Probabilistic Shifts
Alysia helped me realize that I need to describe “probabilistic shifts”, a term I use to describe a certain thought pattern that plays an important part in the way I think of the world. The term came from the way runoff flowing through grass reminds me of a Galton machine.
For an example, see http://www.youtube.com/watch?v=5_HVBhwhwV8.
The Galton machine is a model showing how thousands of instances, each unpredictable, can accumulate into a predictable pattern such as the “bell-shaped curve” of the normal distribution. Each bead cascading down can bounce either left or right at each level, theoretically ending up anywhere along the bottom. However, probability says it will usually end up somewhere near the middle (in the same way that if you flip ten coins, you will probably get between 4 to 6 heads even though 10 heads are possible).
In the same way, rain running off the land is like the marbles flowing down a slope and every grass stem is like a peg that keeps splitting the runoff to left or right. In this situation, however, I’m interested in how the Galton machine-like grass stems, through a series of probabilistic events, spread some of the runoff way over to the outer edges of this slope’s distribution curve, over far more surface area than if the grass stems were not there.
I like to help spread it out even more so that it flows more slowly. By shifting a rock high in the channel, I change the probabilities of how many water molecules will continue along the main path and how many will shift over onto paths more to the side. Though I can’t predict what each water molecule will do as it flows past the shifted rock, I have shifted the probabilities at this point and I can predict with absolute certainty that more water will end up over to the side. I have absolute confidence because I have watched it happen many times. If I move this rock up here, more water flows this way and several seconds later, I can see the water begin to rise in this pool off to the side. If I move the rock back to its former position, the pool soon begins to recede; its inflow has been reduced.
I call these moves “probabilistic shifts” because they are small shifts that simply alter the probability, at this one point, of how much goes one way, how much the other. But I know with absolute certainty that this will shift consequences downstream. This experience creates unstoppable optimism. If I shop at Costco rather than Walmart, more of my money is flowing into the pockets of employees. I can’t track it but I know with absolute certainty it is because of the different employment practices of the two organizations. “More of it is flowing to the 99%.” One does not need certainty to have certainty (if that makes sense). One does not need certainty that every molecule in the flow will be changed by my play to have certainty that the outcome of the flow will change. One does not need every shopper to switch over to organic food to cause changes in how food and money flow within this world. We make little changes, continually, knowing that they will create shifts (albeit small) downstream.
The world is like this flow of water. Each of our lives is like a molecule of water flowing through this world. Millions of choices, conscious and unconscious, are like the splits in the cascading flow shaping our lives and world. These little splits can seem random and therefore insignificant but they can accumulate into significance in an absolutely deterministic way. This is what I am doing when I go around Chrysalis before school starts greeting as many of our students as I can. Creating little shifts in the energy flowing through the emotional atmosphere at the head of the day.
We give up some of our power if we limit ourselves to only a few major decisions or if we don’t act because we think it won’t make a difference. There is a profound difference between “not making a difference” and “making a difference.”
The Width of Water
Both my editors thought I should drop this section but this is me practicing how to communicate part of an idea that I am still wrestling with that I believe might become important. Feel free to skip over it at any time.
A stream’s width is always changing. In some places, the stream narrows and in other places, it broadens out. This change of width, easy to see, connects our mind to many other aspects of the flowing stream.
The easiest place to enter these connections is with the simplistic but still functional equation for stream discharge (how much water is flowing along a stream):
Discharge = width x depth x velocity
Width x depth is basically the familiar formula for finding the area of a rectangle (length x height). Width x depth give us the cross-sectional area of the flow. Velocity tells us how fast the water is flowing through this area. The distance the current can flow in a second gives us the length which when multiplied by the width and depth gives us the volume of a rectangular prism representing the volume of water flowing by in a second. In the United States, the units we measure by are feet x feet x feet/second = cubic feet per second or cfs.
This formula is simplistic because a stream’s depth is not uniform so the cross-sectional area is not rectangular. Furthermore, a stream’s velocity is not uniform. The outside of a river bend flows swiftly while the inside of the bend might actually be eddying upstream. However, this simple formula is good enough to give us a powerful introduction into changes happening within flow.
The key thought experiment is to imagine a stretch of stream where no additional water is joining the stream or leaving it. In that case, the discharge remains the same throughout that stretch. If the discharge remains the same, then that means that width x depth x velocity at one place must be equal to width x depth x velocity at another place along that stretch.
If the stream narrows to half its width at one point, then the product, depth x velocity must double at that point for discharge to remain constant. Depth could double with the velocity remaining constant or the velocity could double with the depth remaining the same. More likely, both the depth and velocity will increase somewhat so that the product depth x velocity doubles.
