Thus far the search for any readily available electronic source of "Avoiding Math Avoidance" has turned up empty. I will have to see if the local library has "Mathematics Tomorrow" or order it for $1.35. I did however manage to find an article by Peter Hilton in the MAA archives that was written at the same time and probably indicates some of what he said.
Math Anxiety: Some Suggested Causes and Cures: Part 1 (June 1980) I am including some snippets along with my own commentary. We agree on some things and not on others. His words are bounded with lines. Any typos are the result of the OCR. He has a point about athority I will address later. - ---------------------------------------- "The learning and the doing of mathematics should fit into the natural intellectual rhythms of the child. That is to say, the mathematics should be clearly seen to be relevant to the life of the child. It should be motivated as responding to the child's natural curiosity and desire to understand and master the child's world." - ---------------------------------------- Statements along these lines always strike me as odd. I have a pretty good memory for detail and the way I remember those years best is that most of my peers did NOT have these natural curiosities for mathematics. I can distinctly remember reciting math stuff and my peers being more than disinterested. I can remember this back to at least 6 years old and that should clearly predate some sort of "mass anxiety". And of course, the question lingers, why are some of us immune to it? I do not disagree with the label of "math anxiety" because anxiety is a real and normal human condition. I do disagree with its hypothesized source. In fact, if I would call anything "natural" it would be the anxiety itself, not the curiosity. I think that is a reality that we must face if we really mean to make math education better. And as a corollary, I would say that a kid that takes to math like a fish to water is abnormal in the truest sense of the word. It doesn't sound pretty but it is exactl! y how I remember it. Having said all of that, even though I think "math anxiety" is normal, I do not believe you should make it worse. You don't cure fear of water by throwing the paitent into the water and letting them drown. - ---------------------------------------- "Mathematics should not be boring since, if it is, it is an unnatural activity for the child. By a similar argument, the teaching of mathematics should not be dominated by the frequent incidence of standardized tests - we will certainly have more to say about this later." - ---------------------------------------- Boring is actually in the mind of the beholder. Naturally we are at odds again because our basic premises are opposite. Peter believes that we are born mathematical and external forces drive it away. I believe that most of us are actually born with an aversion to math, and it isn't math actually, it is an aversion to reasoning. Seriously, most people do not like to think in the same manner that most people do not like to lift heavy objects. I do agree that if you want a kid to learn something well then you must get them interested in it. I am entirely for engaging the unengaged. - ---------------------------------------- "For, if the education is so dominated, then both student and teacher will inevitably conclude that success in such tests not merely provides the criterion, but is actually the objective of the educational process." - ---------------------------------------- A very good observation and I want to think about that more later. We still disagree on the cause though. I think it is a normal reaction (again in the truest sense of the word) and if one wants to change it they will have to un-normalize it. - ---------------------------------------- Rote calculations: These ugly excrescences on the mathematical body are familiar to everybody. Dreary long multiplications and long divisions disfigure every curriculum. Children are drilled in these calculations in the hope that they will be able to execute them accurately. No justification is given for these calculations beyond the gratuitous and utterly misleading assignment to them of the epithet 'basic.' - ---------------------------------------- Yes, he uses the term correctly. He is talking about long division and large arithmetic drill. That age is long over but I do have a question. Why is arithmetic drill "damaging" only because calculators are available? I mean, if doing arithmetic drill is damaging now then wouldn't it have been as damaging in the 1960's? However, he is a staunch advocate of learning arithmetic and the algorithms... - ---------------------------------------- "Let me make it plain that I certainly advocate the mastery of what are called multiplication facts: I also advocate that students be made aware of the existence of pencil-and-paper algorithms and understand why they work. But mere accuracy in the use of such algorithms is no great skill-indeed, real understanding is often revealed by their avoidance." - ---------------------------------------- Now we start agreeing although we still disagree on the causation. - ---------------------------------------- Memory dependence: Children learn early in their traditional exposure to so-called mathematics that they must remember a great deal. They quickly conclude that memory is the only important mental device to employ in doing mathematics. There thus begins an inexorable process in which the reasoning faculty lapses into disuse and the memory becomes overloaded. The effects of this abuse of the system are sometimes slow to reveal themselves, but are very commonly there already by the junior high school level. It is a supreme irony that the measure of progress in any science is its independence of the memorization of brute fact, yet such memorization is endemic in current education. - ---------------------------------------- We are definitely in agreement here. In fact I am facing that right now in my analysis of the AP Calculus exams. Indeed, how many of the new "concept" driven curriculums exploit this unfortunate reflex in children. That is why I am driving PROBLEMS to the front and center, because they expose the hypocrisy in these so called "more understanding" curriculums. He gives me some additional insight into this "memory safety net" by the way he implies that this gets worse and becomes habitual. That makes a fits nicely with my experience with peers. - ---------------------------------------- "It is deplorable that intelligent people do not understand that 'learning mathematics' means 'learning to do mathematics' and not 'learning (skills, facts, proofs) by heart.'" - ---------------------------------------- Strongly Agree. Obviously. - ---------------------------------------- "Unmotivated problems: These abound in the mathematical education of the child. Let me give one typical example. Frequently the child is called upon to execute complicated computations with fractions. There is no indication of any situation in which such a skill would be necessary or even desirable. Moreover there is an intrinsic defect in such skills which damns them. For the child should always be able to check the reasonableness of his answer. For example, an addition involving decimals, 25.37 + 106.86, lends itself to the immediate check that the answer should be more than 120 and less than 140. On the other hand, the apparently equally simple problem 1/7 + 2/5 lends itself to no such check. The child could have no feel for the answer on the basis of an order-of-magnitude estimate. Indeed, even if the child got the answer right, he would not invoke at any stage of the calculation any awareness of approximately how big either the summands or the answer are." - ---------------------------------------- Ok, I don't get this part at all. I mean I understand "Unmotivated Problems" but his example is poor. His specialty is Algebraic Topology and I am sure he than his share of difficult problems just because they were there. You can get more out of arithmetic than computation, much more. I disagree with the example. If he is making an example for a kid that isn't ready then maybe, but when a kid is ready for math it is certainly the right time to do problems just for the sake of math. - ---------------------------------------- Spurious applications: This typical defect or a traditional education is closely related to that above. For, aware that it is today necessary to justify the presence in the curriculum of some mathematical skill on the basis of its utility, and unwilling for a variety of reasons to discard, for example, the addition of fractions, many curricula motivate this fairly useless technique through phony applications. I have met the following problem, "John swims 22 3/5 meters on Monday and 23 4/7 meters on Tuesday. How far has he swum on the two days?" This is intended to motivate the addition of fractions: in any intelligent child it should only motivate the question-how must a swimming Pool be calibrated in order that John could acquire such remarkable information? - ---------------------------------------- Again, he is focusing on something and I am not buying it. We have very different theories about fluency it would seem. Also, I think he is making too much of context. I can see his point somewhat about the problem being somewhat unnatural and they might have just avoided the word problem altogether and just made it a fraction arithmetic problem. But if you have ever done physics problems at any level the only whole number involved is the page number. These two paragraphs about unmotivated problems and spurious applications are not that big to me. He does follow all of this (he has a few more examples) with the following... - ---------------------------------------- Sometimes, spurious applications-of either kind-could be justified by honestly describing them as illustrations of the mathematics, that is, by showing that the mathematics does 'work,' even if the problem were artificial or the mathematics unnecessary to its solution. But this calls for total honesty, and teacher and student must understand that illustrations are different from applications and emphatically do not render applications superfluous. - ---------------------------------------- Well, you know Peter, when I was there (elementary school) I don't remember that being a problem. I recall realizing that the problem was about the math, not the context. I wish you had elaborated more on what you mean by explaining this to the kids. I could see the issue if, what I take for granted, didn't register in the student's, or god forbid, the teacher's head. But it would seem that would take quite a bit of explanation to an elementary aged kid about abstract versus real. But I will think about my stance of taking it for granted for awhile before talking about it. He has some good things to say about tests, or I should say about the complex effect tests have on the whole student/teacher/experience. For example... - ---------------------------------------- All of us teaching at the college and university level know that the problem persists far beyond childhood. Often, when I have been describing a piece of mathematics and have called for questions in order that I might be able to find out whether students are having difficulties, the only question I have received has been "will we get it on the test?" - ---------------------------------------- He should read the AP Calculus board sometimes. I have never mentioned it because I know the teachers are under a lot of stress, but that is an all too common teacher-teacher question now. And then he starts in on multiple choice... - ---------------------------------------- A specially ridiculous type of test is the multiple choice question. I do not suggest that the multiple choice question plays any particular role in generating math anxiety or math avoidance. But I cannot resist this opportunity to declare that, of all types of test question, the multiple choice question is the most absurd. For I take it that a justification for a test item must be that it asks the sort or question which the student is likely to meet when he is doing mathematics. In 35 years of doing mathematics as a professional, I have never yet met a situation in which I knew that the answer was one of four possibilities and I simply had to make the correct choice. Moreover, were I ever placed in such a situation artificially,' it is highly unlikely that I would use as a method of solving the problem the method appropriate to a situation in which I had not been told that the correct answer figured in a set of four potential solutions. - ---------------------------------------- In this article he seems to give a subjective argument against multiple choice tests. I suspect that the argument in the "Mathematics Tomorrow" book is also subjective only because of the quotations I see on the web. But I do not know that because I do not have the book. But regarding his subjective argument, it isn't that compelling. First off, the AMC tests are multiple choice and in those tests I think the format contributes less than 1% to the personality of the test. In other words, the problems are so well worded (and deep) that the last thing you would realize is that this is a multiple choice test. I think the SAT and Achieve problems are fairly well worded but yes, the multiple choice format probably contributes about 5% to the test. But every problem on those two tests could stand on their own as free response problems. Now, the