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Why Americans Stink at Math

math-1+1=3

In a New York Times Magazine feature, Elizabeth Green asks, “Why Do Americans Stink at Math?” It’s complicated, of course, but the short answer is: school.

In the 1970s and the 1980s, cognitive scientists studied a population known as the unschooled, people with little or no formal education. Observing workers at a Baltimore dairy factory in the ’80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

Most American math classes follow the same pattern, a ritualistic series of steps so ingrained that one researcher termed it a cultural script. Some teachers call the pattern “I, We, You.” After checking homework, teachers announce the day’s topic, demonstrating a new procedure: “Today, I’m going to show you how to divide a three-digit number by a two-digit number” (I). Then they lead the class in trying out a sample problem: “Let’s try out the steps for 242 ÷ 16″ (We). Finally they let students work through similar problems on their own, usually by silently making their way through a work sheet: “Keep your eyes on your own paper!” (You).

By focusing only on procedures — “Draw a division house, put ‘242’ on the inside and ’16’ on the outside, etc.” — and not on what the procedures mean, “I, We, You” turns school math into a sort of arbitrary process wholly divorced from the real world of numbers. Students learn not math but, in the words of one math educator, answer-getting. Instead of trying to convey, say, the essence of what it means to subtract fractions, teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed. The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can’t figure out how to apply the strategy for a particular problem to new problems.

What’s interesting here is that, as Green points out, American mathematicians have been at the forefront for decades of innovative ways to teach the subject that are successfully being applied elsewhere—including Japan and China. But, for a variety of reasons, our own attempts to implement these methods are short-lived, inevitably reverting to the way described above that we know doesn’t work.

I went to grade school in the middle of the “new math” craze that was widely lampooned in the media. Yet it actually worked. While there was some rote memorization, particularly of the multiplication tables, the use of set theory, various “base” systems, and learning of the various “properties” (associative, commutative, and distributive) were enormously valuable to me, allowing me to do basic math in my head.

By the time I got to junior high and high school, we were doing “I, We, You.” I remained a solid student, getting decent grades, but learned very little. We were essentially taught to treat math like a recipe to be followed—a series of steps to be applied in a certain order. That was fine so long as the cookbook was available. I could solve the problems every time, so long as I had the recipe in front of me or, as in the case of an exam, had committed said recipe to short term memory. But, not intuitively understanding what I was doing, very little of what I “learned” then is of any use to me.  (I can do basic algebra and geometry in the same way the unschooled in Green’s piece do, by reckoning my way through it. But I’ve long since lost most of the formulas.)

Yet, for some reason, we disdain teaching understanding. Looking up the “properties” to make sure I wasn’t missing one, I stumbled on a math tutorial site that positively sneers at the concept:

There are three basic properties of numbers, and your textbook will probably have just a little section on these properties, somewhere near the beginning of the course, and then you’ll probably never see them again (until the beginning of the next course). My impression is that covering these properties is a holdover from the “New Math” fiasco of the 1960s. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don’t matter a whole lot now.

Why not? Because every math system you’ve ever worked with has obeyed these properties! You have never dealt with a system where a×b did not in fact equal b×a, for instance, or where(a×bc did not equal a×(b×c). Which is why the properties probably seem somewhat pointless to you. Don’t worry about their “relevance” for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I kept track of the properties.

What’s hysterical about that passage is that it’s wrong on virtually every count.

First off, as already noted, understanding those properties makes it easy to do everyday math in your head. A few years back, I positively flummoxed a cashier at a dry cleaners by calculating my bill, including tax, when his register wasn’t working. So, I explained that I just converted the problem to two simpler problems and how it worked. He still thought it was some kind of magic.

Second, the fact that every math system obeys those properties is rather precisely the point!

Now, it’s true that remembering what their names are—and which is which—is unimportant. The key is understanding how to simplify problems so that they make sense to you.

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About James Joyner
James Joyner is the publisher of Outside the Beltway, an associate professor of security studies at the Marine Corps Command and Staff College, and a nonresident senior fellow at the Atlantic Council. He's a former Army officer and Desert Storm vet. He has a PhD in political science from The University of Alabama. Views expressed here are his own. Follow James on Twitter.

