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The Simplest Equation in the World
May 1, 2008
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In an earlier post, I described a bi-directional cloud in terms of one rendering of the Julia Set without really describing what this "Julia Set" was. Let's try here:
The Mandelbrot Set Here, we discussed briefly the Mandelbrot Set, the phenomenal mathematical discovery by Benoit Mandelbrot. In this set, the mapping of data on the complex plane, as opposed to the Cartesian plane, led to many beautiful images, and many non-intuitive findings
The fundamental formula described there was the following:
You may notice a slight discrepancy between this formula and the formula we started the entry with: there is no "c". What happens if we do introduce it? Marvelous shapes like this appear!
A question, then, is: what is the relationship between the Mandelbrot Set and these things called Julia Sets? As I said, our earlier mapping of the Mandelbrot Set looked at the iteration Z←Z2. What happens if we introduce c, and allow it to span the range? Every c generates it's own "Julia Set".
And an enlarged mapping of many Julia Sets looks as follows:
The Value of "Broadness" in the Curriculum "More depth and less breadth" in the curriculum. We hear it all the time. It seems to make sense. However, might the course of fractal history be different if this motto was in place?
Where did the idea come from to plot all these points? To consider iteration of functions about the complex plane? Mandelbrot's uncle had told the young Benoit Mandelbrot of a 1918 paper by French mathematician Gaston Julia titled "Mémoire sur l'itération des fonctions rationnelles". Mandelbrot's uncle claim the paper was a masterpiece and was something to pursue. This was 1945. Why didn't Mandelbrot pursue it? Likely the reason was the paper's theory could not be investigated practically - it required literally billions of calculations to plot a single graph! However, with the introduction of the computer, the investigation of such a theory was now possible!
Reference to the scope of the curriculum may be misplaced here; instead, the value of exposure to a great many things may be the lesson to be learned.
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What is the Meaning of “Sampling”?
May 2, 2008
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The results of sampling and surveys has always bothered me. The President has an “approval rating” one week of 50%, and I am told this has a “margin of error” of +/- 3%. The next week the rating is 49%, and I’m told this approval rating fell! This seems odd to me, because if the first week’s numbers were correct, the “real” rating was somewhere between 47% and 53%, and because the second week was within that range, part of me says the approval rating was statistically unchanged! Another part of me says the number must somehow reflect legitimate change, as things do change over time. How can I make sense of these numbers? I must not understand the nature of sampling, of margin-of-error, etc. What I’d like to do, then, is build my own model of understanding regarding statistical sampling to see if I can better understand these issues. Of course, it does me no good to use someone else’s data for this purpose; I’d like to start with data I’m absolutely certain of. To do this, I’ll create my own data.
My model Rather than start with the USA population, or some huge block of data, I’m just going to start with 100 fictional people, and number them 1-100. To make things as simple as possible, I’m going to assume the first 50 people are boys, and the second 50 are girls. This is about as simple a system as I can conceive. I’ve removed all variability, ambiguity in answers, problems with the questions, etc.
The Goal of My Model The goal of a good sample is to represent some underlying population. Why don’t I just survey the underlying population? Here I’ve only got 100 people, so here it would be easy. What if I wanted to survey the USA population, now approaching 300 million people? Is that feasible? Could it be done? What are the costs in time and money to do this? Or can sampling provide a “reasonable estimate” of the underlying population at a fraction of the resources needed? The Nature of the Problem – the Direction of the Solution
Of course, this does not address my initial concerns! What do the reported numbers mean? Let’s use our “Boy/Girl” model to see.
I’ll start by choosing 50 people at random, and see how many are girls. I’ve got all 100 people in a hat, pull out 50, and there are 27 girls, which equals 54%. Now, I know the answer is 50% because I created the model! But of course I selected the people from a hat at random, so there is some randomness, chance, and variability here. But how much? Let’s do it again and see:
The second time: 25 girls = 50%. Perfect: exactly as I expected. Another … The third time: 18 girls = 36%. Wow! That’s really low. Another … The fourth time: 28 girls = 56% The fifth time: 27 girls = 54%
This process suggests there is certain amount of variation. It’s easy enough to repeat the process: line the 100 people up in the spreadsheet, mix them up, pick 50, and see how many of them are girls. Let’s see:
This is interesting: though I sampled half of my small population, I still got as low as 36% and as high as 60%, though I knew the answer was 50%. It’s true most of the time I got between 40% and 60%, but even these numbers are 10% from the known results.
What is the relationship between this simulation of girls versus boys, and the sampling note starting this article? Let’s look at the second simulation, resulting in 25 girls (or 50%). The goal of a sample is to represent the underlying population, and here it did perfectly. The third simulation, however, came back with an estimate of 36% of the students are girls – well off the actual mark of 50%. What am I to make of this number? Of what use is it? Is it right to say the estimate “went down – from 50% to 36%”, when I know the actual data did not change?
What happens, I wonder, if instead of sampling 50 people, I sample different numbers?
Extending the Model That was people – boys versus girls. What if I extend this process to another familiar example: coins? I know the probability of heads and tails is 50% - given a fair coin. OK: what’s happens if I flip a coin 100 times? Let’s see:
Extending the Model Further So, I’ve got a good idea about random variation. How can I quantify this? The above graphics help me see variation, but where are the probability distribution graphs I’m used to seeing?
Now I’ve got a great idea on what is going on in this “simulation” game with coins in a spreadsheet, rather than 100 coins, I’ll use different amounts. Suppose I start off sampling 500 coins. That seems like a reasonable amount to make sure I get “right at” 50%. Doesn’t it? To make sure I get an accurate understanding of what’s going on, I’ll repeat this process 1,000 times.
This is fairly close to what I expected – the results close to 50% - but even with 500 coins, the results still show a lot of trials at 45% and 55%. By adding a bar chart and grouping the results, I can quantify this data. Amazing! Even with 500 coins, the results were within +/- 2.5% only 72% of the time! What does the variation look like when the results are superimposed over one another?
The Sweet Sight of Predictability Actually, in process management, two elements are relevant in this diagram: are we doing the right things (or wrong things) and are they done consistently (or inconsistently).
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"To Infinity - and Beyond!" May 3, 2008
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Recent (in)action by the Kansas Legislature and Governor regarding the proposed Sunflower Electric Plant demonstrates the tragic example of ineffective thinking when faced with an important dilemma ... On the one hand: promoters cite the demand for electricity, a growing population, and current technology as reasons for building the plant; On the other hand: detractors cite the necessity for clean air and global warming concerns as reasons for promoting alternative energy sources. That is:
Of course, we know - everybody knows - huge fields of giant wind turbines is no solution, so the failure to promote the plant - whether it's nuclear, coal, or hydro - is short-sighted. Recall, we said in order to have a good Kansas now and in the future, we'd like a clean environment plus our electrical needs met. With this strategy, ours will not be met, and consequently, we won't have a good Kansas in the future!
One can sympathize with the governor and the legislature; they're stuck between a rock and a hard place with the electorate who are the real hypocrites here - demanding both right now, and complaining when they can't "have their cake and it eat, too!"
Revisiting the Assumption Regarding "Current Technology" But let's check one of our assumptions above, because it's a crucial one: Sunflower wants to build a coal-burning plant ... why? Because this is the current technology. It's proven. The fuel source is massive. 80 trains a day rumble through my town, delivering coal from the northeast down to Texas and Louisiana as evidence!
In all of these cases, we somehow are trying to harness the immense power of nature to meet the growing needs of man. Are we really exploiting the natural resources well to create electricity? One example where we seem to be doing well is water. A dam, water, gravity, and pressure, and we have electricity! The contribution of electricity to the grid from such dams is huge, and is growing worldwide in popularity. And why not? There's the water. Let's bottle it up, maintain a river flow, and electrically-feed our people! What a great solution! It is a great solution!
And a popular one - look at projects recently
completed, underway, or in the planning stages ...
The Popularity of Water The popularity is clear, as are the benefits. What are the drawbacks? One, for sure, is you need a big river, and there are only so many "big rivers" in the world. Additionally, the resources are immense to create a viable hydro-electric dam, as evidenced by the distance between start and complete dates above. Further, there is a great deal of displacement of the surrounding environment from the water filling the dam basin.
The Popularity of Water Read these two sentences carefully:
1. If we dam rivers, then we can harness the power of water. 2. In order to harness the power of water, we need to dam rivers.
Do they have the same ring? Are they saying the same thing? Are them implying the same thing? Keep those questions in mind, in addition to the following image of the earth:
The first question above makes direct sense; of course, if you dam water, it doesn't mean you're going to use water flow to create electricity. It could be you just want a good fishing hole.
The latter sentence, however, conjures up another image. In order to harness the power of water, then we need to dam rivers? Really? Why? Many other possibilities come to mind.
I put a nozzle on the end of my garden hose, and the spout now emits a skin-piercing stream of water. Alter the technology, and the same water "changes".
What if I alter the water we're considering? What if the focus was, not on rivers, but oceans?
The Power of Water What would it take to harness the force of these waves? After all, the earth's surface is comprised of 70% water! What would it take to harness this power?
Let's save that for smarter folks than I - for the time being, let's assume, if smart people focused on the problem, this source of water can be transformed into electricity.
Instead, I want to focus on Kansas, as that's where I live, and that's where the Sunflower Electric Plant / Wind Turbine dilemma started. Is water-based electricity powered by waves viable here? That seems a ridiculous question, but let's investigate it, nonetheless.
What would it take?
Surely, we'd need a massive "field" of water to make this possible. How big? Who knows, but it would have to be massive. However, Kansas is massive, so geography is no constraint. What would it take to fill it with water - and keep the water? Well, there we may run into a problem. Where would we get enough water to make this doable?
That's a problem.
But is it? Let's think some more. We need a big body of water - of massive proportions - to make this project doable. But why, earlier, have I assumed the water has to be above ground? Water wells for irrigation are abundant, so there must be a lot of water "down there" somewhere. Is there?
This hydro-behemoth, the Ogallalah Aquifer, spans 450,000 square kilometers, and averages in depth 60 meters, though this varies from 30 to 300 meters, as I understand it. Is an underground body of water like this subjected to enough gravitational forces to create waves for the generation of electricity? Are waves even necessary? After all, in my garden-hose example, it wasn't waves or gravity, but rather pressure that changed the water-flow output. Whatever the case, it seems one of the necessary conditions for project success exist! I wonder what else is needed?
A Direction Towards a Solution The April 18-24 issue of "The International Jerusalem Post" contained one of most fascinating (and exciting) articles I have ever read. The article MAKING WAVES describes the technology invented by a gentleman, Shmuel Ovadia, living in Tel Aviv and Managing Director of SDE-Energy. "They say that just 1 percent of the energy in the oceans could power the entire world," Ovadia says, with a raise of the eyebrows and a nod of the head, as if to stave off any "no way" reaction. It is, he assures, a viable goal. Wow.
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Rabbit Seasoning, Eclipses, and Thales
"The Importance of Perspective"
May 4, 2008
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What can these two images possibly have in common? The former is an image of a solar eclipse, the latter the great Chuck Jones cartoon "Rabbit Seasoning" ...
Relevant Dialogue from "Rabbit Seasoning" (about 2:20 in) Bugs: It's true, Doc; I'm a rabbit alright. Would you like to shoot me now or wait 'til you get home? Daffy: Shoot him now! Shoot him now! Bugs: You keep outta this! He doesn't have to shoot you now! Daffy: He does so have to shoot me now! [to Elmer] I demand that you shoot me now! [Elmer raises his gun. As Daffy sticks his tongue out at Bugs, he is shot. Daffy walks back over to Bugs, gunsmoke pouring out of his nostrils] Daffy: [to Bugs] Let's run through that again. Bugs: Okay. Bugs: [deadpan] Would you like to shoot me now or wait till you get home. Daffy: [similarly] Shoot him now; shoot him now. Bugs: [as before] You keep outta this, he doesn't have to shoot you now. Daffy: [re-animated] Hah! That’s it! Hold it right there! [to audience] Pronoun trouble. [to Bugs] It's not "he doesn't have to shoot you now", it's "he doesn't have to shoot me now" [Pause] Daffy: [angrily] Well, I say he does have to shoot me now!! [to Elmer] So shoot me now! [Elmer shoots Daffy again]
Break to "Solar Eclipses"
Here, we talked of the lunar eclipse, which already raises a question: why the need for the adjective "lunar". Let's pass on that for now, and deal with the image above. I've really never understood eclipses from images like this, and today I focused on the question "why". Sure, the image is not to scale. The sun is much, much bigger than both the earth and the sun, but to have a useful image on a screen, there has be some "not to scale" happening, I guess. No, that's not the main source of my ignorance. This ignorance stems from the flow of sunlight to the earth, because I know the diagram above is not right. Instead, sunlight flows as follows:
And since light strikes a broad part of the earth, there is no 'eclipse'. In fact, the illumination of half the earth in the diagram confirms my statement! However, I also know, though that sentence is valid, the whole world does not suffer from mass-delusion - there is an eclipse - at least that's what everybody says - and I can (sometimes) see them myself! What's going on here?
"Solar Pronoun Trouble": A Change in Perspective What's wrong with the above image, giving rise to the contradictory beliefs there's simultaneously an eclipse - and not an eclipse? I'm viewing the image from the perspective of the sun! What's missing from the image is ... me! The image above depicts the vision of the person on earth. That obvious - now - statement has enormous consequences for me understanding what exactly is going on with an eclipse, because now I'm not concerned with the flow of sunlight, but rather what it is I'm seeing! Immediately, I understand everything - merely by shifting my perspective from the sun to - me!
Extending the Issue With a firm grasp of the situation, the next question coming naturally to me is: how can this small object block the sunlight of that huge object so far away? Because one object (the moon) is so close to me and the other (the sun) so far there is an eclipse - that is, the moon blocks all sunlight from my vision.
A quick experiment confirms this: stare at a globe and put a coin between your eye and the globe. There is a distance where sight of the globe is obscured entirely.
This, of course, leads to the next natural question: this all depends on a lot of things: the size of the moon, the size of the sun, the distance the moon is from me, and the distance the sun is from me. Can I find this relationship?
Introducing Thales of Miletus
What did Thales of Miletus do? He recognized something every grade-school kid now using a computer recognizes intuitively. When they resize a shape, they know the relationship between all the parts is the same, and only the dimensions have changed. For example, these shapes are all similar:
as are these:
What did Thales do with such knowledge? He said the obvious (likely after much trial and error and experimentation confirming the hypothesis:
given two similar triangles (triangles 1 and 2): the height of triangle 1 relative to the length of triangle 1 is the same as the height of triangle 2 relative to the length of triangle 2.
That is:
alternatively ...
given two similar triangles (triangles 1 and 2): the height of triangle 1 relative to the height of triangle 2 is the same as the length of triangle 1 relative to the length of triangle 2.
