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#442
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cover article in Time magazine on gifted education
Herman Rubin wrote:
In article , Ericka Kammerer wrote: Herman Rubin wrote: If you understand the concepts, you can do this. If you could compute perfectly in your head, you could do no better. Again, that very much depends, particularly if you're still arguing that one never forgets concepts. I had lots of probability and statistics. For a while, I used it regularly. It's been quite a few years since then. I do not retain enough to properly formulate anything but relatively simple statistical problems. To do more, I would have to go back and study. Did you have the basic concepts? I very much regret that probability is not taught without having the concepts of measure and integral (one does not need calculus for this) first, and in fact it is hidden. Also, attempts to define probability are misleading, and the overuse of "equally likely" makes things harder to learn. Again, it is not about computing answers to simple problems, but knowing how to formulate. Yes, I had plenty of probability and stats (and calculus, and formal logic, and "modern algebra," and so on and so forth). I got it. I simply have forgotten much of it, thanks to disuse. I have also forgotten enough Latin that I can't sit down and read Ceasar easily, enough music theory that I can no longer easily take dictation for more complicated chord progressions, and enough accounting & finance that I'd need to brush up before attempting any serious valuation of a business. It's use it or lose it for most of us. As for statistics, the basic principle of statistical decision making is It is necessary to consider all consequences of the proposed action in all states of nature. If you remember what was taught in the methods courses you probably took, you can show that most of them violate that one sentence on very simple grounds. Of course they do. Living in the real world, however, one does not expect that most, if any, real life studies will achieve that lofty goal. One does the best one can, and then one attempts to recognize the limitations on any results imposed by the failure to achieve perfect methods. Well, I can hardly speak to any of that, as I seem not to have met any of these "educationists" you keep speaking of. Certainly, none of my children's teachers have thought that it was ok for them not to understand concepts or do problems mechanically, incorrectly, or without any understanding. I strongly question whether any of your children's teachers understand the basic concepts of mathematics. Feel free to question. I have felt quite comfortable with most of their approaches, and feel quite comfortable with my children's resulting achievements in math. Do they all have a background identical to yours? Probably not, but I don't find that to have been an impediment. Most of my math teachers in elementary school didn't have that background either, and it didn't seem to lead to any serious issues in my learning (or applying) math either. So, I must say, I'm not terribly concerned about my children's future at the moment. Now, I realize that we are blessed with good schools here, so I'm not suggesting the scenario is as rosy everywhere, but my experience at this point doesn't really lead me to put a lot of faith in the notion that the strategies you advocate are required to teach mathematics. Best wishes, Ericka |
#443
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cover article in Time magazine on gifted education
Herman Rubin wrote:
In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: If you show kindergarteners a bunch of blocks, let them count them and determine that there are 10 of them, and then push some of them to one side and the rest to the other *while they're watching and can see that you didn't remove or add any blocks*, and then ask them how many blocks there are in total, *most* of them will not know that there are still 10 blocks. They're not going to get the notion that a symbol can be a representation for the abstraction that is a variable. What does the above have to do with the concept of variable? A much more derived result of mathematics than the simple concepts is involved here. The fact, that if a set is divided, the number of objects in the two sets together equals the original number is a theorem, which is harder to prove from the axioms than you seem to think if the easier ordinal approach is used. My point is that it is something that is very basic and easily understood and demonstrated by children just a few months older when they are developmentally able to deal with the abstraction required. Up until that developmental turn has been taken, it is difficult even for very smart kids. If they can't get something that simple (they're not being asked to prove it, after all), how are they going to deal with even more abstract concepts? What you think is basic is something I see as having many simple but not yet understood steps. The idea that symbols can represent objects, actions, descriptions, etc., is not of that form. It is pure simple language. No, it's not. The notion of a variable that can represent a wide variety of things is a pretty serious abstraction. At that age, language is much more concrete, usually representing a 1-1 correspondence between the word and that which it represents. Could you prove it? First of all, what would it matter in this context? You asserted that young children (before the age where they're typically understood to have a firm grasp of abstract concepts) can learn abstract concepts easily if only one refrains from attempting to lead up to the abstract concept by way of more concrete examples. I suggested one abstract concept that most kindergarteners demonstrably do not grasp, but typically do grasp a short time later as they begin to move up that developmental curve. Whether or not you or I