Tall, Dark, and Mysterious

7/20/2005

Prerequisites for college math

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 4:09 pm.

Are there any high school math teachers - or, better yet, developers of high school math curricula - who read this blog? I’ve decided to take a break from my usual undirected griping about how woefully unprepared students are to do college math, and divert those energies into something more productive - a working list of what students should be learning in high school math classes in order to prepare them for college (or even, for what they are supposed to be learning later on in high school).

Some months ago Rudbeckia Hirta drily observed that contrary to what one might assume, the college prep track in the high school math system does not, in fact, prepare students for college math. As far as I can tell, high school math in general doesn’t prepare them for damned near anything. In fact, I think that students often learn negative amounts of math in high school: in grade school, they learn how to perform basic mathematical operations and such, and by the time they’ve received their diplomas, they’ve been trained to leave those tasks to their calculators. (See: inability to add fractions, talldarkandmysterious.ca, 2004-present.) The claim that foisting heavy-duty calculators upon students frees them to do more complex and creative tasks is wishful thinking that, in my experience, has no grounding in reality.

When I started teaching, I anticipated that students would be weak in much of the material that they were expected to have learned a year or two earlier. The actual state of affairs was far more dire: many couldn’t do the math that they should have learned a full decade earlier. Most weren’t just weak in math; they didn’t even know what math was. They didn’t know what an equation symbolized, or even that it was supposed to symbolize anything at all: to them an equation was just a jumble of symbols. They looked at me blankly when I asked them to think not only about what steps they needed to perform in order to solve a problem, but also about why they needed to perform those steps. They had no experience, nor understanding, of how to reason logically when presented with quantitative problems. Students threw fits when asked to combine simple techniques in…basic ways” (link via Chris Correa) - no one had ever required them to do that before. In his book Innumeracy, John Allen Paulos’ lamented:

Elementary schools by and large do manage to teach the basic algorithms for multiplication and division, addition and subtraction, as well as methods for handling fractions, decimals and percentages. Unfortunately, they don’t do as effective a job in teaching when to add or subtract, when to multiply or divide, or how to convert from fractions to decimals or percentages.

I disagree somewhat with Paulos, who wrote Innumeracy before calculators were ubiquitous: elementary schools no longer do manage to teach the basic algorithms very well. Other than that, he’s correct. Students’ depth of mathematical knowledge is so shallow that they can’t even figure out when perform basic mathematical operations.

The necessary groundwork for doing mathematics at the college level - or even at the high-school-courses-in-college level - is more basic than anything that students supposedly learning in high school in the most superficial and fleeting manner. And they routinely leave high school without it.

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

  1. Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)
  2. Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.
  3. Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)
  4. Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.

  5. Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x [y] when y [x] is set to zero in the function”).
  6. Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)
  7. More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

What else? I invite comments from everyone with a dog in this fight - college students who’ve taken math classes, college math instructors, curriculum developers, textbook authors. What unwritten (or written!) prerequisites are students missing? What should every student know in order to succeed in the class you took, taught, or wrote the text for?

65 Comments

  1. I’m biting: what do they think cross-multiplication *is*?

    - wolfangel — 7/20/2005 @ 7:20 pm

  2. oh, i can answer this one :)

    they think (semi-reasonably based on the name) that cross multiplication is what you do when you multiply fractions. i teach it by 1) showing that it’s really something they already know (as MS said), and 2) by explaining why i hate the word “cross-multiplying”. my students have invented terms for it like “cross-proportionalizing” and “undenominatorification” once they realize how stupid a word “cross-multiplying” is for something you DON’T do when you multiply.

    i am one of the high school teachers in question, but i teach at a high-achieving college prep private school, and we absolutely teach everything that MS listed above. when the kids complain that the problems on the test don’t look like the problems in the homework, i refer them to an english teacher, who will happily tell them that mere memorization of what was in the book without significant analysis is worth a B- at very best.

    i also agree that high-powered calculators can get in the way of learning (see my earlier comment in this blog on another post). the problem is that good students know how to use it to help their math, and poor students know how to use it as a crutch. but it’s exactly the poor students that we’d like to teach some things on a calculator in case they have to do it in (oh horror!) real life, and their math wasn’t good enough to remember how without a calculator.

    actually, my hat is off to MS for articulating so clearly what the basic goals of a high school math education should be. i especially agree that they have to understand the basic graphical AND algebraic versions of mathematical ideas. i teach this from the beginning in 8th grade by standing on one side of the room and stating a graphical fact (lines A and B intersect at point P) and then standing on the other side of the room and stating the corresponding algebraic fact (the equations A and B are both true when you substitute the coordinates of point P); then i tell them that they MUST understand why those are the same fact, and if they don’t then they’re missing something.

    i’ll tell ya, they’re not leaving MY classroom without knowing these things. i’m with you on this.

  3. Heck, LOW-powered calculators are a hindrance to learning. I never got truly proficient at basic, _accurate_ arithmetic until I was put in a situation where I had no other choice (and that was really, really painful at first).

    I’m one of those old people who handle the mid-life crisis by going back to school, and I’m now in a graduate geology progam. Once I wandered into my favorite lab to study, and found an undergraduate struggling with a “tough math problem”. Now, this young woman was very likely an upper-division student, since the lab is only used for upper-division and graduate classes. Her problem? She had two very large numbers to multiply, and couldn’t get the order of magnitude right.

    So I said something like, “okay, call 3.8673E23 four times ten to the twenty-third, and call 5.369823E15 five times 10 to the fifteenth. So your answer should be on the order of twenty times ten to the thirty-eighth, or 2 times ten to the thirty-ninth.” (I’m making up the numbers, but you get the idea.) She seemed shocked that anyone would tackle the problem this way, and terribly impressed that I could add numbers like 23 + 15 in my head. She also didn’t believe my approximate solution for a moment, and went back to punching resolutely away at her calculator.

    I left the encounter shaking my head. How on earth do you take science classes and do problems without gauging the order-of-magnitude of the answer, to be sure you didn’t make stupid math errors?

    - Karen — 7/20/2005 @ 8:57 pm

  4. I would add estimation and verification to that list. Students should know the difference between a sensible and nonsense answer.

    As you suggested, all of these skills are the skills one needs to have to be able to succeed in high school math. My understanding is that (at least in Ontario) many of these skills are the target of the grade 9 curriculum and it is assumed (likely quite wrongly) that students will have those skills by the time they leave grade 9, if not by the time they have entered it. Perhaps for the lower stream courses this is assumed by the end of grade 10.

    As far as I recall, high school was supposed to teach exponentiation, graphing and graph transformations, manipulation of second and third degree polynomials, conic sections, trigonometry, a little about sequences and series, very basic linear algebra, very basic probability, and very basic calculus. It is not possible that students will master this material without mastering your set of prerequisites.

    I consider high school math to be fairly advanced. The exponentiation and series work could help a high school graduate think clearly about money, but most of the rest of high-school math will never come up in daily life, or if it does, will not be remembered and used. Much of this material, however, is important to students who end up doing trade-work or in a college or university science or engineering class. It’s very, very practical stuff and, at least for some students, does need to be there. My feeling from doing many, many years of engineering is that mastery of the high-school curriculum is enough to allow one to do almost any sort of technical work, though probably not to understand the theory of whatever one is doing.

    My university experience is far from typical (fancy-schmancy engineering “science” program), but it chewed up and spit out students who had not mastered the high-school curriculum. A student who had only managed the seven points which MS outlines would not have lasted a week. Even in a more lenient program, there isn’t the time to repeat high school math to those who had failed to grasp it the first time around.