That is:
The Key to the Idea of Proportions The key, of course, with this formula is, knowing three things, you can find the fourth. It's said Thales measured the height of the Great Pyramid at Cheops with this simple idea.
The Wonderful World of Mathematics by Lancelot Hogben
So too, we can use this simple idea to measure the size of the sun. Doing a bit of internet research, I have the following data readily available:
And using the idea we have similar triangles above, we can readily find the width of the sun:
Closing Thoughts (for now) The width, then, of the sun is 841,004 miles. That's pretty precise, particularly when we said the distance from the earth to the sun was 93,000,000 miles - rounding to the millions! Should this formula be modified to reflect this huge approximation - and, if so, how? We said the moon is 238,857 miles from the earth. As the earth navigates the sun elliptically, so too the moon's orbit is elliptical about us; consequently, this figure is the mean distance. Does this have a bearing on our calculation?
Further, here I was looking for the diameter of the sun, and I found the other three figures on the internet. Could I have found any of them myself? Are there experiments I could do, similar to the simple "eclipse" experiment of the coin and globe above to assist?
Finally, a thought on the theory of proportions, similar triangles, and Thales. From his theory came a couple of equations:
Does listing such equations clarify the idea of proportionality - or obscure it? That is, if you know the idea - via the figure below - you have no use for the formulas. You simply know them. And if you don't know the idea, will you ever memorize the formulas correctly? History and evidence tells us the answer by way of test scores: no.
Just When I Thought I Had Things Figured Out ... The investigation above led me better understand eclipses and proportionality, though the two can be understood independent of one another.
A nagging thought creeps into my mind.
If, what I've said above is true, then the earth's illumination is never altered. Yet I know it is! Might it be the case light emanates from the sun not as I described above, but instead as rays, always perpendicular to the surface of the sun?
More to follow on this one, because the implications are great - not just for correcting my theory above, but for the calculation of the circumference of the earth, and the work of Eratosthenes.
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Celebrating Cinco de Mayo properly
May 5, 2008
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The debate for Spanish as a foreign language requirement brought forth a parent of a kindergartner enrolled in the district DVD-based program. I doubted the claim a foreign language was needed to compete in a global-economy (far better to focus on understanding the foreign countries!). I doubted the claim a foreign language could be taught to a classroom of kids sitting in front of a TV watching a DVD with no adult in the classroom who could speak Spanish. But here came a parent to speak to the School Board. Her child has been in the program, and she was there to speak in favor of it. Paraphrasing: "and we just don't learn the language, we learn the culture. We learn about Cinco de Mayo - Mexican Independence Day." The School Board should have immediately voted "no" on this program, with that direct testimony. Cinco de Mayo is not Mexican Independence Day. But perhaps it should be. And perhaps we should celebrate it, also, as part of our independence!
IN THE MOMENT: Zen and the Art of Logical Haiku ... An excerpt from:
EXTENDED HAIKU From Syllogism to Poetry Interdisciplinary Education and the Japanese Haiku
and an Introduction to Multiple Modes of Expression
CINCO de MAYO: revisited from a “Narrative” Perspective Cinco de Mayo is celebrated on May 5th. What does it represent? For the longest time, I thought Cinco de Mayo represented Mexican independence from Spain in the early part of the 19th century. Now I know it represents a victory over invading France in the latter part of the 19th century! What was France doing invading Mexico? France was owed money by Mexico, and Mexico had temporarily suspended payments to France – and England and Spain. Why were these payments suspended? A 1861 Mexican Civil War created a national debt, which was addressed by temporarily suspending payments to the three countries. Agreements were made with England and Spain, but France rejected the proposal, and instead planned to attack Mexico City, the Mexican capital. The Mexicans, under the command of General Zaragosa with a brilliant cavalry attack, defeated the attacking French army on May 5, 1862.
CINCO de MAYO: revisited from an “Illustration” Perspective
CINCO de MAYO: revisited From a Logical and Haiku Analysis
CINCO de MAYO: revisited from an “Extended Haiku” Perspective
Indebtedness reigns. Ideas have consequences. Lone dissenting voice.
Election baggage. Transatlantic incursion. A Blow to the Heart.
Financial Tussle. Paradigm Shift: Cavalry Thrill of Victory!
United States Independence and Cinco de Mayo? What is the relationship between these two? Is there any relationship? A future issue - or I'll return here to add in the details. In the meantime, you know what to do: HIT THE BOOKS!
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2-Dimensional Cellular Automata Spreadsheet Snowflakes
May 6, 2008
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Reconciling Caution and Abandon - in Literature and Life
May 7, 2008
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THE TWO BROTHERS by Leo Tolstoy Two Brothers set out on a journey together. At noon they lay down in a forest to rest. When they woke up they saw a stone lying next to them. There was something written on the stone, and they tried to make out what it was. "Whoever finds this stone," they read, "let him go straight into the forest at sunrise. In the forest a river will appear; let him swim across the river to the other side. There he will find a she-bear and her cubs. Let him take the cubs from her and run up the mountain with them, without once looking back. On the top of the mountain he will see a house, and in that house he will find happiness." When they had read what was written on the stone, the younger brother said: "Let us go together. We can swim across the river, carry off the bear cubs, take them to the house on the mountain, and together find happiness." "I am not going into the forest after bear cubs," said the elder brother, "and I advise you not to go. In the first place, no one can know whether what is written on this stone is the truth--perhaps it was written in jest. It is even possible that we have not read it correctly. In the second place, even if what is written here is the truth--suppose we go into the forest and night comes, and we cannot find the river. We shall be lost. And if we do find the river, how are we going to swim across it? It may be broad and swift. In the third place, even if we swim across the river, do you think it is an easy thing to take her cubs away from a she-bear? She will seize us, and, instead of finding happiness, we shall perish, and all for nothing. In the fourth place, even if we succeeded in carrying off the bear cubs, we could not run up a mountain without stopping to rest. And, most important of all, the stone does not tell us what kind of happiness we should find in that house. It may be that the happiness awaiting us there is not at all the sort of happiness we would want." "In my opinion," said the younger brother, "you are wrong. What is written on the stone could not have been put there without reason. And it is all perfectly clear. In the first place, no harm will come to us if we try. In the second place, if we do not go, someone else will read the inscription on the stone and find happiness, and we shall have lost it all. In the third place, if you do not make an effort and try hard, nothing in the world will succeed. In the fourth place, I should not want it thought that I was afraid of anything." "The elder brother answered him by saying: "The proverb says: 'In seeking great happiness small pleasures may be lost.' And also: 'A bird in the hand is worth two in the bush.'" The younger brother replied: "I have heard: 'He who is afraid of the leaves must not go into the forest.' And also: 'Beneath a stone no water flows.'" The younger brother set off, and the elder remained behind. No sooner had the younger brother gone into the forest that he found the river, swam across it, and there on the other side was the she-bear, fast asleep. He took her cubs, and ran up the mountain without looking back. When he reached the top of the mountain the people came out to meet him with a carriage to take him into the city, where they made him their king. He ruled for five years. In the sixth year, another king, who was stronger than he, waged war against him. The city was conquered, and he was driven out. Again the younger brother became a wanderer, and he arrived one day at the house of the elder brother. The elder brother was living in a village and had grown neither rich nor poor. The two brothers rejoiced at seeing each other, and at once began telling of all that had happened to them. “You see," said the elder brother, "I was right. Here I have lived quietly and well, while you, though you may have been a king, have seen a great deal of trouble." "I do not regret having gone into the forest and up the mountain," replied the younger brother. "I may have nothing now, but I shall always have something to remember, while you have no memories at all."
The "Dilemma Cloud"
The injection here was a good one, based on the simple assumption the two brothers must always act together. Why? They decided they didn't, and went their own ways. Good for them. The main difficulty I had was with the story's conclusion. Why must each claim they were right? Instead: “You see," said the elder brother, "Here I have lived quietly and well, as I wanted. And you?" "I found the life of excitement, as I wanted." The two brothers sat for lunch and enjoyed each other's company.
A Similar "Dilemma Cloud" You may recall a similar cloud regarding the Donner Party and their movements westward.
What is it about the apparent similarities of this cloud that seem to resonate elsewhere? Is this true? Can this spectrum of application help us solve one - or all - of the "common clouds"? How might solutions differ? How does the consequences of success / failure feed into the decision process?
A Short-Sighted Solution A tempting generalization might be, recognizing in both the Tolstoy short-story and the Donner Party tragedy, group dynamics were in play, a solution to dilemmas like this - moving forward - is to do things oneself. After all:
The falseness of this inference can be seen by fast-forwarding the two scenarios above with only yourself in the picture: you're in the woods and come upon a note: what do you do. You yourself travel west and come "to a fork in the road". What do you do? Choice. And the fundamental choice is whether you will think - or not: To think is an act of choice. The key to what you so recklessly call “human nature,” the open secret you live with, yet dread to name, is the fact that man is a being of volitional consciousness. Reason does not work automatically; thinking is not a mechanical process; the connections of logic are not made by instinct. The function of your stomach, lungs or heart is automatic; the function of your mind is not. In any hour and issue of your life, you are free to think or to evade that effort. But you are not free to escape from your nature, from the fact that reason is your means of survival—so that for you, who are a human being, the question “to be or not to be” is the question “to think or not to think.” “A being of volitional consciousness has no automatic course of behavior. He needs a code of values to guide his actions. Ayn Rand Atlas Shrugged
Triangulating Towards Success Choice indeed, but how to choose? Are we, as Sartre said, "Condemned to be free"? Hardly. We are reasoning animals! But how do we use that reason to solve a problem distant from ourselves? How do we make that problem "personal"?
I guess I could pretend I was one of the brothers, have another member in the class be the other member, and work the problem from that perspective. It's artificial, though. At least to me. Surely there's a way to leverage examples from other domains to help me work the Tolstoy problem.
Let's see.
A plan of attack: in The Two Brothers dilemma above, the choice is "Go or Stay". But it's more than that, because I can face a dilemma of "Go to the movies / Stay home". No, the dilemma above is "Go or Stay", with possible dire consequences. If I'm going to use a real-life example, this critical element must exist. Above, I've used the Donner Party as an example, and that's a good one, because there too the consequences of "Stay on the Known Path / Take a Shortcut" did have dire consequences.
Does this help? Yes. But it's still not personal.
Can I add a story of my own - one with dire consequences? I don't like my job, yet it provides a paycheck. Should I quit or continue?
Which one should I start with to solve? Does it matter? If all three share a common thread, likely the solution is similar in nature!
And the solution, as Sly and the Family Stone said, Takes Us Higher! The solutions above collapse / fold-up to a tetrahedral pyramid of success / understanding!
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Architects of Their Own Future
Chapter 15
May 8, 2008
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Chapter 15 THE MATH ENIGMA
“We need a new type of high school!”, exclaimed Principal Ragnar, exhausted. “I am so tired of hearing that, from Oprah to Gates to the President. Is there anyone here …”, he paused to look around the room, and then continued, “who does not believe that? What a tired cliché! But how do we get there? We’re so caught up in fighting fires we don’t have the time to “reinvent ourselves for the 21st century global economy!” He chuckled at the repetition of rampant clichés dominating the political landscape.
Washington High had finished the prior school year with a “proficient or better” percent of only 36% in math, and NCLB was creeping up on them.
“These kids just don’t have the basic arithmetic background to handle algebra and trigonometry,” bemoaned one teacher. Said another, “Mine just don’t do the homework, and their parents don’t push them.”
Others in the group basked in the glory of the recent advances made in the ACT reading section by a sampling of students. The initial target for this meeting had been to improve ACT math scores, but the momentum had sagged when it appeared there was differences between “math problems” and “textual reading”.
“Let’s look at this differently, can’t we?” said Principal Ragnar, calmly. “We’ve got these kids for a good portion of the day. Let’s work with what we can control.”
He took out a series of completed sample tests from the previous testing session. “I do not understand this,” he said, “pointing at simple problems on the exam. “How can a student answer like this," he said, writing two problems on the board.
"How are we going to achieve any progress if our kids can’t even get these right?”
Mrs. Thompson chimed in. “The frustrating thing is we’ve got a lot of problem solving techniques, so the kids should not be getting them wrong.”
Principal Ragnar thought to himself. Why do you need problem solving techniques for simple problems like these? Techniques, he thought. Isn’t that what lay at the foundation of the reading situation they faced earlier? Kids were taught multitudes of test-taking strategies to pass the test – many of the strategies themselves conflicting with one another. Is there a way to exploit this common characteristic? What if the abundance of techniques is creating more of a problem than leading to a solution? He thought out loud to the group of assembled teachers.
“Can we agree on something: if we’re going to improve this score now, we cannot afford to miss problems like these. This should be low-hanging fruit – but like all hanging fruit, it still must be picked. How do we pick this fruit?”
He pressed on.
“Mrs. Thompson: let me ask you. How do you teach “multiplication of radicals”?
Mrs. Thompson did not like the spotlight on her. Like many high school teachers, she did not have a degree in math, yet she considered herself a good teacher. She was a good teacher! However, like many teachers, she knew how to perform a calculation, but did not know why. She replied: “We actually just apply the rule, ‘when multiplying radicals, multiply the radicand.’”
This seemed to make sense to Principal Ragnar. It also worked with fractions. It seemed anytime you’re supposed to multiply two things, you multiply. How do kids get this wrong? There’s nothing to memorize! He repeated this out loud to the group.
“Not so fast,” interjected the AP Calculus teacher, Mr. Wihelm. When multiplying numbers with the same base, you don’t multiply the exponents, you add them.”
“What about telling the kids how all of this applies to the real world?”, said the Principal, fighting to maintain what he thought was some momentum.
“That would be great,” said Mrs. Thompson, except for one thing.” She continued, quietly. “I honestly do not know how a lot of this does apply to the real world! I’m great at doing the problems, but I don’t know how it applies. And most of the problems in our books are either problem-oriented, like I just said, or they’re real-world problems too technical for me to understand. I have no idea what the ideal gas law pv = nRT means.”
“It’s worse than that with some world problems,” Mr. Wilhelm jumped in. “A lot of the ‘real-world’ problems are so artificial they in fact do not apply. Why do we continue to see problems like ‘if a ship sailed across the river at…’ and ‘if Billy is 4-years older than his sister’?”
“Let’s back up to where we started,” an exasperated Principal continued. We’ve got these kids for a certain time each day, and we need to work with what we’ve got right now. It’s easy to see how these ‘experts’ so quickly jump on the ‘make everything relevant’ band-wagon. But does it help us right now?”
Principal Ragnar continued the discussion. “The thing that gripes me is the assertion change must take place over a series of years. Why does it take years to see the effects of a good idea? Do we have to wait for ‘the experts’ to make things relevant, to reinvent the curriculum, or make other radical changes? Can’t we do something right now?”
