Year: 2012 (page 11 of 14)

April 23, 2012 / 50 Comments

Mathematics education blogs

Here is a list of people who blog about mathematics or mathematics education. Please let me know if you blog about mathematics education, and you’d like your blog to appear in this list. I’d like this list to be exhaustive, rather than exclude people. You can either contact me, or add your relevant information to this form.

So far this list has 347 (!!) blogs. Now that Google Reader is going to be taken offline, I’ve moved the list to a spreadsheet instead of a Google Reader bundle. You can access it by clicking below.

Find the list of blogs here

April 22, 2012 / 8 Comments

Should students learn how to graph functions by hand?

Software to create graphs of all different kinds electronically is ubiquitous. There is no question in my mind that we do students a great disservice if we do not give them opportunities to learn how to use at least a few of these programs. That being said, does the use of these programs potentially make learning concepts related to graphing, or through graphing, more difficult than it would be if the students used traditional paper and pencil graphing techniques?

It should be clear the skill of graphing is different when using technology. Some tasks, like choosing an appropriate scale, which are typically difficult for students are much easier using technology. One can replace time spent learning how to space lines correctly on paper with time spent learning how to choose the space between the lines in the software. In either case, time should be spent on visual design principles and why we might want horizontal lines in the first place.

One problem with using technology for graphing, especially when the purpose is to use the graph to determine a relationship between variables, is that the technology can potentially make the job of graphing too easy. A mind, recognizing that a task is easy, can potentially put insufficient energy into the task, and the mind’s ability to distinguish patterns is reduced. This born out by research on the effect of font type when people learn through presentations, and by Veritasium’s research on effective science videos. A mind insufficiently challenged, either by the task of character recognition, or on it’s misconceptions, is a mind that is less likely to learn.

On the other hand, some very useful learning tasks are so difficult to do when using paper and pencil techniques as to be pointless to do. These tasks can be much more manageable using technology. For example, the standard equation of a parabola, y = ax^2 + bx + c, can be explored through a graphing program. What effect does changing the values of a, b, and c have on the equation? Try this task with paper and pencil and then with technology to see why I ask students to do this task with technology. See the applet below for an example of this (requires Java).

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

There are many graphing tasks which are typically learned very poorly by students. From my experience, there are many middle school students running around with muddled concepts of the equation of a line, wondering what the ‘x’ in the equation is for, and being asked to learn mechanical tasks related to this y = mx + b. I usually find that some exploration of the equation of a line in graphing software typically clears up at least some of these misconceptions.

Some tasks when done with a graphing program can disguise important concepts. There is something to be said for visually placing points on the coordinate plane for understanding coordinate systems, for example. I don’t think it matters if one uses a mouse for this activity or if one draws the point with a pencil, either way, the student has carefully chosen the location of the point. If one types the coordinates into a text box, and just sees the point magically appear, I suspect that one will find learning coordinate systems more difficult. This suggests that the choice of software matters, since some software will let you plot individual points "manually" and some software does not.

I suspect that these issues are less about which technology we use, since paper and pencil is itself a form of technology, and more about how we interact with the technology when learning graphing (or any other mathematical technique). We need to think carefully about what the technology allows us to do, and what underlying concepts we want students to learn. It may be that some concepts that used to be fundamental no longer are with the new technology, and other concepts become more important to learn. I suspect that insufficient research has been done on how pedagogy should change with the use of various technologies in mathematics, particularly ones that change so fundamentally the task of the student.

April 18, 2012 / 4 Comments

Online education is as effective as face to face instruction

Online education is as effective as face to face
(Image credit: ASU presentation)

The research highlighted below the statement "Online is as effective as face to face" was used in a presentation at the ASU Education Innovation Summit to justify students in a k to 12 setting taking online courses.

From the first meta-analysis written by the US Department of Education,

"The meta-analysis found that, on average, students in online learning conditions performed modestly better than those receiving face-to-face instruction…"

Sounds promising, let’s read the rest of the abstract, shall we?

"…An unexpected finding was the small number of rigorous published studies contrasting online and face-to-face learning conditions for K–12 students. In light of this small corpus, caution is required in generalizing to the K–12 population because the results are derived for the most part from studies in other settings (e.g., medical training, higher education)."

