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An opinion essay in The New York Times by Andrew Hacker asked the question "Is Algebra Necessary?", and makes the argument that educators should consider removing the math course from the list of mandatory courses. His main argument: the subject trips up too many students, and prevents them from completing high school or college.
The essay drew hundreds of comments. Some of SchoolBook's contributors went further and wrote their own essays in response, with a parent arguing that the real problem is poor math education and an English teacher posturing that similar logic could be applied to eliminating literature from English education.
What's your opinion? We would especially love to hear from students or recent graduates. Tell us your stories.
I'm not surprised the New York Times would support "set theory" even though most academics discount it as a radical and untenable idea that promotes a specific political agenda. It's not even taught in most Christian schools. Yet no editor's note revealing the inherent bias herein? For shame NY TImes...
Most Christian schools don't teach set theory? You mean their students are not taught the concepts of the union and intersection of sets, or a subset? This is basic math. Why would Christians or the NY Times see set theory as an "untenable idea"?
Most academics discount set theory as radical and untenable? Set theory promotes a specific political agenda? Where on Earth do you come up with such inane garbage? The inherit bias herein is yours alone: you clearly have no clue what set theory is.
Certain fundamentalist Christian schools don't approve of set theory for the simple reason that it reaches conclusions which are unappealing to Christian sensibilities, if not downright contradictory to Christian dogma. Georg Cantor, in the latter part of the 19th century, showed that there exists infinite sets which are "bigger" than other infinite sets. Moreover, he showed that there cannot be a "biggest" infinite set. None of his proofs are beyond the grasp of a fairly bright high school kid. I even think that you, Stephenson Billings, can fellow the arguments. However, Christian dogma has certain hangups with mere mortals making statements about infinity, let alone different "sized" infinities.
None of this is surprising, or new. Galileo was sent to the Roman Inquisition for supporting the Copernican view. Do you, Stephenson, believe now as the Christians all did back then that everything revolves around the Earth, that all the other planets and stars are up in the Heavens solely for our benefit? There are countless examples I can draw from illustrating that when dogma contradicts mathematics and hard science, the former loses every time.
You should read Cantor's little book, "Contributions to the Founding of the Theory of Transfinite Numbers" available for about $12 from Dover Books on Mathematics. It's very readable, and who knows? you might even learn something.
I fully agree with Dr. Billings as to the lack of instruction in set theory in Christian schools. While, as Mr. Kissack points out, the concepts union and intersection of sets, or subsets are important, Christians teach the broader concept of Set Faith.
I shall explain. We all know dinosaurs roamed the earth, and we all can agree humans roamed the earth. If we apply the concept of unions, we can clearly believe humans (such as Jesus) rode dinosaurs as they roamed the earth.
An example used to explain the concept of set intersection can also be easily explained to mini-Christians as such: If we take the set of male humans and intersect it with the set of female humans the intersection is humans who are married; because we all know the definition of marriage is between one man and one woman.
I am a math/science person.
I have done well on all 3 NYS math regents exams, got a perfect score on the SAT math section, and am taking as many AP math and science classes as I can during my upcoming 12th grade year.
True, algebra will not come into your average conversation or be used by a non-STEM major on a daily basis, but neither will a lot of the other things you learn in school. This does not mean that we should simply stop learning them.
Part of being a well-informed and knowledgible person is understanding the basics of multiple subject areas - enough to know whether or not you want to pursue a specific subject further. For this reason, mandating algebra is important especially given the recent emphasis on STEM careers.
Mandated algebra introduces potential STEM majors to the world of math and science. Those not interested need not pursue algebra further than the introductory college course.
A disturbing trend I often hear one of my teachers complain about is the shrinking number of required topics covered on standardized math exams. The number of topics covered on the NYS regents exam and the AP Calculus exams, for example, have shrunk tremendously since their inception. I cannot help but think that eliminating algebra entirely would reinforce this trend.