“Such as”, said Mr. Wilhelm – skeptically?
“Let’s start with these simple problems kids are getting wrong. Let’s not start with the hardest materials – let’s start with the easiest. Can’t we find a way – just us – of improving scores on this material?”
“What do you suggest?” said Mrs. Thompson?
“Let’s do this … for our next meeting, everybody bring in 5 typical problems – easy problems (to us) kids are missing. Don’t bring in elaborate lesson plans, but just a general idea of how you teach these materials now. Let’s concentrate on this one issue.
Since Sputnik, the USA has been on a mission to perform well in math, science, and engineering. Sputnik launched “the new math”, widely considered a huge failure. But didn’t this recognition lead to changes?
Twenty-years ago, we were called “A Nation at Risk”. Now, the motto is “No Child Left Behind”. Have any of these programs done anything to help the students? Why has math lagged behind all other subjects?
Or has it? As soon as this subject is brought up, experts jump into disputes about the data and the meaning of the data. Principal Ragnar is certain of one thing: when you look at the problems these students get wrong, it does not make sense. Not in this age of graphing calculators, specialized attention to math problem-solving, the many techniques to solve problems, etc.
But does all of this help – or hinder – the student? An interesting question.
The Principal arose when the other teachers entered the room. It was a bit before 7:00, and the teachers had agreed to meet early before school, with their assignment to bring a number of problems their students had problems with. He was interested in what they would show up with.
“Good morning, everybody” Principal Ragnar boomed! "I’ve brought some doughnuts and orange juice, so while you get situated, how about we write some of our problems on the board."
Mr. Wilhelm moved forward while the others retrieved their breakfast, and wrote a series of problems.
"I’ve grabbed seven problems from recent tests of my students – problems that really irritate me. We go over these things, we talk about problem-solving strategies, and come test time – WRONG!" The other math teachers nodded in sympathetic agreement. The phenomenon was all too familiar.
Principal Ragnar went to the board: “Look at this problem,” he said, pointing at the ½ fraction problem with frustration. “How can anybody get this problem wrong? The simplest example of “½ of ½ of a pizza is obviously ¼ of a pizza. I can understand how other fraction problems are difficult, but this one?”
“You’re telling me,” said Mr. Wilhelm, equally frustrated. We all use manipulatives now, and many other problem solving devices for the kids, and yet ‘this’ happens at test time!”
“Are you sure they understand the devices,” chimed Mrs. Peterson, an English teacher? “Of course they do,” said Mr. Wilhelm defensively. “Look at how easy this is. This one, in fact, doesn’t even require a ‘problem solving technique’. And let’s not forget how many of your kids don’t know when to use ‘a’ versus ‘an’ in a sentence.”
“Knock it off you two – this was the reason we’re all in this room right now together. How can we improve performance significantly over a short period of time? One thing I was pretty certain of: there probably is not an easy solution, because if there was one, you all would have found it by now. On the other hand, with the reading section, we found there was an easy solution. I see problems like these – math and English – and I say to myself, “This is low-hanging fruit. If we could find a way to solve problems like this – have all our kids score nearly perfectly on problems like this – then we can see massive improvement rapidly. That’s why we’re here!”
“But why can’t the students answer these problems correctly – we give them so many tools?”
“Wait a minute,” said Principal Ragnar, again approaching the whiteboard. “Why do we need so many tools for a problem like this – or a problem like ‘a’ versus ‘an’? If these are so straightforward, why aren’t the kids doing them so straightforward?”
“The pizza / fraction example is an interesting case,” said Mr. Wilhelm. Sure we can cut the pizza into halves and again into halves to get fourths, but how do you cut a pizza into 42nds and then into ‘square roots of 2’? We can make a few problems relevant, but after a bit there isn’t any relevancy.”
“But look – if they didn’t get the ½ x ½ problem right, they didn’t even use the pizza example for that. What good are problem solving tools if they can’t even get a problem like this correct?”
Principal Ragnar sat down. He was frustrated. Everybody was frustrated. They all knew he was right. And they all knew there was not much to be done about it.
He spoke up. "Let’s start with something really simple, and see if we can accomplish something. Right now, let’s drop the notion of “problem solving”, and focus on this simple pizza example. If the kids knew, when working with fractions, to do this, would any of them get this problem wrong?”
Mr. Wilhelm shot up instantly. “If they knew what? If all fraction problems could be sliced like this? Of course they’d get it right! But that’s the problem: most can’t be cut up like this one.” He sat down, frustrated and furious.
“I’m not talking about all problems right now – let’s focus on this one. No other strategies except take a real object in your mind and cut it up. Correct answer. Yes?”
“And what about this one?” Mr. Wilhelm confidently strode to the board and pointed to the exponent problem of 22 x 23. Fractions are easy, especially the cherry-picked one like that we just used. What about a problem like this? Can we do the same thing with this problem?”
The Principal thought about his own method of solving problems, and said, “Do we have to?”
“Do we have to ‘what’?”,
“Are we making this more complicated than it need to be? Of course, hands on materials are good when starting to understand fractions, but there’s a point where you leave the hands-on materials behind, isn’t there?”
“But the point is with all of these problems they’ve not mastered any of the problems where the hands-on materials are relevant?”
“Then how on earth can we expect them to solve these problems? For goodness sakes, we’re wasting the student’s time giving them hard problems if they can’t solve easier problems with 100% accuracy, aren’t we?”
The Principal continued, verbalizing a recurring theme. “Wasting the student’s time, nothing – aren’t we wasting your time as well?”
“But what choice do we have?” the beleaguered math teacher exclaimed? "Do nothing? We’re doing the best we can, developing multiple strategies for solving all these different problems?”
This thought was the launching point for the math revolution. Multiple strategies for solving problems? This has been the manner of teaching math for some time. But is it good? Does this help - or hurt - students? Does this help - or hurt - teachers? Surely, if it helped students, we’d see the results in rising test scores. Do we?
Can we?
Noticing the clock approaching the official school opening, Principal Ragnar dismissed the team. "See you all tomorrow," he said with a grin. The team disbanded. The Principal had a plan.
"Mr. Wilhelm? Would you do us a favor? Instead of bringing in more problems like you did today, would you bring in some actual tests from your students?"
The Principal chuckled - to himself. He was beginning to see a pattern forming.
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Recommended Changes to NCAA Basketball
May 9, 2008
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RULE CHANGE 1 Eliminate the Transfer Penalty
After one season at Kansas State and a promising recruiting year for the 2007-2008 season, Bob Huggins left at season's end, signing with West Virginia for a reported five million over five years with an $800,000 base salary in the first year. Fast forward to Year 2, to this announcement, made hours ago: Bob Huggins' 10-year deal with West Virginia worth $20MMORGANTOWN, W.Va. (AP) — West Virginia basketball coach Bob Huggins will earn at least $20 million in guaranteed income over the next 10 years of his new contract, which includes incentives that could add thousands more.
Of course, Bob Huggins is not alone here. Far from it - he seems the rule and not the exception! What's going on, here? What does "contract" mean, when you not only can renegotiate an existing contract, but also leave an existing contract? But what about the student-athlete, left in the wake of the coaching-boat speeding away to greater dollars? They are not free to leave. Transfer - and they must sit out a year. A coach recruits them with the promise of a great team, leaves, and the player is the one penalized. Does this mean a coach must stay with a university forever? Hardly. Rule Change #1 merely eliminates the transfer penalty. Give the student-athlete the same rights as the coach.
RULE CHANGE 2 Eliminate Charges Inside the Area About the Basket
The NBA has this rule in place now. College needs to adopt it. Why? Last year, in all the games I saw, I can remember maybe one legitimate charge - where the defender actually got into position before the offensive player left the ground. However, a charging call is made frequently - and always wrongly. Instant replay confirms this - always. Take these wrong and frequent calls out of the hands of the officials - have them looking at the position of the feet of the defender - and the call becomes nearly automatic.
RULE CHANGE 3 Time-Outs Can Only Be Called With Definite In-Bounds Possession I'm frankly tired of seeing players roll on the ground for possession of the ball, with several yelling "time-out". I'm even more tired of officials granting it! Most of the time, officially the player calling the time-out is guilty of traveling, because he is avoiding another player to call it. That's the definition of "travel" on the ground - turn to gain an advantage: travel. Call this and I'm half-happy. But I'm not happy with half-happy. Total possession means a standing position with the ball. This also means "in control"; not flying out-of-bounds, yelling in midair: TIME-OUT! Stop this nonsense. A time-out can only be called when a player has in-bounds standing control of the ball.
RULE CHANGE 4 Modify the "Intentional Foul" Designation Fouls at the end of games to send the team to the free-throw line are intentional. That's obvious. What the officials have trouble with now is distinguishing between "intentional - rough" and "intentional - regular foul". That latter distinction points out the folly in this distinction altogether. And this is not a trivial matter. Intentional, right now, implies 2-shots plus the ball, though we know the fouls are always intentional. That comes close to Orwellian Basketball! What's my proposed rule change? Make the distinction one of three: regular: a foul taking place in the course of the game. intentional: a foul taking place in the course of the game. flagrant: a rough intentional foul = 2 shots plus the ball.
As you can see, there is no distinction between "intentional" foul and "regular" foul; and there doesn't need to be, when there is a "flagrant" foul category to fall back on. Is this all we need? Watching the NBA and hack-a-Shaq tells us. No. Simply grabbing Shaq to put him on the line is not basketball, and it should not be rewarded as being basketball. If Shaq has the ball, that's one thing. When he doesn't have the ball, however, it's another. A possible modification: regular: a foul taking place in the course of the game. intentional (off the ball): 2 shots plus the ball. flagrant: a rough intentional foul = 2 shots plus the ball. Does the terminology help? We don't want to call grabbing Shaq's arm "flagrant" because it's not. It is intentional. But it's away from the ball, so the call is "Intentional Foul Off the Ball", awarding the team 2 shots plus the ball. I like it. Notice there is no longer a need to have an "intentional" category, either.
Operational Method of Gauging Official Effectiveness The variability of officiating infuriates good fans. How can games be called obviously different? "So long as the officials are consistent" is the clarion call from many; nonsense: "so long as the officials are consistent and right" is what we're looking for." But what are the right calls? How can we see how officials view calls the same - or differently - or don't see them at all? One problem we as TV viewers have - or at least a difference - is we see the game differently from the officials. They are at ground level in the middle of all the action. We're on couches watching the whole game, with instant replay. Let's take that into account: Let's change places - or at least put them in our position, and vice versa. But not just them reviewing their own game, but everybody reviewing that game. What's called. What's not. What should be? What percentage of officials are calling what is an obvious travel - travel? From the perspective of the TV, likely charging calls will be the opposite of those called during the game. Good. Such recognition of variance will be the starting point in forming operational definitions of many vague items now - block / charge, traveling, carrying the ball, low-post charge, etc.
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An Educational Tribute to Steve Fossett
May 10, 2008
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(this article was written in late September, 2006, when the search for Steve Fossett was ongoing. He has since been legally declared dead. His adventurous spirit lives on, forever!)
I was on hand in Salina, Kansas, in 2005 when Steve Fossett landed the Virgin Atlantic GlobalFlyer, ending the goal of circumnavigating the globe on a non-stop, non-refueling airplane. He did this in 67 hours – a far cry from the 80 days of Phileas Fogg in “Around the World in 80 Days”. Little did I know this was a small part of the adventurous resume of Steve Fossett. He holds 115 documented world records! Further, he has competed in the Iditarod, swam the English Channel, finished the Ironman in Hawaii, and driven in the “24 Hours of Le Mans”. His accomplishments suggest no limits. What are our limits? Evidently, the sky is the limit! On September 3, 2007, he took off from an airstrip southeast of Reno, Nevada, and was not heard from again. The search continues. I hope the best. But I’m also baffled: how can someone simply disappear in Nevada – or anywhere in the United States – for that matter? How hard can it be to find a missing airplane? What’s more striking is initial reports focused on a 600 square mile search area (an area approximately 25 miles by 25 miles). With this focus, I thought surely he’d be found immediately. As days went on with no sighting of Mr. Fossett, the area expanded to a circular radius of 50 miles. How much bigger is this new search area compared with the old?
The search area has been expanded 1200%, so it seems likely the people doing the searching may not know as much as I thought they initially did. Can I help?
The Lay of the Land Let’s get an idea of the “lay of the land”. Let’s start with where exactly Reno, Nevada is:
This gives me an idea of where all the search action is taking place, but I have no idea how big Nevada is. I also don’t know how far his plane could have flown, and therefore, I don’t know the maximum search area, because I don’t know anything about the plane, the fuel, or the average speed of the plane. For example, if the plane’s flight time was three hours and it travels 100 mph per hour, the search area is much different than a plane with eight hours fuel travelling 250 mph. My preliminary plan to get rolling, then: map out the possible search area by finding specific data on the plane, and find out specifically where he flew from.
My Preliminary Plan According to reports, Mr. Fossett took off from the “Flying-M Ranch”, southeast of Reno, in this Bellanca Super Decathlon, a single-engine plane:
(image from www.stevefossett.com)
As I understand it, this plane has an average speed of approximately 140 mph, with a flying time of approximately 5 hours.
The likely maximum range is 280 miles, but what does that mean for Nevada? How big is Nevada, and where precisely did he take off from?
The legend of the Nevada map above ranges from 0 to 100 miles, and the suggested range of the Super Decathlon for a return flight is 280 miles (equaling 2.8 times the length of the legend). That distance runs (considering the northern border as my reference point), from the western border of Nevada to just shy of the eastern Nevada border! This gives me the radius of a circle to apply to a starting point, but what is the starting point? The Flying-M Ranch is located at the following coordinates: Latitude: 38°36′13″N Longitude: 119°00′11″W In fact, zooming in with Google Earth, you can see the actual runway:
With my circle center at these coordinates, and applying my radius as described above, I arrive with the following possible area to search:
It’s no wonder authorities are having trouble finding the aircraft. The possible search site spans nearly the entire state of Nevada, a large part of California, touches upon two other states, and heads out into the Pacific Ocean! In terms of shear area to search, the area has expanded from the initial 600 square miles authorities were searching to 7,850 square miles (50 mile radius) to:
Nearly ¼ million square miles! Clearly, if it took authorities considerable time to search 600 square miles, searching this new sector of land will likely yield little soon. Can this area be narrowed down, despite the fact no flight plan was submitted with aviation officials?
Officials initially said he was flying to check on alternative flat lake bed sites to serve as a site for a land-speed record. This map of these “playa areas” reveals the distribution of these sites: Does this narrow down our search area? As it seems potential areas are spread throughout the state, I’m tempted to say “no”, though the heavy concentration of sites in NW Nevada provides food for thought. Is there anything about these sites making them appealing for land-speed sites?
