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Only one of the authors of this book come from an education background. The other two study and write primarily about business. Their idea is to take the insights they have about the way businesses change and apply them to education.
To me, their most useful insight is that big breakthroughs in innovation first develop in a markets that were not previously being served or were being served poorly. In an example they use a few times in the book, the first makers of the minicomputer didn’t try to market their product to the government or businesses that were already invested in mainframe computers. Instead, they marketed to smaller businesses and individuals who could not afford and didn’t have the space for mainframes. Trying to compete with the mainframes directly would have been a losing cause, since the companies that made the mainframes were already good at it and already had large market shares. In addition, even though the first minicomputers didn’t have all the power and functionality of the mainframes of that day, the early minicomputer users were happy with the product because without it they didn’t have access to computers at all. That is, people in new markets are willing to accept some less than ideal circumstances, because they had no service before. As time goes on, and people become accustomed to the new product, they expect higher and higher levels of service — note that minicomputers got increasingly powerful to match those expectations.
The same is true for educational reform. Educational innovators are going to have a very difficult time being successful while fighting against a well-entrenched system. Unless you serve a new market, your innovation is almost certain to function mostly at the edges and to last only as long as you have “extra” money to pay for it. You aren’t going to truly “disrupt” the system.
The book’s authors believe that educational technology can serve previously unserved or underserved markets and eventually change education as we have known it and they believe that the unserved markets are in places like: small schools that cannot afford to offer “enrichment” courses; credit recovery for students who have to take regular classes, too; providing individualized instruction for students who do not learn well in traditional classes.
The authors think the last example is especially important and suggest that, as the software improves, much of school can be transformed into an individualized environment for students, with teachers guiding and assisting as needed — improving education for all students because they are being taught in the style in which they best learn, rather than in the relatively monolithic, one-size-fits-all methods that most schools employ. Based on their observation of disruptive innovation in other areas, they predicted (when they published the book in 2008) that “by 2019, about 50 percent of high school courses will be delivered online.” Whether or not this is accurate remains to be seen, but the vision they provide is of a student-centric learning environment that is enabled by technology, while remaining very much human and humane.
Disrupting Class is at it’s best when talking about the potential for change and how that change could happen. It’s much less compelling when talking about educational research and policy. I appreciate the authors’ vision, if not always their specific recommendations for moving toward it.
As Kurt Squire notes in the forward to Video Games and Learning, educators and gamers don’t often talk to each other — “game people think educators are lame, and educators think that game people are hyperviolent misogynists destroying all that is good in the world.” Squire and other educator-gamers (see, for example, Keith Devlin) are trying to change that, because they believe that gaming not only has something to teach educators, but that gaming may be a significant part of the future of education. Squire asks not, “Can you learn from games?” but “How do we make good learning games?” and “Can games help transform education?”
Squire is pretty critical of the current educational culture in our country. He argues that it is based in “passive knowledge reception,” with a structure that isolates “students by age ability, filters all information through the teacher, and features few opportunities to interact with experts, much less to become one.” He sees school as all too often full of failure that reinforces the idea that students know little or nothing – stand back and let “the experts” tell you how it is.
By contrast, gaming communities “don’t survive by selling players the fantasy of being incompetent, powerless, forced to follow arbitrary rules, and impotent in the world.” Instead games give the opportunity to “become leaders, teachers, or authors in the domains they are studying.” Games clearly communicate “what players must do to become experts” and regularly give the opportunity to interact with people who are more advanced. They are engaging and motivate learning in a wide range of fields for the purpose of doing something (albeit, usually scoring more points in the game), not just learning because your teacher told you to. Games, Squires believes, can be a meaningful part of, and can influence, a curriculum that also includes, people, books, homework, and all the other things we’ve come to associate with education.
This all might sound pie in the sky. Squire acknowledges that “such a change would require revolutionizing across the entire education system – from the professional development of teachers (who themselves are treated as recipients, not producers of knowledge) to the assessment system.” But he’s not just theorizing; he’s gone out and worked in “low-performing” schools, introducing a gaming curriculum in collaboration with teachers on the ground. Together they have learned what can work and what can’t, adjusting their approach on the fly. The stories of these experiments are some of the most compelling parts of the book.
Squire does not pretend to have all the answers. While he is a reformer, he is not concerned with scaling the interventions he’s helped pilot. He sees education reform as a process, not a product. He wants to spread “reflective teaching practices that engage kids in advanced design thinking about themselves and their communities.” In this he has succeeded.
Research in math rarely generates much controversy, but the recent back and forth over the work of Rochelle Gutierrez and others is something of an exception. That work discusses the cultural roots of math, specifically the way it operates as a gendered space that perpetuates male and white privilege. Even though many professional organizations in math have supported Gutierrez and her work, people continue to demonize her.
Suffice it to say that I stand with Gutierrez. See here for more information.
(For an added bonus, check out The Liberated Mathematician.)
As part of my research, I’ve stumbled across the work of Michelle Pacansky-Brock and what she calls “humanizing” online learning. She has created an infographic summarizing “How to Humanize Your Online Class“:
Pacansky-Brock’s work dovetails nicely with the work of Drs. Luke Wood, Frank Harris III, and Khalid White. In their book, Teaching Men of Color in the Community College (there’s also an online course with the same name), they stress teacher-student relationships that include positive messaging, authentic care, and intrusive interventions. They also point to the importance of high expectations and high support in the context of relevant content, critical reflection, collaboration, and performance monitoring. While that’s a lot to take in, I thought Michael Smedshammer’s approach in his Online Teaching Conference presentation, Five Ways to Leverage Your LMS to Improve Student Equity, put it succinctly: “Put relationships before pedagogy” and use “intrusive pedagogy.”
