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.
Katherine Dunn reportedly said, “All the time I was working on Geek Love, it was like my own private autism.” If so, Dunn’s interior life is subtle and complex and a little creepy. She sees under the simulacrum of “norm” family life that we present publicly to another world of secrecy, manipulation, and pain.
Start with a couple, Al and Lily Binewski, who own and operate a traveling carnival. They have the standard midway games and concessions. Contortionists, sword swallowers, geeks, acrobats, and lion tamers all vie to remove dollars from the pockets of lookie loos that wander to the show. But the big draws have been literally bred by the Binewskis through a pregnancy diet of drugs and other poisons. The resulting birth defects range widely in severity. Some of the babies don’t live; they are preserved in jars, regularly polished, and viewed for a price. The survivors include a legless, armless, “Aqua Boy”; Siamese twins joined from the waist down (they play piano four-hands); and a humpback, albino dwarf (not strange enough for an act of her own, but still useful as a general purpose assistant). Their final child appears so normal they are about to leave him at a laundromat in the middle of the night when they discover his special power.
The family is obsessively private, yet make their living through blatant exhibitionism. With no permanent home aside from a 38-foot trailer, they consider “stuck homes” a trap compared to the freedom they enjoy. Their days are filled with training for their acts, the nights with performing. The surface is all freak. Roiling and dysfunctional dynamics — deep misunderstanding between parents and children, jealousy among the kids — rival any telenovela family. Their insularity is both their power and the undoing.
As horrifying as the idea of purposely creating birth defects is, it’s not the most disturbing thing that happens in the book. Indeed, we learn to sympathize with the children and their parents. We see their humanity and watch with disgust as others react violently to their presence. Conventional notions of normalcy and ethical behavior are called into question.
Beautiful writing, Shakespearean pathos, and fine attention to detail make for a finely-wrought world that — like a voyeur paying to see the freaks — I was fascinated to visit, but happier still to return from to my “normal” life.
It’s hard to say a lot about Neal Stephenson’s latest solo-written novel without spoiling its best ideas and plot twists. Since many of the books pleasures comes from those, I will settle for saying that the palindromic title presages the overall cleverness of the book. Depending on your interests you will probably find the balance of discursiveness versus plot off from time to time, but overall Stephenson’s careful explanation of concepts in the context of a complex social, political, and cultural framework is world building at its best. I don’t know if he has sequels in mind, but there is plenty of potential for it in the rich story he has created.
Every student in the US has to learn algebra. If this statement is an exaggeration, it’s not much of one. Almost all students take at least two years of algebra before graduating from high school and millions take it again in college. In addition, algebra skills are required in most science, engineering, and other course. But as technology evolves and what it means to be an educated person changes, I think it’s time that we think about why we teach algebra and the way we use it in education. In particular, I think it’s time we stop making algebra skills a barrier to success in college.
Now don’t get me wrong – I love algebra. Really. It’s a beautiful achievement, solving problems that challenged humanity for centuries. It’s also fun, and, as a math teacher at a community college, I enjoy supporting people as they learn algebra’s intricacies. I hope algebra is always available for those students who want to study it. However, if we’re honest about the knowledge and skills needed by 21st-century graduates, workers, and citizens, algebra does not rank high on the list. Even in the technical fields, I seriously question how often algebraic skills are actually required.
The issue is especially relevant in the community college setting because large percentages of incoming students are placed into developmental algebra courses, or below. These are the same courses most of us took in high school, but students have trouble retaining the algebraic skills they learned, especially if those skills aren’t related to their majors. As a result, many students struggle to learn algebraic content that, if they’re not going on to calculus, they don’t need for their next courses – topics like factoring polynomials and solving rational equations with variables in the denominator and synthetic division. The data reveal that students who place into algebra or below are very unlikely to ever pass college level math. And because first-generation college students and students of color are placed disproportionately into low-level math courses, the algebra barrier perpetuates educational and economic inequities.
For all these reasons, in 2010 I partnered with a colleague to develop a new course designed to prepare students who were going on to take college-level statistics. The fact is that relatively little algebra is needed to learn statistics and we thought we could help students succeed in statistics using a different kind of course, a course containing only the algebra students would need for statistics. We hoped to help the majority of students who aren’t heading toward calculus and who need statistics to complete their associate degrees and transfer to four-year colleges.
Fortunately, we were not the only ones working on this idea and we learned a lot from professors at other community colleges already trying this approach. (Learn more about the “pre-stat” community at: http://accelerationproject.org/.) With their help we were able to create our course, called Preparation for Statistics, and piloted it in Fall 2011. In the course, we asked students to engage with real data, using statistical ideas in an interactive and constructive teaching and learning style. We even helped them create their own surveys, collect data, analyze the data, and present it to their classmates. It was work to teach this way, but it was also the most fun I’d ever had in class.
Most important, it worked. Data from our college, combined with other colleges teaching similar courses, show that students from pre-statistics courses are successful in college-level statistics and that they are much more likely to complete their math requirements than students that who took the traditional algebra sequence. The evidence also suggests that the courses helped close achievement gaps for underrepresented students. (http://rpgroup.org/system/files/CAP_Report_Final_June2014.pdf) At our college, the evidence was strong enough to expand beyond the pilot stage. Each year we were helping hundreds of students reach and succeed in statistics.
If taking algebra in college is not necessary for success in statistics, what about other math courses? What about science courses? Isn’t algebra the mathematical foundation of modern science?
Questions like these got me thinking about mathematical prerequisites for general education science courses. These are the science courses that non-science majors usually take to satisfy the science requirement for their degrees, things like astronomy, biology, geology, geography, and basic chemistry and physics. I looked for studies of math prerequisites in courses like these, but have yet to find one (if you have one, I’d like to see it). The marked lack of statistical evidence that either supports or refutes the need for math prerequisites in science courses (or any courses, for that matter) is telling. At my college, most of these courses do not have math prerequisites, precisely because they want to attract non-technical majors to the courses (some of the courses advise completion of algebra, but don’t require it).
I did find some unpublished data, collected at my college and two other California community colleges that offer pre-statistics courses. Aggregating the data from all three colleges, students who took pre-stats courses before statistics were almost exactly as successful in their general education science courses as students who took the traditional algebra preparation for statistics (84% vs. 83%). Even disaggregated, the difference between the success of students at each college was never greater than 10 percentage points and the college (my own) with the lowest success rate for pre-stats students in GE science courses was still 72%, compared to 78% success for their traditionally algebra-prepared peers.
These results beg the question of how students without as much algebra are doing so well in general education science courses. One answer, suggested and bemoaned by some, is that instructors of those courses are reducing the mathematical content of the courses to accommodate students who haven’t had algebra since high school. Another potential answer is that, since almost all students took algebra in high school, a little reminding and prompting enables students to use algebra to the extent that they need to solve the problems.
While both of these are possible, I have yet to see any data that support those answers or any other. In the absence of evidence, I think it much more likely that the real skills needed to do well in general education science courses are things like numerical literacy, critical thinking, the ability to connect evidence to an idea, and academic skills like going to class, reading your book, taking good notes, turning in your homework on time, and, perhaps most important, belief in your ability to succeed. All these skills are taught in both algebra and pre-statistics courses; my experience is that more attention is paid to them in pre-statistics courses than in algebra.
But, what if it were true that science instructors have reduced the algebra content of their classes? Would this be a problem? I say, no. From my perspective, science classes exist to teach science concepts, not to test students’ algebraic knowledge. If, indeed, science teachers are making science concepts more understandable for students with less algebra experience, that would be a testament to the quality of their teaching ability. As I like to say, it’s easy to make an idea complicated and hard to understand; the difficult task is to make ideas simple and clear.
We have been making most science and math courses harder to understand by forcing algebra into them, even though it’s not needed or needed only minimally. For example, in a physics class the height of an object thrown in the air can be modeled quite well by a quadratic equation. Understanding of the scientific principle is demonstrated by setting up the equation. Solving the equation is purely algebraic, but most of the time these “physics” problems aren’t correct until the equation has been solved. In a science class, the science concept should be the primary goal. Solving the equation by hand should be less important, especially when computers with powerful solving algorithms are so readily available.
Here’s another example, from a geometry course:
The geometric concept being reinforced is that the sum of the angles in a triangle is always 180°. But, in order to solve the problem, you have to perform some algebra. We don’t need algebra to understand the geometric idea, but if a student can’t do the algebra they won’t get the problem right.
We force students to do similar (and often more complicated) algebraic manipulations in chemistry, biology, oceanography, geography, economics, trigonometry, calculus, statistics and many others. In my experience it is algebra that trips up most students in these courses, not the non-algebra content. Limits, differentials and integrals are challenging ideas in calculus courses, but factoring from beginning algebra is frequently the biggest barrier to completing a calculus problem.
Of course, reinforcing algebraic skills throughout the math and science curriculum is not necessarily a bad thing, but I think too often we do it because that’s the way we were taught, not because of any considered pedagogical reasons. The cost of this decision is high because algebra courses and algebra’s continued use throughout the curriculum is, as I mentioned earlier, so often a barrier preventing students’ success.
And, while algebra can teach attention to detail, mastery of algorithms, symbol manipulation, logic, critical thinking, problem solving, teamwork, numerical literacy, and more, there are other ways to teach those same skills. My experience teaching pre-statistics suggests that we can teach those skills as well or better outside of the abstract context of algebra.
Higher education is changing at an unprecedented pace. These changes are driven partly by increases in the percentage of the population who go to college, partly by pressures from the federal and state governments for more return on their education dollar, partly by employers’ demands for well-prepared, 21st-century graduates, and partly by huge technological advances. In mathematics, the traditional algebra and geometry sequence, familiar to most of us from our own mathematical careers, is being questioned. The algebra sequence, after all, is designed to prepare students for calculus and beyond. But in a world where the most students are not seeking science, technology, and engineering degrees, do we really need to prepare all students for calculus? I don’t think so and I’m not alone. According to the 2015 report Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation, “Alternatives emphasizing statistics, modeling, computer science, and quantitative reasoning that are cropping up in high schools and colleges are beginning to challenge the dominance of the familiar math sequence.” (http://edpolicyinca.org/publications/degrees-freedom-diversifying-math-requirements-college-readiness-and-graduation-report-1-3-part-series) These alternatives are emerging because the knowledge and skills needed by informed citizens of the 21st century can be taught as well or better in other ways and because the cost of continuing to insist on algebra is too high.
I’m open to being persuaded that algebra is as important for college students as we have made it. But, to change my mind, you’re going to need to show that the benefits of algebra are algebra’s alone and that they outweigh the costs of forcing everyone to do it.