Width, depth, and velocity are interconnected in a fundamental way. This malleable interplay between the three is what you see when you hang out around streams. A change in one absolutely requires the others to change. Drop a big rock in the middle of a current, thereby diminishing the cross-sectional area of the current and the water will deepen and speed up as it races around the rock. All three stream variables adjust to the change. More peacefully, if one floats down a river, one notices that when the river speeds up without narrowing, the river bottom is only a few feet away whereas when the river slows down with no change in width, the bottom is way below.
So depth and velocity are the first flow attributes connected with stream width. The connection between width and velocity, however, also connects width with the stream’s gradient. If the slope steepens, the water will speed up. This increase in velocity must create a decrease in the width x depth. The channel will become narrower. On the other hand, if the stream gradient flattens out, the current will slow down which will require that the width x depth to increase. So when we see a stream channel narrow or widen, it often signals that the stream’s gradient is steepening or flattening in those places.
But now, a stream’s velocity plays an exponential role in a stream’s kinetic energy. KE=1/2 mv2. Kinetic energy is one half the mass times the square of its speed. If a flow doubles in speed, its kinetic energy quadruples. Kinetic energy can pick up material and carry it downstream. Kinetic energy gives the flowing water the wherewithal to shape the land which will give rise to several feedback spirals.
For example, if a streambed steepens, the current speeds up, increasing in kinetic energy, allowing the stream to start eroding that area, carrying fragments from the streambed downstream. When the gradient gentles out, the stream slows down, losing some of its kinetic energy. It no longer contains sufficient energy to carry all of its load so some of the fragments drop out, creating deposition. Therefore, steep areas become places of erosion that are worn down and flatter areas become places of deposition that are built up. The steep stream sections grow less steep while the flatter sections grow more steep. The stream develops toward a smooth continuous gradient called stream equilibrium.
This process can be watched in miniature. One place is where a sandy bottom stream opens into a deeper, quiet pool. Sand grains come rolling along the streambed. When they come to the pool, the increased depth slows the water and the sand grain stops moving, raising the streambed so that the sand grain coming down behind it can now roll past that dropped stream grain where it stops rolling. Over time, this process builds an underwater delta at the head of the pool.
I love watching tiny deltas an inch high that are actively growing one sand grain at a time. A sand grain comes rolling along to the abrupt downstream edge of the delta where it rolls over the edge and tumbles down the inch high steep slope and comes to rest at the bottom of the slope. The next sand grain tumbles afterwards and comes to rest next to that sand grain. Those two grains together now allow a following sand grain to come to rest on top of them. A steady stream of sand grains fall over the edge and come to rest on the steep slope below. Gradually, a layer of sand one grain thick builds across and rises up the slope until it reaches the top, to the level of the incoming stream bed. Now the incoming sand grains can roll one sand grain length further before dropping over the edge and starting to build up the next layer (called a foreset bed). Here’s a diagram on the web: http://www.denniskalma.com/river/deltaxsect.gif
This interplay between the current and individual sand grains is a very peaceful process to watch. The change in stream depth changes the sand grain from tumbling to coming to rest but the grain’s coming to rest then causes the stream depth to change which affects every sand grain coming thereafter. Each sand grain follows a different path because of the sand grain that came before it.
But this movement towards “stream equilibrium” affects the other variables. When the stream slows down and drops some of its load, for example, the dropped load raises the streambed, decreasing the depth. Therefore, with both the speed and the depth decreasing, the width must increase. On the other hand, when a stream steepens, speeds up and erodes, it lowers its streambed, increasing its depth. The stream width must decrease. So the oscillations in stream width that we see tell us where the channel is cutting down and where it is building up. It reveals all the small alternating sections of erosion and deposition by which sands and gravels are being moved from section to section over time. When we start seeing a stream in this way, then we start seeing a stream’s widest place being the dynamic beginning in the stream’s narrowing and a stream’s narrowest place as the beginning of the stream’s widening.
The ability of water to shape the land creates feedback spirals. The head of an eroding gully eroding its way upslope is like a black hole, bending runoff towards it, increasing rates of convergence so that even more runoff flows through that section, increasing its mass and kinetic energy. Is the shape of the deepening gully shaping the stream flow or is the shape of the narrowing stream flow shaping the gully?
On the other hand, a widening channel leads to deposition which then fills in channels, forcing the water to flow over an even broader area which slows the flow still more and an alluvial fan begins to build. Water flows broadly and slowly through an alluvial fan so more soaks in here. Plants grow thick, sponging up even more of the runoff, slowing the runoff even more, as even more of the sediment comes to rest among the plant stems. Roots spread upwards into the accumulating sediment; the alluvial fan grows lush green, spreading and slowing the water even more.
