Education Personal: algebra homeschooling math
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Homeschooling My Daughter For My Own Benefit — Math
My kid is 7. She doesn’t need to know algebra.
I am 54. I don’t need to know it either. So why am I studying it?
It’s not that big a time commitment. I have no tests to pass.
But as I thought about my hope for daughter to grow up fully numerate and unintimidated by math-y stuff, I realized that I had to get over my own intimidation in the face of math, which (despite having a whole family full of supernumerate math-lovers) I hated in school.
So I decided to map my own ignorance, with the help of various library books and Khan Academy videos.
How much of the algebra I was exposed to in high school stuck to me? Damned little, apparently. How much of my own incompetence is due to ignorance and/or incompetence? How much of it is due to lingering emotional responses from math trauma in school?
Recently I’ve been playing with lines and slopes. I have absolutely no recollection of ever learning y-b = m (x – a) or anything that looks like it, so my engagement with the formula and its constituents doesn’t seem to have emotional content (unlike, say, quadratics, which are associated in my mind with a terrible homework fight I had with my father somewhere in 9th grade).
The first thing I notice is now many simple mistakes are available for me to make at every step of the way. It is going to take many many iterations before this process is internalized in my (so to speak) mental muscle memory. I am intellectually aware of what’s required to calculate the slope of a given line — but the actual physical process of writing the numbers down in the right positions vis-a-vis one another is fraught with complications.
As a music teacher I’ve got an advantage over some other folks: I never had any musical talent, so I had to build my musicianship from the molecular level up, making every mistake possible. It looks like the same process is happening with math.
Music teachers with “talent” are often ignorant of two key factors in developing mastery: number of repetitions and size of learning increment. It’s not enough to repeat something until your student does it right — once it’s been done right is the time to begin repetitions! And it’s not enough to increment the learning in steps suitable to your own learning style — it’s essential to figure out the increments your student requires, which may be much smaller than what you needed.
My algebra increments are very small. Fortunately, I’m patient. Yesterday I did three or four slope formulas, some several times. I made mistakes in calculating the initial slope; I transposed x and y in my head; I reversed + and – signs; I simply wrote down a 3 where I meant to write a 2. Each of these and more sent me in different wrong directions — since I didn’t figure out what I’d done until later. And that was just in the initial calculation. Once I began trying to plug these numbers into the y-b = m (x – a) formula, a whole new collection of mistakes emerged.
You know what? I’m interested in the mistakes. Getting it right is not the objective here; the “goal” is to figure out as many different ways of getting it wrong as I can.
The fact that my daughter sees me doing this at the breakfast table is a bonus for the homeschooling process. I’m doing it because I’d like to get over my own anxieties.
Education environment Personal Politics: Arctic atmospheric CO2 carbon emissions
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Year 3, Month 6, Day 10: You Thought Y2K Was Gonna Be Bad? Try CO24C.
The world’s air has reached what scientists call a troubling new milestone for carbon dioxide, the main global warming pollutant.
Monitoring stations across the Arctic this spring are measuring more than 400 parts per million of the heat-trapping gas in the atmosphere. The number isn’t quite a surprise, because it’s been rising at an accelerating pace. Years ago, it passed the 350 ppm mark that many scientists say is the highest safe level for carbon dioxide. It now stands globally at 395.
So far, only the Arctic has reached that 400 level, but the rest of the world will follow soon.
Upfucked ungood. Sorry, kids. Good luck with your lives; you’re gonna need it. Sent May 31:
As kids, we clustered around the driver’s seat when the odometer on our family car turned over; Dad would decelerate a bit and we’d call off fractions of a mile. All those zeros were tangible proof of how far we’d traveled. Sometimes we’d celebrate (ice-cream!).
Now we get to watch as another and considerably more ominous number scrolls by. When CO2 is measured at 400 parts per million in the atmosphere over the Arctic, though, it’s nothing to celebrate. Scientists agree that the survival of our civilization hinges on keeping concentrations of this greenhouse gas below 350 ppm, a landmark we crossed decades ago.
