Education ∪ Math ∪ Technology

Month: February 2012 (page 1 of 1)

Let them solve the problem

Arts week poster

When I came in this morning to school, the poster in the photo above had fallen down. I noticed it, and thought to myself, "I should fix that." I went and dropped off my stuff, and when I came back to the poster, one of our students was standing there staring at the poster. He said to me, "It looks like the poster came undone here," and he pointed to a spot on the back of the poster. "I’m going to fix it," he said.

"Do you need any help?" I responded.

"Nah, I can do it. Thanks anyway," he said.

Later that morning, the student said "It’s a bit crooked, but it’s up." "It doesn’t matter if it’s a bit crooked," I replied, "The most important thing is that it’s up."

I could have stepped in and solved this problem for my student, but then he wouldn’t have learned how to solve it himself. "Be less helpful," says Dan Meyer of the math class, but of course the same is true all over the place in education. The objective isn’t to show students how to solve problems, our objective is for students to learn how to solve problems. Sometimes our role is to stand back and let students solve problems for themselves. Will they always come back with the "perfect" solution? Probably not, but they will have learned more in the process.


Sign fallen down again

So it turns out the solution the student tried didn’t work. When I talked to him, he pointed out the flaw in his solution, and suggested a solution. In other words, the problem he solved gave him feedback directly as to whether or not his solution worked.

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.

First student notation Second students notation Third students notation

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.

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.


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.

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?




Profile of a phishing attempt

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:

Another phishing attempt


Facebook app phishing


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.