Comments

  1. JKB says:

    In spite of the fact that schools exist for the sake of education, there is many a school whose pupils show a peculiar “school helplessness”; that is, they are capable of less initiative in connection with their school tasks than they commonly exhibit in the accomplishment of other tasks.
    — How to Study and Teaching How to Study (1909) by F. M. McMurry, Professor of Elementary Education, Teachers College, Columbia University

    My experience with math was similar to yours. I didn’t realize that some the issues with Algebra were due to the idiotic way of teaching it. Algebra was easy for me but I can’t say I own it. The same with Calculus, I can do it, but I can’t really play with it.

    Like or Dislike: Thumb up 2 Thumb down 0

  2. Rick DeMent says:

    A lot of times necessity is the mother of education to paraphrase an old saying.When I was in 11th grade science class, we had three guys who sat in the back and were considered “burnout druggies” didn’t pay attention, didn’t engage in class, always failed the every two week quiz on that section until … we got to the metric system. They could covert grams to pounds, ounces to grams, kilos to ounces, you name it. When something is important to you, you tend to learn it. These guys weren’t dumb, they simply need the practical motivation to learn the topic.

    I think this explains the dairy office worker example. I mean they will go back to doing what they do as soon as the regular person comes back, I mean switch it up and I’ll bet there was math the office worker did that would totally flummox the factory worker. I think that the idea that the unschooled people in the study had some kind of innate superiority in math to the schooled people is the wrong reading of the study. It seems to me that the better reading is that people tend to become good at what they use on a daily basis.

    Like or Dislike: Thumb up 14 Thumb down 0

  3. gVOR08 says:

    Just because someone has to do the old math joke about the illustration – one plus one does equal three for sufficiently large values of one.

    Like or Dislike: Thumb up 4 Thumb down 0

  4. OzarkHillbilly says:

    @gVOR08: Silly man! Everyone knows 1 + 1 = 11.

    Like or Dislike: Thumb up 1 Thumb down 0

  5. Mr. Prosser says:

    @Rick DeMent: Not only what may be done on a daily basis but also what appears to make sense in reality. The druggies can convert ounces and grams and manipulate the decimal and illiterate peanut vendors can add and make a tax percentage in their head but Least Common Multiple (LCM) and Greatest Common Factor (GCF) make no sense to either or to my sixth graders when I was instructing.

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  6. Mikey says:

    @OzarkHillbilly: 1 + 1 = 10.

    (There are 10 kinds of people: those who understand binary, and those who don’t…)

    Highly-rated. Helpful or Unhelpful: Thumb up 21 Thumb down 0

  7. DA says:

    Many people in the US openly brag about how terrible they are at mathematics. I think that students can feel that giving up on math and saying that they’re no good at it is generally acceptable and what they should do, rather than try to do better. I think this is a deeply embedded cultural problem, and I don’t see a solution to it.

    Also, when the curriculum is redesigned to try to be more helpful (i.e., to try to get students to think about what they’re doing, rather than repeat steps mechanically, as James says above), people howl that their kids should learn the old-fashioned way, like they learned (e.g., many of the anti-Common Core math examples that one can find online fit this pattern). But the truth is, as James points out, many people did NOT learn very well by doing it the old-fashioned way.

    Like or Dislike: Thumb up 10 Thumb down 0

  8. superdestroyer says:

    I am always amazed that the people who claim they never use algebra but use spreadsheets all of the time. Once one starts building formulas in Excel, one is doing applied algebra but just does understand it.

    One of the reasons Americans are bad at math is the same reason we are bad at learning languages, school is taught as a series of steps of memorize and then dump. Math cannot be learned that way since you build on everything can came before the same as a foreign language.

    What is also amazing is that all Americans know that they want to be a better athlete is practice but just do not want to practice math.

    Like or Dislike: Thumb up 8 Thumb down 0

  9. OzarkHillbilly says:

    Observing workers at a Baltimore dairy factory in the ’80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.

    Well, that set off my BS detector. Carpenters do math all the time, all day long, and most do it mistake free. And we’re not talking just simple addition and subtraction, we’re talking geometry and algebra and in certain specific situations, higher sub sets of them. Most of them graduated from HS with a C in math of the most basic type. It wasn’t that they weren’t smart enough to do better, it was just that they didn’t care. There are little mathematical tricks one is taught during one’s apprenticeship that make it possible to do these calculations virtually mistake free.

