Who needs calculus?

Calculus is the wrong goal for 90 percent of students, argued Harvey Mudd Professor Arthur T. Benjamin at the Ciudad de las Ideas in Puebla, Mexico.

“For the last 200 years, the mathematics that we’ve learned starts with arithmetic and algebra, and everything we do after that is taking us toward one subject, calculus. I think that is the wrong mathematical goal for 90 percent of our students,” he says. “We’re now living in an age of information and data, and the mathematics that will be most relevant to our daily lives is probability and statistics.” Only some professions require calculus. Everyone reads—and many misunderstand—media reports about health, science, and the environment that contain statistics. Better literacy in probability and stats would benefit everyone.

Most students don’t make it to calculus — or statistics. I didn’t. As a journalist — a notoriously innumerate trade — I frequently had to struggle with statistics to understand reports. I found my arithmetic skills very useful.

The Carnegie Foundation‘s redesign of community college math curricula stresses statistics and quantitative reasoning for students who aren’t headed for STEM careers.

I wonder how high school math would change if students could choose between a STEM-prep or math-for-citizenship track. Would we let students opt out of the calculus track in ninth or tenth grade? How about the kids who keep flunking algebra?

 

About Joanne

Comments

  1. We require the calculus track, for two reasons: 1) to make sure as many as possible of our students can be described as in a college prep program — and the selective colleges certainly do use calculus or at last Advanced Algebra/Trigonometry as a barrier or screening device. This is mostly to the advantage of high schools that are being compared with their peers. 2) we are aware that some undetermined number of students who think they won’t need higher math as adults, might be wrong in choosing non-calculus.

    The kids who keep flunking Algebra don’t end up in Calculus anyway. But they would benefit from a more intense effort to get them solid in K-8 math, which they will definitely need. And then once that’s accomplished, they could take stats lite (with a little Algebra, which you need for stats) or personal finance or another math class that they can see the point of.

  2. Math stretches your mind and helps you to think in formal, logical ways. It prepares your mind for understanding natural phenomena in physics and other hard sciences.

    Well, screw that. If we can’t even learn ‘em algebra, calculus is just out of the question. We oughta get rid of all that fancy chemistry and biology, too. Who ever needs all that stuff? The incredible focus on reading is probably misplaced as well. After all, they can find out just about anything they need to by watching television, right?

    (Joanne: it’s your site and you have every right to do with it as you like, but I’d like to mention that I find video ads with sound that start automatically to be distracting and unpleasant.)

    • I do have some question about the student without a firm grounding on the arithmetic managing to master calculus. I have students who are befuddled with pre-algebra concepts – they just operate at a concrete level, and any moderately abstract concept is capable of causing them trouble.

      Yes, the math can help students learn about logical thinking, but not if it is only taught at a level they really aren’t ready for.

  3. Wayne Bishop says:

    The biggest problem with avoiding calculus as a goal (even if never reached) is that students who are not on the “calculus track” are not getting the preparation that they need for more advanced work in any of the disciplines that require mathematics; e.g., the hard sciences and engineering, even honest economics. That is, and speaking statistically, options are cut off before students have determined whether or not they have the ability and/or the interest in pursuing one of these areas. Even statistics itself, and a deeper level, requires an advanced form of the calculus. Statistics-lite is important and does get taught but often very badly by poorly prepared teachers.

  4. Rob: Try Adblock.

    If you hope to attend a selective college, you’ll need calculus. That choice will produce too clear a demarkation between the college-bound and everyone else.

    • Michael E. Lopez says:

      This isn’t quite true.

      “Need” is overstating the case quite a bit.

      • superdestroyer says:

        If 5000 high school seniors are applying for 1250 places at a selective university, it is very likely that more than 2500 of them have taken and did well in calculus. A student without calculus and the advanced sciences is just another liberal arts types that the university has too many compared to science majors.

      • You can have the conversation with parents whose children were tracked out of the honors math courses. I’ve had several over the past few years, as our local public school is happy to limit access to the honors math & science courses. If you child isn’t in the top courses, he won’t have the (weighted) GPA to compete. His class rank won’t qualify. As high schools determine science courses by math placement, being tracked out of calculus means he’d be tracked into the “science-lite” courses.

