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It might be paranoid, but I think the old style of teaching math -- a few big examples, let the student find their way on their own -- is actually designed to c

It might be paranoid, but I think the old style of teaching math -- a few big examples, let the student find their way on their own -- is actually designed to create an artificial hierarchy and ranking. Rich kids and lucky kids figure out they have to break things down into micro steps, either do that on their or hire a tutor (have a parent) to do it for them, but in class they just seem to be "bright". (I have been that tutor, and totally changed several kids from "I'm/ he is just not good at math" to "Oh -- math is straightforward if you break it down and work your ass off, like almost everything else" This is basically how Law School Confidential thinks law school manages to keep the supply of good lawyers low: fool the students into thinking that talent and intuition teach you law, and let a few students who outline and memorize like crazy "fall up" through the cracks. I say "designed", but in that mostly subconscious way that we "choose" to do most things -- it feels right, it is "obvious" because it is how we learned originally, and it doesnt lead to weird results (a world full of talented math users which might cheapen the supposed "talent" of the lucky ones). In my experience, public school teachers absolutely love to categorize kids into stupid and smart, with disastrous results unless you are one of the lucky ones (which probably has as much to do with good looks, high-prestige parents, and social skills). I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be -- but it's no fun to be an officer without a bunch of stupid enlisted men to boss around, and public school wastes a lot of taxpayers money to create those stupid people.

halostatue 15y ago
I'm not sure what your experience is with public school teachers, but I suspect that very few of them do that. None of the teachers that I know, including the one I'm married to, do what you're suggesting. Inasmuch as any teacher categorizes the students, it's to provide skills-appropriate challenges and assistance. In my experience, most peoples' experience with public school teachers is formed around their own educational experiences, which is essentially no experience at all. Being the parent of a student in public school isn't necessarily a useful experience, either, as the teacher may be trying to tell you that little Johnny isn't as bright as you think he is, but you're still insisting that he's the smartest kid in the class. My wife has had students that simply don't want to learn. They're smart kids, but they spend all their energy coming up with excuses for why they're not doing their schoolwork. My wife has had students that want to learn but are so angry at the world because of what's going on in their home life (abusive parents or siblings or just living in the projects). My wife has had a fourth grader who understood 'quarant' (French for 40) in terms of 'quarantine' and made connections that way. Public school teachers, by and large, love to teach and love to see kids learn. Yes, there's some really bad teachers out there, but there are also really bad designers and really bad software developers. For the most part, though, we (as designers and software developers) don't have to work in thoroughly dysfunctional workplaces.
- forkandwait 15y ago
  I think most public school teachers are soldiers in a system they don't even know is there, which includes "kids who don't want to learn", parents who don't give a shit, administrators who piss off parents, etc, etc. While I took a dig at them in my post, I dont actually think it is their fault, they just go with the flow, and the flow is toward barely conscious re-creation of social hierarchy. My experience of public school teachers has been from volunteering regularly as a tutor, being in the periphery of the educational system as an Upward Bound teacher and tutor. The public school teachers I have met generally taught to the "talented and driven" in their class, did a decent job, but were incredibly naive about the social system in which they did their supposedly idealistic work. And to a one, they all believed in the myth of talent and a wide bell curve (and had never heard of the pygmalion effect...)
  - jacobolus 15y ago
