Before the 13th century AD, math was done in sentences, sometimes called “rhetorical math.” The symbols we currently associate with math began to emerge in the Arab world during the 13th, 14th, and 15th centuries. By the 16th century, French thinkers were developing a fully symbolic system.
The advent of symbolic algebra changed the way we think about, learn, and do math. It also changed the kinds of problems that were doable by the lay mathematician with a basic education. Electronic calculators made arithmetic with large numbers more accessible, but didn’t fundamentally shift the way we think about math or learn it. (There is still plenty of debate in math education circles about the appropriate use of calculators in the curriculum.)
Today, computer technology is slowly altering math and math education, but especially in math education that potential is only beginning to be realized. Much of what we do with computers in math education mimics books, except in more color and with occasional hyperlinks. While there are folks taking advantage of multimedia presentation (think video, interactive sliders, etc. – for instance, the folks at Desmos are doing some great work), I have yet to see computers fundamentally and broadly change the way we teach math in the way that symbolic manipulation on paper did.
One option is to let the internet provide the kind of instructions that we’re used to seeing from teachers. Sites such as Khan Academy and publishers like XYZ Textbooks provide videos with multiple examples worked out slowly and carefully. Students can watch them on their own time, as many times as they want, stopping and starting and rewinding as they need. In class, teachers can clear up misconceptions and extend ideas already developed at home.
This “flipped classroom” model, however, assumes students can access the internet at home, an assumption that is often wrong and disadvantages those with the least (Is Digital Equity the Civil Rights Issue of the Day?). Add to that the fact that desk tops are giving way to small screens and it’s clear we must make sure we are making mobile-native, or at least mobile-friendly, education sites and activities. Even then, folks living in or on the edge of poverty often lose their access.
With this as context, consider Keith Devlin’s Mathematics Education for a New Era. In it, Devlin pulls together a career as a math educator and a love of video games to suggest a way for math and math education to evolve for the 21st century and beyond.
Devlin starts by discussing what he calls eleven principles of an ideal learning environment – like “the learning environment should be as similar as possible to the environment in which people will use what the learn” and “there should be sufficient ‘cost’ to getting something wrong to motivate correction, but not so great that it leads to the student losing heart and giving up” – ideas I think most people would agree with. With that basis, he tries to show how video games fit the principles very closely, even to the point of calling the next chapter “Euclid Would Have Taught Math This Way.” Part of this argument involves discussing the 36 principles of education that go into video games according to James Paul Gee (professor of education at Arizona State University) in his book, What Video Games Have to Teach Us About Learning and Literacy. Devlin goes on to discuss various aspects of math education and finishes by advocating for a math pedagogy that is part “flipped” and all carefully thought out to create optimal learning for each individual student, taking advantage of whatever methods are best for what’s being taught.
I find Devlin’s ideas compelling. Use computers and computer games to do the things they are good at: repetition and drilling (when appropriate); motivation and story. Continue to respect the relationships between teachers and students in a thoughtful system that supports students in the ways that they most need it. He is not arguing that that video games should be the sole way to teach math, or even that it is the best way. Instead, he believes that well-designed math education video games could be a powerful addition to school, home, textbooks, and the rest of the math educational apparatus.
He also makes some useful observations and distinctions for math teachers (like myself):
- The phrase “’do math’ is all too frequently taken to mean mindless manipulating symbols, without the full engagement that comes with genuine mathematical thinking.” In fact, Devlin points out, “skills are much more easily acquired when encountered as a part of mathematical thinking.” But he reminds us, “mathematical thinking is not something the human mind finds natural.”
- Anyone trying to teach math should design situations for students that promote mathematical thinking and expect to need to help them, while always remembering that “attempts to understand what it all means at too early a stage can slow the learning process.” In fact, “full conceptual understanding, while desirable, is not strictly necessary in order to be able to apply mathematics successfully.” Often what is needed in the short term is “functional understanding”:
Calculus is in many ways a cognitive technology – a tool you use without knowing much, if anything, about how it works. For example, few people know how an automobile engine or a computer works, but that does not prevent those people from becoming skillful drivers or computer users. Successful use of a technology does generally does not require an understanding of how or why it works.
I realize all of this is a pretty big pill to swallow for many of us, especially those, like me, raised on endless worksheets of drill, without motivation except a task master with a real or metaphorical ruler ready to slap the idle hand. But computers are changing many aspects of our life, for better or worse, and I don’t think that’s going to stop. Instead let’s figure out how to use them well, for the good of the generations to come. I think that’s what Devlin is trying to do. If it’s not the “right” answer, then it’s a pretty good try.
I’ll leave you with a long quote from the book’s opening chapter that I think captures some of Devlin’s vision and passion:
When people made the first attempts to fly, the most successful machines for transport were wheeled vehicles, and the only know examples of flying creatures were birds and insects, both of which fly by flapping wings. . . but that doesn’t work for humans. The key to human flight was to separate flying from flapping wings, and to achieve flight by another means more suited to machines built from wood or metal. . . .
Putting symbolic expressions in a math ed game environment is to confuse mathematical thinking with its static, symbolic representation on a sheet of paper, just as the early aviators confused flying with the one particular representation of flying which they had observed. To build truly successful math ed video games we have to separate the activity – a form of thinking – from its familiar representation in terms of symbolic expressions.
Mathematical symbols were introduced to do mathematics first in the sand, then on parchment and slate, and still later on paper and blackboards. Video games provide an entirely different representational medium. As a dynamic medium, video games are far better suited in many ways to representing and doing middle-school mathematics than are symbolic expressions on a page. We need to get beyond thinking of video games as an environment that delivers traditional pedagogy – a new canvas on which to pour symbols – and see them as an entirely new medium to represent mathematics.