        Now, if he’s ranked one of the top high school running backs in the nation, it won’t make a difference.

        • Perfectly correct, on both aspects. I am surprised – and pleased – that my kids’ old HS still has honors prerequisites for AP courses. It means that the honors courses are very challenging HS classes and the APs are truly college-level. Although 8th-grade honors algebra (used to be only honors) is necessary for calculus (and AP physics) as a senior and honors sciences may require honors math, it is possible to do honors math and science, plus AP bio or AP chem while starting with honors algebra as a freshman. With no honors-level math, I’m not sure – possibly honors bio doesn’t require it.

      • Michael E. Lopez says:

        If you’re going to be a boring, generic applicant… then sure. You need calculus. Otherwise the other boring, generic applicants are going to kick your ass because you’re not really playing the game that you’re playing well at all.

        But there are other games to play, and other ways to build applications.

        Look, I’m not saying it doesn’t *help*. Of course it does. So does a 4.8 GPA, perfect SATs, and glowing letters of recommendation from people who both are sincere, detailed, and know how to write a good letter.

        But you don’t NEED any of those things, and it’s silly to think that you do. Once you start thinking in terms of what you “need”, you’re back to playing the boring, generic applicant game.

        I mean, sure… go ahead if that’s all you have in your ammo box. Fire away.

        Just don’t forget to have a transformative experience working in a soup kitchen.

        • In our town, the “boring, generic applicants” have all had transformative experiences working in a soup kitchen. And digging latrines in South America.

          They also have AP courses, which include calculus.

          Is calculus necessary for all students? No. On the other hand, when do you want to sort kids into piles? How do you make the decisions?

  5. Even statistics and reasoning skills require a solid foundation in the basics of addition, subtraction, multiplication, division, fractions, and percentages. Without these skills, no student will succeed in programs which do not need calculus.

    I read an article for careers which pay well, and have low unemployment at the moment, and the one which sits at about 2 to 3 percent unemployment is Nursing (LPN and RN), but even for these courses, a working knowledge of algebra is required for admittance into nursing school, and is certainly needed to pass state licensing exams and the NCLEX test.

  6. Bill is right that a working knowledge of algebra is important for nursing-type majors, and a failure to understand basic arithmetic, fractions, and ratios was the biggest cause of failure in the pre-nursing CC classes that I have taught. ‘Preparing’ students for calculus by having them take algebra in 8th or 9th grade, though, doesn’t seem to help them. For some students, a bit more time on fundamentals, maybe statistics, and algebra a bit later might cause them to actually learn it instead of failing it a few times and learning to fake their way through it.

    I can see other arguments for calc-prep (although but 9th grade, most students have some idea of whether their goal is ivy, state U, or CC), but there seems to be a mindset in education that preparing for something harder (like calc) means that you’ve got a better grasp of the pre-req classes, and that’s not necessarily true.

  7. Christina Lordeman says:

    My biggest educational regret is that I never took statistics. Despite my plans to major in international relations (which later turned into German, with the intention of going into teaching), I took AP calculus and I’m glad I did. I’ll probably never need it in a professional setting, but I think that kind of stuff is just good for the brain, even if you don’t really “use” it later on. My university required us to take one course out of a group of courses identified as dealing with significant amounts of statistical analysis (and designed to introduce students to statistics who had never taken it before), but it wasn’t the same as a traditional statistics course. I didn’t realize what a disadvantage that was until I began working on my senior thesis and spent hours with a calculator trying to figure out by trial and error what several graphs, coefficients, etc. were supposed to be telling me. As a teacher with a passion for ed reform, I’m really seriously taking a course… as soon as I have the money to.

  8. Everyone needs real, solid arithmetic, fractions, decimals and percentages, plus basic financial stuff (interest, various charges etc) and a basic understanding of statistics and probability. Most kids, properly taught (HUGE problem – flawed curricula, weak teachers and failure to require effort and enforce grade-level standards) should be able to get this far by the end of 8th grade. Some will get there sooner, to allow starting algebra in 7th or 8th grade. The first group will not get to calculus, but could reach pre-calc or a real statistics course as seniors. I don’t see that as a real problem, since most high schools didn’t even offer calculus until relatively recently, but they should be prepared for a real calc course as freshmen. Obviously, those kids aiming for the most competitive colleges will need it, because the colleges consider it an admissions weeding tool, but that’s not most kids. Make sure they know the basics well and they can build upon that; assigning fancy – and fake – titles to low-content classes helps no one. In the county-wide system my kids attended, it was well-known that the course descriptions were a sham at some schools.

