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Sunday, December 7, 2008

Change of Plans

So there's been a change of plans since the last post. My mentor teacher decided that my lesson on proofs was going to be way over the students heads. I am inclined to disagree, but it's her class, so I'm happy to comply with her wishes. I'm going to teach a lesson on the Pythagorean Theorem instead. As an activity, I made some of those knotted ropes the ancient Egytians used to survey land. We're going to scout out the foundation of a pyramid and hopefully find some Pythagorean triples in the process. After that, I'm going to introduce the actual equation and do the following informal proof.

Each of the four right triangles are of equal size with area equal to


The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C2. Thus the area of everything together is given by:

However, as the large square has sides of length A + B, we can calculate its area as (A + B)2. We can expand this to A2 + 2AB + B2. So

4(½AB) + C2 = A2 + 2AB + B2

2AB + C2 = A2 + 2AB + B2

C2 = A2 + B2


So that's the plan for now. I teach the lesson this coming Friday. I'll let you know how it goes. Right now, I'm off to take the Praxis I. Wish me luck.

Monday, November 24, 2008

My First Lesson Plan

So I am about to teach my very first lesson ever. My mentor teacher is giving me the reigns of her 8th grade math class for one day and she's letting me talk about anything I want. Since I am only teaching one lesson which in no way required to link to their current unit, I have made a crazy decision. When I was back in school, the biggest road block for me in math was the sudden and unforeseen appearance of proofs in geometry. It has always bothered me that we wait so long to introduce this subject. I understand that students may not be cognitively ready to handle this concept until then, but I still feel that if we wait until they are 100% ready, then we've probably waited too long. I think it makes more sense to start broaching the subject as early as possible. To that end, I have decided to do just that. The following is a handout that I have prepared for my lesson. I do not intend to read this word for word, but these are the concepts I want to cover and in this order. If anyone is still reading this, let me know what you think regarding scaffolding, differentiation, and all those other buzz words. (Note: The true/false "quiz" at the end will be handed out immediately after the introductory remarks and before the discussion of statements.)

Mathematical Proof

The word proof means something a little bit different to mathematicians than it does to normal people. This is not surprising, since we know that anyone who willingly chooses to study math is odd, to say the least. For the average person walking down the street, if a statement seems reasonable and comes from a reliable source, then the statement is assumed to be true. Let’s say you turn on the evening news and hear that President-Elect Obama has officially selected his Secretary of State. There’s no obvious reason why you shouldn’t believe the story. We know that the newsroom carefully checks all the facts, and we were expecting Mr. Obama to begin choosing his cabinet soon anyway. So we are willing to accept the statement as true.

Other people would be a little more suspicious. Scientists, for example, test a hypothesis by performing the same experiment many times. If the result is always the same, no matter how many times they repeat the experiment, then they say that the hypothesis must be true

Mathematicians are the most stubborn of all. For them, it doesn’t matter how many times the experiment is run or how many supporting examples there are. There is always the possibility that if you run one more test or check one more example, that the hypothesis will prove false. So instead, what they do is carefully build an argument that proves the statement has to be true in all possible cases. This is similar to the way a lawyer might build a case or a film critic might write a review.

In this lesson, we will take a closer look at how mathematicians build proofs and why it is important for us to learn them.

How do we decide what to prove?

Before we can set about building our argument, we have to decide what it is we are trying to prove. That usually begins with a statement. A mathematician defines a statement pretty much the same way as everyone else. It has to be a declarative sentence. For example, which of the following are declarative sentences?

What are we having for dinner?
Birds have feathers.
I love math.
Bao, please close the door.
The square root of 9 is 3.

The first sentence is a question. Questions sometimes lead to interesting discoveries, but there is no way for us to prove a question by itself. The fourth sentence is a command, also known as an imperative. It tells someone what to do. Again, there is nothing for us to prove. The rest of the sentences are all declarative.

We also require that the sentence have an objective truth value, meaning it is either true or false. If the sentence states an opinion, there is no way for us to prove it true. Which of the following declarative sentences are statements?

I love math.
The square root of 9 is 3.
September 12, 2027 will be a Friday.
Mr. L’s lesson is boring.

The first and last sentences are matters of opinion. There is no way to prove them true or false, no matter how strongly you may agree or disagree. Both of the middle two sentences are statements. The second sentence is obviously a statement, and we know immediately that it is true. What about the third sentence? This is also a statement, because we know that it must be either true or false, even if we don’t immediately know which one.

Now that we know how to recognize a provable statement, let’s look at how to begin building our argument.

How do we build an argument?

There are a lot of different kinds of proof strategies, but the one you will probably use the most is the direct proof. In a direct proof, you start with something you know to be true and build the argument one fact at a time until to reach the desired conclusion. How many facts you use depends on how complicated the statement is and who your audience is. It is a good idea to design your proof for someone who knows less about math than you do.

The facts you will use to construct your proof will basically come in three forms: definitions, axioms, and theorems. A definition is just what it sounds like. Let’s say you want to prove a statement related to a triangle, you must first know what a triangle is. An axiom, sometimes called a postulate, is a statement so basic that it doesn’t need to be proven. For example, the statement “the product of any number multiplied by one is equal to the original number (x · 1 = x) is an axiom. That’s just the way multiplication works. There’s no way for us to prove this; we just have to assume that it’s true. A theorem is a statement that has already been proven, and math textbooks are full of them. Once a statement has been proven true, we are then free to use that statement to prove other things.

The tricky part is deciding what facts to use and in what order to put them. Let’s look at the example of the next page.

What if you were asked to prove that dogs are mammals? Which of the following sentences might you use to build your case and why?

Dogs have a backbone.
Dogs have four legs.
Dogs have either fur or hair.
A mammal is a warm-blooded animal that has a backbone.
Mammals are the only animals that have hair or fur.
All dogs go to heaven.
Dogs can learn over 200 human words.

