A Mathematical Proof 2500 Years Too Late

As a teacher I learned that one of the most important steps in educating someone is understanding how they think through a problem. To best understand how you take in and retain information, you must "unpack" your own thought process. You have to acknowledge every step your mind takes when it studies, analyzes, and solves a problem. It can be difficult to slowly and deliberately review each step of your thinking process like this, but you will be rewarded with improved problem solving skills. It's just like taking apart a machine to understand how it works so you can then make better use of it and take better care of it.

I recently went through this process myself when I got a particular mathematical ant in my pants. I am currently reading a novel that made a mention of the Pythagorean Theorem in passing. The character in the novel had such natural mathematical talents that the Theorem made immediate intuitive sense to her. Reflecting on this savant-like brilliance, I tested my own understanding of the Theorem. Almost everyone who has studied geometry remembers Pythagoras's most famous equation,  "A squared plus B squared equals C squared." If you were a good enough math student, you also remember that this equation describes a natural relationship between the three-sides of any triangle that has one corner at an angle of 90-degrees a.k.a. a right triangle. My problem was I couldn't remember how to prove the Pythagorean Theorem. This seems especially silly to me because Pythagoras originally demonstrated his proof about 2500 years ago.

In mathematics it is never enough to simply trust that an equation or rule works, you must be able to prove it. You can't just demonstrate it inductively either. To prove Pythagoras it would not be acceptable to show that 3^2 + 4^2 = 5^2 (feel free to pause for a second and check that arithmetic if you want) and a triangle with sides measuring 3, 4 and 5 units would have a right angle. Mathematicians don't take a rule that applies to a handful of examples and say, "I assume it works like that all the time."  An effective proof takes established rules and applies them to a generic example so as to create a universally applicable statement.

Lacking the ability to leave questions unanswered, I took my ignorance as an opportunity to test my mathematical thinking. I also decided to write down my process so that I could "unpack" the logical process necessary to prove something in math. Because I have blogged about education topics in the past, I though some segment of my readers might take an interest in unraveling the bundle of thoughts involved. By making my geometric contemplations explicitly clear and simple, I hope to help fellow educators find ways to help their students think through math problems.

An illustrative triangle

To prove the Pythagorean Theorem I needed to use deductive reasoning. So I began by writing down all the facts I could think of that related to triangles. From those points I could begin building my reasons to demonstrate the relationship between the sides of a right triangle.

I began by noting that for any triangle the angles of the corners must add up to exactly 180 degrees. This makes intuitive sense, because you can imagine a triangle as a path that takes three turns to point you in the opposite direction. As we all know, a reversal of direction is 180 degrees -or pi radians if you prefer.

Next I recalled the equation for calculating the area of a triangle, Area= 1/2*base*height. You can see how this works with a simple thought experiment. If you take any triangle and add a conjoined copy like a reflection across one of the sides, you have quadrilateral (i.e. a 4 sided shape) with an area equal to the triangle's base time's the triangle's height. Naturally the area of the original triangle is half the area of this new quadrilateral or 1/2*base*height.

The last fragment of knowledge I drew on was a little more advanced. I knew that there was a relationship between the size of an angle at the corner of a triangle the length of the side opposite to it. This is the basis for those trigonometric functions that probably traumatized most people in geometry class, sine, cosine, and tangent. This makes intuitive sense when dealing with a right triangle. The largest side, called the hypotenuse, is always opposite of the 90 degree angle (which must be, by definition, the largest angle in the triangle).

An example of a 45-45-90 triangle

This is useful because it establishes that in a right triangle with two equal angles of 45 degrees each, a 45-45-90 triangle, the two sides other than the hypotenuse must be equal because they are opposite equal angles.

The reason I focused on the example of the 45-45-90 triangle is because it has some special properties. In the same way we could imagine a quadrilateral made out of two identical triangles, we can create one square out of four identical 45-45-90 triangles. Just picture the hypotenuse as one side of the square with the four 90 degree angles joined in the middles.

One square which contains four 45-45-90 triangles

We can analyze this square to help prove the Pythagorean Theorem (in case you forgot during my babbling about triangles facts, that was the point of this exercise). First, we establish the total area of the square. Since all sides of the square are the length of the hypotenuse,or side C, we can easily calculate its area. The area of a square is the same as any quadrilateral the length multiplied by the height. In a square those lengths are both the same as the length of one of the sides. So area = side * side = side^2. In this square the length of the side is called C so the area is C^2.

We also know that the total area of the square is the same as the combined area of the the four triangles. As discussed above the area of a triangle is given by the equation area = 1/2*base*height. You multiply the area of one of those triangles by 4 and you get this:

4 * area = 4 * (1/2*base*height) = 2*base*height = total area of square

Since the base and the height, which we can call side A and side B, are both the same length we can simplify the above equation for the square's area.

2*base*height = 2*side*side = 2 (side^2) = side^2 + side^2

We can substitute side A or side B in for either side length in this equation. That allows us to rewrite the area of the square as Area = A^2 + B^2 = C^2. This proves the Pythagorean Theorem for all 45-45-90 triangles. 

I started with proving the Theorem for 45-45-90 triangles, because I knew they had special properties that made them easier to work with. Unfortunately you can't create a simple square out of every right triangle. Still, knowing that I had worked the proof for one class of triangles, gave me an insight for working a broader proof. In order to prove Pythagoras correct for all triangles, I had to find a more sophisticated method that I could apply to any type of right triangle. I was able to work from the foundation of my earlier exercise by searching for ways to create squares using the sides of triangles.