    • I have two semesters of college calculus – the “for math majors” flavor, not the softened versions sometimes offered for non-majors. I haven’t used it since. Even in taking two semesters of “for majors” physics, the amount of calculus that could be applied to the lessons was covered in the first part of the first semester of calculus – and for those willing to memorize formulas rather than deriving them it would have been unnecessary.

      I haven’t used calculus since. How often do you use calculus, or even “pre-calc” level math in your daily life? (Or yearly life, or since college?)

      I’m not advocating against calculus – I think kids should be encouraged to learn math to the maximum of their aptitude. I just question its relevance to most careers and, thus, as a litmus test for whether students are learning enough math.

      I agree with those advocating statistics – it’s a more accessible level of math, and if taught well should help people better understand the polls and data pushed at them by the media and interest groups. That has day-to-day, real world value even if it’s not part of your job.

  9. I teach both the AP calculus course and statistics course at our school. The statistics course is not an AP course but students can elect to take it as a dual enrollment course earning credit from a local community college. The students in the statistics course that have not yet had a precalculus course are really struggling. Some are having a hard time with the embedded algebra in the course, others because they are having a hard time making connections, even when I point them out.

    Not everyone can be successful calculus, but those students who can succeed in it are the ones that will do a lot more in life than be the TJ Friday assistant managers.

    See Joann’s

    http://www.joannejacobs.com/2011/11/generation-cupcake-goes-to-college/

    post if you don’t know what I’m talking about.

  10. Addendum: Not all kids should be on a college prep trajectory. Ideally, kids would have solid vo-tech options, starting in sophomore year, that would prepare them for good jobs after graduation, admin assistant, LPN, cosmetology, auto mechanics, surgical techs, medical and nursing assistants, HVAC and many others. Entry into such programs would require solid math skills, as I described above, and the ability to read and write technical material. What used to be the general course was geared toward kids unprepared for vo-tech or unsure of their future plans but it represented solid academics. In my day, while the draft existed, many of these kids – boys- intended to join the military after graduation and acquire specific training there. It still happens, with significant success, partly because the military can do – and does- aptitude testing – which steer kids toward their strong areas.
    At the lowest end of the spectrum, the kids are neither educable nor trainable and should not enter the school system. The next group upwards is trainable and should be in special programs tailored to maximize their abilities. College-for-all cheats a lot of kids.

  11. MomOf4: At many schools, if you were to say what’s in your last post as a teacher in a K-12 school, you’d get fired. Which is why what you said is all correct. :)

    • I know. A relative teaches HS in an affluent suburb and the word is that an untenured teacher would be fired for suggesting that not all of the kids should go to college. Far too many get passed along without ever made any effort and too little has been demanded of them; like the incoming freshmen who could not reliably identify the subject of a sentence with only one noun/pronoun. Those kids won’t do any homework, either.

  12. R. Craigen says:

    I agree with the points Wayne Bishop and “Rob” say above. Let me ask the following:

    What would happen to education if we went through the curriculum in every subject with a fine-toothed comb and asked the question, “when will most students ever need this”? and when the answer was “probably never”, we axed it?

    Ancient greece, classical literature, a second language, most of political science, large swaths of math — trigonometry, polynomials and rational functions, inequalities, most algebra, etc — ALL of music and poetry, most things pertaining to modern fictional literature, most world geography, all laboratory science.

    The answer: We would have a marginally “educated” workforce ready for menial, unskilled labour, and almost completely unable to intelligently parse the world around them every day, little sense of what has transpired to bring them to the present, little imagination for what the future could hold. A society of useful (but not too useful) drones. And completely inadequate to compete in a global socio-political-economic milieu amongst cultures that seed their next generation with this kind of knowledge.

