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Whither Competition?

Now, this may seem like I’m contradicting the opinion of the guest blogger last week. However, I’m not referring to the endless pursuit of rankings and grades.

I’m meaning the fantastic things that can happen when competition is used as an instructional tool. I’m meaning having students race to solve puzzles, or sort number cards into Pythagorean triples. I’m meaning getting a wild energy in class, and having students speak up who never said a word before.

In most of the work on social technology I’ve been reading, it’s paired with the word "cooperation". What happened to competition? Why is it so wrong?

This concerns me because mathematics is particularly suited to competition. Some competitions are downright legendary. Solving problems in high-level math competitions can lead to thinking that shatters the hierarchy of intelligences, creating wonderful things I still don’t fully understand.

What’s wrong with students competing to solve an Internet math hunt, or students challenging each to ever-harder problems?

I simply implore technology coordinators: please consider the possibilities competition can offer.

Goodness, it’s been a week already? I feel like I’ve just nicked the surface of this territory. I’ll try to continue with things I couldn’t fit next week at my blog. I’d like to thank everyone for their comments; I learned much more than I ever suspected possible, and I’ll be closely following this blog and others for new developments.

Jason Dyer, Guest Blogger

Mathematics En Masse

So far I’ve covered a technological and an ideological problem. This one’s logistical.

Specifically, in a discipline where one question can have many answers, it’s easy to set up a traditional forum discussion where every student’s contribution is meaningful and a springboard for further interaction. One student’s opinion does not make moot the opinion of anyone else.

With mathematics, even if the question is applied and relevant, quite often there is only one correct answer. So the traditional forum method can fail, since once one student posts the answer, there is no incentive for further discussion.

This can be solved by splitting — having only individual answers or collaboration in small groups — but this removes the very thing that makes social technology exciting: collective intelligence on a scale above what is possible with in-person interaction. Imagine, for instance, having a problem not just group-solved by your individual algebra class, but by every algebra class in a school.

There’s ways around this, like:

1. Personalized data: Students might figure out the square area of a room in their house. So every student does their calculation on something different, and then statistical methods can be applied to the data as a whole.

2. Required estimates. In a mathematics modeling problem, students try to match real-world data with an equation. While there are "best-fit" equations this can be done by hand, so that there isn’t one correct answer.

Unfortunately, neither method fully taps into the power of the group dynamic. The individual contributions don’t rebound and augment each other like an open-ended discussion. So I’ll ask: what are the best ways to set up a collaborative online assignment in mathematics?

The best solution I can think of is to get students to explain process. The class could essentially write its own textbook with a wiki; the assignment one week might be to finish the online explanation of graphing parabolas, and every student has to pitch in. I have yet to try something like this, so if anyone has, drop a line in the comments.

Jason Dyer, Guest Blogger

The Calculator Wars

The use of a graphing calculator is considered an integral part of the AP Calculus course, and is permissible on parts of the AP Calculus Exams. Students should use this technology on a regular basis so that they become adept at using their graphing calculators. Students should also have experience with the basic paper-and-pencil techniques of calculus and be able to apply them when technological tools are unavailable or inappropriate.

(From The College Board’s AP Calculus calculator policy.)

Yesterday’s post was about a sticky tech issue. This one’s more about ideology.

Fierce debate raged in the 1990s over whether graphing calculators should be used in mathematics education at all. Proponents thought graphing calculators opened new vistas of understanding, as students could play and experiment, and see instantly how functions are affected by different tweaks. Detractors said graphing calculators killed the ability to work things out by hand; that they’re a crux, and true understanding is only obtained by pencil-and-paper repetition of the proper methods.

Arguably, the issue ended (in the United States) with the AP test adoption of the graphing calculator. What resulted was a deeper and more difficult test; not only did students still need pencil and paper methods, but they had to apply the calculator to answer higher-level questions.

(No, not every student will make it to AP classes, but I believe curriculum should be written with the assumption that some will.)

However, the issue never truly died, because there was (and still is) a hidden ideological struggle going on:

Should our primary focus in algebra be on symbolic manipulation, or is visualization and synthesis an important aspect?