can prove that particular mathematical theorem is largely irrelevant to the issue of whether or not this is an abstract concept that a young child can grasp. If you cannot prove that theorem, or even have an idea of how to go about it, are you sure that you can properly present the idea? As I have repeatedly stated, the attempts to teach mathematical concepts to teachers have been extremely unsuccessful, and that includes those who have become high school teachers of mathematics. I do not for a moment believe that one has to be able to prove something in order to grasp a concept. The world is far too full of exceptions to that rule. I will agree that if you can prove something, you likely understand something at a higher level, but not that it is essential to understand everything at that higher level from the get-go. When they come out of high school now, they do not have the development to prove it, or even indicate a proof. I believe that a good program would enable a child who has learned the concepts and what addition is could sketch a proof. I rather suspect that most high school students could swing such a proof if that were something that was taught. I doubt most kindergarteners (or even first or second graders) could. They might produce a memorized proof. Well, I sure as heck didn't produce memorized proofs, since the proofs I was assigned for homework hadn't been given to me previously. Seeing as the neighbor kids seem to have rather similar homework, at least around here, they still seem capable of producing novel (to them) proofs. Now, are there areas where proofs aren't taught anymore? There may well be. As far as I can tell, here isn't one of them. When I started teaching, the binomial theorem, and the derivatives of powers, were proved by induction. Now, the difficulties of teaching induction are so great that this has been dropped. Hand waving, and argument by fiat, are used. So the student gets the idea that calculus methods are to be memorized, and plugged in. Those students, even if they remember all the formulas, cannot do anything but compete poorly against computer packages. Well, my kids haven't been to calculus yet (nor have the neighbor kids), so I can't for sure say what they are teaching in calculus here. I have not denied that a variable is a simple abstract concept. I have said that until children are developmentally ready, they are not going to master even simple abstract concepts. If they cannot understand the notion of a variable, they are in no position to attempt mathematics. I am not even sure that they are ready to read. And yet somehow they manage to begin reading and learning math despite not yet being able to manage more abstract concepts. Again, based on what evidence? You're just basically asserting that something that has worked with 5th graders will automatically work with 3rd graders. How do you know that? Partly because I understand what is in it and what the problems are. I used it to teach my children, one before age 6, and the other somewhat later. And what is your evidence that these two cases are representative? Because it is SIMPLE. Putting it as late as that is because a certain amount of vocabulary is needed. Again, there's a whole body of research regarding the development of abstract thinking. Where is your critique of this literature to say that you are right and it is wrong? Seeing as precision is of interest. If one starts with the view that abstract thinking can only come through the process of abstraction, you will not make any attempts to teach abstract concepts directly. The only traditional mathematics course which made any such attempt is the "Euclid" geometry. I have not said anything about how the teaching of abstract concepts should be approached. I have said that young children are not ready to deal with abstract concepts until they have reached a certain point developmentally. I don't particularly care *how* you attempt to convey the concept. However, there is the game "WFF 'N PROOF", which starts out with versions for small children, which teaches well formed expressions (formulas) and proofs, and everything is symbolic; the notation is Polish, which has no connection to Poland except it was developed by a Pole. It has no parentheses. I'm familiar, thanks. And note that "symbolic" and "abstract" are not the same thing. And there are plenty more sources that teach formal logic, some even in child-friendly ways. Nevertheless, I rather doubt you will find many kindergarteners who are ready for it, nor do I think that if you teach them formal logic that the rest of mathematics will just fall out of the sky and bonk them on the head. Formal logic is not just the sentential calculus. I'm sorry. Did I say that it was somewhere? Arguments with quantifiers are the hard part. In fact, some books teach the sentential calculus through truth tables; whatever method is used, the connectives and quantifiers are the basic concepts. What Aristotle did is NOT adequate. Again, what is the relevance here? You made a claim about formal logic: These books teach formal logic, not any other subject, through the first-order predicate calculus. This is what is needed for mathematics, but does not require mathematics to understand. I said that formal logic was not sufficient for teaching math (nor do I think it is necessary at the elementary level) and expressed skepticism that kindergarteners would hit the ground running with it. Then, you come back with formal logic not being just the sentential calculus. What's your point here? Again, I'm beginning to wonder if these "educationists" are mythical beasts. Proofs are still a core of geometry around here, and were when I took geometry as well. I recall fondly [cough] Mrs. Montagna and her rules about precisely how proofs were to be written up (on white, unlined paper, folded just so, in ink...). How many students now take the proof oriented geometry course? Check in any high school which is