Tall, Dark, and Mysterious


This is why my little college-math-ed blog has so many readers:

File under: Those Who Can't, Home And Native Land, I Read The News Today, Oh Boy. Posted by Moebius Stripper at 4:05 pm.

Because I am Everycollegeinstructor.

In U.S. report released this month, 40 per cent of professors who were surveyed said that most of the students they teach lack the basic skills for university-level work. Further, the survey conducted by the Higher Education Research Institute at the University of California at Los Angeles found that 56 per cent cited working with unprepared students as a source of stress.

Count me among both groups.

The rest of the article is essentially a quantitative, snark-free summary of everything I’ve written about teaching over the past year. Incoming university students, reports the Globe, are woefully unprepared for the demands of university; it’s hard to know how to deal with them; their high school grades mean nothing. Nothing new here, but I’m glad this issue is getting national attention.

Unfortunately, I think the remedies described in the article - more remedial classes! extra help for students who lack basic skills! diagnostic tests for students whose math marks are below 70% or whose English marks are below 80%! - are remarkably short-sighted, and contribute to the unfortunate trend of students paying universities to learn what they used to be able to learn in high school, for free. The main problem, as I see it, is an increasingly incoherent high school curriculum that is quickly diverging from the goals of a university education. And this problem won’t be solved until high school and university educators start talking to one another.

I’ve got a lot to say about this piece, and I’m finding that my thoughts are all over the place, so bear with me. Or don’t, I guess.

First, a quote from Ann Barrett, managing director of the University of Waterloo’s English language proficiency program, that dovetails with the experience of every single college calculus instructor who’s ever taught students who took calculus in high school:

“I have seen students present high school English grades in the 90s, who have not passed our simple English test.”

And the proposed reasons for this?

Some officials blame grade inflation at the high school level. Others say that in this primarily visual world, there’s little focus on the written word. And one professor points to the high school curriculum being so jam-packed with content that teachers have no time to instruct on the basic skills.

My thoughts on these, respectively, are: kind of, but that’s beside the point; give me a break; and ok, now we’re getting somewhere.

Let’s start with grade inflation, becuase it’s the most frequently-cited cause for students obtaining A’s in high school and flunking out of college. I stand by what I wrote on the subject back in January, but I think that the A-students-flunking issue is a lot more complicated than that. If grade inflation were the main culprit, then we could say that a student who gets an A has what may once have been considered C-level understanding of the material; that is, an A in 2005 is equivalent to a C in (say) 1995.

I disagree. My A-minus student does not have a C-level understanding of the grade twelve course that I took a decade ago, the one that prepared me reasonably well for my university math classes. He doesn’t even have a D-level understanding of such material. To say that an A-minus means anything in terms of a student’s understanding of the math they need to succeed in university is to say that there’s any correlation whatsoever between college level math and grade twelve math as it’s taught in BC. And there isn’t.

My student’s A-minus is a in fact pretty accurate reflection of his knowledge. My student does indeed have an A-minus grasp of the material taught in grade twelve math in BC. My student has acquired A-minus-level proficiency at storing formulas in his fucking graphing calculator and memorizing the solutions to homework problems so that he can recall them when he faces the test. He’s quite good at all that, really. It’s just that this proficiency would help him not one whit if he were to take a university-level math class, taught by professors who naïvely expect their A-minus students to be minimally numerate, not to mention vaguely proficient in reasoning mathematically.

Reducing this issue to grade inflation suggests that the problem lies in the evaluation of students, not in the choice or presentation of material. Absolute mastery of BC’s garbage grade 12 math curriculum doesn’t prepare students for university, because BC’s garbage grade 12 math curriculum is virtually disconnected from university. My colleagues and I have griped amongst ourselves about this, but as far as I can tell, there is no communication between high school curriculum developers and university educators. Tweaking grades won’t fix that.