In Search of a Refined Search Area Dry lake beds. I thought, recalling the pictures I’ve seen in The Guinness Book of World Records, land speed records typically were held at the Bonneville Salt Flats in Utah. Apparently, this site was looked at, but not considered because it was believed the terrain was too soft for Fossett’s vehicle. However, interestingly enough, Gerlach, Nevada is the home of the current world land-speed record, and it falls in the NW quadrant we just noted was interesting. But if this is the place where the current world-record is held, why didn’t Fossett choose it? Apparently, it was too rutted to use, and zooming in with Google Earth, one can see this:
Further Refinement and a Possible Scenario Let’s be both realistic and optimistic: the plane was equipped with an Emergency Locator Transmitter (ELT). However, apparently in this model aircraft, I’ve learned the ELT is notorious for failing, which might explain the lack of a transmission. Let’s rule out fatal crash here, and hope he is still alive. If a crash disabled the radio, why then does the GPS not emit a signal alerting us to his position? What might be blocking it if it were operable? Mountains? So a possible scenario: Mr. Fossett flies past Gerlach (a known flat lake bed record-holding spot) in search of non-rutted lake beds. His plan is to circle the high-density area NE of Gerlach, returning in an elliptical route arriving back at the Flying-M Ranch. Something goes wrong, and he is forced to make a crash landing. Perhaps, in flying low to view the lake beds as candidates for his land-speed record attempt, he encounters trouble, and crashes. The ELT, notorious for failing, fails. Mountains block GPS transmissions, and he is unable to signal. Is there any terrain meeting this criteria?
This mountain range rises rapidly from 4,000 feet to 8,000 feet, and seems to meet all the criteria.
Developing Thoughts Since first popularizing this theory and sending it to the folks running the stevefosset.com website, a couple important facts have been disclosed. First, the initial search area of 600 square miles, we’re told now, was the result of “off-the-top-of-the-head number by someone in operations section”, said the Major in charge of the search. “People are so busy trying to focus on the mission at hand, we didn’t have a more accurate figure until last night (when the search area was extended to 10,000 square miles).” “Where could he have gone?” should have been the first question asked and analyzed thoroughly. The tendency of many is to “get out there and do something”, but what? How many resources are wasted being busy? I would not want to be at the bottom of haystack when someone considers me the needle. If we have absolutely no idea where to look, every resource should have been directed towards obtaining information about motive, and narrowing down “the haystack”. He did not file a flight plan with aviation officials. Fine. What do we do? “Better start looking” is not the right answer. “Where was he likely to be?” is a better start. And now even this is in question: we were told initially he was looking for alternative sites for his land-speed record attempt. Later, it was disclosed he was likely on a pleasure flight. This story has gone back-and-forth. The most crucial element in the entire detective process, and it’s up in the air. After many days, we were told a ping was heard SE of the Flying-M Ranch as Mr. Fossett’s last known location. As I understand it, search continues in this quadrant. On what day was this ping detected? Can motive be ascribed to a flight in this direction to narrow the search? Hopefully, these questions were asked, and the search area created with this goal in mind.
A Closing Thought I wish Mr. Fossett well. In always reaching to conquer the unconquerable, he was a model of demonstrating what’s possible. It’s tempting to say only the exceptional are capable of the exceptional, and these feats, extraordinary as they are, fall within the domain of a select few. Is this true? By his own admission, he was not an extraordinary athlete. On the contrary, he was an average person doing above-average things. And thinking this, I’m reminded of a quote from Jonathan Livingston Seagull, and I think: “ ‘Let’s begin with level flight’, and I understood at once that his friend had quite honestly been no more divine than himself.”
How much more are we capable of? The sky is the limit!
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Architects of Their Own Future
Chapter 16
May 11, 2008
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Chapter 16 EXPLOITING THE LEVERAGE POINT IN MATH
(1st draft - a bit of cleaning up to do, still)
The team assembled the following morning to a chaotic site on the lounge table. Saws, hammers, screwdrivers, and levels lay scattered on the table top. Hundreds of nuts, bolts, screws, and nails were strewn everywhere.
The team, rightfully assuming an errant janitor to be the responsible party for the mess, started to assemble at the empty table across the lounge. "No, please. Have a seat at our normal table, would you?", the Principal requested.
"Mrs. Peterson: before you sit down, would you go take a look at that drawer that's always stuck? And Mr. Wilhelm, the magnetic strip on that closet hasn't been holding lately. Could you take a look at it? All of the stuff you need to fix both are right here on the table."
The two, staring in disbelief at Principal Ragnar, put the disbelief to words. "You expect us to really fix those things? I'm speaking for myself - Mrs. Peterson can speak for herself - but I don't know how to do this?"
"But all the materials you need are right here?"
"So what - I'm not a carpenter."
"But these materials are good, aren't they?"
"What does that matter if I don't know how to use them. I know how to use them, of course, but I don't know how to detach a magnet (without damaging the closet), and put on a new one."
"So more good things doesn't help?"
Mrs. Peterson understood immediately - she saw the connection with the reading section of the ACT. "You're getting pretty good at this 'Socratic' thing, Principal."
"What do you mean?", knowing what she meant.
"This is exactly the same thing that happened with the reading section of the ACT. We teach the kids to scan, to skim, to outline, to read thoroughly, to go to the questions first, blah blah blah", and we end up with a ton of rules for "problem-solving". Many of them are in direct conflict with one another. And though we agreed they were good strategies, they were only good in a certain context. In our context, they distracted the students; hence, our scores were always bad."
"Indeed. Thinking over our conversation yesterday, it sounded exactly as you just described - as well as some conversations I've had with Coach!"
"So what are you saying," Mr. Wilhelm said, interrupting the conversation. "Our problem-solving strategies are bad?"
"Are they helping right now?"
"Not really."
"But you're sure they're good?"
"Absolutely."
"But how can they be good if they're not helping?"
Mr. Wilhelm thought that over for a moment: "The problem is the kids aren't sure when to use one strategy over another."
"I think you're right. Is that any different from you now knowing how to fix the closet, despite all the tools being right in front of you?"
Principal Ragnar continued. "I'll bet if I take one of your test papers from that stack in front of you, pick a page at random, and go to the first wrong answer, I'll find what amounts to chicken-scratch on the paper. Care to take me up on it?"
"No bet - but do it anyway."
The Principal did as described, and found the problem 3/4 divided by 5/9. There were a couple of ways of doing the problem, one answer first circled and then erased, a second answer circled, both wrong.
"How do you teach this problem?", the Principal requested.
"Well, for the division of fractions, we simply give them a rule: invert and multiply."
"I remember the rule well. Students now-a-days aren't doing so well with it?"
"That's for sure - but what alternative do we have? It's not possible to use manipulatives on a problem like this. You can't break something down easily into ninths, and you sure can't use this in a division problem."
"I didn't say you had to."
"What are you saying?"
Principal Ragnar thought a moment: it was no use asking Mr. Wilhelm a question on math - he would know all the answers. He thought of Mrs. Peterson and threw out a hard problem
"Mrs. Peterson: we don't want to leave you in the cold: could you tell me what this equals?" He wrote x4 x-2 on the board.
She thought - and thought. It seemed an eternity - to her. It had been so many years. How do you solve a problem like this? Multiple exponents? Add them? What does the negative sign do to the problem?
"x-8?", she answered in the tone of a question.
The Principal held his hand in Mr. Wilhelm's direction, indicating he not say anything. "That was a guess, wasn't it, Mrs. Peterson?"
"But was it a right guess?"
"Let's find out, because I'm not sure myself. Let me ask you, to see if we even know what exponents are, what is 32 ?"
"Nine," she responded, more confident in this answer.
"It's 'nine, because 32 = 3 x 3, right?"
"Right."
"Then you also know what 31 is, right?"
"Of course: 3."
"It's 31 = 3, right?"
"Right."
"Then if 32 = 3 x 3, and if 31 = 3, then what is 32 31?"
"27."
"Because 32 31 = 3 x 3 x 3 = 33?"
"So if that's the case, what is the rule when multiplying exponents?"
"You add them."
"Now, let's go back to our original problem: what is x4 x-2?"
"I see now: x2. I guess I would have got that one wrong."
Principal Ragnar, happy with his dialogue, was not pleased with that dismissive answer. "Yes you would have gotten that one wrong. But don't you see how easy it is NOT to have gotten that one wrong?"
"Now wait a minute. Are you telling me I have to do that for every problem?" Mr. Wilhelm himself was not convinced.
"No. I'm saying all of this is the job of the student."
"OK, Mr. Smart-Guy: you picked a pretty easy problem, one you can reduce down to a small amount of numbers. Let's see you do x14 x-253 the same way?"
"Fine. Mrs. Peterson: what is 32 ?"
"Nine."
"It's 'nine, because 32 = 3 x 3, right?"
"Wait a minute - you're not doing this problem?"
"Are you sure?'
"What do you mean?"
"Look at our example above: we don't care about 3s and 9s. We care about finding the general rule. In that case, the simple example worked well. It helped us find the general rule. Once I have that general rule, there isn't a problem I can't solve!"
Mr. Wilhelm picked up on the challenge. "OK - what is 4/5 x 2/3?"
Principal Ragnar again picked on Mrs. Peterson: "Mrs. Peterson, let's suppose I take a pizza and cut it in half. How much is here?"
"One-half?"
"And if I eat one-half of what's left, how much is left?"
"One-fourth."
"So it looks something like this?"
"Right."
"So 1/2 x 1/2 = 1/4?"
"Right."
"So if 1/2 x 1/2 = 1/4, then how do you multiply fractions?"
"You multiply the top and the bottom."
"If the general rule for multiplying fractions is multiplying 'tops' and 'bottoms', then what is 4/5 x 2/3?"
"8/15."
"Indeed."
"You're right - but just as the tools on this table are good tools, they're also no good unless one knows when and how to use them."
"So you're saying ... what?"
"All of the problem-solving techniques you teach have, to be effective, need to be directed towards the one end goal."
"Which is?"
"Finding the general solution."
"And that's it?"
"It was for these two problems, wasn't it?"
"Will this ensure the kids don't leave a mess on their papers? After all, one of the things we find is students have a hard time checking their work."
The Principal thought about that issue. It was a great issue. The language above was fine when it was language, but how do you make it work on a piece of paper? How do you allow the student to focus on thinking and not writing? How do you allow the student to accurately check their work?
"This part seems clear: our goal is to find the
general solution, apply it to our problem, and get an answer.
Right?" "Yes."
"Suppose we organize those three elements like this?" He drew the following on the board:
"That's the easy part," said Mr. Wilhelm. But what do we do next? Here's where we've got all of our problem solving techniques. You yourself just used two. You used algebra to solve the exponent problem, and you drew a picture to solve the pizza problem. There's many more."
"Why don't we leave it at that - there's lots of possibilities here, right? Why don't we simply leave a box here with something like "whatever it takes", or "any means necessary" that helps you."
"You're telling me to have the students write all of this for every problem?"
"No - but let's not forget they're only scoring about 50% on these state tests. Is there any sense answering all questions and getting half-wrong? My guess is, if students did write this for the problems they didn't know how to do, they could at least get 1/2 of those right."
"What happens if they can't think of anything to put in our 'wonder-box'? Our "Whatever it Takes?" box? What then?"
The Principal Ragnar smiled: "Wouldn't you just love that!"
"What do you mean?"
Mr. Wilhelm smiled.
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Dedicated to Those Who Desire Not the Fish But the Ability to Fish
May 12, 2008
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Ordering the Randomness The above graphics were randomly selected. The "Degree Displacement" Diagrams, in order, are as follows:
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May 13, 2008
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Here, I described my luck at finding a tanker truck at the local
station just as I pulled in. From this experience came an
investigation of the amount of gas in the vehicle, how many people per
year a truck served, and the volume of the truck.
With the price of both oil and gas rising rapidly, let's change the focus to the relationship between these two fluid. The apparent causal relationship seems to be: as oil rises, gas too rises. Why? Let's save that until we, at a minimum, confirm this guess. But in the process of doing this, let's document our process, including modifying graphs and the reasons for this modification.
Graph 1: The Price of Gas
It seems a straight-forward chore: graph the price of gas. Is this a good graph? The default for Excel is a zero value at the bottom of the y-axis. This leaves half our graph with no data! Additionally, the y-axis gridlines have the same density as the graph line. Is this good? Or does this "sameness" detract from us reading the graph? Further, there are no x-axis gridlines? Why not? What does our graph look like with them included? Why do we need a gray background? Why do we need out graph outlined? Let's take another look at our modified graph, including all the changes noted above: Graph 2: The Price of Gas (modified)
That was gas. Now, let's look at oil. There's no need to replicate the errors made in "graph 1" above; let's implement the changes in our first "oil" graph. Graph 3: The Price of Oil
The theory seems confirmed - a direct relationship between the prices of oil and gas. But is there? We've got two graphs here, so it's tough to make a direct comparison. In fact, this is the exact same problem we saw here with the KC Star graphics. Let's not duplicate that error; instead, let's use the same solution here we imposed there Graph 4: Combining the Two Graphs
Of course, legends are necessary when two-or-more datasets are graphed. Otherwise, how would we tell one set from the other? But what happens when we place the legend away from the data, which we must do by definition? It has our eyes glancing back and forth - from data to legend to data. Is this necessary, or can the graph be modified to avoid this problem? Let's try: Graph 5: Combining the Two Graphs (modified)
There we have it - the relationship is confirmed. Why? I still don't know, and I still don't know why the price of oil is rising. I don't know where we get our oil from, and I don't know what happens when we do get the oil. I don't know how the oil becomes gas, and all I know about gas getting to my station is seeing the truck above. But there is something missing in the above graphic. Though there seems a direct relationship between the two variables, my "oil-scale" increases by a factor of 3 (120 / 40), while my "gas-scale" only increases by a factor of 2. What happens if I make them both a factor of 3?
Graph 6: Combining the Two Graphs (modified again)
This shows the relative price of oil far exceeding the price of gas. But have I violated one of my rules for data-display - namely, allowing the scale to exceed the graphed values? Where did that rule come from? Is such a rule set in stone? Of course not - it's a helpful guide, useful sometimes, other times not. How, though, to capture "when it's useful - when it's not" in documentation form? The Audible Ready Tree ...
Concluding Thoughts Our relationship holds, but now a new question arises: why is the gap between these two elements increasing? More questions to investigate! For example, look at this graph. Instead of graphing the direct price of gas and oil, this graph is the ration of the price of oil / price of gas.