AP Calculus exams on the other hand are rife with "fact" problems that could not exist outside the context of a multiple choice test. And ! Peter already told us what he thinks of "fact" problems. Could it be that he is presenting his case against multiple choice questions in that light? I still stand by my earlier statement that when a student gets past 600 on the SAT they have worked the majority of the problems and the format plays a very small role. I would love ask him his thoughts more specifically to actual examples of multiple choice tests, such as the AMC. I wonder if his distaste is simply multiple choice (seems like it) or the typical "fact" based multiple choice. I will have to look in the local libraries for the book or I guess I will have to order it. It is only $1.35. Kind of a shame I spent more on the "Seeing Mathematics" books, but not much more. My feeling is that it will be more of the subjective style of argument seen here. Speaking of math anxiety... I can certainly say that I have not experienced math anxiety, probably the opposite whatever that may be. And I don't know of a subject that ever invoked anxiety in me. English was certainly my most disliked subject but the feeling there was mostly disappointment and frustration, not anxiety. I remember one time in high school we were assigned book reports and I got "Don Quixote". I read that thing three times and still couldn't make sense of it and then all of the sudden I got it and it felt like it must feel for another kid when they get "completing the square". I remember reading it a couple more times to make sure I got it and I wrote my paper with new found enthusiasm and handed it in, spelling/grammar errors and all. And I got a "C" because of the spelling/grammar errors. That was certainly disappointing and not the last time either, but I kept the enthusiasm and endeavored to persevere. I was actually quite good at spelling through 4th grade or so but I it seemed to g! et in the way and I lost interest. I certainly don't remember any teacher causing it. The only thing that could have saved it was if I could put other interest (distractions) at bay and re-discipline myself with exercise. And such is life. |
Robert,
Thanks for posting Peter Hilton's article on math anxiety. Do you have a link to this article so that interested in saving a copy can do so? Here are some questions regarding your thoughts about this article: 1. Why do you believe that the classroom environment plays only a minor role, if any at all, in producing math anxiety or boredom in kids? Do you believe that bad mathematics textbooks don't play a role either? What about our anti-mathematical society and the scores of adults who struggle with mathematics? Mathematics, especially in K-12 (even more so in K-5), is taught in such a way that it isn't supposed to make sense to most kids and is taught as something boring because few teachers try to make math interesting to their students. Math is boring to kids because most teachers make it boring! And most K-5 teachers don't like math and don't understand it, so they project that boredom and confusion about math onto their students. "If my teacher doesn't like math and finds it confusing, then it must be boring and confusing!" or some variant of that probably runs through their minds either consciously or unconsciously. And I'm sure that most K-5 students don't realize that their teachers are completely different from mathematicians and scientists and other professionals in their understanding and views of what mathematics is. Could I be wrong about that last statement? Perhaps. Suppose I am wrong. Now let's see where that leads us. Then the students see their teacher as more of an ordinary user of mathematics than those professionals who make heavy use of mathematics in their careers and lives. If math is supposed to be boring and incomprehensible in the sense that "the rules" don't make any sense as to where they come from or why they are followed to an ordinary user of mathematics, then they naturally conclude that mathematics is supposed to be that way except only for the professionals in mathematics. Their future teachers often contribute to that kind of thinking because they are not much different from their previous teachers. If their parents and most other adults they know are as confused about mathematics as most of our population is, then they have even better reasons to believe that. Why should they not believe that if they see almost all ordinary users of mathematics being baffled by the subject? It's then clear that this kind of thinking will lead to grossly negative opinions about math. Now let us suppose the student sees the teacher as a professional in mathematics and not as an ordinary user of the subject. If they see a professional in mathematics being bored by the subject and not able to explain where "the rules" come from or why they are followed, then students will naturally believe that mathematics is supposed to be that way for everyone because it is that way for the experts, the ones who know what mathematics is. Add to that the majority of adults they see who struggle with math. The problem is that most K-12 students don't have any experiences with mathematicians and other mathematics professionals and only with ordinary users of mathematics who struggle greatly with the subject. So society plays a major role in giving students strongly negative impressions of mathematics. The classroom environment in most math classes further contributes to this crap by making math boring and nonsense for the kids. 2. Standardized tests and the craze for grades play a major role in creating math boredom and anxiety. Have you considered that idea? When grades and tests place huge stress on kids "to get the math now," they lose interest in solving genuine problems and struggling with them. They want to get math now and and want it to be as simple as possible so they can avoid punishment. And that punishment tells them that if they can't get math "in time," that's a bad sign. And most students are taught in such a way that they can't get it "in time." In short, it's okay to struggle a while in getting math or coming up with a solution to a problem, but schools don't tell students that and their inflexible schedules with tests and grading indirectly tell students the opposite of what I just said. 3. Why do you believe that unmotived problems and bad applications presented in school aren't a big deal? Problems don't have to motivated for their applications in the real world because mathematics is fascinating as a subject in its own right. I think that getting kids interested in mathematics for its own sake will cause them to be more interested in mathematics than otherwise (that's not to say that they shouldn't learn to be interested in mathematics applications as well; my primary interests in mathematics are in pure mathematics, but I do enjoy applied mathematics as well). But most problems aren't motivated at all, not even from the viewpoint of pure mathematics; they are just sitting there for no apparent reason. Bad applications hurt because they make students think that mathematics isn't really useful after all. If they are intelligent enough to see that these so-called "real-world applications" are really problems completely unrepresentative of problems in real life, then what are they supposed to think about real world applications of mathematics, especially if they never see a problem that really is representative of real-world problems? "If this is the best they can come up with in how mathematics is used in the real world, then math must be useless after all" is probably running through their minds. And if they see no reason to study mathematics for its own sake either, then it's obvious what happens later. 4. Peter Hilton makes a good case against multiple-choice tests, especially the use of them as sole or primary ways to assess students' understanding of mathematics. One reason he doesn't include is that these tests ignore the students' ability to construct his own explanations of concepts and why mathematics works. And the lack of such ability makes the students' understanding of that mathematics being tested on the exam incomplete for a student at that level. What evidence do you have that these tests you cite are not really multiple-choice tests in the sense that the multiple-choice format plays only a minor role in how students take the test? Perhaps you are correct, but I don't know because I don't see any evidence. Even then, I have a problem with not seeing the students' reasoning. I don't doubt that multiple-choice questions asking for recitation of facts and formulas are bogus questions. I agree with you in that. In fact, these questions are bogus even as free-response questions. Perhaps such questions don't have to be chucked completely, but they should play only a minor role--if any--on a math exam. 5. One final comment: I agree with your comments that Peter Hilton's suggestion that fractions should be discarded from the math curriculum is ridiculous. However, he does make a good point that fractions might not be motivated very well. The example Hilton gives about a so-called "application" of fractions is not necessarily bad in itself as a word problem, but he does make a valid point in that it is a phony way to show how fractions are used in the real-world. Furthermore, the metric system doesn't use fractions (except decimal fractions), so that fact there contributes further to the application's phoniness. Jonathan Groves On 3/1/2010 at 11:34 pm, Robert Hansen wrote: > Thus far the search for any readily available > electronic source of "Avoiding Math Avoidance" has > turned up empty. I will have to see if the local > library has "Mathematics Tomorrow" or order it for > $1.35. I did however manage to find an article by > Peter Hilton in the MAA archives that was written at > the same time and probably indicates some of what he > said. > > Math Anxiety: Some Suggested Causes and Cures: Part 1 > (June 1980) > > I am including some snippets along with my own > commentary. We agree on some things and not on > others. His words are bounded with lines. Any typos > are the result of the OCR. He has a point about > authority I will address later. > > - ---------------------------------------- > "The learning and the doing of mathematics should > d fit into the natural intellectual rhythms of the > child. That is to say, the mathematics should be > clearly seen to be relevant to the life of the child. > It should be motivated as responding to the child's > natural curiosity and desire to understand and master > the child's world." > - ---------------------------------------- > > Statements along these lines always strike me as odd. > I have a pretty good memory for detail and the way I > remember those years best is that most of my peers > did NOT have these natural curiosities for > mathematics. I can distinctly remember reciting math > stuff and my peers being more than disinterested. I > can remember this back to at least 6 years old and > that should clearly predate some sort of "mass > anxiety". And of course, the question lingers, why > are some of us immune to it? I do not disagree with > the label of "math anxiety" because anxiety is a real > and normal human condition. I do disagree with its > hypothesized source. In fact, if I would call > anything "natural" it would be the anxiety itself, > not the curiosity. I think that is a reality that we > must face if we really mean to make math education > better. And as a corollary, I would say that a kid > that takes to math like a fish to water is abnormal > in the truest sense of the word. It doesn't sound > pretty but it is exactly how I remember it. > > Having said all of that, even though I think "math > anxiety" is normal, I do not believe you should make > it worse. You don't cure fear of water by throwing > the paitent into the water and letting them drown. > > > - ---------------------------------------- > "Mathematics should not be boring since, if it is, it > is an unnatural activity for the child. By a similar > argument, the teaching of mathematics should not be > dominated by the frequent incidence of standardized > tests - we will certainly have more to say about this > later." > - ---------------------------------------- > > Boring is actually in the mind of the beholder. > Naturally we are at odds again because our basic > premises are opposite. Peter believes that we are > born mathematical and external forces drive it away. > I believe that most of us are actually born with an > aversion to math, and it isn't math actually, it is > an aversion to reasoning. Seriously, most people do > not like to think in the same manner that most people > do not like to lift heavy objects. I do agree that if > you want a kid to learn something well then you must > get them interested in it. I am entirely for engaging > the unengaged. > > > - ---------------------------------------- > "For, if the education is so dominated, then both > student and teacher will inevitably conclude that > success in such tests not merely provides the > criterion, but is actually the objective of the > educational process." > - ---------------------------------------- > > A very good observation and I want to think about > that more later. We still disagree on the cause > though. I think it is a normal reaction (again in the > truest sense of the word) and if one wants to change > it they will have to un-normalize