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#431
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cover article in Time magazine on gifted education
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#432
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cover article in Time magazine on gifted education
In article .com,
Beliavsky wrote: That is a sign of a deficient education, almost as bad as not knowing the alphabet. I think kids who have not mastered their addition and multiplication tables should not be permitted to join middle school. Maybe there should be elementary school exit exams. We don't have middle school, so I don't know what age you are talking about. I was probably at about 80% accuracy by the end of 6th grade. What would you rate as a pass? Funnily enough, I had no problems with mathematical concepts, techniques or anything else, once calculators were allowed in Year 8. In 6th grade, I would have told you that I was "bad at maths" because I didn't know that arithmetic mathematics. Actually, I am bad at arithmetic. I am rather good at maths. In a school in India attended by a niece, children learn the multiplication tables at age 5. Poor kids. I have already mentioned my son's classmate, who had addition facts memorised at age 5. Unfortunately, she had no idea what to do with them apart from recite them in order. IMO that is not mathematical thinking! -- Chookie -- Sydney, Australia (Replace "foulspambegone" with "optushome" to reply) "Parenthood is like the modern stone washing process for denim jeans. You may start out crisp, neat and tough, but you end up pale, limp and wrinkled." Kerry Cue |
#433
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cover article in Time magazine on gifted education
In article . com,
Beliavsky wrote: While you take a few seconds to reason out what 8 x 9 is, and perhaps get it wrong (a process requiring more steps has a higher chance of error), someone who has it memorized has progressed to something else. In many math exams, time is a factor, and the person who has it memorized will be at an advantage. In my high school maths exams, marks were always given for correct working. When I was not allowed access to a calculator, I still did well, as only small proportion of marks was allotted to correct computation. If your maths exams beyond primary school are really putting a higher premium on accurate computation than on correct working of problems, which is what you are implying, you guys are stuffed. -- Chookie -- Sydney, Australia (Replace "foulspambegone" with "optushome" to reply) "Parenthood is like the modern stone washing process for denim jeans. You may start out crisp, neat and tough, but you end up pale, limp and wrinkled." Kerry Cue |
#434
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cover article in Time magazine on gifted education
Chookie wrote:
In article . com, Beliavsky wrote: While you take a few seconds to reason out what 8 x 9 is, and perhaps get it wrong (a process requiring more steps has a higher chance of error), someone who has it memorized has progressed to something else. In many math exams, time is a factor, and the person who has it memorized will be at an advantage. In my high school maths exams, marks were always given for correct working. When I was not allowed access to a calculator, I still did well, as only small proportion of marks was allotted to correct computation. If your maths exams beyond primary school are really putting a higher premium on accurate computation than on correct working of problems, which is what you are implying, you guys are stuffed. In my experience, higher level math classes often allow the use of calculators or give at least partial credit if a correct approach was used. Best wishes, Ericka |
#436
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cover article in Time magazine on gifted education
Chookie wrote:
In article , Ericka Kammerer wrote: And what is your evidence that if they'd just been exposed to these things earlier, they'd have grasped them easily? What's to say that they wouldn't have been equally confused earlier? Well, my Dad insisted on teaching me Boolean logic at age 7 because he said that if I were any older, I wouldn't be able to understand it! ;-) But that's not a particularly abstract concept, and I suspect that to some degree it is true that it's easier to get before you've had too many years of assumptions built up. Nevertheless, I don't think the notion that there is no such thing as a developmental curve holds water. Some things simply aren't grasped well until one is developmentally ready, and when one *is* developmentally ready, they come quickly and easily. Flogging them before that point is a waste of time. Best wishes, Ericka |
#437
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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: Do not confuse the ability to solve with understanding. In my book, it's a rather poor sort of understanding that doesn't lead to an improved ability to *do* something. It leads to the ability to formulate problems in the appropriate language. Only if you remember quite a bit more than the basic concept. Not much more, if that. If you understand the "real world" problem in precise terms, and if those concepts are sufficiently close to the concepts of the mathematical system involved, you can formulate the problem. You might not be able to solve that formulated problem, and it may be that nobody can do better than find an approximation. In many situations, even that cannot be done. But what good would it be to know how to solve if one does not have the basic concepts to formulate?l If you are faced with a medical decision, you need to formulate your preferences in quantitative terms, together with your probability evaluations, both of which may take computing. Then using the information from the medical people, you can evaluate which procedure should be followed. I would have difficulty doing this with my abilities to calculate and my knowledge of probability and decision making without going to a computer to take my evaluation and tell me what I would consider to be the best result. The best I can do with the help of the computer would be an approximation. 