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Architects of Their Own Future
Chapter 17
May 14, 2008
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Chapter 17 REGULAR SEASON THOUGHTS "You should be home tending to your garden. What are you doing here?" It was Saturday morning, and Principal Ragnar had gone to school to retrieve his weekend reading material he had inadvertently left the night before. "See what you've done to me?", the coach bellowed. "What I've done to you? What have I done to you?" The coach showed him basketball DVDs he had ordered from Amazon. "My wife gets mad every day the mailman shows up, so I have to have these mailed here." "What are they for?" "We're 11 and 5 right now. Who would have thought it during the preseason. 11 and 5! Doing what? In focusing on how to really play the game. We have a chance to make the state tournament this year - something we've never done before! Our kids are playing better than they ever have!" He paused ... "And ...", said the Principal, enticing Coach Thompson to continue. "I realize just how little I actually did know about basketball, but how much fun it is to really learn how to coach this game." He continued: "Here's an example. John Wooden was often ridiculed for having his kids put on their socks a certain way. He was labeled a stickler for details, relentless about them." "I would think the same thing if I heard that about socks." "But why would he do it? It's because he realized a lot of his players were getting blisters on their feet because their socks weren't worn correctly. Align the seam with the toes, and suddenly you have so more blisters - and thus, better players!" "Amazing - who would have thought it?" "HE DID! And it's a simple thing - if you think about it. Who thinks about it, however?' "Apparently, not many people." "So I was thinking about other simple tasks like that. Take 'stretching', for example. Why do we stretch? I'm a PE teacher, and I took kinesiology and physiology in college, so I should know the answer. I know how muscles work, but I've never really given much thought to what constitutes 'good stretching'." "And ..." The coach pulled a DVD from his duffle bag. "We start this morning. Actually, I might take this one home and watch it tonight on my computer when everyone's asleep." "During the last game, I saw a lot of the players run back into the hallway just prior to going in to the game. What was that all about?" "You were paying attention! Imagine yourself - on the bench, when you get the call from me: 'Principal: in for Jones.' You're stiff, you're cold, and you're expected to immediately go in, and play at full speed. No other sport does that. Why do we? Since most of our substitutions are on a rotation basis, I know when the kids are to go in, and I have them go get ready." "Any change? Do they play any better?" "It's hard to tell - but it sounds reasonable to me. Certainly a lot more reasonable than having them go in stiff and cold!" The Principal smiled. "So what does the next two weeks hold for the team?" "I think we've got a real shot at going 14-6. West-Central will be our toughest game, especially with their inside player 6'10", while our tallest player 6'2"." "Maybe you can intentionally foul him, like the pros do with Shaq? Or pretend to take a charge, like many pro and college players do." Coach Thompson's facial expression changed. Was it something the Coach had said? Was it the fact the coach had said anything at all? Why had he blurted out what the coach should do when that was the coaches job. These thoughts took but a moment, but the damage was obvious. He tried to correct himself. "But I'm no coach - you're the basketball mastermind. What do you plan on doing?" "I've thought of both those - particularly since their inside player is a terrible free-throw shooter. Why not simply grab him? That's not basketball, to me. We've spent the whole year really finding out what basketball is all about. I'm not about to throw all that away and 'win at any cost'." "Good luck with the games this week - and next!" Principal Ragnar exited the coaches office, retrieved his own books, and headed towards the car. He reflected on the turn the conversation took. Why had he jumped in? It wasn't a matter of pride, or stealing someone's thunder. He hadn't intentionally jumped in to show how smart he was. On the contrary, his verbal blunder was accidental. Nonetheless, the hurt was clear. Why had he done it? How often had it done it before? What will it take not to let it happen again?
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The Circumference of the Earth
May 15, 2008
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We return to Eratosthenes and his calculation, 2200
years ago, of the circumference of the Earth. In
Part 1, we left off with the following
diagram. Who am I? I am Iatosthenes, modern version of
Eratosthenes.
An Introduction I've spent so much time describing the math and science behind my endeavor, I've neglected to tell you a bit about myself. I am the librarian in the great library of Alexandria, and the year is approximately 200 BC. Of course, you know that last part is only right, as I can't know it's "BC".
Our great library of Alexandria lies on the coast of the Egypt, with a beautiful view of the Mediterranean Sea. While organizing the tremendous volumes of scrolls in the library, I came upon an interesting fact: the sun, on the summer solstice, shines straight down to the bottom of a well in Syene. Later generations will know this town as Aswan, and, perhaps not coincidentally, this writer wrote about the great Aswan Dam and the flow of the Nile!
I got to thinking about the phenomena of the well while rereading Euclid's Elements outside one day as the great library cast a shadow upon me. What might this shadow mean? I know the great Thales of Miletus measured the height of the Pyramids by way of the shadow and similar triangles. Might such an idea help me?
What kind of help? Towards what end? As I picture the sunlight going straight down that well, I can imagine what it would do after that - if it only could. I'd go straight to the center of the earth. Now there's a thought. How deep is that? How big is this thing we live on called earth? Let's take that ambitious target and see what it would take to achieve it - finding out how big the earth was: if, at the summer solstice, I could calculate an angle where I live, I could then determine (thanks to Euclid) the angle intersecting at the center of the earth. Knowing this and the distance from Syene to Alexandria, I could then figure out the circumference of the earth. Easy enough. The day is approaching: summer solstice. I've got to be ready to measure the distance of the shadow cast by the rod, perpendicular to the earth's surface. If I know both the height of the rod and the distance of the shadow, I can find the ratio of these two figures, and then consult our "tangent table".
The Results are In My 57' rod cast a short shadow of only 7'. I guess that's to be expected, given we're so close to Syene. If we lived far away, the shadow would be huge.
Now what? The ratio of the rod to the shadow, or the rise/run, is slightly over 8.1. Consulting our "tangent table", I see this angle must then be 7˚.
What to do with My Results Seven degrees! This internal angle, of rays emanating from the center of the earth towards both Syene and Alexandria, is seven degrees. Twice now I've used the geometry of Euclid.
But what to do with seven degrees. I need the distance from the two cities to do anything. I know there's been lots of speculation as to how I figured this distance out: professional marchers, camels, many others. For the sake of this discussion, let's assume it is 461 miles.
More Thoughts (for the reader) I've left out a number of the details and assumptions in this brief write-up, and I hope the reader will forgive the author. On the other hand, I hope the reader will not forgive the author, and will instead seek out explanations for themselves. My assumption, for example, required both cities to be on the same meridian. Is this true, and, if so, how did I know it? Even if I knew about the well in Syene, being nearly 500 miles away, how could I know it was noon, and the sun was directly overhead there? If I had not known of the well, could I have still found a way to estimate the circumference of the earth?
Should you think the calculations mere hypothetical and academic, keep in mind the implications of my good friend, Ptolemy, and his calculations. Ptolemy, of course, changed my estimate in future years, believing the earth to be significantly smaller than I. His estimates survived the ages, and, in the 15th century, Columbus set sail for India, believing it was this shorter route!
How might the world be different had he not set sail!
Ignorance is bliss?
Never!
The Tangent Table What is the "tangent table", you may ask? You've used tables, dictionaries, encyclopedias, etc., all your life. In many instances, the tables exist because, in solving one problem, you do a technique. Another problem, another calculation. A third calculation - like the first - and, well I wish I had written that down. So we invented a "rise / run" table to allow us to have access not just to problems we have solved, but problems we've not yet even encountered!
So we divided the circle into 360 equal parts, and, at each point along the circle, dropped a line to the x-axis, and then stretched that line to the origin. This gave us two distances: rise and run. The ratio was then calculated and recorded, along with the degree measure.
We didn't stop at one degree - we did this for all 360 degrees! Sure, it took time now, but in the long run, the time savings was immense! Also, now our geographers could simply carry around the small table in the pocket, rather than having to do a lot of measuring each time they needed to!
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Architects of Their Own Future
Chapters 18 & 19
May 16, 2008
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Chapter 18 COMPLEX vs. SIMPLE SYSTEMS: In Search of Inherent Simplicity The Fourth Chautauqua
(1st draft - a bit of cleaning up to do, still)
The basketball team was doing well. Students were doing well. The ACT was going well. Principal Ragnar walked his neighborhood. He was not. He resumed his nightly peripatic stroll, this time about the baseball fields located down the street from his house. He'd always been a baseball fan. Though not particularly good as a player, he enjoyed the numbers of the game, the rules of the game, with every play providing one to the ability to think - before and after the play was over. He strolled by a young boys team practicing. They were no more than 10. "Two things on that plan, guys: when there's less than two outs, and runners on second and third, when the ball is hit to the second baseman, let's go for the out at first. Runners: remember, you don't have to run, but if the ball is hit to the right side, then you can advance." Principal Ragnar looked at the faces of the young boys. "Right, coach - they understood every word you said." Likely they didn't understand any word! The Principal reflected on the coaches lengthy verbal narrative: it was right - but there was something not right about it. He continued to watch the practice. It didn't take long for the next winded-explanation to come. "OK, guys: with no outs and runners on first and third, when it's a fly ball to the outfield, we want the runner on first to go halfway to second, in case the outfielder drops the ball. Then you run to second base. We want the player on third to stay on the base. That way, whether he catches the ball or drops it, they can run home." Again, thought the Principal, the explanation is technically right, but there was something odd about it. He'd never been taught a rule like that before. You just "get it" playing baseball enough. Well, that wasn't exactly right. You learn a few basic rules of the game, and then play the game! All the scenarios come up, and from the few rules you know, you play the game right. How else could practically every adult male know so much about baseball, without being taught all this. The Principal hated to think of the coach explaining the infield fly rule, or the dropped third strike rule to these kids! He chuckled to himself. The he stopped. Is this any different from what was being done in the math classroom? In some instances, we teach rules. In other problems, we teach the students to figure problems out with examples. And, of course, the old favorite for order of operations: PEMDAS. Parenthesis / exponents / multiplication / division / addition / subtraction. Of course, PEMDAS was but one of about 10 acronyms for order of operations. He thought of this coach sitting in on the math classroom. He could envision the coach chuckling at the classroom activities! The same problem? What is the problem? The Principal had already described it regarding baseball: the coach was making things far too complicated. Teach a few rules and the rest becomes clear. Is the same true with math? He thought back to their "solution" for math problems earlier: of requiring students to complete the "conduction" template, as it came to be called. This made sure students didn't miss easy problems. But it did another thing as well: it focused the students on one specific idea, rather than a hodge-podge of problem-solving techniques. Indeed, that's exactly what they had seen, though at the time they hadn't put words to it. Is a direction of massive improvement simply to "simplify"? That was too easy: who doesn't know that? Well, he thought, this baseball coach doesn't. A well-intentioned gentleman who knows a lot about baseball, but doesn't know how to present it to kids. We can excuse him. But this was our job? How come we don't see it? That's ridiculous. Of course we see it. There's probably one teacher in a thousand who wouldn't agree with some motto like "teach principles", or "make it simple", or something like this. OK - then why don't we do it? He thought of the classroom - not just math, but science, english, art - any classroom. The students, every one different, every one at a different level of ability. Is this the reason why .... why what? He thought back to the baseball example. What was it that frustrated / humored him there? It was examples instead of rules. Specifics instead of general principles. Is this why we teach many specifics instead of general principles? Perhaps it's one reason. The Principal wasn't satisfied with the examples. Just because there are lots of kids in the class mean you teach lots of specifics. He thought of the upcoming state assessments. Even charter schools were under the gun to perform well on these tests. There was a lot riding on the performance of the kids. What happens if there's a lot riding on the kids' performance, and test time is coming? What does the teacher do? Memorizing river names, chemical formulas, and mountain ranges? State capitals and general's names from the Civil War? Rules for multiplying and dividing fractions? But this all seems like the baseball-talk above? How did we end up here? In order to have a good school, we want to teach general principles. We want to show the relationship between curriculum. We want to show kids how to think, how to solve problems, how to infer, how to reason. On the other hand, in order to have a good school, we must do well on assessment tests. And in order to do that, we have to .... resort to memorizing river names, math rules, etc. Is this the dilemma we're in? Likely there are other problems besides the testing environment causing problems in education. After all, education existed long before NCLB came along, and one reason for NCLB was because of poor performance in schools. Indeed, the clamor over NCLB between the two parties isn't whether NCLB is good or not, but rather what does it mean to fully fund it! Back to the dilemma. Principal Ragner paced the sidewalk heading towards school. Is it the case tests force us to memorize at the expense of thinking? Again, it was probably more complicated than that, but let's assume this is involved somehow. He seated himself at a bench facing the pond at the school. Picking up a stick, he started to draw figures in the dirt. Likely Archimedes was doing this same thing thousands of years ago - and he was killed for doing it! He thought back to an earlier walk he had made, talking about wasting time in the classroom. Surely, this was taking place here too, though unintentionally. Might it be the case, in attempting to improve scores by forcing kids to memorize myriad facts, we hurt test performance because kids don't memorize them? And if that's the case, aren't we really wasting time doing this? By being placed in a dilemma between "simplistically" and "complexity", we've chosen complexity with the hopes of doing well now. We haven't. And in not performing well, we .... what? At the next round of tests, we have to ... do the same thing? We're caught in an endless spiral of teaching rules, instead of teaching general principles! A horn honked in the background. It was Mr. Stephens, the science teacher, driving his truck towards school. The truck was loaded with what looked like big pieces of sheetrock. The Principal turned, waved, and yelled, "What are you doing here so late, and what's all that in your truck?" The excitement in Mr. Stephens' voice took the Principal by surprise. "Get in," the teacher hollered. "I want to show you something!"
Chapter 19 THE SCIENCE OF SCIENCE
"What's all the sheetrock for in the back?". "It's not exactly sheetrock, but we'll talk about that later. Mr. Stephens pulled around to the back of the school, backing his truck up to the service entrance. "We'll get those later. Come on! Let's go!" He dashed off, the Principal in hot pursuit. He had never seen the teacher as excited as he was right now. They approached the science room. "Welcome to the Exception Squad!" "The what?" "That's just our informal name for now, but look." The two men stepped into the science room. It looked like any ordinary science room, with the exception the chalk boards around the room were filled with writing, pictures, and diagrams. "What is all this?" "Do you want the long version or the short one?" "Why don't we start with the short one." "OK: this is thinking - real thinking." "That's it? That's all the explanation I get? You'd better give me the long version instead!" "Yesterday, we were talking about the spectrum, prisms, and colors. We did experiments with light hitting the prism, and seeing the colors of the spectrum. At the end of class, one student asked if this was what happened to cause rainbows." "A good observation", noted the Principal. "Too good." "What do you mean by that?" "Ever since we created the science materials for the ACT, it's impossible for me to take such sentences at face value. The old 'me' would have agreed with the student. The new me wrote this on the board:"
"Looks good to me," said the Principal, struggling to read the teacher's handwriting. Reading this as a complete sentence, I have, 'If sunlight strikes the rain, then I see a rainbow.'" "Would you forget the handwriting for a moment! It looks good, does it? Let me ask you a question: every time it rains, do you see a rainbow?" The principal thought for a moment. "Probably not - in fact, they're so rare in the sky you usually point them out when they do appear." He continued before Mr. Stephens jumped in: "I see where you're going with this: how come we don't see a rainbow every time it rains?" "Not exactly - but close. Above, we've accounted for 'sunlight' in our explanation. A lot of times, during or after, there is no sun. Therefore, there can't be rainbows, right?" "Right." "But there are times when there is sunlight, but at the same time there is no rainbow, right?" "Right." "But above we've said EVERY time sunlight strikes rain, we see a rainbow." "OK ..." "Don't you see - above, we claimed to know something about the relationship between sun, rain, colors, and rainbows. But we've just came up with an example where what we said wasn't true." "I see now - that's why you call it the 'exception squad'." "Kind of - you said you wanted the long version, so now you have to have some patience!" "When you're looking for exceptions like this, it's very easy. But what do you do with them? It's like back-seat driving - anybody can do it. But does it help? Does it advance anybody's learning? So we devised this slick little diagram to voice your objection."