it. > > > - ---------------------------------------- > Rote calculations: These ugly excrescences on the > mathematical body are familiar to everybody. Dreary > long multiplications and long divisions disfigure > every curriculum. Children are drilled in these > calculations in the hope that they will be able to > execute them accurately. No justification is given > for these calculations beyond the gratuitous and > utterly misleading assignment to them of the epithet > 'basic.' > - ---------------------------------------- > > Yes, he uses the term correctly. He is talking about > long division and large arithmetic drill. That age is > long over but I do have a question. Why is arithmetic > drill "damaging" only because calculators are > available? I mean, if doing arithmetic drill is > damaging now then wouldn't it have been as damaging > in the 1960's? > > However, he is a staunch advocate of learning > arithmetic and the algorithms... > > - ---------------------------------------- > "Let me make it plain that I certainly advocate the > e mastery of what are called multiplication facts: I > also advocate that students be made aware of the > existence of pencil-and-paper algorithms and > understand why they work. But mere accuracy in the > use of such algorithms is no great skill-indeed, real > understanding is often revealed by their avoidance." > - ---------------------------------------- > > Now we start agreeing although we still disagree on > the causation. > > > - ---------------------------------------- > Memory dependence: Children learn early in their > traditional exposure to so-called mathematics that > they must remember a great deal. They quickly > conclude that memory is the only important mental > device to employ in doing mathematics. There thus > begins an inexorable process in which the reasoning > faculty lapses into disuse and the memory becomes > overloaded. The effects of this abuse of the system > are sometimes slow to reveal themselves, but are very > commonly there already by the junior high school > level. It is a supreme irony that the measure of > progress in any science is its independence of the > memorization of brute fact, yet such memorization is > endemic in current education. > - ---------------------------------------- > > We are definitely in agreement here. In fact I am > facing that right now in my analysis of the AP > Calculus exams. Indeed, how many of the new "concept" > driven curriculums exploit this unfortunate reflex in > children. That is why I am driving PROBLEMS to the > front and center, because they expose the hypocrisy > in these so called "more understanding" curriculums. > He gives me some additional insight into this "memory > safety net" by the way he implies that this gets > worse and becomes habitual. That makes a fits nicely > with my experience with peers. > > > - ---------------------------------------- > "It is deplorable that intelligent people do not > t understand that 'learning mathematics' means > 'learning to do mathematics' and not 'learning > (skills, facts, proofs) by heart.'" > - ---------------------------------------- > > Strongly Agree. Obviously. > > > - ---------------------------------------- > "Unmotivated problems: These abound in the > e mathematical education of the child. Let me give > one typical example. Frequently the child is called > upon to execute complicated computations with > fractions. There is no indication of any situation in > which such a skill would be necessary or even > desirable. Moreover there is an intrinsic defect in > such skills which damns them. For the child should > always be able to check the reasonableness of his > answer. For example, an addition involving decimals, > 25.37 + 106.86, lends itself to the immediate check > that the answer should be more than 120 and less than > 140. On the other hand, the apparently equally simple > problem 1/7 + 2/5 lends itself to no such check. The > child could have no feel for the answer on the basis > of an order-of-magnitude estimate. Indeed, even if > the child got the answer right, he would not invoke > at any stage of the calculation any awareness of > approximately how big either the summands or the > answer are." > - ---------------------------------------- > > Ok, I don't get this part at all. I mean I understand > "Unmotivated Problems" but his example is poor. His > specialty is Algebraic Topology and I am sure he has > done his share of difficult problems just because > they were there. You can get more out of arithmetic > than computation, much more. I disagree with the > example. If he is making an example for a kid that > isn't ready then maybe, but when a kid is ready for > math it is certainly the right time to do problems > just for the sake of math. > > > - ---------------------------------------- > Spurious applications: This typical defect or a > traditional education is closely related to that > above. For, aware that it is today necessary to > justify the presence in the curriculum of some > mathematical skill on the basis of its utility, and > unwilling for a variety of reasons to discard, for > example, the addition of fractions, many curricula > motivate this fairly useless technique through phony > applications. I have met the following problem, "John > swims 22 3/5 meters on Monday and 23 4/7 meters on > Tuesday. How far has he swum on the two days?" This > is intended to motivate the addition of fractions: in > any intelligent child it should only motivate the > question-how must a swimming Pool be calibrated in > order that John could acquire such remarkable > information? > - ---------------------------------------- > > Again, he is focusing on something and I am not > buying it. We have very different theories about > fluency it would seem. Also, I think he is making too > much of context. I can see his point somewhat about > the problem being somewhat unnatural and they might > have just avoided the word problem altogether and > just made it a fraction arithmetic problem. But if > you have ever done physics problems at any level the > only whole number involved is the page number. These > two paragraphs about unmotivated problems and > spurious applications are not that big to me. > > > He does follow all of this (he has a few more > examples) with the following... > > - ---------------------------------------- > Sometimes, spurious applications-of either kind-could > be justified by honestly describing them as > illustrations of the mathematics, that is, by showing > that the mathematics does 'work,' even if the problem > were artificial or the mathematics unnecessary to its > solution. But this calls for total honesty, and > teacher and student must understand that > illustrations are different from applications and > emphatically do not render applications superfluous. > - ---------------------------------------- > > Well, you know Peter, when I was there (elementary > school) I don't remember that being a problem. I > recall realizing that the problem was about the math, > not the context. I wish you had elaborated more on > what you mean by explaining this to the kids. I could > see the issue if, what I take for granted, didn't > register in the student's, or god forbid, the > teacher's head. But it would seem that would take > quite a bit of explanation to an elementary aged kid > about abstract versus real. But I will think about my > stance of taking it for granted for awhile before > talking about it. > > > He has some good things to say about tests, or I > should say about the complex effect tests have on the > whole student/teacher/experience. For example... > > - ---------------------------------------- > All of us teaching at the college and university > level know that the problem persists far beyond > childhood. Often, when I have been describing a piece > of mathematics and have called for questions in order > that I might be able to find out whether students are > having difficulties, the only question I have > received has been "will we get it on the test?" > - ---------------------------------------- > > He should read the AP Calculus board sometimes. I > have never mentioned it because I know the teachers > are under a lot of stress, but that is an all too > common teacher-teacher question now. > > And then he starts in on multiple choice... > > - ---------------------------------------- > A specially ridiculous type of test is the multiple > choice question. I do not suggest that the multiple > choice question plays any particular role in > generating math anxiety or math avoidance. But I > cannot resist this opportunity to declare that, of > all types of test question, the multiple choice > question is the most absurd. > > For I take it that a justification for a test item > must be that it asks the sort or question which the > student is likely to meet when he is doing > mathematics. In 35 years of doing mathematics as a > professional, I have never yet met a situation in > which I knew that the answer was one of four > possibilities and I simply had to make the correct > choice. Moreover, were I ever placed in such a > situation artificially,' it is highly unlikely that I > would use as a method of solving the problem the > method appropriate to a situation in which I had not > been told that the correct answer figured in a set of > four potential solutions. > - ---------------------------------------- > > In this article he seems to give a subjective > argument against multiple choice tests. I suspect > that the argument in the "Mathematics Tomorrow" book > is also subjective only because of the quotations I > see on the web. But I do not know that because I do > not have the book. But regarding his subjective > argument, it isn't that compelling. First off, the > AMC tests are multiple choice and in those tests I > think the format contributes less than 1% to the > personality of the test. In other words, the problems > are so well worded (and deep) that the last thing you > would realize is that this is a multiple choice test. > I think the SAT and Achieve problems are fairly well > worded but yes, the multiple choice format probably > contributes about 5% to the test. But every problem > on those two tests could stand on their own as free > response problems. Now, the AP Calculus exams on the > other hand are rife with "fact" problems that could > not exist outside the context of a multiple choice > test. And Peter already told us what he thinks of > "fact" problems. Could it be that he is presenting > his case against multiple choice questions in that > light? I still stand by my earlier statement that > when a student gets past 600 on the SAT they have > worked the majority of the problems and the format > plays a very small role. I would love ask him his > thoughts more specifically to actual examples of > multiple choice tests, such as the AMC. I wonder if > his distaste is simply multiple choice (seems like > it) or the typical "fact" based multiple choice. > > I will have to look in the local libraries for the > book or I guess I will have to order it. It is only > $1.35. Kind of a shame I spent more on the "Seeing > Mathematics" books, but not much more. My feeling is > that it will be more of the subjective style of > argument seen here. > > Speaking of math anxiety... > > I can certainly say that I have not experienced math > anxiety, probably the opposite whatever that may be. > And I don't know of a subject that ever invoked > anxiety in me. English was certainly my most disliked > subject but the feeling there was mostly > disappointment and frustration, not anxiety. I > remember one time in high school we were assigned > book reports and I got "Don Quixote". I read that > thing three times and still couldn't make sense of it > and then all of the sudden I got it and it felt like > it must feel for another kid when they get > "completing the square". I remember reading it a > couple more times to make sure I got it and I wrote > my paper with new found enthusiasm and handed it in, > spelling/grammar errors and all. And I got a "C" > because of the spelling/grammar errors. That was > certainly disappointing and not the last time either, > but I kept the enthusiasm and endeavored to > persevere. I was actually quite good at spelling > through 4th grade or so but I it seemed to get in the > way and I lost interest. I certainly don't remember > any teacher causing it. The only thing that could > have saved it was if I could put other interest > (distractions) at bay and re-discipline myself with > exercise. And such is life. |
In reply to this post by Robert Hansen
Robert Hansen posted Mar 2, 2010 10:04 AM:
> Thus far the search for any readily available > electronic source of "Avoiding Math Avoidance" has > turned up empty. I will have to see if the local > library has "Mathematics Tomorrow" or order it for > $1.35. I did however manage to find an article by Peter > Hilton in the MAA archives that was written at the same > time and probably indicates some of what he said. > <snip> Try searching for "Overcoming math anxiety" - there are plenty of links to sites available, some of them with what seems to be useful material. I too like Jonathan Groves would like very much to have a link to the whole article. At first reading of what you've excerpted, it seems the best that could be done. GSC |
In reply to this post by Robert Hansen
I had to buy the article through JSTOR. I was too impatient to go to the college and get it. If your school has a subscription to JSTOR than you can pull it for free.