That sounds a lot like the authors of the meta-study specifically recommended against using this meta-analysis as support for online learning in a k to 12 setting. I wonder why the presenter used this study?

We cannot draw conclusions on the effectiveness of online learning (as opposed to blended learning – wherein a student learns from a mixture of online and classroom activities) for k to 12 students based on the effectiveness of online learning in a post-secondary setting.

The motivations of k to 12 students and post-secondary students are different.
Many post-secondary classrooms do not represent the most effective pedagogy, so it may be easier for online learning to be equivalent or superior.
Access to resources necessary to be successful in an online setting (like a computer) are more prevalent in a post-secondary setting.

April 16, 2012 / 5 Comments

A problem with digital books

A bookshelf full of books
(Image credit: Harold Bakker)

When I was growing up, my house had a 50 foot wall full of books which our family had inherited from my grandfather with the house. Whenever I was bored, and the weather was ugly, I would pick a book off the shelf and read it. I devoured books from the shelves, some of which were probably inappropriate for my age, and some of which are considered classics. The library available in my home helped me become a better reader because I never ran out of something to read.

This is an experience my son has now, because we have a library of books in his room (albeit a much smaller library). However, as books become digital, it will be much more difficult for our library to be visible and accessible to my son. Parents will tend to make more choices of what their children should read. Licensing on books that prevents them from being copied means that children will likely have to explicitly ask for permission for every book that they read from their parent’s collection. Books will be selected less at random, and because we will be more likely to select books entirely based on our own interests, we will be less likely to be exposed to new information. Note that it would be relatively straight forward to set up "book servers" in houses that could act as personal libraries, and serve the same function, but current digital rights management on books makes this impractical (not impossible, every DRM has its weaknesses).

We are headed toward a society where books are not visible and accessible in our houses. If the books are invisible, they might as well not be there.

April 16, 2012 / 2 Comments

Separate science history from science inquiry

Zombie Feynman on Science
(Image credit: XKCD)

It occurrs to me that we have two goals for science education. One is to teach students what existing science is known, and how it can be applied to our lives, or how it is interesting to us. I call this first purpose, "Science History." The other goal is to teach the process of doing science, of thinking scientifically. This purpose, I call "Science Inquiry."

I think we should separate these two purposes into separate courses or domains, because the purpose of the first is diluting the effect of the second. Many children finish school thinking that science is a collection of facts known about the world, and do not spend enough time learning how those "facts" were derived.

Labs are a good start to learning science inquiry, but many experiments done in labs have issues.

The labs are rarely designed by students. This leads to underestimate the difficulty in designing a good experiment, and to over-emphasize the paperwork portion of science.
The labs rarely take more than 30 or 40 minutes to complete. Students rarely have to repeat a lab because of experimental error. They learn from this experience that laboratory science can be easily parcelled into sitcom-like episodes.
Students do not learn enough about the reasons why we have designed lab reports, and think of the portions of a lab report as blanks to be filled in. Quite often they will fake data so as to complete the boxes faster.

A Science Inquiry course would focus on the process of doing science, and less on the students learning existing scientific knowledge. Obviously students would be likely to find connections between the Science History course and the labs that they design, and hopefully they will also see connections between their Science Inquiry and other domains of knowledge.

A Science History course would focus on what existing scientific discoveries we have made, who made these discoveries, and what are the stories around these discoveries, and how these discoveries impact our lives. It might cover some principles behind the philosophy of science, as well as the connections between science and other domains of knowledge (like math for example).

Right now, many schools see Science Inquiry as optional or even an inconvenience. This suggests to me that some people think that thinking scientifically is an inconvenience or too troublesome to teach, and this scares me. While "thinking scientifically" isn’t the only way to think, it’s an important one, and certainly, we should all learn how to do it. I also think that Science Inquiry is indistinguishable from thinking scientifically. If you remove the inquiry, then it isn’t really science.

April 12, 2012 / 3 Comments

Tinkering for students

I watched Caine’s Arcade yesterday (see below) and while it is a bit sad to me that an amazing kids’ endeavour has turned into an opportunity for a film-maker, the movie itself is very touching. I recommend watching it.