Choosing the right facts to use in your argument is trickier than it might seem. Sometimes you will come across a true statement that is related to the topic, which doesn’t help you prove your point. For example, the last sentence is true. Scientists have demonstrated that some dogs can respond to more than 200 different commands, but that doesn’t help us decide if they are classified as mammals. The same is true of the fact that dogs have four legs, since we know that lizards have four legs but are reptiles. Each line of your proof should move you closer to your goal, but don’t be afraid if you hit a dead end. When that happens, just go back a step or two and take a different path.

Once you have some idea of the kinds of facts you will need, you have to figure out where your proof should begin. Sometimes that will be obvious by looking at the statement you are trying to prove. For example, look at the statement “If the student gets an A on the final exam, then the student will get an A in the class.” In order to prove this is true, we can begin by assuming that the student gets an A on the final exam. But sometimes, the starting place is not immediately clear. In that case, it becomes a judgment call. Where you choose to start may depend on your audience. If you are proving something to your teacher, you might choose to start differently than if you were proving something to a classmate. Generally, it is a good idea to assume that your audience knows almost nothing about math.

Let’s look at an example. What if you were asked to prove that ice cubes float in water? Where might you begin? It would probably be a good idea to make sure that you explain what we mean by the word float. So a good starting point would be the following statement: “When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.” Once we have that, we can start adding more details. The following sentences, when placed in the correct order, will complete the proof. In what order do you think they should go?

1. When we say an object floats in a particular fluid, we mean that if we were to drag that object to the bottom and release it, it would be pushed to the surface.
2. Since 917 is less than 1000, we know that ice cubes float.
3. The density of liquid water is 1000 kg per cubic meter.
4. For objects to float, the object must be less dense than the fluid.
5. The density of water in solid form (ice) is about 917 kg per cubic meter.

We have already said that the first sentence will be our starting point, so we can set that one aside. The second sentence restates what we are trying to prove, so it makes a good conclusion. Now we just have to worry about the other three. According to the fourth statement, in order to prove that something floats, we must first prove something about its density. We don’t even need to know what density is, just that the object’s density is less than water. Statements three and five give us key information about the density of both water and ice. So we may conclude that the statements belong in the following order: 1,4,5,3,2, You could make the argument that statements 5 and 3 could be flipped, which is certainly true. However, it is a good idea to match the natural order given in the previous statements. Since both 2 and 5 mention the density of ice first, we should put statement 5 before statement 3.

What does any of this have to do with math?

So far we have been avoiding dealing with actual math. Now that you know a little more about proofs, we can look at an actual mathematical example. The following proof deals with a topic that is not normally presented until Algebra II. DO NOT BE ALARMED!!!
None of what we have been talking about is usually covered until Geometry, so you’re already way ahead anyway.

Before we can deal with the proof, we must first look at the following theorem.

Theorem A: For any number x , xa · xb = xa+b.

Remember, a theorem is a statement that we already know to be true. Somebody has proven it for us and put it here for us to use. But we still need to make sure that we understand what it is saying. Let’s look at an example.

Example: x3 · x4 = x3+4 or x7

We know that x3 = x · x · x and x4 = x · x · x · x
So we are saying that x3 · x4 = (x · x · x) · (x · x · x · x) = x · x · x · x · x · x · x = x7

Once you understand what the theorem is telling you, it isn’t that scary. You have already been working a lot with x2, and even though these powers are larger, the idea is exactly the same. Now that we understand how to apply the theorem, we can use it to prove other statements. For example, let’s look at how to prove the following:

For any number x, x0 = 1.

You are probably starting to get a headache. You know what x3 means. That’s just x multiplied by itself three times. We can do the same with x4 and x5, but what in the world do we mean by x0? DO NOT BE ALARMED!!! This is exactly the same question that mathematicians once asked themselves. Then they answered it with the following proof:

To Prove: For any number x, x0 = 1.

We are told that x is a number. This tells us that we can apply Theorem A to the problem since that theorem is true for all numbers. Begin by raising x to some arbitrary power, say x7. Then using the theorem, we can rewrite x7 as x7 · x0 since

x7 · x0 = x7+0 = x7.

We know by equality that x7 · x0 = x7. In other words, if we multiply x7 by x0, we get back x7. There is only one number that when multiplied by another number returns that second number. That number is the number 1, therefore, we have proven that x0 = 1. ▪

The little ▪ symbol at the end of the proof shows that you are finished. You do not have to use that symbol. You could simply write “The End” if you like, just as long as your reader knows that you are finished proving whatever you are trying to prove.

So far, all of our proofs have been written in paragraph form. Although there is nothing wrong with this, in school we commonly use an organizational device known as the two-column proof. As you’ve probably guessed, they have two columns. On one side you have a list of statements and on the other you have the reasoning behind them. The following is an example of what the previous proof would look like as a two-column proof:

Argument Reason Why

1. The unknown x is a number. 1. We are given this in the problem.
2. x7 · x0 = x7+0 2. By Theorem A.
3. x7 · x0 = x7 3. By addition.
4. x0 = 1 4. By the multiplicative identity. ▪

Are we done yet?

The short answer is “yes.” You now know some of the basic concepts of mathematical proofs, and even though you won’t see any of this again until high school, you can start using some of the ideas now. Whenever you are asked to solve a word problem, you can practice using some what you have just learned to defend your answer. You can also use the idea of proof outside of math class. When you are writing essays in language arts or social studies, you can use the same steps to build your argument.

The remaining pages of this handout are full of examples for practice. You will be asked to pick out provable statements, put statements in proper order, and even to build a few simple proofs. The more you practice now, the easier this will be in the future.

Name__________________ Sequence___

1) Give an example of a statement. (Remember that for it to be a statement, it must be provably true or false.)