For me, this means it’s the teacher’s responsibility to reach out to our students. We cannot just offer to meet with students or put the invitation out there. We need to schedule times with our students and seek them out if they don’t show up. This is especially important for those students who need the most help and may be most embarrassed to asked for it.
In my own experience as an online and hybrid teacher building the relationships is the most difficult part, particularly because it’s one of the strengths I bring to my face-to-face classroom. Online and hybrid instructors typically see our students much less often than our in-the-classroom colleagues. We have to create human presence through the internet. When I started teaching online, I not only didn’t know how to be “present” remotely, I wasn’t very comfortable with trying. It felt completely alien and weird and the opposite of personal. I had to learn to get over myself. No matter how accomplished I was in the classroom, I had to adapt to the new environment. I had to get out of my comfort zone and do the things that my students needed. The resulting experiments have not been perfect, but I’ve learned a lot.
Here’s some concrete ways you can make your presence more real for your students online. You can use these to show who you are and to demonstrate your passion for your subject and for teaching. (Many of these ideas are gleaned from Pacansky-Brock and Smedshammer, as well as Fabiola Torres):
- Embed video of yourself in your syllabus
- Use icebreakers
- Offer online office hours using Zoom or Google Hangouts
- Use a texting service like Remind:
Email is old-fashioned. There’s no better way to reach your students than on their phones. I now require my online and hybrid students to sign up to receive texts from me. Even though I give them the option to talk to me about why they don’t want to get those texts, not one student has asked to get out of the requirement. Far from being intrusive, most students seem to appreciate my texts and see it as sign that I care about their success.
- Post weekly or more frequent videos in your course:
Encourage students and let them see your normal or even geeky self. Torres calls these “learning nudges.” You can use QuickTime or iMovie or any number of other applications.
- Respond to student work with a video — this is really easy inside Canvas. There should be a button to press under the box for assignment comments that allows you record and post a private video right inside the LMS.
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 they 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: email@example.com.
Questioning the status quo has always been fraught, even deadly. The furor around Andrew Hacker’s, The Math Myth, is no exception (though as far as I know Hacker has not been physically attacked or threatened). The accepted truth Hacker challenges is the sequence of math courses that almost all US high school students take – commonly called Algebra I, Geometry, and Algebra II – and which a slightly smaller number retake, as remedial or developmental courses, when they enter college.
These courses are designed to lead students toward calculus, a worthy goal as one of the great scientific and mathematical achievements of the last 500 years, but one that, to be fair, is not crucial to function effectively as a citizen of the 21st century. Instead, this math curriculum is the result of a Sputnik-era concern over the threat of Soviet competition in space and science more broadly.
As such, Hacker’s book asks us to reconsider our lock step requirements for all students in math and offers an alternative based in the thinking of a numerically literate social science professor. Here in essence is his argument, as I see it:
- Currently, the US requires all students to take math leading to calculus.
- This curriculum teaches skills and knowledge that are not used in most people’s everyday life.
- This curriculum teaches skills and knowledge that are rarely used, even by scientists, engineers, computer scientists, actuaries, or any other work we typically think of as needing mathematics.
- This curriculum is not improving the quantitative literacy or reasoning of our society.
- The transfer of math skills and thinking to other fields, as is often claimed, is unproven at best.
- Mathematical proof is abstract and unrelated to the way we in fact establish truth in the world, for example scientific proof or legal proof.
- The cost of forcing all students into the same math curriculum is too high, in terms of preventing too many otherwise talented students from completing their studies and entering the professional workforce.
- Therefore, we should offer rigorous alternatives to the current math curriculum that promote improved quantitative literacy and reasoning.
Along the way, Hacker includes some thoughts about why the status quo is what it is. Tradition is a big piece of it, as is using math as a surrogate for precision and rigor, something I have often observed. In addition, our math curriculum represents a de facto form of tracking for students, keeping out the “unwanted” from professional careers. You should read that as African-American, Latino/a, and other non-white students who are disproportionately stuck in the math pipeline. The status quo also serves mathematicians by giving them many jobs teaching all the students forced into those classes. Finally, Hacker argues that preventing students in the US from completing their degrees keeps the flow of foreign-born workers, often willing to work for less money than their US-born counterparts, open and strong.
Whether you agree with Hacker’s premises or not, he presents an array of evidence that is not easily dismissed. In fact, critics of the book mostly do not attack the ideas I’ve outlined above. Instead they focus on Hacker’s use of terms, which admittedly is not always careful from a mathematical perspective. That said, in no serious critique of the book have I seen anyone disagreeing with the basic premise that teaching math as we currently do in the US is costing our society the loss of many talented students who excel in many areas, but are denied access to college degrees because they do not complete the math requirements.
Keith Devlin, an educator, Mathematical Association of America-sponsored columnist, and a voice I respect, explicitly agrees with Hacker that “Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.” Devlin’s critique of Hacker draws a distinction between what is taught in US schools as “algebra” and algebra as it was historically developed and currently practiced by mathematicians. This distinction is useful as a defense of algebra as a whole, but not as a critique of Hacker’s work, precisely because Hacker’s argument is about how algebra is taught and used by our educational system. I say, for those that are concerned by Hacker’s use of “algebra” as a convenient metaphor representing “the current state of math education in this country,” substitute the longer phrase.
From my perspective, The Math Myth is titled provocatively for the purpose of creating controversy and selling books. Hacker does not attack the importance of math overall, but does question the current math establishment. As a thoughtful voice from outside the discipline, we should listen, broaden our thinking, and be open to the constructive message he brings. It is the students, as Hacker points out, who pay the price for our insistence on the status quo.