    An easy one is how to square up a house (or any rectangular object): Measure the diagonals. If they are equal it is square. Ask a carpenter about the Pythagorean formula and he will look at you like you have ostriches coming out of your ears, but every damn one of them knows the 3-4-5 rule.

    The working world is full of such little tricks, only the educated** are surprised by them.

    **I do, of course mean some of the educated, maybe even most, but certainly not all.

    Like or Dislike: Thumb up 9 Thumb down 2

  10. OzarkHillbilly says:

    @Mikey: HA! I’ll have to remember that one.

    Like or Dislike: Thumb up 3 Thumb down 0

  11. OzarkHillbilly says:

    @DA:

    I think that students can feel that giving up on math and saying that they’re no good at it is generally acceptable and what they should do, rather than try to do better.

    And This. I was always good at math, even found it fun. My Trigonometry and Analytic Geometry is the one class I never skipped my Junior (?) year. But my little sis “just didn’t get it.” and like so many others gave up trying.

    Remedial Math is a curse upon our nations high schools.

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  12. Tillman says:

    I learned the Pythagorean theorem in fifth or sixth grade (and could apply it just fine), but I honestly didn’t understand it until tenth because this one teacher was the first to show a drawing of three squares arranged around a triangle. “Squaring” something wasn’t just multiplying it by itself, there were actual squares involved!

    Frankly, I think the steps method of teaching math is widespread because most math teachers don’t know the subject much better than their students. College math was easier to learn for me because the profs could explain something down to basic principles if need be. Weirdest thing when you can be flummoxed by basic arithmetic at times, but number theory and other advanced math is intuitive.

    Like or Dislike: Thumb up 4 Thumb down 0

  13. A post mentioning “New Math” without a single mention of Tom Lehrer? I am disappointed.

    https://www.youtube.com/watch?v=UIKGV2cTgqA

    Like or Dislike: Thumb up 1 Thumb down 0

  14. James Joyner says:

    @Tillman:

    I think the steps method of teaching math is widespread because most math teachers don’t know the subject much better than their students.

    Yes. The piece actually goes into this a bit:

    At most education schools, the professors with the research budgets and deanships have little interest in the science of teaching. Indeed, when Lampert attended Harvard’s Graduate School of Education in the 1970s, she could find only one listing in the entire course catalog that used the word “teaching” in its title. (Today only 19 out of 231 courses include it.) Methods courses, meanwhile, are usually taught by the lowest ranks of professors — chronically underpaid, overworked and, ultimately, ineffective.

    Without the right training, most teachers do not understand math well enough to teach it the way Lampert does. “Remember,” Lampert says, “American teachers are only a subset of Americans.” As graduates of American schools, they are no more likely to display numeracy than the rest of us. “I’m just not a math person,” Lampert says her education students would say with an apologetic shrug.

    There are, of course, elementary and high school math and science teachers who love math and science and excel at teaching it. Most of them, though, just like kids but have no especial aptitude for the subject matter.

    Like or Dislike: Thumb up 2 Thumb down 0

  15. Tyrell says:

    Children today are not required to do much measuring, estimating, sorting, counting, predicting, or thinking with numbers- this before school.
    In school, card games can teach a lot of math skills. There are several card games that would help and be better than some of the math activities that I have seen them doing.
    Teachers are complaining that Common Core leaves out learning multiplication facts, so you are getting kids going into high school that don’t know 3 x 4. Their teachers are aghast.
    “If it isn’t on the test don’t teach it”
    Students should be taught stock investing.

    Like or Dislike: Thumb up 0 Thumb down 5

  16. Stan says:

    I haven’t checked the statistics, but I have a strong feeling that students at New Trier High School in the wealthy suburbs north of Chicago do a lot better in mathematics than students in the city’s impoverished south and west sides. Part of this is due to better home environments, part (if you believe Charles Murray) to better genes, and part is due to better teaching. New Trier teachers have an average salary of $84,151, compared to $55,558 in the rest of Illinois. You get what you pay for.

    American public education is like American health care. It’s great for the wealthy, poor for the poor. We’re the only advanced country that spends more per pupil on the education of affluent students than on children from struggling families. To compete with China and other rising powers we have to invest in our future, and nothing is more important to our future than education. Yet governors like Scott Walker in Wisconsin and Sam Brownback in Kansas base their political programs on trashing public education. I bleed red, white, and blue, but I’m afraid that our best days are behind us unless we come to our senses and realize what’s really important.

