libraries.
Please do not misunderstand, though: I’m not talking about taxpayer money, nor union dues. My idea need not, and should not, affect the budget of any school district, nor union budget. All that need happen is for individual people — teachers and administrators — to go home, look at their own bookshelves, and help students directly, by donating some of their already-paid-for books to school libraries.
While I make no claim to represent any organization, I am a teacher, and a member of the NEA (the National Education Association) in the United States, as well as my state and local NEA affiliates. In an effort to start this new, helpful way to compete, I will give books to the school library where I teach, next week, which is the second week of the new school year. That’s a lot easier than, well, putting men on the Moon.
This is something we can all do. All of us in the education profession, after all, already agree that we want students reading . . . and this is something we can easily do, to work together towards that goal. School libraries need hardcover books which are student-friendly, meaning that they appeal to a young audience, on a wide variety of subjects. Both fiction and non-fiction books are helpful.
Lastly, in the hope that this idea catches on, I will simply point out one fact: helping turn this idea into a reality is as easy as sharing a link to this blog-post.
This is my 22nd year teaching. This year, I teach in only one department. This is nice; I’ve spent much of my career in multiple departments, simply because I am certified in multiple subject areas. This year, in my building, I am one of three science teachers. Our high school has become so large that the 9th grade has been “spun off” to a new freshman campus, while remaining part of the high school, and I’m one of the teachers who gets to go to the new campus. This provides my students, my colleagues (especially at the new campus), and myself the opportunity for a fresh start, to a greater degree than is usually the case when a new school year begins.
My students are in just two subjects, this year: Physical Science, and Pre-AP Physical Science. I don’t want the students in the class without the “Pre-AP” prefix to feel that they are in a “lesser” class, in any sense of that word, so I am renaming “Physical Science,” slightly: “High School Physical Science.” It is my hope that this change in wording serves to communicate high expectations, and 9th grade is the first year of high school — which, in the USA’s public school systems, means 9th grade students must pass courses to earn credits toward graduation, usually for the first time.
In the other class, Pre-AP Physical Science, I am teaching that version of the course for the first time, but I feel well-prepared by the extensive training I had this Summer, and last school year, through my university, the school where the Summer training was held, and the College Board. Both classes will challenge students, but it is also true that the two classes will be different, for Pre-AP Physical Science have to leave students prepared to function effectively, later, in other Pre-AP and/or AP science courses.
Physical Science is an introduction to two sciences: physics, and then chemistry, at least in my school district. It helps me that I have experience teaching both subjects as higher-level, “stand-alone” classes. In this class (both versions), we also touch on some other sciences which are also physical sciences, such as geology, astronomy, and the science of climate change. However, those sciences do not dominate these courses, as physics and chemistry do. The image above is from chemistry (and was created with Stella 4d, which you can try here), and shows a model of a sixty-atom all-carbon molecule called buckminsterfullerene, one of a class of roughly-spherical carbon allotropes called fullerenes. Mathematicians call this particular fullerene’s shape a “truncated icosahedron,” and, in sports, this same shape is known as the (non-American) “football” or “soccer ball.” Physical modes of this shape may be made with molecular model sets of various kinds, Zometools, and other materials. In both versions of my science classes this year, building models of this molecule will be one of many lab activities we will do; one of my goals this year is for my students to spend a third of their time doing labs. The legal requirement for science class time spent in lab, in my state, is at least one-fifth, so more than that is fine. Science classes helped me learn both science and mathematics, but what I remember the most is the labs. I don’t think that’s just me, either; students learn more effectively, I have observed, by conducting scientific experiments themselves, than by being “lectured at” for extended periods of time.
I’m looking forward to a good year — for all of us.
Liquid mercury, in schools, poses three major problems:
It is extremely toxic,
It has a high vapor pressure, so you can be poisoned by invisible mercury vapor leaving any exposed surface of liquid mercury, and
Playing with liquid mercury is a lot of fun.
These are compelling reasons to leave use of mercury to those at the college level, or beyond. In the opinion of this science teacher, use of liquid mercury in science classes, up through high school chemistry, inside or outside thermometers, is a bad idea. If the bulb at the bottom of a thermometer, as well as the colored stripe, looks silvery, as in the picture below (found on Wikipedia), then that silvery liquid is mercury, and that thermometer should not be used in labs for high school, let alone with younger children. Your local poison control center can help you find the proper thing to do with mercury in your area; it should definitely not just be thrown away, for we do not need this serious environmental toxin in landfills, where it will eventually reach, and poison, water. Red-stripe thermometers without any silvery line, on the other hand, are far safer, although broken glass can still cause injury.
I turned ten years old in 1978, and, by that time, I had already spent many hours playing (unsupervised) with liquid mercury, pouring it hand-to-hand, etc., so I know exactly how irresistible a “plaything” mercury can be, to children. Luck was on my side, and I suffered no ill effects, but I can state from experience that children should not be tempted with highly-toxic “mercury as a toy,” for it’s not a toy at all. Mercury spills require special “hazmat” training to clean up safely; anyone encountering such a spill who does not have such training should simply notify the proper authorities. In the USA, this means evacuating the area immediately, and then calling 911 — from far enough away to keep the caller from breathing invisible mercury vapor.