While we always came home at the end of a family drive, it now looks as though industrial humans may have driven too far. The Earth we grew up on is irreversibly behind us, thanks to the past century’s profligate consumption of fossil fuels. No cheering this time.
Warren Senders
Education environment Politics: analogies sustainability
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Year 3, Month 5, Day 13: Ain’t No Place A Man Can Hide
The Barnstable Patriot discusses the ways climate change is affecting Cape Cod, Massachusetts’ own vacation paradise:
Climate change is costing Cape Codders. It is eating at our shorelines, causing storm surges to overrun our beaches and houses. It is raising the price of our homeowner’s insurance. Our vulnerable sandy habitation, 10 miles wide, is part of a global system of weather that affects us locally, according to four experts who spoke at a climate change forum sponsored by the League of Women Voters at the Harwich Community Center April 28.
The takeaway message is that while belief in climate change is falling, the reality of it is increasing via accumulated science from real events, according to Dr. Eric Davidson, executive director of the Woods Hole Research Center, which looks at climate science from the Amazon to the Arctic. Davidson warned that hard facts prove the dangers of rising global warming. He said that since the world focused its attention on this issue at Rio de Janeiro in 1992, emissions have been lowered in some nations, but by and large, little has been accomplished.
Unless we mitigate, adapt and change now, Davidson said, there will be increased suffering from heat, violent weather extremes, famine, drought and flooding, all of which, data collected, measured and sifted over time show, will increase exponentially. He added that actuarial information from insurance companies supports the data.
Describing global warming as the “parked car effect,” Davidson said that heat from the sun comes through the window, but in re-radiating back out it becomes trapped, heating up the car. The earth’s atmosphere is the same, trapping rising methane, carbon dioxide and other gases from fossil fuel use in a big puffy blanket of molecules that prevent the heat from getting back through the “car window.” Since Scripps Institute of Oceanography in California, the National Oceanic and Atmospheric Administration and others have been keeping records, from 1960 to now carbon dioxide has increased from 320 parts per million to 380 parts per million. (The Arab oil embargo of 1973 diminished greenhouse gas emissions briefly by lowering usage.) Davidson says that La Nina and a sun spot cycle actually are cooling the planet somewhat now, but when the solar cycle changes and we enter El Nino, warming will accelerate. Best scenario, the Cape will have a mid-Atlantic-states climate in the future; worst, a climate like South Carolina’s.
This is a generic letter, but one that makes a useful point. I’m going to do a few more on this theme today (May 4) if I get the time.
We often hear that combating climate change will require a “new Manhattan project” or a “new Apollo program.” But both of these analogies are inexact. America’s development of the atomic bomb was kept under wraps until the devastation of Hiroshima and Nagasaki — but successful climate technologies must be transparent and accessible to all. While the race to the moon was no secret, there was little ordinary citizens could do beyond sending pennies to NASA — but preparation for global warming’s consequences has to happen in our daily lives, not just in the top echelons of government.
Mounting a robust and enduring response to the burgeoning greenhouse effect is not in itself a goal, like making an explosion or returning safely from the moon. Rather, it is an essential transformation in the way we collectively understand our responsibilities to the environment and to our posterity. If we are to survive and prosper in the coming centuries, our species and our civilization must change our focus to the long term. And, perhaps paradoxically, we’ve got no time to waste.
Warren Senders
Education environment: agriculture Gardening IPCC scientific consensus
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Year 3, Month 5, Day 12: Not Phrenology, Phenology
USA Today runs an article on phenology. Ominous:
As the climate warms, many plants are flowering 8.5 times sooner than experiments had predicted, raising questions for the world’s future food and water supply, a new international study concludes.