    In MS’ case, the attempt to re-teach middle and high school is part of a cruel scheme to encourage as many students as possible to have their dreams of business school smashed in a calculus course (if not in one of its pre-reqs). One can put all the blame one wants to on bad curriculum or poor teaching, but it’s not clear to me why students who managed to spend that many years not reading their math textbooks or deciding that maybe they should find out what was happening up at the front of class should be empowered to make investment decisions.

    For the truly keen, there are already adult night school courses reteaching this material at about the right pace. There are also textbooks in the college library and ads for private tutors plastered all over campus.

    IMHO, the business school should also be doing its own dirty work by running its weeder courses. It might also make sense to make those weeder courses be tailored to the math business graduates would be likely to use (?computer-aided financial math and statistics?) rather than calculus.

    My view is that this is not a university level problem, it is an elementary school and grade 9 problem. It must be a problem to which high-school teachers close their eyes. Numeracy, like grammar and spelling, is not on the curriculum in grade 11, and will only be addressed by exceptional instructors.

    I do not believe it is difficult to address, any more than poor spelling or grammar are difficult to address. In high school, only some of each period is used for lecture. Much of it is time devoted to in-class work. Instructors could use some of that time to work with students (likely in groups) on their weaknesses. For those fond of standardized exams, one way to encourage this would be to have a standardized exam at the beginning and another at the end of the year. Both count towards student grades, while only the difference counts towards the instructor’s “grade”. While we’re entertaining my crackpot ideas, why not put “test-taking skills” into the curriculum?

    I talked today with a friend of mine who got her first “B” on an essay. She is about to enter her last year of an anthropology BA. She has never learned grammar, has always made it up as she went along, and had never found it to be an issue until today. Neither spelling nor grammar were part of the curriculum in her Orange County schools, and no teacher had decided to put them on or even to tell her that it would be a good idea for her to learn how English works. Now that it’s an issue, she’ll be learning it on her own. Even so, I wonder. When those green lines appear on the screen, what does she think?

    I have sometimes been that student who didn’t have any clear idea of what the course was about and what its general structure was, and it has seldom worked out well. In my few stints TAing, I found that many of the students I saw had somehow lost track of what we were trying to accomplish, and I spent as much time explaining bigger picture issues as I did going over smaller techniques.

    I guess I would want every student to know to look for the organization of the material and to know there’s a problem when that picture isn’t clear.

    - Jordan — 7/20/2005 @ 11:34 pm

  5. Oh, come now, Karen.  You know the answer only too well:  you do it poorly, with many errors.

    - Engineer-Poet — 7/20/2005 @ 11:46 pm

  6. I somehow managed to avoid the math classes with people who didn’t Get It, but having suffered through classes in other fields where a lot of students had somehow missed learning the essentials, I think this quote from _Zen and the Art of Motorcycle Maintenance_ applies to the question you’re asking:
    “[T]here is no manual that deals with the real business … the most important aspect of all. Caring about what you are doing is considered either unimportant or taken for granted.”
    If you don’t care about actually learning (or teaching) something, no list of “What you should know at the beginning/end of this” is going to help.

    - dave — 7/21/2005 @ 11:15 am

  7. It’s kind of implied in your post, but I think one thing that really, really, really needs to be ingrained in math students is the fact that math is cumulative. Anything you learn in one year will almost certainly be used the next year, and the one after that, and the one after that. Moreover, once they ‘get’ that each new topic in math was produced by the logical extension of previous topics, and see the interplay between them, it becomes easier to understand what equations are, and how to manipulate them.

    While taking trig in high school, I recall one classmate who complained that a section of a test was unfair because it involved unit circles, which we did ‘last year’.

    - Independent George — 7/21/2005 @ 11:47 am

  8. When I was in grad school in math, one of the often-quoted adages was “You never understand course N until you’ve taken course N+1.”

    (Hey, it’s really hard to be sure I’ve typed my e-mail address right when I can’t see it)

    - Linda Seebach — 7/21/2005 @ 1:25 pm

  9. For IG (and not MS, as she’s heard this one several times):
    I once taught a freshman calc class, where about half claimed to not know how to calculate the area of a circle, and thus my related rates problem was unfair.

    Needless to say, they didn’t get very far with that argument.

    And even if they did know the formula, they thought so little about math, they probably would have tried taking the derivative of pi.

    - meep — 7/21/2005 @ 1:40 pm

  10. Wolfangel - they think, among other things, that (for instance) to solve the equation

    x/3 = (2x+1)/4 + 1

    they need to “cross-multiply” to get 4x=3(2x+1)+1. Notice how the 1 just sticks around, unaffected by the magical operation of cross-multiplication.

    Jordan - re estimation and verification - oh, god, YES, and I can’t believe I left that one out, because I’ve definitely mentioned it before in this space (and doubly so, in my teaching). Indeed, the very notion that an answer even can make sense or not seems to be foreign to many of my students: you do the work, and come up with an answer, and maybe it’s right and maybe it isn’t and that’s that. I think that some of my students are just so relieved to be done with a question that the idea of doing more work to check their answer - which they won’t even necessarily get any marks for! - is unthinkable.

    And yes, I realize that my list of prerequisites is inadequate for those contemplating going into a Serious University Program. A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part. The students in the service courses I taught last didn’t need to know exponentiation, solving quadratics, matrix multiplication, or much else on the list you provided. No, the reason for their lack of success in my courses had to do with them just plain not understanding what it meant to do math. Some of them could go through the steps of solving an equation, but most of those saw those steps as a magical procedure handed down by some Smart Mathematicians that happened to do the job they needed to do. Few of them understood the “why” of any of the algorithms they committed to (short-term, alas) memory in high school.

    But, Jordan, you say - For the truly keen, there are already adult night school courses reteaching this material at about the right pace.

    Don’t be so sure: in a moment of frustration last term, I had a meeting with the department head that consisted largely of me flailing my arms around and wondering what the hell my students did in the prerequisite courses in which they had presumably received marks of C’s or better. They needed C’s in grade 11 in order to take my class, and they were lacking basic math skills. I remember my grade 11 class, and I can’t imagine a student half as clueless as many of my precalc students getting a C in it. Students who couldn’t, say, solve linear equations would have failed my grade 11 class. Department Head explained: he told me that many of my students, particularly the older ones, were victims of the local adult night school, which taught grades nine through eleven math over the course of one single year. Most of those students hadn’t taken a math class in five years or more. Yet, in this adult ed program, 80% of students get A’s; 10% get A-’s, and 10% get B’s. Which is why I’m somewhat divided over whether colleges should offer math at the remedial level. Ideologically I think, hell no, that’s not the role of the university; but from a practical standpoint I realize that if they don’t learn this stuff in university, there is literally nowhere else for them to learn it.

    Independent George, in response to your experience of While taking trig in high school, I recall one classmate who complained that a section of a test was unfair because it involved unit circles, which we did ‘last year’ - last term I spoke to the precalc 2 prof, who had inherited many of my precalc 1 students from the previous term. One was a girl who tried so very hard, but…well, you know. Anyway, he was explaining conic sections to her, and mentioned finding equations of the asymptotes - “in the form y=mx+b”. She looked at him blankly, and he reminded her, “From precalc 1, remember?”

    And she said, completely seriously, “Well, yeah, we did that last term. Not this term.”

    I don’t expect my students to retain everything I teach them; that would be unrealistic. If this girl had said, “oh yeah, I recall doing that, but I don’t remember it very well - I’ll have to go over my old notes,” I’d have understood. But this student - and so many others - didn’t even seem to realize that one might be expected to use material gleaned from a previous course.