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. 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. Here again, learning how to get answers makes if very difficult to learn how to aks the questions. Also, even without the memory issue, I don't buy that there is no relationship between understanding a concept thoroughly enough to formulate solutions well and being able to solve the problem. Obviously, there are problems that are essentially unsolvable by hand due to their complexity, but my experience is that working through at least some problems by hand (in a somewhat simplified version, if necessary) generally is very useful in helping people understand what they're learning. No one has argued that every problem should always be solved by hand. The educationist using statistics puts his data into a computer program. If he knows how to do it by hand for a simple problem, it becomes no easier to get the results. If he uses a poor formulation of the problem, the computer may well give him a poor answer; the computer is a super-fast sub-imbecile, and does not think. 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. -- 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 |
#438
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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: 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. 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. 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. 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. The concept of variable is an abstract concept. Do not make it an abstraction of something else: even though the idea may have evolved from less, it is easier to understand it as the SIMPLE idea than to try to build it up. Again, you still have not backed this notion up with anything other than your personal assertion. I do not see that someone who has difficulty recognizing that variable is a simple abstract concept can get the point. 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. 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. 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. When I say that the biggest problem is the use of vocabulary which a third grader (or even a fifth grader) would have difficulty with, I have some idea of what that means. Also, there is a matter of presentation; the notation in Suppes and Hill, which left out a particular part, is harder to understand than that in the college book by Suppes, which I used with that for my children, and the notation in my late wife's book is simpler. 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. 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. The rest of mathematics will not fall out of the sky, but they will have the language to be able to see what is and what is not a proof. Ok. And? Euclid came close for his geometry, and the educationists have changed the geometry course to learning facts and computations instead of 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. BTW, I object to the rules about what paper to use, etc. I know (second hand; the person who told me had it first hand) that when the group asked by SMSG to produce a high school geometry book did an excellent job, with all the axioms left out by Euclid added, they were told that this book would be put on the list, but that they should produce an easier one, using algebra and computation. Recall that Euclid could not use algebra; he could use symbols for geometric objects, but nothing else. Even a half century ago, it was known that the only real mathematics course in high school was this geometry course. You can call it real or not as you please. I don't know that I buy the assertion that all students ought to learn only "real math" by that definition, or even that following your approach is the best way to teach the math that they do need to know. 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. -- 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 |
#439
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cover article in Time magazine on gifted education
In article ehrebeniuk-6C9865.22235707092007@news,
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. I have to disagree. It is the gifted who have the ability to generalize, and even most of them have problems in getting rid of the garbage of the special cases. It is still the case that most measure theory courses start with measures on the real line, and cause problems thereby. I do not think that anyone has difficulty with the concrete after the basic abstract ideas are learned. But we cannot teach abstract ideas the way we teach facts and methods. One does not understand an abstract idea until the "light bulb" goes on. Examples may be needed to do this, but the general concept has to be there, and kept in mind. If the examples are learned and used before the abstract concept is presented, problems often arise. I have seen far too many good students who seem to run into a stone wall when they hit abstract concepts. In set theory, and I do not mean set algebra, it is necessary to take "set" as undefined! -- Chookie -- Sydney, Australia (Replace "foulspambegone" with "optushome" to reply) -- 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 |
#440
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cover article in Time magazine on gifted education
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. Even a weak learner can go from theory to practice. 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. It may still be difficult to compute; good mathematical computation is NOT taught. -- 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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