Thursday, May 18, 2006
On the one hand, it’s one of the foundational disciplines, and some level of numeracy is clearly necessary just to be a functionally-educated person in our society. I don’t just mean arithmetic, either; if you don’t have a basic grasp of percentages and fractions, say, much of everyday economics (20% off!) must be mysterious. A basic grasp of statistics is essential to any kind of business or management position, as well as an informed appreciation of baseball. (To the extent that I can do math quickly in my head, it was mostly because of an obsession with earned run averages in my youth. Yes, I was a nerd.) Concepts like “compound interest” or “statistical significance” or “exponential growth” require a decent grounding in math.
All of that granted, I haven’t had to find the cosine of anything outside of a math class. Although I got through calculus 1, I’d be hard pressed to tell you what it was about, what it was supposed to convey, or just why the hell I had to take it at all. (There was something about integrals, whatever those are, and something about derivatives, whatever those are. And I get credit for the class!) And I still think that foisting geometrical proofs on innocent children is child abuse.
I’ve heard math defended as the ultimate expression of logic, usually with the implication that people who aren’t good at math aren’t good at thinking. The argument strikes me as arrogant in the extreme. As a student, I remember intensely disliking the language of math. It seemed to be designed specifically to defeat understanding. (At some level, I’ve long suspected that the affinity of math/science geeks for hobbits and spaceships has to do with a taste for arcana. Either you have that taste or you don’t; intelligence has little to do with it.) At least with history or literature, I could picture what we were talking about. The stranger came to town, the countries went to war; got it. To this day, the concept of a negative exponent leaves me in a fetal position, quaking.
(I was that weird kid who actually preferred word problems to equations. At least with word problems, there was something to picture, no matter how banal. A train leaves Chicago...)
At my cc, and at most colleges I know, every student has a math requirement. And on the occasions when I’ve had access to the data, I couldn’t help but notice that math courses always have the highest attrition rates.
It isn’t just calculus, either. Sometime after I went to college, someone got the idea to repackage 9th grade math as something called “College Algebra,” which I still think is an oxymoron. College Algebra has high attrition. As much as I hated calc, I was fine with stats. But even statistics has high attrition. Hell, the remedial course in arithmetic – yes, arithmetic – has high attrition.
I don’t claim to understand that. Is it an American thing? Is it a generational thing? Is it a function of suspect pedagogy? I’m honestly stumped.
Wise and generous readers, I ask your input. Why does math stump so many more students than any other discipline?
P.S. I loved word problems.
I was lucky in that I had really terrific math teachers in high school. Too many teachers try to teach math by just having students do sample problems over and over again, which is a boring way to learn math. A good teacher shows you the pattern -- and then you can grasp the concept after one problem.
I do think the kind of analytical skills used in math are transferable to many other disciplines. When I am revising a poem -- moving words around, trying different ways to balance it -- the process feels much like solving a math problem.
I teach formal logic in the philosophy department. We fill 6 sections per semester because it fulfills the math requirement in the transfer system. Our attrition rate is high as well -- because it requires a high level of discipline to learn the concepts necessary to solve the problems.
I think the root of the problem is that society does not reward intellectual achievement, clear and systematic thinking and it does not require such thinking in order to be successful. Our students have very little motivation to do the hard work necessary to train their minds. No wonder they drop out of these courses.
I assume this idea is the bane of many teachers' existence.
However, I do think this actually does create a selection bias - in a math class, you pretty much immediately see whether you are on top of things. Whereas you can take an entire term of literature, think your insights are more than profound, and only later, when it is too late to quit the course, find out that, really, there *is* a wrong and a right way of thinking about it.
So in my opinion these so-called "soft" courses tend to have as many people who are stumped, they just don't know it yet...
Seemed like that when I was in college, anyway.
I assume the second sentence is given as evidence for the claim made in the first. I have a theory about this kind of thing, which goes something like this. (ahem) If you're interested in more than one thing, the skills you develop doing those things will merge together organically and come to seem (to you) like the same thing.
Prior to developing this theory, I was often (somewhat naively) baffled when encountering people with only partial overlap in interests and skills. If they're interested in math (e.g.), why aren't they also interested in (e.g.) precise use of language? Aren't these the same thing? (They are, in me; but not in everybody.)
The transferability of skills you perceive between mathematical and poetic work might be of this type -- more to do with your experience doing both, and less to do with their intrinsic similarity.
Because the feedback on their success (or lack of it) is completely objective and immediate. No illusions that last-minute cramming can overcome deficiencies.
You may not know integrals and derivatives today, but you must have been able to perform them once to have received credit for the course.
Another problem, of course, is lack of interest. I am teaching
Business Math so it's easy to provide relevant examples. It would be difficult to find cool examples with advanced trig, for example.