My breakthrough came when I realized that any right triangle can be used to define a square area. (This was important because, as I would learn later, earlier generations had used this as the basis for other proofs of Pythagoras.) You can form a square by arranging four identical copies of so that their right angles form the corners of the square.

The square defined by the combines sides of  four copies of a triangle

In order to keep the sides equal length we'll combine all the triangle legs of length A against all the legs of length B. This gives us a square where the length of each side is A+B. Since we get the the area of a square by multiplying the length of a side by itself, we can calculate the are of this square in the following way:

Area = (A+B) * (A+B) = A^2 + 2AB + B^2

When you arrange the triangles in this manner, the hypotenuse of each triangle forms the side of another smaller square with the square discussed above. Since all the sides of this square have the length of the hypotenuse, which we call C, you can define the area of this square as C^2. The area of the square with sides of A+B can be redefined as the area of the hypotenuse square plus the area of the four triangles. Each of the triangles has an area of 1/2*A*B.

Area = C^2 + 4(1/2*A*B) = C^2 + 2AB

We know from our earlier definition of the larger square's area that the area is equal to A^2 + 2AB + B^2. Therefore

Area = A^2 + 2AB + B^2 = C^2 + 2AB

Because the two quantities are already equal, we can perform any mathematical operation to both sides and they will still be equal. In this case we can subtract the quantity 2AB from both sides because in exists in both expression. This leaves us with A^2 + B^2 = C^2. This proves the Pythagorean Theorem for all right triangles.

Just for clarity, I do not recommend any math teacher tries to explain the Pythagorean Theorem using my meandering thought process. Instead I hope teachers of all kinds appreciate that though my method may not seem perfect it achieved the desired result. The purpose of unpacking is often to find the twists and kinks in a person's reasoning, but it can also demonstrate that differing methods of thinking can be acceptable. Students need to know that there is no one right way to go about thinking. Every teacher should encourage students to unpack their thinking, so they can see that -however they go about reasoning- they can always find ways to sharpen their thinking. You can always improver your problem solving  even if it means solving a problem 2500 years too late.

Why did a rumor tell me more about teenage culture than eight months of observation?

1.
Around the turn of the twentieth century, a new kind of research was brewing on the shores of Lake Michigan. Several social scientists working at the University of Chicago developed new methods of studying human behavior on both the social and individual level on a more intimate and in depth level than had been attempted before. This form of investigation became known as the “Chicago School” of sociological and anthropological research and focused on applying scientific principles to fieldwork practices. The chief tool of a “Chicago School” researcher was participant observation, wherein the researcher would actively join in with the group she was studying over an extended period of time. Once the researcher integrates herself into the culture she is able to gain better insights into the workings of the group, and is able to gain freer access for discussions, interviews, and data collection. Over the last 70 or so years since the “Chicago School” has become a commonly accepted research practice a stunning amount of information has been gained about different sub-cultures and groups in our society. A terrific example about a grad student at U of Chicago who spends years living with a crack gang in the Chicago housing projects is discussed by Steven Levitt and Scott Dubner in their book Freakonomics. This example demonstrates how a participant observer can bring subtle details of how a group, in this case a fairly successful street gang that traffics crack cocaine, organizes itself, supports itself, and communicates both internally and externally. I personally have taken a great interest in these research practices as I have been completing my student teaching and I have theoretically been a participant observer for the last eight months.

The most obvious question to ask me as I complete my teacher training is what I learned during my observations? Unfortunately the obvious question isn’t necessarily the best question with which to start this conversation. First, you have to consider which group I am meant to be studying. High school buildings are homes to two distinct sub-cultures, one of teenage students and one of the slightly older teachers. Given that the purpose of my time as a student teacher is to teach me what it takes to become a full blown teacher, the answer once again appears clear; I should be observing the “teacher culture” to learn what makes a successful teacher. Yet, that simple answer belies the complexity of the student teacher interaction. My training has continually emphasized that ineffective teachers focus classroom activities on what the teacher likes to do, while effective teachers base their instruction on what best engages and interests the students. So to be an effective teacher I have to learn both how a quality instructor delivers information, and how a quality student processes that information. By understanding the students I can spread good learning habits, and activate the mental processes my students already use.

Thus, as I undertook my student teaching I had to keep one eye on my mentor teacher and which of her practices seemed effective, and my other eye on the students and what educational strengths and needs they exhibited. Over the past school year I feel I have become a sufficiently skilled teacher. I was able to incorporate a number of practices I observed from the teachers with which I work and my mentor teacher and instructors have given me plenty of positive feedback. I would have to say that my observation of the teacher sub-culture had born terrific results.

Until a few weeks ago I thought I had achieved similar results with my observation of the student sub-culture. My students had responded well to the instruction I designed; they were highly engaged and seemed to retain the information well. I was able to relate to the students, since I am still young enough to catch most of their pop culture references. I felt I had a good handle on how to communicate to students and gain their cooperation. Then, last month, I heard about a rumor that passed amongst my students that taught me more about teenagers’ and their culture than I had gained in seven months of observation.

2.
The story begins about a month ago, in early May 2006, in my U.S. history class. My mentor teacher had assigned a major research paper as the last significant assignment of the school year. They were first given the assignment about five weeks before the final paper was due on May 18th. They had to create an outline for their paper and turn that in after two weeks. From there on out though, they had nothing to do but work on their final draft. They were told all of these facts in very clear terms when the paper was first assigned and given assignment sheets for each student so they could have a resource in case they ever needed to remind them of this, and of course they all knew they could talk to either my teacher or me if they had any questions. My mentor teacher and I both felt that the students had been well guided and prepared for this particular lesson. Our confidence about this should emphasize just how little we understood about the students at the time.