On to the next idea - we live in a visual world, with little emphasis on the written word, so no wonder Johnny can’t read - am I missing something here? Did our world become significantly more visual in the last decade - a time during which universities have reported tremendous increase in unprepared students? The high school texts I’ve seen are jam-packed with the written word.

What I do see is this: I see students calling me over to their desks to ask about a word problem, and half the time me reading the word problem aloud to them is enough to answer their question. I see students skimming over paragraphs of text (not that I blame them) and then asking me what they really needed to read in order to solve the problem. I seldom see any indication that students are reading their textbooks beyond skimming over the examples so that they can match them to the homework questions. I’ve lost track of the number of students I’ve tutored, or fielded during office hours, who did not avail themselves of the indices of their textbooks. The reason they couldn’t show that two events were mutually exclusive was because they didn’t know what “mutually exclusive” meant, nor did they think to look it up.

When I was in high school, my English teachers routinely gave marks for producing drafts of essays. Producing the draft was worth half marks; the rest of our marks came from the quality of the actual essay. An incoherent essay could easily earn a B if the writer produced a draft. When I was in grade twelve, we had to submit one or more essays every week. There was plenty of emphasis on the written word; our ability to use it well, however, was virtually irrelevant.

Things have gotten worse in my home province, according to a former camper of mine. This camper was a brilliant math and science student; by his own account, he was “average” in English - and he wanted to improve. But he was having trouble doing so, because he was never assigned essays as homework. A few years earlier, he told me, teachers were reporting a rise of internet plagiarism. The school board’s solution: stop assigning essays for homework. In 2002, the only essay-writing experience that high school English students had, consisted of sitting in class for eighty minutes or so and producing an unedited, unresearched paper. It’s not hard to imagine a student who excels at writing those sorts of papers, flunking out of a class that requires long, researched papers.

There’s plenty of emphasis on the written word. There’s virtually none on developing the skills required to use it effectively.

Moving along - I am a lot more sympathetic to the third proposed explanation for the increase in unprepared university students: the emphasis on content over skills. Erin O’Connor, from whom I pilfered the original link, puts it well:

What [the article strongly implies] is that the problem stems in no small part from an ideology of progressive education that is famously hostile to skills acquisition (which requires such child-stifling practices as memorization, drill, repetition, and so on).

This certainly rings true in math, where I labour endlessly to disabuse my students of the notion that if only they memorized more formulas, more examples, they’d be doing a lot better in my class. The idea that there is a smallish set of basic skills that, solidly understood and correctly applied, will carry them through more difficult work, is alien to them. Pointing out that they can use material in Chapter n-k to solve a question in Chapter n risks an uprising. (True story: the precalculus 2 prof last year had a student in his office ask how to find a hyperbola’s asymptote. The prof reminded the student how to find equations of straight lines, and was met with a blank stare. “We did that last term,” she explained earnestly. “You didn’t show us how to do it this term.”) Last April, I talked to my then-department head to suggest completely reworking the curriculum for the terrible precalculus class. He was more than receptive, and took notes as I ranted. One idea that came up: teaching half the content, but taking time to make sure that students had a solid grasp on everything that was taught. It interests me, thought it doesn’t surprise me, that Erin and the English professor quoted in the article have come to similar conclusions about the courses with which they have experience: those courses too display an emphasis on content to the exclusion of skills that can be more broadly applied.

High school curricula are disjointed. We get a topic here, an application there - and we get nothing to tie them together. There’s no overarching theme for any course, no concept to unify the incredible mass of content. Students are understandably hard-pressed to recall any skills they learned in high school. And I can’t blame them for wondering, on occasion, “what’s the point of all this stuff?” I’m not even sure the people who designed their courses know.

At the end of the day, we’re left with two facts that are increasingly troubling, and increasingly at odds with one another:

1. High school students are discouraged from pursuing post-secondary options other than university; but

2. A high school education does not prepare one for university.

The first of these is seldom challenged among high school teachers and guidance counselors; the second is addressed at the university level alone. Unless high school and university educators start working together to figure out what they’re trying to accomplish, and how best to accomplish it, we’re still going to have unprepared students scrambling when they enter university, and we’re still going to have short-staffed universities rushing to endow them with the skills they should have acquired in high school. That’s not education; that’s damage control.