    - Moebius Stripper — 7/21/2005 @ 2:04 pm

  11. Math(s) Education

    As I’m certain plenty of people do, when I take up a new subject, I tend to seek out as much information on it as possible. It’s a bit obsessive, I understand, but at this point in my life, I’m not going to fight it. That said, along with books, cla…

    - Truck and Barter — 7/21/2005 @ 2:14 pm

  12. MS, is there anywhere you can raise the question regarding why this night-school program is allowed to claim they are actually teaching something for the money they get, when it is obvious that they are not?

    Or doesn’t anyone (including the public) care?

    - Engineer-Poet — 7/21/2005 @ 8:06 pm

  13. They’re probably not claiming they’re teaching something for the money they get; they’re probably claiming they’re giving you a mark in the class for the money they get. Important difference.
    And yes, a big part of the reason is that “people who care” is a vanishingly small set.

    - dave — 7/21/2005 @ 8:32 pm

  14. as a parent how can I tell if my child has mastered these things. would a 760 on SAT math, 30 act math and a 5 on the AP calculus exam indicate that the student is a least prepared for college?

    he plans on taking variable calculus for his senior year. i know it is a little late to worry but am asking any way.l

    - cajunrjb — 7/22/2005 @ 9:32 am

  15. When I took the SAT math it only tested through algebra.  Unless the test has been extended I’m not sure a 760 score would indicate complete readiness for calculus, but anything that high shows enough general aptitude that it wouldn’t take much to get there.

    - Engineer-Poet — 7/22/2005 @ 11:48 am

  16. Interestingly enough, no matter how much we fight it, education will fail.

    Students don’t see a point to it, eventhough it will affect the rest of their lives. Even if you say, “this is important, it will come up again later” it doesn’t stop people from forgetting it.

    I get people all the time that say, “I haven’t done math in years” and I always respond something to the effect of, “that was then, this is now” meaning you need to stop thinking in that frame of mind and work on catching up what you’ve forgotten.

    People just aren’t dedicated to doing work. I don’t recall if I’ve posted about a student I had a class with, at another university, that complained about reading 20 pages of philosophy. I couldn’t believe it, considering that I frequently have to read 100 pages within a night.

    Falling behind on work is understandable, but I think the most frusterating thing for everyone is that the students complain and don’t want to do any work. I deal with students who frequently come into the tutoring center complaining that the class is “hard” and the prof is “unfair” about the way they grade.

    I could rant for pages on how upset the students you have make me and how education is disappointing, but I know you don’t want to hear all that and I can certainly do it on my blog if you want more.

    - Vanes63 — 7/22/2005 @ 11:50 am

  17. I forgot to add that as long as a student wants to work at it, no matter what they scored on the SAT they can do well.

    Most upper-level students make more mistakes on simple things (air-head type mistakes like forgetting negative signs, etc.) than anything. So if they work hard at it, it doesn’t matter what they scored.

    I took the AP Calculus exam, got a 4 and did very well in my college math courses. I wouldn’t be looking into Ph.D. math programs if I didn’t.

    If they are looking into competitive college programs, I’d just say remember to be competitive and work hard. Studying never hurt anyone.

    - Vanes63 — 7/22/2005 @ 11:53 am

  18. It depends what one wants to do — a 5 on either of the calculus AP exams would indicate one is ready to do college level math (the AP tests haven’t been dumbed down… I saw last year’s tests). You could do integral calc if you took only the AB exam, or multivariate calc if you took the BC exam. I would say that’s good enough for hard science majors (e.g., physics, chemistry), math, or engineering.

    However, the SAT doesn’t tell you anything about math level - it’s an aptitude, not an achievement, test. I got 720 on the math portion of the SAT while I was taking algebra 1 (in 1987), and I sure as heck wasn’t ready for calculus at that point. I wasn’t even ready for trig at that point.

    - meep — 7/22/2005 @ 12:18 pm

  19. People just aren’t dedicated to doing work.

    This seems to be worse nowadays than it was a few decades ago (somebody correct me if I’m wrong).

    If that’s true, I suspect there is a memetic evolutionary phenomenon involved. As technology improves, we need less and less people (in relative terms) truly dedicated to producing. Supply decreases (again in relative terms) to better match demand. Parents who don’t work that hard are unlikely to drive their kids to do so, even if the latter are “genetically capable”. Millions of instances of this seems to have given rise to the bizarre “I shouldn’t need to do homework to succeed in this class” entitlement meme. Thanks to this, some math teachers are faced with the burden of transmitting what amounts to a foreign culture in their classes, in addition to covering the course material.

    - Dog of Justice — 7/22/2005 @ 1:20 pm

  20. Vanes63 - hey, misery loves company. I’ll read.

    Engineer-Poet asks:

    MS, is there anywhere you can raise the question regarding why this night-school program is allowed to claim they are actually teaching something for the money they get, when it is obvious that they are not?

    This is part of the larger problem of there being no communication between the high schools and the universities. In general, we should really raise the question about why regular high school students aren’t learning what they’re supposed to be learning when they get to us. With the night school, part of the issue is that a lot of programs require students to [pass | get C’s in | get A’s in] some upper-year high school math class, but then those students will never see that math ever again. So, there’s no great harm in giving those students C’s or above: they’ll get into the college program they want, and their inability to do basic math will never bite them in the ass. The real problem is that those aren’t the only students in the night school, though they do comprise the majority: the rest are actually going to have to use the math they supposedly learned in the course.

    Incidentally, the kid I’m tutoring is struggling for a C in grade 12 math, which he needs in order to get into a two-year diploma program. He has friends in that program, and they report that the only mathematical skill they ever use in their math classes is plugging numbers into calculators. It shouldn’t surprise anyone that when high school math is usually used as a hoop to jump through rather than as an opportunity to learn important and potentially useful skills, actual learning is low priority and tends to fall by the wayside.

    - Moebius Stripper — 7/22/2005 @ 1:33 pm

  21. MS: Let me add one more to your list of requirements:

    8. Students should understand what math is as a field and why people study it. Mathematics is concerned with understanding the abstract connections and patterns of all kinds of things. Specifically, they should realize that mathematics is much more about understanding than “getting the right answer.” A student in my class who gets a correct answer on a lucky guess isn’t a good student, whereas a student who works hard but makes a minor mistake may understand things much better and is doing better. Getting the right answer is the kind of thing that should happen because of understanding. Students should also understand that problems are just example scenarios in which to test your understanding, not the ultimate goal.

    (I think this is a major problem, especially for the weaker students. Part of the reason, I suspect, that many of your students are hopelessly behind is that they fell behind somewhere around fractions in 6th grade and managed to tread water in class by studying hard enough to get the answers right, and they sort of kept doing that while their understanding was left far, far behind.)

    - Moses — 7/22/2005 @ 9:21 pm

  22. Maybe this all explains something that I saw a few years ago, in an introductory Ring Theory class.

    We were talking over our midterm which, not surprisingly, a good chunk of the class had done poorly on. In particular, some people had problems with a question that involved matrix multiplication. One older student was upset because we hadn’t gone over the material in class at all, to which the professor countered that a prereq for the course was a basic linear algebra course. The student then replied something to the effect of “I took that course two years ago. What do you expect me to do, go back and read my old notes for it?”.

    It just baffled me. Why did he think that there are prereqs for courses if not to assure the professor that we know the material contained in them?

    - Simon Rose — 7/23/2005 @ 7:57 am

  23. This is a great thread and I think the specifics of the list are excellent. However, a very important aspect of all of this is whether the TEACHERs actually understand mathematics. And this probably requires all of the teachers, not just the math teacher. Whenever math comes up it should be explained correctly. Anda teacher should be able to follow a student’s reasoning. Numbers, fractions, percents, estimates come up all the time.

    I think math is unusual in that it is pretty much useless unless you understand what you are doing. I would quantify this by adding to the list that one should be able to do the same problem several different ways.