So as I sit, in my student persona, in the College Trig class I will need to know (not just understand) how I can use the knowledge of how to find a cosign value. If not, then I will learn the steps just long enough to pass the quiz/test. Anything longer will be lost. Am I lazy? Of course not. What I am is exigent…if the new material does not present itself to me in a way that I can integrate it into my larger life, then I will do what it takes to "learn" it short-term and move on. There are other factors in my life that need my attention: some "worthy," some not so much.
As educators, then, it follows that merely hoping that "logic," "discipline" or "modalities of thought" or any other such hallowed chestnuts of academia will derive from exposure to a discipline (like academic radiation) results in poor success: students will drop, fail or forget.
Answer: non-major presentation of the material in creative and demonstrably applicable lessons. Leave the arcane to the majors (which should include the hobbyist).
Mind you, this is coming from someone who went to a college which did not have any distribution requirements (only requirements for the major), and I now regret that a little bit. I probably should have taken some math.
If the math instructor is doing their job, students can't fake it in math like they can in so many other subjects.
Plus, in the words of Russell, "Many people would sooner die than think. In fact they do."
Fourth grade, I wept through problems with pennies, sixth grade got my last A ever in math, seventh had an algebra teacher with an intolerable lisp, ninth grade my trig teacher told me if I would just slow down and be careful I'd do better, 12th grade AP calculus where I never, ever got things fully correct.
Freshman year of college, had a teacher who filled up board after board with Greek and then yelled at us for not being interested, and then FINALLY my last quarter of my freshman year I had a teacher who was clear, organized, friendly, encouraged us to form study groups, and made infinite series interesting. I got an A that quarter!!!
I now have a PhD in a hard science, so that may tell you something about my ability to struggle through.
I'd like to take a really great math course again, just for fun, but the only one that exists at our university seems to be taught by my husband. And taking it from him would be a disaster! ;)
Logic is a branch of mathematics and
mathematics is necessary for precise thinking, so yes, if you don't understand mathematics, your thinking is limited. Words are both imprecise and inaccurate (their meanings aren't the same when read by different people), so the mathematical notation that you dislike is necessary.
Mathematical notation was devised to make thinking about mathematics much easier and it succeeded beyond our greatest hopes. Far more people can do algebra today than could before the relatively recent innovation of algebraic notation, and even non-intelligent machines can do it. However, the notation for each class is a new and different thing to learn for students, much like a foreign language though smaller in scope.
(I was that weird kid who actually preferred word problems to equations. At least with word problems, there was something to picture, no matter how banal. A train leaves Chicago...)
Precision is difficult, but there is one other essential type of thinking that mathematics teaches: abstract thinking. Students start with the abstract idea that the 3 in 3 apples is the same as 3 rocks or 3 people, then move onto the ability to deal with quantities without explicit numbers, denoting them by x in algebra, and eventually realize that algebra is a plural word and that you can solve problems by coming up with appropriate algebraic systems.
Abstraction is a powerful thinking tool. I think you'd agree that someone who couldn't abstract the concept of 3 from 3 apples and apply it to other groups of 3 would be limited in their thinking. The same type of expansion in your ability to think happens when generalizing from elementary arithmetic to basic algebra and from basic algebra to abstract algebra.
I don’t just mean arithmetic, either; if you don’t have a basic grasp of percentages and fractions
That said, I think that mathematics is taught very poorly in the US. For example, percents are fractions with a denominator of 100 and a slightly different way of writing them, despite being taught in a separate unit as a separate concept in many elementary schools. Likewise, fractions are basic arithmetic. If you understand quantity, you don't need to memorize any special rules for adding or multiplying fractions.
Part of the problem is that American textbooks are much larger and cover more topics than textbooks in other countries. Singapore covers 8-9 topics a year in K-12 math, while the US covers over 30 a year. However, students in Singapore learn that ratios, fractions, and percents are the same thing and work with these concepts together instead of artificially separating them.
The other problem is that the methods for learning math are different from the methods of learning many other subjects. Students attempt to learn math through memorization, which simply doesn't work. There are literally an infinite number of things that you could memorize in mathematics, but only a relatively few that are useful to memorize. We need to do a better job of convincing students to give up memorization and also do a better job of showing how to think mathematically and approach mathematical problems (I'd like to see Polya's How to Solve It become a core part of the math curriculum.)
Many arguments made against FL yesterday seem to apply here as well:
a) students won't become "fluent" in math in a semester or two, so why bother?
b) it probably won't apply to their major so we should let them take what they want
c) it will be an unnecessarily difficult course and likely bring GPAs down, along with morale, retention, etc.
d) that taking a semester or two of calculus (Right on DD! When did "college algebra" become college-level math?) in order to be exposed to a new way of thinking ignores the practical, service-oriented education desired by many studnets and instructors
Few seem to oppose the reason for including a math requirement (unlike the FL requirement). The idea of it is a good thing. The problem exists in implementation, at the curricular- and classroom-levels.