1. Why do you believe that the classroom environment plays only a minor role, if any at all, in producing math anxiety or boredom in kids? Do you believe that bad mathematics textbooks don't play a role either? What about our anti-mathematical society and the scores of adults who struggle with mathematics? What I said was that an aversion to math is normal, just like an aversion to reading is normal. So the aversion is already present and you must overcome that. Regarding classrooms, I said that doesn't mean you make it worse. I gave the example of someone afraid of water. You don't overcome that by throwing them in water and letting them drown. I am enormously for helping kids past that aversion. But once they are past, get on with the math. "Mathematics, especially in K-12 (even more so in K-5), is taught in such a way that it isn't supposed to make sense to most kids and is taught as something boring because few teachers try to make math interesting to their students." My post later tonight will reveal more of what I think about that. But I think that traditional methods are exactly the right way to teach math to those kids they fit. But you are right, they do nothing for the aversion and for the others they are like throwing them into the water and letting them drown. I would like to point out though, that these David Copperfield type horror stories do not exist much anymore. My district uses more engagement type teaching although they use it for too long with all the kids in my opinion. "2. Standardized tests and the craze for grades play a major role in creating math boredom and anxiety. Have you considered that idea? When grades and tests place huge stress on kids "to get the math now," they lose interest in solving genuine problems and struggling with them." I had occasional nightmares more than 10 years after my last college final exam. Always the same plot. I am sitting down to the test and realize I haven't attended one class. Such a thing never happened for real but it certainly seemed real in the dreams. It is a competitive world. What do you want me to say? 3. Why do you believe that unmotived problems and bad applications presented in school aren't a big deal? Problems don't have to motivated for their applications in the real world because mathematics is fascinating as a subject in its own right. I will try to address it in my later post. I am not against motivated problems but when they are used too much they are actually detrimental. And yes, if mathematics is not itself fascinating in its own right, then you not getting the picture. I don't mean drop dead fascinating, just that you need to recognize math for what it is if you have any claim of understanding. In fact, that question coming from students "what is this for" is an indicator that they are indeed not connecting to the math. You will buy some time using more interesting problems as a proxy, but you still must connect the student to the math. The fascination with real world problems you describe is actually called "Physics".:) "What evidence do you have that these tests you cite are not really multiple-choice tests in the sense that the multiple-choice format plays only a minor role in how students take the test? Perhaps you are correct, but I don't know because I don't see any evidence. Even then, I have a problem with not seeing the students' reasoning." What do you mean "evidence"? Just look at the test. When I say "role" I mean if you were taking the test, how often would you look at the choices and what effect would that have on your solving the problem. The problems on an AMC test are so hard and cryptic that you will get discouraged looking at the choices and you actually avoid looking at them at all. On an SAT you look but they don't often play a role IF you are scoring high. and there are the distractors. If you are scoring low then I suppose you concentrate on the choices more. But that is kind of moot since they obviously are not helping. |
In reply to this post by Robert Hansen
<< SNIP >>
> 5. One final comment: I agree with your comments > that Peter Hilton's suggestion > that fractions should be discarded from the math > curriculum is ridiculous. > However, he does make a good point that fractions > might not be motivated > very well. The example Hilton gives about a > so-called "application" of > fractions is not necessarily bad in itself as a word > problem, but he does make > a valid point in that it is a phony way to show how > fractions are used in > the real-world. Furthermore, the metric system > doesn't use fractions (except > decimal fractions), so that fact there contributes > further to the application's > phoniness. > > > Jonathan Groves > Lets not presume hegemony for the metric system, nor any other of weights and measures. Conversion constants make the world go around. We'll have those eight ounces in a cup, four cups in a quart, four quarts in a gallon, as long as people watch Julia Child clips on Youtube or whatever. Probably another four centuries at least. Of course an important way to present fractions is in the context of slicing and dicing some important polyhedra, sure to be encountered again and again if enjoying a technical career. The ratio of volumes to volumes will always be an excuse to introduce p : q i.e. p/q. http://mathforum.org/kb/thread.jspa?threadID=2047973&tstart=15 Note: some math teachers have some anxiety about the spatial volumes thread, as that's not what's in their textbooks and they feel like fish out of water without that kind of authority to sustain them. Others take to spatial volumes like fish to water, can't wait to get back from the workshop to tell their students about the volume six rhombic dodecahedron, never mind it's not in Discovery or whatever tome (anyway, this might be Borneo, so closer to Singapore in that sense -- not saying Seattle is always first for every new teacher training). Kirby |
In reply to this post by Robert Hansen
<< SNIP >>
> Speaking of math anxiety... > > I can certainly say that I have not experienced math > anxiety, probably the opposite whatever that may be. > And I don't know of a subject that ever invoked > anxiety in me. English was certainly my most disliked > subject but the feeling there was mostly > disappointment and frustration, not anxiety. I > remember one time in high school we were assigned > book reports and I got "Don Quixote". I read that > thing three times and still couldn't make sense of it > and then all of the sudden I got it and it felt like > it must feel for another kid when they get > "completing the square". I remember reading it a > couple more times to make sure I got it and I wrote > my paper with new found enthusiasm and handed it in, > spelling/grammar errors and all. And I got a "C" > because of the spelling/grammar errors. That was > certainly disappointing and not the last time either, > but I kept the enthusiasm and endeavored to > persevere. I was actually quite good at spelling > through 4th grade or so but I it seemed to get in the > way and I lost interest. I certainly don't remember > any teacher causing it. The only thing that could > have saved it was if I could put other interest > (distractions) at bay and re-discipline myself with > exercise. And such is life. I think you're quite a lucky man to be able to say all this. You've been given a gift, don't take it for granted. A few chairs over, another kid is truly suffering, bitter taste in the mouth, slight tremor from fear. Mathematics is truly threatening, as that sense of failing is also panic. Will society value me? Capitalism is cruel to those it thinks it doesn't need. As a curriculum designer, we're responsible for looking from various points of view. The gifted student viewpoint is one of many. Designing everything to advantage her or him, at the expense of all others, is likely going to end up backfiring. I think this 'off your duff' mathematics is going to help change the playing field in some positive ways. Kids who don't relish the math test, may nevertheless excel in the ropes course. You might think the latter has nothing to do with math, but it does. Tying knots is a branch of topology, not just sailing. For other students, the mathematics of the catenary are combined with real time working on a real railroad -- for academic credit, not as "slaves of the system". Some will sign up for longer terms, as much about railroading is inherently fun to some temperaments. Kirby |
In reply to this post by Robert Hansen
Robert,
At least we both agree that math anxiety is a real problem for many students, and I can tell that you believe that school does make it worse. Is it natural for some students to have math anxiety? Will some be intimidated by mathematics, no matter what we do? I think so. Much of learning to avoid anxiety for any subject--mathematics or not--is learning to enjoy challenges. We can encourage students to try to enjoy challenges and to try to find something about mathematics or any other subject worthwhile and enjoyable. But we can't force that in anyone; the student must either want that or at least be open to learning that. And we can encourage students to try to use their anxiety to either help them (for example, even experienced speakers and lecturers normally feel some anxiety in giving a speech, but they know how to make that anxiety work in their favor or at least know how not to let it hurt them) or at least not to ruin them. So we can encourage students to try to learn these strategies, but ultimately it is their decision--not ours--to be open to learning that. Furthermore, I believe math anxiety comes in different forms and different levels of intensity: Some math anxiety is so severe that it becomes crippling. Other cases of math anxiety are not crippling. Students with the worst cases of math anxiety are intimidated by any explicit mention of the word "mathematics." Other students are not intimidated by some kinds of mathematics and are intimidated by others. The Spring II term at Argosy University had started yesterday, and several students who have introduced themselves to the class have said that they don't fear arithmetic and perhaps even enjoy it--at least somewhat--but are intimidated by algebra. Concrete numbers make sense to them but not variables. Others might feel comfortable with precalculus mathematics but not calculus. Still others feel comfortable with the math that engineers or scientists use but not with pure mathematics. Even mathematicians can feel some anxiety about math (I remember reading that somewhere but don't remember where). I do remember reading in Marcus du Sautoy's book "The Music of the Primes" that the Riemann Hypothesis is an intimidating problem and that many mathematicians dare not touch it. So I don't think your claim about math anxiety being natural at least in some students is not entirely wrong--especially when we consider these observations about different kinds of math anxiety. However, K-12 math educators need not be concerned about anxiety for post-calculus mathematics or pure mathematics, and that kind of math anxiety is clearly not being addressed here. But is it really that natural for all the droves of students whom we see who have crippling or near crippling math anxiety for K-12 mathematics? I seriously doubt it. There may be no practical way to prevent that kind of math anxiety in all students, but I do believe that far more students suffer from math anxiety than what really should be the case since I believe that some natural elementary math ability exists in all of us. The ability to learn pure mathematics that we see at the graduate level and beyond I'm sure is a different story. The school environment and our culture's beliefs about mathematics and students' experiences with many adults who struggle with middle school or even elementary school mathematics create a highly complex mixture of experiences that generate math anxiety. The mixture is so highly complex that we cannot attribute the causes of math anxiety to any one particular cause--not even for any particular student. For any particular student, it is very difficult--perhaps impossible in some or many cases--if the direct cause is actually natural and that school and other similar experiences I have mentioned have simply worsened the student's fear or if the student has learned to fear mathematics because of school and these other experiences. If there is a way to sort all this out for a particular student, I believe we would need the help of a psychologist. I don't have any proof that your claim is wrong, and I don't have proof that mine is correct. But because of our cultures' negative attitudes towards mathematics, the bad classroom environments that many students suffer through, the bad textbooks that reduce mathematics to a bunch of meaningless rules, etc., I