    It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.

    We need to define and expect something similar as far as being numerate.

    And, definitely, no calculators!

    - Susan — 7/23/2005 @ 10:22 am

  24. As a high school math teacher, i can assure you that each of the prerequisites you propose is taught ( or, at least, presented ) to every student, college bound or not, in my district starting in the eighth grade. I teach the same material to ninth, tenth, eleventh and twelfth graders when I’m not trying to teach them material they should have learned in grade school.

    My district increased its math requirement this year from two years to three. So now, when a student shows up in a college math course, she will have had three years of math beyond the elementary prerequisites you have proposed.

    And still.. you will find your classes include students without the least of skills.

    - djac — 7/23/2005 @ 10:40 am

  25. I have to second what Moses wrote about understanding what is going on being more important than getting the right answers for problems. Students get by, by memorizing correct answer- generating-algorithms for specific situations which they promptly forget once the exams are over the same way they would if they memorized a list of historical events for a medievel history course.

    Teachers usually include a few ‘tougher’ questions to test understanding but these just sort the A’s from the B’s and C’s. The B’s and C’s are sorted from the failures by the memorization work. So if you understand you can get a A, if you’re willing to work a little you get by and only if you aren’t willing to work a little do you get left behind (if then, even).

    I well remember an occasion back in high school with probably the best math teacher I ever had. The topic was dividing fractions and the student at the board had successfully solved a routine problem. The teacher asked the student why they had solved the problem the way they had and the student responded with the rhyme:

    ‘When in doubt, don’t ask why / just invert and multiply.’

    The teacher (an excitable type) lost it, and went into a big rant about how we should *always* ask why.

    Of course, there are some things which simply need to be memorized, the times table being the one which is most often overlooked.

    - Declan — 7/23/2005 @ 5:31 pm

  26. ‘When in doubt, don’t ask why / just invert and multiply.”

    Wow, that’s appaling. I’m going off topic, but it reminds me of this mnemonic poem for how to extract a square root from 1857.

    - Sanctimonious Hypocrite — 7/23/2005 @ 7:32 pm

  27. Here in Australia, first-year university mathematics courses generally require that students have studied
    at least single-variable calculus at the high school level, and preferably a (slightly) more advanced multi-variable calculus course.
    Sometimes a university will offer a calculus course for students who for whatever reason didn’t study it in high school. The sitaution is similar in the UK if I am not mistaken.

    When I was in high school, the only people who didn’t study calculus were those who had decided not to study any subject requiring mathematics at university level.
    As to good textbooks, I’ve recently started reviewing basic mathematics for pleasure as preparation for reading, again for fun, serious mathematics texts in analysis and algebra.
    There’s an old U.S. high school text, Modern Introductory Analysis by Mary E. Dolciani et al., which I quite like. It omits some proofs (e.g., the fundamental theorem of arithmetic), but over-all it’s well written and includes proof-related exercises suited to p

    - Jason — 7/24/2005 @ 1:31 am

  28. The worst thing about that fraction rhyme is that you just know that student learned it from a previous teacher, likely the one who taught fractions in the first place. It’s not the sort of rhyme kids make up.

    Also, I love teachers who are so committed to the subject they teach that they take it personally when their students show a lack of respct for it. I had an English teacher like that back in grade school; his enthusiasm was contagious.

    - Moebius Stripper — 7/24/2005 @ 3:23 pm

  29. That’s why it’s so rare!  School administrators heard about it being “contagious” and they must’ve started treating it like a disease.

    - Engineer-Poet — 7/25/2005 @ 5:03 am

  30. Moreover, they should understand that the horizontal bar in a fraction denotes division.

    As a high school maths tutor, I have to point this out to someone every day. Note that I only have 30 students total, and I have been doing this gig for many more than 30 days.
    Because fractions are taught so separately from division, it takes a lot of effort to help students realise that they are actually the same idea.

    I’ve had a lot of success with getting students to put or imagine brackets around the denominator and numerator of any algebraic fraction, that seems to help.

    - Ronald — 7/26/2005 @ 2:04 am

  31. I remember my math courses as being very disconnected. Only sometimes did we really need to remember something we had covered in the previous years. For instance, basic algebra required a knowledge of fractions and basic arithmetic, but nothing else. Plane geometry came a year later and didn’t require anything at all! Then came intermediate algebra, which required basic algebra, and trig which relied on basic arithmetic again. Then advanced algebra, which required intermediate algebra, and solid geometry which required basic arithmetic plus only a little bit of solid geometry. Calculus was the first course which combined algebra, trig, and some geometry. Linear algebra required basic arithmetic and modern algebra didn’t require anything (it was all theory). Differential equations required calculus, as did probability and sadistics. Theory of complex variables supposedly required calculus, but you couldn’t prove it by me.

    So, where is this supposed relationship between courses to be found?

    As an aside, I didn’t know that the bar in fractions stood for divide until we learned about REMAINDERS! What a revelation that was.

    And estimating the “correctness” of an answer came about my senior year of high school when some of us took an after school course in how to use slide rules. What I simply LOVE about calculators is that you don’t lose track of the darned decimal points!

    - Rex — 7/26/2005 @ 11:57 am

  32. La Carnival

    Jenny D’s got the latest Carnival of Education up, and it’s a beaut. Don’t miss Tall, Dark, and Mysterious’s post on what kids should learn in high school math. It’s an eye-opener, not least because it’s appalling to realize just…

    - Number 2 Pencil — 7/27/2005 @ 6:36 am

  33. “It’s kind of implied in your post, but I think one thing that really, really, really needs to be ingrained in math students is the fact that math is cumulative. ”

    Independent George hits the nail on the head. It’s not like English Lit, where if you hated _Silas Marner_ you can let that information dribble out of your skull before moving on to other books. Every time I’ve taught statistics, I’ve emphasized the importance of doing every homework assignment and coming to every class, but there’s always a kid or three who thinks they can blow off a chapter or two and be just fine. Not.

    I also second Moses’ suggestion that students understand why people study math. I find that crucial to teaching statistics as well - of course, it’s a very applied course, so it would be silly to teach it without explaining the why.

    - Kimberly — 7/27/2005 @ 6:44 am

  34. Jenny’s carnival

    Jenny D is hosting the Carnival of Education, which features Moebius Stripper on what high school students should learn about math before they show up in her college classroom and Math and Text on the fatal phrase, “My kids won’t…

    - joannejacobs.com — 7/27/2005 @ 3:30 pm

  35. Some commenters have mentioned what I think is the most important thing: students have to understand WHY they’re studying math. Then they have to understand how it all works, big-picture: that is, what, exactly, they’re studying. Neither of these is ever explained.

    There’s a scene in the movie Sliding Doors where the Gwyneth Paltrow character is doing little doodles. They’re stick people, and it’s taken for granted — and considered cute — that she (and probably the audience) can’t do any better. Erm, I react to that as most of you react to math deficiencies: how can you get all the way through school and not understand how to draw what you see? Isn’t it common sense? Isn’t it a useful skill, at least as useful as taking the derivative of a function?

    Math, beyond the basics, isn’t a given: you have to justify it as well as I’d have to justify compulsory drawing lessons. And you have to put as much thought into teaching it as I’d have to put into teaching non-artists to draw, including the theory, the bigger goal — not just the conventional sticks-and-circle symbol for a person; not just which equation to use for which kind of story problem. This was missing from my math education, and it’s why I had such an awful time of it. My high-school math didn’t end up being a prereq for much college math.

    - kate q — 7/27/2005 @ 6:22 pm

  36. As a high school math teacher, I have to say that your list is very good. In fact it is covered in the exam that all of our students must pass in the state of Texas in order to graduate.

