I have been reading article after article arguing against algebra and for algebra over the past month or so. I felt compelled to dispute the original New York Times piece until someone did it rather competently the next day. But then yesterday I read this piece on the Washington Post that I would not accept as an argument from any person. It is poorly thought out, poorly supported, and just in general reflects a poor grasp of careful logical reasoning. While I agree with some of the author's points, I seek to refute each and every argument which he points out discredit algebra's position in the curriculum.
"Why this religious zeal over algebra? It helps students learn how to think, people claim. Really? Are mathematicians the best thinkers you know? I know plenty of them who can’t handle their own lives very well."
First of all, this is an ad hominem attack on those mathematicians who are referred to by the author. But more importantly, the claim that mathematicians are good thinkers implies that they would handle their lives well is a logical fallacy. This is an example of "hasty generalization". There are so many possible missing assumptions. For example, one possible underlying assumption is that if a person can think well, then they will always act in such a way as to "handle their lives well" - whatever that means. I mean this is analogous to the idea that all investors are rational and will act in ways that are rational; a claim that is hotly disputed by behavioral finance. Additionally, the author apparently fails to realize that mathematicians are human beings, and human beings are not perfect. This is also an example of an emotional fallacy called flattery. When the author says: "Are mathematicians the best thinkers you know?" he is really asking people who for the most part might not be mathematically inclined to agree with him. He is flattering them.
"Reasoning mathematically is a nice skill but one that is not relevant to most of life. We reason about many things: parenting, marriage, careers, finances, business, politics. Do we learn how to reason about these things by learning algebra? The idea is absurd."It is hard to understand how the author can simultaneously claim to understand how to reason mathematically and at the same time be able to say that it is not relevant to most of life. Let me clarify, in order to be able to make this claim as an objective fact that we would be able to take seriously, then we would have to be convinced beyond doubt that the author is capable of reasoning mathematically with as much fluency as anybody: including mathematicians. Then, and only then, would we then be able to agree that such a person would be capable of analysing "parenting, marriage, careers, finances, business, politics" from both a mathematical point of view as well as from a non-mathematical point of view. Only after making such a comparison would such a person then be able to tell us that the mathematical perspective did not assist them in making sharper decisions. However, the author is not a mathematician and is therefore probably not one of the persons who is capable of making such a statement credibly.
On the other hand, mathematical reasoning is certainly central to business and finance today, and I can think of no business professional today who would dispute this. Whether that business person posesses mathematical fluency or not. Insurance, for example, is probably the most mundane application of mathematics that there is. There are people called actuaries who design mathematical models which permit each and every one of us to protect ourselves from risk. Questions that arise in business can overwhelmingly be attacked better in a mathematical way, otherwise companies would not be willing to pay these individuals as much as they are paid. Inventory planning, investment management, risk management, portfolio management, budgeting, and many aspects of business are best dealt with using the tools that mathematics provides. Politics can also benefit from such tools. For example, statistical analysis(which falls under the umbrella of mathematical reasoning by the way) can be used to identify those geographic areas that are the best candidates for advertisement by a politician to convert them one way or the other. The statisticians will have already determined those geographic areas that are winnable and those that are unwinnable. By targeting in this manner, the politician will be able to make their dollars work harder, thus allowing for a smaller budget to lead to a stronger campaign against an opponent who might not be willing to use such a strategy. The same kind of technique could be used in any marketing of course. How many of which kind of crop to be planted on a field of a certain size and given certain other constraints in order to generate the greatest profit or to minimize costs can best be solved with mathematical reasoning than without(such a problem is a typical textbook example of linear programming by the way).
The other two situations marriage and parenting might not have problems that are amenable to mathematical reasoning, but that does not disqualify any of the value that mathematical reasoning has brought to this world. Every convenience that we have today was concieved by a human, and some of those ideas have been aided by a liberal application of mathematical reasoning.
"Yet, we hear argument after argument about the need for more STEM education (pretending we don’t have lots of unemployed science PhDs). Everyone must study chemistry, memorize plant phylla and do lots of trigonometry."
This can be refuted in many ways. At the most basic level, the same argument that the author is making could be used about the necessity of high school. In other words, if people who have high school diplomas are unemployed, then we do not need high school. On the other hand, in any fruitful endeavor that people are going to do in life, it is difficult to know ahead of time what their ultimate achievements are going to be. No one knows whether a PhD student is going to simply become a PhD or is going to become someone who is going to utilize their knowledge for some incredible feat. PhD committees cannot know this ahead of time, and it is for this reason that they pick the most promising candidates for their PhD program. They want to make sure that they maximize the potential positive effect that their program will provide to the planet. Some of those PhDs go out and make things happen, and some don't. That's just reality and it does not discredit any of the arguments in support of STEM.
"The argument for algebra rests on the transfer from math to other areas of life, something that has never been proven despite the claims of people such as University of Virginia cognitive scientist Daniel Willingham."Who cares if the intellectual benefits of algebra do or do not transfer to other areas of life? Algebra has fascinated human beings for thousands of years. Common algebra problems as well as a rudimentary form of the quadratic formula has been found on Babylonian cuneiform tablets that were used essentially as textbooks for students of that time. Algebra has been taught therefore for far longer than just a few centuries. Algebra is also intrinsically beautiful, with its own way of assigning meaning to the world around us, and as a problem solving language that has proven so useful to humanity. Algebra and mathematics generally, like other highly developed human pursuits including Art and Literature and Music, are what make us human and separate us from all other living things on this planet. I think that I can speak for most of humanity when I say that Algebra, and math generally, is something that we are very proud of having.
