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.
Jay Fitger is a middle-aged professor of English and creative writing at Payne (pun intended) University, by his own admission a “second-tier,” liberal arts college. Though fully-tenured, he is sagging under the pressures of a writing career in neutral, a personal life in reverse, and the general neglect of the liberal arts in academia.
Nevertheless, numerous students and colleagues request letters of recommendation from him for things ranging from administrative positions at the college to jobs in paintball emporiums to $400 scholarships. Fitger completes these letters and does mention the recommendee, but feels free to add whatever else he wants. Personal issues, critique of the institution to which he writes, and praise of the traditional letter as opposed to the electronic form are common themes.
A representative sample:
Dear Committee Members,
This letter recommends Ms. Stella Castle to your graduate institution in the field of public policy. And to begin the recommendation on the proper footing: no, I will not fill out the inane computerized form that is intended to precede or supplant this letter; ranking a student according to his or her placement among the “top 10 percent,” “top 2 percent,” or “top 0.000001percent” is pointless and absurd. No faculty member will rank any student, no matter how severely lacking in ability or reason, below “top 10 percent.” This would be tantamount to describing the candidate in question as a witless beast. A human being and his or her caliber, intellect, character, and promise are not reducible to a check mark in a box. Faced with a reductionist formula such as yours, I despair for the future, consoling myself with the thought that I and others of my generation, with its archaic modes of discourse, won’t live to see the barren cyberworld the authors of your recommendation form are determined to create.
Even if you sympathize with Fitger’s sentiments, it’s hard to see how such a letter can help Ms. Castle regardless of what praise he heaps on her after this first paragraph.
In my own life as an academic, I have known my share of Fitgers. They are pompous, self-centered, and not very nice. The (mostly unacknowledged) privilege they enjoy is matched only by the degree to which they feel persecuted and powerless. It is a potent brew. In Dear Committee Members, Julie Shumacher has captured them in all their delicious irony. I see it as a cautionary tale: there but through compassion, love, and hopefully humility go I.
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.
Philip Marlowe, Raymond Chandler’s most famous character, is something of a mystery. He muddles his way through investigations, doing his best to make sense of an unfair and world, but what we know about Marlowe the man fits in an ash tray in a two-bit bar somewhere in the L.A. basin. Trouble Is My Business pulls together four stories that, read together, uncover some of Marlowe’s character.
In the title story, Marlowe is hired to reveal the gold-digging nature of a woman chasing a rich man’s son. Though he doesn’t trust her, he’s also sympathetic to a woman trying to survive in a complicated world — and by the end things get way more complicated than they first appeared. In the second story, Marlowe agrees to help a friend make a little money using a quasi-legitimate scheme. His friend ends up dead and Marlowe’s framed for the murder. The third story sees Marlowe working with another friend, this time a woman on the trail of some insurance money for stolen pearls. Finally, in “Red Wind,” on impulse he helps a woman escape the police and ends up shielding her from her dead lover’s perfidy.
The moral code that emerges from these vignettes is one that says you do a lot for friends — even when it sounds like a bad idea. It assumes women are innocent or at least deserving of protection, never asks for more than fair compensation, and often gets less. Sometimes Marlowe’s pronouncements clang against modern ears, but I don’t think he (or Chandler) cares. He lives his life as he sees fit and expects no sympathy.
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.
The Maxie family is English, landed, and cash-poor. The son works as a doctor in London. The daughter is divorced and living at home with her mother. Together with a long-faithful servant, they can barely keep up the house and nurse the invalid father, near death. When they hire an unwed mother with an infant son and some attitude above her station, it’s not hard to see who will die. Who did it is another story.
With more red herrings than a fish market and P. D. James’ cheeky, understated prose, it’s a pleasure to read. If, in the end, she implies that the victim is to blame, I put that down to a previous generation’s sensibility (the novel was published in 1962) and the class differences that so often pervade English literature. If you like British mystery shows like Father Brown or Midsomer Murders, you’ll enjoy the book.