Here at Mediocre U., the chair of the Math dept got tired of the kind of jokes that DD related. He got tired of having to defend the first-year calc requirement, which ultimately turned inot a defense of his discipline. He got tired of the very high attrition and heated e-mail from students about their classes.
And so, he did something I've never heard of before. He reassigned all the TAs to courses after calculus and had every professor and post-doc take two sections of calculus. He sold it as a one-year experiment. (You can imagine the political firestorm that ensued.)
It worked. It was brilliant. Retention rates were at an all-time high. Drop-and-retakes were the lowest ever. A couple of fulls in that dept have even rededicated themselves to first-year calc and are (re)building a helluva program. And as those students are now filtering up into other majors, they are reporting great things about their calc experience.
The lesson learned: implementation. The requirement is a good one. It just needed to be addressed in a different way, with greater attention to the teaching, the pedagogical practices. Academic integrity doesn't need to be compromised to do this.
I think there's also an attitude in the US that math is for smart people; smart people are nerds; and the general teenager/college student would rather die than be labeled as a nerd.
Also, there's no encouragement for math. For example: There's a book called "Innumeracy" by J.A. Paulos; the title comes from the idea of illiteracy being applied to numbers and math. While parents are encouraged before their child's birth to read to the child to prevent illiteracy (and encourage love of reading & supposedly learning), there's no such attitude about math.
That may be the problem: you're not lazy enough. In algebra, you just need to memorize three things (linear equation, quadratic equation, and the binomial theorem), then it's all do the same thing to both sides of the equation. It's trying to memorize all the possible types of problems so you can solve them by applying pattern matching techniques is what makes mathematics hard.
I see the same lack of laziness in computer users. When you watch someone repeating a set of steps over and over again, it looks like the computer's programming the human instead of the other way around. I was watching a colleague renumber his references by hand throughout a paper for the tenth time this week (and who knows what time in the decades he's been writing) instead of taking a few minutes to learn a tool to automate it for him.
Unfortunately, laziness requires self-examination to realize how much effort the way you're doing things requires and to see how much easier it would be if you would take a moment to think instead of rushing into a task with your old methods.
I assume this idea is the bane of many teachers' existence.
But I think this is reinforced by grading schemes I saw as a double major in math and women's studies. I was an above-average college student, but not fantastically bright, and yet I could whip up a 10-page Lit or History paper that would form a good chunk of my grade in a single day and never get lower than a B+. But I could never blow off my math courses until the last few days before the midterm, because those professors would actually give bad grades.
I can understand how frustrating it must be for people in the arts and humanities to have people think that their discipline is all about spouting off one's opinion, no matter how unfounded; clearly, that's not the case, and I've read history scholarship that's a carefully rigorous as a math proof. But as far as coursework goes, it always seemed that as long as those opinions were coherent and spell-checked, no one got worse than a C.
That's a great point. (I'm the WS and math major from above.) It's funny, but whenever it comes up that my dad and I used to work on math workbooks when I was little (in the ages 3 to 6 range), the reaction is far from the "how adorable!" reaction that a child and parent reading a book gets. Instead, people assume he was some over-bearing, "Sports Dad" type whose goal was producing a math genius, not just spending time with his child on a stimulating activity. I assume that this stems from people's idea that math can't possibly be fun, and therefore subjecting a child to math is plain cruel.
As many others have said, maybe math classes have high attrition because a student can't charm his way out of failing an exam. It's right there: 35%. I have often been shocked by how many students who cannot write sentences that even approximate sense are passed through composition. This does no one, least of all the student, any good, and in my lit classes they are miserable because they never got the skills they needed to begin with. Maybe the problem is not high enough attrition in the humanities?
I was also going to mention Paulos and innumeracy, but someone beat me to it. He writes a column for ABC you might be interested in:
The problem with math (and organic chem) for me was this: you cannot memorize it and you cannot BS it. Sure, you can memorize the formulas. I was fine with that. But you can't memorize an exact problem and spit it back out for an exam. The numbers change. You can put an example of a derivative on the board and I understand it. Start adding fractions or exponents though, and I'm lost. Same with orgo: I can follow you through a synthesis example, I can have all the rxns memorized, but give me different pieces to stick together and...I got nothing. And you can't fake that. You get a zero.
It may be that I never learned to think critically that way. It may be that my brain just isn't good at that type of thinking. But I struggled through 2 semesters of calculus because it was required. I have little to no use for it now.
But I CAN calculate percentages and fractions in my head!
Whereas those e.g. history and literature classes have been so useful. Why was I, a math major, required to waste my time on that???
Other posters have made related comments, but I want to spell out: math is different because
1. One can have the last word. Once the proof is out there, noone's going to take that fact away.
2. There is no arguing from authority. Either you can prove things or you can't.
#2 leads to interesting differences between math conferences (which I, a math professor, attend) and others; nobody dresses to impress. It's a problem when teaching math to business majors; suddenly nobody can pay any attention unless the professor has a goddamned tie! and one must remember to account for this.
