Archives (page 41 of 97)

February 22, 2012 / 0 Comments

Exploring matrices

I introduced matrices to my students last week. Together, we worked out the algebra necessary to find the inverse of a 2 by 2 matrix, and developed the idea of the determinant of the matrix. The algebra was hard for my students, and we focused on looking for patterns. I showed students how a matrix can be applied to solving simultaneous equations, so they understood that there is some context for matrices. Next week, I intend to show some of the applications of matrices to game theory, and have students explore the consequences of this application.

Today, I talked about the inverse of a 3 by 3 matrix, which was easily found on a calculator. However, for good or for ill, students are expected to know how to find 3 by 3 matrix inverses "by hand" in preparation for their IB Math SL exam. So we needed a way to understand how the inverse of a 3 by 3 matrix is formed. I showed students that one can construct 2 by 2 matrices within the 3 by 3 matrix, and that the determinants of these 2 by 2 matrices "magically" appear in the inverse of the 3 by 3 matrix. The proof of this (for 11th grade students) is hard.

So instead of going through the proof, I decided that students should explore the relationships between the positions of the 2 by 2 embedded matrices, and where their determinants appear in the 3 by 3 matrix. I don’t myself have an intuitive sense of exactly where these determinants will show up, but I know there is a pattern, and that my students will find it.

What was fascinating to me is the different ways students represented the notation necessary to show these patterns.

Examples of student notation. Click to enlarge.

Each student came up with their own notation to represent the patterns they were finding. They realized (some of them with some guidance) that it was pretty critical to include the location of the 2 by 2 matrix in their notation, to make it easier to find patterns. The actual notation they use doesn’t matter to me, as if they do continue with matrices, they’ll learn the appropriate notation later. What was critical for me was that they could come to some understanding of how to find the inverse of a 3 by 3 matrix.

What I found most interesting about this activity is that there is room for exploration in learning matrices, which suggests to me that it is very likely that any mathematical topic has some opportunity for exploration.

February 21, 2012 / 13 Comments

What is the purpose of social media?

I’ve begun to question the use of social media. I am finding Twitter to still be a valuable tool for connecting with other educators, but over the past couple of years, I have noticed that the #edchat channel has become more and more cluttered with advertisements and links, and there appears to be less discussion occurring.

When Clifford Stoll suggested that computers had no place in education, he said:

“Every voice can be heard cheaply and instantly. The result? Every voice is heard. The cacophany more closely resembles citizens band radio, complete with handles, harrasment, and anonymous threats. When most everyone shouts, few listen.”

To be clear, I don’t agree with Clifford Stoll’s assessment of the use of computers in schools. Computers can be powerful tools for education. Are they always used for the most productive purposes? Definitely not, but they have that potential, provided we (as educational technology enthusiasts) provide appropriate support and guidance, and that the teachers using the technology are thoughtful in its use. However, Stoll’s observation that there is an awful lot of noise in the Internet is totally true.

Neil Postman had the following to say of our information age:

But what started out as a liberating stream has turned into a deluge of chaos. If I may take my own country as an example, here is what we are faced with: In America, there are 260,000 billboards; 11,520 newspapers; 11,556 periodicals; 27,000 video outlets for renting tapes; 362 million TV sets; and over 400 million radios. There are 40,000 new book titles published every year (300,000 world-wide) and every day in America 41 million photographs are taken, and just for the record, over 60 billion pieces of advertising junk mail come into our mail boxes every year. Everything from telegraphy and photography in the 19th century to the silicon chip in the twentieth has amplified the din of information, until matters have reached such proportions today that for the average person, information no longer has any relation to the solution of problems.

When we post endless links after each other in Twitter (in what seems to be an effort to increase our own online profile?) and forget the social aspect of social media, we contribute to the noise. I can remember going through a phase myself where I was using scheduled tweets so that I could be posting all day and night, and fortunately, it did not take me too long to see the error of my ways; I too was contributing to the noise of the Internet.

While the regular #edchat discussion was happening today, I noticed that the stream was littered with off-topic links, mostly by well meaning people looking for some exposure for their product, service, or exciting news from their part of the world. These posts are inevitable as we want to share what we are doing, but we also need to remind ourselves of purpose of social media; it’s not about attention, it’s about communication and collaboration.