On May 4th, three weeks after we had originally assigned the paper, as one of the students paused at the door as she left class. She turned around and asked very sheepishly whether the research paper was due that day. My mentor teacher wasn’t in the room at the time, but I knew the students still had at least two weeks before they needed to turn in the final paper. I looked at the student with a puzzled expression and then explained the situation. She gave me a relieved smile and said that she had spent most of the previous night working on it. I asked her how she had come to believe that the paper was due today. After she gave me the details of the rumor she had been swept in, I realized I was on top of something special, and I asked her to write it down for me. What follows is her account with the students’ names replaced to protect their identities.

Student #1’s experience with the rumor:
While I was at a café after school, I got worried that the paper might be due tomorrow. So, I called Student #2. Student #2 said the paper wasn’t due until May 18th. Then later in the evening I received a call from Student #3 saying that the paper was NOT due tomorrow, and that he was making phone calls to clarify. A little later I went online on an instant messenger and Student #2’s away message said the paper was not due tomorrow. Later on when chatting, Student #4 asked me if it was due tomorrow and I told her it was not. I heard that Student #5 thought it was due and that Student #6 actually wrote her paper.

I was fascinated by this for a couple of reasons. The rumor was actually so convincing that someone wrote the paper two weeks in advance of its due date. I knew the rumor had to be responsible for this flurry of activity, because I had learned from experience that the consequences for turning in work late had never inspired such dedication on their own. Moreover I was struck that at least six different students had heard this rumor and NONE of them had attempted to contact either my mentor teacher or me to clarify the issue.

Having read Malcolm Gladwell’s The Tipping Point which discusses how social memes like rumors are spread and where they gain their power, I decided to dig into the story a little deeper. I wanted to gain a better understanding of how this rumor had started and spread. I also was curious to determine what gave it the power to make a student write an entire research paper in one night. Rumors being so ethereal and temporary by nature, I needed to quickly gather more information while it was still fresh and available. I sought several of the students who took part in the above story and had them share their account of how the rumor spread and how they reacted to it. The following examples provided special insight.

Student #4’s account:
That evening I got a call from Student #5. She said the paper was due tomorrow and that Student #1 and Student #7 were already working on the paper. So, I started working on my paper. A while later I called Student #2 to figure out if the paper really was due today or not. Student #2 said it definitely was due, so I kept working on it. When I was almost finished I called Student #3, who told me the paper was NOT due. I decided to still go ahead and finish it since I was almost done anyway.

Student #3’s account:
What happened was Student #5 misread the assignment sheet, and she told a bunch of people that she thought it was due. So Student #2 called me to see if it was due or not and I told her it wasn’t. Then a while later Student #4 called me to ask if I knew whether the paper was due. I told her no. So, I called Student #5 to let her know that it wasn’t due, so she wouldn’t tell other people it was due today. Then I went ahead and started calling lots of people in case any of them thought it was due today.

These three accounts alone provide a robust amount of information that we can use to investigate how rumors and other comparable social memes form and travel. The insights may not seem especially groundbreaking, though that does not limit my interest. This one rumor revealed a level of complexity and autonomy to teenage culture that I had never before thought existed and which I can use to great effect as a teacher.

Let’s begin with how the rumor traveled between the seven or so students who became wrapped up in it. Since each of the student accounts above had their own sequence of events, I’ll lay out a general timeline to clear up the muddle.

First, Student #5 somehow, reportedly through misreading the assignment sheet, came under the impression that the paper was due the next day. She may have contacted multiple students, but the most important person she contacted was Student #2. Notice how Student #2 appears in all the accounts at some point spreading the rumor. It seems that most people who began working on the paper were prompted to do so by Student #2. As the rumor spread, to at least seven students, though I suspect more, it eventually reached Student #3 who seems to have been the first student to encounter the rumor who had enough confidence in his own accurate knowledge to begin countering the rumor. Many students who heard the rumor apparently were only convinced of its falsity after hearing from Student #3. While Student #3 did not seem to convince Students #4 and #6 that they did not need to complete their papers, he still made the effort to contact them and many others.

3.
In The Tipping Point Gladwell describes the kinds of people social epidemics like rumors require to spread and to be made believable. He creates categories for these people like connectors, who can contact a great number of diverse people, mavens, who others rely on to provide important information, and salesmen, who can convince others of an ideas value. In the rumor I have recorded, I believe no one person falls neatly into any of these categories, though several display attributes of some.

Like mavens, Students #5 and #3 were trusted by their classmates to be sources of information. Other students trust them so much that even upon hearing the rumor the others don’t check their own assignment sheets, instead relying on the word of Student #5 or Student #3. It may even be possible that some students don’t worry about keeping track of information and things like assignment sheets, because they know someone like Student #5 who they trust to remember that stuff so they don’t have to.

Student #3 also served a role as a connector along with Student #2. Both these students seemed to be directly responsible for either spreading or dislodging the rumor. The accounts shown above clearly show the lengths to which both Students #3 and #2 will go to share a piece of information they feel is important. Part of the story of this rumor which the accounts don’t reveal, is how dissimilar the students who encountered it were, or at least how dissimilar they seemed to me as the classroom teacher. The students involved had class with me at different times, they were different races, they came from different backgrounds, and they seemed to belong to different social circles. Yet Students #3 and #2 were both capable of connecting most of the people in this loose social network that tied them all together.