2) Put the following statements in order to prove that water is a liquid at room temperature.

Room temperature is defined as 70° F.
The temperature at which a liquid becomes a solid is called the freezing point.
There are only three states of matter: solid, liquid, or gas.
Since 70 is greater than 32 and less than 212, water must be a liquid at room temperature.
The temperature at which a liquid becomes a gas is called the boiling point.
The freezing point of water is 32° F.
The boiling point of water is 212° F.

3) Fill in the blanks with the appropriate letters.

To Prove: If 4x + 10 = 38, then x = 7.

We begin by assuming that 4x + 10 = 38. By the subtraction property of equality, we can subtract 10 from both sides of the equation. (_____) Then using the (______), we can divide both sides of the equation by 4. (______) Therefore, we have proven that if 4x + 10 = 38, then x = 7. ▪

a. multiplicative identity
b. This gives us 4x = 28.
c. This leaves us with x = 7.
d. division property of equality
e. This gives us 4x = 38.

4) Convert the finished proof from Exercise 3 into a two-column proof.

Argument Reason Why

1. 1.
2. 2.
3. 3.
4. 4. ▪

5) Prove that if x + 3 = 15, then x = 12. (Hint: Look at Exercises 3 and 4.)

6) Prove that the area of the inner square is exactly half the area of the outer square.
(Hint: Don’t be afraid to draw more lines if it will help.)

7) Congruent angles are angles that have the same measure. Prove that if the two horizontal lines are parallel, then angle 1 is congruent to angle 8. ( Hint: Use alternate interior angles.

8) Prove the three angles of a triangle sum to 180°. (Hint: Draw a line through C that is parallel to line AB, then think about supplementary angles.)

Name_______________________ Sequence_____

For each question, circle the correct answer.

1) Give an example of an integer. True or False.

2) How do we simplify 5x + 3y – (-2x) ? True or False

3) Portland, Maine is the best city in the US. True or False.

4) The first three questions are impossible to answer. True or False.

Tuesday, November 18, 2008

The Other Maine: Faces of Homelessness

The following is a reflection on a series of articles recounting tales of homelessness in the state of Maine.

This is the sixth time I’ve started this reflection. I’m too angry to know exactly how to begin, but the due date is fast approaching, so I’d better get something on paper. The articles on Maine’s homeless children have induced a state of frustration in me so powerful as to dampen my regular flow of wit. The situation seems hopeless to me. One of the best and worst things about being human is that our ability to act deliberately in total disregard for natural instinct makes us think that we can “fix” laws of nature. Try as we might, we can not legislate away the fact that all systems have selection pressures and that not every member of that system is going to survive.

When I moved here to Maine, I had to get a new driver’s license. I also had to get a dog license, which I had never had before, despite have pets my whole life. In the same office, I saw applications for hunting licenses, fishing licenses, business licenses, and marriage licenses. You need a license to start a fire in city limits, to broadcast on a radio station, or to practice law. But anyone with a working set of genitalia can have children. We don’t get to license that. It’s a natural right endowed upon us by the universe. But it’s a right that carries with it a tremendous amount of responsibility and therein lays the problem.

The children are blameless in this struggle. As a social species, we feel compassion for them, knowing full well that they are the victims of their parents’ bad choices. We want to help, so we pass laws or enact assistance programs. But nothing works. The system is simply too big with too many cracks. The price of progress is that our family group is just too extended for us to help one another anymore. We are largely on our own and some of us are bound to fail.

It is a question of inheritance and education, really. For example, by monetary standards, I am neither rich nor poor. I live from paycheck to paycheck, and though I am comfortable now, I am one disaster away from having to renegotiate. But I have a huge safety net underneath, because no matter what happens, I always have my inheritance. Don’t misunderstand; there is no money to be had. I am not to be the beneficiary of some familial fortune. My inheritance is the power than comes from a superior education. It is my firm belief that if I were to be stripped naked, blindfolded, and dumped anywhere on the globe, that I would have the requisite skills to quickly rebuild a life. Nothing short of massive head trauma can steal that from me.

It occurs to me that my exception has proved the rule, that even an eternal “have” like me could be turned into a “have-not” with a quick crushing blow to the occipital lobe. So how do I respond to that realization? Can we chalk these scenarios up to bad luck and count our own blessings or are we obligated to help in any way we can? I suspect our humanity obligates us to the latter course. As admirable as that instinct may be, it dooms us to a certain amount of frustration and failure. I have chosen to combat this social ill, along with all other systemic malfunctions, in the only way I know how- as an educator. A good education can provide a measure of relief that no government assistance program can. It is the only solution I see as being effective, so that is how I choose to do my part.

Monday, October 20, 2008

Whose History?

Those who can not learn from history are doomed to repeat it.

Same shit, different day.
-Steven King, Dreamcatcher

Why do we study history? Scholars assure us that history informs both our present and our future, that while the universe is vast with possibility, mankind tends to tread familiar and well-worn paths. Great generals painstakingly recreate battles waged beyond living memory to better understand the nature of warfare and to prepare for future engagements. Scientists use data gleaned from the past to predict tomorrow’s reality. To be sure, the past constantly nips at the heels of the present. But is that really why we so dutifully record our stories for the historians of tomorrow? Perhaps, it is. Maybe humanity, the only species on the planet known to understand its own mortality, compiles these complex annals for practical reasons. But I doubt it. It strikes me that a far more primal imperative is at work. In short, we love a good story.

Since before history was history, men and women have been telling stories. We tell stories about the gods and about the heavens and about the creatures of the earth. But mostly, we tell stories about other people. In evolutionary terms, we have been singing our own praises since we strayed from the safety of the trees and started roaming the African savannahs. History has mostly been an oral tradition, passed down from generation to generation as campfire tales and bedtime stories. Apprentice bards learned their craft from tribal elders as tales were honed and polished to suit the tongues of the tellers. It was very much more an art than a science, and a certain creative license was expected and encouraged. Then along came the written word, and suddenly that which was ethereal and fleeting achieved a degree of permanence that changed the discipline forever.