    All this is not to say that calculus is uniquely, absolutely essential in order to be educated, but to point out that the question being asked, if taken to its logical consequences, is a dangerous one. Getting educated is not a matter of “can you foresee when this will be used”, as much as “is this a critical piece in the construction of a properly educated mind for the 21st century?” I would be very careful about tossing out what was probably the central mathematical artifact responsible for dragging the Western world out of the dark ages and into the scientific and industrial movements that have so transformed the world, without something similar with which to replace it. (If you’re interested, I have some suggestions.)

    The value of a given discipline or curricular subject-matter is found not as much in which identifiable practical uses we can point to, but in the variety and complexity of its connections to other essential parts of knowledge. In this respect, few mathematical fields stand out as prominently as trigonometry and calculus, which may be the two most likely to succumb first to the blade of the “Will I ever actually use this?” axe.

    This is before one considers the most important reason for learning along the precalculus/calculus thread: It is the central vein in the mine of developing abstraction skills. While other parts of public school mathematics are rich in abstraction (not to mention a few fields outside math — owing largely to the mathematical tools used therein) … this is the motherload.

    Is abstraction important? In the 21st century, our information-driven economy and social fabric I would argue that abstraction is critical. From a more philosophical perspective, abstraction is the thing that separates human from beast. Every human being is born with a capabicity for it far beyond the already-impressive cerebral skills of apes and other animals with otherwise robust mental capacities. It enables us to manipulate the world around us in a mirror-world of symbols, enabling us to solve complex, apparently intractable problems using tools passed down that facilitate almost miraculous economies of thought and action. Strip a human of his or her capacity to function abstractly, and you have a beast that wears clothes. Every civilization throughout history that has laboured to bring higher levels of mathematical sophistication to a broad swath of its citizenry has risen above surrounding and preceding societies, and I believe it is for this reason. It is the magic ingredient of a successful, forward-moving society though its influences are hard to trace; it lifts society in holistic ways by developing certain mental muscles in their only natural setting.

    Society doesn’t need every child to learn calculus, or to read the classics, or be able to articulate central themes of Western history and how they bear on current events. Or, for that matter, to understand what a “95% confidence interval” signifies. But unless a large and robust core does learn these things, our schools are cultivating a field lacking the very nutrients essential for a healthy modern civilization. Then, welcome to the dark ages.

    • Roger Sweeny says:

      I teach 11th grade honors physics, and even though my students are tops in the class, they have difficulty with abstraction. For many, their brains are still not ready. The people commenting here are so used to abstraction it seems easy to us, as easy as sitting on a toilet and evacuating the bowel. That is also easy to us. But try to teach a one and a half year old to do that and you will have months full of sorrow. Try to teach a two and a half year old and you’ll both enjoy it.

      • Michael E. Lopez says:

        I think there might be multiple “windows” for some of this sort of stuff.

        My best results teaching young people formal logic (outside of college) have been with 5th graders and high school seniors. I have ridiculously small sample sizes and this is all anecdotal, but the 12-16 year-olds just don’t seem to get it, nor do those under 10.

        • Roger Sweeny says:

          It’s an interesting thought: that as puberty kicks in, some abilities are lost, perhaps to reappear later.

  13. I think it’s undeniable that everyone show know arithmetic (i.e. the addition, subtraction, multiplication, and division of multi-digit numbers, percent, fractions, decimals, and so on).

    On the other hand, I’m not convinced that every single student needs to know Algebra and beyond. Why? Because every student will not pursue the same career path. Different career paths will require different levels of mathematical knowledge. Some career paths will require Algebra and beyond. For students who are in such career paths, they should definitely have knowledge of those levels of mathematics. For students who are not in such career paths, it is pointless for them to take Algebra and beyond. On the assumption that a student knows what career path he/she wishes to pursue and it is not related to algebra or beyond, studying such subjects is a distraction from those subjects which are more relevant to their desired career.

    For my case, I will become an elementary teacher. I will only be an elementary teacher. For me, learning algebra was pointless. Why? Because what I was taught about algebra in high school is not something that my students in elementary school will learn. At best, they will learn about communicative and distributive properties (i.e. commonly associated with the acronym BEDMAS). But, nothing much beyond that. Let me be more specific again. I will never use the quadratic formula. I will never need to be able to find the angle of a hypotenuse (such as in geometry). And, so on and so on.