In other words, are graphical methods and applications just an afterthought?

While this doesn’t seem to relate directly to social technology, the interesting enhancements computers can offer don’t attract the interest of a symbolic-manipulation-only teacher. Alternately, graphing calculators can be thought of as the gateway application — take a teacher comfortable with them, and it’s easy to hook them on related Internet apps. For the Internet to be truly useful, teachers need to see there is a world beyond factoring binomials.

So, if you’re a technology coordinator with a resistant math department, there’s one question you might ask: are there teachers who haven’t taken their class set of calculators out of their boxes? If so, there might be more going on than mere tech resistance.

Jason Dyer, Guest Blogger

On the Typesetting of Equations

Don’t get me wrong: there are options to put equations on a computer. Most computers have at least Equation Editor if not MathType. Scholarly papers in mathematics often use LaTeX and it’s also possible to blog LaTeX equations using WordPress. I have used these before (my Pre-Calculus students are currently working on blog posts with LaTeX equations) and they are fine in their own context.

However, I consider these typesetting solutions. What I mean is most closely analogous to the difference between a word processor and a desktop publisher. In a word processor the text can be played with, experimented with, refurbished and remodeled in a manner easier than even pencil and paper. The layout of the text, however, is secondary. With a desktop publisher the placement of text is key, or at least a distraction to a student still trying to figure out their sentences need verbs.

In the course of a normal math class, parts of equations need to be crossed out, circled, arrowed, underlined, and spaced out. When I laid out my solution to a general conic, I couldn’t use just an equation editor; I had to manipulate the parts in an image editor.

To put it another way, it’s difficult to think the equations directly to the computer screen; sometimes I have to work things out on paper first and then translate them visually. That’s hardly conducive to a live collaboration. It’s certainly still possible, but the pragmatic teacher has to ask if the loss of fluidity is worth the gain of remote interaction.

(Before you post about your brilliant online class and how everything is done on computer, keep in mind I am simply trying to explain where objections are coming from. I’m hoping it’s clear at least why the hurdle for mathematics is different than for language-based curriculum. Now, if you’ve seen these problems and have found neat ways for getting around them, post away.)

There are ways with technology to translate writing directly to digital form; on a SMART Board or a digital writing device. Some teachers will upload their SMART Board presentation to the Internet immediately after class, including any writing that happened. If anyone has done a particular lesson with these tools, feel free to share your ideas.

The ideal would be for the word processing equivalent of an equation editor. I’m not sure how that’d work; somehow it have to include the crossing out, circling, arrows and so forth we have students do in an entirely natural manner. There wouldn’t just be the ability just to write equations line by line as in a textbook, but to drop a single row and move a variable by using the inverse without any worry of typesetting. Take a look at my general conics example again; if that sort of layout could be done easily, with just natural typing, mathematics teachers would flock to your door.

Jason Dyer, Guest Blogger

A Clarification on Mathematics Technology

Before I go on, I need to clarify something: I am addressing the use of social technology, not technology in general. (Although an aversion to technology in general is related to my graphing calculator point.) My example of a tech cynic is not a technophobe by any means. I am referring to what is supposedly the "new wave" of blogs, Twitter, wikis, and so forth. Additionally while it could be argued math teachers in the United States don’t use enough of it, I am not discussing project-based curriculum. That’s been around for a long time and while it can be augmented by new technology, being a technophobe (by any definition) doesn’t mean shunning it either. So while a comment like (courtesy Glenn)

If the something that you’re asking them to do is take a test. Well. Never mind. Sorry, didn’t mean to waste your time.

might apply to some teachers, I am presuming a faculty past that point but yet fussy about all the new gadgets being thrown at them. I am trying to explain why they’re fussy, and that it’s not just because their teaching isn’t Modern enough in an overall sense.

Jason Dyer, Guest Blogger

Idealism and Reality in Mathematics Technology

So, you’re making your technology pitch to the school. You’ve just been to the conference and still feel the warm buzz of The Future, and you want the teachers to embrace the blogs, the wikis, the collaboration with schools in different cities, different states, different continents.