Once a story gets written down, it is much harder to edit. Books can be burned and edicts decreed, but some vestige will always remain. With this realization, the study of history took an egocentric turn. To the victor go the spoils, and no spoil is of greater importance than the ability to calcify one’s own version of the tale. This incontrovertible truth lies at the heart of every multicultural historical debate. Truth, like beauty, is in the eye of the beholder. The history textbooks in our classrooms typically have a lot of ground to cover. The authors are limited to one version of each story, and too often, that version is the only one most students will ever hear.

To truly understand history in a way that will move mankind toward enlightenment, we must be willing to listen to all sides, to construct our truth from all the facts at hand. As educators, it is our duty to teach our students to think critically and take nothing for granted. But that does not mean throwing out the historical baby with the bath water. Historians like Howard Zinn would have us simply substitute one half-truth for another. It is the responsibility of a scientific observer to remain as impartial as possible, to acquire objective evidence before drawing any conclusion or backing any agenda. I have read Zinn’s A Peoples History, and objective it is not. There is a clear and unapologetic agenda of tearing down heroes and championing the downtrodden. If history is a popularity contest, Zinn is backing the kid with coke-bottle lenses and acne.

According to Alejandro Segura-Mora, teachers are “cultural workers” who “can, and should, challenge white supremacist values.” I agree with this in as much as I believe we should teach our students to question everything, including ourselves. It is unfortunate that students expend so much effort parsing out the “right” answers. I am lucky enough to be training in fields like mathematics and physics, where subjectivity rarely comes into play, and “right” answers are even possible. For the rest of the world, such a concept is meaningless. For historians and poets, if you think you know the answer, you probably don’t. Here in America, where we argue black and white, it is especially easy to overlook the myriad shades of gray. As the current bearers of the mantle of imperialism the stretches back for millennia, we must force ourselves to take a hard look in the mirror. To those who mock opponents of the PATRIOT Act as overly dramatic, I offer the Alien and Sedition Laws and the Japanese Internment. To those who celebrate the low prices of Wal-Mart, I reply with sweat shops and abusive child labor. These comparative histories are no less powerful for my having heard them before. They are essential to our understanding of civilization and the stories must be told.

What we should not do, what I refuse to do, is to promote any one set of values of another. My job is to teach mathematics, and it is not an easy one. I have no problem with demonstrating how mathematical problem solving can be used to inform cultural debates, but it is not my place to inject my own politics, even if they be the politics of multi-cultural awareness and equality. Our goal should be to strip away bias, not to replace it with our own. As much as I despise the dogma of “white supremacy,” I would prefer that they at least be white supremacists that can model linear equations, follow statistical arguments, and demonstrate abstract reasoning ability. To that end, I will resort to whatever strategies I deem effective, including allegedly “sexist, racist, culturally insensitive, and contemptuous” games like Oregon Trail.

There is little doubt in my mind that this popular simulation exhibits each of these qualities, but then so does history. I freely admit to spending many hours attempting to cross the Great Plains in my digital wagon train, although in the interest of full disclosure, I never made it to the Oregon frontier. On the occasional attempts where I avoided falling victim to dysentery, my adolescent male fascination with violence sidetracked me toward extended squirrel hunting expeditions. Even so, I enjoyed learning through the game for no other reason than that it was fun. That’s why we play games after all-for fun. Part of that fun comes in the challenge of winning. If the game can not be won, the wind slips out of our sails. No one would want to play a game called Trail of Tears. Slave Trade 2 won’t be flying off the shelves. Imagine spending countless hours watching your computer avatar lying side by side with countless others in the underbelly of a slave ship with the ultimate goal of arriving triumphantly in the cotton fields of the agrarian American south. How is that supposed to facilitate the love of learning? It can’t. So I guess we’ll just have to settle for the “contemptuous” Oregon Trail and remember that it is only a supplement to a balanced curriculum, not a primary source.

The issue of cultural bias is unavoidable in history class, but it exists in one form or another in all subjects. Though we can never eradicate those prejudices completely, by discussing them openly and honestly, we can significantly curtail their influence. As teachers, we should content ourselves with sparking that debate, knowing that it is often more important to ask questions than to find answers.

Sunday, September 28, 2008

In Quotations

The following is part of a class assignment. Quotes are pulled from Ordinary Ressurections by Jonathon Kozol.

“Liberals may think that they can contradict a stereotype,” he notes,” by
walking off into a neighborhood where they do not belong” and may believe
they’ll “be protected” by their ideologies or sentimental loyalties. “It
doesn’t work like that,” he says, “and you’d be unwise to believe it.” (p.167)

There is a famous story in mathematical circles concerning the death of Pythagoras. It is said there was an uprising against his mysterious cult, but that the venerated leader escaped the original attack only to be cut down later, due to his refusal to cross a bean field. It seems amidst their reverence for number and theorems, the Pythagoreans harbored beliefs in the most extraordinary of things. Because of its passing resemblance to human sex organs, the legume was deemed indecent. Pythagoras, according to this legend, was killed because of his steadfast refusal to be touched by such “vile” plants.

Nowadays, our intellectual elites, our modern-day Pythagoreans, are sometimes slain because of where they are afraid not to go. For fear of appearing racist or bigoted, they disregard stereotypes with celebrated flair. With the kind of passionate ignorance of reality that stems from early sequestering within the walls of the Ivory Tower, they literally and metaphorically walk where they do not belong.