Higher carbon dioxide emissions from burning fossil fuels can affect how plants produce oxygen, and higher temperatures can alter their behavior. Shifts in natural events such as flowering or leafing, which biologists call “phenology,” are obvious responses to climate change. They can impact human water supply, pollination of crops, the onset of spring (and allergy season), the chances of wildfires and the overall health of ecosystems.
To better understand this, scientists from 22 institutions in Canada, Sweden, Switzerland, the United Kingdom and the United States studied 1,634 species of plants across four continents. They compared how plants responded based on historical monitoring data and on small-plot experiments in which warming was artificially induced.
Jeez. Nobody saw that coming, did they? Sent May 3:
It’s unsurprising that researchers studying the responses of plants to increased atmospheric CO2 found their predictions nearly an order of magnitude too low. The uncomfortable fact is that almost without exception, scientific forecasts have underestimated the magnitude, speed and significance of climate change and its effects. There are two important reasons for this disconnect.
The first is that scientific language is inherently conservative, striving for accuracy without emotion. A phrase like “statistically significant correlation” doesn’t immediately trigger anyone’s adrenalin — even when it’s linking greenhouse gas concentrations to a warming planet. The second is that scientific research is usually specialized, thereby minimizing the effects of interacting factors in a complex situation — and if any situation deserves the term “complex,” it’s global warming.
America and the world must mount a robust and meaningful response to the climate crisis, if we are to avoid a future full of unpleasant surprises.
Warren Senders
Education environment: assholes idiots media irresponsibility scientific consensus
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Year 3, Month 5, Day 11: How About Putting In Some Animatronic Reindeer?
Why don’t I watch TV? Oh, right. Now I remember:
Forecast the Facts, the activist group that first confronted GM about its support of climate change doubters the Heartland Institute, now plans to muster a public campaign targeting the Discovery Channel. The purpose: to get Discovery to acknowledge the scientific consensus on man-made climate change in its programming.
The flap follows the recent airing of the final episode of Discovery’s lush exploration of the polar regions, “Frozen Planet.” The last of the seven-hour series, “On Thin Ice,” was devoted specifically to presenting evidence of climate change including discussion of the challenges facing polar bears, collapsing ice shelves, diminishing habitat, and naturalist David Attenborough (Alec Baldwin is the narrator and host of the series) saying, “The days of the Arctic Ocean being covered by a continuous sheet of ice seem to be past. Whether or not that’s a good or bad thing, of course, depends on your point of view.”
Strangely missing from the narration, however, is any mention of the causes of climate change, even presented as theory. An April 20 story in the New York Times revealed that the producers made a deliberate choice not to present this material, anticipating criticism from the small minority of viewers who do not accept scientific opinion about human causes of global warming.
Series producer Vanessa Berlowitz told the New York Times that including the scientific theories “would have undermined the strength of an objective documentary, and would then have become utilized by people with political agendas.”
Whores. Sorry. That’s a libel on whores. Sent May 2:
It is unfortunate but unsurprising that the Discovery Channel has chosen to soft-pedal any mention of the human causes of climate change in their “Frozen Planet” series. In the decades since Ronald Reagan’s deregulation of media ownership, the influence of corporate ownership on news and opinion programming has increased, invariably to the detriment of the truth.
The notion that discussing the facts of anthropogenic global warming would allow the series to be “utilized by people with political agendas” is utterly disingenuous. By omitting the facts of climate science from the documentary programs, the producers had already allowed their work to be “utilized” by corporations — whose political agendas are firmly anchored in the profit motive.
The scientific agreement on climate change is extremely robust. To characterize thousands of dedicated researchers as “people with political agendas” is both journalistically and morally irresponsible. Let us hope the Discovery Channel finds its conscience.
Warren Senders
atheism Education Politics: sheesh
by Warren
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I’d Like Tautology Dressing On My Word Salad, Please.