Fortunately, there is a safe alternative which can give students a chance to experiment with a room-temperature metal: an alloy of three parts gallium to one part indium, by mass. Gallium’s melting point is between normal human body temperature and room temperature, so it can literally melt in your hand (although a hot plate is faster). Indium, on the other hand, has a melting point of 156.6°C. For this reason, I will not buy a hot plate unless it can reach higher that that temperature. (Note: use appropriate caution and safety equipment, such as goggles and insulated gloves, with hot plates, and the things heated with them, to avoid burns.)
Once both elements are massed, in the proportions given above, they can then be melted in the same container. When they melt and mix together, they form an alloy which remains liquid at room temperature.
Some might wonder how mixing two elements can create an alloy with a melting point below the melting points of either of the two ingredients, and the key to that puzzle is related to atomic size. Solids have atoms which vibrate back and forth, but don’t move around each other. In liquids, the atoms are more disordered (and faster), and easily slip around each other. In solid, room-temperature gallium, all the atoms are of one size, helping the solid stay solid. Warm it a little, and it melts. With pure indium, this applies, also, but you have to heat it up a lot more to get it to melt. If the two metals are melted and thoroughly mixed, though, and then frozen (a normal freezer is cold enough), the fact that the atoms are of different sizes (indium atoms are larger than gallium atoms) means the atoms will be in a relatively disordered state, compared to single-element solids. In liquids, atoms are even more disordered (that is, they possess more entropy). Therefore, a frozen gallium/indium alloy, with two sizes of atoms, is already closer to a disordered, liquid state, in terms of entropy, than pure, solid gallium or indium at the same temperature. This is why the gallium-indium mixture has a melting point below either individual element — it requires a lower temperature to get the individual atoms to flow past each other, if they are already different atoms, with different sizes.
Those who have experience with actual liquid mercury will notice some important differences between it and this gallium-indium alloy, although both do appear to be silver-colored liquids. (This is why mercury is sometimes called “quicksilver.”) For one thing, their densities are different. A quarter, made of copper and nickel, will float on liquid mercury, for the quarter’s density is less than that of mercury. However, a quarter will sink in liquid 3:1 gallium-indium alloy. To float a metal on this alloy, one would need to use a less-dense metal, such as aluminum or magnesium, both of which sink in water, but float in liquid Ga/In alloy.
Other differences include surface tension; mercury’s is very high, causing small amounts of it on a floor to form little liquid balls which are difficult (and dangerous) to recapture. Gallium-indium alloy, by contrast, has much less surface tension. As a result, unlike mercury, this alloy does not “ball up,” and it will wet glass — and doing that turns the other side of the glass into a mirror. Actual mercury will not wet glass.
The most important differences, of course, is that indium and gallium are far less toxic than mercury, and that this alloy of those two elements has a much lower vapor pressure than that of mercury. Gallium and indium are not completely non-toxic, though. Neither indium nor gallium should be consumed, of course, and standard laboratory safety equipment, such as goggles and gloves, should be worn when doing laboratory experiments with these two elements.
People have been trying to figure out what intelligence is, and how it differs from person to person, for centuries. Much has been written on the subject, and some of this work has helped people. Unfortunately, much harm has been done as well. Consider, for example, the harm that has been done by those who have had such work tainted by racism, sexism, or some other form of “us and them” thinking. This model is an attempt to eliminate such extraneous factors, and focus on the essence of intelligence. It is necessary to start, therefore, with a clean slate (to the extent possible), and then try to figure out how intelligence works, which must begin with an analysis of what it is.
If two people have the same age — five years old, say — and a battery of tests have been thrown at them to see how much they know (the amount of knowledge at that age), on a wide variety of subjects, person A (represented by the blue curve) may be found to know more, at that age, than person B (represented by the red curve). At that age, one could argue that person A is smarter than person B. Young ages are found on the left side of the graph above, and the two people get older, over their lifespans, as the curves move toward the right side of the graph.
What causes person A to know more than person B, at that age? There can be numerous factors in play, but few will be determined by any conscious choices these two people made over their first five years of life. Person B, for example, might have been affected by toxic substances in utero, while person A had no such disadvantage. On the other hand, person A might simply have been encouraged by his or her parents to learn things, while person B suffered from parental neglect. At age five, schools are not yet likely to have had as much of an impact as other factors.
An important part of this model is the recognition that people change over time. Our circumstances change. Illnesses may come and go. Families move. Wars happen. Suppose that, during the next year, person B is lucky enough to get to enroll in a high-quality school, some distance from the area where these two people live. Person B, simply because he or she is human, does possess curiosity, and curiosity is the key to this model. Despite person B‘s slow start with learning, being in an environment where learning is encouraged works. This person begins to acquire knowledge at a faster rate. On the graph, this is represented by the red curve’s slope increasing. This person is now gaining knowledge at a much faster rate than before.
In the meantime, what is happening with person A? There could be many reasons why the slope of the blue curve decreases, and this decrease simply indicates that knowledge, for this person, is now being gained at a slower rate than before. It is tempting to leap to the assumption that person A is now going to a “bad” school, with teachers who, at best, encourage rote memorization, rather than actual understanding of anything. Could this explain the change in slope? Yes, it could, but so could many other factors. It is undeniable that teachers have an influence on learning, but teacher quality (however it is determined, which is no easy task) is only one factor among many. Encouraging the “blame the teacher” game is not the goal of this model; there are already plenty of others doing that.
Perhaps person A became ill, suffered a high fever, and sustained brain damage as a result. Perhaps he or she is suddenly orphaned, therefore losing a previous, positive influence. There are many other possible factors which could explain this child’s sudden decrease of slope of the blue “learning curve” shown above; our species has shown a talent for inventing horrible things to do to, well, our species. Among the worst of the nightmare scenarios is that, while person B is learning things, at a distant school, the area where person A still lives is plunged into civil war, and/or a genocide-attempt is launched against the ethnic group which person A belongs to, as the result of nothing more than an accident of birth, and the bigotry of others. Later in life, on the graph above, the two curves intersect; beyond that point, person B knows more than person A, despite person B‘s slow start. To give credit, or blame, to either of these people for this reversal would clearly be, at best, a severely incomplete approach.
At some point, of course, some people take the initiative to begin learning things on their own, becoming autodidacts, with high-slope learning curves. In other words, some people assume personal responsibility for their own learning. Most people do not. Few would be willing to pass such judgment on a child who is five or six years old, but what about a college student? What about a high school senior? What about children who have just turned thirteen years old? For that matter, what about someone my age, which is, as of this writing, 48? It seems that, the older a person is, the more likely we are to apply this “personal responsibility for learning” idea. Especially with adults, the human tendency to apply this idea to individuals may have beneficial results. That does not, however, guarantee that this idea is actually correct.
I must stop analyzing the graph above for now, because the best person for me to examine, at this point, in detail, is not on the graph above. He is, however the person I know better than anyone else: myself. I’ve been me now for over 48 years, and have been “doing math problems for fun” (as my blog’s header-cartoon puts it) for as long as I can remember. This is unusual, but, if I’m honest, I have to admit that there are inescapable and severe limits on the degree to which I can make a valid claim that I deserve credit for any of this. I did not select my parents, nor did I ask either of them to give me stacks of books about mathematics, as well as the mathematical sciences. They simply noticed that, when still young, I was curious about certain things, and provided me with resources I could use to start learning, early, at a rapid rate . . . and then I made this a habit, for, to me, learning is fun, if (and only if) the learning is in a field I find interesting. I had absolutely nothing to do with creating this situation. My parents had the money to buy those math books; not all children are as fortunate in this respect. Later still, I had the opportunity to attend an excellent high school, with an award-winning teacher of both chemistry and physics. To put it bluntly, I lucked out. As Sam Harris, the neuroscientist, has written, “You cannot make your own luck.”
At no point in my life have I managed to learn how to create my own luck, although I have certainly tried, so I have now reached the point where I must admit that, in this respect, Sam Harris is correct. For example, I am in college, again, working on a second master’s degree, but this would not be the case without many key factors simply falling into place. I didn’t create the Internet, and my coursework is being done on-line. I did not choose to be born in a nation with federal student loan programs, and such student loans are paying my tuition. I did not create the university I am attending, nor did I place professors there whose knowledge exceeds my own, regarding many things, thus creating a situation where I can learn from them. I did not choose to have Asperger’s Syndrome, especially not in a form which has given me many advantages, given that my “special interests” lie in mathematics and the mathematical sciences, which are the primary subjects I have taught, throughout my career as a high school teacher. The fact that I wish to be honest compels me to admit that I cannot take credit for any of this — not even the fact that I wish to be honest. I simply observed that lies create bad situations, especially when they are discovered, and so I began to try to avoid the negative consequences of lying, by breaking myself of that unhelpful habit.
The best we can do, in my opinion, is try to figure out what is really going on in various situations, and discern which factors help people learn at a faster rate, then try to increase the number of people influenced by these helpful factors, rather than harmful ones. To return to the graph above, we will improve the quality of life, for everyone, if we can figure out ways to increase the slope of people’s learning-curves. That slope could be called the learning coefficient, and it is simply the degree to which a person’s knowledge is changing over time, at any given point along that person’s learning-curve. This learning coefficient can change for anyone, at any age, for numerous reasons, a few of which were already described above. Learning coefficients therefore vary from person to person, and also within each person, at different times in an individual’s lifetime. This frequently-heard term “lifelong learning” translates, on such graphs, to keeping learning coefficients high throughout our lives. The blue and red curves on the graph above change slope only early in life, but such changes can, of course, occur at other ages, as well.