Yet they should take note of one of mathematics less superstitious methodologies- the field of logic. It is one thing to say that not all group members fit a stereotype, and another entirely to say that none do. Behind every cultural stereotype lies a probabilistic heuristic that has proven true enough to survive. Our protection comes not from naively ignoring those stereotypes, but from being able to see past them.

He doesn’t just “sustain” the difficulties of existence. He steers around
them in inventive ways that give him the defense he needs…He never seems like
someone who’s agreed to be defeated. (p. 236)

There is much talk of the “resilience” of children among those schooled in the social and psychological sciences. The term conjures images of the downtrodden, of those who are forever being peeled from the tread of society’s boot. We marvel at how these young people are able to persevere in spite of such a relentless attack. In reality, we give them too much credit, and at the same time not enough. We are right to be surprised by their ability to rise up after being hit by such continuous volleys. No human could be expected to “sustain” those difficulties. Yet while we overestimate their recovery strategies, we underestimate their ability to avoid the attacks altogether. Their “resilience” comes not from standing strong before life’s frontal assault, but from their keen awareness of the likely places of ambush and the ingenuity to escape around the danger. In so much as they manage to avoid the obstacles, they demonstrate their ability to play outside the rules. They employ strategies that boggle our minds, because for all our well-meaning ideologies and good intentions, these children are the ones actually playing the game. We are merely spectators.

I know nothing of theology; but it occurs to me that modest hesitations-normal
ones, like those in ordinary conversations- may allow a bit more space than a
relentless speaking style does for people in a congregation who may feel the
world has tried to clip their wings and that the powers and the principalities
of their society might actually prefer it if they didn’t fly too high. (p.246)

Regardless of the chosen medium, a true artist knows that there is as much power placed in what is not said as in what is. The blank patch of canvas, the silent rest signs of a great composer, the soft pause of a seasoned orator- these are the nothings and nowheres that make the lasting impressions. In that little white spot, the artist invites the audience to fill in the blank- to insert a piece of themselves into the work. This is a lesson of which we as educators would do well to take heed. People are willing to be lead, but will steadfastly refuse to be pushed. We must remember that each lesson belongs not just to us as teachers, but the students as well. In order for our artwork to be remembered through the passing of time, we must not be afraid to put brushes in the hands of our audience.

Monday, September 15, 2008

Profiling 101

Over the weekend, I had to create an academic profile for myself using data gleaned from a series of questionnaires. One was called the GREGORC inventory, which I had never heard of before. Another was regarding Gardiner's multiple intelligences, which I had heard of before. The rest were short answer questions about learning styles and preferences.

I scored comfortable to very comfortable on 6 out of the 8 intelligences. I just missed Bodily-Kinesthetic, but I am very deficient in IntraPersonal. Evidently, I don't care very much about people's feelings. On the GREGORC, I came up Abstract Sequential, which means I value thinking and analysis above all else.

Surprisingly, I found the learning styles sheet the most telling. It consisted of a flowchart of sorts, which laid out all the different learning environments. (noisy, quiet, visual, aural, etc.) I feel quite comfortable with any and all environments save for one- Kinesthetic Mobility Tactile. That category includes such horrors as role playing, mime, and immersion. You can put me in a dark, loud, hot room and I can still concentrate, but five minutes of mime and I'm contemplating murder-suicide.

Well, that's me in a nutshell. (How did I get in this nutshell?)

Friday, September 5, 2008

Where I'm From

So my first assignment for Culture and Community is to right a poem. It is supposed to be modeled on Where I'm From by George Ella Lyon. Here is a rough draft. Feel free to critique.


I am from the streets
(of suburbia.)
From circles to courts and cul-de-sacs.
I am from Big Wheels and lemonade stands,
from broken bones to broken homes,
and always at Pizza Hut.
I am from where the street only goes in one direction.

I am from t-shirts and Bible Belts,
from funnel cakes and Rocky Top.
(Man, I hate that song.)
I am from Frisbee and tenor clefs,
from matinees and tender feet.
I am from sadness and euphoria.

I am from behind the scenes to on-stage,
where everything is backwards.
I am from perennials and postholes,
from what do you want to be when you grow up
to you can't get there from here.
I am from the library.

I am from GA to ME
on foot over mountains.
I am from spotty dogs and smelly socks,
from sunsets to morning dew,
from heavy hearts to ultra-light.
I am from the do-overs to the birds ablaze.
Where the street now goes in both directions.

Wednesday, September 3, 2008

Culture and Community

I had my first education class yesterday, CPI211I: Culture and Community. I thought you'd like to hear about it.

Let's begin the story by explaining that my university has two campuses, 8 miles apart. I think they began as separate colleges that merged at some point. I typically attend classes here in Portland, which is within walking distance from my apartment. But the EDU department is on the other campus in Gorham, ME. Being the eco-friendly fellow that I am, I decided to take the bus that the university offers between campuses, especially since I help fund that bus as part of my student fees.

I hopped on the bus along with 40-50 other people. Every seat was filled and the aisle jammed full of late arrivals. The subsequent 8 mile ride set the tone for the rest of the day. It took 40 minutes to make the trip, which was compounded by the miserably sweltering heat. As I felt my internal temp rising far above comfort level, I read with eyes squinting away the sting of sweat pouring down my forehead, a sign that proclaimed the bus a WiFi Zone. You see, at USM, we have the technology to allow students to receive wireless internet while rolling down the highway at 60mph, but lack that required to build a bus with functioning windows.

I arrived with minutes to spare at took one of the remaining seats in the classroom, which regrettably, was just barely cooler than the bus. It is not hyperbole to say that knowing how that class went, I would have preferred to remain on the bus for that 3 hours.