I agree wholeheartedly with Matthew Levey and Steven Koh. There is also a logical fallacy here. We have an author pushing on math education and its utility. But the fallacy is that if you focus on any single slice of academic education you can make the same argument. Lawrence Summers in the NYT in January selected some slices for disparagement, including foreign language and history. http://www.nytimes.com/2012/01/22/education/edlife/the-21st-century-education.html?pagewanted=all
But focusing education only on utilitarian subjects is not the answer. We learn by being exposed to intellectual riches. The French and Chinese languages, Geometry, and European History are all incredibly rich grounds for exploration. Any one subject area can be attacked on the logic of utility. But collectively, they are all valuable lessons in how multifaceted and complex our world really is. And by the way, I have an MBA in Finance from the University of Chicago, and am in business, but I find that people who are bilingual and have electrical engineering backgrounds are great hires. They get complexity in a way that econ majors don't.
Algebra should be required. Students who do not have a background in this area are forever closed off from the science, technology, engineering and math careers that are so critical to the success of America.
Few people in high school know what their career paths will be, or what the jobs available will be 30 years from now. I bet there are lot of people in today's market who surely wish they had the math skills to take on the challenges that are now being filled by H1B immigrants who DO understand algebra.
Don't sell America short on math skills. We already have one of the worst math programs in the developed world. Making it weaker is not progress.
It's not about understanding algebra. It's about being able to think abstractly. Most people can't. If they could they'd be physicists or mathematicians. Our society needs people to function in other areas and higher math isn't integral in most other areas.
When I was in high school I had a hard time thinking abstractly, hence I did not do well in algebra or other math-related classes. It wasn't until I returned to college as a adult that I overcame this math block and "got" algebra. Often the difference in being able to understand abstract concepts is having a teacher who can do an adequate job of teaching the concepts in a way that students can understand. Bottom line was that in high school I had a math teacher who really didn't engage her students, whereas in college I had an instructor who was an excellent teacher.
Thank goodness our culture still considers illiteracy something of an embarrassment; we sure don't feel that way about innumeracy. George's example of 'practical' math, figuring what 30% off of $16 would be, is optimistic - I'm betting most US adults would have some trouble with that. I teach at a community college where something like 68% of students entering from local high schools in Bergen County, NJ (per capita, one of the wealthiest counties in the nation) certainly can NOT figure out such problems, requiring 3 or more semesters of remediation before they take their first college-level math class. If over two-thirds of our graduating HS seniors, coming from wealthy suburban high schools, cannot figure percentages, something is VERY wrong! I have sat with students who - when this problem is posed: "The web site shows that if you walk for 30 minutes you will burn 230 calories, so what if we walk for another 30 minutes?" - literally become paralyzed with fear. That's just very sad.
Bill,
I just read that the 1999 NAEP, asked 8th grader how many millions were in a billion. There were five multiple choice responses and just 22% got it right. So basically not a lot better than random guessing would yield.
Indeed the math issues are deep and profound, and seen most readily in the nation's community colleges, where most of our high school students end up. Here in NY the data indicate that if kids have to take remedial math in college they have something like a 20% chance of ever graduating college.
If that doesn't bother us, and force us to look hard at how we instruct in k-12, I can't imagine what will.
In today's meager job market, we need imagination to spur creation. We need a new Bill Gates, a new Steve Jobs, and a new John F Kennedy. We need imagination, because if it can be imagined, it can be created. And what we have now is neither sustainable nor will it lead to a future worth living for.
We need to stop teaching kids facts, because for the most part there is no such thing as facts. Facts can change over time. We need to stop focusing on memorization, and focus on critical thinking. Our public education system has created a generation of mindless drones, who might be great at reciting facts, but can't think for themselves. Look at our political discourse. It isn't about substance, it's about name-calling and lies. Why? Because recent elections and polls show that these work on most Americans. Why? Because most Americans lack the critical thinking skills to read between the lines and lack the basic research skills to look deeper into issues. What we should be eliminating are standardized tests that rely on filling in circles. We waste too much time and money on teaching to a test, because the test results determine funding. We need to drop political correctness and start challenging students. If a student gets offended(within reason), don't look at it as a bad thing, that's a good thing. Students should be pushed past their boundaries. Kids will fail, but that's not bad. What's bad is when kids fail, we don't encourage learning, we encourage blaming the teacher, or society, or the boogyman in the closet. We need an education system that teaches problem solving, and teamwork.