    Like or Dislike: Thumb up 9 Thumb down 0

  17. rodney dill says:

    @Mikey: to be or not to be = FF

    Like or Dislike: Thumb up 2 Thumb down 0

  18. steve says:

    Way too much homework and not enough real problem solving. It would also help if math was not taught in such a compartmentalized fashion. It should be integrated. (Pun sort of intended.)

    As an aside a lot of what you are talking about is arithmetic. That should be doable by everyone. Just a matter of motivation. Real math skills will be limited by natural aptitudes. We can’t all be writers and few of us will score well on the Putnam no matter what we do.

    Like or Dislike: Thumb up 3 Thumb down 0

  19. Just Me says:

    The math program our school uses works very hard to teach math logic-the why behind the algorithm and the program actually teaches various algorithms and clearly states that students can use any algorithm that works.

    I do think math is a skill that is good when used daily. I have very good mental math for things like adding, subtracting, multiplication, division and estimating. I’m also very good with cooking fractions. But I’ve forgotten much of my algebra and some aspects of geometry although a little review usually pulls the memories from the recesses of my brain.

    I do think some people have an instinctive understanding of math-when my son was 5 he explained to me why multiplying zero with anything results in the answer zero. In second grade he subtracted left to right (it was an odd algorithm but it worked and his teacher didn’t make him do it the right to left method they were taught). Even now when he does things in his head he can’t really tell you how he does it and he says what he does in his head isn’t always what the teacher is teaching them. I think he has been blessed with teachers who allowed him some freedom to work with what works for him rather than insisting he follow the algorithms the way he is taught.

    Maybe part of the problem is we try to hard in education to box people into thinking of math in only one way rather than being open to exploring other ways.

    Like or Dislike: Thumb up 4 Thumb down 0

  20. JKB says:

    @OzarkHillbilly:

    Precisely. Remedial math should include shop class. Not as a substitution but as a teaching method. After the kid gets good at doing the “on the job” math, a competent teacher shows them how the same thing is done in math class. In shop class, it isn’t just numbers. Your mistakes are ugly and on display for all to see, so the kid is eager to avoid math errors that magnify in the physical.

    One growing problem now is that fewer and fewer kids have experience in the physical world anymore. When kids came off the farm, they had out of school experience. Now days, most of the nitty gritty is automated, hiding the math and mechanics from view. Just 100 years ago, all but the most sheltered girls had to interact a lot with the world just to get to school, such as handling horses, understanding wagons and buggies, fetching water, etc. Now, few have to do little more than dash from magically air conditioned homes to air conditioned autos, where they are consigned to the back seat, then into air conditioned schools with automatic everything.

    Like or Dislike: Thumb up 3 Thumb down 0

  21. Part of it is also social. Being illiterate is generally considered an indication of a lack of intelligence. If someone can’t read, they’re unlikely to flaunt that fact. If they do mention it, it’s likely going to be treated as an embarrassing secret.

    Yet for some reason, people are happy to openly flaunt their innumeracy and generally think it’s amusing.

    Like or Dislike: Thumb up 9 Thumb down 0

  22. @Ryan Caldwell:

    Funniest part of that song to me as someone born in the late 70s is that supposedly confusing part of the song make perfect sense to me, but it took me quite a while for me to figure out what he was doing in his supposed “normal” version of how to solve the problem.

    Like or Dislike: Thumb up 1 Thumb down 0

  23. JKB says:

    @Stormy Dragon:

    Well, for a long time, and in a lot of schools, being good at math will get you beat up. Our schools are a weird culture where actual learning is socially disrespected. Being a good athlete, being in a play, doing art, all don’t hurt you on the social scene but being a studious student is the path to social pariah. Students are happy when classes are interrupted, even when they are taking out large loans for the instruction.

    We take intellectually curious 5 yr olds and by their 3rd year on the school job, they are loath to think about work when they are off the clock. And they are on the whole constitutionally unable to think creatively about how to accomplish their job. This latter isn’t new, I saw the same report from some research published in 1886.