Almost immediately, we were asked to leave our seats and form a mosh pit in the center of the room. We would spend the next 2 hours performing various getting-to-know-you tasks, beginning with the "human atom." We were asked to move about the room as though orbiting a fictional atomic nucleus. At random intervals, a call to "freeze" was given and further instruction given. At each stopping point we were to grab a new partner and perform a task. The first time, we were to join elbows with someone and introduce ourselves. Next time, it was the knees. The next time, we were to invent a three-part secret handshake. Next, find a common letter between your names and think of three ways to form that letter with your body.

It went on, and on, and on.

At the end, we formed a "truth circle," where we commented on how we felt. People stepped forward to proclaim how relaxed they were and how great it was not to be confined to desks. I stepped up and announced that I was far more tense than I had been before we started.

First impressions are important, and my impression of this class is that it is a complete waste of my time and money. It isn't that I don't get the point of these exercises. In fact, I may have been the only one in the classroom that did. The point was to make connections. Before we can teach someone or learn from someone, we must first establish a connection. That connection will be based on mutual experience or memory or goals or history or whatever. That kernel will provide the foundation for the relationship that will allow for learning to occur. Many of those relationships form a community. This kernel need not be some huge thing; it only has to be real.

Nothing about these artificial activities serve that purpose.

Tuesday, September 2, 2008

So Far, So Good

Just got back from a 2 mile run. I know that isn't much, but I hate running and I haven't moved that fast in a long time. Anyway, I thought I'd blog a bit to celebrate.

As I mentioned, I have my first two education classes this semester. I've read both of the textbooks already (insert nerd joke here) and I think I'm really going to like one, and really hate the second. The first is all about how to better integrate the knowledge coming out of laboratories with the day to day functions of the classroom. It discusses recent breakthroughs in neural networks, epigenetic theory, and a host of other interesting fields that may one day be useful to teachers. Hopefully, that day will come sooner than we think.

That book is entitled How People Learn.

The second class is more touchy-feely. It seems to be about how cultural differences embedded in society affect the achievement of students in the classroom. I certainly understand how this knowledge will prove useful to me as an educator, but based on the textbook, the class seems to go far beyond that. There is definitely a subtext of massaging and molding the culture via education to curb some larger social ills, and I don't think that is the primary job of schools.

But we shall see. Perhaps my opinions will change throughout the semester. For now, I have to pack my lunch in my A-Team lunchbox and head off to campus.

Monday, September 1, 2008

Promises, Promises

Look, I know you have no reason to trust me. I've made promises like this before. I'll do better, I say. I'll write more often, I swear. And for a while, I do. It's just like old times. We talk, we laugh. It's like when we first met. But eventually, things change and I'm back to my unreliable ways.

No more, I say. It's a new semester and the buck stops here. I have two education classes and I expect much debate, discussion, and reason for intellectual reflection. I vow to share every thought, every argument, every tiny detail with you.

And I'm going to start jogging as well.

Friday, July 25, 2008

Unequal Pay

It seems that many school districts are having a hard time attracting qualified math teachers. Gee, I wonder why that is? Could it be that those potential educators are having a hard time turning down the six figure salaries that Google or AT&T are offering them? I seriously doubt there's nearly that kind of disparity between professional historians and social studies teachers.

I can't even count the number of times I have advocated raising teachers salaries. All teachers need to be paid more for what they do, but we ought to bear in mind the realities of the situation. Marquee players bring in a larger chunk of revenue and so they get paid a larger salary. Well in the world of education, STEM grads are the marquee players. I'm not saying this is fair; I'm saying "fair" is irrelevant. If the goal is to better educate young people, then those kids deserve to get the best teacher money can buy.

In the case of math teachers, that price may just have to be a bit higher.

Thursday, June 5, 2008

Word Problems

This is what word problems sound like to mathphobes.

See more funny videos at CollegeHumor

Tuesday, March 25, 2008

Exceptions and Higher Standards

I'm not quite sure how I feel about this. An art teacher in Florida is in danger of losing his certification because he cannot pass the minimum general skills requirements due to diagnosed discalcula. His argument is that he doesn't see what math skills have to do with art.

I guess it depends on your perspective. I certainly feel bad for him, but truly, all things being equal, wouldn't you rather have an art teacher who is good at math? I know I would.

Wednesday, March 19, 2008

Classroom Etiquette

I have heard it said that there are no stupid questions, and while this may be true, I think it deserves some clarification. A well crafted question designed to invoke a particular response can never be stupid, and nor can the person asking it. The very fact that the question was raised implies that the student has identified a specific gap in knowledge and wishes that hole to be filled. A stupid person doesn't even know what they don't know. Even broad questions like "Can you explain that again?" or implied questions like "I don't understand" can be helpful in determining where the explanation ought to begin. As a teacher, I will strive to maintain patience, addressing all questions no matter how outlandish or repetitive.


There are certainly inappropriate questions. I believe that students ought to keep the following things in mind when asking a question in class.

1. Determine how specific your question needs to be. If you know the specific point that you need explained, feel free to cut to the chase. If you are completely baffled, just say so.

2. Remember that in most cases, you are not the only person in the class. If you feel that you are monopolizing the teachers class time, perhaps you should seek help outside of the regularly scheduled class.

3. Attempt to gauge your needs against those of your peers. If you feel that others are likely to have similar queries, then the question will benefit the whole class. If your are the only one who is confused, save the question for another time.

4. If you know for a fact that the question has been asked before, it should probably be held until after class.

5. Remember that good teachers have a prepared lesson plan that includes not only topics to be presented but the order in which they are to be presented. Do not ask questions about a topic that has not yet been covered. This disrupts the flow of the lesson and can negatively affect the ability of your peers to learn.

Most importantly.

6. Remember when I said there were no stupid questions? Well I lied. There are. There are really, really stupid questions. You should desperately fear being the person who mistakenly asks one of these questions. Before you even think of raising your hand or uttering a single syllable, ask yourself what your classmates will think of you. Will they be grateful that you share their confusion and that you have bravely stuck your neck out to obtain clarification? Or will they scoff, roll their eyes, or berate you.