see less reason to believe your claim than mine. How can you or I say for sure that this particular student has a natural fear of mathematics when the student had been taught by bad teachers who reduced mathematics to arbitrary rules, had been using textbooks for years that did the same, that the teachers didn't care much for teaching or for mathematics, who have been passed along by their teachers so that they soon fell too far behind to catch up? Even worse, many of these students are never really told that struggling to learn mathematics is not a sign of disability in mathematics and that not learning mathematics quickly enough to stick with the class schedule is not a sign of disability either. So many of these students begin to believe that they are stupid because they cannot learn as quickly as they are expected to. But few people can learn the algorithms and procedures and other "rules" of mathematics without knowing why these work and why these rules are followed and why they make sense. I'm not stupid, but I sure as heck have lots of trouble learning lots of rules in which I cannot see why they work, why they are chosen--in other words, rules that seem to me to have no rhyme or reason to them. So I believe those who struggle with learning mathematics without learning at least some reason as to why mathematics works the way it does are not necessarily stupid when it comes to mathematics. But schools either explicitly or implicitly tell students otherwise. Talking to the student can help, but lots of students probably underestimate their abilities to learn math and don't remember all their previous experiences and might not be able to determine if their fear is actually natural and just worsened by these experiences or if these experiences actually caused their fear. Some students definitely suggest that they believe their fear of mathematics was caused by terrible experiences in school. Several of my students at Argosy for this new term have mentioned in their introductions some terrible experiences with bad teachers. One of them had mentioned an elementary school teacher she had for several years who punished her by embarassing her in front of the class when she didn't learn like the teacher had wanted her to learn. The teacher had belittled her many times in class by calling her stupid. Another one recalled a high school algebra teacher who hated teaching, wouldn't explain anything, resisted students' questions, and essentially reduced math class to having students read the book and figure it out for themselves. "Read the darn book and leave me alone!" he implictly said to the class. I plan to comment in a later post about your most recent comments here on unmotivated versus motivated problems and standardized tests. Jonathan Groves On 3/3/2010 at 8:00 pm, Robert Hansen wrote: > I had to buy the article through JSTOR. I was too > impatient to go to the college and get it. If your > school has a subscription to JSTOR than you can pull > it for free. > > > 1. Why do you believe that the classroom environment > plays only a minor role, if any at all, in producing > math anxiety or boredom in kids? Do you believe that > bad mathematics textbooks don't play a role either? > What about our anti-mathematical society and the > scores of adults who struggle with mathematics? > > > What I said was that an aversion to math is normal, > just like an aversion to reading is normal. So the > aversion is already present and you must overcome > that. Regarding classrooms, I said that doesn't mean > you make it worse. I gave the example of someone > afraid of water. You don't overcome that by throwing > them in water and letting them drown. I am enormously > for helping kids past that aversion. But once they > are past, get on with the math. > > > "Mathematics, especially in K-12 (even more so in > K-5), is taught in such a way that it isn't supposed > to make sense to most kids and is taught as something > boring because few teachers try to make math > interesting to > their students." > > My post later tonight will reveal more of what I > think about that. But I think that traditional > methods are exactly the right way to teach math to > those kids they fit. But you are right, they do > nothing for the aversion and for the others they are > like throwing them into the water and letting them > drown. I would like to point out though, that these > David Copperfield type horror stories do not exist > much anymore. My district uses more engagement type > teaching although they use it for too long with all > the kids in my opinion. > > > "2. Standardized tests and the craze for grades play > a major role in creating math boredom and anxiety. > Have you considered that idea? When grades and tests > place huge stress on kids "to get the math now," they > lose interest in solving genuine problems and > struggling with them." > > I had occasional nightmares more than 10 years after > my last college final exam. Always the same plot. I > am sitting down to the test and realize I haven't > attended one class. Such a thing never happened for > real but it certainly seemed real in the dreams. It > is a competitive world. What do you want me to say? > > > 3. Why do you believe that unmotived problems and bad > applications presented in school aren't a big deal? > Problems don't have to motivated for their > applications in the real world because mathematics is > fascinating as a subject in its own right. > > I will try to address it in my later post. I am not > against motivated problems but when they are used too > much they are actually detrimental. And yes, if > mathematics is not itself fascinating in its own > right, then you not getting the picture. I don't mean > drop dead fascinating, just that you need to > recognize math for what it is if you have any claim > of understanding. In fact, that question coming from > students "what is this for" is an indicator that they > are indeed not connecting to the math. You will buy > some time using more interesting problems as a proxy, > but you still must connect the student to the math. > The fascination with real world problems you describe > is actually called "Physics".:) > > > "What evidence do you have that these tests you cite > are not really multiple-choice tests in the sense > that the multiple-choice format plays only a minor > role in how students take the test? Perhaps you are > correct, but I don't know because I don't see any > evidence. Even then, I have a problem with not seeing > the students' reasoning." > > What do you mean "evidence"? Just look at the test. > When I say "role" I mean if you were taking the test, > how often would you look at the choices and what > effect would that have on your solving the problem. > The problems on an AMC test are so hard and cryptic > that you will get discouraged looking at the choices > and you actually avoid looking at them at all. On an > SAT you look but they don't often play a role IF you > are scoring high. and there are the distractors. If > you are scoring low then I suppose you concentrate on > the choices more. But that is kind of moot since they > obviously are not helping. |
In reply to this post by Robert Hansen
I don't think what I am talking about is "math anxiety", I think I am talking more about "math dislike". There is a big difference. But if you do not deal with "math dislike" early on then it could become "math anxiety". My son had "vegitable dislike" and I dealt with it before it became "vegitable anxiety". He also had "math dislike" and I'll be honest, the first "Oh man!" response I got out of him in response to having to do a math lesson, everything I read or heard about "math anxiety" ran through my head. He now fully understands that 4 licks is more than 3 licks. Lol, just kidding, he's had a couple spankings before, but not for anything academic. But I couldn't help thinking if that had something to do with his rapid understanding of cardinality and order. Maybe it would make a nice activity for Bill Marsh's curriculum.:)
But I stood firm and at the same time gained some paitence. We made it through that phase and the other day when I saw him make change I couldn't contain my smile nor he his. I don't think you are looking at the whole picture. When we go to the science fairs or other school activities, it is always the same group of parents. I bet if you look back into these students' lives I bet you will find that the missing link wasn't the teacher. It will be something closer to home. I wouldn't be a public elementary school teacher if you paid me because of what is being asked of them. They are really in a tough spot right now. "But is it really that natural for all the droves of students whom we see who have crippling or near crippling math anxiety for K-12 mathematics? I seriously doubt it." Have you ever looked at kids and reading? We could move this discussion over to reading and be in the same boat. Kids hate reading. Not all kids of course, but most. Getting them to read a book is like getting them to clean up their room. This is where parents come into play. I guess with reading you can just kind of do it. You can't do that with math because math has problems to solve. You either do it or you don't do it. Yes, you can do arithimetic and not algebra, or algebra and not calculus. Also, in reading you don't ask the kid to read the greatest books and understand them. But in math you do. If this was art you would be asking every kid to understand and paint like picaso. "Some students definitely suggest that they believe their fear of mathematics was caused by terrible experiences in school." Americans believe all sorts of weird crap. We don't have to believe what is true because we've been on top of the world for awhile. It's been decades now with this reform and there hasn't been any real progress. I base that on these algebra test results coming back. In one district 90% of the students failed the test. That's no joke. And you could go to just about any district and get that result. If I was going to fix it, first I would disconnect entirely from the later math reforms and remediation (algebra etc) becaause they are way too late in the game to be applicable to where the problem actually lies. I would focus on the early years. When a kid arives in algebra class and can't get it, then common sense tells you that it isn't the algebra they lack, but something before algebra. They can't lack what they have showed up to learn. What they lack is fluency in the previous math and they lack it because the only real exposure to it was what occurs that one hour with 20 other kids 5 days a week. And if most of that hour is spent "engaging" them then I seriously doubt they got much math in what tiem was left. |
In reply to this post by Robert Hansen
Jonathan Groves wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=7001933 > [...] several students who have introduced themselves > to the class have said that they don't fear arithmetic > and perhaps even enjoy it--at least somewhat--but are > intimidated by algebra. Concrete numbers make sense to > them but not variables. Others might feel comfortable > with precalculus mathematics but not calculus. Still > others feel comfortable with the math that engineers > or scientists use but not with pure mathematics. This reminds me of something Isaac Asimov once wrote concerning his math experiences. He began by saying how math was effortless to him in his early school years. So was algebra, although there were signs of possible limitations when he was in high school. For example, he wasn't on the math team -- probably because he had to work in his father's shop after school -- and he had nagging feelings that the people on the math team had reached heights well beyond his capabilities. Then, in his first semester of college, he took differential calculus, and suddenly math started requiring some work to master. He was able to get an A I think, but it was no longer an effortless A as had been the case up to that point. Then, in the next semester, everything fell apart when he took integral calculus. Things no longer made sense immediately, and sometimes not even after a lot of study and reflection. He struggled (relative to his earlier experiences in math), and only managed to eek out a grade of B, and never took another math course again. This was the introductory 1.5 pages of one of his essays on some mathematical topic, but I don't remember which essay/topic it was. Asimov said in these introductory remarks that it seemed most people have their own personal "math barrier" (my phrase; I don't remember what Asimov used), except for a very few who go on to become mathematicians, and then he said that fortunately his barrier was high enough to allow him a reasonable amount of room under which to wander around and write about certain math topics, such as the topic he was using this introduction to introduce. [Asimov always introduced his essay topics with a roughly 1.5 page personal note that more or less (often "less", but it would still nonetheless be very interesting) led into the topic he was going to discuss.] The only quote I found on the internet about this is the following [1], which I think isn't from the essay itself but from his autobiography: "After years of finding mathematics easy, I finally reached integral calculus and came up against a barrier. I realized that this was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way." [1] In typical internet fashion, I found this same quote on dozens of web pages, as if someone initially posted it somewhere and then everyone and their brother just copied it, with no one anywhere bothering to look it up and say exactly where it came from or posting the far more extensive comments that Asimov made in the essay I'm thinking of (which I have, but it's at home somewhere). Dave L. Renfro |