Caine is not an unusual child, but he has had a number of unusual circumstances which have allowed him to create his arcade. First, he has dedicated space that has been created for him to pursue his interest. This is unfortunately extremely uncommon for many children today. He has parents who completely support their son’s exploration, and act as mentors to assist him. He has access to the materials and tools he needs. He is not judged on his creations by a rubric, or by a test, but rather his creations are seen as part of a description of who he is. The creations Caine has made are only physically distinct from him; they can be seen as an extension of himself.

Tinkering to me means, the ability for kids to create and explore in ways not bound by the current rules and structures of schools. The original explorations of programming using Logo to me were more about tinkering than computer science. Unfortunately, over time, tinkering on a small scale looks like a useful activity for students to do, and so it gets scaled up. For example, this page about using Logo in learning now includes sample projects with "step by step" instructions. When creation becomes scripted, it ceases to be creation, and becomes assembly.

So instead of scaling up tinkering, what I’d prefer to see is individuals given freedom and resources to produce what they think tinkering space should look like. Instead of thousands of identical tinkering spaces, each space should be unique and suited to the community in which it exists. The explosion of art work created in community centres has happened not because we tried to create a formal structure for people to do art in society, but because we provided spaces and resources for these centres to develop.

I’d like everyone to have the opportunity Caine has, to explore the world and build extensions of themselves in it.

April 12, 2012 / 1 Comment

Computing in schools

A possible future for computer science in schools

Gary Stager posted "Dumbing Down" a few days ago, which is a passionate plea for computers to be used for computing in schools. He writes:

Although I’m only 48, I have been working in educational computing for thirty years. When I started, we taught children to program. We also taught tens of thousands of teachers to teach computer science to learners of all ages. In many cases, this experience represented the most complex thinking about thinking that teachers ever experienced and their students gained benefit from observing teachers learning to think symbolically, solve problems and debug. There was once a time in the not so distant path when educators were on the frontiers of scientific reasoning and technological progress. Curriculum was transformed by computing. School computers were used less often to “do school” and more often to do the impossible.

Gary’s argument is (mostly) sound, and indeed, I argued for almost exactly the same thing in the keynote I presented in Alberta in February. Of course, both of our arguments have a flaw.

Our purpose in introducing computing is to both use the full power of a computer in schools. Indeed, the way we currently use computers in schools is much like using cars for their heaters; sure the cars will keep us warm, but it completely misses their potential as transportation devices. This analogy is somewhat apt, since the computer (as Dr. Papert has pointed out) can be used to transport kids to Microworlds.

Peter Eden said:

“I am interested in your comment that the power of computers as a tool is almost superficial until you learn to program. Others peoples’ programs are like using the computer to do something that can be done by other means. Word processing is really just typing. Many maths programs are really just calculators. A database is really just a record keeping system. But once you begin to program a vast new world opens up. Everything you program becomes a new tool. Because its your tool you can modify it. You modify it in ways you never imagined when you began.” (Peter Eden, personal communication, January 29th, 2012)

So we agree that learning how to program is an excellent endeavour, and one that basically everyone should learn how to do. What I think Gary and I disagree about is whether or not this particular learning should happen within the formal structure of schools.

Gary points out in his article that they had "tens of thousands" of teachers involved in learning how to program so that they could teach their students. Tens of thousands is a lot, but there are millions of teachers world wide. Tens of thousands is a drop in the bucket compared to the number of people who are teachers.

If we were to teach them all (or a sizable subset of all of the teachers) how to program so that they could teach their students, we’d have to institutionalize learning how to program, and I think that this would be a disaster. We’d end up with benchmarks, prescribed curriculum, and standardized testing.

I did a mathematics degree, and one requirement of the degree was that I take a course in computing, which I think is a perfectly sensible requirement. The problem was, I had to do a 1st year computer science course, and this course was ungodly boring. It was so boring, that despite attempting twice to finish a 1st year computer science course, I gave up, and did a "Computing for Mathematicians" course instead. Of course, I knew how to program already, so the programming skills themselves were not very useful to me. However, what I learned from this experience is that it is tremendously easy to take something full of life and turn it into something deadly dull. If every student was forced to endure the same kind of learning I experienced during that 1st year computer science course for 12 years without the opportunity to opt out, none of them would ever touch a computer again.