In response to a facebook meme that said, “Study one religion and you’re hooked for life. Study two religions and you’re done in an hour” an old colleague who’s gone Xtian posted the following:
Religion that is focused on and filled with spirituality, like life itself, does not and need not make sense in the human mind as it seeks to address the deep desires and highest aspirations in people who perceive themselves to have a soul. The human institutions of religion are imperfect expressions of human superconsciousness, and to the extent that any repetitive mental effort seeking to understand realities beyond its intended purpose mostly serves to confound intelligibility, I agree that studying multiple imperfect expressions of superconscious truisms must by definition produce an irrational and pointless result.
Does this mean anything?
Some Nuts And Bolts Of Music Theory
When I started learning music seriously I was a teenager. I’ll turn 54 in a couple of weeks, and I’m still figuring out all this stuff, despite (or perhaps because of) being a professional musician and music teacher for three decades. Having a 7-year-old daughter is an enormous help.
In high school I took my first music theory class. The teacher’s name was Mr. March, which should have been a clue. The first day, he said to the class, “I’m going to test your musical ears.” He told us to take out a piece of paper. Then he said, “I’m going to play two intervals on the piano. You write down which is bigger, the first or the second.”
Then he turned his back to us and pressed some keys on the piano.
I did not have a freakin’ clue what was going on.
I did not recognize that he was hitting two keys simultaneously. What I heard was a series of sounds. What did he mean by “which one is bigger”? I’m pretty sure I just gave up on the exam.
Mr. March was operating under some default assumptions that were never stated. This is not uncommon in teaching, and it’s practically a given in music teaching, where teachers are distressingly likely to start where they are, rather than where their students are.
Here’s what I tell students who want to learn about music theory.
Musical sound concerns itself with vibration within the frequency range that our ears can perceive. Vibrations outside that range don’t get picked up by our ears, so we won’t talk about them.
Some vibrations have periodicity. Others do not. An example of the first kind is a tone played on a flute; an example of the second kind is crumpling a sheet of paper.
While musical performance uses both types of sounds, the study of harmonic relationships is only concerned with periodic sounds — the ones with identifiable frequencies, usually measured in cycles-per-second. Sounds with identifiable frequencies are called tones. If you take a series of rhythmic impulses and speed them up, they will turn into tones.
If you have two tones with the same frequency, they are in a very specific relationship. Their numbers match; they are in a 1-to-1 ratio. The musical term for this relationship is unison.
If you and I sing the exact same note, our vocal chords are vibrating at the exact same speed, and we are singing in unison. If we’re almost but not quite at the exact same speed, the frequency ratio between our voices changes from 1:1 to something more complicated. 189.235147 : 193.772121 is almost the same as 190:190 (which reduces to 1:1) but it’s a more complex relationship — and it’s perceived by our ears as “out of tune.” Obviously there are a lot more ways to be out of tune than to be in tune!
If you have two tones in the frequency ratio 2:1, their numbers no longer match, but their relationship is still simple. One vibration moves twice as fast as the other. The musical term for this relationship (in Western musical tradition) is octave.
Notice that the term “octave” means “eight,” which has absolutely nothing to do with the actual mathematics involved.
To our ears, the frequency of any power of 2 seems to have the same “quality” as any other. Notes an octave apart are given the same name in nearly every world musical system that goes so far as to name the notes in the first place. This means that experientially, 2:1, 4:1, 8:1, 16:1… are all identical 1:1.
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Musical intervals can be quantified in various ways.
Keyboard or melodic distance simply measures how far you have to move your finger to get from one member of an interval pair to the other. From the lowest A on the piano to the highest is a finger distance of about a meter and a half. From “middle C” to the C-sharp immediately above it is a finger distance of about a centimeter. By this measure, the first interval is significantly “bigger.”
Ratio size just addresses the distance between the two numbers, and it maps nicely onto the melodic distance measure. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 (n.b., if you know this already, you also know that on the piano, thanks to the baffling miracle of equal temperament, this statement is untrue. Bear with me for the purposes of discussion, ‘k?). 128 to 1 is a bigger jump than 16 to 15, so the first interval is significantly “bigger.”