There is some room for sharing resources and links, but we need to be mindful of what the ratio of noise to conversation is at, and limit ourselves to sharing only that which is most valuable, and ideally share it outside of times people are using a particular hashtag to have a discussion. Obviously a link can extend the conversation, and where possible, we should post links which extend or challenge our thinking. We need to post a few less links, and have more discussion.

Howard Rheingold says, “If we decided that community came first, how would we use our tools differently?” The purpose of social media is to connect to other people. Let’s remember that when we post, please.

February 20, 2012 / 3 Comments

Taking advantage of the mobile nature of a mobile device

An iPhone (or any other smart phone / tablet) is a mobile computing device. Applications that are designed for a mobile computing device should take advantage of the mobile nature of the device. Too many educational (cr)apps are designed simply as better flash card systems. They rarely take advantage of the most important affordances of the mobile devices they are on, and are easily replicated without using the technology.

Not only can you take pictures and videos of the world wherever you are able to travel, with a little bit of hardware, you can turn your iPhone into a mobile microscope, allowing you to view the microscopic world, which combines the mobile nature of the smart phone with its computing power. Your GPS in the iPhone allows you to participate in Geocaching.

Have an idea on the go? Use your smart phone to record a note about the idea, or a create a podcast on the fly. You can use the Internet capability of your smart phone to collaboratively keep track of data (or anecdotal observations) when out in the field. Heading out bird watching? Keep track of your GPS, a photo of the bird, and any other anecdotal evidence you need with that one device in your pocket.

The point is, try and find the educational uses of Smart phones which actually take full advantage of the capabilities of the phone, rather than limiting kids to using the phone as an extremely small computer screen.

February 14, 2012 / 5 Comments

Questions about physics from kids

My son has some questions about the world. I wonder how often these kinds of questions are asked by children? My suspicion is that almost all children ask questions like this at some point, and that how we react to their questions has a lot to do with their willingness to experiment and figure out the answer for themselves.

Update: I showed some of these videos to a colleague at work, and she told me that her kids ask the same kinds of questions. I had a thought; does anyone have a project where they keep track of questions about science (or mathematics) that kids ask?

February 13, 2012 / 0 Comments

Presentation: Programming in Mathematics Class

This presentation is based in part on the TED talk Conrad Wolfram gave a couple of years ago, and on some insights gained at the Computer Based Math summit I attended in November. The below presentation is slightly abreviated to make it easier to share on the web.

February 13, 2012 / 0 Comments

Presentation: Where is the technology?

Here are the slides for a presentation I did at the University of Alberta Technology Fair.

February 2, 2012 / 5 Comments

Profile of a phishing attempt

(Click the photo to view it larger.)

Last night, I got another one of the many direct messages I receive each day via Twitter telling me that someone has written something horrible about me. Since I was using my older computer, and I have Ubuntu installed on it, I decided to click on the link provided in the direct message, deciding that the risk of accidentally downloading a virus was minimal.

The page took a while to load, almost 3 or 4 seconds, and then this page showed up. I was a bit surprised for a second, and thought, hrmm why am I back at Twitter, and why am I not logged in? I reached for the keyboard and was about to type in my password, when I stopped myself and thought, "I should check the URL first." I’m glad I did.

If you look closely, you’ll notice that the URL for this site is not quite right. The word Twitter has an extra i and v in it that shouldn’t be there.

I realized that this was a very clever phishing attempt, and that I had almost fallen for it, even though I knew in advance that the link was very likely to lead to trouble.

My recommendation is to be very suspicious of links you receive via social media and email. If the link seems out of context, or you aren’t expecting someone to be sending you a link, don’t click on it. If you do click on it, DO NOT enter your password or other information on the site. Instead, navigate by yourself to the appropriate website, and enter your login information there.

Please share this information with others so that we can curb the link baiting and phishing going on now through Twitter, email, and other similar services.

More examples:

Update: If you happen to get caught by one of these phishing attacks, don’t panic. You just need to change your Twitter password. If you cannot access your Twitter account, you should still be able to request a new password be sent to your email here.