    The only way that colleges can be sure that high schools are teaching what needs to be taught is to hold kids accountable. And that means standardized tests.

    One of the good things from the No Child Left Behind initiative here in the US is the requirement that all states have in place an assessment to show whether the kids have mastered what needs to be mastered. The test we give is given in 4 separate parts for each of the 4 content areas, and if the students don’t pass all 4, they don’t get a diploma! The math test covers the curriculum of Algebra I and Geometry - the minimum courses that all high school students should take. In order to receive a “college bound” diploma, the student must have also taken Algebra II.

    As soon as these tests are implemented nationwide, I think the colleges will see a student body that is a lot better prepared. The colleges, however, also need to stay up with the times and stop outlawing calculators. Colleges need to move into the 21st century and realize that calculators are here to stay. Yes, test the kids to make sure they know the basics, but use the calculator to enhance your teaching; don’t just assume it is a technological crutch for every kid who owns one. I teach Pre-AP and AP math courses, and we use the calculator a lot to solve problems that are not as contrived as the ones I solved in high school without one. Having taught with the calculator and without, I’ll definitely keep my calculator, thank you.

    PS - A 5 on the AP Calculus exam DOES say that a student is very prepared for college math - the test is given with both calculator and non-calculator parts, and there is no way a kid can make a 5 and not have these listed prerequisites under his belt.

    - Jill — 7/27/2005 @ 9:06 pm

  37. Eh? I’m a grown adult and I still don’t know why I had to take so many years of useless math courses. The best reason I came up with was “To train your brain to be able to use numbers.”

    For instance, fractions. Everyone on this thread has been pointing out that fractions are simply mini-equations, and you don’t get why students don’t know that. They don’t know that because it’s useless knowledge.

    If you were teaching them to multiply fractions, would you allow them to convert to decimal and multiply? No, you wouldn’t. So there is no incentive to learn mathematics well, because every class is “I will teach you to jump though this hoop. You will be tested on your ability to use the exact method I taught you.”

    I’ve only had to use math above an 8th grade level *once* in my life. It was for a game, to calculate shield strength for a spaceship. That’s it.

    Unless you’re getting into one of the engineering or science fields, anything above algebra isn’t needed.

    - ErikZ — 7/27/2005 @ 10:05 pm

  38. Weird… are trackbacks and comments all jumbled together on wordpress? Anyhoo:

    Kate Q - I agree, but the problem is that there is a gap between understanding the ‘why’ of math, and gaining the skills requied to do it. When you start with still-lifes, and gradually move on to more compicated shapes and lighting conditions, you can at least see your progress. With math, it’s absolutely essential to nail the basic skills before you move on to the applications, but the connection between basic skills and the advanced ones are not so obvious in math until after the fact.

    Why do I need to know 6 x 9 if I can just enter it into a calculator? Because you’re eventually going to need to factor out a quatratic. What’s a quadratic? Don’t worry - you’ll get to it in another five or six years. And then, it’ll be another two or three years before you learn why you need to understand quadratics, and another year or two after that before you can finally apply it to something useful in life. So just trust me and do your homework, dearie… Looking back, it all makes sense, but the view from below is an entirely different story.

    I think that’s why calculators are so tempting. When you see physicists and engineers whip out calculators for all their problems, it’s perfectly logical to conclude it’s ok to skimp on the basics if these math people are using them. What’s lost is understanding why those countless hours of mind-numbingly boring drills are necessary prerequisite to the fun stuff.

    - Independent George — 7/27/2005 @ 10:13 pm

  39. The items MS has listed are the exact things that I try to teach in my PreAlgebra (for those not ready for Alg I) and Transitional Algebra (for those still not ready for Algebra I) classes. I agree with most of the posts here and am glad to see that I am not alone in my views. There is still one point that I have not seen mentioned yet, how do we convince a student to care if he or she learns or not? Most students, even the ones with above average ability, just do not want to put in the drudge work required to truly master math.

    And yes, Jason, the old Dolciani is, I think, the best math test ever, but it fell victim to the move to the pretty pictures, bright colors, and shiny objects (and lack of coherence) that are found in textbooks today.

    - decaf — 7/27/2005 @ 10:27 pm

  40. It would also help if they would drop non-sense topics from the curriculum (or maybe these topics are simply misnamed?). I have bachelors degrees in mathematics and physics and a masters in physics and still have no idea what synthetic division is, leading me to suspect it’s not particularly important. When I read the initial mention of ‘cross-multiplication’ I had no idea what was being refered to. Admittedly, this may have something to do with having dropped out of high school early to go to college, and thus having missed out on the culture of useless high school math terms, but why then, are high schools spending time and energy on them?

    - quadrupole — 7/27/2005 @ 11:36 pm

  41. Oh my god! I had forgotten about synthetic division and how much I came to hate it when I was teaching pre-calc!

    Despite my students showing me how to do it about a bazillion times, I still can not remember how to do it. Even my own father suspects that I am stupid because I can’t do sythetic division because (he says) it is “easy.”

    Here is what I do know: It is a compact notation to divide p(x) by x-a.

    And in reference to the state testing: Doesn’t help. To get out of high school in my state, one must take a very comprehensive algebra test. The passing score must be super-low or the students forget everything after the test (inclusive or).

    - Rudbeckia Hirta — 7/28/2005 @ 4:05 am

  42. Why do you have to learn any subject? Why do we have to learn Shakespeare or chemistry or ancient history? (I can assure you that I haven’t titrated any acids lately, no Assyrians have come knocking at my door, and nowadays you’d send Hamlet to watch some Woody Allen movies (though I don’t know if that would increase his murderous inclinations.))

    You learn these things because they provide mental discipline. Because they make you more flexible and able mentally. Because one can think more deeply about the world and reality. Because you learn different ways to think and to learn. And for pure entertainment (though I’m probably lonely on this one. I really enjoy reading Dickens for fun, not for any particular literary analytical purpose.)

    For example, I’ve got a math background waaaay beyond what is needed for my job. However, many others here have only just enough math to do their jobs. So what happens when there’s a change? When new solutions need to be found? I’ve got a lot more tools in my possession than others. And I’ve found that gets me the interesting, new stuff to do, while other people just get to do the same old stuff. Pretty cool.

    More to the point: how does anybody know what they’re going to need in the future? Industries and jobs get created and destroyed all the time. Whaling was a way of life for lots of guys, a way of life that got destroyed when gaslight, electric lights, petroleum products, etc. replaced what whales got you (and also destroyed with the rise of the environmental movement.) If one trained only for whaling, what do you do? Complain about today’s society not providing any jobs for whalers?

    Likewise, career-focused education is good, but one needs to have a strong foundation - knowledgeable in a wide variety of subjects and skills - to be able to retrain if need be.

    - meep — 7/28/2005 @ 7:28 am

  43. right; presenting “synthetic division”
    is a very bad idea since one needs to know
    “long division” anyway and memorizing conventions
    about where signs go & suchlike is anti-mathematics.

    since different people use the phrase “cross-multiply”
    to mean differnent things (and indeed, sometimes
    the *same* person’ll use it to mean different things!),
    teachers should probably never use it all (to avoid confusion).

    - vlorbik — 7/28/2005 @ 9:14 am

  44. Synthetic division is really rather nice. But beyond the recreational factor (ha! factor…) it’s of limited utility now that we have calculators. It’s kind of like being able to sew; It’s handy once in a while, but you’ll never recoup the cost of learning it. Nobody’s going to hire you to do synthetic division, and you don’t need to learn it to get through calculus.