It is helpful to understand what factors can affect learning coefficients. Such factors include people’s families, health, schools and teachers, curiosity, opportunities (or lack thereof), wealth and income, government laws and policies, war and/or peace, and, of course, luck, often in the form of accidents of birth. Genetic factors, also, will be placed on this list by many people. I am not comfortable with such DNA-based arguments, and am not including them on this list, for that reason, but I am also willing to admit that this may be an error on my part. This is, of course, a partial list; anyone reading this is welcome to suggest other possible factors, as comments on this post.
In the map above, the People’s Republic of China (PRC) is shown in red, while the Republic of China (ROC) is shown in yellow. “Barbarian” nations (from the point of view of the Han, or the ethnic group we call “Chinese” in English) are shown in orange, and both oceans and large lakes are shown in blue. The third (and only other) majority-Han nation, the island city-state called Singapore, is not shown on this map, as it is too far to the South to be seen here. From the point of view of the Han, “barbarians” have been, historically, those humans who were not Han, while “the Han” can be translated as “the people.”
This historical xenophobia I just described among the Han is hardly unique; it is, in my opinion, simply human nature. The British rock band Pink Floyd explained this, quite well, in the following song, “Us and Them,” from 1973’s classic Dark Side of the Moon. This album, in the form of a cassette tape which had to be purchased by my parents (for I would not let go of it in the store we were in), happens to be the first musical album I actually owned, back when it was newly-released (I was born in 1968). If you choose to listen to this song, please consider this idea of xenophobia, as simply being a human characteristic, while it plays.
Ancient Greeks had the same “us and them” attitude about those who did not speak Greek, and the English word “barbarian” is derived from Greek, with a meaning which parallels what I have described in China. Eurocentrism, in general, in the study of “world history,” is well-known. Moving to another continent, the people where I live, the USA, are famous for learning geography one nation at a time . . . as we go to war with them, of course. Only a tiny percentage of Americans knew where either Korea was located until we went to war there, and we (as a people) did not know where Vietnam was until we went to war there. More recently, Americans learned — twice! — where Iraq is, though many of us still, inexplicably, confuse it with Iran. This list of xenophobic nations is far from complete, but these examples are sufficient to make the point.
When, in 1939, British Prime Minister Winston Churchill uttered the famous phrase, “It is a riddle, wrapped in a mystery, inside an enigma,” he was referring to the Soviet Union (or USSR), although the proper noun he actually used was “Russia.” However, this quotation applies equally well to the PRC, which has one indisputable advantage over the USSR: the People’s Republic of China still exists, while the Soviet Union does not. In the last post here, I began an ambitious series, with the goal of explaining China. I promised, then, that my next post in the series would explain my qualifications to write on the subject of the PRC, the ROC, Greater China, and the Han — so that’s what I need to do now.
I am currently working on my second master’s degree, in an unrelated field (gifted, talented, and creative education). However, my first master’s degree was obtained in 1996, when Deng Xiaoping, while no longer the PRC’s “paramount leader,” was still seen as its most prominent retired elder statesman. It was Deng Xiaoping, primarily, who made (and defended) the decision to send the tanks in, and crush the pro-democracy demonstrators in Tiananmen Square, in Beijing, in June of 1989, which I watched as they happened, on live TV. I was horrified by those events, and this has not changed.
During the early 1990s, I began studying the economic reforms which made the era of Deng Xiaoping so different from Chairman Mao’s China, trying to figure out the solution to a big puzzle: how so much economic growth could be coming from an area dominated by a huge, totalitarian, country which, at that time and now, was one of the few remaining nations on Earth which still claimed to be Communist. This study was done during the time of the “New Asia” investment bubble, as it was called after it “popped” (as all investment bubbles do, sooner or later). New Asia’s economic growth was led by the “Four Tigers” of Hong Kong (still a British colony, at that time), Singapore, Taiwan, and South Korea. South Korea is, of course, Korean, but the other three “tigers,” all had, and still have, majority-Han populations. What money I had, I invested in the Four Tigers, and I made significant profits doing so, which, in turn, led to a general interest in East Asia.
Motivated by simple human avarice, I studied the Four Tigers intensely, leading me to focus (to the extent made possible by the course offerings) on 20th Century East Asian history, during the coursework for my first master’s degree. There was a problem with this, though, and I was unaware of it at the time. My university (a different one than the one I attend now) had only one East Asian history professor, and he was very much a Sinophile. Sinophiles love China uncritically, or with the minimal amount of criticism they can get away with. When we studied the rise to power of Mao Zedong, and the PRC under the thumb of Chairman Mao, I heard it explained by a man who viewed China, and Chairman Mao, through rose-colored glasses, even while teaching about others who made the same error, to an even greater degree. I had already read one book about the Cultural Revolution, earlier in the 1980s, so I was skeptical, but he was also my only professor. The result was confusion. This was the book I had already read, along with a link to a page on Amazon where you can purchase it, and easily find and purchase the Pink Floyd music posted earlier, if you wish to do so. This is Son of the Revolution, by Liang Heng and Judith Shapiro, and you can buy it at https://www.amazon.com/Son-Revolution-Liang-Heng/dp/0394722744/ref=sr_1_1?ie=UTF8&qid=1468869380&sr=8-1&keywords=son+of+the+revolution.
