Math choices for seniors

Students who take three years of high school math often forget what they’ve learned by college and end up in remedial classes. So more high schools are requiring seniors to take a math class — but not necessarily trig, precalculus or calculus, reports Education Week.

The most popular emerging courses include statistics and discrete math . . . as well as classes in quantitative reasoning, math modeling, and math in business and finance.

The academic demands of those courses vary. Many target students who have completed Algebra 2, but who do not want to take precalculus or fear they would find it too much of a strain. Some are being drawn up for elite students who have flown through all the math classes available to them.

Many students are surprised to discover that math can be useful in real life. It’s not just designed by sadistic adults to torture teenagers.

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  1. “Many students are surprised to discover that math can be useful in real life. It’s not just designed by sadistic adults to torture teenagers”…students would understand this better if math textbooks & curricula made a serious attempt to talk about applications. Generally the word problems (“two trains leave stations at 10:00 AM, one going east at 50mph and the other going west at 70mph”) are obviousy contrived and don’t have the feel of reality about them.

  2. Barry Garelick says:

    Problems about trains and such are included in math courses because of the concept of rate and how it is applied. Understanding how such problems are solved generalizes to a wide class of problems that crop up in science courses. The so-called “real world” problems that are an attempt to make math relevant are often special case problems that do not generalize and do not teach much math, if anything at all.

  3. Barry Garelick ,

    I agree with you, but classes on how to compute compound interest, loan payments, rates of return and leases involve real math and are real world problems.

    david foster ,

    Even today determining when trains meet is very relevant, but railroads are invisibe to most people today.

  4. Barry Garelick says:

    I agree with you, but classes on how to compute compound interest, loan payments, rates of return and leases involve real math and are real world problems.

    I don’t disagree. Those things should be covered and generally are in 7th and 8th grade. Algebra courses and higher, however, get into the types of problems of related rates. They need not be confined to trains–related rates present themselves in a variety of forms. For example, the familiar “work problem” of two people who mow lawns at constant rates, so that one can do the job in 20 minutes and the other in 30 minutes, and asking how long it takes to do the job together, uses the same underlying principle of rate as the following problem:

    John has enough money to buy 30 oranges or 20 apples. How much of each can he buy if he wants to buy equal amounts of both?

  5. IMHO, typical exchange with struggling algebra student:

    Q: Solve this equation: x + 1 = 11 A: What are you talking about?

    Q: What number do I have to add to 1 to get 11? A: 10

    I’ve seen this many times in tutoring algebra students. The “strange” terminology of algebra, even in very simple form, seems to confuse students. The skill that seems to be lacking is the ability to translate the symbols into logical thought. The students that eventually figure this out do well. But at least in CA this skill is not taught very much as the textbooks have mostly given up on word problems. And most math teachers seem to be relying heavily on the textbook, which isn’t surprising.

  6. “John has enough money to buy 30 oranges or 20 apples. How much of each can he buy if he wants to buy equal amounts of both?”

    I saw a better (more relevant) problem that compared the rates for two cell phone plans. It looked at two different base rates for different amounts of minutes, with different rates for “extra” minutes. Students had to figure out the most advantageous plan given their usage.

  7. Barry Garelick says:

    I have nothing against relevant problems, provided that they are not the contrived type that have no generalizable qualities to them. What I’ve noticed in tutoring students, however, is that once students know how to solve problems, they enjoy solving them, and the relevancy of the problem is of lesser concern. Some see the relevancy is a motivator. It might be, but in my experience, a bigger motivator is providing proper, clear, and logical instruction which the commenter “pm” is getting at.

  8. I think this is a great idea! I always did fine in math, but I didn’t enjoy it much, so I opted not to take it my senior year and got through college with just one, measly math class. As I start looking at the GRE, it makes me wish I’d taken more math!

  9. gbl3rd is correct about the continued importance of the “when do trains meet” problem, but the normal formulation of the problem in textooks does not give any clue of its real-world importance. The problem could be made more interesting by including a sample track diagram (very often, in today’s world, single track with passing sidings and occasional stretches of double track) and a couple of paragraphs about the work of the Train Dispatcher.