Additionally, Student #2 seemed to have the skills of a salesman, she was able to state a case for the rumor so convincingly, that some students actually went forward with the paper. All students had been informed to the contrary previously by my mentor teacher and I. On some level they should have known Student #2 was misinforming them. Yet, Student #2 must have stated the rumor in such a persuasive way as to compel many of the students to take some kind of action. Furthermore Student #3 seemed to lack the skills of a salesman. Though he supplied numerous students with the correct information, he still couldn’t dissuade some of their fears. Clearly a social epidemic requires some convincing power to become contagious or to stick around long enough to make an impact.

4.
All of these skills have been identified by Gladwell and others and crucial components of any information sharing network. Yet prior to the day I heard of this rumor neither I nor any teacher with whom I’ve spoken had thought that high schools students would have developed them so completely. To some degree it seems obvious; we are talking about the children of the information age. They likely have been surrounded by advanced information/communication technology since birth. For them forming complicated and sophisticated networks probably comes as naturally as farm work came to our ancestors. With this in mind imagine, then, what schools could accomplish by including the students more in their plans.

I have witnessed many occasions during staff meetings or other school events when the teachers fretted about how best to get some important piece of news to the students or the parents. The task always seemed challenging because schools are still committed to old fashioned top-to-bottom methods of spreading information. This always requires time and planning to ensure the full population is covered by whatever announcement method is used. Think about how much simpler it could be for schools if they didn’t have to worry about contacting every student, just those that act as information banks for the general population. Consider how much quicker news could be spread when schools feed announcements to those students who can connect a wide variety of students instead of trying to use an endless stream of pamphlets, phone calls, and P.A. decelerations. Any of these tools can be powerful aids for schools, but I have never seen or heard anyone in education attempt these methods.

One important fact underlies all of this: the students formed this information network independently. I cannot emphasize how surprising this was too me enough. I really shouldn’t have been surprised though. I have heard of everyone from marketing firms to government agencies delving into the teenage subculture. Why then has no one attempted to make use of this in education? The students’ information networks already exists, it’s as much a part of their culture as anything else I observed during my student teaching. It may even be one of the more fascinating aspects of student culture. The fact that the teachers’ culture is so ignorant of the students’ networks marks the need for better participant observation by educators. The lesson for me is clear to best deliver information to students more educators have to learn how to share in their culture.

A dazzling display of teaching and technology.

This week I got a little bit of a gift (in keeping with the season I suppose), because I had the privlege of watching several very high quality people give presentations on the different methods of employing technology in education they had practiced themselves. I was dully ipressed with the broad array of lessons and projects the students undertook. They used technology in a variety of ways and found so many special tool that I wished I made notes of all of them. I thought everyone who presented did an incredible job, and i really wish they had been alloted more time so I could have heard more from all of them. I was especially interested in the different methods the presenter had found to encourage students to interact or communicate in new ways. The ideas presented varied from an online international poetry guild to a large scale role play exercise meant to generate debate and conversation on a number of controversial issues. Those who used these applications seemed to find the level of enthusiasm from the students quite high, and their investment in the work they put into it just as high. It made me think that it seems to be a common misconception that web creations are some how not as "real" as corporeal items. A website isn't as good as magazine article. Wikipedia isn't as good as the Encyclopedia Britanica. I think this has commonly translated into a resistance to web based lessons by older generations of teachers, because they never felt the material was as serious once you involved the Internet. For the students I heard about, and those I have seen in my own experience, the work they create on the web is every bit a significant a that which they turn in on paper. I felt encouraged by these presentations. They seemed to confirm my longstanding theory that computers can model anything, and that a good teacher can draw a lesson from anything. I want to extend my thanks here to all who presented and to those who organized the event. I had a great time, and I feel I learned something too.

A (Partial) Follow-Up

Yes, the rumors are true. My ongoing quest to provide quality educational material over the inter-web strives on. My last post speaking directly to the topic sounded a little downbeat, but I actually reamin fairly optimistic. I and my coworkers are marching onward for the betterment of the students. While difficulties remain, we have decided not to let that deter us from producing the a website of the highest quality. Even if the exact contents are yet to be determined, that doesn't me we can forge some fabulous packaging. I have taken to trying to piece together elements of the visual lay out. If I was more familiar with how one posts visual images on a blog I may share some of my work with you. For right now though I can tell you I feel fairly optimistic as the group has a god cohesive theme in place to define the website. We hope to have at least a partial site up soon ("partial" only in the sense that we have had to scale back some of our original goals, it should still be fully operational and accessible). Watch this space for future updates, as I am sure to link to it in a subtle form of bragadacio. Until then I leave you to ponder over the scatter brained thoughts of my scatter brained life.

Comics and Education Converge!

Lynn Johnston may have just become my hero. She writes and draws the popular daily comic strip For Better or Worse about the various highlights and misadventures of the Patterson family. Like G.B. Trudeau Johnston has used her daily platform in hundreds of newspapers to raise awareness of various issues. She will writes comics about solemn and important topics like the death of a parent with grace and candor. Her willingness to raise these issues has long drawn attention, and her deft skill at dealing with them with sensitivity and insight have long earned her admiration well beyond the comic community. Recently she turned the gaze of her strip on the difficulties disabled children face in education. She even mentioned some of the technology they use to help them handel daily tasks. Hopefully this widespread exposure will help raise awareness of just how many challenges with which the disabled must contend. I always have believed that comics have a special power to get into our minds through their combined visual and lingual presence. When both the words and pictures team up to bounce around your head, it boosts the chances of your subconcious mind remembering and your concious mind being forced to digest it. In my current projects I am trying to do as much as I can to accomodate students with disabilities, but true change won't occur until more people realize just how much we take for granted. If you want to check out the strip go here.