  1. I couldn’t agree more. A problem is that this “jamming every concept under the sun” into a course is not restricted to the high schools. Several different folks at my school teach freshman/sophomore courses in which they attempt to cover evrey conceivable topic “so that the student will have seen it” (an actual quote) and the argument goes, be better able to actually understand it the next it they see it. I refer you to your comments about the asymptotes in your post ;)

    Being that our school is not Harvard or MIT, I try to cover far less material, focusing on basic skills and problem solving and try to show consistent themes (building up one’s mathematical ability) through the course. I have no evidence to prove that my method is better (other than anecdotal evidence showing that Professor X’s students learned nothing from the alternate approach), but I feel at least 1 student per term will learn something real from me.

    - William — 9/28/2005 @ 5:20 pm

  2. I remember that the first math course that I honestly enjoyed (I had done well in them before, but didn’t really care too much) was our pre-calculus class in high school - because for the first time, there was a point to the material. Instead of learning one thing, then moving onto something completely unrelated, we used what we had just learned to build up to the next thing. We learned limits so that we could learn about continuity; we used both of these (well, sort of) so that we could talk about derivatives. It was great - for the first time ever, Math had a purpose.

    And how here I am, studying to get my MSc in Mathematics. I can honestly say that my desire to continue studying math stems from that class, and this sense of connectedness that suddenly showed up.

    - Simon Rose — 9/28/2005 @ 5:39 pm

  3. No kidding, William. And the “jamming” is hardly limited to high school. As part of a curriculum development project for my daughter’s charter school, I reviewed the state mathematics core curriculum. They are supposed to be teaching probability in Kindergarten (you know, before a kid can even grasp that it’s not likely there are real monsters under their bed) and they’ve got all kinds of algebra crap crammed in as early as first grade, with kids using cute little shapes instead of variables; but for some odd reason they can’t seem to find the time to actually teach the kids how to add, subtract, multiply, and divide. Go figure!

    - Wacky Hermit — 9/28/2005 @ 7:12 pm

  4. >>What [the article strongly implies] is that the problem
    >>stems in no small part from an ideology of progressive
    >>education that is famously hostile to skills acquisition
    >>(which requires such child-stifling practices as memorization,
    >> drill, repetition, and so on).

    >This certainly rings true in math, where I labour endlessly
    >to disabuse my students of the notion that if only they memorized
    >more formulas, more examples, they’d be doing a lot better in my

    I’m not following you. It rings true that anti-memorization “progressive education” is a cause of unpreparedness, expressed by your students’ frustratng fondness for memorization?

    - Susan — 9/28/2005 @ 7:32 pm

  5. Susan - oh, good point. Gah, I do need to proofread. Let me attempt to explain/backpedal: The hostility to skills acquisition in general - not memorization in particular - was what rang true for me. Many of my students seem never to have memorized the very basics - multiplication tables, and such - and they didn’t drill nearly enough on the basic skills that should by now be natural to them. Consequently, these students tend to see math as a disjoint collection of facts that they need to memorize. Had they been exposed to less content, but required to learn it more deeply, (and, yes, had they been required to memorize a small collection of facts) they wouldn’t be seeing every example as a topic unto itself. They’d have the tools to attack new problems based on the small but solid roster of skills they’d acquired. Also - my students are not fond of memorization by any means. They just don’t see any alternative to memorizing formulas and examples - that’s the only way they know how to “learn” math. Memorization is the most primitive conscious use of the brain, and my weaker students never got past that stage.