These simple guidelines ought to ensure that your questions improve the overall quality of the class, rather than detracting from it. Teachers desperately want participation, but there comes a point when enough is enough. And as a student, you ought to know where that point is.

Tuesday, February 19, 2008


Recently, I was forced to add a text messaging package to my cell phone plan. My incoming texts have skyrocketed in the last few months, as more and more of my loved ones rediscover the joys of the telegraph. I must admit, there are situations where I find the technology useful. Perhaps I'm sitting in class and I can't talk or I'm at work and I shouldn't talk. Yet even as I send my conveniently packaged alphanumeric messages sailing through the stratosphere, I am not a real texter.

I write my messages in plain English. Occasionally, I will exchange 2 for to or too, or 4 for for. (That was fun to write.) But that's pretty much it. I never LOL or talk to my BFF. It isn't that I look down on the abbreviation process. Far from it, actually. I admire its speed and efficiency. Fluent texters can condense essays into a few acronyms. Despite what many language mavens or old fogies might think, there is nothing inherently wrong with streamlining communication. Oh, and if you currently find yourself disagreeing with me, I'm going to go ahead and point out your hypocrisy. Laser, robot, sonar, scuba, TGIF, snafu, RSVP. If you have ever used any of these words, heck, if you've ever used a contraction, you too are butchering the Queen's English.

If you are a mathematician, you certainly have no room to talk. Imagine what math would be like without all the symbols and notation. (This isn't rhetorical, I actually want you to imagine it.) There was a time when symbolic algebra didn't exist. There was an era when quadratic equations were expressed plainly in words. For example, I could ask you to draw a square such that the magnitude of its area added to the magnitude of a single side is equal to six. That is equivalent to saying x squared plus x minus 6 equals zero or x^2 +x-6=0. (The square would be 2x2.) In recent years, we've taken to calling these word problems. There are no symbols to manipulate or mathematical abbreviations to remember. It's simply a question written out in our language of choice. Ironically, these simple problems cause math students the greatest trouble.

The difficulty stems from the down side to abbreviation. Sure it's fast and efficient, but you do lose a certain something in the process. Imagine reading a Shakespearean sonnet in text messaging. It probably wouldn't carry the same meaning. Actually, it would probably never happen. The commonly used texting lexicon, while great for casual conversation, is not designed for expressing new or interesting ideas. Symbolic math suffers from the same problem.

Last semester, my most challenging class turned out to be Intro to Statistics. My professor was Egyptian, and there were hidden language difficulties. You wouldn't notice it at first. His accent was thick, but not impenetrable. He was perfectly fluent, at least when it came to his discipline. After awhile, I noticed something. He was essentially reading the math straight from the board, simply pronouncing the rhetorical equivalent to the symbols he was writing. There were no metaphors or personal anecdotes. No stories or comparisons. Because of this, I struggled when it came time to apply what I was allegedly learning. I grasped the equations, but not the math behind them. In effect, I was struggling with the word problems.

And you're telling me this because? (Refer to title of post for irony.) I'm simply saying that the same breakthroughs that made long division and calculus possible, also make math too easy to compartmentalize. It becomes an oversimplified model, so efficiently streamlined that it no longer represents the real world, which is exactly what students complain about. Of course, they complain about the word problems, too, but then those little buggers are never going to be completely happy.

So what's the point of all this? Simply that we have been the architects of our own demise. And the only solution is this. More word problems. Less symbols. And for god's sake, if you have something to tell me, quit doing calisthenics with your thumbs and just talk to me on the phone.

Wednesday, January 30, 2008

Practice Makes Perfect

When I volunteered to lead a study group for my physics class, I wasn't reallly expecting anyone to show up. Not a single person asked me for help last semester, even though it was obvious by my test scores that I was significantly ahead of the curve. So I was planning on essentially having a study hall after class each day where I could get other work done. Surprisingly, I have had several customers and at least one new person each day.

I am becoming more and more certain that I love teaching. It gives me such a high when a student leaves the room feeling more confident about the material than when they arrived. In many ways, they are helping me more than I'm helping them.

Wednesday, January 23, 2008

The Joy of Cooking

While on your bi-weekly excursion to your local grocer, you notice that there is a new species of mushroom in stock. You've never seen it before, and you're not sure how to prepare it or what exactly it will taste like, but something about it's aroma appeals to you. You place it in your cart an continue on your way.

You run to Target for some personal items, and as you scurry around a child screaming in the center of an aisle, your eyes fall on a birthday card with a curious drawing. You open it, read, and chuckle. Though no one close to you has an upcoming birthday, you buy it and save for an appropriate occasion.

Realizing that your couch desperately needs to be either reupholstered or thrown away, you embark on a trip to the fabric discount store. You find the shade of red that you are looking for, but you also find a bolt of green that you can't resist. It's a real bargain, and it finds it's way home with you.

What do all of these stories have in common? Well, for starters, I know people who regularly do these things, and I'll bet you do, too. People who buy a dress without having anywhere to wear it or squirrel away the styrofoam packing from a computer purchase. These aren't packrats that I'm portraying. They don't save everything indiscriminately. Rather, these are people whose life experience allows them to judge the potential usefulness of a brand new item with reasonable accuracy. They are experienced cooks with a new ingredient, accomplished seamsters with a new cloth, and stylish socialites with closets full of perfect ensembles.

These characters are to their area of expertise what mathematicians are to mathematics. Why do these spmetimes bespectacled bookworms play around with formulas and numbers and patterns, seemingly without purpose? Why do they care about things that don't relate to the "real world?" They play for the same reason we all play. Because it's fun. They care because the shelf life of a piece of mathematics is a whole lot longer than a mushroom. It is not uncommon for a discovery to collect dust for centuries before someone makes a connection or draws a comparison, and suddenly that dusty function springs to life, providing just the thing needed to complete the recipe.