A (Partial) Summation

As I reach a crossroads in my work developing a web site for a high school in my area, I find myself reflecting on my work up to this point as a means of guidance for the future. I would really like to see something strong and positive come out of this. Yet, more than a few complications have prevented the project from developing optimally. As such it seems that I will only have the means and time to create something of partial utility for the school and students. Still I feel even a limited success can have a strong impact. As I have tried to make clear I am a strong believer that a well designed web resource can be an incredible guide and portal to other information and resources. I know one tool that I wish to make available to students is a means of express their own thoughts and reflections on the program of which they can be proud. I think if I set up some type of blog that's held by the community it might give the students a chance to communicate with each other and the outside world in a way that can be comfortable and encouraging. Also, I hope that for all the joy I have taken in creating this blog, the students themselves may take some pride in seeing their work published on the web.

True Accessibility in Technology

We have all heard the warnings on the various TV news magazine programs about why you have to be careful on the Internet. Nothing is as it seems on thw World Wide Web, you can't trust anyone. If someone claims to be a Nigerian prince, or a potential hot date, or a government agency, odds are they are in in reality a con artist out to steal your vital information. We all know why this is an issue in the Internet. The Web is abstract it's "out there" in the ether where no one has to follow the physical rules of our universe. Which means any sleezy character can take on some new fictionalized identity and suddenly become someone new.

I raise this because it demonstrates the commonly held belief that, while the Internet exists parallel to our world, it doesn't actually conform to it. All the rules a different in cyberspace and we are all in some way liberate by this. It doesn't matter if we can't go to a store because the Internet can bring the store to us. If we aren't comfortable meeting people in person, the web allows us to meet them virtually. Once you give someone technology they are reborn out in the ether as a new entity with freedoms and powers they never had before. This all works well in theory, but I recently had my eyes opened to a very important flaw in the Internet's underlying thinking. It claims to create equal oppurtunity through equal access, but the truth is modern computer technology isn't equally accessible.

I recently met with someone working on bringing computer technology to disabled students. He demonstrated some of the difficulties those with handicaps face in using computers. For instance the keyboard I am typing this with relies on an assumption that I am not dyslexic. The screen I am using may be much harder to read if I were color blind. This is to say nothing of the assumption that I have control over my hands or any vision at all. These problems are so rarely recognized by all of us who are currently abled that many of the resosurces we have created to be accessible by computers don't even take it into account. These challenges can be true barriers for a disabled person to the treasure trove computers have opened, requiring additional work and cost to overcome. I raise this issue because I feel compelled to make some statement that it should not be their burden. If those of us who design and distribute computer tech did so with more consideration of making it truly accessible we could save many people a tremendous amount of hardship. Those of us who wish to use technology as a teaching tool could benefit from this as well, as it would allow us to guide our students to various resources with much more ease and confidence that the Internet would provide truly equal oppurtunities.

An Ownership Society

I fear the repercussions of posting my personal information on this blog. Naturally I don't mean accidentally telling you all my cel phone number, or where I buried the bearer bonds. I just worry because I feel I work on some sensitive projects. While spewing my missives into the electronic abyss may help me think about these things, I still must guard against accidentally giving you details of a person or situation of which I have knowledge that those involved may wish to keep private. I have so far tried to be as veiled as possible in my writings while still allowing for those I am try to communicate with know of what I speak.
The question here is one of ownership. I have access to information about others, but I do not own the information. It's simply not mine to give away. Yet so many issues involving information technology revolves around these questions. Can anyone own information posted on the Web? If not, then how can someone feel safe in posting anything on the Internet that may be of value to them, but since its out here in the "free domain" or "market place of ideas", they can't control what happens to it? If someone can own information then how do you protect it? Or, perhaps to dray a sharper point, then why share it at all, given the well known risks of the Internet?
As I have said before information technology can model anything. We can use it to create whatever suits us, and then transform into something entirely different in the next second. Nothing is impossible out here amongst the ether. But then I am reminded of Roger Ebert's criticism, (here I am paraphrasing) "If nothing is impossible, then does anything matter?"
As an educator I think about these things and wonder if this impacts the utility of teaching with technology. I think of the profound transformative effect technology has on everything it touches, and I think how in the midst of this maelstrom we can only guess at what may or may not be of future value to students. So to some degree, educators simply have to find a way to get their students to find meaning in their educational experiences with technology.
Educators agree that students need to have something they can take away from a classroom for their education to have value. Normally that is mean figuratively, but all to often we don't consider the literal implications of such. The papers that hung on the refrigerator door, those trophies from field day, those art projects that all ended up looking like ash trays, those meant something to us as children. Yet a student who's education comes through the form of information and networks, what do they take home?
In our increasing focus on a technology centric education sometimes I have to ponder how you can add substance, real physical presence to what a student learns. On one hand they own their skills and experiences in the truest sense. No one can take away a student's memories and make them her own. The skills and knowledge can only be used by their rightful owner. Still I think we need more. I have recently seen the power of what happens when an educator gives a child something they can literally own as a product of their education in technology. I cannot go into details, but the effects are astounding.
Additionally, I will reveal that I come from a long line of educators. Once my mother gave me some advice. She said "When a student becomes the proprietor of their own knowledge and learning it changes everything about them. It changes how much they'll work for it, how much they want to retain it, and most of all how they improve their behavior because of it."

