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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.
I am currently on sabbatical till January, 2018. During my sabbatical my primary work-related responsibility is to complete a research project.
In my research project I’m trying to pull together three areas that I have worked in over the course of my career as a community college math teacher: math education, multicultural education, and online education. My initial research has found that, while there is literature in the overlap of pairs of these (math and multicultural, math and online, multicultural and online), there is little where the three areas intersect.
If further research confirms that little or no work has been done in these area, then this niche needs to be filled. The importance of better math education is well-documented. As our college student population increasingly diversifies, the need for the still majority-white teaching profession to understand how to better communicate with students of all backgrounds is more crucial than ever. And, though I don’t think technology is the answer to all educational problems, we would be foolish to think that online education is going away; on the contrary, the private-sector is pushing that way, legislatures have visions of the savings it can produce, and students are demanding the flexibility of learning on their own time and from where ever they happen to be.
I’d love to collaborate with others on what I think is a critical confluence of research and practice. If you’d like to work together, or if you know of work in the intersection of math education, multicultural education, and online education, I’d like to here from you. Please comment here or contact me at: firstname.lastname@example.org.
When we are online, every like, every follow, every click is recorded and analyzed by the corporations, large and small, that rule the internet. They use these terabytes of data to market their products, to predict how new products will sell, and more. Exactly what other uses they make of the data, most of us don’t think much about, but the corporations own it and we give them permission to collect and use it when we agree to their terms of service.
The fact that most of us don’t think about someone watching our online behavior is a central assumption in Christian Rudder’s book, Dataclysm, made explicit by the subtitle Who We are (When We Think No One’s Looking). Using that premise, Rudder analyzes the clicks, messaging behavior, and survey results from the online dating site OkCupid, as well as few others. He has access to this data because he is a founder of the site and knows other people in the field. He leverages this privileged information into a book length speculation about what the data means.
Some of Rudder’s observations are well-considered and interesting. Some are less profound. At times I think Rudder jumps to erroneous conclusions and I’d wager a significant amount of money that any thoughtful reader of the book will agree with Rudder sometimes and disagree at others, depending on the specific context. Probably most readers will be occasionally offended by the book. But despite the fact that his ideas are often not fully supported by the data, they are also not fully contradicted by the data. So, even when you disagree with his conclusions, you have to admit he could be right. We just don’t know.
Overall, that makes for a provocative book that opens the imagination for the kinds of knowledge we could gain with careful analysis of the vast quantities of data we, as a global internet society, are collecting.
But beyond agreeing or disagreeing with Rudder, I have a more fundamental issue with Rudder’s approach to the data. He writes,”As far as I know, I’ve made no motivated decision that has bent the outcome of my work.” With this sentence he claims that he uses no theory to reach his conclusions, as if, somehow, he just lets the data talk and listens carefully, transcribing the data’s proclamations accurately.
I don’t think Rudder is naive, but I can only take him at his word. As any scientist or thinker knows, it is impossible to be theoryless. So, to claim explicitly to be theoryless means either he doesn’t know what theory or theories are guiding his decisions or he refuses to tell us. Either way, it is a deep flaw in the book that the reader doesn’t know the theoretical approach taken by the author.
Read the book for some interesting applications of descriptive statistics (and, typographically, for some great use of the color red!). But read with a skeptical mind.
Last year, I started using Twitter as an extra credit tool in my classes. I give students homework extra credit after every class for one of three options:
- A summary of class
- A question about class
- An answer to someone else’s question
The benefits of this are many:
- Students make community with each other.
- I get to see what students thought class was about and if there are questions that need answered.
- Students practice summarizing.
- When students miss class, they can check the Twitter feed to see what they missed.
Often students summarize concepts very nicely and creatively. It’s great to see students learning from each other and forming community, which usually spills out of the virtual world and into the physical.
As much as I’ve liked it over the last year, my students this semester have already raised the Twitter community to a new level. Below are several examples of tweets from my students — none of which satisfy the criteria for extra credit — which are fun and reveal a relationship with our class that I can only hope to reproduce in future semesters:
As a bonus, here’s an exchange with a student in my statistics class this Labor Day: