In a NY Times editorial, by Andrew Hacker, a Professor Emeritus in the Department of Political Science at Princeton University, argued that Algebra should be removed from the mandatory high school curriculum in the United States. I believe this is wrong. But, the frustrations and limitations Hacker outlines can and should be addressed if we are to improve education and the opportunities for students in the future. First, Hacker does acknowledge the technical necessity of the subject and does not argue that we should not be teaching Algebra at all, only that it should not be a universal requirement. Hacker believes that only those interested in the university-level study of hard science or engineering should take Algebra. His argument as to why students should not be indiscriminately forced to study this subject breaks down more or less as follows:
- Algebra is more difficult and abstract than other Math subjects like Geometry and Arithmetic, and discourages students from completing high school or basic 4-year college degrees. As evidence, for the Algebra portion of state exit exams, in Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia. And, at the City University of New York, 57 percent of its students didn't pass its mandated algebra course.
Algebra is unnecessary in everyday occupations like factory machine tool operation, business management, personal finance, or even veterinary medicine, for other applied scientific practitioners.
Mathematics should be made more accessible and taught as a Liberal Art, with courses in the history and philosophy of mathematics, and its applications in art, music, as well as political and economic statistics.
Young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.
Edward Frenkel at Slate has already published this account, which brings to light some of the limitations of Prof. Hacker's proposals. These include the fact that many of the examples of practical applications of Mathematics that Hacker cites like the calculation of the Consumer Price Index, do in fact, require Algebra and often higher Mathematics as well to understand. With this particular issue already well covered, I want to address other parts of Hacker's argument, and outline what I believe we need to do about it.
First, let's address the purpose of and need for Algebra in today's world. While evaluating abstract expressions like (x² + y²)² = (x² - y²)² + (2xy)² seems to be at the heart of everyone's concept of Algebra, that is not the real essence of the discipline. Algebra at its heart is an extension of arithmetic that allows us to uncover an unknown value of interest to us, when it is something that doesn't appear at first blush to be reducible to a 2 + 7 = ____ type of problem. It helps us discern how hard we have to study for that final exam to earn a B in the course; it helps us determine how much we need to set aside from every paycheck to afford that down payment; it helps us design the most affordable deck for the back yard; it helps us trade off risks and rewards in our retirement savings.
Facility with the techniques and tools of Algebra is absolutely essential to understanding statistics (a basic necessity of citizenship in today's world), being able to program a manufacturing machine, choose the best investments and major purchases, and code computer applications or understand their likely errors. And, that is not all. While Algebra is imminently applicable to everyday problems beyond the obtuse "train travelling from Cleveland.." word problems packed into textbooks, our courses tend to focus on the Mathematician's appreciation of the beauty and symmetry of abstract Math combined with the machine-like drilling useful for preparing in advance of standardized tests. While I currently use Algebra to do all kinds of simple problem solving, I don't recall ever seeing such an application as a high school student--that fact lies at the crux of the problem.
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Next, the problems with the success rate of students in Algebra courses could be a result of poor preparation by earlier courses, lack of motivation due to poor course design, and insufficient teaching skills in the classroom, among other things. This is in addition to Math requirements generally being increased considerably over the past 20-40 years, while English requirements have remained the same--many parents have been less able to help their children in this area. It will be difficult to pick out what the most easily solvable part of that problem is. I believe that Mathematics curricula should be redesigned to highlight and have students experience the very real everyday applications possible with each added algebraic tool. Students should see at every step how each new skill enables them to more easily solve an actual problem faced by real people.
The abstract nature of Algebra that Hacker bemoans is not limited to that subject. Geometry, which apparently Hacker sees no problem with, is also struck by a similar mix of antiquated skills (bisecting angles or inscribing pentagons by hand with a compass), and abstract logical proofs. The reason why Geometry doesn't illicit the same reaction from Hacker and educators probably has more to do with the multiple avenues for successful learning and demonstration of skills that exist in Geometry and are lacking in standard Algebra instruction. Unlike the construction of a good argument in a History essay, the answer to an Algebra problem is either 100% right or 100% wrong.
I do agree with Andrew Hacker that we could do better in how we educate students with regard to quantitative disciplines. They should be less distilled into sub-specialties and more intertwined. We need to redesign our Math courses to clearly build on complimentary skills and emphasize the inter-relatedness of Algebra, Statistics, Chemistry, Physics, Geometry, Engineering, Economics, and even Calculus (and sure, Dr. Hacker, even Music, Poetry, History, Art, Architecture, and Philosophy).
Computers and calculators can now solve symbolic problems like (x² + y²)² = (x² - y²)² + (2xy)² as easily as they can perform arithmetic, and have been able to for the past decade or more. Curricula still lag behind this development in most places. Long division (which Hacker says is a requirement) is less emphasized today, but still taught so that we can understand what is happening behind our calculators' faces, and get by when we happen to find ourselves without one. We still need to maintain the foundations of those skills so we can recognize when we've made a mistake in punching things into our solution box. But, most importantly, we all need to understand how these skills are applied in solving real problems. Evidently, long division passes this test with many, but Algebra does not (why would there not be resistance to a subject we haven't all been taught how to properly use, only how to execute it?).
So, what do we need to do about it? The real issue is that we need to focus Mathematics more on real contemporary problem solving at every level from early elementary through early college classes. Engineers, scientists, statisticians, accountants, financial managers, etc. need to be setting mathematics curriculum in equal shares to the mathematicians.
Most of us will not be mathematicians. Most of us do not need to become scientists, engineers, or computer programmers, either. But, it is also clear that we live in an increasingly complex world. One where generalizable skills can prepare us better for future success than training on a particular model of robotic machine would be. Sure, we could learn enough to get by in that factory without much foundational understanding about how the underlying computational systems work, but one might also get by without being fully literate in English and without having learned to analyze the themes in the works of Charles Dickens.
In the world where Algebra and the skills therein are learned by only a few, how would we not end up increasingly at the mercy of the labor saving systems and their creators when they are supposed to serve us. The choices we have to make as a society involve questions of statistical economic models and mathematical functions governing our energy supply. We don't all have to solve these problems personally, but we should be able to recognize when we're being taken. Those critical quantitative reasoning skills we all acknowledge modern citizens to need do in fact require algebra, even if we choose to no longer refer to the subject with that name. Our culture is a technological one, and Mathematics is an integral aspect of that culture. In order to fully participate in our society we need to understand Math as much as we need to understand the Constitution, History, Biology, and English Literature. We can do better by the students of tomorrow. We have to.