Life’s ability to do a Galton machine spreading and thereby slowing runoff alters the entire stream equilibrium of a drainage. By spreading and slowing the runoff, the plants allow the slopes to rise more steeply against the forces of downward erosion. The land rises steeper and higher – or rather the land is worn down more slowly.
The land, its soil, the stream, its width, its depth, its speed, its energy, its load, its steepness, and life – all fit together to form a writhing brown and green serpent of flowing mass and energy slithering over the land, with cause and effect undulating back and forth.
Circular Drainages
Somebody drove onto a several-acre muddy field on the Chrysalis campus and spun around, creating loops of tire tracks that act as channels for unoff. That led me to go out during a rain storm to see if I could help heal some of the damage. It’s become a quite delightful challenge, trying to apply principles derived from nature’s converging drainage systems to a human-created looping system.
Cranes Calling Overhead
A couple of weeks ago, a flock of about one hundred sandhill cranes, heading north, circled low over the school for fifteen minutes. I announced it over the PA and most of the Chrysalis classes came outside to watch. To me, a flock of sandhill cranes make one of the most primeval sounds of them all and I was delighted that our youngest students were able to hear a sound that I never heard until my twenties. (If you have never heard sandhill cranes, please listen to this audio file because the sound is an important part of the story. http://www.allaboutbirds.org/guide/sandhill_crane/sounds – a great bird site if you haven’t discovered it already.)
About the same time the next morning, another flock circled low over the school for ten minutes on its way north. So the next week at our school-wide Tree Assembly, Sandhill Cranes were our Species of the Week. They inspired me to give a talk that went something like this:
I want to share a story with you. It’s a story about sandhill cranes but it’s also a story about why Chrysalis is a nature school. Some people ask why we take time to take kids out into nature when they should be learning reading and math. What’s so important about nature?
This story is from when I was a young man in Alaska. It was early September, which in Alaska is autumn. Days are dramatically growing shorter. The stars can be seen again. Frost at night. The tundra plants have all turned reds and yellows and orange. I was hiking to the Wickersham Wall, the greatest vertical rise in the world, three miles from base to height.
I hear, high above, the sound of sandhill cranes. I look up but I can’t see them. I keep walking; the sound of cranes grows. I look up with my binoculars and finally I see them, way up there, a flock of about a hundred cranes flying in loose V’s. I love their wild cries calling down to me in this beautiful, wild land. But their calls, they sound broader than that flock. So I keep searching with my binoculars and I see another flock, and then another, and another. There are about ten flocks forming a giant V of a thousand cranes calling high overhead. By now, I’ve taken off my pack and am sitting down looking up at this spectacle. The sound is so vast. So vast. I keep looking around and, my god, there is another giant V of a thousand. And another. And another. These giant V’s form a vast wing of ten thousand sandhill cranes stretched across the sky. I gaze upward as they pass over and gradually recede to the southeast.
I sling my pack back on and continue walking. About 15 minutes later, I hear the sound of cranes approaching. And the same experience unfolds. Ten thousand cranes majestically, exuberantly streaming overhead in a great wing made of giant V’s made of flocks and fading into silence.
I continue on. 15 minutes later, I hear again the vast sound of ten thousand cranes drawing near. This time, after sweeping over the vast flock with my binoculars, I direct my gaze beyond this flock in the direction that the cranes have been coming from and behind this flock, I see the next great wing of cranes following and beyond them, smaller in the great distance, the next great wing, and beyond them, another great wing, and beyond them, remote but coming this way, yet more great wings of V’s of flocks of cranes from the wild.
A great river was flowing overhead. A great river of cranes was converging from its headwaters which lay in millions upon millions of acres of tundra from which arose each year a new generation of cranes to join this river that had flowed for millions of years. This is what I was part of. I was part of a world where one hundred thousand cranes pass overhead fueled by the same oxygen that I was breathing. This amazing world, this is what we are part of. We are surrounded by other living things doing amazing things, showing us that we, too, are capable of amazing things. And that is why we take you out into nature, so that you can be reminded that you are part of something far greater and you have been given this wonderful opportunity to play a part in it.
Scientific Notation – size of the universe.
My eighth grade math class got to the mathematics of scientific notation, especially how huge multiplication problems can turn into simpler additions of exponents. When the curriculum shifted over to addition with scientific notation, a deviation from the students’ expectations led them, in response to a problem involving adding planetary distances, to an answer larger than the size of the universe without them even wondering if their answer made sense. This led me off into the relative scale of things. I used this link (http://htwins.net/scale2/?bordercolor=white) and started moving outward. We discovered that the entire observable universe was three orders of magnitude smaller than the answer they had put down
This led me into a series of thought problems each evening that I shared with them the following mornings. The final thought problem was this. So, a hundred years ago, the best astronomers in the world thought that our galaxy was the entire universe. Everything observable was within the “confines” of our galaxy. Our galaxy is 10 x 1021 lightyears in diameter so that is the dimension within which the best astronomers of that time thought. Now, we are thinking the observable universe is 10 x 1027 light years across. How much bigger is this universe than the one a century ago?