"I'm sorry to say that the subject I most disliked was mathematics. I have though about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it."
Malcolm X, quoted in The American Mathematical Monthly, vol. 102, no. 4, April 1995.
You mentioned not having to find the cosine of anything. What is your view on the purpose of a college education? Should students only be taught that which they reasonably might need to use later in life? If this is the case then colleges will need to offer hundreds of mini courses lest a student be exposed to something they don't need. How many students will it take for such a mini course to be offered? I am not necessarily opposed to this idea but is this what society really needs and/or wants? I think this is where we are headed but that is another topic.
I have tried a wide variety of teaching strategies in my classes and the result is always the same. Around 50% of the students fail or drop. I have talked with senior faculty about this and in our department the failure rate has been an increasing function over the years. This while their tests have gotten easier. (They claim their tests have gotten easier and I believe them because in my 5 years my tests have gotten easier.)
So why the increasing failure rate? I have come to the conclusion that the reasons are cultural. Americans, in general, do not pursue learning for the sake of knowledge. Learning mathematics is work and few are willing to work to learn the subject. Increasingly my students are of the opinion that merely attending class ought to be sufficient effort to learn the subject. That simply isn't the case.
My students are not willing to spend 20 - 30 minutes trying to figure out a single problem. If they can't solve it within 20 seconds then they give up and come to class the next day and have me do the problem for them. There is no patience on the students' part.
I also hear students complain that they don't know how this "stuff" is used. I do lots of examples of "practical" applications. These problems are called word problems and they are almost universally hated by the students. Their claims of wanting to know the relevance of the subject are hollow and are mostly a rationalization for not wanting to do the work required to learn he subject.
Finally, my question is why are the failure rates in other subject so low? I have seen the writing of students who pass composition courses and it is generally atrocious. I have talked with students who pass American History and don't know the decade that the Civil War was fought or other basic information. I believe that the failure rate in mathematics is not high enough. I routinely pass people who don't deserve to pass in the name of retention. I fear what the administration would say if I gave students what they really deserve.
In my state the Board of Regents wants to institute an exam that all students must pass in order to receive a college degree. This suggests to me that we faculty are passing too many students. The degrees we award are suspect because too many have gotten by without being held to rigorous academic standards.
Despite the legislature's desire not everyone is cut out for college. The more people we let in the higher the attrition rate is going to be.
Mathematics is the most cumulative field, requiring a mastery (not just familiarity) of all of your previous math classes. You only have a limited amount of working memory, and you need it all for the math you're learning now. You don't have to master British literature to study American literature, and you can even do well in chemistry as long as you're just familiar with physics, but you simply can't really understand calculus without mastering algebra, which in turn requires mastering arithmetic.
That's one of the reasons why students find mathematics so hard--they didn't practice their previous subjects until they mastered them. The attitude described by one poster above--learning the material for the exam then forgetting it--ensures that you will fail in mathematics. At some point in the sequence of math classes, there will simply be too much cumulative material to learn for the exam.
To summarize the thread so far, there are three reasons why mathematics is hard:
1. Mathematics is cumulative. You must master previous courses to succeed in your current one.
2. Mathematics is precise. It's obvious when you don't understand the subject, and it's harder to think precisely than to think fuzzily.
3. Mathematics is abstract. It's easier to think in terms of 3 rather than x, and it's easier to think in terms of x rather than in new algebraic systems.
Teasing aside, several of the posters raise useful points. Here are some observations from somebody who struggled with arithmetic and algebra in the common schools yet mastered enough mathematics (these three things are different) to on occasion publish economic theory in decent journals.
Maggie May is correct to suggest that teaching (at the elementary and secondary grades?) matters. Perhaps those teachers have degrees in pedagogy, with limited exposure to applied arithmetic and algebra, and the notions that cryptanalysis and insurance make use of these things just never occurred to them. There are other tricks one can use to help students. (In arithmetic, some drill is unavoidable. But there are tricks, e.g. "to multiply any number by 9 tack on a zero, then subtract the number;" I have a huge repertoire of these.) For many students the first stumbling block is algebra, which crosses the line from drill to logic. It might be as simple as allowing students to name variables, e.g. in the "Juan is twice Marias's age" problem the expression J = 2M might be more transparent than the canonical x = 2y.
App Crit has identified a pet peeve of mine, the academy's willingness to leave some of the most important courses (the core courses) to beginners and part-timers. (Imagine a Navy in which seamen who just completed their training at Great Lakes immediately become drill instructors.) The senior professors are more likely to be able to distinguish the profound but ill-posed question from the clueless one and perhaps seasoned enough to treat each tactfully.