Bill, I couldn't agree more. Your thoughts and analysis are correct. As a teacher; however, the basic facts are the foundation to critical thinking that are being overlooked in the schools. Many students don't know their multiplication facts or literary elements by middle school. I believe education has to go back to "drill and skill" in the younger years to even try critical thinking once students enter junior high. As you stated, "...and lack the basic research skills to look deeper into the issues." Without the facts, the students don't know how to do research, multiply, read with the elements in mind, etc. If we bring back the learned facts, we can start to apply and think critically.
Bill and Gretchen,
I could not agree more too.
When you're struggling to remember that 12 x 12 makes 144, factoring a polynomial with 144 as a constant is a real drag. We need both automaticity on facts, and insight about the process.
As my friend Robert says "facts are like the vegetables; you don't get dessert (critical thinking, opinions) until you eat 'em"
Yes, Algebra should be mandatory. There are two core intellectual reasons for this:
1. As Matthew Levey notes, learning algebra is an essential component to developing a capacity for abstract, analytic thought, maybe even the most important one.
2. It's in and of itself interesting, even fun, when well taught. It's full of new perspectives, puzzles, and knowledge in a way that the arithmetic that precedes it for most kids is not. You can't solve algebra problems via memorization. You have to think your way through them. That may be hard for some kids, but the solution is to make curricula more flexible not to eliminate the subject altogether.
There's also enormous practical and vocational utility to algebra for a very large portion of people. People doing any sort of quantitative work use algebra (whether they recognize it as such or not). And that's not just engineers, scientists, and financial analysts. It's "knowledge workers" of almost any description, crafts people, clerical workers, small business owners, and more.
You want to be able to judge what a politician says about the budget? You're better off knowing some algebra. Judge whether the bottom line on personal financial documents like tax returns, mortgages, or credit card statements are reasonable? Figure out how much money you need to save for your retirement? Find a mistake in that spreadsheet you're working on at the office? Figure out whether A-Rod or Jeter is a more productive player? Algebra. It's woven into countless aspects of life, as important and useful as arithmetic for people who actually want to understand what they're doing beyond the most rudimentary level.
To say that people don't need to learn this is to consign them to an unacceptable and tragic level of ignorance. That doesn't necessarily mean that everyone needs two years of it, complete with high-stakes tests, but it does mean that everyone should have at least some measure of exposure to it. Plane geometry and statistics, too by the way.
As Matthew Levey notes, learning algebra is an essential component to developing a capacity for abstract, analytic thought, maybe even the most important one.
I disagree with this assessment. I believe the ability to perform abstract and analytic thought is either already there, or it will never be there. 'There' being in the student's mind. If it isn't part of the natural ability of the student, trying to 'teach' analytic and abstract thinking is wasted effort and a lesson in futility. Perhaps there are tests to screen who should take higher math courses and who should stick to basic math. Same goes for understanding poetry which also is a case study for abstract thinking. Standardized testing of standardized curricula makes dummies out of students whose ability lies elsewhere. Teaching out of the box leads to thinking out of the box but certain courses such as mandatory higher math classes are simply wasted on the math challenged and cause poor overall grade averages and lead to frustration and early exit of formal schooling.
Kishmir,
You state: "I believe the ability to perform abstract and analytic thought is either already there, or it will never be there. 'There' being in the student's mind. If it isn't part of the natural ability of the student, trying to 'teach' analytic and abstract thinking is wasted effort and a lesson in futility."
However, the brain is incredibly plastic, and intelligence is not fixed. It is influenced by many factors, including how much effort (and how efficient that effort is). Please inform yourself about this by reading up on recent research on the subject. I truly hope you are not in education, because these kinds of beliefs can be tremendously damaging, whether they are held by the teacher or the student.