    Wacky Hermit - probability in kindergarten? Do elaborate! Morbidly curious minds want to know. However, I have long supported algebra in first grade - though not before students learn how to add and subtract actual numbers. I just think that there’s something terribly wrong when first graders understand that two dogs plus three dogs equals five dogs, and then six years later they’re completely bewildered to see that 2d+3d=5d. And then, six years after that, they still don’t get that you can collect the d’s.

    - Moebius Stripper — 9/28/2005 @ 7:46 pm

  6. One thing I notice (particularly in perusing your exam
    horror stories
    ) is that many students have gotten to university without picking up something I would consider a fairly basic ability: critical self-evaluation. The ability to look at your paper, see that you’ve written “-2 = 7″, and get beyond “but I thought that was how you’re supposed to do this kind of problem” to be able to think: “But -2 can’t possibly equal 7, I must be doing something wrong.”

    I think it’s partly a consequence of the memorize-the-formula approach that many students aren’t looking at their work in a practical “Does what I’m doing even make sense?” way. Of course, this is rashly assuming that any of them even re-read their answer once they finish writing it.

    - Geoff — 9/28/2005 @ 9:57 pm

  7. And if you really want to have cold sweats in bed at night, all it takes is to dwell on the fact that these pathetic examples of “progressive” education are the ones who made it past the college-entrance hurdles….

    I can see what colleges and universities could do about this; they could rate high schools based on the abilities of the students who come through their doors.  In the extreme cases, they could refuse to accept students from schools which produce remedial-quality graduates.  That would tend to get the attention of parents.

    Unfortunately, ElHi teaching is going to stink for quite some time because the teachers stink.  They’re products of a society which pays most other skilled occupations better (and siphons off most of the smartest) and subjects them to dehumanizing working conditions.  Their general skill requirements for graduation are less than should probably be required for entrance to their programs.  They can’t teach grammar, writing, or math because they can’t tell a quadratic from a quintile, an adverb from an adjective, and wouldn’t know a dangling participle if it bit them on the ass.

    People (including students) have contempt for these teachers because many of them are contemptible.  Getting the current crop of losers to actually teach the skills required and assign grades reflecting reality means having them instruct in things that can barely do themselves.  Rotsa ruck!

    - Engineer-Poet — 9/28/2005 @ 11:24 pm

  8. Well clearly no one uses basic skills in life, so they aren’t worth it. However since no one in real life actually uses graphing calculators either I conclude we should start teaching kids MATLAB and technical programming in 8th grade since technology is important.

    - Mike — 9/29/2005 @ 12:52 am

  9. I wonder whether the influence of our culture of instant gratification might not be a strong influence. Students don’t want to put effort into acquiring knowledge and skills, they just want to be told what to do and press a few buttons on a calculator.

    I hasten to add that not all students are like that, I just think it’s a growing trend amongst students in general.

    - Sharon — 9/29/2005 @ 2:20 am

  10. Well, E-P, I wouldn’t agree with your characterization of teachers, especially as I’m related to a whole bunch of them. Teachers aren’t necessarily treated like crap — it depends on your location… and if you add in stuff like pension and health benefits, the total compensation of teachers (even forgetting that they could take on another job in the summer, if they wanted) is pretty decent. So let’s not start the “throw money at it” issue — private school teachers are generally =worse= paid than public school teachers, and do at least as well as their public counterparts. There may be a couple reasons private schools can get better teachers: they can differentially pay by subject matter, they don’t require education degrees or teaching certificates, and they can throw out disruptive students and improve the school environment.

    Once upon a time, the teachers, even at the elementary school level, were pretty knowledgeable in their subject matter…such as when I was in school in the 80s. Most of those teachers came of age when there were 3 possible careers for a woman: teacher, nurse, secretary (being a nun isn’t a career… it’s a calling — and many of those nuns were teachers and nurses). Some of the men I had as teachers in high school and middle school were laid-off engineers (first round of downsizing) or guys who took early retirement from a different career.