This book was read for an undergraduate sociology course, at my first college, during the Reagan years. The important thing to know about Liang Heng, the book’s primary author, is that he was, himself, of the Han, as well as being from the PRC itself. The professor for this course wanted us to see the horror of a mass movement gone horribly wrong, and she chose this insider’s view of the Cultural Revolution, during which I was born, to do that. What I heard from my East Asian history professor did not mesh well with what I was taught by my East Asian history professor, and so I left that degree program confused. This professor’s argument, in a nutshell, was Chairman Mao was a figure of tremendous importance (true) who had good intentions (false), and tried to do amazing things (half-true, and half-false by omission, for these were amazing and horribly evil things), but had them turn out wrong (true), with many millions of his own people dying as a result, over and over (definitely true; Mao’s total death total exceeds that of Hitler or Stalin, either one). The “good intentions” part was what confused me, of course, for Mao was a monster, yet, from my later professor, I was hearing him described as a Great and Important Man.
I would have remained in this confused state, has I not also read this book, also written, primarily, by a person of the Han: the amazing Jung Chang, who has her own page on Amazon, at http://www.amazon.com/Jung-Chang/e/B00N3U50ZO/ref=sr_tc_2_0?qid=1468870698&sr=8-2-ent. (On that page, I notice she has a newer book out, which I have not read, and she is such a fantastic author that I am buying it now.) This, by contrast, was her first well-known book, and the one I read as an undergraduate:
Wild Swans tells the story of three generations of Han women: Jung Chang’s maternal grandmother (who had bound feet, and could barely walk, for that reason), then the author’s mother, and then finally Jung Chang herself, who found herself a Red Guard during the Cultural Revolution at the age of 14. This book tells their story, and is riveting. It has nothing nice to say about Chairman Mao, and contains much criticism of “The Great Helmsman,” as his cult of personality enthusiastically called him, yet he is not the focus of Wild Swans. The author’s family, over three generations, is.
I did my master’s degree work from the Sinophile professor described earlier, and then, later on still, I encountered Sinophobes. The opposite of Sinophiles, people who have Sinophobia have nothing nice to say about China, nor the Han. They hate and fear things Chinese because they fear the unknown — in other words, Sinophobia is a more specific form of xenophobia.
So, first, I read Liang Heng, and then, later, I started reading Jung Chang. Next, I heard the Maoist viewpoint explained quite thoroughly by my Sinophile professor, while my reading of Liang Heng and Jung Chang had exposed me to an anti-Mao, but non-Sinophobic, point of view, which is a direct consequence of the fact that both authors were actually of the Han, and had direct exposure to Maoism. Later came the Sinophobes, and their written and spoken, anti-Chinese, case for . . . whatever. (Actually, the Sinophobes never make a case for anything, unless one counts hating and fearing China and the Han as being “for” something. I do not.) Later still, one of my close friends studied ancient Chinese history and philosophy extensively, and we had (and still have) many talks about both ancient and modern China, including Chairman Mao, and the silliness of the Sinophobes, but this friend is more interested in talking about, say, Confucianism, rather than Maoism, or Mao himself. I was primed to learn the truth about Mao, but had to wait for the right opportunity.
Think about this, please. How many books have been written that accurately describe Stalin as a monster? How many exist about Hitler? I should not have had to wait so long to find out something about Mao I felt I could believe, and that described him as the monster he was, but wait I did, for no such book existed . . . until Jung Chang came to my rescue, with her next book, after 1991’s Wild Swans. All 800+ pages of it.
It took her many years to write this tome, and it was published in 2005. She grew up under Mao, having been born in 1952, not long after the revolution of 1949, which established the People’s Republic of China. Chairman Mao finally died in 1976. Two years after that, Jung Chang was sent to Great Britain as a college student, on a government scholarship. Being highly intelligent, and not wanting to return to China, she went on to become the first of the Han to receive a Ph.D. at any British university. This book, focused on Mao’s formative years, rise to power, and tyrannical rule, all the way to his death, is, as its subtitle states, “The Unknown Story” of this historical period. Jung Chang was uniquely qualified to write this story, having lived through so much of the events described in her book. She knew how expendable people were to Mao, having witnessed it, and survived. To the extent possible (and she was quite resourceful on this point) she used primary sources. This is why I give her much credibility.
These are the ways I have learned about China: from three books by two of the Han, long talks with a personal friend, and two college professors with different points of view on China, and Mao in particular. I have rejected the points of view of both the Sinophiles and the Sinophobes, and now I try to learn what I can from other sources, especially sources who are, themselves, of the Han — although I am weakened in this respect by the fact that I am only bilingual, with my two languages being mathematics and English, in that order. If you think this approach makes sense, I hope you will read my other posts, past and future, about China and the Han.
An interesting phenomenon in physics, and physics education, is the simplicity of symmetric situations, compared to the complexity of similar situations which are, instead, asymmetrical. Students generally learn the symmetrical versions first, such as this static equilibrium problem, with the hanging masses on both left and right equal.