It is common for instutions like schools, to take endeavours which are exciting and interesting on a small scale, and attempt to bring that exact same experience to everyone. Unfortunately, most often these endeavours pick all of the wrong parts of the activity to "scale up." In scaling up mathematics education, we took an experience where people mostly played around with ideas, and turned it into fill-in-the-blank worksheets, completely destroying the purpose of learning mathematical thinking. Computer science in schools would fall into the same trap as science education has, which is that people think the purpose of science education is to teach facts about science, instead of a way of thinking.

What I would prefer is for space to be created outside of instutions for this type of thinking to occur. Much like we have community centres for art, and for physical activity, we could have recreational centres for computing. Instead of instutionalizing (and eventually centralizing) the learning of computing, I’d like to see it de-instutionalized. I’d like to see a thousand different models for learning computering rather than the inevitable staleness that would occur if it were introduced en masse to schools.

April 6, 2012 / 0 Comments

The language of technology

Technology has a language, a history, and it shapes our culture. While the focus of this article is on the language of technology apparent in Microsoft Word, every technology we use has similar traits.

Check out the "ribbon" (or menu bar) of Microsoft Word.

First, when you examine Microsoft Word, and many other programs like it, you’ll notice that there are many visual cues within the program as to how it works. What is less obvious is that each of these visual cues relies on the person viewing it to understand what the cue means. These visual cues are a form of language, and it is often this language which poses a significant barrier to using the technology. If you don’t understand the language being used, then every function of the program you want to use requires you to memorize the sequences of steps needed for that function.

Look at these examples of how language is used in Microsoft Office. It’s probably somewhat obvious what B, I, and U stand for, but what does the little triangle to the right of them mean? My mom didn’t know what the ~~abc~~ meant, and I know some people don’t know what the x₂ and the x² mean either.

Formatting

The question mark might be obvious, and it might not. It doesn’t look like a button, so one might not know that one can click on it for additional information. Further, buttons themselves are a form of language, and so even if this were shaped as a button, someone could conceivably not know it was something to be clicked on.

Help icon

In the margin bar, there are a couple of interesting 5 sided polygons lined up on top of each other. These are intended to indicate the different types of margins and indents you can apply to the document. Why are these icons chosen?

Margins

The origin of the icons chosen for the shape of the margin/indent icons in Microsoft Word appear to be very similar to the shapes originally used to represent the same functionality on a type-writer. Why those shapes were chosen for the type-writer, I don’t know.
Typewriter
(Image credit: Awkward Science)

More examples:

Tabs

Triangle drop down

As a final example, look at the Save file icon. Virtually none of your students will know the origin of this icon since none of them likely grew up using a computer with a floppy disk drive. Still, they know that they click on it to save their work, but this is language they’ve learned from us.

Save icon

Instead of thinking of teachers or children as being digital immigrants or digital natives, we should think of their exposure to the language of technology, and how knowledge of this language influences their ability to use technology.

March 30, 2012 / 7 Comments

Another alternative to the traditional conference

(A typical conference presentation – Image credit: Emmanuelvivier)

I’d like to propose an alternative to the typical conference model. Chris Wejr got me thinking after he sent me a message suggesting that we host a conference sometime in 2013 that he called a ‘hybrid conference’ and this post by John Burk also influenced my thinking as well.

A typical conference

Some of the problems with a typical conference for many people are:

They don’t know anyone at the conference before they attend it, and so connections they could potentially make at the conference are not made,
At the conference itself, too much time is spent by presenters talking, and not enough time is spent by participants assimiliating what they learn,
Most conferences have no follow-up after the conference.

The best parts of a conference (in my experience) are:

The ability to meet and discuss ideas with other people in the same field as myself,
Being inspired by people doing amazing projects, and who give awesome presentations/keynotes,
Learning about ideas outside of our own personal areas of expertise, in other words, being pushed by others to improve ourselves.