    I understand that in pre-computer days, dispatchers sometimes identified meet points by stretching string across graph paper, with the angle of the string representing the planned speed of the train. Mention of this could be an interesting way to introduce graphical solution methods for the problem.

  10. There are two separate issues here: having a year of no math before diving heavily back into math in college, and motivating students to value math.

    The first problem is important, but a course in statistics as a senior might not be the best solution. It would be better to plan ahead so that students take their last main-line (?) math course as a senior. If you force kids to take an extra math course as a senior (4 years of math), then that is a change to the curriculum. Something else has to be taken away.

    Trying to fix the problem of motivation often leads to lower expectations. This is hidden when the material is taught from a real-world or top-down approach to the material.

    Solving word problems is very important. You have to learn the basic skills of understanding the problem, drawing pictures, labeling unknowns, defining equations, and solving them. The problems are often simplified to make learning basic skills easier. For DRT-type problems, the classic one is where two trains start at the same time and head towards each other at different speeds. So, is the problem that students can’t relate to this, they don’t like math, or that they have never been directly and carefully taught the skills to solve the problem? Would people be happy if the problem involved the walking speeds required for two kids to meet at the game store somewhere between them at the same time?

    How about this for a problem. Wayne and Garth live in Pittsburgh and want to go to a Grateful Dead concert in Detroit that starts at 7:30 PM. They need to pick up a friend in Dayton, OH, but this friend is never on time. They will probably have to waste an hour there. Then, they have to have spend probably an hour to park and get into the arena. Wayne’s wreck of a car can’t go over 60 and they will have to stop for gas and munchies, which should take 15 minutes, but it will probably take a half hour. If they add 20 percent more time for unexpected delays (to look for excellent babes), when do Wayne and Garth have to leave home to get to the concert on time?

    Can you just see this kind of class? The kids are broken into groups to solve this real-world problem. They think it’s a hoot. They talk about the concerts they’ve gone to. It looks like there is a lot of active learning going on. Finally, someone gets to work. A whole class goes by and maybe, just maybe, the teacher will finally pull the plug and go over the problem. It’s hard to imagine that the teacher could bring everyone to the point where they could do a similar problem by themselves. Will any of them be able to handle all of the variations of DRT problems where the objects are moving towards each other, away from each other, or moving in the same direction? What about the cases where a different variable is unknown? You can’t possibly cover all of the possibilities with large, real world problems. Then you have to cover mixture problems and other weighted average problems.

    This isn’t about relevance. It’s about low expectations and a fun, top-down approach to mastering basic skills. It never happens, and you cover less material.

  11. Barry Garelick

    Those things should be covered and generally are in 7th and 8th grade. Algebra courses and higher, however, get into the types of problems of related rates.

    I had to learn this on my own much later in life.

    That is great if it is being absorbed, but…
    We must not be doing a very good job considering all the people “taken advantage of” by mortagage brokers.

    david foster

    I think a course in mental arithmetic and shortcuts, estimation, graphical methods, analog computation and digital computation could be useful.


    I agree real world problems can be too big and complicated for high school. They might get an “answer”, but it would not be worth the loss of time that could be spent on simpler problems that illustrate basic principals. I have never felt very good about teams tackling problems at the high school level. It did not work when I was in high school, but I would like to think things have changed.

    On the original issue:
    I would have thought taking physics or chemistry would provide enough math retention for those not taking math their senior year.

  12. Okay, I took five years of math jammed into four years of high school, then took a bunch of math courses in college and eventually got an engineering degree, which requires a lot of applied math.

    But I don’t work as an engineer.

    Other than simple arithmetic (checking account, measuring for cooking, shopping in grocery stores, etc.), what part does math play in my life? Sure, I can understand statistics and know what a margin of error is. But how does any of my knowlege of calculus or linear algebra or modern algebra or trig etc. play any role at all?