Read what it says here, for instance (different context, but still relevant to what I am saying):
http://www.ascd.org/publications/educational-leadership/oct07/vol65/num02/The-Perils-and-Promises-of-Praise.aspx
I taught college-level algebra for 5 years, and I can tell you that I have witnessed students' abilities for abstract thought increased dramatically when said students were willing to put in the time and effort. They also had to be smart about how they used that time and energy, of course. And that is part of the invaluable lesson from being presented: you can master something that initially seems impossibly hard and in doing so you can improve your mind.
What Dr. Hacker may not realize is that not understanding algebra makes most of science difficult, if not impossible to understand. Just think of f-ma. My students who do not understand how to rearrange an equation have a difficult time working through simple physics problems because they do not understand how to rearrange a formula. What our students need is a better understanding of all aspects of math, not less math.
Ben Lewin
Again, the ability to conceptualize abstractly is not inherent in all students. It's not just a question of better understanding. Higher math is like music. Sit two bright people in front of a piano and one will start playing shortly, with minimal instruction and the other will simply stare at the keys and accomplish nothing.
Dear Ben,
C.P. Snow, the renowned British physicist and author, famously asked why it is that we're quick to label someone as ignorant if they don't know Shakespeare, but we're quite comfortable if they have no idea about the Second Law of Thermodynamics.
It's a longer story than I had space to explain, but in my reference to the conservation of energy, I was trying to suggest that knowledge is also intrinsically good.
Whether I major in physics or not, aren't I in a better place to understand the world around me if I grasp some basic concepts about energy, matter, and chemistry.
And who's to say, when I'm just 13, that they know which 'track' I'm on, and whether I'll end up in your physics class or not, right? In America k-12 is supposed to be about opening options, not foreclosing them.
Dear Matthew,
I think that we are pretty much in agreement. Rather than categorizing subjects as "essential" or "non-essential," we need to expose our students to as much knowledge as possible, in order to give them as many opportunities as possible.
When I have a student ask me in my science class, "Why do I have to know this?" my answer to them is almost always the same. The more knowledge you have, the better decisions you will be able to make as an adult, and the more choices you will have after you graduate.
Sir, you are most certainly preaching to the choir!
Algebra should be optional. I've taken the course before in high school and have never had to use any of that fluff in my daily life. If a student chooses to pursue a career in science or the like, then algebra is essential. As it is, it's a nearly impenetrable subject with little real world application.
it's impenetrable to you because your mind doesn't think abstractly while it may be functionally optimally in other areas. If you had the ability to think abstractly you would utilize higher math and find real world applications. So I would agree with you but not because higher math is fluff. It's just not for you.
The only topics that are absolutely required by 100% of the population to be able to live are the art of breathing and eating. The plumber may not need to be able to write, the singer may not need to be able to count, and the mathematician may not need to locate Russia on a map. But that does not mean eating and breathing should be [the only] topics covered at school. The fact that you think you've never had to use algebra is not really more relevant than people who think they don't need to know how to write to fix cars.
@Kishmir: What rancid nonsense. The capacity for abstract thought is as inherent to humans as the ability to speak -- everyone has it, absent some debility.
@Lisa: The entirety of the built world -- from the computer you use, to the building you live in, even to the clothes on your back, are all built with and on top of algebra. There is not a single made object you have ever touched that did not encode algebra in some profound and subtle way. Engineering, science, economics, and business are all incomprehensible without access to algebra. It seems clear that you had poor teachers, but that is hardly an excuse for such a deeply mindless comment.
If an item costs $16 and is 30% off ,what is the final price?If you pay with a $20 bill,how much change will you receive?THAT is about all the vast majority of adults know.Unless the student has a direct application of using algebra,there is no real world reason to require it.