    One of the biggest things that school systems can do to hire a better level of teachers is to change what the barrier to being hired is: need to pass a test of basic skills (a high school exit exam(s) would be a good start, and then exams in one’s subject matter if middle school/high school teacher. An AP exam would be good there.), and removal of teacher certification requirements.

    Other ideas: No raises for getting higher degrees in education. No tenure (long-term contracts with performance requirements ok). Differential pay for positions, as needed (this would likely be volatile depending on the economic markets — and think, I hear IT/engineers bitching all the time about outsourcing… I bet lots of those people could teach useful high school subjects. One of my more interesting math teachers had been laid off from Black&Decker, I believe…. of course, there would be smoothing once people teach for a while, but entry-level pay should be dependent on the job market for that paarticular knowledge.)

    - meep — 9/29/2005 @ 3:34 am

  11. I =somewhat= like E-P’s idea of blacklisting high schools, but I think the best bet all around is to rely =more= on standardized tests. When I went to NC State (an engineering-centered university), I had to take the regular SAT (SAT-I to you whippersnappers) and three achievement tests (currently SAT-II tests). This was for admissions purposes.

    I think that if the universities told students that they would be throwing out their high school report cards completely, and looking only at the results from these normed, standardized tests…. that would =really= get the attention of the kids and parents.

    - meep — 9/29/2005 @ 3:44 am

  12. The problem with relying on standardized tests is the typical one: that high schools get into a “standardized test arms race” with one another, and focus exclusively on the material and question style of what will be on the tests.

    - saforrest — 9/29/2005 @ 6:48 am

  13. “The idea that there is a smallish set of basic skills that, solidly understood and correctly applied, will carry them through more difficult work, is alien to them.”

    This concept has been proven in many cog sci studies. There has been lots of discussion on it at Kitchen Table Math. And you’ll definitely want to read this post.

    - cog sci — 9/29/2005 @ 8:10 am

  14. Mmmm, sure. I took the AP exams in calculus, comp sci, chemistry, and physics. We did some practice exams, but we actually had to know material to do well on the exams.

    Likewise, I’m currently taking actuarial exams, the first four of which are multiple choice, with calculators, statistical charts, and formulas of moment-generating functions to boot (perhaps that has changed). I promise you that you cannot “game” the actuarial exams. If you don’t know life contingencies, you won’t pass the third exam.

    I have no problem with “teaching to the test” if the test is substantive. “Test tricks” can’t get you anywhere with some exams, even multiple-choice ones.

    - meep — 9/29/2005 @ 9:36 am

  15. Yeah, MS, probability in Kindergarten. Look for yourself:
    Utah Kindergarten Math Core

    Scroll down to Standard V, Objective 2. It seems pretty reasonable until you remember that these are kids who can barely tell the difference between fantasy and reality– and they’re supposed to comment on the likelihood of events and whether they’re possible or not possible. And in first grade, they’re required to rank events as more or less likely. In practice, though, Kindergarteners and first graders are given four-color spinners because they’re “cool probability manipulatives”.

    As for algebra, I’m in favor of teaching them algebra. I would start, however, with making sure they understand the commutative, associative, and distributive laws, instead of making them fill in number sentences with cute little shapes. These are given extremely short shrift in every math textbook and curriculum I’ve seen, mostly because teachers don’t grasp how important these are to algebra (which is, at its most basic level, arithmetic with numbers you don’t know). I can’t tell you how many of my students are just shocked, shocked to learn that there are actually laws that numbers follow– laws that don’t tell you how to move the numbers around the page, but express actual properties of numbers– and that all their algorithms are just shortcuts to do faster computations with those numbers.

    The thing that pisses me off the most about some math curricula, though, is an extreme overemphasis on students developing “number sense,” to the exclusion of actual math activities. There is, of course, such a thing as “number sense,” but these curricula are written as if they expect students to get it from being bitten by a radioactive number or something. Certainly they’re not supposed to get it by, you know, actually doing enough problems that they might get some kind of idea of what the answer ought to be; they’re supposed to get it by pulling guesses of the answer out of their butts, on the presumption that the answer is a single-digit integer.