Then you meet resistance.

You up your game: you set up workshops, seminars, buy new software, buy new hardware, try to convert a few followers hoping entire departments will follow.

Maybe you grab the English department and suddenly have every student in school with their own blog. Perhaps the technology people are putting student-produced videos on YouTube. You hear the foreign language folks are using Skype to call Mexico and work with a network of 5 classes.

(Ok, ok, idealism here. But these things are at least possible.)

But hit the math department –

And all I’m saying is, look, I’ve got some math to teach over here. And until I can count on two fingers the number of math teachers who are building a meaningful practice out of tech, until this stuff begins to approximate the importance of a cash register to a grocery store checker . . .

which is the good reaction. (Quote from Dan Meyer.)

A lot of technology-coordination-types seem puzzled by this – why should math be any different from other subjects when adopting modern tech? – but it is, and there are (at least) three major reasons why.

The difficulty in working with equations on a computer

Sure, high school math teachers do statistics, and graphs, and even the occasional art project, but the meat of the content is working with equations. Even something easy to type like

Solve for x: 2+x = 3

can be a bear to demonstrate the steps on, and maybe could be done with fixed fonts and a fiddle like

2+x=3
-2  -2
  x=1

but this is a post-understanding sort of kludge, and it’s not simply possible to work with things as easily as paper.

Students 3.0

The first graphing calculator was introduced in 1985, and it’s been a battle ever since.

One camp is fundamentally opposed to the notion of using graphing calculators, while the others think graphing calculators should be used at every level. This is still ongoing even though the AP Calculus tests require a graphing calculator and college expects students to arrive with the skills. (At one university I know of, they give incoming students a list of calculator tasks and say “if you don’t know how to do these, figure it out, because we’re not going to teach you.”)

Because the acceptance of graphing calculators is nearly a prerequisite for many modern math apps, arguments get stalled at the door. In other words, upgrading to wikis and the like is version 3.0, and 2.0 is still in beta.

Sometimes there really is only one right answer

With disciplines where multiple viewpoints are all equally valid, it’s easy to have a collaborative discussion where every contribution is valued and important. When solving a problem with only one right answer, snarls can hit. Maybe one student dominates the discussion, or things shut down too early, or everyone is stuck in a way that requires massive teacher intervention. These issues often aren’t discussed, and the edict to focus on process rather than solution gets messy in practice. (Although it’s a start, and even if there’s only one solution there may be multiple ways to get there.)

How this week will roll

I’m going to switch between general assessments of what’s going on and specific examples. I’m going to make a wish list for what I’d like to see in modern technology, because my sentiments match closely with the quote above.

I’m going to need your help. Some things I’m wanting really don’t exist, but I’m hoping there’s hidden gems out there I haven’t come across yet. If nothing else, maybe a developer will take notice and fill the gaps.

Jason Dyer, Guest Blogger

Where I’m Coming From

My run officially starts tomorrow, but I wanted to get my standpoint up.

I’m a high school mathematics teacher, and I focus on my class. I spend most of my time thinking about curriculum, not theory. I take a pragmatic approach and always ask, primarily: does it work?

However, I’m willing to try anything once. So I’m going to break open this week and take on one of the Big Issues: why is it so hard for mathematics teachers in particular to use social technology, and what’s needed to fix the problems? Mathematics teachers are often frustrated, because the generalizations about social technology don’t answer the question: so what do I do with it? There’s a lot of technology coordinators out there (greetings!) and I want to bridge the gap, so you understand where we’re coming from.

I believe both tech optimism and tech pessimism are dangerous. Too much optimism can blind one to failure, and too much pessimism can cause something to be discarded after only a single failure (when all it needed was a retooling). I’ll be aiming at the middle road, trying to balance practical reality and unrealized potential.

I’ll be out of my element, but feel free to compliment or criticize or add or subtract. I’m not going to have all the answers alone, but maybe together we can work this out.

Jason Dyer, Guest Blogger