not of the honors variety; you will find it small. Well, I have no idea what it is like everywhere. I can tell you what it was like when I was taking geometry (plenty of proofs, thank you very much). In my county, proofs are a required part of geometry, according to county standards (including for non-honors courses). BTW, I object to the rules about what paper to use, etc. Well, so did I, but Mrs. Montagna was a very old- fashioned teacher and she did believe in such things. While it was annoying, I don't think it was particularly harmful. Every teacher has his or her peccadillos. I'm willing to spot 'em a few as long as they don't interfere with the learning. If we wait until the teachers understand the basic concepts of mathematics, they will never learn the basics, and only the geniuses will have a chance to understand them. The math that you think they need to know can be done for them, and more and more is. Understanding concepts and formulating are what can be human; the rest is merely mechanical. So far, I have yet to see that that lack of understanding is pervasive here. Perhaps it is elsewhere. I recall a study a few years ago comparing advanced high school calculus students from Japan and the US. IIRC, the both groups of students performed equally well on more conceptual questions, but (given that the test did not allow calculators), the Japanese kids beat the pants off the US kids when it came to problems requiring more challenging computation (with many of the US students not being able to solve the problems at all without a calculator). Doesn't sound like there's a huge emphasis on plug'n'chug to me. In addition, isn't the whole controversial "reform calculus" (and reform math in general) supposed to focus more on concepts and less on mechanics? Best wishes, Ericka |
#444
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cover article in Time magazine on gifted education
Herman Rubin wrote:
In article , Ericka Kammerer wrote: Chookie wrote: In article , (Herman Rubin) wrote: Abstract ideas are NOT merely abstractions of more concrete ones, but exist by themselves. Done that way, children can understand them. Only if they are developmentally ready. Claiming that they exist independently does not suddenly make them less abstract and more accessible. It makes them MORE abstract, and hence more accessible. The abstract idea, when understood, is simpler than what it is an abstraction of, if presented that way. I thought I'd mentioned this earlier, but apparently it is a distinctive of gifted people that they work more easily from the abstract to the concrete, from theory to practice. Average learners go the opposite way. But even gifted kids have to scale the developmental curve, and will not be ready for higher level abstractions until they're ready for it. That might be a bit sooner than for others, but it's not instantaneous. Also, there's a difference between abstract concepts and general/theory vs. specific/practice. What is a "higher level abstraction"? Generally, the more abstract, the easier, IF one does not make a big issue about what it means. A higher level of abstraction means more removed from the concrete. And young children do not think the same way adults or older children do, particularly in terms of how they are able to reason about abstractions. So, while it may be the case that your assertions hold true for those who are developmentally able to deal with abstractions, I seriously doubt it holds true for those who are not yet at that point. Even a weak learner can go from theory to practice. I'm not sure that assertion holds up. Some studies at least suggest that weaker learners do better the other way 'round. Personally, I don't have an opinion there--haven't looked at the issue enough. *I* prefer to go from theory to practice when learning known material, but don't know that's representative. If one understands something, and I do not mean knows the words or even knows how to prove the theorems, it is easy to apply. Well, sure. I don't believe anyone has disputed that. (Though I have suggested that believing one understands something is not always the same as understanding something, and thus it is necessary to test one's understanding through application.) Best wishes, Ericka |
#445
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cover article in Time magazine on gifted education
"Herman Rubin" wrote in message ... In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: ................. No, it's not. The notion of a variable that can represent a wide variety of things is a pretty serious abstraction. At that age, language is much more concrete, usually representing a 1-1 correspondence between the word and that which it represents. They do know about pronouns, and the ambiguity in their use. They also know of ambiguity in common nouns, and there are quite of few of them such as boy, girl, table, chair, raindrop, dog, cat, rabbit, and enough more for them to realize that this is not the case. They can handle a story in which rabbits are named Flopsy, Mopsy, Cottontail, and Peter. How hard is it to get across the idea that they can have any other set of names. It is exactly this which can make it difficult. Very young children overgeneralize. If something's furry, they may call it "doggie" or if they see a woman, they may call her "mommy". Later, they add specifics. That doggie is actually a cat, and his name is Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And everything is in relation to the child. To a very young child, EVERYTHING is a variable. As they grow up, they start getting the idea that some things are fixed, can be trusted, can be depended on and which can't. Because that dog is NOT a cat, and he's not a horse either, even though all three are items in the set of "furry animals". Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're not interchangable parts. And, no, the world doesn't stop when you're not there, and no, it doesn't revolve around you. Until a child is through this stage, which continues into early elementary school, I don't think the concept of "they can have any name you give them" is going to have the right effect-because young children already believe this, and are slowly but surely learning that this ISN'T the case for most of the things they encounter in day to day life. |