    - Wacky Hermit — 9/29/2005 @ 12:41 pm

  16. Cog sci - thanks for the link. I’d actually wanted to say something about the “confessions of an engineering washout” article - I said some of it here, and I don’t know if I’ll get a chance to say more.

    saforrest and meep - I’d like to see the issue of standardized tests evolve from should-we-or-shouldn’t-we to evaluating the actual tests. I’d rather write the tests for the kid I tutor than have him write the standardized ones, because the standardized tests produced by the school board BLOW GOATS. Gaming them is child’s play, which is why my kid, who just the other day asked me what mark he’d need on his next test to get a B, has an A-minus in grade 12 math right now.

    Wacky Hermit - good lord, that’s ambitious. And I’d be a lot more open to the “developing number sense” objective if there were any evidence that bypassing the basics were in any way conducive to developing it. What we end up with instead are kids who don’t know their times tables, and don’t blink when they find out that their five year, $1000 investment at 5% compounded monthly will net them $250,000.

    Speaking of which -

    Geoff - it’s not just a lack of critical self-evaluation (though that’s certainly part of it). The reason my students can write things like “2=7″ is because they don’t know what the equals sign means. They are wholly incapable of relating real life to mathematics. They understand that “equal rights for women” means “women have the same rights as men”, but they don’t know that x+7=2x-5 means that “x+7 is the same number as 2x-5.” Honestly: I’ve explained the equals sign, in so many words, to students who have graduated high school. They’re always so amazed. That’s what equals means?

    - Moebius Stripper — 9/29/2005 @ 1:49 pm

  17. i agree with MS that one of the big problems is relating math to the real world. not in the “trains leaving at 50 miles per hour from San Francisco” sense, but in the sense that math (and its symbols) are in place to mirror the way we think about patterns in the world.

    i’m thinking of a number and then i add 5 to it, and i end up with 20. you automatically start thinking about what number you can add 5 to and get 20. maybe you explicitly subtract. even poor math students can figure out some way to come with “your number was 15″. but what they don’t get is that:

    x + 5 = 20
    - 5 -5

    x = 15

    isn’t just following some rule with x’s and =’s. it’s the math-language way to write what they were thinking already.

    that’s how it starts…with that kind of disconnection. then once they already don’t understand chapter n (and relate it to some way they can already think), they have no choice but to just memorize the rules in chapter n+1.

    i just taught my (admittedly very bright) 8th and 9th grade students about the formal properties of relations. i’m about to write a blog post about it if anyone is interested. the upshot was to try to point out how 7=7 is not what the reflexive property is all about using relations other than “equals”; relations that they can connect to their lives so that they see that the formal names are just the math-language way of describing what they were already thinking.

    math as a language…i’m more and more convinced that this is a good way to get around a lot of these problems.

    - polymath — 9/29/2005 @ 2:31 pm

  18. uhhhh, that spacing didn’t work out in my equation, but i think y’all know what i meant…subtracting 5 from both sides.

    - polymath — 9/29/2005 @ 2:33 pm

  19. I would be very nervous about a kindergarten teacher teaching my (purely hypothetical) child probability. Two of my math professors (married to each other) told me the story of how when their son was in kindergarten, the kids had a geometry lesson with a bunch of cardboard cutouts of different shapes all over the ground. The teacher told the kids to go stand on a rectangle. My professors’ kid was the only one standing on a square. The teacher told him he was wrong. Even when both Ph.D.-holding matheamticians tried to explain to her why their son was correct, the teacher didn’t understand.

    If a teacher can’t understand the concept that if the definition of a rectangle is “four sided figure of all right angles” then a square qualifies, I’d be terrified of what s/he’d teach my kid about four-sided spinners. :)

    - KimJ — 9/29/2005 @ 5:55 pm

  20. Polymath - one of the things I loved about teaching statistics is that I could get away with a lot of questions that began, “interpret your answer” and “explain, in words…” I really felt confident that my tests were measuring genuine understanding. That class wasn’t the hardest I’ve ever taught, but you really had to know the material cold in order to do well in it. (Corollary: in this class, moreso than in others, I had no sympathy for students who tried to pull the “I understand the material, it’s just that I am not getting the marks I deserve” line.)