Inside the Philosophy Factory hits on a problem that comes up in the transition from algebra to calculus. Much of algebra -- particularly the "finite math" version -- is still reasonably mechanical. Calculus crosses the line into logic and analysis, the guts of mathematics proper. There's a different set of skills in use there.
The earned run average figures in the development of at least one prominent mathematician. It's either John Allen Paulos or Henry Petroski (I read a lot of this stuff but can't remember as well as I once could where I saw it) called out a high school teacher of his who asserted an earned run average was bounded above by 27. (Not if you give up one run without any put-outs it isn't!) Sines and cosines ... when I did a hitch on race committee at Lake Geneva I'd estimate distances to the next mark at which the proper sightings to the start mark and the windward mark were satisfied. a*a + b*b - 2*a*b*cosu.
I think I struck a nerve with this one...
Yes, I remember dittos. When the teacher handed out fresh ones, we'd all breathe deeply. Kids today miss out on that.
I didn't want to go into critiques of math pedagogy, since I really don't know enough about it to say anything intelligent. My suspicion is that the standard approaches aren't working, but I don't know what might work better. The idea of having the senior faculty teach the intro course is certainly tempting.
The point about grading standards strikes me as the elephant in the room. It's true that the threshold for passing is much lower in the 'softer' disciplines, though I would certainly say that the best work in the 'softer' disciplines exemplifies intellectual rigor.
The line about cosines was really a left-handed acknowledgement that, were it not for boneheaded transfer policies based on the sanctity of academic silos, I'd gladly institute a much more laissez-faire approach to gen ed. But that's not the world we live in, and the financial obstacles to that are considerable.
Still, I'll freely admit a basic skepticism towards most gen ed requirements. Were it entirely up to me, and were funding not an issue, I'd follow something closer to the Brown University model.
As someone who has taught in a 'softer' discipline, I've often envied the finality of answers in math. The commenter who noted the difficulty in distinguishing between idiotic sloganeering and honest differences of opinion has a point, especially at the freshman level. (That becomes less true as you progress in the major.)
Maybe the real problem is too little attrition in the softer subjects. But I'd get drummed out of the profession for pushing that.
Yet I've used my advanced math all the time. I've used trig in building and laying out gardens. I've used trig in *knitting* for crying out loud. Last week I used an integral to satisfy my curiousity about my progress through a complicated knitting pattern. I use algebraic formulae regularly to figure out all kinds of things - what kind of milage I'm getting, how much of a difference it would make to drive out of my way to save 5c/gallon. I multiply fractions every time I pull my bread machine cookbook out and size portions for my family. How can you cook decently (well, bake anyway) without a firm understanding of fractions?
Take what maggiemay said about her approach to math, what jo(e) said about his, and what inside the philosophy factory said about his/hers. Then ask yourself which, if any, of those students the average college math course serves.
If you're serious about this, you might want to check out the National Center for Academic Transformation (http://www.thencat.org/). Several of their projects have centered around completely overhauling intro college math courses, with impressive results. It seems to take a lot of top-down effort to make it work, though.
This simply isn't true. I have calculus textbooks from one and two generations ago and both are substantially different from each other and from current texts. Before you even open the texts, you notice the tremendous increase in size (which is the true for all types of math texts) from generation to generation. Unfortunately, the quality of the contents as decreased as the quantity has increased. In instructional methods, we've had the new math and the new new math. If anything, there's been too much change in mathematics, not too little.
I have students who can set up problems but not solve the algebra. I have students who can do algebra but then have no idea what the number they derived means. And I have students who can do neither. But the students who can do both are relatively rare.
I totally agree.
I also second whoever said that there's too much focus on breadth, not enough on depth. We face this problem even in PhD programs in math; why should the first three weeks of a spring semester class that's the second in a series be spent reviewing the material from the first class in the series that ended a month ago? It wastes time, because we were never expected to master it in the first place. And a three-week review of an entire semester doesn't aid in mastery, either.
I can understand review in first-year grad classes, because incoming students have wildly varying background. Yes, math is cumulative, but the review can get a little excessive.
First, full disclosure: I am a mathematician by training, and a mathematics teacher by vocation. I am also an avid learner, who has learned thousands of things in his life without worrying about when exactly I'm going to use them.
A friend of mine who is a mathematician and a soccer coach makes the analogy this way: Your soccer coach doesn't tell you, "In the third game, early in the second half, you're going to want to take a left-footed banana kick from the corner..." They just don't know that. They try to teach you to do a good left-footed banana kick, so that, when the opportunity arises, it will be in your repertoire. The larger your repertoire, the harder it will be for the other team to stop you.
For my money, that's why you need to take a variety of courses. So you'll have a wide variety of ways of thinking, when you need them. I wouldn't say that people who aren't good at math "aren't good at thinking." They just have one fewer mode of thinking than people who are. And people who are lightning calculators but can't read or write are missing some important tools, too. They can do the left-footed banana kick, but they can't dribble.