Dewey, Teetchum & Howe

John Dewey did for education what Einstein did for science. If you do much research into the theories and frameworks of education you are bound to sooner or later run into John Dewey's work. He redefined the terms people used to think about education and likely set the tone for the next century of educational reform. recently I had to consider what his views might be on an ongoing project of mine. As I described earlier on this blog, I have been working on creating a website for a specialized program at an area highschool. I have been struggling to determine what educational value this website can add to their experience that they don't get out of the classroom. Thankfully I have the writings of John Dewey to consult. He describes the purpose of schooling as two fold. First, we want to provide the students with knowledge and training they can use in their life, obviously. Second, educators try to socialize students so they can be prepared to deal with the expectations and demands of society as a whole. I had the oppurtunity to view the students for whom this website is intended at their school. What I took away is that like most teenagers, they prefer not to be exposed to society as a whole, but rather to select a small group of their peers with whom they spend most of their time. Now I am not certain that there is anything we can do to end this practice in general. Within this website though, I believe I can do something to improve the socialization process. John Dewey says that school, shouldn't be more demanding on students than the outside world. I feel that these artificial social barriers created by students do add unneeded complications. Worse still the Internet is used all too often as a device to limit our exposure to the outside world to those things with which we already agree. So I am trying to think of ways that this website will break down the cliques and allow students to experience the full social experience their school experience should allow. Once again I will be sure to keep you updated on my progress and welcome any recommendations in the comment area.

How can a website be educational?

Before you read this blog entry you should know that I will be discussing an ongoing project of mine in education. So if you aren't interested in issues involved in education, don't fear I will soon be posting some more stuff about comic books and basketball soon (Infinite Crisis and the NBA preseason are upon us, I don't know which geeks me out more). Also it should be mentioned that for various reasons I can't go into great detail about the who's and where's of this project. For those of you in the know I'm sure you can fill in the blanks. For those of you not in the know, this may be a little frustrating to read. I'm sorry about that, but maybe you should wonder why I haven't let you in the know.

I recently began working with a high school in my area to help design and create a website. This is a fairly common place task these days. Many professionals find thmselves increasingly involved in some sort of web design project at their own place of work. I really am involved in this project instaed of your basic web consultant, because I come from a perspective of education which the high school appreciates. However, this project is a little more complicated than I had anticipated. First, I am not designing a website for a particular school, but rather for a specialized program within the school. Second, there a remultiple teachers at the high school involved in this project who, in theory, are directing my team in what to do. Except, they can't seem to agree amongst themselves about some of the major goals this website is meant to meet, or how it should go about using them. This was on the points that they did have a specific vision for, and not those for which they were expecting our team to sort of generate some kind of purpose Athena-like from our brows. Third, the website is meant to be for a huge audience, including parents and students who are interested in the program, those currently involved in the program, the alumni of the program, and the teachers themselves. All in all, I was a little unsure how we should go about this.

In spite of all of this I determined that I should indeed sally forth and explore this program with my own eyes. I have to say that the program (which combines classes in the social sciences and humanities under the auspices of a few teachers, and a lot of interesting ideas in how to teach the material) is actually quite impressive, and I would think something that could lend itself well to the flexibilty of the Internet. I was really intersted in learning what the students woul like out of this website. I thought that a generation of students raised under the influence of the Internet would surely be interested in the oppurtunity to include a website into their education. I was astonished by the actuall reaction of the students. I was honestly told by more than one student that creating such a website would be impossible.

They felt that the program was so special and so self-contained that it couldn't be transferred onto a website. They couldn't imagine what a website could provide them that they didn't have already. I understood that this program is set aside within the schools, to the extent that they percieved the whole program existing in a kind of pocket universe, that doesn't touch upon the rest of their world. I watched these students use other web resources with ease. I know they use the web as a major source of entertainment. I came to think that they might view the Internet, the same way I view comic books. I love reading comic books, and I can see many more uses for them than the average person would, but if you ever told me a comic book would be incorporated into a class I was taking I would still balk at the idea. So I realized the challenge for myself would be determining what I could bring to a web site that would give it special educational value for these students.

My first thought is that I should use the adabtibility of Internet to emphasize the lelements that the program provides, but that websites normally lack. In my past experience I have found that what the Net adds in convenience it tends to lose in context. Meanwhile the program is steeped in context, as the teachers try to link their lessons around common themes and materialsl. Sso the question become how to I create a context rich website? I will keep you posted on any ideas I may have, but I will also welcome suggestions, in the comment section.

A pronouncement, on the educational value of Wikipedia

Wikipedia does more than provide a new location for information. It provides a whole new way of thinking about, and processing information. I say this not on the basis of my bubbly optimism for internet resources, though my apparent glee in previous posts may suggest it. Rather this comes from my own observations of how individual's interactions with Wikipedia vary from those with other resources.

The other day I was observing a group of high-school age students preparing for a research paper in a library. They had access to an abundance of traditional print resources (I should compliment this library on their magnificent taste in resources. There were many great books to be found there.) as well as computer terminals to access the internet. The students did seek out many books and spent a lot of time doing the traditional sort of research (taking notes from a desk, cross-referencing, that sort of thing). So, I felt satisfied that their schools, must be teaching them something about the old ways to find out information. That reminded me of medieval scholars who had to work so hard for their knowledge and wisdom, that their learning was seen as all the more noble for having been hard won.

Then something interesting happened. One of the students asked me if I knew who Levi Strauss was. She had found his name among a list of topics she would like to research. I told her that Levi Strauss was the Levi of Levi's Jeans. She became really excited and set off to learn more about him. I became concerned that I might have made a mistake, that maybe Levi Strauss didn't have a thing to do with jeans. So I went to the reliable Encyclopedia Britanica. I didn't find anything under S for Strauss. I didn't find anything under L for "Levi's brand jeans", the entry on denim was not very helpful. After ten minutes I decied to consult Wikipedia. One simple search and there was all the information you could bear on ol' Levi and his company and a plethora of links to even more information. Then I noticed that while the students did use the books to answer deep questions, they were relying on resources like Wikipedia for all the basic questions. For them knowledge wasn't isolated and static, like it was for those medieval scholars working by candel light. They saw information as dynamic and instantaneous, just as Wikipedia presents it with information flowing on and off the pages.