#446
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cover article in Time magazine on gifted education
In article ,
Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: ................. No, it's not. The notion of a variable that can represent a wide variety of things is a pretty serious abstraction. At that age, language is much more concrete, usually representing a 1-1 correspondence between the word and that which it represents. They do know about pronouns, and the ambiguity in their use. They also know of ambiguity in common nouns, and there are quite of few of them such as boy, girl, table, chair, raindrop, dog, cat, rabbit, and enough more for them to realize that this is not the case. They can handle a story in which rabbits are named Flopsy, Mopsy, Cottontail, and Peter. How hard is it to get across the idea that they can have any other set of names. Could you prove it? First of all, what would it matter in this context? You asserted that young children (before the age where they're typically understood to have a firm grasp of abstract concepts) can learn abstract concepts easily if only one refrains from attempting to lead up to the abstract concept by way of more concrete examples. I suggested one abstract concept that most kindergarteners demonstrably do not grasp, but typically do grasp a short time later as they begin to move up that developmental curve. Whether or not you or I can prove that particular mathematical theorem is largely irrelevant to the issue of whether or not this is an abstract concept that a young child can grasp. If you cannot prove that theorem, or even have an idea of how to go about it, are you sure that you can properly present the idea? As I have repeatedly stated, the attempts to teach mathematical concepts to teachers have been extremely unsuccessful, and that includes those who have become high school teachers of mathematics. I do not for a moment believe that one has to be able to prove something in order to grasp a concept. This is certainly true, and I have made the point quite often here and on the mathematics and statistics groups. However, I do not think that one can understand the concepts of the integers without having some idea of the simple proofs by induction, even if the details are not remembered. The world is far too full of exceptions to that rule. I will agree that if you can prove something, you likely understand something at a higher level, but not that it is essential to understand everything at that higher level from the get-go. Being able to prove something does not guarantee the understanding of the underlying concept or concepts, although it is more likely than being able to compute answers. There are, in fact, simple theorems for which the simple, but not so short, proofs give far more of an understanding of the theorem than short proofs using high-powered results, and I am quite capable of both. When they come out of high school now, they do not have the development to prove it, or even indicate a proof. I believe that a good program would enable a child who has learned the concepts and what addition is could sketch a proof. I rather suspect that most high school students could swing such a proof if that were something that was taught. I doubt most kindergarteners (or even first or second graders) could. They might produce a memorized proof. Well, I sure as heck didn't produce memorized proofs, since the proofs I was assigned for homework hadn't been given to me previously. Seeing as the neighbor kids seem to have rather similar homework, at least around here, they still seem capable of producing novel (to them) proofs. That is what the goal of teaching should be. And often these novel proofs are much better than the ones previously known. Now, are there areas where proofs aren't taught anymore? There may well be. As far as I can tell, here isn't one of them. There are, and in many, even if those courses exist, not all good students get an exposure to it. When I started teaching, the binomial theorem, and the derivatives of powers, were proved by induction. Now, the difficulties of teaching induction are so great that this has been dropped. Hand waving, and argument by fiat, are used. So the student gets the idea that calculus methods are to be memorized, and plugged in. Those students, even if they remember all the formulas, cannot do anything but compete poorly against computer packages. Well, my kids haven't been to calculus yet (nor have the neighbor kids), so I can't for sure say what they are teaching in calculus here. I have not denied that a variable is a simple abstract concept. I have said that until children are developmentally ready, they are not going to master even simple abstract concepts. If they cannot understand the notion of a variable, they are in no position to attempt mathematics. I am not even sure that they are ready to read. And yet somehow they manage to begin reading and learning math despite not yet being able to manage more abstract concepts. Are they learning math? Or are they learning to calculate? Again, based on what evidence? You're just basically asserting that something that has worked with 5th graders will automatically work with 3rd graders. How do you know that? Partly because I understand what is in it and what the problems are. I used it to teach my children, one before age 6, and the other somewhat later. And what is your evidence that these two cases are representative? Because it is SIMPLE. Putting it as late as that is because a certain amount of vocabulary is needed. Again, there's a whole body