"And how does this read?" "OK - you're the student: you come up to the board and read the left side. 'If sunlight strikes the rain, then I see a rainbow.' Now you're thinking what we just thought above. I don't always see a rainbow. So we write it like this: 'if sunlight strikes the rain, then I'd expect to see rainbows every time it rained and there was sunlight.' Knowing you don't, you put an 'X' through it, showing something is wrong somewhere." "The student writes all that?" "Well, we just started, so there's probably lots of refinements coming, but don't you see what follows?" "Go on." "Somewhere our little theory we thought was so right is suddenly wrong. But where? How do we fix it? How about this?" The teacher added a word to the initial diagram.
"That begs the question, doesn't it? After all, just saying 'sometimes' doesn't help, does it?" "Think about it and actually read the boxes, would you." The Principal read the words, but was immediately interrupted. "Out loud, please." "Fine. 'if sunlight strikes the rain, then I sometimes see a rainbow.'" "I see what you're saying. I want to know when 'sometimes' is. But what's 'reading out loud' have to do with anything?" "It's just another thing we're playing around with. We don't read everything out loud of course, but for most of our logic boxes, we do. This has a different 'feel' to it." "OK - so what do you do now? Who can fix the statements on the board. And what's with the sheetrock in your truck?" "LONG VERSION, remember!" "Anyone can fix the statement - correct it - modify it - change it. But they must be written down." "Let's say it's just me and you. In doing this, I start to wonder when 'sometimes' is. I find somewhere it's the angle of the light hitting the raindrops that causes the rainbow. Maybe I know the angle, maybe not. So maybe I write the following:
"Now you come along. What do you think?" "Well - I'd want to know what angle this is, of course." "Me too. Let's assume generally it's right. Is that it?" "OK - yes, I'm happy with it now." The teacher scribbled another diagram on the board, and also crossed out 'rain', replacing it with 'raindrops'.
"What do you think?" The Principal gazed at the board: "I guess I never thought of it like that before, but you're right. Why don't we see billions of small rainbows, instead of one huge rainbow?" "Now the sheetrock?" "Almost! As I said, the initial question came at the end of class. That was Thursday. You can imagine my Thursday night, thinking about this simple question with a not-so-simple answer. So I started writing things down. What's amazing is how hard it is to actually write things down - in complete sentences. It's also amazing how wonderful 'a little bit of knowledge' is. It masquerades as knowledge, but it's really not." "So we spent the whole class period Friday talking about the process above. We started simple: three things pretty obvious: rainbows, earthquakes, and snow. And this is the result." His arm swept out a path around the room of the three chalkboards covered with diagrams. "And do the students go up in order - how does that work?" "What we did today was allow anyone who had a constructive thought they could put in words could go up." "And how did that work?" "It depends on what you mean by 'work'? Did everyone go up? No. Did some people monopolize the board time? I think so. But another thing happened. To make sure everyone was doing something, I made everyone turn in a piece of paper with at least one thought on the topic." He went to his desk and pulled out a stack of papers. "Here's my most quiet kid. Look at this paper." On the paper was a diagram of light hitting a prism, and then the spectrum. coming out the opposite side. Next to this was a raindrop with the same pattern. Underneath were the words: "How come light comes from both a prism and a sphere?" "What a great question!" exclaimed the Principal. "You bet it is - one I've never thought of. And look at the drawings. This kid's not only shy, but a brilliant artist." "You should run your whole class like this!", announced the Principal in an energetic voice. "Hold on." The smile eroded from the teacher's face, and the Principal at once recognized his error. "Don't forget," Mr. Stephens said, now subdued. "I've got a certain amount of material I've got to get through each class. That's what we're tested on. What I was going to play around with was using these materials the last 10 minutes of class." "Now the sheetrock?" "OK - end of the long story. It's not sheetrock. It's a material called melamene-coated hardboard. You can write on it, erase it, and put it anywhere. As you can see, we only have three smaller chalk boards in the room, so I thought I would quadruple our writing-area! We've come a long way since our Science-ACT materials, haven't we? It's funny, thought the Principal to himself. In this day of computer-aided projectors, here were writing-boards being put forth as a new means of learning. "Indeed you have", he congratulated his teacher. The Principal assisted in the carrying in the boards, thanked and congratulated Mr. Stephens on a job well-done, and resumed his walk. Earlier, he had rained on Coach Thompson's parade, interjecting a solution to the Coaches problem. You could see the excitement drain from the face of the coach, not replaced with emptiness, but a form of resentment and anger. The Principal had recognized the error immediately, and vowed not to repeat it. Shortly thereafter, he had done the same thing to Mr. Stephens! What was going on? He wasn't doing this intentionally; in fact, he was incredibly supportive of his peers. Yet there he was, offering solutions to their problems. It's one thing if he was asked, but he wasn't. How many more times has he done this? How many more times have others done this to his peers?
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Visual Remainders: Pascal and the Mod Squad
May 17, 2008
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When we last talked with our good friend, Pascal, we created an impressive lineage spanning a 1000 years. For now, let's focus on the triangle of coefficients he's widely known for discovering:
Given our tendence to create diagrams, let's see what can be done with this triangle. At first glance, not a lot. There's just a lot of numbers. Something that could be done is recognize which ones are odd, which ones even. Let's suppose we color just the odd ones. What happens?
Of course, the triangle extends far beyond this. This is just a few of the rows. What happens when I extend it?
This is an unexpected response - and one suspiciously similar to our findings here regarding Zeno and The Chaos Game. Now our lineage has gone back, at least analogically, another 1500 years!
Continuing On But why stop at even and odd? Another way of defining "even" is "evenly divisible by two". "Odd", then, becomes a number when, divisible by 2, there is a remainder. What happens when I divide by 3, 4, 5, etc., and ask myself the same question: is there a remainder: yes or no?
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Submitted to the South Dakota Poetry Contest
May 18, 2008
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May 19, 2008
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In 2003, we visited Napa Valley and met Robert Mondavi. I knew little of wine, even less of wine making. Our tour opened my eyes to the unbelievable variables involved in the process of transforming the grape to wine. The grape - the simple grape - is not so simple!
Thomas Oldcastle
What happens, specifically, from grapes to wine? What does different soil contribute to the process? How does the grape "age"? What are the implications of "stressing the vines" for other crops? How does it all work? I left with many questions, and left the following invitation with Robert Mondavi. I sadly did not follow-up. I wish I had. On May 16th, Robert Mondavi passed away. I shall follow-up. To Kalon! (To Kalon is a vineyard owned by the Mondavi Winery. From Harvests of Joy: "In Greek, To Kalon means 'highest quality' or 'highest good.' To me, that meant, simply, The Best."
January 11, 2007 Michael Round Center for autoSocratic Excellence 13234 Long St. Overland Park, KS 66213
Robert Mondavi Robert Mondavi Winery Highway 29 Oakville, CA 94562
Mr. Mondavi: I write with “congratulations” for your wonderful and inspiring book “Harvests of Joy”. I visited Napa Valley four years ago, toured your winery and Opus One, and have since reread your book – with fascination, a thought, and a challenge. What do you think our educational establishment – and our children – would feel if they – our kids – had the same passion for learning as you do wine? Imagine moving from the drudgery of “learn this / learn that” to one of “stressing the wines”!
MY GOAL The Center for autoSocratic Excellence is developing a series of mini – Math / Science textbooks spanning a wide range of subjects demonstrating the ease “complex” subjects can be learned easily, well, rapidly, and with joy. All children – our children – can and should be eagerly pursuing math and science with enthusiasm! I have enclosed copies of two of these books for your review and for you to have a reasonable idea of what I think a “New Kind of Textbook” might look like – no more textbooks hundreds of pages long, strapped on our kid’s shoulders like concrete blocks, but rather brief and insightful issues engaging the student. Why do I write? Many fine organizations would answer simply “donate money to our cause”. My motto is more concrete: “provide value in exchange for value”. Become a sponsor of a mini math-science textbook. Have your company name on the cover of quality educational materials at the fountainhead of the rebirth of educational excellence! Demonstrate concretely education and industry can run side-by-side in making our communities and our country better – now and in the future. What might such a textbook look like? Is this a textbook to teach young children about wine? Is this what wine is – to you? Wine is about understanding nature – and working with nature! Can there be anything more educational than this? I would be more than happy to discuss how I would envision your specific book looking, getting your thoughts, and talking further about my goal of publishing 12 of these books this year, and the marketing of them. What companies am I looking to for sponsorship to get this project up and running? There are only two criteria: I. there has to be a great deal of “scientific intrigue” – to me. Understanding nature is what this is all about; II. there has to be a business philosophy of “life is wonderful” – such that I can say, “I would love to work there”.
A relevant story to all of this: I am giving a presentation in Athens in a few months, and the topic is very similar to what I’ve discussed here. My opening example talks about the Spartan commander, Leonidas, heading off the Persians at the Pass of Thermopylae 2500 years ago. The numbers differ, but generally it was 300 Spartans against hundreds of thousands of Persians – and the freedom of the world was at stake! One can visit the Pass now, and see a commemoration marker. With no knowledge of the past, it is simply a marker – something to take a picture of. However, knowing the history, it is a place deserving of the utmost respect, appreciation, and awe! That’s what education is to me – knowledge of the world, leading to an unbelievable respect for the world. I look forward to hearing from you. Contact me directly at (913) 515-3911 if you would like.
To Kalon!
Michael Round Center for autoSocratic Excellence
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Architects of Their Own Future
Chapter 21
May 20, 2008
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Chapter 21 A GOOD SEASON "52 - 45. A tough way to end the season." Principal Ragnar had, of course, talked with coach Thompson immediately after the game-ending loss. He had talked briefly in the halls of school, and again briefly at lunch earlier that day. But this was the first chance he had had to sit down and talk with the coach in depth. "14 - 6. Not bad for a team predicted to finish in last place in the conference." "I've coached for a lot of years, but I've never felt more proud of a team than after that game. I've never seen a team play that hard for the entire game." "Any regrets?" "You mean about not employing some gimmick to stop their center - to get him to the foul line? None." "You just decided to play good defense, huh? It worked. What do you hold him to? 14 points?" "14 points, yes, but 'good defense'? That doesn't begin to explain what we did." Coach Thompson, recognizing the Principal's silence as a request to continue, did. "OK - you've got a great post man. What do you do? Play in front of him? Behind him? Double team him? What?" "All three make sense to me. Which did you choose?" "You're missing the key question!" "Which is?" "Who is he getting the ball from?" "OK - the guards and forwards." "Exactly - and that's where we focused our efforts. Play incredible defense on the ball, so those players have trouble passing in to the post." "It worked. How many turnovers did West Central have?" "23 - you bet it worked. It worked so well I think that's one of the reasons our shooting percentage was down. We were so tired from playing great defense." "And the postman?" "We decided to play behind him - for this reason. One - don't let the passer get the ball to him. If the ball does get to him, make him shoot a short jump-shot. Also, if we're playing behind him, we're in a much better position to rebound. You noticed, of course, he was in foul trouble in the 4th quarter?" "Of course." "A simple thing: 'play behind him', and many good things come from it." "But you wouldn't do that if the post player was shorter - or wasn't very good, right?" The Coach looked hurt - and angered. "Of course we wouldn't. What has this year taught us about the game? There's a lot of thinking behind everything! Play good defense? Nonsense. Set a pick? Meaningless. Ballfake. Useless. The kids must be shown how to do all these things. The same is true of defense. Of everything!" "This season became like a chess-game for me, not because you're always thinking of moves, but because you're always thinking. And it's not thinking about how to come up with gimmicks to address certain situations, but rather thinking about what really matters in the game - and how to achieve this." "What's next for you?" Principal Ragnar didn't need to remind the Coach the future of the school was still in doubt, meaning the future of the team was also in doubt. "That's a good question. Since the game, I've had calls from three schools, one small school and two larger schools across the state." "What are you going to do?" "I'm going to help my two seniors, that's what!" "What do you mean?" "There were a number of basketball recruiters in the crowd Friday. Of course, they weren't there to see us - they were there to scout West Central and their post player. But a few of them came to the office after the game, wanting to talk about our players! They said they had not seen such a well-played and hard-fought high-school game in a long time." "Scholarships?" "At least partial, and possibly full rides. Division II schools, but scholarships!" "And their academics?" "If you had asked me last summer, I would have thought 'not a chance', even if they were offered a scholarship. Sure, their grade point averages were above 2.0, but they were not really good students." "And now?" "And now, one's carried a 3.2 this year, the other a 3.5, and they both scored well - of course - on their ACT; a 26 and a 29." The Principal thought back to the loss. "Any issues with fans regarding the loss?" "A few. Most people, as you could tell, were great. They cheered after the game, and the homecoming was wonderful." "And others?" "There were a few who said things like, 'had I only fouled him', or 'play zone against a big man', and a couple other suggestions." "You don't let it bother you?" "Of course it bothers me! If I were them and this was last year, I probably would have done those things! But then I began to realize what sports were - at least at this level. You don't win at any costs. There are necessary conditions to be met to play the game well." "So what bothers you?" "I once felt as they do, but don't now. They're our fans. How do I get them to believe what I'm saying - what I'm doing?" "A good question - for next season." "To the future!" Coach Thompson handed Principal Ragnar a bottle of water, and the two bumped plastic bottles in the air.
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ANNOUNCEMENT The 2008 goal of finished pages was 2,500. On May 21st, unofficially, the 1,000-page barrier has been breached! |
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May 21, 2008
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I've defined prime numbers from a geometric perspective, and graphed them from a spiral perspective, but not addressed the question: is there a biggest prime number? Euclid's proof regarding this question is considered to be the essence of conciseness in proofs, so let's see:
Euclid's Proof "There are an Infinite Number of Primes" Assume there are a finite number, n, of primes, the largest being pn. Consider the number that is the product of these, plus one: N = p1...pn+1. By construction, N is not divisible by any of the pi. Hence it is either prime itself, or divisible by another prime greater than pn, contradicting the assumption.