    HOWEVER, as I said earlier, if it is known that a student will definitely use that information in their future career, then that student ought to take the levels of mathematics pertaining to that career. Otherwise, essentially, they are forced to take algebra (and possibly, beyond) on a guess that they will need it. Their academic time is being gambled with and potentially wasted.

    • Michael E. Lopez says:

      HOWEVER, as I said earlier, if it is known that a student will definitely use that information in their future career, then that student ought to take the levels of mathematics pertaining to that career.

      School’s not career training, or it shouldn’t be. It’s life training, and happiness training. Now you can buy a lot of happiness with high economic productivity, but let’s not confuse instrumental and ultimate ends.

      Also — and more importantly — no one — no matter how certain they are — “knows” what their future career is, ESPECIALLY in a modern economy and DOUBLE ESPECIALLY when they’re in pre-collegiate school.

      And if you’re really going to be an elementary school teacher, then you’re a fool for not thinking that understanding algebra and geometry is going to help you work with your students.

      If all we really needed was elementary-level knowledge in our teachers, we could have the 6th graders teach 4th grade and save a lot of money. But that would be silly.

      • tim-10-ber says:

        Mchael — high school has been and should be again work training. We don’t need everyone going to college…we need true voc-tech back in all high schools with programs that kids excel in and leave high school with a certificate saying they are qualified for work in X field. We use to have them…why did we stop? Now we have the academies but they don’t appear to have a true voc-tech component to them. Just a parent’s perspective

        • Even if high school changes to have more of a voc-tech focus for many students, so many careers require algebra. Most community college programs require completing College Algebra(seems to be at most Algrebra 2).
          The bottom regular (perhaps not special education) high school plan of classes should ensure that students don’t require math remediation and are at a minimum ready for College Algebra. I would really prefer that students also have a competency in Algebra because it seems a waste of money to pay tuition for a high school level class.

        • Michael E. Lopez says:

          Mchael — high school has been and should be again work training. We don’t need everyone going to college…we need true voc-tech back in all high schools with programs that kids excel in and leave high school with a certificate saying they are qualified for work in X field. We use to have them…why did we stop?

          We probably stopped (1) because some idealist fool decided that we should have college for everyone; (2) because there aren’t just a couple of basic career tracks anymore; and (3) because many professions (automotive repair comes to mind immediately) are far more complicated than they used to be.

          Still, I don’t disagree with you about work training or the practical nature of K-12 education. We’re somewhat on the same page, there. There is a difference, though, between work training, or “training for the world of work”, which is an important part of the life training I was talking about, and training for a particular job. It’s not, in my opinion, the high school’s job to help little Timmy learn to be a plumber. (Of course, one need look no further than the discussion CrimsonWife and I had a few weeks ago to see that people’s opinions on this issue differ dramatically.)

          It might be both possible and proper for schools to have classes in things like applied electronics, basic mechanical engineering theory, computer science, or good old fashioned wood and metal shop. I’m of the mind that getting rid of those programs on a large-scale basis was a mistake, and it seems we agree on that, too. (I’m actually of the mind that more “college track” students should take these courses.) But that’s still not the same as preparing someone for a specific field. That just doesn’t seem practical to me — there are too many fields, changing too quickly, and the labor market is so fluid that one really doesn’t know what one is going to be doing in a few years.

          If Timmy wants to be a plumber, he can go talk to a plumber, who will be impressed with the knowledge of mathematics, physics, the capacities for logic and reason, his facility with tools, and the strong and consistent work ethic that the school has given the little bugger.

          He could take that same set of assets to an electrician, an auto mechanic, or a construction company.

          (Or if he wants to be a plumber, there’s an internet filled with “how to” videos and articles that his education will have prepared him to understand.)

          • J. D. Salinger says:

            Calculus as a bolster for a college application is one thing. But the path towards calculus is one that is necessary to keep options open for STEM type fields per Wayne Bishop’s note above.

  14. tim-10-ber says:

    On calculus — to be able to take high school calculus kids must have algebra I in 8th grade. Sadly this is no longer encouraged…besides the math foundation needed to do this and do it well in 8th grade algebra I is sorely lacking in so many elementary and middle schools math teachers.