A different conference model

First, we would assign people to cohorts (based on their interests, or on questions they answer during registration) after they register, and setting up email lists (since most people are more comfortable with email than with other social media, and it would automatically provide records of the partiicpant conversations) for those cohorts, along with a facilitator for each cohort. The job of the facilitator is to provide information to the cohort about the conference coming up, and to encourage conversation and introductions between participants before the conference. These cohorts would also be sent links to video presentations (which should be broken into small chunks and include searchable transcripts of the video) that they can watch in advance of attending the conference. Ideally, presenters would be part of these cohorts.

The people would then attend the conference, and potentially move around through their sessions (which would have to be scheduled in advance, like a typical conference, but with input from the registrations) as a cohort, with sufficient opportunities during the sessions to connect and discuss the ideas, or at least between sessions. Ideally each session would be run more like a workshop, rather than a lecture, since most (if not all) of the people in the cohort would have already listened to the presentation. In some of the more advanced sessions, participants would produce a product as a result of their time together.

Social media could be used during the presentation as a back-channel, so that people from outside of the conference could learn from the participants, and share their ideas back to the conference.

Naturally, most participants attending the conference would know some other people there. They would have conversations, and they could choose to eat together. Obviously, if one wanted to continue through the conference as a solo participant, this would still be supported by this model, one would just choose not to interact with their given cohort.

After the conference, the cohort email lists could be used for follow-up, as well as other social media. People would be expected to continue to ask questions and discuss ideas, as well as share their successes (and failures) back to the group after attempting to implement whatever strategies, techniques, or resources they learned about through the conference. Provided the participants made the effort to seek follow-up, they would have an avenue to receive it.

This conference format would help mitigate some of the problems with the typical conference format, while not taking away any of the benefits. It would further have the benefit of allowing people who could not afford to attend the conference in person to still participate in many meaningful activities related to the conference itself.

It would definitely require more work from participants than is typically expected for a conference, and I’m sure this would turn some people away from attending this conference. That being said, those people I think rarely get very much out of typical conferences anyway, and I’d rather not build a new conference model based on the lowest common denominator.

Do you see any flaws with this model? Can you think of any ways of improving it?

March 20, 2012 / 4 Comments

Math in the real world: Gardening

My uncle called me today, and asked me a math question. Normally, I get called and asked technology related questions, but occasionally people remember that I have a mathematics background and call me in to assist.

My aunt wants to build a raised garden bed with a very particular shape. My uncle has been tasked with building it. She wants 3 of the sides of the shape to be 4 feet long, and the 4th side to be three feet long, and the whole shape should form a trapezoid (with a line of symmetry down the middle of the trapezoid). It took a little bit of chatting on the phone to get this to be clear, and I can see how being able to send each other pictures would have been really useful. To be able to build this shape as accurately as he would like, he needs to know all of the angles of the shape, so he can cut the pieces of the wood with the angles in the right position using a miter saw.

Trapezoid garden bed

I looked at the shape and decided that the fastest solution would be to build the shape in Geogebra, and measure the angles, which resulted in this.

Not the exact solution, but close enough that my uncle would be able to use the miter saw (which has a maximum accuracy of 1 degree, according to my uncle) and cut the wood for his shape. It took me about 3 or 4 minutes to draw the shape in Geogebra and measure the angles.

After my phone call with my uncle was over, I decided that I should double check this solution though, and verify that I knew how to solve it.

I drew an imaginary line across the shape, and labelled that side x. This allowed me to create a pair of equations using the Cosine law, and I ended up with the following equation to solve:

First equation

which simplifies to:

Second equation

and finally leads to this calculation:

Third equation

On my calculator, that leads to a value of the smaller angle of about 82.8° and a larger angle of 97.2°, which means that my diagram that I drew for my uncle is fairly close. Wanting to be sure that my answer was correct, I also checked it using Wolfram Alpha, and on my graphing calculator.

After I told my uncle the solution, he told me that my aunt had suggested drawing the diagram carefully on a piece of paper and measuring the angles with a protractor, but he had complained that solution wasn’t "mathematical enough." Of course, this leads to a discussion of what it means to do mathematics, anyway.

Does it matter which way I solve this problem for my uncle? Which of these techniques would you classify as "mathematics"? All of them? None of them?