    Public school teachers I know (both those teaching in middle class schools and those teaching in harsh conditions in poorer areas) absolutely know that the system is there, but it’s not something trivial to change as an individual, and many of them are doing all they can just to keep their heads above water. Teaching 20 or 30 students in a class – whether you have the same students all day as in elementary school, or whether they’re in 5 hour-long batches as in middle/high school – is a tremendously demanding job under the best conditions. If most of those students are dramatically behind where the standard curriculum says they’re supposed to be (discouraged because they’re failing all the official tests), with heart-breaking home lives or time-consuming jobs outside school. Add administrative overhead and so forth, and teachers in many cases just don’t have the time and energy to give every student what he needs. It’s a risky career move and a heck of a lot of work for a teacher to decide that the state mandated curriculum is a load of crap and independently push off in a new direction. Learning math requires intense focus, and a class of 30 people who don’t want to be there makes everything really really difficult, not at all comparable to a 1-on-1 or small group tutoring situation. The “talented and driven” get some attention because they are easier to teach. But I don’t think your generalization is fair to teachers. The ones I know dump their hearts and souls into reaching the worst-off students, and go home and cry about it when they don’t feel like they’re getting through.
simpleTruth 15y ago
To play the devils advocate: IMO, it's the focus on the micro steps that's the problem. HS level math is vary simple. You can write a single textbook that covered the full range of math from preschool to calculus, but the focus on 3 days of instruction, a day of review, and then a quiz or test slows things down. It can be easier to skip ahead and then go back and review than try and approach math in tinny nibbles. It's like spending a full year going over cement foundations before you mention that the goal is to put a house on top of that flat slab. Personally, I used to do other classes homework assignments in my math classes. I can recall getting in about 4 seconds a new topic that the class spend a full week going though in minute detail. It was so bad I once accidentally did the next chapters review vs the assigned homework and did not even notice at the time. PS: I am all for better instruction, but perhaps we could consider going a little further. We could probably get the average 10th grader to really understand Calculus, but I think the goal should be to dive into DifEq and number theory etc.
- tokenadult 15y ago
  IMO, it's the focus on the micro steps that's the problem. It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems." PROBLEMS VERSUS EXERCISES I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount). Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves. "It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy. "It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician." Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix. I have since read about this distinction in several other books. "Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics: Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . . Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . . Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . . Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . . Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems." Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73. This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)" "First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . . "A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it." Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4. ". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.' "The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises." Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii. "It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance." Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.
boboblong 15y ago
"I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be" You mean "narrower" or "skinnier", not "flatter". A flatter bell curve would mean the opposite.
timwiseman 15y ago
but it's no fun to be an officer without a bunch of stupid enlisted men to boss around I do not know how you intended this to sound, but it comes off in a way that is horribly inaccurate of both officers and enlisted men. The majority of officers I know have gone out of their way to help the enlisted learn and did what they could to ensure they had access to colleges. Many officers were ready to help any enlisted person with a real desire to do so to get their education finished and get a commission themselves. On the other side of it, most of the enlisted men I knew were at or above average intelligence. Especially in certain fields such a communications and intelligence it was very common to find people who already had degrees when they enlisted or finished them quickly after enlisting.
- forkandwait 15y ago
  I don't know if you will read this, but I was just being flip, and meant nothing specifically about officers or enlisted men beyond the fact of a hierarchy that might be more based on opportunity and family than anything else. I have immense respect for all the military professionals I have knows (not that many, but still).