I do think it is important to teach from books, and show students how to learn from books. One of my favorite facts about books is that if you want to get rid of the ideas they contain you have to burn them. Wikipedia, and any of the information it contains, can be deleted, gone forever, in an instant. That sort of resource may be handy, but it cannot replace traditional encyclopedia's. For a book says that the information contained therein is so important, someone chose to write it down, someplace safe, someplace permanent, so that we should never forget. Information should flow and change, but any knowledge you think is worth keeping should be written down on a physical piece of paper, as if to say, "here are my thoughts for all to see, and I stand by them". Knowledge than can be destroyed with a keystroke will never have the same meaning as that which has to be burnt.

The Wiki Witch of the Web

I have done some research into wikis in general and wikipedia in specific these last few days. From some of the literature I have attained it seems that there's an awful lot of talk that wiki's are either an incredible revolution or the worst resource on the web. I am more than a little befuddled by all of the apparent polemic effect of such a simple concept for a web application.

Years ago, when the internet was just rising into the public concsience, and I had only a dim understanding of what could be done when you let computers talk to each other, I sort of envisioned the web as a collection of wiki-type sites. I just assumed that the most useful thing someone could do with web communication was to set up a commonly held and maintained source of information and entertainment. Later when I found out how websites really worked, I was a little disappointed, then I heard of the idea of a "killer application". I guessed that meant that people were trying to tap the secret of how to create the kind of internet I envisioned. A while later the whole idea of the internet became a little mundane for me (that's kind of sad now that I reflect on it, it's a little like taking the stars for granted) and I learned from an urban geography professor that e-mail remains the best example of a killer ap and I guess I agreed at the time. So dim had my hopes for the web become that when some one told me about wikipedia, I thought "Oh, somebody wants to make the Hitchhiker's Guide to the Galaxy." That thought alone should have sent me leaping into the air except I only conceived of the site as a sad rip-off instead of the full blown realization of a dream that now exists. (Sidebar: For those of you unfamiliar with the science-fiction/comedy writings of Douglas Adams, the Hitchhiker's Guide to the Galxy is a ficticious book created by aliens to accumulate all knowledge in the universe including facts, cultures, trivia, and the bizarre. The Guide is compiled by many independent reporters who can add to it or access it via a shared network. This is obvious to us now but when the book first came out in 1979 it still qualified as science fiction. By the way I recommend everyone read the novel for themselves.)

It took me a while to realize just how fantastic a resource wikipedia could be, and , as some of the research I've done shows, a fantastic object for future study itself. The way the website develops, what kind of information can you find there, and how are conflicts over the entries resolve, are all questions worthy of deep study by a whole variety of social scientists. I have found some enteries that discuss how teachers can form whole entire lesson plans on wikipedia, not the information in the enteries, but the website itself. I contend that historians, anthropologists, sociologists, behavorial psychologists, and many more could benefit greatly from researching wikis. They create microcosms of human activity that serve as perfectly sized samples for detailed and reliable investigation. I was reminded of another idea by Douglas Adams, when he explained his fascination with computers. He contended that what made computers great wasn't so much anything we already used them for, but what they represented, which was the means to model anything. A computer can model a typewriter, a calculator, a television, the Sears catalouge, an ecosystem, a beating heart, anything. While so far computers have been models for either basic tools or advanced scientific concepts, I have been giving thought to how they could model something of use to social sciences and the humanities. I believe wikis are just computer models of cultures, total cultures worthy of study, with their own rules, traditions, beliefs, conflicts, and historical record, all of it constantly shifting exactly as any real world culture would. Thoughts like this have made me terribly excited about the possible uses of wikis. I know realize a wiki isn't a killer ap. It's so much better than that; it's a living application.

Even more teacher resources

I have recently come under a veritable bombardment of information concerning how information technology can be utilized in a classroom. Last week I heard from several people who each had found websites full of educational content, well packaged for the use of either teachers or students. All of these websites had distinguishing features and aspects that made them irreplacable for certain uses. The individuals had also come up with rather detailed plans as to how specific lesson plans could be drawn from their respective websites. I was especially fortunate in that the people I heard from shared an interest in social studies, something I clearly value. This made me think that perhaps the web's greatest asset t teachers is it's vitality. Their is a unique stigma associated with text books commonly used in school. No matter what one's background or use for the material, text books are declared dull, dry, cumbersome, and often outdated for classroom needs by almost universal agreement. While one must be discerning when scanning the web for reliable information, the simple fact is that no shortage of great ideas exist online. More are being added everyday, so no teacher can claim not to have fresh material from which to draw lessons. I believe the web could greatly accelerate the rate at which new ideas and new methods are introduced into classrooms, reducing the chances of children becoming bored with their classroom experience.
I also had the honor of hearing from one of the leaders of the Internet Public Library. This resource offers an banquet of reliable, helpful informatin for anyone intersted in learning. They are also responsive to public input, so they can best adapt the services they provide. I saw many examples of great work, however, the skop of a project like, just seems like more than I could every handle.
In thinking about all of this I was again reminded of the stunning speed with which our youth are adapting to the information age. It brought to mind some questions I have had over whether a teacher can truly hope to stay as technologically adept as her students, especially given her other responsibilities. If I have the oppurtunity to speak to a young teacher who has found uses for technology in the classroom, I'd ask if they'd yet encountered a situation where the students were teaching them, and if they have thoughts on how to deal with such an event.