What does this mean? Why can fellow mathematicians praise this as marvelous when I can't picture what's being said? Surely, these aren't Euclid's exact words - his words would be in Greek. This is someone's translation of Euclid.
Maybe it would help to go to the source.
Euclid and The Elements The relevant passage comes from Book IX (Applied Number Theory), Proposition 20. Let's see for ourselves what Euclid had to say, and go to the source:
Obviously, as English is meaningless to Euclid, Greek is meaningless to me. There is one noticeable difference, however. Euclid used diagrams in his proof. The translation above didn't. I wonder why.
A More Direct Translation OK - let's try another translation: a "literal" translation of Euclid's words, side by side, and see if this helps me:
Sadly, I'm still in the dark. I think my understanding is clouded by my initial reading. Why would I consider N = p1...pn+1? Where did this number come from? What is the relationship between this product and Euclid's line?
I shall have to fight myself to see the light.
So let the fight begin!
In Search of a Starting Point Let's simply enumerate some primes: 2 3 5 7 11 13 17 ... What about 1,244,679? What about 42,693,888,221? It's clear there are a lot of numbers out there. I could never count them all. So a proof must proceed indirectly; that is, I'll assume there are a finite number of primes, and, in the course of the proof, hope to expose a contradiction. In doing so, I'll invalidate my claim (finite number of primes), thus proving the infinitude of primes.
A Finite Number of Primes If there are a finite number of primes, that implies there is a greatest prime. Let's call it n. Let's construct a special number greater than n and see if it's prime.
Interesting Implications: In Search of a Contradiction
The Contradiction Revealed The logical implications of our process leads us to conclude we've found a number greater than n which is prime. How can this be if what we said earlier was the case; namely, n itself was the highest prime? The implications of this contradiction? Our initial claim (there are a finite number of primes) is invalidated. Therefore, there are an infinite number of primes.
Putting it all Together
Thoughts on My Proof It's certainly not as concise as Euclid's or other interpretations. In the process of trying to flush out the meaning of the proof, a number of thoughts floated to the surface.
1. The distinction between prime and composite numbers. It was necessary to know all numbers can be reduced to prime number factorization. Did Euclid take this into account in The Elements? Indeed. This issue was addressed in Proposition 31 of Book VII;
2. Where did the idea come from of multiplying primes together, and then adding '1'. Could I have done that myself? After all, this is the crucial link in the "proof-chain";
3. My logic is via branches above. Euclid's was a combination of narrative and lines. Other interpretations are narrative only. What other possibilities exist? For example, how can the idea of "rectangularization", or a new definition of prime numbers, assist?
As always, many new questions!
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May 22, 2008
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How Can the Horrors of a "Lack of Freedom" be Visualized?
A Consistent Philosophy What is the role of government? Politicians simultaneously hold positions where we have no business meddling in the affairs of Iraq, yet demand we intervene in Sudan. We're told diplomacy commands we engage the UN to deal with North Korea, yet act unilaterally with Iran.
The existence of such contradictions suggests the failure of politicians to hold a consistent foreign policy. What constitutes such a policy? How - if at all - should a government intervene in the affairs of foreign countries? Is war the inevitable state of relations among men - or is it a symptom of a disease? What are the roots of war? If men want to oppose war, it is statism that they must oppose. So long as they hold the tribal notion that the individual is sacrificial fodder for the collective, that some men have the right to rule others by force, and that some (any) alleged “good” can justify it—there can be no peace within a nation and no peace among nations. Ayn Rand “The Roots of War,” Capitalism: The Unknown Ideal
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Architects of Their Own Future
Chapter 22
May 23, 2008
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Chapter 22 FREEDOM TO FOCUS ON THE CORE PROBLEM The Fifth Chautauqua
The school year was extraordinary. Students' ACT scores were at the 99th percentile nationally. The basketball team missed the state tournament by one game. Classroom learning was up, by all measures. But how do you measure "happiness"? How do you quantify "joy in learning"? How does a report card capture the renewed energy and passion of not just the students, but the teachers? He thought of the innovations many of the teachers had employed as the school year progressed. Drama and Science getting together to write a screenplay on Archimedes? Math and Literature collaborating to re-write a crucial scene from The Adventures of Tom Sawyer? Unheard of, in the past. But was it revolutionary? Interdisciplinary learning? The idea was not new. Far from it. In fact, the idea was as old as education itself. Everything is interrelated, after all - why not teach it that way? But if the sense is common, why is the practice not? He rounded a blind corner on the walking path in time to avoid a speeding bicyclist and hollered "Slow Down!" to the youth. Why is what not common practice? A great teacher knows everything is interconnected. If one knows what is right, you do what is right. Right? Since most teachers don't, why not? Principal Ragnar stopped by a opening to a slow-moving creek, and tossed several pebbles, one by one, into the creek, hearing the voices of teachers. They came easily. "I was trained to teach math - how can I integrate this with other courses?" "And even if I could, where do I get the time to meet with the other teachers?" "And even if I had the time, how is this going to show up on my state assessment scores?" Several other complaints came to mind, the result of all indicating reasons why interdisciplinary education is the exception and not the norm. The teacher has been placed in a tremendous dilemma:
The Principal recognized the familiar tone of the dilemma. It was similar to the one he had shown at a public meeting months earlier regarding foreign language and the dilemma the school faced. The public feedback had been astounding. Sure, he still fielded complaints from parents demanding French, Spanish, and Chinese be taught. It was the volume and tone of the sympathetic letters which most startled him. Not only did these people say they now understood the problem schools were in, but they said they faced the same problem in their occupations. From computer programmers to civil engineers to a restaurant waitress, they all gave wonderful examples of how they were "between a rock and a hard place" and "why doesn't anyone understand". What was clear, in the teacher dilemma above, is unless the constraints of the teacher in the classroom are recognized, it served no purpose providing them "more great things" to help with their job. New curriculums, new "downloadable lesson plans", new teaching methods, and yes, innovative methods to make the curriculum interdisciplinary served no purpose to the teacher striving to improve test scores within a rigid curriculum created at the district level. This, of course, explained why so many good initiatives were not used in more places. The quality of a bottle of wine is irrelevant when the glass is already full! This did not explain, however, how Washington High had developed and used many new initiatives! Or did it? The Principal strode along the creek's edge, marveling at the diversity of life, of designs, of nature. The acorn. The blooming flower. The flowing stream. It was all marvelous. What had Washington High done to get to where they were now? They were now doing many things from an interdisciplinary perspective, but this ability was the effective of a lot of hard work - not the cause. It would have been ridiculous to ask teachers to coordinate lesson plans when performance was low. It was only because performance had improved they were able to do now what they were doing. What did they do first? They recognized the obvious, of course. Unless one improves significantly, there is no free time to do what they were doing now - everybody would be too busy "putting out fires". The ACT had taught them this, initially. There were so many easy points missed on the exam. Let's find a simply solution to ensure this "low hanging fruit" did not go to waste. But in the course of their investigations, they also realized things were made artificially hard for the students. Myriad "test-taking" tricks were offered to the student under the guise of "help", but they really did the opposite. Was there a way to simplify the processes? That's what they had done - with the reading section, the math section, and the science section. Allow the student simple techniques to establish a "foothold" in the material, and performance increased exponentially! This focus on a "lever point" had, of course, done something unexpected. They knew, for example, with the ACT real learning had not been accomplished, despite the immense increase in scores. Hopefully, they had exposed the ACT - and other such tests - as irrelevant measures of intelligence and knowledge. But while exposing the "ACT-emperor" as having no clothes, something else extraordinary had happened. The focus on the lever-point had so improved performance - from a test-taking perspective - teachers now did have the time to really develop lesson plans with deeper meaning. Rather than focusing on grading math quizzes with the same mistakes, for example, teachers had time to better understand temperature scales, the derivation of the relationship between Fahrenheit and Celcius - algebraically and geometrically. They had the chance to work with the science department to understand what it was that was rising in the tube. Such interaction would have been impossible had students not been performing as they had. The conflict, then, between interdisciplinary and mono-disciplinary, wasn't so much a conflict between these two. This was a core-conflict giving rise to the core problem and many other undesirable effects in the system. The conflict and the constraint dealt with time. And now everybody had more of it.
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The Viability of the Computational Universe?
May 24, 2008
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One-Dimensional Elementary Cellular Automata Single-Cell Starting Point
One-Dimensional Elementary Cellular Automata Single-Cell Starting Point Interesting Results: Cell-Colors Inverted
RULE 30 Random Start with Probability of Life: 50%
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May 25, 2008
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THE "CLASS 3" ABERRATION Two brothers raised on the science-fiction genre of The Matrix, the 13th Floor, and Contact search for evidence of the computational world. What they find transports their emotions from eagerness to despair, and back to exhilaration when their search for proof of the digital world culminates instead in the discovery of the meaning of life! Stay tuned for this 3-part science-fiction mini series.
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An "Alfred Nobel" Siting - in Nebraska / 2008
May 26, 2008
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Nitroglycerine was invented in 1846 and used as an explosive. Unfortunately, it was very volatile. The question: what do you mix with nitroglycerine to maintain its explosive force, yet reduce its volatility? The answer is "Alfred Nobel in Nebraska".
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Function Maximums and Minimums
May 27, 2008
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A common calculus question is: what is the maximum value of the function f(x) = 2x2+3, or any such function. Calculus students readily and quickly go through the process, but what is the rationale of the process? Let's see. One thing I know is, when there is a maximum (or minimum), the graph changes direction. It either was going up - and now it's not, or vice versa. Drawing a tangent line at this specific point where direction changes results in a horizontal line. Tangent, we know, means literally "to touch". So, to find this "highest point" requires me to find where the tangent line is horizontal. And since tangent is really "rise / run", I'm looking for the point where there is no "rise"; that is, tangent = 0. Let's take a function and see how this would work: consider f(x) = x - x2
Carrying Out the Procedure So, to find this "highest point" requires me to find where the tangent line is horizontal. And since tangent is really "rise / run", I'm looking for the point where there is no "rise"; that is, tangent = 0.
Further Thoughts Questions I have for the Calculus Student (high school or college):
1. What are the x-values called that give rise to maximums and minimums?
2. A function changes directions (and hence, has a slope = 0) at BOTH a maximum and a minimum. How do we differentiate between the two?
3. Does every function have a maximum and a minimum?
4. The function y = x^2 has neither a maximum nor a minimum, when defined over all values of x. If I restrict the domain to [-2,2], how do I calculate the function maximums and minimums?
5. If a function has a maximum or a minimum, the slope of the function is zero. If the slope of the function is zero, does this mean there is a maximum or a minimum at that value?
6. How would we modify the above statements to take into account a function where the derivative does not exist? What is an example of such a function? What is an example of a function with a maximum occurring where the derivative does not exist?
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May 28, 2008
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The poet, Edna St. Vincent Millay, wrote EUCLID ALONE, presumably as
praise for achievement of unparalleled proportions. As Euclid’s
investigations into prime numbers are often sited as a work of brilliance,
I’d like to investigate his work, and do some of my own.
Euclid Alone by Edna St. Vincent Millay
Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone.
Are there an Infinite Number of Primes? Euclid is known as the father of geometry, written in The Elements. However, this work includes not just geometry, but many other mathematical items, including number theory. In this section, Euclid proves many things about prime numbers. His proof on there being an infinite number of prime numbers is considered a wonderfully elegant proof. We’ll see later. However, I’m not concerned about that now. I’d rather see for myself just how many prime numbers there are – and how to find them. I’ve not calculated any myself – outside small numbers under 100. How would I go about developing an algorithm to find prime numbers? Let’s be clear what a prime number actually is. Consider the integer 10. How many ways can I multiply two positive integers and get 10? 5 x 2 = 10 2 x 5 = 10 10 x 1 = 10 1 x 10 = 10 Because there’s more than two ways to reach 10, 10 is not prime. The definition of “prime” is “the only factors of the number are the number itself and 1”. That’s what I see above. Let’s build this into a programming algorithm. The Making of the Algorithm
This seems to capture well what I did above when checking to see if 10 was a prime or not. But I see a problem immediately. Let’s say I start at ‘1’, and check to see if ‘13’ is a prime. I calculate 13 / 1 = 13, see there’s no remainder, and, by the above algorithm, assume it’s not prime. The same is true for the upper end of the checking process. If I calculate 13 / 13 = 1, I arrive at the same wrong conclusion. Therefore, to avoid this problem, instead of looping from 1 to ‘my number’, I’ll loop from ‘2’ to ‘my number - 1”.
Refining the Process My process above works perfectly. Now, how can I improve it? What else do I know about integers? I know all even numbers are even because they are divisible by 2 with no remainder. I could check immediately to see if the number is even: if it is even, I know instantly the number is not prime. But more than this, if I have eliminated all even numbers, I don’t need to check to see if ‘my number’ is divisible by an even number. That is, I can skip-count by 2s starting at 3, and essentially eliminate ½ of the checking!
Refining the Process: A Continuation Well, I’ve cut my calculations by more than ½ with these simple modifications. Is there anything else to be done? I notice my table above, showing the factoring of ‘10’, two of the sets of factors are “2 x 5” and “5 x 2”. I notice if I check one combination, I do not need to check the reserve combination. But what is the point at which I can stop? Where do the factors reverse themselves? Once I pass the “square root” of “my number”, the number I’m checking for primeness, I’m through. Is ‘121’ prime? Once I get to 11 x 11, I do not need to continue! Adding this into my logic, I arrive at the following structure:
How does all of this logic accelerate the calculations? Suppose I was checking to see if 100 was prime. Initially, I made 100 calculations. Now, I immediately say ‘no’, because it’s even. What about odd numbers? Suppose I was checking to see if 121 was prime? Initially, I made 121 calculations. Now, knowing the square root of 121 = 11, and skipping the even numbers, I check only 3, 5, 7, 9, and 11. Instead of 121 calculations, I now only make 5! Applying this algorithm, let’s see how fast I can find all the prime numbers between 1 and 10,000. Prime Numbers from 1 – 10,000 “less than 1 second in Excel”
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May 29, 2008
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One-Dimensional Spherical Cellular Automata Single-Cell Starting Point
One-Dimensional Spherical Cellular Automata Single-Cell Starting Point Interesting Results: Cell-Colors Inverted
One-Dimensional Spherical Cellular Automata Random Starting Point Interesting Results: Cell-Colors Inverted
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The Second-Most Misunderstood Rule in Baseball
May 30, 2008
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An INFIELD FLY is a fair fly ball (not including a line drive nor an attempted bunt) which can be caught by an infielder with ordinary effort, when first and second, or first, second and third bases are occupied, before two are out. The pitcher, catcher and any outfielder who stations himself in the infield on the play shall be considered infielders for the purpose of this rule. When it seems apparent that a batted ball will be an Infield Fly, the umpire shall immediately declare “Infield Fly” for the benefit of the runners. If the ball is near the baselines, the umpire shall declare “Infield Fly, if Fair.” The ball is alive and runners may advance at the risk of the ball being caught, or retouch and advance after the ball is touched, the same as on any fly ball. If the hit becomes a foul ball, it is treated the same as any foul. If a declared Infield Fly is allowed to fall untouched to the ground, and bounces foul before passing first or third base, it is a foul ball. If a declared Infield Fly falls untouched to the ground outside the baseline, and bounces fair before passing first or third base, it is an Infield Fly. Rule 2.00 (Infield Fly) Comment: On the infield fly rule the umpire is to rule whether the ball could ordinarily have been handled by an infielder—not by some arbitrary limitation such as the grass, or the base lines. The umpire must rule also that a ball is an infield fly, even if handled by an outfielder, if, in the umpire’s judgment, the ball could have been as easily handled by an infielder. The infield fly is in no sense to be considered an appeal play. The umpire’s judgment must govern, and the decision should be made immediately. When an infield fly rule is called, runners may advance at their own risk. If on an infield fly rule, the infielder intentionally drops a fair ball, the ball remains in play despite the provisions of Rule 6.05 (L). The infield fly rule takes precedence.