In your case, Dean Dad, you haven't needed integrals in the career path that you took. Some other people in that class became physicists or engineers, and use integrals every day. Other may not use integrals much, if at all, but did learn some logical or abstract thinking that served them well somewhere along the line.
P.S. Is there such a thing as a left-footed banana kick, or did I make that up?
Well, let me see if I can't clue you in, Dean Dad.
Many people love viewing works of art. Even those who aren't art finatics, those for whom art has relatively little impact in their lives, are (for the most part) reasonably tolerant of art. Most people would find the notion of destroying all works of art to be odd at best and psychotic at worst.
Similarly, many people like history to some extent (hell, the entire discipline has a channel!). Even those who don't find that history is for them are, usually, tolerant of it. Again, those who wish to dismantle history as a discipline are (for the most part) viewed as kooks.
However, no other discipline seems to be as feared (nor anywhere near as misunderstood) as mathematics. In fact, it is so loathed and misunderstood that many educated people (even deans!) are proud to express their fear and loathing of mathematics in public venues.
Do you want to increase retention in mathematics courses? Defeat the prevailing meme of willful innumeracy that is all too common nowadays.
While it's a cute joke, it illustrates both the cumulative nature of the subject and the fact that mathematicians don't fundamentally think of math as being "for" something. (Not that you can't solve lots of interesting applied problems, but that's just a bonus).
It's math -- you learn basic algebra so you can do calculus -- you learn calculus so you can study analysis. You learn a subject in mathematics so you can learn more exciting math that you couldn't do before.
Asking a mathematician "what is this good for" is silly; obviously, it's good in and of itself, it doesn't have to be *for* anything.
People that think math needs to be good for something go off and become engineers and physicists.
I can't add much to the thread, but I wanted to raise a few more points.
I think the high attrition rates exhibited by mathematics classes say more about the students than it says about the subject of mathematics. When I was a college student, I was drawn to mathematics because I liked the challenge of it and I appreciated that there was so much less bullshit than I experienced in other classes. Apparently many students today, including many of my own, don't care to be challenged, and seem to prefer bullshit. My institution, for example, just established an AA degree in a practically worthless ethnic studies field. This was supposedly because of popular demand. I was far from being an excellent student as an undergrad, so maybe I'm some kind of freak, but I actually wanted to learn something when I was a student.
Aside from the confluence of immaturity and human nature, I'm not sure there is any one common cause of the high crash & burn rates one often observes in math courses that are taught at the college level (remedial or otherwise).
Certainly a significant contributing factor is the SOP in K-12, where many math teachers are not actually qualified to teach the subject. When I was a high school teacher, at one time or another more than half of my math department coworkers had no significant formal background in the subject. I'll never forget one coworker, who had a degree in music, admitting that he "never really did understand all those proofs" in his geometry book. My heart went out to the kids in his classes, yet this guy won awards from the district for some of the nonsense projects he had his geometry students doing-- popsicle stick and toothpick models-- and none of the parents seemed to even notice that their kids knew practically no geometry whatsoever. When these kids came to my precalculus class or my calculus class and were flunking out, all too often I was the heartless bastard, or the racist ogre, or whatever. As one of my then-administrators wisely observed, it is not easy to make chicken salad from chicken shit.
At the start of my last year as a high school teacher, one of the principals told us at the meeting before school started that more than 40% of our freshmen class was social promotions-- kids who had flunked their way through middle school and were being passed along to us anyway.
Some of those kids will grow up and later seek additional education at a local community college like the one I teach at, and I don't imagine that many of them will have come very far, academically, from their days as a flunkie in middle school and high school. We take 'em all, so long as they can pay (or borrow) the tuition money.
Another contributing factor to the poor performance of many students in college math classes has to do with their ability to ignore the obvious: since you can't fake it in math, you need to put in a reasonable number of hours studying. When you come to class, late, after missing the last week of class, and you realize that the lecture makes about as much sense as the lyrics of a Sugar Ray song, the proper response requires you to start doing some serious work. I tell my students "It's not like you're going to wake up one morning and all of a sudden find that you're a failure in algebra/precalculus/calculus/statistics/whatever. Success, like failure, happens day by day, hour by hour." At some point, a proper lecture presupposes the students are doing a certain amount of preparation for class:
Yet, when I incorporate a product like ALEKS in my remedial classes and build my grade so that up to 30% of the credit can be obtained by just working with ALEKS for two hours per week, the students piss and moan like my one-year old when I refuse to pick him up out of the playpen or something.
After one functional illiterate famously obtained a degree from the state university system here in California, they decided to require all graduates to pass a writing proficiency exam. Personally, I think they ought to have a quantitative literacy exam as well. I don't care what a person majors in, I can't see how it would be bad if their degree provided a reasonable assurance that they can at least communicate effectively in English and do a few moderately complicated computations.