halostatue 15y ago

I'm not sure what your experience is with public school teachers, but I suspect that very few of them do that. None of the teachers that I know, including the one I'm married to, do what you're suggesting. Inasmuch as any teacher categorizes the students, it's to provide skills-appropriate challenges and assistance. In my experience, most peoples' experience with public school teachers is formed around their own educational experiences, which is essentially no experience at all. Being the parent of a student in public school isn't necessarily a useful experience, either, as the teacher may be trying to tell you that little Johnny isn't as bright as you think he is, but you're still insisting that he's the smartest kid in the class. My wife has had students that simply don't want to learn. They're smart kids, but they spend all their energy coming up with excuses for why they're not doing their schoolwork. My wife has had students that want to learn but are so angry at the world because of what's going on in their home life (abusive parents or siblings or just living in the projects). My wife has had a fourth grader who understood 'quarant' (French for 40) in terms of 'quarantine' and made connections that way. Public school teachers, by and large, love to teach and love to see kids learn. Yes, there's some really bad teachers out there, but there are also really bad designers and really bad software developers. For the most part, though, we (as designers and software developers) don't have to work in thoroughly dysfunctional workplaces.

forkandwait 15y ago
I think most public school teachers are soldiers in a system they don't even know is there, which includes "kids who don't want to learn", parents who don't give a shit, administrators who piss off parents, etc, etc. While I took a dig at them in my post, I dont actually think it is their fault, they just go with the flow, and the flow is toward barely conscious re-creation of social hierarchy. My experience of public school teachers has been from volunteering regularly as a tutor, being in the periphery of the educational system as an Upward Bound teacher and tutor. The public school teachers I have met generally taught to the "talented and driven" in their class, did a decent job, but were incredibly naive about the social system in which they did their supposedly idealistic work. And to a one, they all believed in the myth of talent and a wide bell curve (and had never heard of the pygmalion effect...)
- jacobolus 15y ago
  Public school teachers I know (both those teaching in middle class schools and those teaching in harsh conditions in poorer areas) absolutely know that the system is there, but it’s not something trivial to change as an individual, and many of them are doing all they can just to keep their heads above water. Teaching 20 or 30 students in a class – whether you have the same students all day as in elementary school, or whether they’re in 5 hour-long batches as in middle/high school – is a tremendously demanding job under the best conditions. If most of those students are dramatically behind where the standard curriculum says they’re supposed to be (discouraged because they’re failing all the official tests), with heart-breaking home lives or time-consuming jobs outside school. Add administrative overhead and so forth, and teachers in many cases just don’t have the time and energy to give every student what he needs. It’s a risky career move and a heck of a lot of work for a teacher to decide that the state mandated curriculum is a load of crap and independently push off in a new direction. Learning math requires intense focus, and a class of 30 people who don’t want to be there makes everything really really difficult, not at all comparable to a 1-on-1 or small group tutoring situation. The “talented and driven” get some attention because they are easier to teach. But I don’t think your generalization is fair to teachers. The ones I know dump their hearts and souls into reaching the worst-off students, and go home and cry about it when they don’t feel like they’re getting through.

simpleTruth 15y ago

To play the devils advocate: IMO, it's the focus on the micro steps that's the problem. HS level math is vary simple. You can write a single textbook that covered the full range of math from preschool to calculus, but the focus on 3 days of instruction, a day of review, and then a quiz or test slows things down. It can be easier to skip ahead and then go back and review than try and approach math in tinny nibbles. It's like spending a full year going over cement foundations before you mention that the goal is to put a house on top of that flat slab. Personally, I used to do other classes homework assignments in my math classes. I can recall getting in about 4 seconds a new topic that the class spend a full week going though in minute detail. It was so bad I once accidentally did the next chapters review vs the assigned homework and did not even notice at the time. PS: I am all for better instruction, but perhaps we could consider going a little further. We could probably get the average 10th grader to really understand Calculus, but I think the goal should be to dive into DifEq and number theory etc.

tokenadult 15y ago
IMO, it's the focus on the micro steps that's the problem. It would be very problematic, and this happens, if teaching consisting of showing micro steps is followed up by homework and assessments that also focus narrowly on micro steps. That serves up exercises for students, but it doesn't allow students the learning opportunities developed by working on actual problems. The rest of this comment is a FAQ file I send to families of new students in my math classes about the distinction between "exercises" and "problems." PROBLEMS VERSUS EXERCISES I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount). Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves. "It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy. "It is impossible to overstate the importance of problems in mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician." Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix. I have since read about this distinction in several other books. "Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics: Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . . Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . . Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . . Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . . Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems." Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73. This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)" "First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . . "A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it." Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4. ". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.' "The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises." Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii. "It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance." Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.

boboblong 15y ago

"I wouldn't say there is NO bell curve, but it could be much, much flatter if it we wanted it to be" You mean "narrower" or "skinnier", not "flatter". A flatter bell curve would mean the opposite.

timwiseman 15y ago

but it's no fun to be an officer without a bunch of stupid enlisted men to boss around I do not know how you intended this to sound, but it comes off in a way that is horribly inaccurate of both officers and enlisted men. The majority of officers I know have gone out of their way to help the enlisted learn and did what they could to ensure they had access to colleges. Many officers were ready to help any enlisted person with a real desire to do so to get their education finished and get a commission themselves. On the other side of it, most of the enlisted men I knew were at or above average intelligence. Especially in certain fields such a communications and intelligence it was very common to find people who already had degrees when they enlisted or finished them quickly after enlisting.

forkandwait 15y ago
I don't know if you will read this, but I was just being flip, and meant nothing specifically about officers or enlisted men beyond the fact of a hierarchy that might be more based on opportunity and family than anything else. I have immense respect for all the military professionals I have knows (not that many, but still).