The problem is to find the measures of the three angles shown above, with values given for all three masses. Here is the setup, using physical objects, rather than a diagram.
The masses on the left and right are each 100 g, or 0.100 kg, while the central masses total 170 g, or 0.170 kg. Since all hanging masses are in static equilibrium, the forces pulling at the central point (at the common vertex of angles λ, θ, and ρ) must be balanced. Specifically, downward tension in the strings must be balanced by upward tension, and the same is true of tension forces to the left and to the right. In the diagram below (deliberately asymmetrical, since that’s coming soon), these forces are shown, along with the vertical and horizontal components of the tension forces held in the diagonal strings.
Because the horizontal forces are in balance, Tlx = Trx, so Mlgcosλ = Mrgcosρ — which is not useful now, but it will become important later. In the symmetrical situation, all that is really needed to solve the problem is the fact that the vertical forces are in balance. For this reason, Tc = Tly + Try, so Mcg = Mlgsinλ + Mrgsinρ. Since, due to symmetry, Ml = Mr and λ = ρ, Mr may be substituted for Ml, and ρ may be substituted for λ, in the previous equation Mcg = Mlgsinλ + Mrgsinρ, yielding Mcg = Mrgsinρ + Mrgsinρ, which simplifies to Mcg =2Mrgsinρ. Cancelling “g” from each side, and substituting in the actual masses used, this becomes 0.170 kg = 2(0.100 kg)sinρ, which simplifies to 0.170 kg = (0.200 kg)sinρ, then 0.170/0.200 = sinρ. Therefore, angle ρ = sin-1(0.170/0.200) = 58°, which, by symmetry, must also equal λ. Since all three angles add up to 180º, the central angle θ = 180° – 58° – 58° = 64°. These answers can then be checked against the physical apparatus.
When actually checked with a protractor, the angles on left and right are each about 53° — which is off from the predicted value of 58° by about 9%. The central angle, of course, is larger, at [180 – (2)53]° = 74°, to make up the difference in the two smaller angles. The error here could be caused by several factors, such as the mass of the string itself (neglected in the calculations above), friction in the pulleys, or possibly the fact that the pulleys did not hang straight down from the hooks which held them, but hung instead at a slight diagonal, as can be seen in the second image in this post. This is testable, of course, by using thinner, less massive string, as well as rigidly-fixed, lower-friction pulleys. However, reducing the error in a lab experiment is not my objective here — it is, rather, use of a simple change to turn a relatively easy problem into one which is much more challenging to solve.
In this case, the simple change I am choosing is to add 50 grams to the 100 g already on the right side, while leaving the central and left sides unchanged. This causes the angles where the strings meet to change, until the situation is once more in static equilibrium, with both horizontal and vertical forces balanced. With the mass on the left remaining at 0.100 kg, the central mass at 0.170 kg, and the mass on the right now 0.150 kg, what was an easy static equilibrium problem (finding the same three angles) becomes a formidable challenge.
For the same reasons as before (balancing forces), it remains true that Mlgcosλ = Mrgcosρ (force left = force right), and, this time, that equation will be needed. It also remains true that Mcg = Mlgsinλ + Mrgsinρ (downward force = sum of the two upward forces). The increased difficulty is caused by the newly-introduced asymmetry, for now Ml ≠ Mr, and λ ≠ ρ as well. It remains true, of course, that λ + θ + ρ° = 180.
In both the vertical and horizontal equations, “g,” the acceleration due to gravity, cancels, so Mlgcosλ = Mrgcosρ becomes Mlcosλ = Mrcosρ, and Mcg = Mlgsinλ + Mrgsinρ becomes Mc = Mlsinλ + Mrsinρ. The simplified horizontal equation, Mlcosλ = Mrcosρ, becomes Ml²cos²λ = Mr²cos²ρ when both sides are squared, in order to set up a substitution based on the trigonometric identity, which works for any angle φ, which states that sin²φ + cos²φ = 1. Rearranged to solve it for cos²φ, this identity states that cos²φ = 1 – sin²φ. Using this rearranged identity to make substitutions on both sides of the previous equation Ml²cos²λ = Mr²cos²ρ yields the new equation Ml²(1 – sin²λ) = Mr²(1 – sin²ρ). Applying the distributive property yields the equation Ml² – Ml²sin²λ = Mr² – Mr²sin²ρ. By addition, this then becomes -Ml²sin²λ = Mr² – Ml² – Mr²sin²ρ. Solving this for sin²λ turns it into sin²λ = (Mr² – Ml² – Mr²sin²ρ)/(-Ml²).
Next, Mc = Mlsinλ + Mrsinρ (the simplied version of the vertical-force-balance equation, from above), when solved for sinλ, becomes sinλ = (Mrsinρ – Mc)/(- Ml). Squaring both sides of this equation turns it into sin²λ = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(- Ml)².