On Fri, Mar 5, 2010 at 1:00 PM, Dave L. Renfro <[hidden email]> wrote:
> << snip >> > The only quote I found on the internet about this is > the following [1], which I think isn't from the essay itself > but from his autobiography: > > "After years of finding mathematics easy, I finally reached > integral calculus and came up against a barrier. Â I realized > that this was as far as I could go, and to this day I have > never successfully gone beyond it in any but the most > superficial way." > << snip >> > Dave L. Renfro I'll respond to this in part because Isaac Asimov was a big influence in my life. I was one of those kids who "fell for" his trick, which was to get us hooked on science and math through science fiction. That's exactly what happened in my case. From reading his novels I developed a healthy appetite for his non-fiction writings. One of Hansen's recent posts was about the impossibility of ever getting more than 25% through a Dolciani type pre-calc / calc program, no matter how good the teaching. We're going to lose 75% due to whatever factors ("home life", "lack of aptitude", "math anxiety"... a long list of factors take the blame). My question is why assume the precalc / calc pipeline as the one and only. That's like going to a water slide park and having only the one water slide (typically a pipe), or going to a roller coaster park (like that one near LA) and finding only the one roller coaster. Surely mathematics is a more varied playing field, and if we're going to admit up front that 75% won't make it through calculus, then why can't we offer other rides? In my 2008 talk at Pycon, I inveighed against what I call "Calculus Mountain", which is precisely this killer hill that is used specifically and by design to "weed out" those who can't hack it. http://www.youtube.com/watch?v=hbeHdg8mtdc (almost 2,500 views -- more than most journal articles). Those making it over this mountain feel proud of themselves, glad to be gifted in whatever way, but is this really all that great a design in the first place? Many posters here have questioned this status quo over the years. It's not really heresy. Enter discrete math and its relatives ("digital math" still on the back burner as not well defined, except maybe by me on Wikieducator). If we had the budget (not saying we do, given economics is not well under- stood), we could easily design a track for "calculus refugees" that sampled a lot of other connected topics and (drum roll) actually prepared students with technical skills for careers that (drum roll) don't involve much if any calculus. These career paths actually exist. Sometimes calculus gets to be more motivated when there's real physics involved, and I've seen past postings to this archive where the view was: lets leave "calculus" to the physics department and concentrate on something "more pure" for the "real mathematicians" (segue to "real analysis" at this point). More than just feeding the snobbery of "pure math" aficionados, one could see this is a useful concrete suggestion: leave calculus to the physics department at the high school level. Those wishing a "pure math" approach will elect to pursue this later. We have a lot of history explaining how the calculus managed to insert itself as the road hog and singular "gateway discipline" it has become in today's (manifestly broken) design. It simplifies things for the education industry, to just have this one pipeline, never mind the 75%+ attrition rate. A vast army of calculus teachers gets steady employment. It's a known territory, well explored, a status quo, a comfortable regime, for those on the inside. I've taught it myself for pay, another calculus mercenary. Granted all that, is it high time for a coup? Shall we keep the precalc / calc track as an option, but not pretend it's the one true realistic ticket to a technology career? Should we stop treating calculus the way some people treat Christianity, as the one true straight and narrow, as in "my way or the highway"? Why give it such monopolistic status? Complacency breeds corruption, we all know, and here we have a manifestly broken system that rarely questions the self-interest of those most vested in its perpetuation -- a classic case of this very syndrome? Who dares to protest the hegemony of AP Calc indulgences (= less time in purgatory if you kowtow to the high priests)? The competing track or tracks, which may contain more group and number theory, more about primes and composites, more about polyhedra and vectors, Euclid's Algorithm for the GCD, more about crypto- graphy and cartography, more on hexadecimals, unicode, spherical trig, GPS/GIS, simulations, more computer programming... might also contain some calculus, just nowhere nearly as much. Some stellar documentaries would cover the basics. Lots would get previewed. We believe in whetting the appetite more than drilling or killing at this age. We're more like Asimov in that way. We'd even do some history, explain about the Newton- Leibniz contention and how Bishop Berkeley, for whom UC Berkeley was named, tried to blow calculus out of the water because of its sloppy proofs (in those days, pre lots of fancy formalisms). Telling that story (among others): http://mathforum.org/kb/message.jspa?messageID=6990127&tstart=0 Sharing history is in keeping with our more general philosophy of sharing lore, telling stories, not keeping math teaching bereft of an historical dimension, a hold-over from divide-and-conquer management strategies that are anti-liberal arts in their objective: to keep everyone so narrowly specialized so they won't be in a position to question or second guess their orders (keep the overview, the view from a height, for an inner circle of generalists **). Hansen was agreeing with me that Mathematics for the Digital Age by Gary and Maria Litvin was a worthwhile text. There's a lot to pack in to just one year in that book, and by adding topics, branching out, including more polyhedra, even more truly object oriented programming, one could imagine fleshing that out to two, three, even four years of material. Might we serve the calculus refugees in this way? This gets back to my AM versus DM track proposal. AM = analog math = pre-calc / calc. DM = digital math = still mostly under-developed, and therefore a potential growth industry, could be huge. I think it works as shorthand. The Computational Thinking course they want on math-thinking-l, edu-sig and places would fall under the DM heading. They'd even like a Discrete Math AP test, although Gary Litvin explains why that's not happening (doesn't address the real problem e.g. that 75% attrition rate with disproportionate performance by zip code area). Kirby ** per 'Operating Manual for Spaceship Earth' (recommended earlier, on my reading list at Princeton), the great pirate managers instituted this divide-and-conquer strategy, making everyone under them extremely narrow, and then were themselves unable to follow the action when the sciences took off into the invisible realms (radio, quantum physics). The great pirates died out and now we're stuck with everyone too narrow to see a big picture, so no one in charge, everyone semi-paralyzed, perhaps blaming some non-existent cabal of supposed insiders. Asimov wrote a similar essay entitled 'View from a Height' which made similar points, about how challenging it is to develop big picture views, given the knowledge explosion that engulfs us, a confusing morass of information if we don't keep hammering on those heuristics. |
In reply to this post by Robert Hansen
Kirby,
Thanks for your post here. I believe that offering a track for discrete mathematics and its related topics for those who do not want to take calculus or feel that they cannot do so will serve these students well. Much of these examples you have mentioned here provide students with ample opportunites to learn some good mathematics and to gain any necessary mathematical tools they will need without having to learn calculus. When I consider your comments here, I see no reason to deny these students this option for them to pursue a career in technology. I think that there must be some way to raise that 25% passing rate. The passing rate for algebra is so low because students lack understanding of arithmetic when they try to take algebra. To make this mess even worse, students are often evaluated for their understanding of arithmetic in meaningless or inaccurate ways. Yes, we do want students to know how to perform arithmetic computations. But that ability in itself does not mean that students really understand arithmetic well enough to give themselves a chance to succeed in algebra. Many students have been taught arithmetic with little, if any, time devoted to understanding arithmetic conceptually, to understand why the algorithms work, how to apply arithmetic to problems in which formulas are not given explicitly, etc. And not to mention that most of them do not know what the spirit behind mathematics is, that mathematics cannot be studied via memorization without understanding, what problem solving really is (yes, there are procedures they can use and should be taught, but genuine problem solving is not reducible to following recipes), etc. Learning algebra requires learning how to generalize, and students are given few--if any--opportunities to learn to generalize before they take algebra. The success rate for algebra is low because the pre-algebra classes are designed to guarantee failure for the vast majority of students. Whether this design is purposeful or not is another question, but the answer to that question does not influence the reality of what is happening. Until these problems are fixed, the success rate for algebra will continue to remain this low. In short, students need much more than arithmetic to succeed in algebra, but all that the students need are not given to them in pre-algebra classes. Even if we were currently seeing a much higher success rate than we actually see now, I would still support these additional options that Kirby suggests offering. These options are good ones to offer, regardless of what the current success rate is. Jonathan Groves On 3/5/2010 at 6:35 pm, Kirby Urner wrote: > On Fri, Mar 5, 2010 at 1:00 PM, Dave L. Renfro > <[hidden email]> wrote: > > > > << snip >> > > > The only quote I found on the internet about this > is > > the following [1], which I think isn't from the > essay itself > > but from his autobiography: > > > > "After years of finding mathematics easy, I finally > reached > > integral calculus and came up against a barrier. I > realized > > that this was as far as I could go, and to this day > I have > > never successfully gone beyond it in any but the > most > > superficial way." > > > > << snip >> > > > Dave L. Renfro > > I'll respond to this in part because Isaac Asimov was > a > big influence in my life. I was one of those kids > who > "fell for" his trick, which was to get us hooked on > science > and math through science fiction. That's exactly > what > happened in my case. From reading his novels I > developed > a healthy appetite for his non-fiction writings. > > One of Hansen's recent posts was about the > impossibility > of ever getting more than 25% through a Dolciani type > pre-calc / calc program, no matter how good the > teaching. > We're going to lose 75% due to whatever factors > ("home > life", "lack of aptitude", "math anxiety"... a long > list of > factors take the blame). > > My question is why assume the precalc / calc pipeline > as the one and only. That's like going to a water > slide > park and having only the one water slide (typically a > pipe), or going to a roller coaster park (like that > one > near LA) and finding only the one roller coaster. > > Surely mathematics is a more varied playing field, > and > if