    I had a student, otherwise mediocre, who had mastered synthetic division and long division of polynomials. He used it on every problem whether it worked or not. I always enjoyed grading his homework. It’s really remarkable how many problems can be solved this way.

    So why use/teach synthetic division? Like meep said above; because it’s good training, I guess; because it’s fun; And because I can.

  45. So many great comments, many of which respond to others in a way that I’ll happily let stand. Just a few additional remarks: kate q, you say

    . Erm, I react to that as most of you react to math deficiencies: how can you get all the way through school and not understand how to draw what you see?

    Actually, I don’t react that way to math deficiencies. Yes, I am good at math, and I’m not going to pretend that if only my students worked as hard as I did, they’d be just as good at the subject; I recognize that I have some innate ability. However, I’d be an irresponsible teacher if I were to say, “Stupid students, how could they not get this stuff?” It’s my job to teach them. No, my reaction is more of the form, “How lousy is our math education system that after ten years of seeing this stuff, students don’t even know the basics?” It’s as though Gwyneth Paltrow’s character had taken ten years of art classes (did she? I never saw the movie) and was still making stick figures.

    Re synthetic division - if any of you folks who hate the stuff (and I’m right there with you) ever find yourself instructed to teach from the text I used for precalculus, throw a fit. Quit your job, if necessary, impale yourself of your slide rule, anything: the entire third chapter is synthetic division. Synthetic division to find roots! Synthetic division to bound zeroes of polynomials! Synthetic division to count them! I’m serious. And of course, the book doesn’t even try to justify the magical synthetic division methods; the proofs, naturally, are “beyond the scope of this course.” Interestingly, I was one of three instructors teaching this course with a mostly-common final exam; we tended to teach from the book to the extent possible, as that’s the best way to prepare for a semi-standardized test when communication among instructors is less than it should be. But without even consulting one another, we decided to skip synthetic division entirely. (Of course, some students “learned” it in high school. They insisted on using it, because “they didn’t understand [my] method” - which, as RH correctly suggests, is by far the more intuitive one.

    - Moebius Stripper — 7/28/2005 @ 11:47 am

  46. I am strongly in favor of more math training as good mental discipline. My point is, there are stupid things that are taught in math by primary and secondary math curriculums that leave actual mathematicians and practitioners of the art scratching their heads and wondering what the hell you are talking about. I followed the link provided above on synthetic division. Wow, that’s SOOO much harder than simply extending the concept of long division to polynomials, and it offers far fewer opportunities to understand the similarities between polynomials and number bases. Why must children be vexed with harder, more confusing, more mechanical, less enlightening methodologies?

    As to the question of how does anyone know what they will need in the future, a good occams razor is: does anyone in the real world use this. Include in the real world practicioners of the art. If the only place a concept is ever used is high school, it’s probably not worthwhile. Please note, when I say include practitioners of the art in the real world, I mean things like: do mathematicians use it, do grammarians use it, do historians use it, etc.

    - quadrupole — 7/28/2005 @ 11:52 am

  47. As a graduate student in physics who is perennially roped into tutoring students taking the introductory sequence (especially the intro physics course for non-science majors), I come face to face almost daily with this failure to learn any form of basic math. This is despite the fact that not only must students have graduated high-school in order to enroll in the course, they must also have either passed or tested out of the remedial math curriculum of the college. And yet, somehow, I am continually faced with those who do not understand the concepts presented in the course because they do not grasp the basic math underlying those concepts.

    One student, in particular, comes to mind. After roughly an hour of work setting up a simple one-dimensional kinematics problem, she had finally managed to reduce the problem to a single equation, with one unknown. Thinking the problem solved, I moved on to another student, only to get called back some fifteen minutes later by this same student working on the same problem. Our conversation went as follows :

    “You have one equation, with one unknown, right? So, just solve for x.”

    “What’s an unknown?”

    - Jason B — 7/28/2005 @ 12:36 pm

  48. Invert and multiply and cross-multiplication are important theorems. They can and should be proved at the middle school level.

    If I were to list primary three deficiencies of my pre-calc students (at university), they would be 1) a poor grasp of symbol manipulation skills that should be taught at the Algebra I level, 2) lack of knowledge of fundamental facts of geometry and 3) poor language skills.

    The symbol manipulation problem originates in the fact that they were never required to master fractions. We do not teach fractions so that our students can do calculations, we teach fractions so that our students can do algebra. Unfortunately, we are not teaching operations with fractions to mastery. This is a massive barrier to learning algebra.

    Regarding language skills, my students cannot and will not deal with any text that requires them to read carefully for precise meaning. They cannot and will not make arguments that require rational justification. I think this has as much to do with the way English is taught as with how math is taught.

    The bottom line is that at no point prior to college do we require students to master college prep mathematics to move on. Sure there are problems with teachers and curricula but no one is willing to say that a student who can’t factor a simple quadratic or solve a simple proportion problem or distinguish between perimeter and area or make a coherent logical argument has no business in college.

    Re: calculators, these tools allow students to deal with all kinds of interesting problems that would otherwise be impossible. Who wants to invert a 9x9 matrix by hand? Unfortunately, calculators are primarily used as a crutch to avoid mathematics. It doesn’t have to be so but it will be this way as long as calculators are used on exams -

    - which raises the final topic. The only way to get accountability in middle school and high school math is for someone other than the classroom teacher to write the exams. The temptation is just too great for teachers and administrators to pass on students who just don’t have it.

    - charles R. Williams — 7/28/2005 @ 4:28 pm

  49. The bottom line is that at no point prior to college do we really do education in this country. And increasingly, in all to many subjects, we don’t do it in college either.

    - quadrupole — 7/28/2005 @ 8:14 pm

  50. To answer the question “why do I have to learn this stuff?” and “when will I ever use this again in the real world?” I often give this answer to my students…

    Yes, you may never use this type of mathematics ever again. But mathematics is the one course in high school that can teach you to think critically in ways no other course can. We as educators are trying to teach you our material - we are also trying to teach you to think critically… to analyze situations, to look for errors, to think things through. And no matter what career you go into, you will need to have those skills. Since many of you don’t know what you want to do for a career and I can’t teach all of you things directly in your career field anyway, I will use the tools I have been given to teach you to THINK. I often call my math classes “brain calisthenics.”

    They can’t argue with this when it comes to the usefulness of my class.

    - Jill — 7/28/2005 @ 9:01 pm

  51. When I sartes teaching 21 years ago, only 2math credits were required for graduation, and those 2 could be Consumer Math and General Math (which will be all that 80%, or so, of the populace needs). When the state school board decided to require Alg I for everyone, we math teachers said that the failure rate will go up, which it did. We got pressured to lower the failure rate and we said that the content of the course would be diluted, and it was. The state DOE has since put in 2 courses (PreAlgebra and Transitions to Algebra) to help prepare students to handle Algebra I. The rigor of the classes has improved due to standardized subject area testing, but the failure rate and drop-out rate are increasing. Now, with NCLB, students must pass a test that covers Alg I and Geometry.

    I question whether it is wise to treat all students as if they are, each and every one, college bound in a technical field. We cannot predict what our students will be doing this weekend, much less what they will do in the future. All we as teachers can do is improve the odds of their success by looking at the successful people of the past and try to give our students the same tools. I am not sure that the one size fits all approach is best for everyone. Some people are just not mathematically talented. Must we put them through the stress of being required to study a subject they are not successful in? Especially if they are convinced they will never need or use it. I tbelieve we need to rethink the question of what should be required.