I can't cook. I wouldn't know a porcini from a shitake. But I love to eat, and I appreciate the experienced chefs who are willing to try new ingredients and new combinations. Nor will I ever gain fame as a mathematician, but I still love what it is they do. And you should, too.

Wednesday, January 16, 2008

Cutting the Cord

Finally, I'm wireless. Let the games begin.

Here We Go Again

Second verse, same as the first. Almost. This semester, I only have one new professor. The other classes are continuations of last Fall. In a way, that's good, since I know exactly what to expect. I would like a bit more variety, though. I think USM might be too small for that kind of diversity. The Physics Dept only has three professors, if that gives you any idea.

Speaking of physics, my professor walked into class yesterday with a cast on his right arm. Evidently, he had attempted to close a window in his office on Monday and fell, breaking the bone fairly dramatically. He is being forced to revamp his teaching style, since he can't really write on the board very well. I am attempting to turn his misfortune into my advantage by offering to be a pseudo-TA. I would help prepare class materials (handouts,slides, etc.) and gain valuable experience in exchange, and maybe even some work study money. It's still up in the air right now, but I'm hopeful.

In other news, as of about 4pm, I should be the proud owner of a new Dell notebook. AmongHopefully, this should make it easier for me to keep up with my blogging. Perhaps I'll even toss in a vlog or two.

Thursday, January 10, 2008

A Steel-Driving Man

I've been using my holiday break to catch up on my pleasure reading, which to the lay person, would be largely indistinguishable from school assigned reading. I just polished off one about econometrics called Super Crunchers. Econometrics, as near as I can tell, is what actuaries do, only hepped on on some powerful digital 'roids. Evidently, as Moore's Law continues to hold and computing power explodes, decision making that was once left to the "experts" is now being given over to fairly rudimentary mathematical formulas backed up by a whole lot of terabytes.

Chapter after chapter chronicled the successful usage of econometrics to predict things as diverse as good baseball players to Hollywood blockbusters to medical diagnoses. See as it turns out, those experts aren't really all that expert. Time after time, they fail to beat the predicting ability of simple equations, equations which boil down all of life's subtleties to bare bones. As I read, I found myself nodding in agreement. It is well documented how poorly doctors perform on tests of statistical reasoning. Why should patients trust in their abilities to prescribe treatment if they don't really understand the odds. If the use of a simple algorithm can help save lives, then why shouldn't doctors swallow their pride and admit defeat, as it were?

Of course, it wasn't my ox in the grinder. Until the chapter on education. Then suddenly, I was appalled. How can the process of teaching be boiled down to scripted lesson plans? What kind of robotic rote learning could possibly come out of that?

I've since come to my senses. These computer programs are not a threat, but a blessing. They allow us to do the things that we, as fellow humans, are uniquely suited to do. If a doctor is free from having to analyze symptom after symptom, if a diagnosis is less than a Google search away, then that medical professional can focus more on the healing process. They can spend a few extra minutes holding the hand of a scared little girl, or explaining the treatment details to her mother. Besides, the programs are still guided by the garbage in, garbage out principle. They are only as accurate as the information passed into them, and that will still require skilled human doctors. The same is true for teaching.

I have no desire to sit and read from a script all day, and frankly, it's silly for me to do so, even if it proves to be the most effective teaching method. The same computers that analyze the data behind these lesson plans can easily conduct the lessons as well. Sit the kids down in front of some powerful learning software. That's fine by me, because I know it doesn't render me obsolete. It empowers me to do all the "extra" things that I wouldn't ordinarily get to do. I would have the time to really get to know each of my students. I could allow them each to progress at their own pace, and have the ability to work one on one with each of them.

No one alive today would take offense at being beaten by a machine in a test of speed or strength. What John Henry learned the hard way is second nature to us. Yet we cannot seem to accept that certain intellectual feats are now better performed by cousins of those athletic machines. We feel as though our very humanity being stolen, when in fact, the machines are helping to teach us what being human really means.

Saturday, January 5, 2008

Don't Cry for Me

My love of math and science is no secret to those who know me. Friends call me up at all hours, asking me to answer questions or settle bets regarding all manner of things. I am the Phone a Friend. At least, that's how it works with those who have come to love me. The reaction from strangers is quite different.

I've noticed lately, and especially at work, that the discovery of my mathematical predilections is normally accompanied by a wince and/or head tilt. As soon as I mention that I am majoring in math and physics, I am treated as though I've announced a death in the family. "Oh, I'm sorry," they say. Or my personal favorite, "So you're one of those." Usually, I laugh it off, and use it as an opportunity for research. I ask why they feel that way. Why is they're fear and loathing of math so complete that someone else's involvement causes them pain? I find the conversations fruitful, if not more than a bit repetitious.

The respondent almost always remembers loving math as a small child. They can usually pinpoint an exact year or teacher which soured them on their studies. Often times, they remember being told by a teacher that math was simply not for them. It is at that point that I am able to commiserate. In my junior year of high school, after having taken all honors math classes, my teacher told me one day that I simply lacked the "flare for math." It galls me that people who would say such things are allowed to teach any subject at all, let alone such a notoriously tricky one.

I know now, and I continuously attempt to impart to my friends, that math class is not terribly different from shop class. Both are all about tools and toolboxes. Math class is no more about mathematics than wood shop is about craftsmanship and design. Just because you can hammer a nail does not make you an architect, nor does hating long division mean you are cosmically predestined to avoid math. This is a fact that is lost on most students, and too many teachers, and it is one that bears constant reminder. Students must be given a glimpse of the horizon so that they have something to journey toward. Otherwise, we are asking them to practice for a championship game that will never come.