Answering the questions you pose requires algebra. You must know what computation to complete, and that is done by setting up an equation (in your head, usually, so you don't realize you're doing algebra). You could also ask if the original price is $20 and the sale price is $15, what percent discount are you receiving, which could also be relevant. You need to understand how to arrange the equation in your head so that you can complete the correct computation (using arithmetic).
I am shocked that a college professor would think that Algebra should be "dropped in favor of more practical math," whatever that means. If he refers to just having students study arithmetic, I can say that as a graduate with a bachelors degree in mathematics and a masters in Applied Mathmatics that this is ridiculous. Algebra is a fundamental part of mathematics and is used in science, technology, engineering, and, needless to say, in higher mathematics.
Furthermore, we use alebra everyday to be able to calculate real life problems, such as calculating the monthly payment on mortgage at a certain interest rate. Our education system is already lacking and to say the we should "drop Algebra" is just another problem to add to the long list of ways our education system is failing our teachers and our children.
For a smart person this argument goes over your head. You have a background in higher math therefore you see the need for higher math in everyday life. But you're not functioning in the real world. If you lacked the ability to think abstractly, thus not being able to utilize your brain to handle higher math, would you still think it essential in your everyday life?Higher math is like anchovies. They're not for everyone.
Johanna,
My dad (Cornell '64) remembers Hacker as a very good professor of government back when.
I think he's just gotten a bit beyond his area of real expertise, and it shows.
Imagine if we said that since only 40% of citizens participate in the average election, we should drop US History and the Constitution from the curriculum of the average elementary school, in favor of practical classes that teach one how to apply for a driver's license or a zoning variance. "They're not gonna vote, so why stress them out with memorizing useless facts about Hamilton, Jefferson, or how a bill becomes a law?"
Schools only teach students formulas. How the heck are we supposed to understand where to apply these things? I agree, we should be taught their practical purposes as well, and be given more problem solving examples. Colleges, GMAT tests, SATs, real life will only you require real life PROBLEMS. Once you have a problem, you can identify the concept and start analyzing how to solve it, right?
Well middle schools, high schools, they don't do that well. When they do, it's too confusing. Some students want to learn it, some don't. Most don't... I know a girl who is starting high school this year, she's been hating math for years. Most common argument? When am I supposed to need this sh*t?!
Well, there you go, they just don't get math, I don't blame them. I was an A student in math after 9th grade, but do see most of it as "blah" now. I would have preferred more problem solving types of questions in relation to these equations.
What does Lisa Lin think is the purpose of education, and what criteria she suggest for deciding what high school courses should be optional as opposed to required? Literature could be criticized for having even less in the way of real world application than algebra. For many students, science is nearly impenetrable; for others, a language other than English is. Among other things, we are better readers, writers, and thinkers for having studied literature, just as we are better problem solvers, shoppers, drivers, vacation planners, voters, and thinkers for having studied algebra. We are better adults, better world citizens, and better role models for having studied all of these things. The education system we have isn't perfect, but algebra deserves no special whipping-boy status among what we as a society have come to consider part of a high school education.
Thanks for this post, Matthew. Very well said on so many levels. I wrote about this on my blog as well and would love for your thoughts: http://karenmahon.com/2012/07...
Thanks!
Karen
Neither 'Drop Algebra' nor 'Fix Math Education' makes sense. Whats wrong with 'Algebra'? Whats wrong with 'Math Education'?
There is nothing wrong with algebra. It is truly helpful in building the minds of students. It improves the thinking and analytical abilities of students tremendously. I will rather it was made mandatory
I believe this is an important area for discussion. Two observations:
1. I think this piece and this piece and the comments neglect Hacker's main argument, which is that Algebra acts as an artificial filter, weeding out many students who would otherwise be fully capable of completing a college-prep high school curriculum.
2. There's an enormous gap between knowing how to solve a first order equation, understanding orders of magnitude, and being acquainted with probability, skills which I agree every college bound student and most non-college bound students should have, and knowing how to factor a polynomial, using the binomial theorem, doing analytic geometry, or solving an equation with multiple roots, all of which I believe are still part of the Algebra curriculum, and which very few adults ever use under any circumstances. As for set theory, which the writer mentions, I agree that is useful, and might even engage the interest of some HS students: do they even teach it anymore in HS Algebra? I thought it went out with the New Math.