January 31, 2012 / 6 Comments

Exploring multi-user online text spaces for collaborative game building

(Screenshot from the opening screen of Arctic MUD)

I have a confession to make. When I was in university for my first degree, I spent way too much time playing games, and not as much time as I should have on my studies. One of the games I played was called a multi-user dungeon (or MUD).

When I was asked recently by a colleague to help her find a typing game for her students, I was reminded of my experience playing MUDs. A MUD is a game written entirely in text, and their use predates the Internet itself. You enter the game through a specialized client, and type in commands, and read the results of your commands on the screen. Whole worlds can be created in muds. One of the MUDs I used to play had thousands of rooms players could move through, each of which had a description of the room. These descriptions were important, and because there was information in the room descriptions associated with mini-quests in the game, you had to read them thoroughly, and pay attention to details.

I took a typing course in high school, but none of the typing skills took. The course was incredibly boring, and my memories of it are of typing asdf asdf asdf over and over again. I did not learn typing from the typing course I took, I learned how to type as a result of spending hours and hours playing a MUD in school.

Every command I sent the mud had to be sent via the keyboard. There were no options to click on buttons to make the game work, you had to type. As a result, I had a strong incentive to learn how to type faster, because by typing faster, I made the game go faster. Typing faster gave me more control over the game, and this was amazingly empowering. Typing accurately was important too. The game moved fast, and if I typed something wrong, my character in the game died, or the game just sent back an annoying ‘Hunh?’ when it didn’t understand my command.

The next thing I realized is that the different parts of the mud (called zones) are each stories. When I move my character through the zone, it is like I am moving through a story, and controlling the action and pace of the story. This is part of the reason kids are so excited to play video games today; they have some control over the pace of the story, and the flow of the game. Today’s video games are excellent places for students to explore narratives, but they aren’t text narratives, and it is still important to understand text narratives.

Mud story
(Screenshot from a zone on Arctic MUD called "Forest of Haven", created by Draknor)

The zones in a mud include things called mobs, which are non-player characters controlled by the computer. These mobs can be considered to be bit actors in the stories of a mud, and the players are the main characters. Imagine that instead of just reading about Tom Sawyer and how he tricked the other boys into whitewashing the fence, that you could join him in this mini-quest in the story, and interact with him and the other characters in the book. How would that change how you viewed the book?

I thought that muds could be a way that students could read, access, and see an interpretation of a story, but they could also create interpretations of stories. One of the advantages of muds has always been that the source code for many muds is free to download and modify, and that the resource required to run a mud are fairly minimal. The editing systems for most muds is reasonably archaic, but its not too complicated to create simple effects, certainly well within the reach of most students.

A group of university students created an online interpretation of Dante’s Inferno as an exercise for one of their literature classes. When the mud was active, you could literally walk around the layers of hell as described by Dante. Can you imagine what it must have felt like to have created Dante’s Inferno?

I see some interesting things one could do with a mud, but I need some feedback. Is this just a crazy idea?

Update: I just found out about Moose Crossing, which is a decades old project to do exactly what I am suggesting but with more control given to students than I had originally envisioned. Awesome.

January 30, 2012 / 0 Comments

On the evolution of schools

Wired Magazine recently reported an interesting story about a study suggesting that massive changes in the size of a species could taker as long as 24 million generations to occur. This process is very slow, and part of the reason it takes so long is that there is no one directing the evolution of the species; it happens as a result of a series of incremental (and sometimes contradictory) selective adaptations of the species to its environment.

Schools have taken glacially long to adopt changes, with some classrooms still approximating those of the 19th century. Maybe the problem has not been a lack of leadership, but a lack of patience? If we really wanted the classroom to evolve, maybe we would place external pressures on classrooms which would encourage a different direction, and then be patient as the process of cultural revolution occurred. Perhaps this is the role of the computer; not to itself force change in education, but to change the environment around schools.

January 30, 2012 / 2 Comments

Where is the technology?

The following is an excerpt from a keynote I will be giving on February 13th for the University of Alberta.

There is a technology our students use that is destroying their ability to learn actual mathematics.