    Like or Dislike: Thumb up 2 Thumb down 0

  24. Ron Beasley says:

    I did OK in math, OK but not great. I got better at it after I started my work as an engineer since I had to use it everyday to do my job. This was fortunate for my two sons because in Jr High and High School their math teachers new much less than I did and I would teach them at home at night. Frequently their friends would come over for the evening sessions.
    One of my oldest sons’s math teachers would videotape his first class in the morning and just play that tape for his other classes for the rest of the day. After a several month battle we were able to get him fired.

    Like or Dislike: Thumb up 6 Thumb down 0

  25. Tyrell says:

    Some of the factors (no pun) in math education in our schools: school systems seem to jump on the latest teaching fad like it is the map to the 7 cities of gold. Pacing guides have students moving from one skill to the next whether mastered or not. Teachers are not given a lot of input or involvement in decision making when it involves curriculum. Teachers are spending a lot of time supervising (early bus duty, traffic duty, late bus duty, breakfast, dances, carnivals, games), and fund raising. There is a lot of research and extra help for students with reading difficulties, few concerning students who have problems with math. There are many free on line math programs (Kahn Academy is great). Probability is a fascinating math topic: students can learn a lot of math by playing poker, rummy, 21. Testing drives everything: practice tests for practice tests, testing workbooks, test practice homework, test skills practice (how to pass multiple choice tests).
    “Teach what you test” is the order of the day. More time is devoted to test practice then actual instruction.
    I am still trying to figure out this “absolute zero ” stuff. I do know that my savings account is stuck on absolute zero !

    Like or Dislike: Thumb up 1 Thumb down 1

  26. Jc says:

    It is a simple problem of doing something instead of understanding something. Most people can do math, but fewer truly understand math. I believe Common Core (benchmarked based on international systems) attempts to get kids to do this. We are so set in our alogrithms in math, we don’t think outside the box, then when we are asked to we get all upset and say “that is not how I learned it!” or “this is ridiculous!”, when we never truly learned anything to begin with. Meanwhile kids in other countries are using multiple ways to learn and understand mathematics, and performing better than our kids in math…yet we continue to beat a dead horse because of fear of change or trying something new

    Like or Dislike: Thumb up 4 Thumb down 0

  27. Just Me says:

    Tyrell-pacing is a huge one.

    Back to one of my objections to Common Core-having standards to difficult/complex for the develolmental stage of the students. There is a trend in current education to push higher learning down with the belief that this will make kids better students and leave time to teach more complex things but the average kid isn’t going to really get he more complex things if he never mastered the foundations and previous material. Math especially builds on prior learning. If a student doesn’t master earlier steps they become frustrated and likely end up with gaps.

    Also, having kids practice and learn math in shop class is a great idea-shop class provides a hands on application of math skills (and one question students almost always ask is why they have to learn X because they are certain they won’t ever use X outside of school). Problem is many schools in a move to cut costs have eliminated most shop type classes. Our high school now only offers a very basic wood shop class and a drafting class. Thirty years ago they had several levels of wood working, a metal shop, an auto shop and several other intro classes for exploring trades.

    Oddly enough-the trades are seen as something students do who can’t go to college but the trades actually use math more than many other occupations (when I worked in social work the extinct of my math use was adding up my hours for my time sheet). I think a lot of students learn better when exposed more to the real world application and hands on practice. Sitting in a classroom at a desk means they get the idea it is uninteresting or isnt useful.

    Like or Dislike: Thumb up 2 Thumb down 0

  28. PD Shaw says:

    By focusing only on procedures — “Draw a division house, put ’242′ on the inside and ’16′ on the outside, etc.” — and not on what the procedures mean, “I, We, You” turns school math into a sort of arbitrary process wholly divorced from the real world of numbers

    I don’t know what she’s talking about. I’ve reviewed by kids’ math homework and tests, and I think generally on a topic like this, they would be asked to start by considering a group of 242 blocks that need to put into 16 baskets. Why would they need to do that, because they have 242 treats to hand out to 16 kids. They may make them draw lines through a group of 242 blocks. They might add 16 plus 16 until they reach 242. There are a lot blocks and block drawing in the introductory stage to some new type of arithmetic. A lot of story problems about snacks and treats.

    And the books, the tests, and the worksheets appear to be standard fare.

    Like or Dislike: Thumb up 1 Thumb down 0

  29. Rob in CT says:

    @Just Me:

    Fair enough, though I’ve read some complaints about “common core math” (my understanding is that it’s not really “common core math” b/c common core is about curriculum more than method. Am I wrong about this?) that were complaining about what looks to me like a superior way of teaching math. Basically, the new way looks like the way my father taught me how to do math in my head – break things down to easily digestible parts, get the answers, and tally them.