I use trigonometric functions regularly, both personally and professionally. I have been looking at buying a mid to high end bicycle and thus looking at frame geometry. Yesterday I did several calculations to figure out seat and head tube angles. I also design lights for the theatre as part of my job. If I get the algebra wrong, circuits get overloaded. I use several aspects of trig to figure out exactly what lighting instrument I want where.
Math is cumulative in the big picture, but also on a class by class level as well. If you miss a class or two, it can be very difficult to catch up, and often you don’t have the necessary information to understand the current lecture. This contributes to the high attrition rates. It is easy to fall into the trap of “Oh, I‘ll only miss this one class. Crap, I don’t understand the homework, so I’ll look like a fool if I go to class….”
Math requires a great deal of practice. For three 75 minute class periods per week, I worked ten to twelve hours of homework and studying per week. That is a huge commitment of time to succeed. There was a direct connection between good grades and just doing the work. In any subject there are always some wunderkind who can excel without working the books, but they are very rare in math.
There are some amazing math teachers/professors out there. I’ve been lucky to have studied under several over the years. Many instructors get there degrees in mathematics, which trains them to solve math problems, not in mathematics pedagogy, like a MAT (Master of Arts in Teaching) in mathematics, which trains them to teach math. I have to attribute some of the problems to just bad instructors.
Math is also a foreign language. It has a way of viewing the world that can be useful in certain situations. Learning another language might help someone understand a different world-view, or at least understand there are other ways of seeing the world. The golden ratio is used in design applications all the time. It just looks right to the eye. The Fibonacci series describes many biological functions.
There are also some fairly well known anecdotal correlations between a love of math and other professions. Many math majors also are into music. Maybe it’s the way music can be described with math. The octave is a perfect doubling of frequency and a perfect fifth interval is a multiple of one-and-a-half of the frequency. Many hedge-fund managers use principles based on work with game theory.
Mathematics can define perfection, unlimited by the limitations of our physical world.There is no such thing as a perfect circle and yet math can describe one. I have heard that physics is applied mathematics, and engineering is applied physics. As a world of pure ideas, mathematics is very powerful, both in the ability to theorize, and in sheer beauty. I am awed by the primordial beauty of Mandelbrot fractals.
Two of my best friends are mathematicians. They both argue, correctly I think, that cutting edge mathematics is where most of the current advances in philosophy are happening.
Nobody will be able to excel at every subject. This is not a put-down on anybody; rather it is a tribute to the breadth of our knowledge. There is just too much information out there for any one person to be able to master. Sometimes students discover their bliss by taking GE courses. That’s how I found theatre.
Well, I think a lot of it is because of the way it's taught. Much mathematics is about learning how to understand naturally occurring patterns in the world. It's not a bunch of formulae and equations that were handed down to us by a diety on a mountain - it's ways of looking at things - finding patterns in the world and figuring out ways to describe those patterns. I was NEVER taught math this way and I doubt that most kids are.
As an example, I was taught that pi is 3.14.... While this is true, it wasn't until I was an adult that I got to do an activity that showed me why it is 3.14 and where that comes from. Most of the 3rd-8th grade teachers with whom I work haven't seen this before. There's NO chance their students have.
Too many people (including teachers and professors) think you 'know' math when you can use a summation sign, or flipt and multiply to fivide fractions, or spit back 25 formulas. That's the way math has been taught (also with WAY too many things covered in unconnected ways).
I think a major part of the reason for our students (K-20) failing or avoiding math is this bad approach to teaching that doesn't allow them to understand how math works. Much less where this stuff happens in everyday life.
Another part is absolutely the low expectations our culture puts on knowing math. Someone pointed out in a talk I was in a while ago that in the US no one would readily admit they can't read, but if they say 'I can't do math' everyone around them chuckles with them. For too many people, it's a funny thing to not be able to calculate tips, etc. It is acceptable not to succeed. And, that's a huge problem.
Musey_me (blogger isn't letting me log in for some weird reason)
"Maybe the real problem is too little attrition in the softer subjects. But I'd get drummed out of the profession for pushing that."
Well, shouldn't you be drummed out of the profession, dean dad? In fact, shouldn't you be forced to clean out your office and resign today?
What? Why aren't you laughing, dean dad? Not so funny when people are laughing about *you* being shitcanned, is it?
If I were to *ever* hear from a dean (or anyone else) at any kind of meeting the suggestion that a math department isn't needed, I'd file a complaint so quickly it'd make his/her head spin.
If instead you're looking for a theory of everything, I think that a lot of my students' problems started in K-8. If my students came to college skilled at working with fractions, life would be easier for all of us.
I am inclined to agree with the commenters who see mathematics as a formal logic course, although there's a significant amount of foreign-language competency built in there, too.