    The answer is: None. Nada. Zilch. Zip. Zero. Math is simply as irrelevant to my life as a foreign language.

  13. It is very important that students understand how math is used in practical situations, but it is not necessary for this to happen in a math class. I homeschool my kids so I don’t know what happens these days in schools, but I recall from college that it was in chemistry and physics class that I got the best understanding of how math is useful, rather than just interesting in its own right. Maybe schools do this already or maybe they cannot get this organized, but it seems that an educated person should take a physics course, and a physic course will require some geometry, algebra, and trigonometry.

  14. Rex,

    I have met quite few high school graduates who could not balance their checkbook or use a ruler. I have known a few college graduates who could not use a ruler.

    There are quite a few people who run away from a problem involving simple calculation or conversion of units screaming,”I don’t do math”. Some are proud they cannot handle it. The proud ones were all college graduates

    I would like this to become much less frequesnt. My experiencee has been the more math you take the less trouble you have when you encounter these simple problems.

    You are right that calculus and beyond don’t have much to do with this problem. Some of my friends who work as engineers tell me they do not use the higher math the learned in college. I got the impression the math is needed if they do graduate work.

  15. Other than simple arithmetic (checking account, measuring for cooking, shopping in grocery stores, etc.), what part does math play in my life? Sure, I can understand statistics and know what a margin of error is. But how does any of my knowlege of calculus or linear algebra or modern algebra or trig etc. play any role at all?

    The answer is: None. Nada. Zilch. Zip. Zero. Math is simply as irrelevant to my life as a foreign language.

    Ditto. Every once in a while in my last paid position I would encounter a problem that I *COULD* solve using calculus. But there was an alternative way of solving the problem that did not require the use of calculus. I happened to personally prefer the calculus-based method but most of my colleagues did it the other way & still got the correct answer.

  16. Mark Roulo says:

    Other than simple arithmetic … what part does math play in my life?

    Isn’t this true of almost *all* academic subjects? What part does literature play in most people’s lives? Or history? Or science? Or geography?

    Sure, engineers often use some of the engineering classes they took. Ditto for accountants.

    But my guess is that for most people in college, very little of the material they study is ever used.

    So … I don’t quite get where this is supposed to lead.

    Is the point that we shouldn’t teach/require math as a pre-req. for getting into college? Or for graduating college? Or what?

    -Mark Roulo

  17. As someone who mainly advises art history majors and first year students who have self-selected as future humanities majors I am sadly familiar with the problem of trying to get folks to take a math course before they forget what little they learned in high school. Eventually they realize that they DO have to take a math course – but by that point it’s sometimes been 4 full years!

  18. “So … I don’t quite get where this is supposed to lead.”

    It’s not leading anywhere, Mark, as far as I can tell. Many degree programs have already reduced their math requirement to a minimum. Departments were losing students because they couldn’t pass the math courses, and losing students is not good for the health of a department. It sounds like there is a desire to reduce the math requirements for engineering too.

    This reminds me of the Everyday Math argument about whether people really need to know how to divide fractions when they get out of school. The answer might be “not many”, but you also don’t know which ones they are. Can you hear the educational doors slamming shut?

    Actually, I use a lot of calculus in the programs I write, from line integrals to derivative techniques in nonlinear optimization. By the way, for the statics and dynamics class taken in the early undergraduate years in engineering, you better know how to integrate. Engineering means math. If it doesn’t have math, then it’s vocational school.

  19. I’ve got a few thoughts on problems and math on my website. Here’s a link.

  20. Is there a reason to believe that these students taking 3 years “forgot” what they learned, and needed remediation, as opposed to the OBVIOUS hypothesis: that they never learned it in their math classes, despite passing grades?

    4 years of bad math instruction is not much better than 3 years of bad instruction.

  21. Rex, If you can’t see how knowing more than 7th grade math–including simultaneous linear solutions, or differential equations would help you to understand derviatives, credit default swaps, counterparty risk, how to judge deflation, how to determine the present value of a discounted loan, and you can’t imagine how knowing any of that would affect your life, then I say you can’t even make an informed judgment about legislation congress votes on. you aren’t able to perform as a citizen.