This technology is so advanced that you can use it to find any formula. In fact, almost all of mathematics we know is available via this technology. Some students, lacking the resources at home, are able to go to the library and use this technology and learn all of the mathematics we teach ahead of time, thus destroying our lessons during the year. Other students can even access this technology from home!

When using this technology, students don’t need to think. It can record every step they do during a problem so they don’t need to actually remember where they are. In fact, if they have done a similar problem before, they can just copy the steps from the previous time.

Students are using this technology to share ideas outside of class; plagiarism and cheating is rampant because of the ease of sharing provided by this technology. This technology is inequitable because not all students have equal access to it at home. Some of our homeless students, for example, do not have access to this technology at all.

Further, the production of this technology is damaging our ecosystem, and we throw millions of tons of it away each year. It is an environmental catastrophe in our schools, and something must be done.

Yes folks, it’s time. We need to stop using paper in our mathematics classes.

Pencil writing on paper
Photo by Simon Stamm

Sarcasm aside, this objection to paper, pencils, and writing in education is a real one, at least in the age of Socrates, who (according to Plato) said,

"…this discovery of yours will create forgetfulness in the learners’ souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves."(Dialogues of Plato, Phaedrus, pp. 275).

It is true that pencils and paper require the person using them to know less, and to allow the medium of pencil and paper to hold information for them. Pencil and paper can be thought of as a solid state storage device with a low ratio of information to volume capacity, and a terrible indexing system. Pencil and paper are like a hard-drive for the mind.

However, the fact that pencil and paper require less memorization of information does mean that they result in less thinking. In fact, our entire civilization’s technological advances rest on the fact that we first created a very portable, reusable, and reasonably durable medium upon which we could record our discoveries for the next generation. What will happen to our society when the fundamental medium of communicating ideas changes?

The photocopier did not change education. It made the lives of teachers easier, but very little about the pedagogy the teachers used with their teachers changed. Teachers taught lessons, and students practiced what the teachers taught through their worksheets. Penmanship suffered, since students were no longer required to carefully transcribe the worksheet from the blackboard, but the trade-off was that students and teachers got more time to discuss the ideas being presented.

The Internet, on the other hand, will change education. As we become better connected as schools through the Internet, good educational practices which once took decades to take hold in the education world, will begin to spread more rapidly. We will find new ways of connecting our students with content, and it will become less important to attempt to contain every important fact about the world in our heads. Our memorization skills will suffer, but the trade off will be that we will have access to a much larger set of information as a result. Knowing facts will still be important, but knowing how to learn new facts will increase in importance. The computer, as a tool to interact with the Internet, will become a necessity for all schools.

Unfortunately for education, some of the ways computers have been introduced into schools have been somewhat ridiculous, to say the least. Seymour Papert used the analogy of the pencil to show just how nonsensical some schemes for introducing computers into schools are. He said,

"Imagine (if you can) that we lived in a world without writing-and, of course, without pencils, pens and books. Then one day, somebody invents writing and the pencil, and people say, “Wow, this would be great for education. Let’s give these things to all the children and teach them to write.” So then somebody else says, “Hey, wait a minute. You can’t just do that. You can’t just give every child a pencil. You’d better start by doing some rigorous experiments on a small scale. So, we’II ‘put one pencil in a classroom and we’ll see what happens. If great things happen, we’ll put two pencils in a classroom, and if greater things happen, then we’ll put in more…" (Papert, S. 1984.)

Our generation has a new technology, the computer, to replace paper and pencil, excepting where paper and pencil is still more efficient or more educationally sound. A reasonable question to ask is, to what extent should the use of a computer replace the now traditional use of pencil and paper in schools? An even more reasonable question to ask is, how does school change with the introduction of computers?

There will come a day when our descendants will not look at computers as a type of technology anymore than most of us see pencil and paper as a form of technology. In that time, some of the concerns we have today will be seen as antiquated, and even bizarre.

References:

Papert, S. (1984). “New Theories for New Learnings.” School Psychology Review, Oct. via http://dailypapert.com/?p=755

Plato,. (1875). The dialogues of Plato. 2d ed. Oxford: Clarendon press.

The Reflective Educator