we're going to admit up front that 75% won't make > it > through calculus, then why can't we offer other > rides? > > In my 2008 talk at Pycon, I inveighed against what I > call > "Calculus Mountain", which is precisely this killer > hill > that is used specifically and by design to "weed out" > those who can't hack it. > > http://www.youtube.com/watch?v=hbeHdg8mtdc > (almost 2,500 views -- more than most journal > articles). > > Those making it over this mountain feel proud of > themselves, glad to be gifted in whatever way, but is > this really all that great a design in the first > place? > Many posters here have questioned this status quo > over the years. It's not really heresy. > > Enter discrete math and its relatives ("digital math" > still on the back burner as not well defined, except > maybe by me on Wikieducator). If we had the budget > (not saying we do, given economics is not well under- > stood), we could easily design a track for "calculus > refugees" that sampled a lot of other connected > topics > and (drum roll) actually prepared students with > technical skills for careers that (drum roll) don't > involve > much if any calculus. These career paths actually > exist. > > Sometimes calculus gets to be more motivated when > there's real physics involved, and I've seen past > postings to this archive where the view was: lets > leave "calculus" to the physics department and > concentrate on something "more pure" for the > "real mathematicians" (segue to "real analysis" at > this point). More than just feeding the snobbery > of "pure math" aficionados, one could see this is > a useful concrete suggestion: leave calculus to > the physics department at the high school level. > Those wishing a "pure math" approach will elect > to pursue this later. > > We have a lot of history explaining how the calculus > managed to insert itself as the road hog and singular > "gateway discipline" it has become in today's > (manifestly broken) design. It simplifies things for > the education industry, to just have this one > pipeline, > never mind the 75%+ attrition rate. A vast army of > calculus teachers gets steady employment. It's a > known territory, well explored, a status quo, a > comfortable regime, for those on the inside. I've > taught it myself for pay, another calculus mercenary. > > Granted all that, is it high time for a coup? Shall > we > keep the precalc / calc track as an option, but not > pretend it's the one true realistic ticket to a > technology > career? > > Should we stop treating calculus the way some > people treat Christianity, as the one true straight > and > narrow, as in "my way or the highway"? Why give > it such monopolistic status? Complacency breeds > corruption, we all know, and here we have a > manifestly > broken system that rarely questions the self-interest > of those most vested in its perpetuation -- a classic > case of this very syndrome? Who dares to protest > the hegemony of AP Calc indulgences (= less time > in purgatory if you kowtow to the high priests)? > > The competing track or tracks, which may contain > more group and number theory, more about primes > and composites, more about polyhedra and vectors, > Euclid's Algorithm for the GCD, more about crypto- > graphy and cartography, more on hexadecimals, > unicode, spherical trig, GPS/GIS, simulations, more > computer programming... might also contain some > calculus, just nowhere nearly as much. Some > stellar documentaries would cover the basics. Lots > would get previewed. We believe in whetting the > appetite more than drilling or killing at this age. > We're more like Asimov in that way. > > We'd even do some history, explain about the Newton- > Leibniz contention and how Bishop Berkeley, for > whom UC Berkeley was named, tried to blow > calculus out of the water because of its sloppy > proofs (in those days, pre lots of fancy formalisms). > > Telling that story (among others): > http://mathforum.org/kb/message.jspa?messageID=6990127 > &tstart=0 > > Sharing history is in keeping with our more general > philosophy of sharing lore, telling stories, not > keeping math > teaching bereft of an historical dimension, a > hold-over > from divide-and-conquer management strategies that > are anti-liberal arts in their objective: to keep > everyone > so narrowly specialized so they won't be in a > position to > question or second guess their orders (keep the > overview, > the view from a height, for an inner circle of > generalists **). > > Hansen was agreeing with me that Mathematics for > the Digital Age by Gary and Maria Litvin was a > worthwhile text. There's a lot to pack in to just > one > year in that book, and by adding topics, branching > out, > including more polyhedra, even more truly object > oriented > programming, one could imagine fleshing that out to > two, three, even four years of material. Might we > serve the calculus refugees in this way? > > This gets back to my AM versus DM track proposal. > AM = analog math = pre-calc / calc. DM = digital > math = still mostly under-developed, and therefore a > potential growth industry, could be huge. I think it > works as shorthand. The Computational Thinking > course they want on math-thinking-l, edu-sig and > places would fall under the DM heading. They'd even > like a Discrete Math AP test, although Gary Litvin > explains why that's not happening (doesn't address > the real problem e.g. that 75% attrition rate with > disproportionate performance by zip code area). > > Kirby > > ** per 'Operating Manual for Spaceship Earth' > (recommended > earlier, on my reading list at Princeton), the great > pirate > managers instituted this divide-and-conquer strategy, > making everyone under them extremely narrow, and then > were themselves unable to follow the action when the > sciences took off into the invisible realms (radio, > quantum > physics). The great pirates died out and now we're > stuck > with everyone too narrow to see a big picture, so no > one > in charge, everyone semi-paralyzed, perhaps blaming > some non-existent cabal of supposed insiders. Asimov > wrote a similar essay entitled 'View from a Height' > which > made similar points, about how challenging it is to > develop > big picture views, given the knowledge explosion that > engulfs us, a confusing morass of information if we > don't > keep hammering on those heuristics. |
In reply to this post by Robert Hansen
Kirby, I actually spoke with Litvin some and his discrete math. What is confusing sometimes is how you use the term "digital math". Remember when you first showed me Litvin's text and I responded that it would fit perfectly on my shelf next to my Numerical Recepies book? To me, all of that is a an authentic branch of mathematics.
After I understood more of what he was trying to do I liked it. I suggessted it as Algebra III but I don't think he fully understood my reasoning. Basically, towards the end of (authentic) Algebra II you have several lose ends where (to put it simply) the degrees of the polynomials become large or the intuitive models are simply too computational to visualize. Probability and statistics result in complex combinations and permutations. Or actual algebra resluts in higher degree polynomials. One of those loose ends leads to Calculus, but the others are kind of just left there, to be picked up later in college when needed. That seems to me the natural place where the branch of discrete math begins. It would be hard to displace AP Calculus but I do think Litvins course would be a great course for an elective in there somewhere. But the CollegeBoard is driving some sort of AP exam taking frenzy and I don't think you will get enough kids to sign up if it isn't AP discrete math. But you know what would happen to it if AP got their hands on it. Be careful what you wish for. |
In reply to this post by Robert Hansen
I see my 25% figure being thrown about. Let me clarify. This was what I suggested was probably the highest you could get the percentage of kids that do well in Algebra. That is not the current rate.
1. Around 6% of the kids currently take AP Calculus. 2. About 25% of those actually do well. 3. That means out of the calculus pipeline, about 1.5% of the kids "get it". 4. Others will get it in college as well, but the % is low. 5. I estimate that 5% of the kids currently get Algebra, and is being very optimistic. You will never get all the kids, and probably not 25% even, because kids like to do many different things. When I say "get" algebra, I mean like an engineer or a mathematician. Some do that and go on to do something else and some just go on to do something else. You don't see people wanting every kid to be a writer. You really don't need that much math for all these other jobs. How many times do I have to tell you that the most used math in the world is Excel! |
In reply to this post by Robert Hansen
Quoting Robert Hansen <[hidden email]>:
> Kirby, I actually spoke with Litvin some and his discrete math. What > is confusing sometimes is how you use the term "digital math". > Remember when you first showed me Litvin's text and I responded that > it would fit perfectly on my shelf next to my Numerical Recepies > book? To me, all of that is a an authentic branch of mathematics. > > After I understood more of what he was trying to do I liked it. I > suggessted it as Algebra III but I don't think he fully understood > my reasoning. Basically, towards the end of (authentic) Algebra II It's really too bad that the word "authentic" isn't banned from these conversations, along with the banning of "obvious" and its various forms from actual mathematics texts and classrooms. What's meant when Robert, Wayne and some others use that word is, I suspect "stuff I like and recognize from my own math education and experiences, with nothing I don't recognize or dislike." Just as when math teachers use "obvious" they mean, "It's obvious to me, hence it must be obvious to you, and if it's not, you're stupid." Well, maybe they don't MEAN to communicate that. They merely succeed in communicating it to many, many students. Which is a tiny part of what I suspect G.S. Chandy and many others are thinking about when they suggest that bad mathematics teaching practices make kids hate and fear math. It's not math itself that does so for the most part. Similarly, I find that I hate in advance anything that is described here as "authentic" (unless I have a lot of trust for the person using that word in the context s/he's using it - sorry, but it's pretty hard not to start associating the endorsements with the endorsers after a while here) until I have some reason to do otherwise. One might just as well say, "nice" rather than "authentic," as it communicates as little that is specific without a lot that remains to be demonstrated. No one seriously sits down to design an inauthentic course or write an inauthentic textbook. Given the well-known criticism of US curricula as a mile wide and an inch deep (yet, oddly, we find no criticism from that perspective of Saxon Math books that appear to shotgun spray upon the table of contents a wide variety of thinly explored topics), one would think that a few folks would be waking up to the idea of picking some big ideas and running with them for a reasonable amount of time, giving kids the experiences, challenges, and tools needed to get a handle on a deeper understanding of those big ideas and the time to explore them. Yesterday, I saw a wonderful 32 minute or so video from a 9th grade math classroom in small school (public) in Queens in which the teacher gave what clearly was a second class period spent on a problem (no way to tell what part of the PREVIOUS class was spent on it, of course, but the problem had obviously carried over since the students already had gotten to a clear-cut point with it, but needed to go further). It was remarkable how much valuable thinking and reasoning emerged strictly from three main sources in that video: student conjectures and conversations (whole class, in pairs, and in whole class again), a good problem that clearly "problematized" something that the students already believed was unproblematic (how to graph on the Cartesian coordinate plane the solutions of a linear inequality derived from a simple but realistic real-world situation), and the teacher's absolute ability to ask lovely and on-point questions that never gave anything away, to redirect inquiry back to the students, and to NEVER tell them anything directly. This wasn't apparently any special class or specially selected 9th grade. Of course, it also wasn't a video from September. A lot of work on the part of the teacher, other teachers and staff in that school, and the willingness of those kids to buy into whatever that work must have been, created a classroom culture in which those conversations were not only possible but