    Which brings us back to “why do so many Americans suck at math.” Well, in part because their parents did too. Also in part b/c culturally saying “I suck at math” is totally acceptable or even laudable. This is part of, but worse than, the general “nerds suck” culture that has, it seems to me, diminished since my childhood. When I was in school – in an affluent suburb with excellent schools that sent 95+% of our students to college – it was DEEPLY uncool to like school, participate in class, and so forth. That’s seriously messed up.

    Anyway, I was fine through algebra 1 (I loved it, in fact). But as we got into geometry, algebra II and calculus, I checked out more and more. I still got ok grades (somehow!), but none of it sunk in. I do suspect the teaching method was part of the problem. It was terribly abstract to me. Connecting it to real-world scenarios might have helped.

    Like or Dislike: Thumb up 2 Thumb down 0

  30. PD Shaw says:

    @JKB: re shop;

    I attended a large high school with one of the largest vocational centers in the state at the time. And I was curious last year about how it was doing, and there was an article from an instructor complaining about politicians emphasizing the college curriculum and No-CHild-Left Behind sucking away resources. He said that:

    We do calculus over here. I’ve invited the college prep kids to come see how we do it, and they’ve never taken me up on it. We have machines that would blow their minds. (paraphrasing)

    I have no idea what machines he’s talking about; I’m one of those pencil necks, but seems like we do not have enough machinists and this might be the reason why.

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  31. grumpy realist says:

    @OzarkHillbilly: No, no. 1 + 1 = 10.

    Also, it’s very easy to think of a system where A x B is NOT B x A. Rotations…

    Like or Dislike: Thumb up 1 Thumb down 0

  32. PD Shaw says:

    New Trier teachers have an average salary of $84,151, compared to $55,558 in the rest of Illinois. You get what you pay for.

    Average teacher salary (2012)
    Chicago: $76,000
    Other than Chicago: $69,969
    New Trier: $107,493

    Chicago teachers get paid better than average in the state. The lowest salaries are in downstate.

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  33. grumpy realist says:

    If anyone is still flummoxed by calculus and wants to understand how it all works, I highly, highly recommend Prof. E. McSquared’s Calculus Primer, if you can track down a used copy. Great cartoon book and is quite hilarious with the in-math jokes.

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  34. beth says:

    I have spent probably a full year’s college tuition on math tutors for my daughter starting in third grade and continuing on through high school. We finally found a local math teacher who’s been teaching math for 40 years and he was the only one I can say really made a difference in her understanding of math. The other tutors got her through the school year and enabled her to pass that class but did nothing to further her knowledge of numbers and how they work. They all used different methods and they all had their own little worksheets and tricks they used. So I feel quallified to say (based on my purely anecdotal evidence) that while there may be many people who understand math and many people who are good teachers, there are precious few who can combine both skills. To steal a phrase from James Carville – it’s the teacher, stupid! We really need to focus on finding the right people (and paying them enough) to teach math.

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  35. Pinky says:

    Many years ago I read “Please Understand Me”, the bible of the Myers/Briggs Personality Type movement. It talked a lot about education, both from the teaching and the learning side. It said that teachers are largely divided between Organizers and Feelers – very low representation of the creative or analytical types. All academic reforms play out the ongoing rivalry between the organizer’s and feeler’s way of thinking. No reform ever gets fully or properly implemented, because half the teachers will be fundamentally opposed to it. Then the tide turns, the other side regains the upper hand, and contrary theories and policies are all the rage.

    Interesting stuff. I’d recommend the book to anyone – the education stuff is only one portion. I was lucky as a kid; I never looked to teaches to inspire me, and they didn’t. I was just interested in learning things. But my point is, it’s very difficult for people to understand other people’s thinking or learning processes. We’ve definitely reached a point in our educational system where we need to restore the integrity of the high school diploma, and that’s going to require standardization. But we have to remember that standardized levels of results don’t imply standardized methods of teaching.

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  36. Rob in CT says:

    All academic reforms play out the ongoing rivalry between the organizer’s and feeler’s way of thinking. No reform ever gets fully or properly implemented, because half the teachers will be fundamentally opposed to it. Then the tide turns, the other side regains the upper hand, and contrary theories and policies are all the rage.