There are now two equations solved for sin²λ, each shown in bold at the end of one of the previous two paragraphs. Setting the two expressions shown equal to sin²λ equal to each other yields the new equation (Mr² – Ml² – Mr²sin²ρ)/(-Ml²) = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(- Ml)², which then becomes (Mr² – Ml² – Mr²sin²ρ)/(-Ml²) = (Mr²sin²ρ – 2MrMcsinρ + Mc²)/(Ml)², and then, by multiplying both sides by -Ml², this simplifies to Mr² – Ml² – Mr²sin²ρ = – (Mr²sin²ρ – 2MrMcsinρ + Mc²), and then Mr² – Ml² – Mr²sin²ρ = – Mr²sin²ρ + 2MrMcsinρ – Mc². Since this equation has the term – Mr²sin²ρ on both sides, cancelling it simplifies this to Mr² – Ml² = 2MrMcsinρ – Mc², which then becomes Mr² – Ml² + Mc² = 2MrMcsinρ, and then sinρ = (Mr² – Ml² + Mc²)/2MrMc = [(0.150 kg)² – (0.100 kg)² + (0.170 kg)²]/[2(0.150 kg)(0.170 kg)] = (0.0225 – 0.0100 + 0.0289)/0.0510 = 0.0414/0.510 = 0.812. The inverse sine of this value gives us ρ = 54°.
Having one angle’s measure, of course, makes it far easier to find the others. Two paragraphs up, an equation in italics stated that sinλ = (Mrsinρ – Mc)/(- Ml). It follows that λ = sin-1[(Mrsinρ – Mc)/(- Ml)] = sin-1[(0.150kg)sinρ – 0.170kg)/(-0.100kg)] = 29°. These two angles sum to 83°, leaving 180° – 83° = 97° as the value of θ.
As can be seen above, these derived values are close to demonstrated experimental values. The first angle found, ρ, measures ~58°, which differs from the theoretical value of 54° by approximately 7%. The next, λ, measures ~31°, also differing from the theoretical value, 29°, by about 7%.The experimental value for θ is (180 – 58 – 31)° = 91°, which is off from the theoretical value of 97° by ~6%. All of these errors are smaller than the 9% error found for both λ and ρ in the easier, symmetrical version of this problem, and the causes of this error should be the same as before.
To begin this experiment, I first purchased two refrigerator-sized Fractiles-7 sets (available athttp://fractiles.com/), and then, early on a Sunday, quietly arranged these rhombus-shaped magnets on the refrigerator in our apartment (population: 4, which includes two math teachers and two teenagers), using a very simple pattern.
Here’s a close-up of the center. There are 32 each, of three types of rhombus., in this double-set, for a total of 96 rhombic magnets, all with the same edge length.
The number of possible arrangements of these rhombi is far greater than the population of Earth.
The next step of the experiment is simple. I wait, and see what happens.
It should be noted that there is a limit on how long I can wait before my inner mathematical drives compel me to play with these magnets more, myself — but I do not yet know the extent of that limit.
As a teacher, I have had variants of this conversation many times. The specific details, however, are fictional, for this changes, somewhat, each time it happens.
Student: Guess what? It’s my birthday!
Me: Congratulations! How old are you?
Student: I’m seventeen!
Me: Well, happy 18th birthday, then!
Me: Look, on that one day, 17 years ago, when you were born, that was your birthday. That day has a better claim on being your birthday than any other day, because it’s the day you were born. That was your first birthday. But you weren’t one year old yet. You turned one year old a year later, on your next birthday . . . your second birthday. A year later, on your third birthday, you turned two years old. Do I need to continue?
Student: So I’m 18? I can buy cigarettes without a fake ID, and vote, and stuff?
Me: No, not for another year, because you’re only 17 years old — but you have had 18 birthdays. Say, here come some of your friends. Use this bit yourself, if you want to, and have fun with it.
Student, to other students: Hey, guys, it’s my birthday! I’m 18 today!
…At least I try. Also, sometimes, the educational outcome is better than in this fictionalized example.
Soon, the Arkansas Democrat-Gazette will run my mother’s obituary. However, it would not be right for me to allow the obituary they print to be her only one.
Mom’s name when she was born, on January 4, 1942, was Mina Jo Austin. Later, she was known professionally as Mina Marsh. However, I chose to legally change my last name to her maiden name, in 1989, after my parents divorced. I did this so that I could have a last name I associated only with my good parent, for I only had one — the one now in this hospice room with me, as I write this, with little time remaining to her.
This is an old photograph of her, and her two younger sisters, taken when my mother was a teenager.
Her father, whom I knew (all too briefly) as “Daddy Buck,” taught her many things, very early in life, just as Mom did, much later, for me. He taught her about justice, and its opposite, using as one example of injustice the internment camps for Japanese-Americans which were then operating, here in Arkansas, when my mother was a little girl. Even in the wake of Pearl Harbor, and in complete disagreement with the masses, my grandfather thought it an obscenity that people had been herded into these camps simply because of their ethnicity, and, in a world where evil does exist, he decided his daughter needed to know about it. Only with knowledge of evil can one stand up to it, oppose it, and speak truth to it, even when that evil is mixed with power, as happens all too often. He instilled in her a strong sense of justice, and taught her courage, at the same time.