Harmonic distance, on the other hand, measures the complexity of the ratio involved. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 — but 128:1 reduces to 1:1, and 16:15 doesn’t reduce. An eight-octave jump has a harmonic distance of zero, while a “semitone” has a much greater harmonic distance. So when we use this measuring system, the second interval is “bigger.”
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All harmonic intervals can be described as frequency ratios. Here are some of the ones we use most often:
3:2 is described in Western musical terms as a “fifth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol / 1-2-3-4-5), which has nothing to do with the actual mathematics involved.
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4:3 is described in Western musical terms as a “fourth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa / 1-2-3-4), which has nothing to do with the actual mathematics involved.
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5:4 is described in Western musical terms as a “Major Third.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi / 1-2-3), which has nothing to do with the actual mathematics involved.
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5:3 is described in Western musical terms as a “Major Sixth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol-La / 1-2-3-4-5-6), which has nothing to do with the actual mathematics involved.
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As my little videos demonstrate, rhythmic impulses turn into pitch when you accelerate them. If you record yourself tapping 2-against-3 for an hour, then accelerate the recording by multiple orders of magnitude, you’ll wind up with two tones a fifth apart.
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You don’t need to know about frequency ratios to use them effectively (just listen to the Beatles and you’ll hear some dynamite frequency ratios rendered with exquisite fidelity by people who never gave the math a moment’s thought). Most composers don’t know. Most musicians don’t know.
So why bother?
Speaking personally, I can say that learning all this has transformed my experience of music. I can spend a long time perfecting the tuning of a single interval — precisely because I have learned to perceive it as a source of deep experiential insight into simple mathematical relationships. Why bother? Because it’s cool; because it’s beautiful; because it’s universal.
Okay, that’s all for today.
Education environment: assholes denialists idiots media irresponsibility
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Year 3, Month 5, Day 8: I’d Like A Triple Oy With Vey-iz-Mir Sauce…
The denialist outlets are all a-twitter over James Lovelock’s recent remarks. The New York Daily News is a fine example:
Talk about an inconvenient truth: One of the scientists who most forcefully sounded the warning bells of climate change now says his predictions were a bit overheated.
Back in 2006, British environmentalist James Lovelock declared that “before this century is over, billions of us will die and the few breeding pairs of people that survive will be in the Arctic where the climate remains tolerable.”
Lovelock and fellow believers helped lead Al Gore to become the Earth’s most famous climate warrior.
But, in an interview with MSNBC, Lovelock admitted that his dire predictions were, excuse us, hot air.
“The problem is we don’t know what the climate is doing,” he said. “We thought we knew 20 years ago. That led to some alarmist books — mine included — because it looked clear-cut, but it hasn’t happened.”
Almost wistfully, he noted: “We were supposed to be halfway toward a frying world now.”
Assholes. Idiots. Sent April 29:
Yes, climate scientist James Lovelock, now 92, has drawn back a bit on his earlier apocalyptic forecasts. But it would be a very bad mistake to assume that climate change has now turned out to be a myth. That’s not what he said, that’s not what he meant, and that’s not a sensible response either to his words or to the climate crisis that is unfolding around us.
That Lovelock thinks we probably won’t face gigadeaths in the next few decades doesn’t mean climate change isn’t real. He, like other climatologists, is essentially a “planetary physician.” While it’s good news if your oncologist tells you that a tumor hasn’t spread as far or as fast as the worst-case predictions, that doesn’t mean you should start smoking again. Lovelock’s statement suggests simply that we have a tiny bit more time to change our ways before things get dangerously out of control.