of research regarding the development of abstract thinking. Where is your critique of this literature to say that you are right and it is wrong? Seeing as precision is of interest. If one starts with the view that abstract thinking can only come through the process of abstraction, you will not make any attempts to teach abstract concepts directly. The only traditional mathematics course which made any such attempt is the "Euclid" geometry. I have not said anything about how the teaching of abstract concepts should be approached. I have said that young children are not ready to deal with abstract concepts until they have reached a certain point developmentally. I don't particularly care *how* you attempt to convey the concept. At what age are children ready to understand the concept that sequences of symbols can stand for ideas? Or that one can attack a sequence of symbols by using rules of pronunciation? However, there is the game "WFF 'N PROOF", which starts out with versions for small children, which teaches well formed expressions (formulas) and proofs, and everything is symbolic; the notation is Polish, which has no connection to Poland except it was developed by a Pole. It has no parentheses. I'm familiar, thanks. And note that "symbolic" and "abstract" are not the same thing. And there are plenty more sources that teach formal logic, some even in child-friendly ways. Nevertheless, I rather doubt you will find many kindergarteners who are ready for it, nor do I think that if you teach them formal logic that the rest of mathematics will just fall out of the sky and bonk them on the head. Formal logic is not just the sentential calculus. I'm sorry. Did I say that it was somewhere? Arguments with quantifiers are the hard part. In fact, some books teach the sentential calculus through truth tables; whatever method is used, the connectives and quantifiers are the basic concepts. What Aristotle did is NOT adequate. Again, what is the relevance here? You made a claim about formal logic: These books teach formal logic, not any other subject, through the first-order predicate calculus. This is what is needed for mathematics, but does not require mathematics to understand. I said that formal logic was not sufficient for teaching math (nor do I think it is necessary at the elementary level) and expressed skepticism that kindergarteners would hit the ground running with it. Then, you come back with formal logic not being just the sentential calculus. What's your point here? The predicate arguments are the harder ones. Many logic books do not even do sentential calculus from scratch, but use truth tables. This cannot be done for the first-order predicate calculus. There is a big difference between being able to understand what a proof is, which should be required of all, and being able to produce proofs. Again, I'm beginning to wonder if these "educationists" are mythical beasts. Proofs are still a core of geometry around here, and were when I took geometry as well. I recall fondly [cough] Mrs. Montagna and her rules about precisely how proofs were to be written up (on white, unlined paper, folded just so, in ink...). How many students now take the proof oriented geometry course? Check in any high school which is not of the honors variety; you will find it small. Well, I have no idea what it is like everywhere. I can tell you what it was like when I was taking geometry (plenty of proofs, thank you very much). In my county, proofs are a required part of geometry, according to county standards (including for non-honors courses). In that case, your county is quite unusual. BTW, I object to the rules about what paper to use, etc. Well, so did I, but Mrs. Montagna was a very old- fashioned teacher and she did believe in such things. While it was annoying, I don't think it was particularly harmful. Every teacher has his or her peccadillos. I'm willing to spot 'em a few as long as they don't interfere with the learning. If we wait until the teachers understand the basic concepts of mathematics, they will never learn the basics, and only the geniuses will have a chance to understand them. The math that you think they need to know can be done for them, and more and more is. Understanding concepts and formulating are what can be human; the rest is merely mechanical. So far, I have yet to see that that lack of understanding is pervasive here. Perhaps it is elsewhere. I recall a study a few years ago comparing advanced high school calculus students from Japan and the US. IIRC, the both groups of students performed equally well on more conceptual questions, but (given that the test did not allow calculators), the Japanese kids beat the pants off the US kids when it came to problems requiring more challenging computation (with many of the US students not being able to solve the problems at all without a calculator). Doesn't sound like there's a huge emphasis on plug'n'chug to me. Quite a few years ago, I taught a probability course with the full calculus sequence as a prerequisite. This course satisfied the probability requirement for a teaching major, as did a lower course with fewer prerequisites, and it was not intended for those. To make a long story short, on a take-home part of the final (they never could have handled it on an in-class exam), only 5 of the 21 such had any idea how to set up problems involving calculus similar to the example problems or homework problems, and these were discussed in detail in class. In addition, isn't the whole controversial "reform calculus" (and reform math in general) supposed to focus more on concepts and less on mechanics? To do this, you have to go "all out". Doing it part way achieves little. But I know of no such calculus courses at the college level; the physicists and engineers want their students to be able to solve applications using calculus yesterday. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558 |
#447