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On the other hand, one of the most important points that the author does make, a point that I fully agree with and support with every fiber of my being is "to begin teaching people to reason well enough to make sensible political and life choices." We need people to learn how to think clearly, to ask probing questions, to think critically, and we need them to be able to communicate those thoughts in a clear systematic fashion so as to engender dialogue with their neighbors for the advancement of whatever cause that humans collaborate on(preferably a good activity). For example, people need to understand what it is that they want politicians to do and have the capacity to research the relevant facts of the competing politicians in order to pick the politician that will best serve their own needs and desires. People need to be able to evaluate their own lives and their own goals fully so as to determine how to go from where they are to where they want to be.
With regard to algebra my personal feeling is that it is absolutely necessary to teach high school students algebra. I agree when the author talks about having people pursue their own interests, but I disagree with regard to timing. High school is meant to provide a general foundation for future academic studies. Therefore it must prepare future artists and engineers equally well. How many of us can honestly say that we knew when we were freshmen in high school -when most students take algebra - that we knew what we were going to be when we grew up, or even what we would study in college. Danica McKellar, author of Math Doesn't Suck and other mathematics books, even talks about how her love of mathematics came late. Therefore, I think that young people who probably are unsure of what they are going to become deserve the best foundation possible for whatever choice that they are going to make. Therefore, algebra is absolutely a necessity. And for those who are talented, Calculus as well. I think that this will help those students who are going to become engineers and scientists start off with a solid mathematical foundation. For those who are not talented, tough. The reality is that like the PhD committee above, none of the administrators or any of us can predict what the future of individual students will be: therefore we have to prepare them as strongly as possible with the basic general foundation necessary for success in any endeavor. Since I think we can agree that high school students, do not know where they are headed academically, we must choose a curriculum that provides the foundation for any endeavor, in particular engineering, math, and science.
The author also claims that "the average person never does abstract reasoning." That is perfectly ok, not everyone has to engage in abstract reasoning. However, I can personally say that abstract reasoning is second nature for me now after studying mathematics for a long time, and it has enriched my life. Abstract reasoning allows one to extrapolate and allows for something that occurred once to take on meaning that can have applications to other situations. I think that the ability to make connections among different things is an important skill. To see how two seemingly different things or situations are fundamentally the same is something that has value. It allows an individual to make stronger, more richly argued arguments. It also makes life more interesting. Abstract reasoning also has very strong applications to the real world, most computer programming requires that you understand what the program is going to do in individual cases and write it in the abstract so that it can handle case after case after case but get written only once. It takes abstract reasoning to understand that different symptoms of an ailment can be caused by a single cause.
A huge point that I also want to bring up is that Algebra has intrinsic value and could quite justifiably be taught for its own sake irrespective of its applicability or any other argument that may be put forward in its support. Algebra is part of our heritage as human beings and is a tapestry that has been handed down for millenia, tracing its roots back to Babylon and taking turn after winding turn throughout history through to the present. It has been used to solve practical problems throughout history, but has also flourished theoretically often without any question in that development as to whether those theoretical advances would have applications also. As we have come into the twentieth century, those theoretical advances have found utility in computer science and cryptography but emerged from purely theoretical considerations. Algebra, I am arguing, can be taught just for its own sake just like art and literature. It is simply one of the most beautiful creations that man has created.
Deviating slightly, I also believe it deserves mention that would the author's ideas as laid out in this short article be taken seriously, then they could as easily apply to literature, history courses, and chemistry courses, and most other subject matter courses. He says:
"It isn’t just mathematics that is the problem, of course. Why do we all learn to balance chemical equations or memorize homilies about U.S. history? Because back in 1892, the president of Harvard University designed curriculum and said that those subjects should be the basis for high school classes."
In other words, if we take the author's quotation to its logical conclusion, we would be creating a high school curriculum that would be devoid of chemistry which would hurt potential chemists and medical doctors, devoid of history and letters which would impact our future politicians and the character developing stories that are discussed, devoid of biology, etc. Would literature courses also be stricken? Art courses? Is education only about cognitive abilities and not about passing on our culture? Again, as adults, we have to be the ones that are going ultimately to decide the curriculum that our children are going to be educated by. If it were up to me, I would recommend a curriculum that is going to prepare our children for any endeavor that they are going to ultimately end up choosing. Additionally, each of these courses when properly taught fosters the critical thinking skills that each of us needs as adults to make sound decisions and live well.
In conclusion, I firmly agree with the author that we have to have a curriculum that engenders critical thinking skills and that grows cognitive abilities in every single student, but I believe that a good high school curriculum has to lay a solid foundation for every student regardless of their future academic endeavors. We have also seen that algebra, and more generally mathematical reasoning, is very important to the development of so many of the things we take for granted. We have also discussed the fact that Algebra needs no excuse to be taught other than that it is part of our human heritage. In short, Algebra should definitely be taught in high school as part of the curriculum.