This turns into: how many times bigger is 1027 than 1021? We find this by dividing the first number by the second. Because of scientific notation, this turns into 10(27-21) which is 106. That is 1 with 6 zeroes after it. It’s 1,000,000 times further to the edge of the universe now than astronomers thought a hundred years ago. However, space is a volume and since we measure space in three dimensions, the width must be cubed to give us the volume. The kids had learned that when a power is raised to a power, you multiply the exponents so one million cubed is (106)3 which is 1 x 10(6×3) which is 1 x 1018. That is 1 with 18 zeroes after it. So we have found the answer to our question in a couple of minutes. The universe that the astronomers of our present time think about is 1,000,000,000,000,000,000 – one quintillion – times larger than the universe that the best astronomers of a century ago thought about.
Eighth Grade Analogies
Years ago I read an article stating that all the pre-college tests like the SAT were not correlated with achievement in college and beyond. Only one test did predict it and that was the Miller Analogy test. Perhaps analogical thinking is important in some fundamental way so I started giving my eighth grade students analogies to complete on their Weekly Pages. This quickly gave way to having kids create their own analogies. A few years ago, I started sharing the better ones in Chrysalis’s weekly newsletter. Below are a selection of analogies I’ve recorded from the last two and a half years. I share them for three reasons. One is in hope you find them as delightful as I do. Second is because they serve as windows into the eighth grade mind. There is a lot going on in there; that’s why I love teaching eighth grade. (There’s a common image that junior high kids are unpleasant. That’s not my experience. I think that what people find unpleasant is seeing the unpleasant things within our culture beginning to be practiced by adolescents. They mirror their environment. In a caring environment like Chrysalis, they are very, very wonderful.) The third reason is to contemplate why an analogy test would be correlated with achievement better than other standardized tests. What was going on within the students as they created these? (If you don’t get one of them, think about it for a while.)
Minion is to villain as partner is to hero. Josiah
Amelia Earhart is to planes as Michalengelo is to paintbrush. Sean
Orca is to wild as goldfish is to domestic. Jake
iPhone is to iPhone case as acorn is to shell as organs are to body. Hannah
Flag is to country as mascot is to school. Caitlin
Rock is to paper as scissors are to rock. Josiah
Cast is to broken arm as ice cream is to broken heart. Sean
Pandora’s box is to mystery for Pandora as Odysseus’s bag is to mystery to his crew. Will S.
Stain is to clothes as crack is to windshield. – William S.
Sword is to close range as crossbow is to long range. – Lukas
Pop is to quiz as unprovoked is to attack. Cameron
Door is to out as ladder is to up. Damian
Black hole is to matter as censorship is to free speech. Will H.
Elevator is to stairs as first class is to coach. Josiah
Kermit the Frog is to Miss Piggy as Ken is to Barbie. Mackenzie
The Constitution is to “Whatever the King says” as democracy is to monarchy. Caleb
Toothbrush is to teeth as Q-tips are to ears. Austin
Black Death is to fleas as West Nile is to mosquitoes. Sierra C.
Tired is to sleep as flat tire is to air. Ellie
Hunch is to investigation as spark is to fire. Andrew
Soy sauce is to Chinese food as ketchup is to American food. Kaela B.
Hatred is to heart as gallons of oil are to the ocean. Kendra
Tears are to the soul as a broken dam is to a river. Kendra
Beauty is to art as butterfly is to butterfly net. Brianna-
“You can do it” is to encouragement as “Santa Claus doesn’t exist” is to disappointment
Gladius (a Roman sword) is to cavalry saber as stab is to slash. Marshall
Ticket is to concert as passport is to country. Ellie
Bad sound effects are to sci fi as sparkles are to fantasy. Marshall
A rumor is to young friends as a crashing wave is to a beach. Violette
ab is to ba as 6+4 is to 4+6. Bella
Words are to language as snow is to silence. Isabella
Shark is to deep waters as tiger is to tall grass. Violette
The Magna Carta is to England as the Constitution is to America. Jack
Hammer is to build as sledgehammer is to break. Ian
U.S. is to U.S.S. as U.K. is to H.M.S. Jack
Rain is to clouds as leaves are to trees. Violette
Just learning how to play with all of this.
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