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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?
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