    I have one son who made it through pre-calc in high school as he took algebra I in 9th grade and is in a college program where only one year of general math is required. Not sure I am truly wild about this…

    The other son is in a college program where all kids are required to take college algebra and calc as part of the gen ed requirement. His degree requires him to take statistics. He will do fine but i wish he had a stronger math foundation (meaning much, much better teachers) from k-12 in traditional, academic magnet and private school.

    I did not take high school calc.trig but was required to take college algebra and calculus to get a BS degree. I struggled because of a horrible math foundation in 1-8 but ended up doing well in college. Yes, I use calc sometimes in my career…

    But I strongly agree with Momof4′s comments earlier…

    • Michael E. Lopez says:

      You are SOOOOOOO right about the grades 1-8 math preparation being key.

      I pretty much ignored math in high school because our department was awful. I mean GOD awful. (At least the parts that I encountered.) I simply gave up after Algebra 2 and figured I’d play catch-up later. It’s not like Trig or Calc was on the SAT, after all.

      But I had a great foundation through Algebra 1, and that made learning Calculus, Linear Algebra, and Combinatorics much, much easier in college.

  15. GoogleMaster says:

    I *thought* I recognized the name. Art Benjamin is a self-styled “mathemagician” and author of a book by that name. He does mental math. http://en.wikipedia.org/wiki/Arthur_T._Benjamin

    Back to the topic:

    Calculus builds on algebra. Go back to the beginning where you learned about x plus delta-x, y plus delta-y. If you don’t “get” algebra, you’ll never succeed in calculus.

    And algebra is just arithmetic with symbols. If you don’t “get” basic arithmetic including fractions, or if substituting in an x instead of a 3 really throws you for a loop, then you need to sit through basic remedial training until you can do algebra. I use algebra in daily life — at the grocery store, in the stock market, planning to purchase a house, etc.

    There is probably a significant population who will never understand calculus, and these people shouldn’t be made to study it. However, everyone of at least normal intelligence should know enough basic math to be able to comparison-shop. That still leaves question of whether we educate to algebra the top 68% (IQ 85+) or the top 95% (IQ 70+).

    Calculus gives you options. It’s a good idea for those who can do the work to take calculus so that they can avoid eliminating entire fields of study and employment from their future.

    I have an electrical engineering degree. I had no idea what I wanted to do with that degree, but I have worked exclusively in software development since graduation 25 years ago. Programming requires algebra, even for the UI guys. (If this rectangle at point (x,y) on the screen is w by h pixels and I want to double the area and move the rectangle halfway across the screen, what are its new coordinates and size? Does that place the rectangle off the visible portion of the window?)

    My first job was at a defense contractor and required that I be able to read and implement block diagrams in software. Other programmers were working on equations of aerodynamics, radar systems, and other highly mathematical topics. We hired far more engineers than computer scientists.

    My current gig (12+ years) involves financials, linear algebra, dynamic programming, and whatever the OR/stats guys at our company come up with. I’ve been sent a 150-page document full of Greek symbols and worked-out examples to implement. That was the most fun I’ve had so far!

    Bottom line, if you study calculus, you may never use it, but if you need it, it’s there.

  16. Actually, the highest math class my high school offered for regular students (we were the first in the state to have an I.B. program) was ‘Analysis’ which was a term used for pre-calculus.

    Our math department (1978-1981 when I attended) offered general math, consumer math, algebra I/II, geometry, trig, and statistics/probability, along with computer math (yes, there were mainframe computers and modems in those days, albeit at 110/300 baud) :)

    Lu-Lu is also correct that a lack of fundamentals is why so many students who would like to major in nursing never make it past the preliminary stage, and it would appear that poor math preparation in elementary school (grades 1-5) is the main culprit, as a student who hasn’t mastered the basics I mentioned above using only pencil and paper and through hard and thoughtful work, isn’t going to succeed in higher level math (which is algebra or higher).

    I took algebra I as a 9th grader, and the statement which I remember most today is how our teacher used to say:

    You guys and gals have no problem doing algebra, you simply cannot add., subtract, multiply, and divide correctly :)