Mom started college at Harding University, in Searcy, Arkansas, and demonstrated her courage, and refusal to tolerate injustice, there, during the 1960 presidential election campaign. The assembled students of Harding were told, in chapel, that it was their duty, as Christians, to go forth on election day, and cast their votes for Richard Nixon, because allowing John F. Kennedy, a Catholic, to become president would be a horrible, sinful thing to do. She found this offensive, in much the same way that her father had found America’s treatment of Japanese-Americans offensive during World War II. On principle, therefore, she withdrew from Harding, and transferred to the University of Arkansas (in Fayetteville) to complete her college coursework. She also, later, left the denomination associated with Harding, eventually becoming a member of the Episcopal Church. I am grateful to her church here in Fayetteville, Arkansas, for the many comforts they have given her over the years. They even went so far as to raise the funds needed, in 2010, for her emergency transportation, by air, to a Mayo Clinic in Minnesota, where surgery was performed to save her from a rare adrenal-gland tumor called a pheochromocytoma. Without this help from them, her life would have been shortened by over five years.
Mom is survived by two children. I came along in 1968, and my sister (who had three children herself — my mother’s three grandchildren) was born the following year. Mom is also survived by three step-grandchildren, and two step-great-grandchildren. Mom began to teach both my sister and myself, as early as she could, what her father had taught her, early in life. Strangely enough, one of my earliest memories of her doing this also involved Richard Nixon, for the first news event I clearly remember seeing on television was Nixon’s 1974 resignation speech. At that young age, and with my parents clearly disgusted with America’s most disgraced president to date, I blurted forth, “I wish he was dead!” Mom wasn’t about to let that pass without comment, and did not. I remember the lesson she taught me quite well: there was nothing wrong with wishing for him to lose his position of power, as he was doing — but to wish for the man to die was to cross a line that should not be crossed. One was right; the other was wrong. It is my mother who taught me how to distinguish right from wrong. From this point forward, I now have a new reason to try, in every situation, to do the right thing: anything less would dishonor my mother’s memory.
It was around this time that my sister and I started school, and to say Mom was deeply involved in our experiences at school would be to understate the issue. In a conservative state where many schools openly (and illegally) do such insane things as teach young-earth Creationism in “science” classes, and anti-intellectualism is sometimes actually seen as a virtue, our entry into the school system was not unlike entering a battleground. At this time, education specifically designed for gifted and talented students simply did not exist in Arkansas. Mom had already had some teaching experience herself, although she had since moved on to other work. She was often appalled by the inane things that happened in our schools, when we were students, such as this from the fifth grade, and this (also from elementary school), and this especially-awful example from the seventh grade. Never one to tolerate injustice, Mom was deeply involved, from the beginning, in the formation of an organization called AGATE (Arkansans for Gifted and Talented Education), which fought a long, uphill, but ultimately successful battle to bring special programs for the education of gifted and talented students into the public schools of our state. She did this for her own two children, true — I consider forcing someone (who already understands it) to “practice” long division, year after year, to be a form of torture, and she was trying to save me from such torture — but she also did it for thousands of other Arkansas students, and tens of thousands have since benefited from her work in this area.
Mom was never content to fight in just one struggle at a time, for there is too much important work to do for such an approach. She was also a dedicated naturalist, a Master Gardener, and served as the Deputy Director of the Arkansas Natural Heritage Commission for 25 years, seeking ways to protect and preserve areas of natural beauty, and scientific significance, in our state. After retiring from that position, she later served on the board of directors of the Botanical Garden of the Ozarks, and also became the Development Director of the Ozark Natural Science Center.
My mother affected the lives of a great many people in her 73 years of life, including many who do not even know her name — but neither gaining credit, nor fame, was ever her goal. She will be deeply missed.
# # #
[About the rotating image: the picture of the banded agate, a reference to AGATE, the organization, on the faces of Mom’s dodecahedron, at the top of this post, came from here. The rotating dodecahedron itself, which the ancient Greeks associated with the heavens, was created using Stella 4d, software available at this website.]
In the Summer of 2014, with many other science teachers, I took a four-day-long A.P. Physics training session, which was definitely a valuable experience, for me, as a teacher. On the last day of this training, though, in the late afternoon, as the trainer and trainees were winding things up, some of us, including me, started getting a little silly. Physics teachers, of course, have their own version of silly behavior. Here’s what happened.
The trainer: “Let’s see how well you understand the different forces which can serve as centripetal forces, in different situations. When I twirl a ball, on a string, in a horizontal circle, what is the centripetal force?”
The class of trainees, in unison: “Tension!”
Trainer: “In the Bohr model of a hydrogen atom, the force keeping the electron traveling in a circle around the proton is the . . . ?”
Class: “Electromagnetic force!”
Trainer: “What force serves as the centripetal force keeping the Earth in orbit around the Sun?”
Me, loudly, before any of my classmates could answer: “God’s will!”
I was, remember, surrounded by physics teachers. It took the trainer several minutes to restore order, after that.