even likely. Getting back to "curriculum," one thing that was crystal clear (and I have no idea where this problem came from, what book, if any, this teacher uses or if she pulls problems from various sources including on-line and print materials, her own reworking of such problems, etc.): there was a lot more to think about in this problem than probably the kids expected, and they were given adequate time to do that thinking. The results were impressive. And sorry, no standardized test results were shown. I just listened to what the kids said and what they came up with and the sorts of questions the teacher asked (almost all of which were of the sort "Why?" "How do you know that?" and so on. [Note to Wayne and Company: the video not only identifies the school and its location, but the name of the teacher as well. But I've not yet asked permission to share any of that information or the video and can't do so without it, since it's available at the moment only at a website that isn't public. I'll check to see just what I'm free to say beyond what I've already said (small school, public, in Queens, NY, and the basic topic of the lesson. When the usual nonsense is offered up, I'll consider offering more details about the actual content of the problem and the things students and teacher did, which frankly is all that interested me. Since I wasn't being asked to buy anything, I didn't feel a driving need to check if the students were actually plants from Bronx Science or having their voices dubbed in from students at Westinghouse High or Cass Tech).] I wish I didn't have to be so disparaging of your take on so many things (save on computer stuff, where I bow out as unqualified and inexperienced; I trust Kirby to keep you sober and responsible in that arena), Robert, but I can't help it. You just say things that seem so disconnected from real kids in real schools, the kids who aren't likely to be asked to join your hypothetical clubs and teams filled with mostly white and Asian "talented" hot shots. When you tell us that people who've written really good books don't "get" your reasoning, it's tongue-biting time. People generally are able to write books like the Litvin's book without worrying about what Robert Hansen thinks or wants to do with them. And they will continue to be able to do so. Thanks Darwin for small favors. - -- ************************** Michael Paul Goldenberg 6655 Jackson Rd Lot #136 Ann Arbor, MI 48103 734 644-0975 (c) 734 786-8425 (h) [hidden email] rationalmathed.blogspot.com It was when I found out I could make mistakes that I knew I was on to something. - Ornette Coleman ************************** - -- ************************** Michael Paul Goldenberg 6655 Jackson Rd Lot #136 Ann Arbor, MI 48103 734 644-0975 (c) 734 786-8425 (h) [hidden email] rationalmathed.blogspot.com It was when I found out I could make mistakes that I knew I was on to something. - Ornette Coleman ************************** |
On Mar 6, 2010, at 9:18 AM, Michael Paul Goldenberg wrote: It's really too bad that the word "authentic" isn't banned from these This statement is, well, authentically obvious. --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State College of Denver |
In reply to this post by Robert Hansen
Hi,
Over a year ago, members of this forum helped me by completing an online survey that I distributed to collect data for my dissertation. I was able to complete my work and defended just over a month ago. Thank you! In the comments areas, as well as during phone interviews, many participants asked for a summary of the findings. It took me some time to follow up, but now they are available in case you are interested. I have posted a summary at http://www.surveymonkey.com/s/Calc_PreCalc_Survey_Results Many thanks, Kostas D. Stroumbakis. |
In reply to this post by Robert Hansen
Dave L. Renfro wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=7002318 > This reminds me of something Isaac Asimov once wrote > concerning his math experiences. [snip] > The only quote I found on the internet about this is > the following [1], which I think isn't from the essay > itself but from his autobiography: > > "After years of finding mathematics easy, I finally reached > integral calculus and came up against a barrier. I realized > that this was as far as I could go, and to this day I have > never successfully gone beyond it in any but the most > superficial way." > > [1] In typical internet fashion, I found this same quote on > dozens of web pages, as if someone initially posted it > somewhere and then everyone and their brother just copied > it, with no one anywhere bothering to look it up and > say exactly where it came from or posting the far more > extensive comments that Asimov made in the essay I'm > thinking of (which I have, but it's at home somewhere). I looked up the Asimov comments I was thinking of. I was generally correct in what I said about them (see URL for earlier post if you want to compare what I said with what Asimov actually said), with two main exceptions. First, it wasn't from the introduction to one of his essays, but rather the comments were in one of the chapter introductions in his book "Opus 100". Second, I wrote things as if he took diff. calculus his 1st semester of college and integral calculus his 2nd semester of college, but it seems he took analytic geometry his 1st semester, diff. calculus the next semester, and then a 2-semester sequence on integral calculus his 2nd year (the 2nd year course probably covered double and triple integrals, with diff. calculus covering partial differentiation, at least this is how early 20th century calculus texts sometimes ordered the topics). I remember somewhat when I got this book, and can place the date fairly well (Fall 1970), since I got it right after the paperback version came out. There was this "candy kitchen / bookstore" in Charlotte that we used to visit often when I was growing up (I was 11-12 and in the 5th grade during Fall 1970), and I got a lot of popular science paperback books there. I had already begun reading some of Asimov (I think I was introduced to him through his book "Is Anyone There?" that my father had gotten a year or two before then), so when I saw brand new copies of "Opus 100" on one of those metal circular-swinging book racks, I just had to get it. [Perhaps my father mainly got it, but I remember looking at it in the car the rest of the day, while we did some other shopping in Charlotte and then drove the 60 some minutes back to where we lived.] By the way, that candy kitchen closed a few years later and, as far as I know, nothing like it existed elsewhere in that region (NC and SC). In the late 1960s and up until maybe 1971 or 1972, I got quite a few issues of "Erie" and "Creepy", plus lots of comic books (besides superman type stuff, one that I remember quite well and liked was "Magnus, Robot Fighter 4000 A.D."), and naturally none of them survived my leaving for college in Fall 1977. (The standard "mother cleans out son's closet" story.) What follows is the relevant Asimov quote. As far as I can tell, none of the following passages are on the internet (at a freely available and google'able web site). Any bets on how long after I post this before it starts to show up on people's blogs and web sites? My guess is 2 to 3 months . . . *************************************************** *************************************************** Isaac Asimov, "Opus 100", 1969. From the October 1970 Dell Publishing Company version, pages 89-90. When I was in grade school, I had an occasional feeling that I might be a mathematician when I grew up. I loved the math classes because they seemed so easy. As soon as I got my math book at the beginning of a new school term, I raced through it from beginning to end, found it all beautifully clear and simple, and then breezed through the course without trouble. It is, in fact, the beauty of mathematics, as opposed to almost any other branch of knowledge, that it contains so little unrelated and miscellaneous factual material one must memorize. Oh, there are a few definitions and axioms, some terminology---but everything else is deduction. And, if you have a feel for it, the deduction is all obvious, or becomes obvious as soon as it is once pointed out. As long as this holds true, mathematics is not only a breeze, it is an exciting intellectual adventure that has few peers. But then, sooner or later (except for a few transcendent geniuses), there comes a point when the breeze turns into a cold and needle-spray storm blast. For some it comes quite early in the game: long division, fractions, proportions, something shows up which turns out to be no longer obvious no matter how carefully it is explained. You may get to understand it but only by constant concentration; it never becomes obvious. And at that point mathematics ceases to be fun. When there is a prolonged delay in meeting that barrier, you feel lucky, but are you? The longer the delay, the greater the trauma when you do meet the barrier and smash into it. I went right through high school, for instance, without finding the barrier. Math was always easy, always fun, always an "A-subject" that required no studying. To be sure, I might have had a hint there was something wrong. My high school was Boys High School of Brooklyn and in the days when I attended (1932 to 1935) it was renowned throughout the city for the skill and valor of its math team. Yet I was not a member of the math team. I had a dim idea that the boys on the math team could do mathematics I had never heard of, and that the problems they faced and solved were far beyond me. I took care of that little bit of unpleasantness, however, by studiously refraining from giving it any thought, on the theory (very widespread among people generally) that a difficulty ignored is a difficulty resolved. At Columbia I took up analytical geometry and differential calculus and, while I recognized a certain unaccustomed intellectual friction heating up my mind somewhat, I still managed to get my A's. It was when I went on to integral calculus that the dam broke. To my horror, I found that I had to study; that I had to go over a point several times and that even then it remained unclear; that I had to sweat away over the homework problems and sometimes either had to leave them unsolved or, worse still, worked them out incorrectly. And in the end, in the second semester of the year course, I got (oh shame!) a B. I had, in short, reached my own particular impassable barrier, and I met that situation with a most vigorous and effective course of procedure---I never took another math course. Oh, I've picked up some additional facets of mathematics on my own since then, but the old glow was gone. It was never the shining gold of "Of course" anymore, only the dubiously polished pewter of "I think I see it." Fortunately, a barrier at integral calculus is quite a high one. There is plenty of room beneath it within which to run and jump, and I have therefore been able to write books on mathematics. I merely had to remember to keep this side of integral calculus. *************************************************** *************************************************** Dave L. Renfro |
In reply to this post by Robert Hansen
Dave L. Renfro wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=7004907 > I remember somewhat when I got this book, and can place > the date fairly well (Fall 1970), since I got it right > after the paperback version came out. There was this > "candy kitchen / bookstore" in Charlotte that we used > to visit often when I was growing up (I was 11-12 and > in the 5th grade during Fall 1970), and I got a lot of > popular science paperback books there. I had already > begun reading some of Asimov (I think I was introduced Ooops, I was in the 6th grade in Fall 1970. This makes more sense also, since I was well into reading Asimov and other popular science stuff in the 6th grade, but not so much in the 5th grade. For example, I learned about tesseracts in the 6th grade and how to draw them in two ways (which I got from somewhere in the first half of George Gamow's book "One, Two, Three ... Infinity", plus around the 6th grade I also read "A Wrinkle in Time"), complex numbers (from Asimov's "Realm of Numbers"), and other math topics that I didn't encounter in school for several years afterwards. I think we were working with fractions and decimal long division and other things in 6th grade, this being 3 school years before we could take algebra 1, and I distinctly remember not thinking of complex numbers, square roots, distance formula in R^2, R^3, and R^4, etc. as actual math, but rather as something that was an off-shoot of my interest in speculative science and science fiction. Dave L. Renfro |
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