Warren Senders
They published it, albeit in a highly edited form:
Medford, Mass.: Re “Hot air on climate change and the end of the world” (editorial, NYDailyNews.com, April 29): Yes, climate scientist James Lovelock has drawn back from his apocalyptic forecasts. But do not assume climate change is a myth. That Lovelock thinks we probably won’t face gigadeaths in the next few decades doesn’t mean climate change isn’t real. His statement simply suggests we have a bit more time to change our ways before things get out of control. Warren Senders
Education music Personal Warren's music: homeschooling
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Music at home…
…Daughter and I have exchanges about music theory. She calls them “wacky questions,” and enjoys it when I give her puzzles about harmonic relationships. “If A is ONE, then what is the TWO chord? The FIVE chord?” “Spell a G major triad.” Etc., etc.
Recently we began moving into questions about harmonic sequences. “In the key of C, what is a I-IV-VI-V-I progression?”
She’s seven. I don’t have any huge expectations about this; it’s just a fun game we play. This is way out of her league.
Or is it?
At tonight’s guitar practice I was coaching her into a D-minor chord (the standard one at the bottom of the neck). She started playing a sequence, not too adroitly…and when I tried to steer her in the direction of something I had planned, she said, “Stop! I want to play my own progression!”
Then she dictated: “D minor, A minor, C, A minor, D major, G, A major, D.”
I did a little on-the-spot voice-leading to make two harmony parts and we sang through them. Cool. My daughter’s composing her own chord patterns.
Then she told me to “write it down, so we don’t forget it.”
I think it’s time to show her more about notation.
atheism Education environment Politics: apocalypse armageddon denialists eschatology evangelicals religion theology
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Year 3, Month 5, Day 2: A 50-Watt Bulb?
The faithful are opening their eyes. Or are they? The News Virginian reports — you decide:
In “The Global Warning Reader: A Century of Writing about Climate Change,” Dr. Bill McKibben presents “The Evangelical Climate Change Initiative,” a 2006 document signed by 86 American Christian evangelical leaders. Signers include: Rick Warren (“The Purpose Driven Life”); W. Todd Bassett, National Commander of the Salvation Army; Ron Sider, President of Evangelicals for Social Action; and advisors and columnists for Christianity Today magazine. “In the name of Jesus Christ, our Lord,” they said, “we urge all who read this declaration to join us in this effort” of teaching and acting on the following four claims.
1. “Human-Induced Climate Change is Real.” Among the evidence the signers studied was that collected by the Inter-governmental Panel on Climate Change (IPCC) whose 1988-2002 chairman, John Houghton, is a committed Christian. They remembered that the science was settled enough for the Bush Administration to state in a 2004 report, and then at the 2005 G-8 summit, that humans were responsible for “at least some of it (climate change).” The IPCC, however, holds that human activities are responsible for “most of the warming,” according to the evangelical leaders.
2. “The Consequences of Climate Change Will Be Significant, and Will Hit the Poor the Hardest.” The signers emphasized the impact of even the smallest increases in human-caused world-wide temperature upon people in poor countries: tropical diseases, hurricanes, flooding, reduction in food crops, famine, and the vulnerability of refugees to exploitation and violence, even internal and external military oppression. “Millions of people,” they wrote, “could die in this century because of climate change.” They also noted the destruction it could bring to “God’s other creatures.”
I’m not going to take this one on faith. Sent April 23:
The rejection of climate change has long been a shibboleth of political conservatives, who have a record of denying inconvenient facts and expertise that goes back at least fifty years. Why, then, are evangelicals — one of the most consistently conservative voting blocs in the country — beginning to accept the scientific reality of global warming? While some may be encouraged, I am less sanguine about the motivations behind the faithful’s abandonment of long-held denialist positions.
Environmentalists are interested in the long-term survival of the planet; talk to a “tree-hugger” and you’ll hear someone whose worries about humanity’s future in the year 3000 motivate them to conservation and the wise use of resources. By contrast, evangelicals eagerly anticipating the End Times may have little reason to practice sustainability. Is climate-change acceptance among conservative Christians accompanied by a growing conviction that industrialized humanity needs to change its ways to avoid catastrophe? Or are they cheering on the burgeoning greenhouse effect, assuming that the souls of the faithful will be providentially rescued from a disaster of Biblical proportions?
Warren Senders