  22. I’m one of those ones who thinks that the main value in studying maths is the option value. The more maths you study at high school and university, the more options you have at university. I know several students who had to change what subjects they were studying because of lack of enough of a maths background, and it’s so hard to pick up maths in a hurry too.

  23. griefer,

    I use my HP12C to get the present value of a discounted loan. Those other things are peculiar to very constrained parts of the finance industry, and are done by the back-room boys & girls, not by any managers or salesmen. No politician in the world understands them, nor do the vast majority of investment bank managers. But more people understand them than understand the lack of real science behind the global warming religion.

    I took math because I enjoyed it, and it was necessary to understand physics and engineering. But because I am not working as an engineer, I don’t use any of the higher math skills in my everyday life (other than my knowledge of statistices, which I alluded to in my first post). I think I’ve used simple algebra (which I learned in 8th grade) maybe once a year. Basic physics I use more often, basic chemistry in cooking, biology never, but earth science a lot–at least the section about understanding the weather.

    I happen to think that everyone should be educated in the basic liberal arts, most of which has no bearing on jobs or hobbies later in life, but the reasons for being exposed to basic liberal arts (which includes math, English, foreign language, history, philosophy, science, art and/or music) are (1) you never know, you might really like it, and who knows where that might lead, and (2) it stretches your mind.

    But don’t pretend that much of it has to do with living in the real world.

  24. I recall long ago (well over a decade ago) when variable rate mortgages were first introduced there was a small ruckusl because some banks were not computing the payments accurately. It is a good idea to know finacial math. The back room made “mistakes” that benefited the bank. Should you trust the agent who profits from overcharging you to compute your bill properly? Money is numbers and numbers are math.


    I agree the the problem is frequently poor math instruction and not the amount of it. No one agrees on what good math instruction is because it can only be judged by results. Usually instruction is judged by some philosphical or psychological concept.

  25. It wouldn’t be a bad idea to include a course on statistics. Journalists are particularly public in their ignorance. I am tired of reading that an increase from 50% to 60% is a 10% increase.

  26. I think that all U.S. high school students should have 4 years each in Math, Science, Social Studies, and English. Those four subjects are essential to undertsanding the world we live in.

    Even if you don’t use one of those four subjects in your college major or at your job, you need them for those times when you’re serving on a jury, or voting for your Congress member and/or the President based on what legislation they drafted and/or signed or vetoed.

    Lawyers love juries that don’t understand Math or Science, and politicians love voters that don’t understand Math or Science. Those people are so easy to fool!

    I think the standard Math curriculum in the U.S. should be:
    8th Grade: Algebra I
    9th Grade: Geometry
    10th Grade: Algebra II
    11th Grade: PreCalculus (Trig, Logarithims)
    12th Grade: either Calculus or Statistics

  27. You are so right about journalists and statistics, Margo/Mom. Years ago, a numerate reporter started a circular at the Mercury News to explain things like what 100 percent means and how to describe an increase from 50 percent to 60 percent. It was very basic math. Reporters read it avidly and gratefully.

  28. What percentage of trials in this country would require jurors to have an understanding of Trigonometry and/or Calculus?

    Basic math up through simple algebra, I definitely agree. But more advanced stuff is highly unlikely to be relevant to serving on a jury…

  29. I only did litigation for two years, but never in that time (or since) did I ever observe or hear that lawyers love juries who don’t know any math or science. If a case depends on math or science, that’s when the lawyers bring in expert witnesses.

    I also disagree that graduating from high school with a strong academic curriculum makes you any better a citizen, let alone making you able to understand the world we live in. Maybe I am prejudiced by knowing some very “smart” people who teach at Cornell who don’t have the slightest clue about how the world works.

    Basic psychology, basic economics, and basic statistics would be much more useful tools.