    - decaf — 7/28/2005 @ 9:35 pm

  52. I think to a large extent students forget. I once tutored economics at uni, and spent a fair bit of time with mature students who had done their schooling well pre-calculators (they weren’t used most of the time I was at school), and yet didn’t understand these matters. I think the best time was when I’d worked with a mature student through a whole problem, and we came to an answer of 5/4. To answer the question, we needed to know if 5/4 was greater or less than one, which meant either that quantity demanded increased or it decreased. So I asked: “Is 5/4 greater or less than 1?” Out came the calculator.

    Plus there’s limited time to spend on maths at school, which means limited time for really drilling things into people’s heads. And in particular drilling in those understanding issues you described. If something hasn’t been drilled in, I tend to forget it. Especially after a gap of a couple of years.

    - Tracy W — 7/28/2005 @ 10:08 pm

  53. “When the state school board decided to require Alg I for everyone, we math teachers said that the failure rate will go up, which it did. We got pressured to lower the failure rate and we said that the content of the course would be diluted, and it was.”
    See also: credential inflation. If you have a Masters in reading comprehension (or whatever level is equivalent to what we used to call “minimally competent”) you’ll have no trouble around here finding rants on, if not precisely that, closely related subjects.
    Funny, though, I think the department I work in is split more or less evenly between Ph.D.s and people with no formal credentials at all, just Clue and demonstrated competence in the field; the only one I can think of who has any formal credentials but no Ph.D. is the manager. So it’s not that that degree that HR says you need is actually required to *do* the job, just to get it.
    Corollary: The more people get a given credential “because you won’t be taken seriously without it”, the less useful it is in determining whether to take somebody seriously.

    - dave — 7/29/2005 @ 6:15 am

  54. Required Math Skills

    Tall, Dark and Mysterious:
    Are there any high school math teachers - or, better yet, developers of high school math curricula - who read this blog? I’ve decided to take a break from my usual undirected griping about how woefully unprepared student…

  55. “The bottom line is that at no point prior to college do we really do education in this country. And increasingly, in all to many subjects, we don’t do it in college either.”

    That is false. My son got an IB diploma with a score of 32 out of an urban public school. The motivated student in the typical public high school can get an excellent education if his parents are supportive and if he is willing to make the huge effort required to do it. It is true that our colleges hand out many worthless degrees that suggest that the bearer might be ablt to read and write but again, for the student who will make the sacrifices and is prepared our universities do a fine job.

    Education is not something that is done to the student. It is something the student does in the setting of the school. Immigrants from cultures that value education do exceptionally well in our schools.

    On a different topic. Perhaps 80% of our students will not use algebra in their careers. Nonetheless, a student who wants to master a technical field must be in a rigorous math program from the middle school on. The minute we give up teaching a middle school student fractions and hand him a calculator so he can do computations, technical careers are no longer an option for him. Now do we have the will to track college prep students starting in the 6th grade so that the others will not be burdened with the unpleasantries of algebra. Or will we insist that all students graduate with what they need to succeed at college in technical fields. Unwilling to face facts and make hard choices, we muddle through in a dishonest way. We “prepare” all students for college on paper but not in reality. We do de facto tracking while denying that this is happening. We require more and more high level courses from our students and then water them down.

    - charles R. Williams — 7/29/2005 @ 10:00 am

  56. If you want to know why children are not prepared for college, you only need to pick up a math textbook and read it, not from a mathmatician’s point of view, but from a student’s point of view. If you can do this, you will find that the material will follow one line of thought to an end point and then pick up another, different line of thought, only somewhat related to the first line, and follow that to an end point, and so forth. This, I have been told, is spiral learning. It also is disjointed.

    While you are doing that, pay close attention to the language. You will find that mathmaticians have adopted more than one word for the same thing. For simple example, divisor, dividend and quotient for long division, but numerator and denominator and mixed number (I think) for fractions, and percent for decimals. For another example, look how easily the author of the initial post wrote this: “For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x [y] when y [x] is set to zero in the function”).”

    I took algebra I and II, plane geometry, trig, and statistics and probability in high school in the early ’60s, just before that New Math experiment. I just finished two years of helping my daughter work through pre-algebra and algebra I. There is little similarity between the way I was taught (linearly, I suppose) and the way she was taught (spirally). While helping her, I started the following list:

    Mathematics

    First Axiom: Given two or more equally correct explanations of a mathematical operation, a textbook writer will always prefer the most complex.

    Corollary 1: The more basic the operation, the longer the explanation.

    Corollary 2: The more basic the operation, the greater the number of pages of the textbook answer key.

    Corollary 3: If one name for a simple concept is good, two names are better.

    Second Axiom: The term “real life” means a completely imaginary world when used in a mathematics textbook.

    I do not now remember what prompted me to start, nor do I remember why I stopped. However, I can assure you that there was plenty more material in the textbook with which to work. And one of those things would have been “the big picture.” You can be absolutely sure that, when a student is trying to learn how to factor, for example, the least helpful thing the student can be told is to “remember the big picture,” which apparently means using factoring in some mathmatically delightful torments yet to come.

    One interesting concept is: “Anything you learn in one year will almost certainly be used the next year, and the one after that, and the one after that.” This is true, of course, but it is true of most all knowledge. However, it fails to take this into account: If you learn the equation for the area of a circle in one week and never do it again until surprised two years later by the need to remember the equation to answer a test question, you are likely to miss the answer. Why? Because our minds are not like a Windows program with little folders containing little folders in series. Oh, the equation may be there somewhere, and a glance at it would remind us of what we once learned. But why should a mind that retains the formula believe that all other minds should retain the formula? In my real life, I have never used it. Because I need to know the square footage of a wall when I paint it or the square footage of a garden when I fertilize it, I use the formula for the area of a rectangle reasonably frequently and remember that formula. When I travel, I remember d = rt. But don’t expect me to remember how to figure the surface area of a cone, however good I may have been at that when I was actively working that kind of problem.

    Another interesting concept is: “You learn these things because they provide mental discipline.” Well, yes. . . . But that is precisely what they used to say about learning Latin.

    - L. Ashley Higgins — 7/29/2005 @ 3:59 pm

  57. (@ L. Ashley Higgins)

    You should remember that the goal of mathematics textbooks is not to teach mathematics. It has NEVER been to teach mathematics, really, it has been to provide a reference that helps reinforce a classroom. Thus, things like including multiple terms are essential, because teachers may use more than one term to refer to a concept. Besides, textbooks are a distant second in importance. Students may try to read them, but if they don’t learn from someone actively teaching them, they’re not going to learn much from a book.

    Students who feel they need to turn to the textbook to learn their mathematics either have extremely bad teachers, or only realize that their passive learning didn’t work in class and hope that a little passive reading will fix everything. If a student actually ever read a math textbook as an interactive exercise, I’m sure it would help a lot.

    As for “all knowledge” building on itself like math does, that’s hogwash. Some does, certainly, but as someone who actually teaches, and whose classes have prerequisites, I can say that people who don’t do well at the prereqs are going to suck in my class because they have to learn twice as much. This is not true of, say, history or art classes: while there may be some theoretical backbone that students get that will help them between topics, the specific information from one class may be totally irrelevant to the rest. (For instance, does it matter who the early explorers of America were when you’re studying 20th century history? Maybe a little bit, but not really. Not like it matters that you know how to do algebra before you can take derivatives.

    Oh.. and multiple terms for simple mathematical concepts? I doubt it’s really that confusing, baffling, enigmatic, difficult, obscure, cryptic, opaque, inscrutable, or unfathomable. Come on, math classes only ask you to learn about 10 words a year, I think people can cope.

    - Moses — 7/29/2005 @ 6:56 pm

  58. Thank you, Charles R. Williams. Too often we forget about the students who do well and suceed in their studies. Yes, there are too many students who do poorly, but if the schools and teachers were as universally bad as most news media say, then how do we explain the successful students? Students must make an effort to learn. If they make no effort, then no one can teach them anything. If they are willing to put in the effort, then a good education is available in most schools.