This is quite a rule! Unfortunately, it's size hides a lot of the rationale as to why the rule exists, expands, is refined, etc. Actually, this is the story of all rules!
Let's play with this to see if we can uncover all of the crucial elements of the rules, so we could call it correctly, if we were the umpire. In fact, let's assume we're the umpire, since our judgment apparently governs! So judge well!
A typical scenario: runners on 1st and second, 1 out. A pop-up is hit. The runners, knowing it will be caught, stay close to their bases. The pitcher, knowing this, catches the ball, intentionally drops it, throws to second for one out, and back to first for another. An easy double-play.
There must be a rule to do away with "easy double-plays", so the thought was in 1895.
But is that a correct sentiment? "Easy"?
There are lots of easy double plays in the game. A line drive is caught by the pitcher, and a runner on first, running with the pitch, is easily doubled off first.
Should we do something about this?
There is a difference, though, isn't there? In the latter case, the hitting team had a choice, while in the former case, the runners had no choice. If they run, they're out for not tagging up. If they stay, they're doubled-up with an intentional drop.
A thought, as we're now both umpires and on the rules-committee.
We can't reward a team for "actions contrary to good baseball".
How's that for a general principle?
If a team drops a ball intentionally, that's not good baseball - don't reward them.
But what to do about it?
Protect the runners by declaring the batter out, whether the ball is caught or not. That is, runners on first and second, one out, and there's a pop-up. The umpire immediately calls: Infield-Fly: Batter's Out".
This takes away the force play if the fielder drops the ball.
What's the rule, then? First and second, one out?
What about bases loaded, one out? Bases loaded, no outs? There are lots of other scenarios. What are the key elements?
We're trying to remove the automatic double play, so there needs to be less than two outs. Also, in removing the double play, there needs to be a force at third.
Is that our rule? When there's less than two outs and a possible force at third, and there's a popup, the umpire calls: Infield Fly, Batter's Out.
Is that it?
Can the runners ever run?
What happens if they do run, and the ball is dropped?
The umpire's ruling merely provides them safe-haven at their own bases. All other rules still apply. They can run, but must tag up. If they run without tagging up, and the ball is dropped, they're fine - stupid, but fine.
But who is catching the ball? ANYBODY - provided, in the umpire's judgment, there is grounds to believe there will be a simple double-play with the ball intentionally dropped.
If the ball is hit to an outfielder in the outfield, yes, the umpire can call "infield - fly: batter's out".
Why don't we heat this latter call often, then?
Because balls hit to the outfield allow the runners to slightly get off their bases in the event the ball is dropped. If the ball is dropped, surely one runner will be thrown out, but likely not two. Therefore, no advantage would be gained by the defensive team, and consequently, no call is made.
The Infield-Fly Rule: The second most misunderstood rule in baseball.
What's first, you might be wondering?
Interference!
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Architects of Their Own Future
Chapters 20 & 23
May 31, 2008
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Chapter 20 A SCIENTIFIC FOOTHOLD
Mr. Stephens proudly looked at his classroom, now surrounded by writing boards and logic diagrams. "How will his students perceive the change? How had they perceived the change?"
It had been quite a year.
He had not liked Principal Ragnar stepping into the Principal's job. He himself had applied, and been angered the school board had hired Principal Ragnar. Sure, thought Mr. Stephens, Principal Ragnar had experience. He was also retired.
Mr. Stephens had also not liked the initial focus on the ACT. He disliked focusing on tests, test-taking strategies, and other such gimmicks. He liked doing experiments, teaching kids about how science and nature works, and the lab. How he loved the lab.
Now, in these early morning summer hours, he had instead found himself in front of math and English teachers trying to figure out how to improve scores on the science section of the ACT, led by a man he now disliked.
He had disliked the new Principal even more now.
When two non-science teachers had scored poorly (22 out of 40, 46 out of 40) on the science section of the ACT, Mr. Stephens had attempted to show them how to read the graphs to understand the questions when Principal Ragnar had interrupted. Mr. Stephens remembered the exchange well:
Principal: "Are you saying to do well on these questions, all we need to do is be able to read a graph well?"
Mr. Stephens: "The graphs summarize the science experiments, so 'yes', you must be able to read the graphs."
Principal: "But the graphs could be summarizing anything, right?"
Mr. Stephens: "I guess so."
Principal: "So you really don't need to know much science to answer these questions correctly? You just need to be able to read the graphs? Is that what you're saying?"
Mr. Stephens: "Wait a minute: science is a lot more than that!"
Principal: "Don't get mad at me - get mad at the ACT!"
He had thought a lot about that exchange over the past several months. A good portion of the ACT, he had found out, required little knowledge of science. After all, all of the data necessary to answer the questions were right in front of the student. One with the ability to read graphs well and understand text could perform remarkably well.
But there was more than that. He found, investigating the graphs, the presentations themselves were horribly poor visual depictions of the data under consideration. He found the legends, for example, had his eyes bobbing back and forth, from dashed line to legend back to dashed line, trying to make sense of the several lines embedded on the graph.
He had also found a commonality with the English Department in this regard: the student was dropped into the middle of an experiment, with no background of the study, the experiment, or the explanation, and told to "make sense" of tables of data, various experiments, and narrative explanation. It was no wonder most students performed so poorly on this section of the test.
Performance could hardly be said to be dependent on one's scientific background. More than likely, he thought, kids with a good science background did well simply because they were generally smarter!
Unfortunately, he was at a disadvantage with his material as compared with his literature-counterparts in working with the ACT. Whereas the reading section consisted of four passages of 10 questions each, the science section consisted of several passages, each with merely 5-6 questions each.
This had meant more blocks of data to understand.
Yet, he still fought a fight of 40 questions in 35 minutes.
How to make sense of it all?
The goal: get the student quickly grounded in the material. A foothold. Someplace - anyplace - that allows the student a starting point.
And what a journey it had been for them and him!
Sadly, he had quickly realized how little science the students did have to know to do well on the science portion of the exam. Excitedly, he realized the simple methods he had devised to assist in taking the test also helped in his classroom.
Most important was his new mindset on his classroom. He could always tell when few, several, or most students did not understand a topic well. He had always had difficulty in knowing what to do at this point. Lots of techniques were at his disposal, of course, and this myriad of methods had not seemed to help.
Now there was a goal: the establishment of a foothold - anything - to get the students started. Most often, he had found, it was a metaphor. Other times, it was simply the repetition of the same experiment. Some methods had worked well - some failed miserably.
But his class was alive with learning!
Why?
He knew why.
He spent little time answering mundane questions from his students. He spent even less time regrading papers on repeated topics. Those days seemed long ago.
He spent much more of his time researching, understanding, and discussing extended topics with his students.
He himself was alive with learning!
Chapter 23 A TEACHER GET TOGETHER
Principal Ragnar was in the process of re-introducing the invited guests, Charles Jones and Michael Anderson of the ACT, to the teachers assembled for their weekly staff meeting, when Mr. Stephens, walking across the floor with his cup of coffee, said loudly, "Remember to tell them there is such a thing as a dumb question - in fact, most questions are dumb!"
Stunned, Mr. Anderson asked, "I'm sorry - what do you mean by that?"
"See! There's one right there!"
"Mr. Stephens," replied Ms. Hendrickson, the English teacher. "What is going on with you today?"
"All right." Contained in those two words was an apology, albeit meek. He turned to Principal Ragnar. "Do you remember our conversation about, gravity, weight, and the solar system? About radio signals penetrating buildings but not mountains surrounding valleys? About these questions now coming from my students? These weren't questions in addition to the usual questions I use to field; these were questions replacing the trash I use to answer."
"And your point is 'what'?"
"My point is: 'what happened to the questions they once needed me to answer'?"
"And?" "The answer is 'I don't answer them anymore.'"
"Why not?"
"Because the questions don't get asked anymore."
Mr. Anderson lurched into the conversation with a statement posing as a question: "They don't ask them anymore because they already know the answer, right?"
"There's another one. No, it's not because they know the answer, it's because they can figure the answer out."
"What's the difference?"
"Another one!"
"Another what?"
Mr. Jones chuckled, making Mr. Anderson more angry. "This wouldn't be so funny if you were the one talking."
"But I'm not the one doing the talking, you are!"
"Do you understand what he's talking about?"
"Nope - but I'm sure not going to ask him with you on the stage!"
Mr. Stephens relented: "Look, guys: let's suppose I've got a physics problem with a truck weighing 3 tons. However, I need to convert this to kilograms to insert the number into my equation. How do I do it?"
The looks on their faces told Mr. Stephens something he already knew. "You're not sure weather to divide or multiply - and by what, right?"
"Do you think my students were any different?"
"Do you know how often I was asked 'Mr. Stephens? Do I divide or multiply by 2.2? Mr. Stephens? When I'm given kilograms, I multiply by 2.2, right'?"
"And now they remember the formula, right?" He immediately withdraw his question, and responded with an "I see." He waited patiently for Mr. Stephens to continue.
He was granted relief. "No, they don't necessarily remember the formula. However, we created little algorithms to make sure they could test to see whether what they were doing was right or not."
"And with this algorithm, the students now don't ask any of the questions they use to. They don't get any of these problems wrong like they once did. And this same method applies to all things we do with units, in addition to other places."
"So why are most questions dumb? There, I asked another one!"
"That's just become my mantra, because I tire of the
old adage, 'There's no such thing as a dumb question'. Well, look at
all the questions the students were asking, which they don't now.
They now have the ability to figure out the problems for themselves.
You tell me: what does that make the questions they once asked?" "Does Stephens get to monopolize the meeting?", balked Mrs. Frederickson? I'd like to talk about my visit to the museum.
"Fine - Fine - if you two want to talk further, let's wait a bit. Mrs. Frederickson, you have the floor."
"Is this the museum visit for the Impressionist tour? I thought you talked about this at last month's meeting, and decided you weren't going", said Coach Thompson.
"That's what I thought at the time. But the reason I thought as I did came about by thinking of the results of other visits my classes have made in the past. Had any kids actually learned anything? Maybe. Have they even had a good time? Maybe. And so when we've been focusing on not wasting time, and effective classroom time, I realized this was wasted time."
"So you didn't go? I thought you did go!" blurted Principal Ragnar.
"Would you men let me finish? I thought about not going, but then imagined the uproar if I canceled." Principal Ragnar silently concurred. He had thought the same thing earlier. He was glad it wasn't his call. Now he wondered what call Mrs. Frederickson had made. She continued: "So I gave the kids the choice, and had their parent's sign off on the choice."
"You gave the kids ..." Mrs. Frederickson raised her hand to cut off Mr. Stephens. "I gave them the choice, but with one additional thought: they had to write a paper on something of the Impressionist movement if they were going."
"You mean if they weren't going?", Mr. Stephens interrupted again.
"Now why don't you stick to your mantra. Here me out! As I said earlier, I've had many museum visits - or trips in general. We all have. Let's be honest. How much do the kids really learn? Or take in? And compound this lack of genuine learning with the hassles of logistics, and I wonder why we even do them. But let's just stick to the museum. There's so much there to learn - to see - the kids - anyone - is easily overwhelmed by it! And what happens? There's so much to see you actually see very little. Maybe those aren't the right words. You see a lot, but it has no meaning. So my thought was, 'How can I get the kids to get to the museum and focus on something specific?' That's where the idea of doing a report before-hand came from. With so much research available on the internet anymore, doing a research project takes little time - it takes even less to see the actual art-works anymore."
"So when they got to the museum, they had something to focus on - be it works of Van Gogh, or Degas, or a specific piece of Monet. They all had something they could go to and see - to focus on."
"And what of the students who opted out?"
"I had 5 out of 42 who opted out. I had them sit in a study hall and work on their own homework and assignments."
Mr. Anderson chimed in: "Didn't having 37 reports to grade take up a lot of your time?"
"Who said anything about grading them? I didn't even look at them! I just checked off whether they had done them or not. Remember my goal: get the kids focused on something, so when they go to the museum, they had something to start with."
Mr. Jones asked with incredulity: "What is it you're all doing here?"
Principal Ragnar responded, sincerely and immediately: "Were learning, Mr. Jones, and having fun doing it!"
"Do you remember your first visit here? I was in a history class? That was one of our first experiments. What we asked was something like, 'if the teacher is really doing a good job, they ought to be able to explain the material back to the teacher.' So we tried a few things, and didn't like how they worked. The problem was the teacher who had taught the material was too biased in the dialogue - they dominated it too much. So we thought, 'what would happen if we brought in a different teacher for the class, a teacher who knew nothing about the class materials but could ask good questions. As the teachers were busy, I volunteered for the job."
"And?"
"And we found a couple of things: we found where there were tremendous gaps in teaching and learning. We found where things were working very well. We found it was very easy to get kids to talk when the teacher was different than the one presenting the materials. It's amazing how good the students can become in presenting, debating, discussing, and merely talking in front of people when they're given the task of doing that."
"Do you do this all the time?"
"No. I don't have time for this, and our teachers don't either - at least right now. However, what we have done is have all of our substitute teachers do this. Rather than spend class time working on materials already prepared by the regular teacher and taught frantically and chaotically, we have the substitutes merely go over old material, and see how well the class knows that material. The substitute then reports back to the regular teacher on items maybe needing additional work."
"What else have you tried?", said Mr. Jones.
"I think Mr. Stephens was right about questions. Listen to you. We've just given more concrete examples that have been actually been tried than many schools do over the course of a couple years, and you ask 'What else?' You should be asking, 'How have you done this?'"
"OK - how have you done this?"
"You mean, 'OK - how have you done this without a single dime extra being spent', right?"
"That's right."
"To paraphrase a commercial line, we've recognized time, like the mind, is a terrible thing to waste."
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