    KimJ - oh, good point. And that reminds me of another family legend: me, three years old, watching my grandmother draw shapes for me to name. She drew what she thought was a circle, but it obviously wasn’t a perfect one, and I was insisting, “Oval! Not circle, oval!” Except that I couldn’t pronounce the letter “v”, and my grandmother didn’t understand what I was trying to say, so she kept saying, “No, honey, it’s a circle,” and I was getting really agitated.

    - Moebius Stripper — 9/30/2005 @ 9:03 pm

  21. KimJ — I’ve been trying to teach my 2-yr-old that all squares are rectangles, but not vice-versa, and she doesn’t quite get it yet.

    At least she’s not saying “But infinity is infinity!” to me (which I heard the first time I tried Cantor’s diagonal argument on a bunch of gifted adolescents.)

    - meep — 10/1/2005 @ 10:38 am

  22. When I was teaching college psychology (stats was one of my courses) I had my share of unprepared students. I always reviewed a bit of algebra at the beginning of the stats course to get them ready for the formulas. They were generally annoyed that I made them solve stats problems by hand rather than only on the computer.

    But I also feel that I have to add to this discussion that in my son’s 3rd grade class (he’s now in 4th), they regularly took timed tests in which there were benchmarks they had to pass. They would have to do 100 problems in 5 minutes and get 95% right to pass. These were tests of addition, subtraction, and multiplication. We did a lot of practice with flashcards at home. So some schools are focusing on basic skills! His 4th grade teacher also tells me that they are focusing more heavily on writing skills now to prepare them for 5th grade. They have to write drafts of (very short) papers and then correct them to turn in a final version.

    My daughter in kindergarten will be using a phonetics-based learning-to-read program that is very drill-based. The teacher said it’s very boring, particularly for the teachers who have to read the scripts pretty much verbatim, but it works. No probability yet!

    - Psychmom — 10/6/2005 @ 8:13 am

  23. The thesis that what is required is skill development as opposed to memorization or an over abundance of content is to trivialize the problem. It turns out to be the case that over the past 3 decades, mathematical skills levels have not changed. 80% of students who leave school, for whatever reason, are mathephobic. This result is a symptom of the axiomatic approach to mathematical teaching that has gone on for the past 12 decades. What is not recognized is that much of the mathematics taught in elementary and high school makes no sense, not even to mathematicians if they would take the time out to ponder to any depth their mathematical utterances. A simple example will suffice: Students are taught that a circle divided into 4 pieces with 1 selected can be represented by 1/4 which is considered to be a rational number. Now from a mathematical point of view it is true that 1/4 + 1/4 = 1/2. But is this really true? Suppose that the second 1/4 represents a piece of a square instead of a circle? If you consider this a problem, just what would these two objects multiplied together look like? 1/16 of what? Now of course the mathematician will argue that you can’t add apples to oranges thus no problem. However, do numbers really come in different flavors? Any understanding of science requires the ability to add, subtract, multiply and divide apples and oranges.

    - AKnowNothing — 10/6/2005 @ 5:45 pm

  24. Psychmom - thanks for your comments. It’s especially refreshing to hear from social science college instructors who think that mathematical literacy is important. Last year, I found it really discouraging that apparently the social science profs - whose students were required to take my stats class - never used even a little of what I taught.

    AKnowNothing - Axiomatic approach to mathematics that has gone on for the past 12 decades? I don’t know where you live, but ’round these parts, elementary / high school math is anything but. (Not that that’s an entirely bad thing - there’s such a thing as too much formality.)