    Sounds a bit like politics. ;)

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  37. PD Shaw says:

    I don’t know that anybody has answered the question of the post, including me, but here it is:

    Why Americans stink at math?

    It doesn’t pay.

    Higher salaries in the U.S. go to those with higher verbal scores. High math scores can be a plus, as they suggest strong analytic skills.

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  38. Just Me says:

    @Rob in CT:

    Common Core is a set of standards for each grade level that students are expected to know. It isn’t a curriculum or even a teaching method. It is a “by the end of 7th grade students should demonstrate knowledge of X, Y, Z.”

    My beef with Common Core is that some of these standards are set at age groups where students may not be developmentally ready to grasp them. There is a push as well to have 7th graders learning algebra (not a Common Core standard but more just a weird educational push with the idea that schools can get more math in). While some 7th graders can learn and master algebra many just aren’t ready yet. There is no reason to push algebra in 7th or 8th graders who may need to spend more time mastering the concepts algebra builds from and even more so developing better head math.

    The math curriculum our school uses in elementary up to 7th grade tries to teach math logic and encourage head math although it also fails in that it moves way too fast for some kids and hose kids never fully master a concept before moving on and when it spirals back to the concept it is at a more difficult level. Many students can’t do basic facts-and to adequately do head math you have to know addition/subtraction/multiplication and division at a basic level. Also mastering the base 10 system (although the curriculum does use base 10 a lot but it is kind of useless if students still need to use fingers to add and multiplication charts to multiply in middle school).

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  39. Tyrell says:

    @PD Shaw: Beginning teachers around here make just a little more than minimum wage. Put in thirty years and you get $45,000, give or take. But teachers do get paid health coverage (but to add a spouse or family member is expensive), and get a decent retirement package if they make it long enough. Most beginning teachers have a second job.

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  40. PD Shaw says:

    @Tyrell: In Illinois, a teacher making the average salary can retire in 30 years and expect to collect a $75,000 per year pension, and in 10 years COLAs will make it $100,000 per year. Link Illinois and Chicago tend to be listed as among the highest in the nation, which makes me skeptical that more money answers anything. The amount of money guaranteed to teachers here is beyond the ability of the tax base to support. Average salaries are going to continue to increase because no new teachers are being hired, the youngest teachers are being fired, so we are left with fewer, higher-salaried teachers, who might just be burnt out and waiting for the 30-year clock to run out.

    In any event, I have my list of policy changes, but the only one really relevant to this thread is that (a) there is no evidence that teachers with masters or PhDs are better teachers than those without, (b) but there is some evidence that teachers with masters or PhDs in a math field are better teachers than those without, and (c) schools should compensate accordingly.

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  41. Grewgills says:

    First off, as already noted, understanding those properties makes it easy to do everyday math in your head.

    These tactics are the ones that I see people complaining about on facebook under the impression that they are part of common core.

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  42. Grewgills says:

    @superdestroyer:

    I am always amazed that the people who claim they never use algebra but use spreadsheets all of the time. Once one starts building formulas in Excel, one is doing applied algebra but just does understand it.

    A friend of mine’s employee at an insurance company told him, ” I don’t need to know math to do statistics” when he was trying to explain how to use the macros in their actuarial spreadsheets.

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  43. Grewgills says:

    @PD Shaw:

    I don’t know that anybody has answered the question of the post, including me, but here it is:

    Why Americans stink at math?

    It doesn’t pay.

    I think more than that, as some have pointed out, innumeracy is accepted in our society in a way that illiteracy is not. At least weekly I hear someone say, ”I’m just not good at math” or some variation on that. I can’t remember hearing anyone say to me, ”I’m just not good at reading.”

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  44. Grewgills says:

    @Just Me:

    My beef with Common Core is that some of these standards are set at age groups where students may not be developmentally ready to grasp them.

    That is huge. They are expecting children to come in to 1st grade knowing how to read. A friend of mine teaching in small town Georgia was expected to have her 1st graders writing in paragraphs by the end of the year. Most of these children had no support from home. Add to that, they were not allowed to flunk a student in any grade level more than once. She is now teaching 4th grade and has a couple of students that have taken every grade twice an only moved forward in any of them because of that rule.