Dear Peter,
Hacker's arguments about elite schools and how they screen were indeed a big part of his essay. But his proposed solution of 'real world' math involved just as much algebra, a fact pointed out by practicing teachers like Patrick Honner (see http://tinyurl.com/curmsa8) and Dan Willingham (http://tinyurl.com/cpca6wg). Furthermore, the SAT math section only goes as far as Algebra I; so unless Hacker's proposal involves re-writing that test too, we're back the point that kids who want to attend college (even as lit majors) will need to learn algebra.
In the interest of a shorter response, I tried to address the broader question of why algebra is important, regardless of ones college choice.
The examples you refer to as relevant all seem to involve, to one degree or another, abstraction from a general rule and an ability to use variables in place of numbers. So we seem to agree that if the 'problem' is that most kids can't pass algebra, the solution is to teach the subject well. Not drop it from the curriculum.
Lastly, I see that set theory is formally considered to be part of logic. I'd have loved it if my junior high teachers could have taken me beyond A U B to the barber's paradox. But alas, instruction was hit-or-miss mack then too.
In case you didn't see this earlier post, here's how one math teacher in the city had successful results with the AP Calculus course -- or, rather, how his Bronx students had success. http://www.nytimes.com/school...
It's interesting to see how many responders have strong opinions about algebra, but do not see the humor in the false names of other responders. Perhaps we are too focused on our own ways of life and opinions and not enough on what is going on in the rest of the world. Not only is there humor, but also life without algebra, or diagramming sentences, or literature, or a foreign language. Why not broaden the high school curriculum and make most of it elective?
The dumbing down of a society begins with the removal from society's ideals of the importance of creativity. The destruction of a society begins with the non-memory of the concept of creativity...
I barely passed Algebra I in H.S., and didn't bother myself further with analytical mathematics. I'm not a math person. On the other hand, I write intricate novels with multi-dimensional characters who mirror life as we all know and live it (or should, or should never wish it on our enemies). My analysis of literature comes from the interrelationship between facts, emotions, story, reality, truth, and fiction. Believe me: I don't miss the loss of algebra.
Mark Beyer
Author of "The Village Wit"
and "What Beauty"
blogs at www.bibliogrind.com
The curriculum being narrowed to cover just the tested ideas is very disturbing Stephen. Whether in math or history, no?
Perhaps the better question for you is whether you see value your literature classes in the same way that a non-STEM classmate ought to see the value in Algebra and Statistics?
Here is a simple reason why we need algebra to be taught in schools. Recently, I looked at a box of "healthy" cereal, and noted that the nutritional values (including serving size) on the box were based on 2,000 calorie diet. As a math/science guy (and teacher), I was able to solve the fairly simple algebra problem: If the average pre-teen should consume 1,500 calories a day, what would be an appropriate serving size for this cereal?
Those of us who have had a "traditional" math education probably do not even realize anymore when we are doing basic algebra, but we use it any time we try to figure out an unknown amount.
To me, learning Algebra is as necessary as learning how to read. As a new york city public high school student, I can find many flaws with the structure of mandatory curriculum, largely the lack of depth and understanding each class at the most basic level requires. The mandatory Algebra II/Trig regents I took this year could be passed by using only a calculator. The New York State board of ed continues to lower the amount of curriculum necessary to pass the regents exams, and continues to curve these exams to make them easier and easier to pass.
high school is where students can master an understanding of logic and the basics of a broad area of different topics, in order to take one tract further in college and professional life. So yes, "taking algebra" may not be so important to many students, but fully understanding algebra, and applying it to daily life and more advanced math will be important to any student, no matter what they study in college. Dropping algebra from mandatory high school curriculum would only increase an ignorance and dependence on technology that this generation has. Instead of dropping algebra, the state should improve and expand on it.
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