I'm a biologist (and Web weenie) by profession, but I'm what's called a "high-verbal"; I think naturally in words. I attended an engineering college, and the only way I survived was by actually translating my calculus work into word problems: if I could verbally describe the relationship symbolized by an equation, I was fine, since I can do logic. Unfortunately, this is an EXTREMELY inefficient and imprecise method. I've since improved in my ability to understand mathematics "natively", so to speak, but that was self-taught.
The math instructors I've had seem to have fallen into two types: the ones who don't understand math either, or the ones that think naturally in that way and who have trouble understanding people who don't. In my case, math that was taught as if it were an unusually logical foreign language would have been a great help.
Okay, I grant that maybe you no longer are ready to sit down with a piece of paper and solve for dy/dx when y=(some simple or nasty function of x).
I'm sure that the underlying concept of derivative, i.e. derivative is the rate of change, has stuck, though. We need more students in the CC; let's attract more to enroll:
enrollment rate = d(Student Population)/dt
Another example is attrition rate, which is part of -d(Student population)/dt. Graduation is the other part, and with that we'd be getting into differential equations:
d(# students)/dt = enrollment rate - graduation rate - attrition rate.
While this may seem like common sense, having the formal math background can potentially help you recognize when a quantitative argument does or doesn't make sense.
How about an application of sine, rather than cosine? Hiking at Half Dome in Yosemite National Park there were some parts of the trail where you could choose a different steepness (think angle and walking along the hypotenuse) to attain the same ultimate change in elevation (think opposite side of the triangle). My friends and I felt less impressed with our 8-mile hike when we were having lunch at the top and met a party who chose the much shorter 90 degree (vertical) ascent.
Yes, engineers and physicists do look for applications of math equations. I've found that teaching student engineers to convert a word problem into simple equations is a difficult task in itself, but that would be a different post.
Pluses: Excellent books available, resources such as computers, software available, excellent research about how to teach math available, funding is available if some body wants to pilot the program with latest teaching methods.
Minuses: Highest paying jobs that uses math such as programmers, data analyst,actuary, statistician, applied mathematician are going to foreigners. Unionize teaching profession and don't throw teachers who doesn't know how to teach. Think that teaching math to kids is an abuse(like this so called CC Dean) and torture.
Way to go fool Americans.
B thesis: math gets harder because it is so cumulative. You build upon what you previously build upon before. If a person forgets any component, they are lost.
Despite lots of graduate classes in statistics, I've never met a statistician (stastics is one kind of applied math) that could teach effectively at all.
I work with middle schoolers that struggle with math (whose futures concern me since our adult lives revolve around personal finance, data (political, etc), etc). I had to use everything I knew about differences in learning styles, Maslow's Hierarchy of Needs, personality types, etc to build connections with my students that would then allow me to get past the math resistance that they had developed as the result of years of being told (through scores or people's comments) that they couldn't do it. Some of my kids could construct a model from random materials or program a game from nothing but the symbols on a keyboard or create a budget for a business plan that would rival adult versions I had seen, but they were convinced that they couldn't do 'math' because they could never pass the standardized test. (Another concern: are the standardized tests that run our k-12 curriculum really designed well enough to guide math competency? Other countries refer to MathS (plural). Are we giving all forms of math literacy equal weight? But, I digress.)
Perhaps those in social sciences/humanities and seasoned educators (rather than TAs) are more likely to make the critical person-to-person relationships that carry students through the most dreaded material. When I taught college level Psychology courses, students would ask for my assistance with fundamental skills like math and writing. Why? I don’t know. Maybe they did so because I was willing to meet them where they were and had every confidence/hope that I could get them where they needed to be (even if that meant I preferred to make a referral to my "good friend in the tutoring lab" with a promise that “if it doesn’t work out, come back and see me”).
Everything in math has an application, even cosine. And, yes, some students work with the end in mind. Big thinkers tend to do this, as do those who place high value on the reward components of the motivation equation. Certainly it isn't hard to find something/anything where a professional would use the mathematics. If truly "nobody" uses it, ever, than I would agree that the content probably falls in the realm of "hobby" or "expert specialization" and it should be removed from the general curriculum. I, however, have yet to find that 'unused' concept in math. I use cosine in my carpentry.
So who are these struggling students? The students in my class were largely students who don’t have a lot of math exposure at home or limited life experience, barriers to learning (ex. language), and/or artistically inclined. I had to spin my lessons for these unique qualities in my students.
I already know that some will disregard my reply because I am a psychology major, because I am referring to struggling math learners in middle school, and/or because I am not being academic in my response by citing a lot of research; but I truly believe that the differences between adolescents and adults who are struggling with basic skills are not significantly different.
I am taking measures to qualify to teach math at the developmental/community college level. Even adult students can overcome math anxiety and learned helplessness. I see struggling math learners as victims of their own youthful choices and their beyond-their-control educational experiences. When I view them as victims looking for hope, I treat them very differently than when I'm upset that they haven't learned it yet.
These are just some musings from a different perspective.