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  45. Pharoah Narim says:

    Silly Doug, School isn’t for learning. It’s for teaching (the flip side of the experience paradigm) and socialization. The best, longest-lasting learning occurs when the self teaches the self…usually driven by the catalyst of necessity or love of learning itself. We can shuffle the deck as many times as we want….it’s always going to come back this foundation. And as long as that foundation can’t be sufficiently monetized….we’ll keep shuffling. They’re trying their best I suppose.

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  46. Tyrell says:

    @Grewgills: The goal for first grades should be a sentence. Paragraph structure can start in second. Developmental appropriate is important. You can’t build a building if you do not know how to read a set of blueprints.
    Retaining students is not a magic solution and can create more problems. Often it worsens behavior. And what parent wants a fifteen year old in their child’s fourth grade class?

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  47. grumpy realist says:

    @Grewgills: Which is really, really bad. Everyone should be able to do a back-of-the-envelop (or in your head) calculation so you can take a look at what the machine is spitting out at you and see if it makes sense or not.

    Blind trust in spreadsheet calculations can get you into quite a pickle….

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  48. @James Joyner:

    The article is pure fluff. It speaks only about math problems that come up before grade 6. Even then, it gets everything wrong. The most egregious example: Students who were confused about the location of 5/6 vs. 5/12 on the number line engaged in prolonged discussion to clear up the confusion. Afterwards, a student asked to prove that 3/12 = 1/4 confidently states
    “Three sections of 12 go into one fourth.” Apparently, the quoted math educator accepts this as a proof. It is anything but. “Section” is presumably the student’s way of saying “one twelfth.” Therefore the student is saying “Three twelfths go into one fourth.” This is hardly a proof that “Three twelfths equal one fourth.” There are many age-appropriate proofs, but this is not one of them. Rather, it is indicative of the lack of content knowledge of some math educators.

    Stanley Ocken
    Dept of Mathematics
    The City College of the City University of New York.

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  49. DrDaveT says:

    Disclaimers:

    I have a PhD in applied math. My secondary school math curriculum (1970s) was an extreme New Math program — 6 years of “Unified Mathematics”, which I supplemented with some traditional Trigonometry because that had been skimped.

    I have been an assistant professor in an engineering school of good repute, teaching many courses at both undergraduate and graduate levels. I have tutored students from grades 6 through 12 in basic arithmetic, algebra, calculus, probability and statistics. I have tutored undergraduate students in probability and statistics, operations research, and logic.

    My perspective: yes, Americans suck at math. Badly. And I agree with those who have said that this is a cultural issue. Being good at math is seen by the culture at large as a net negative, like being able to speak Klingon or recite the names of all of Dr. Who’s sidekicks over the years. Until we fix that, America’s technology advantage will continue to erode steadily until it vanishes. We’ve been riding the coattails of Asian immigrants and their children for years now; before long, they will stop coming here to be educated and keep their skills and attitudes in Asia.

    It starts very young. The most important thing I’ve learned in tutoring algebra is that none of my students really has trouble with algebra. What that have is a complete lack of basic arithmetic skills, which is crippling when you’re trying to learn algebra. There is simply no way that you’re going to learn to factor polynomials when you have to think long and hard to figure out what 60 divided by 15 is — even after your coach has asked you how many minutes are in an hour.

    When the culture feels the same way about not knowing your basic multiplication tables that we do about not knowing the alphabet…

    When we feel the same way about not knowing basic probability that we do about not being able to speak in complete sentences…

    When we do not permit teachers of elementary mathematics to be people who hate math…

    We might make some progress.

    (End of rant.)

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  50. @Grewgills:

    In response to: Why do people say: “I’m just not good at math,” but not “I’m just not good at reading.”

    First, I wonder what dataset underlies such an assertion.

    Second, assume that it’s true. The difference between math and reading is that math answers are either right or wrong. The people who say “I’m not good at math” have objective evidence to back up their statement: they know that their grade, so to speak, is zero.

    Trouble with reading is much less clear cut. People who are poor readers may feel clumsy, but they don’t feel wrong. There’s a big difference.

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  51. Um… Actually, you were probably using the Order of Operations to solve your issue with the vendor. The particular type of numbers you were using (being rational) was irrelevant to the issue.

    …which was kinda the point being made in the article which offended you. Sorry for the confusion.

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