The Wide World of Education Online

Having examined a few websites on my own, and witnessesed a wonderful presentation, -plus of course, I have been inspecting the blogs of others to hear about some resources they've found- I am beginning to take the Internet seriously as resource for education. I especially like all of the websites that provide a means to extend the classroom community outside of the school building and our already to limited class hours . I have been giving some thought lately to the issue of classroom time, and how limited a chance a teacher has to truly make an impact in the lives of her students. Thus I believe that the more a teacher can do to extend the learning outside her classroom walls and free up time for actual instruction in the class, then the more students will benefit from this use of technology. I also appreciate the sites that offer teachers either tools to ease the daily routines, such as those that help you create rubrics, or that can provide them with ideas for lesson plans or methods. I realize how taxing teaching can be, and any time a teacher can spend just preparing for class. Any tool that allows for more thought on education and less on dull routine is worthy to me.

Internet Education Resource

While exploring the web for resources that teachers can utilize, I began looking for web sites that could be useful to both teachers and students. Since, I focus on teaching social studies, and as the name of this blog would indicate, I decided to look for historical resources online. One site that has really attracted my attention is the University of Evansville's website on ancient world cultures, found at http://eawc.evansville.edu/. After exploring this site a little I believe it could have many positive uses in a lesson plan, but it does have several problems. The site seems to have been assembled by academics, which means that some of the material offered could seem dull or confusing to high school students. The internal navigation is not immediately understandable. Some of the links are not labled clearly, and many times it is unclear exactly what information a given link wil direct you towards. Many other links are dead, and I don't know how often such problems are corrected. Though since I have been visiting the site over several days, it would seem unlikely that the web site is updated often. Additionally, the website lacks much decor or interactivity, which may bore some students. Having said all that, I found the website contained an abundance of helpful historical information. They have stored many documents of historical significance, which students might not otherwise be able to find. I feel many lessons could incorporate these documents. The sight itself also has some helpful passages and quizzes that teachers could consider when designing lessons, though as mentioned earlier these might be a turn off for students. The website also presents examples of ancient artwork, and other cultural artifacts, to help students gain an appreciation of ancient cultures. Lastly, the website compiles essays of historical thought and research, which seem much more academic and instructor oriented in nature. I would suggest that a teacher, might what to only expose the student's to these, in easily to digest selections of the most compelling material. I will continue to explore this website and others to determine tha best use of this resource for an educator

Copyright v. Teachers

In considering the many weighty legal issues with which teachers must contend, I'll admit I'm at something of a loss. I had never realized the many different ways that one could violate laws invovling intellectual properity. It seems to me, that I have never had a teacher who did not violate these rules in some way. I realize that not all these violation would necessarily result in the teacher's dismissal or a law suit for the school. Yet, I also wonder if the problem has gotten so large, especially with the Internet, that the various companies and publisher who produce material to suport teachers are actually in financial jeopardy. I certainly would want to be as informed as possible, not only for my own leagal protection, but because I feel teachers have a moral obligation to serve as model citizens for their students. I believe I should do everything possible to incorporate my students into the information age. I fear that in pursuing that goal and opening up as many resources to my students as possible I am bound to run afoul of some copywrite issue. The articles we read noted that only a trained expert could really contend with all the complexities of intellectual properity laws, and I worry what might happen if I don't have such aresource available. I also feel that I will never be as informed as I should be because I am more likely to focus on complying with those laws that are meant to protect the children. I will be mindful of laws like CIPA, because as much as I value protecting copywrite, I'm goint to focus more on protecting my students from possible threats. I guess the best you can do is to stay as informed as possible, do your job, and plead ignorance in the case of any violations

re:blogs in the classroom

As I consider the possibilities of using a blog as an educational tool, I realize I should probably first address the many uncertainties I still have. First, I have had very little experience using a blog or even similar internet tools, such as message boards or chatrooms or whatever else is available. I will therefore need to spend some time understanding what is special about this particular tool, what capabilities it allows and which ones it limits, before I could even begin to evaluate its utility in education.
I understand the ability to have a shared space were students can share thoughts and ideas does have a special promise. As some of the articles on the topic have noted students will feel more likely to express themselves on a blog than they will in a classroom. While I want to encourage as many ways for students to communicate with eachother and the teacher I worry that the blog may replace normal classroom conversation to some degreee, and effect the classrrom atmosphere I desire. While I would hope the class blog would be a venue to extend the mutually supportive and enjoyable classroom climate, I fear it could also become an arena of anonymous abuse. I enjoy the idea of creating projects for the entire class to cooperate upon using the blog. Perhaps students could create a large paper where individuals students would be responsible for creating sections of the work, and also with coordinating with eachother to create a web based presentation.
On a different level I believe using tools like blogs will become as important as poster or presentation projects. Schools and teachers have long accepted that the safety of a classroom enviroment would be a good starting point to help teach students skills such as public speaking or display design. We agree to this idea because we realize that the professional world will likely expect them to communicate information in this manner. With the increasing use of information technology in corporate America I do believe that this type of web based presentation skills will be just as desirable at some point. I therefore believe a proactive teacher and educational system will construct support structure to develope and harness these skills from an early stage.
For the time being I like the idea of occasionally incorporating a blog into classroom projects on an experimental basis. After all it is only through exposure that we can truly gain a proper understanding of such things. In my classroom I feel it will be a while before I start using such tools, not due to reservation just my own need to build up my skills and understanding so that I can most effectively guide my students learning.