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cover article in Time magazine on gifted education
Donna Metler wrote:
"Herman Rubin" wrote in message ... In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: ................. No, it's not. The notion of a variable that can represent a wide variety of things is a pretty serious abstraction. At that age, language is much more concrete, usually representing a 1-1 correspondence between the word and that which it represents. They do know about pronouns, and the ambiguity in their use. They also know of ambiguity in common nouns, and there are quite of few of them such as boy, girl, table, chair, raindrop, dog, cat, rabbit, and enough more for them to realize that this is not the case. They can handle a story in which rabbits are named Flopsy, Mopsy, Cottontail, and Peter. How hard is it to get across the idea that they can have any other set of names. It is exactly this which can make it difficult. Very young children overgeneralize. If something's furry, they may call it "doggie" or if they see a woman, they may call her "mommy". Later, they add specifics. That doggie is actually a cat, and his name is Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And everything is in relation to the child. To a very young child, EVERYTHING is a variable. As they grow up, they start getting the idea that some things are fixed, can be trusted, can be depended on and which can't. Because that dog is NOT a cat, and he's not a horse either, even though all three are items in the set of "furry animals". Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're not interchangable parts. And, no, the world doesn't stop when you're not there, and no, it doesn't revolve around you. Until a child is through this stage, which continues into early elementary school, I don't think the concept of "they can have any name you give them" is going to have the right effect-because young children already believe this, and are slowly but surely learning that this ISN'T the case for most of the things they encounter in day to day life. In addition to that, while there are similarities between variables and things like pronouns, that doesn't necessarily make pronouns all that abstract. Young children are still using pronouns to refer to fairly concrete things-- people, objects, etc. The ability to deal with abstraction isn't just about understanding that there are "catch all" names for concrete things that would otherwise be referred to by a specific name. An abstract concept, on the other hand, is abstract regardless of how it's referred to. Best wishes, Ericka |
#448
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cover article in Time magazine on gifted education
In article ,
Donna Metler wrote: "Herman Rubin" wrote in message ... In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: Herman Rubin wrote: In article , Ericka Kammerer wrote: ................. No, it's not. The notion of a variable that can represent a wide variety of things is a pretty serious abstraction. At that age, language is much more concrete, usually representing a 1-1 correspondence between the word and that which it represents. They do know about pronouns, and the ambiguity in their use. They also know of ambiguity in common nouns, and there are quite of few of them such as boy, girl, table, chair, raindrop, dog, cat, rabbit, and enough more for them to realize that this is not the case. They can handle a story in which rabbits are named Flopsy, Mopsy, Cottontail, and Peter. How hard is it to get across the idea that they can have any other set of names. It is exactly this which can make it difficult. Very young children overgeneralize. If something's furry, they may call it "doggie" or if they see a woman, they may call her "mommy". Later, they add specifics. That doggie is actually a cat, and his name is Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And everything is in relation to the child. To a very young child, EVERYTHING is a variable. As they grow up, they start getting the idea that some things are fixed, can be trusted, can be depended on and which can't. Because that dog is NOT a cat, and he's not a horse either, even though all three are items in the set of "furry animals". Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're not interchangable parts. And, no, the world doesn't stop when you're not there, and no, it doesn't revolve around you. Until a child is through this stage, which continues into early elementary school, I don't think the concept of "they can have any name you give them" is going to have the right effect-because young children already believe this, and are slowly but surely learning that this ISN'T the case for most of the things they encounter in day to day life. Are you sure? This has NOT been tried, and I did not even try it at all with my first child, and only did it with my second child later. The difference in their abilities was great, and they both would have benefited from programs which went at their mental levels. The first one got it without my teaching, from the books I have mentioned before which use difficult vocabulary, when in kindergarten. The second got in in connection with learning the order of the alphabet; variables for the various problems were used in alphabetic order, with no regard to what was being discussed. As I have said, the material being used for teaching was not intended for the age at which they were taught, and there were other drawbacks with them. Children can learn abstract ideas if they are taught. How hard it is to teach them is not at all clear, but the idea that the process of abstraction, which is quite difficult, is needed to understand abstract ideas, is just plain false. Another point; it is not necessary to be able to prove theorems, or to carry out calculations, to understand the underlying principles or the results. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558 |
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