    I do have one slight disagreement with your views. You say your son got his IB diploma out of a publis school. This is a great achievement to be proud of, but the danger in programs like IB or AP courses is that the achievement becomes the goal to the exclusion of what is really important in education, the knowledge itself. By this I mean that students will do what is necessary to get the desired grade without necessarily understanding and learning for the long term. They will turn in every assignment and pass every test, but not truly learn. I try to tell my students, “Forget the grade; get the Knowledge”. Of course, if you have the Knowledge, the grades will follow. And having the knowledge if and when you need it is always better than needing it and not having it.

    - decaf — 7/29/2005 @ 8:03 pm

  59. 連立一次方程式

    More cracks showing in Japan’s post-bubble educational system. (For once, the English article isn’t much thinner than the original

    - The White Peril 白禍 — 7/30/2005 @ 5:58 am

  60. Susan asks: “whether the teachers understand mathematics?”.

    I’m sad to say that the empirical evidence says no. Currently 29 states plus the District of Columbia require prospective high school teachers to take the Praxis II test (10061) in mathematics content knowledge. In 1999 The Education Trust did an analysis of teacher licensing exams (available at their website called “Not Good Enough …”).

    For the 10061 they found: this test contained only 8 of 50 questions that were at college level. The remaining 42 of 50 were high school level math. 70% of the questions were classified as “simple” – a term they define in the report.

    How did teacher candidates do on this exam? - terrible. The report largely bemoans the low passing scores that states set It misses the larger point - schockingly few teachers do well, i.e. get the equivalent of an “A” or “B” on this exam. The report also doesn’t adjust the raw scores for the fact that the Praxis does not penalize for wrong answers. The raw scores should be adjusted downward, a significant adjustment when the scores are this low.

    When the report was issued 2 out of 13 states had (unadjusted) passing score below 50%, a point that was specifically mentioned in a subsequent report by the Teaching Commission. In fact, when properly corrected, 12 of the 13 states set passing scores below 50%.

    That was 1999, have things post-NCLB gotten better? In 2004, I reviewed the state data and one could see an improvement at the bottom end. The percentages of teachers with very low scores had decreased. However, 27 of 30 states that now use the tests still set (adjusted) passing scores below 50%, eight of the 30 have passing scores below 30%!! These teachers are then said to be “highly-qualified”.

    That’s bad enough, but if you do an analysis of the high scorers you find that fewer than 2% of the candidates would earn an “A” on this exam, another 3% would get a “B”, 6% get a “C”. Almost 90% would get a “D” or an “F”.

    So Susan I’m afraid too many of our teachers do not know math, and these results don’t take into account out-of-field teachers.

    Why is it we conduct TIMMS and OECD-PISA studies comparing students internationally, but we don’t compare teachers? We might learn something important (like Liping Ma’s work)

    Decaf asked: “if the schools and teachers were as universally bad as most news media say, then how do we explain the successful students?” The answer is that the successful students succeed despite the bad teaching. (They are also more likely to get the better teachers.) Some students can actually read and largely understand without the need for much teacher intervention. Think of them as self-home-schooled.

    The college I wanted to attend (The Cooper Union) would only accept the BC exam for advanced placement credit. My high school only offered the AB course. I studied the extra material, took the BC exam, and got a 5. Which teacher should I credit for my success?

    - Walt — 7/30/2005 @ 6:27 pm

  61. Doing well on the Praxis 2 test requires a very solid grasp of high school math through pre-calculus. Most of the questions are “C” level high school problems that have to be worked very quickly. The college level stuff is relatively simple. Questions about very simple finite groups, the definition of equivalence relation, etc. Because of the speed factor, the test is a relatively good measure of content knowledge required to teach through pre-calculus. I have no opinion about the cut-offs. I will say that many people hoping to teach high school math have a great deal of difficulty with this test.

    Walt,

    Congratulations. Maybe you would like to teach. Half of your students have not mastered the prerequisites for your course. Most of your students will not do the work required to master the material. Some of your students take vacations with their parents during the school year. If you do not allow them to make up the work, their parents call them in sick. The administration will not let you flunk a student without a time-consuming bureaucratic process. Managing unruly students is your problem - don’t expect any help from the assistant principal - don’t expect students to manage themselves. And don’t forget, your students all have to pass the proficiency test which covers material in the middle school curriculum from two years ago. One more thing, you had better not give the daughter of the PTA President a “B” unless you can prove that’s all she earned.

    - charles R. Williams — 7/31/2005 @ 3:11 pm

  62. Charles you’ll get no debate from me that there is more to mathematics teaching than knowing math. But it would seem to me that knowing math is a necessary, if not sufficient, condition for being an effective teacher. The Praxis II results demonstrate rather conclusively that far too many teachers are demonstrably lacking in this knowledge.

    The extent of the problem is well hidden from public view and I doubt that many policy makers are aware of it. How can Arkansas possibly certify someone with a 116 on their Praxis II? Do they hold their noses and reason that they at least they got 40% of the questions right? Do they realize that in reality they only knew the answers to 20% and got one fourth of the remaining 80% right by guessing? A parent in Arkansas will be told that their kid’s teacher is “highly-qualified” by virtue of “passing a rigorous test of mathematics content knowledge.” They won’t be told that they scored far lower than any mathematically competent high school graduate. If the public ever finds out I think it will be considered a scandal.

    - Walt — 7/31/2005 @ 6:07 pm

  63. The listed prerequisites for college math are all directly tested with the SAT or ACT college admissions tests. Calculators won’t help–in fact, taking the SAT without a calculator won’t have a noticeable impact on most scores.

    Anyone who denies this fact is either unfamiliar with the tests in question or speaking with some sort of ideological bias.

    Since these tests do provide insight into how high school students master these concepts, it’s quite easy to determine how well high school students are prepared for college math. Just look at the median scores for the SAT and ACT. Remember that the lower end of the spectrum is somewhat distorted by school districts that require their students to take the test whether they are college bound or not (at least one midwestern state requires all its students to take the ACT in order to graduate).

    I won’t bore you with the details of statistical evidence demonstrating that no one posting here has the foggiest idea what they’re talking about. You can look up the data yourself.

    Ironically, all you folks spouting off about the value of math seem perfectly content to back up your assertions with useless anecdata.

    There’s no question that American schools do a dismal job of instructing poor black and Hispanic students. But the middle is doing as well as it’s ever done, and the top tier students are doing far more difficult work now than at any time in the past 50 years.

    - Cal Lanier — 7/31/2005 @ 8:16 pm

  64. L. Ashley Higgins wrote:

    If you learn the equation for the area of a circle in one week and never do it again until surprised two years later by the need to remember the equation to answer a test question, you are likely to miss the answer. Why?

    I don’t believe anyone who has once understood the formula for the area of a circle will forget it. You know it’s area is proportional to the square of the linear dimension, because that’s what area is. If you haven’t understood area to begin with, you’re not going to understand the area of a circle even if you memorize the formula.

    If you have understood area, and you have understood pi, you will know what the area of a circle must be. And anyway, how could anyone ever forget “pi are square?”

    our minds are not like a Windows program

    As I’ve aged, my mind has gotten more and more like a Windows program: unexplained system hangs, lock-ups, and random data loss. I hope the rest of you have better luck.

    “You learn these things because they provide mental discipline.” Well, yes. . . . But that is precisely what they used to say about learning Latin.

    They were right.

  65. I don’t believe anyone who has once understood the formula for the area of a circle will forget it.

    Especially if you got there by integrating rdθ .

    - Whatever — 8/1/2005 @ 3:52 pm

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