    And there are no different flavours of [real] numbers, but there are different things whose quantities are represented by numbers. And “times” means “groups of”, or “of”. So for the circles and squares case, 1/2 times 1/2 means “1/2 of 1/2 of a square (or circle).” Trust me, mathematicians have taken the time out to ponder this…we just didn’t find it necessary to dwell on it for terribly long.

    - Moebius Stripper — 10/6/2005 @ 10:13 pm

  25. Any understanding of science requires the ability to add, subtract, multiply and divide apples and oranges.

    Okay, I’ll bite, whats an apple times an orange?

    - djfatsostupid — 10/7/2005 @ 10:35 am

  26. “Okay, I’ll bite, whats an apple times an orange?”
    Simple - it’s one apple*orange.

    - dave — 10/7/2005 @ 11:29 am

  27. I don’t think Cantor’s diagonal argument or other arguments regarding
    cardinalities of infinite sets are intuitively obvious.

    - gregbo — 10/9/2005 @ 9:03 pm

  28. I’m with you there, gregbo, as I’ve actually tried presenting the Cantor diagonal argument to different groups of people. You definitely have to be prepared to understand it…. but most importantly, you’ve got to be ready to accept the conclusion.

    “But infinity is just infinity!”


    - meep — 10/10/2005 @ 1:16 am

  29. Maybe what’s needed is non-standardized testing, akin to the British A-Level or the IB. Non-standardized tests predict university grades slightly worse than high school grades as far as I know, but they help keep high schools on their toes.

    In another thread, people talked about teaching dot and cross products in high school. In Israel the level of math teaching is below average for a developed country, and yet every student needs to pass an exam that has questions about dot products in order to graduate high school. In Britain it’s not mandatory to take mathematics beyond grade 10, but if you do then you have to know how to use dot and cross products, how to find eigenvectors of matrices, and how to work with hyperbolic functions.

    - Alon Levy — 10/10/2005 @ 5:59 am

  30. Argghhh! Consider, please, these problems and issues as potential explanations for the disconnect the article raises:
    (1) More than ever before, we funnel children into college. Children who, in the past, would never have set foot on a campus. We spend billions each year making this possible thought loans, grants, etc. Is it any wonder that many kids entering college are unprepared?
    (2) Even many of the best high school students are disconnected from learning, and let’s not even bring up their parents. I’m convinced that most of my students are orphans as I see little or no evidence of any parentel involvement. And don’t even think about asking parents to hold their kids responsible for academic performance. Why, if they tried, their children might not “like” them anymore! They might not be their homies.
    (3) Students today, even the GT/AP kids, are non-readers. From this failing, all manner of failings spring (see #2 for the genesis of this problem). Every year, I have the unsettling experience of having GT kids tell me that they have only read–cover to cover–a single book in their 16-18 years. Some have not accomplished that.
    (4) Our school stills suffer to a ridiculous extent, from the disabilities imposed by the “self esteem” movement. In my school, no student may receive less than 50% on their report card, even if they do nothing at all. I’ve students I’ve seen only a few days in a semester due to being suspended, yet they still receive at least 50% credit. And while we preach educational excellence, our principals–responding to pressure from above-let us know that no student should fail.

    On one hand, I sympathize with my college teaching brethren–I’ve also taught college–but on the other, we must deal with reality. I must deal with kids coming into high school with far lower than high school level skills and take them as far as I can take them. So too must those exalted inhabitants of the lofty, ivy-covered academies.

    - Mike — 10/12/2005 @ 7:59 pm

  31. Teaching Carnival II is here!

    Doodles of a fifteenth-century student, University of Aberdeen Library (link from Derek Hughes on C18-L) Can one start a…

    - scribblingwoman — 10/15/2005 @ 10:52 am

  32. Why College Students aren’t ready for Math

    How much of the problem with disparity between expectations and reality is due to inadequate K-12 education, and how much is due to the simple fact that so many more people are going to college?

    - DrStat — 10/18/2005 @ 11:53 am

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