    Reading and becoming self-informed, along with the ability and understanding to separate fact from opinion, are crucial to understanding the world we live in. And the ability to separate fact from opinion is not taught in high school, nor did I learn it in college. It took several years of law school to become adept at that particular skill (although the basics of it are easy, it takes constant practice to readily recognize BS when you come across it).

    Look at any newspaper article and see if you can separate fact from opinion in the article. One hint: look at the adjectives, and if they are positive or negative, rather than neutral, you are looking at the writer’s opinion and not fact.

  30. In his first comment greifer brings up the idea of poor math instruction versus the amount of math instruction. I tend to go with the poor instruction hypothesis. From what I see teaching math to college freshmen I feel there is a lot of evidence that elementary math, particularly fractions, is not being taught well in recent years. This seriously compromises students’ ability to learn algebra. However the problem seems to be well hidden. In the past year I have become more and more aware that apparently students manage to pass math courses, with a C at least, in spite of serious deficits in the basics. This past year I started giving a fractions quiz on the first day of class in my lower level algebra course, and the results are indeed impressive, and that is not good news. I have described my observations and conclusions in some detail at

    In a comment gbl3rd says, “No one agrees on what good math instruction is because it can only be judged by results. Usually instruction is judged by some philosophical or psychological concept.” I certainly agree with the second sentence in this quote, but I’m not so sure about the first sentence, or what we might infer from that first sentence. Results may be what ultimately counts, but I think there is a lot to be gained by looking very carefully at the process. I further think that the field of education has done a very poor job at looking carefully at the process. Indeed I wonder if our goal of being “data driven” has actually worked against our ability to look closely at what actually happens in the classroom. I have more thoughts on this at

  31. Well, the unethical lawyers would, I’d think. Like if they were trying to help a guilty party get away with something (whether that be a guy trying to get away with murder, a corporation trying to get away with poisioning its customers, etc.) wouldn’t they want a jury they could easily fool with conflicting evidence? You don’t think a basic knowledge of Science or Math wouldn’t help in those cases?

    Also consider that, if the jury doesn’t know any better, an ‘expert’ could stretch the truth beyond belief, and be believed by default because he/she is an expert.

    I would also think cases where money was misappropriated would be best served with juries with basic Math skills.

  32. Brian Rude,

    I never really thought about it before but there is a lot to be said for studying processes. Management theorists, Frederick W. Taylor and W Edwards Deming, both heavily emphasize studying processes and measurement. I do not think educators think the work of these men add anything of value to evaluating instruction.

    One of Deming’s Management Obligations is Drive Out Fear. Other wise the researcher/manager would never learn what is really going on. I think that is a big obstacle in our schools.

  33. Chynell S. says:

    From my experience, my school required 3 years of math and an optional 4th year for seniors. I wanted to stay on track so I took a general math course. It covered fractions, interest rates, percentages, and other useful math topics for real life issues. My freshman year in college I took a math placement test and I was placed in fundamental math courses before I could take the standard Algebra for Calculus. I excelled in Alegbra I, Geometry, and Algebra II throughout high school. The fundamental math courses were not challenging at all for me. I have talked to several other students who were in the fundamental (basically remedial) math courses who were decent students that went all the way up to Pre-Calculus in high school. This confuses me because I do not know if it is the way we are being taught in high school or if there is a flaw with our university. This goes along with greifer’s hypothesis: “they never learned it in their math classes, despite passing grades?” I think it is a mixture between not having enough practice and how well they learned it in the first place, not one or the other. I had an excellent Algebra I teacher and although I haven’t done Algebra I math problems I still can recall how I would do them and that was 7 years ago.

    I also agree with Tracy W. More high school and college students need to value math education because it provides a window of opportunities throughout one’s life that you can never anticipate. An art major may think they will never have to use math, but even basic principles like loan interest rates, etc. can help you save money and not be conned into investing in something that is truly a rip off. Also, career and major changes are so common these days that a math education will make you a well-rounded person. I think it should be required all 4 years. If they are not taking math their senior year then it should be other math-related courses such as economics.