[ARCHIVED THREAD] - What is Math? (Page 1 of 3)
Posted: 7/18/2011 11:22:02 AM EDT
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Moving a discussion from another thread here, as it deserves its own thread.
It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG This is not GD. Argue the point, or troll elsewhere please. |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG This is not GD. Argue the point, or troll elsewhere please. I'm not arguing, I am stating a fact. Math is not problem solving. Math is the application of memorized rules to determine a solution. I've stated my position. You have stated your position. You want links, instead of comments about how I *feel* about math? http://www.tutorfi.com/Math/rulesinmathematics http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx http://www.enotes.com/math/q-and-a/application-lagrange-rule-235653 http://www.sosmath.com/matrix/determ2/determ2.html http://mathworld.wolfram.com/LHospitalsRule.html http://betterlesson.org/lesson/7641/prime-factor-gcf-lcm-divisibility-rule-application Your move, Euler. TRG |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG This is not GD. Argue the point, or troll elsewhere please. I'm not arguing, I am stating a fact. Math is not problem solving. Math is the application of memorized rules to determine a solution. I've stated my position. You have stated your position. You want links, instead of comments about how I *feel* about math? http://www.tutorfi.com/Math/rulesinmathematics http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx http://www.enotes.com/math/q-and-a/application-lagrange-rule-235653 http://www.sosmath.com/matrix/determ2/determ2.html http://mathworld.wolfram.com/LHospitalsRule.html http://betterlesson.org/lesson/7641/prime-factor-gcf-lcm-divisibility-rule-application Your move, Euler. TRG So, you posted links to some rules in math, and that proves that math is defined as "application of rules"? Application of L'Hopital's rule is not the definition of math. You've proved nothing with your links, other than that you can find some math rules, and some grade-school level tutoring sites that say to follow the rules to solve math problems. That's nice, if you're in grade school and learning how to count. It's useless when you attempt to derive the formula for the surface area of a sphere, or how a natural language works, or solve a computer science problem. Math is not rule application. Math is the study of, and the development of, rules that define natural phenomenon. What rule, for instance, did Sir Isaac Newton (and Leibniz, for that matter) use to develop the theories of calculus? What rule did Albert Einstein use to discover the principles of general relativity? Interestingly, you bring up Euler –– what rule did he apply to develop the Euler-Lagrange equation? You view math as an end product, and ignore the work that real mathematicians do to create the product you see. |
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Also, are mathematical rules, relations, and operations invented or discovered? I'd like to know if 'purple' is a color that is perceived equally by all. If a color blind guy is fair to say 'that is not purple' based upon his perception of the shade. IMHO, rules, and relations are defined/discovered. Operations are invented. TRG |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG This is not GD. Argue the point, or troll elsewhere please. I'm not arguing, I am stating a fact. Math is not problem solving. Math is the application of memorized rules to determine a solution. I've stated my position. You have stated your position. You want links, instead of comments about how I *feel* about math? http://www.tutorfi.com/Math/rulesinmathematics http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx http://www.enotes.com/math/q-and-a/application-lagrange-rule-235653 http://www.sosmath.com/matrix/determ2/determ2.html http://mathworld.wolfram.com/LHospitalsRule.html http://betterlesson.org/lesson/7641/prime-factor-gcf-lcm-divisibility-rule-application Your move, Euler. TRG So, you posted links to some rules in math, and that proves that math is defined as "application of rules"? Application of L'Hopital's rule is not the definition of math. You've proved nothing with your links, other than that you can find some math rules, and some grade-school level tutoring sites that say to follow the rules to solve math problems. That's nice, if you're in grade school and learning how to count. It's useless when you attempt to derive the formula for the surface area of a sphere, or how a natural language works, or solve a computer science problem. Math is not rule application. Math is the study of, and the development of, rules that define natural phenomenon. What rule, for instance, did Sir Isaac Newton (and Leibniz, for that matter) use to develop the theories of calculus? What rule did Albert Einstein use to discover the principles of general relativity? Interestingly, you bring up Euler –– what rule did he apply to develop the Euler-Lagrange equation? You view math as an end product, and ignore the work that real mathematicians do to create the product you see. All they 'did' was develop the rule. Calculus did not need Newton to create it. Newton sussed out the rules, wrote them down, and taught others how to apply them. If math is a language, then your premise fails. Newton used existing rules to determine more complex rules in which to apply them. In Newton's world acceleration is based upon other, lower rules, reapplied to make a more complex rule. He did not invent a new dialect, he simply wrote a more complicated sentence. If math were an actual language, then parts of it would become outdated, disbanded, antiquated and inaccurate. Instead, it is a collection of agreed upon rules that we all use to transmit ideas using an agreed upon standard of rules. People refer to math as 'problem solving' and accept it as such. In reality, it is rule application, nothing more, nothing less. http://www.tutorfi.com/Math/rulesinmathematics To belabor the point, even the symbols used are agreed upon standards, along with base 10 numbering. When I give you a problem to 'solve' such as 7x7, we have already made several agreements. Additionally, although you may 'solve' this by quickly determining the answer to be 49, this is based upon the rule of addition. 7+7+7+7+7+7+7 So, did you really 'solve' this problem, or did you learn the rule of addition? Or did you simply memorize the multiplication table and apply a memorized 'solution' to the problem. You might see yourself as a 'problem solver' and you might see math as 'problem solving' in reality, it is not. It is memorization of rules, application of those rules in agreed upon structures, and determining the 'correct answer' from that agreement. I'm in Comp Sci, I used math ALOT when I was working for The Center for Cognitive Research back in the early 90s. I met alot of interesting psychology students, faculty and researchers. The brain is a weird thing. It likes rules. It craves them. It also loves problem solving. Rule application uses an entirely different portion of the brain from problem solving (with an unknown/novel solution). The brain, when confronted with a new stimulus will attempt to apply rules (from the memory portion of our brain). When this fails, other areas of the brain attempt to make new rules and connections. I saw it first hand in multiple research studies using the program that I wrote to present the stimulus. Math problems stimulated memory areas. When that failed, the brain activated other areas to make connections. As an aside, sign language is stored in a deaf person's brain differently than in a person who learned it later in life. Along with the associated memory/language responses. Auditory areas were active in the hearing person, since they also associated a 'sound it out' component in their memory. It glowed on the brain maps. "You only truly solve a problem once, after that it is recalled rule application." Dr. Holmes circa 1992, loosely paraphased, by me, from a lecture in the course on Human/Computer Interaction I was taking from him. So, unless YOU are inventing a new connection in math, you are applying rules. TRG |
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Language.
Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Language is used to communicate ideas/thoughts/concepts between people. Language also helps us put thoughts into coherent form for ourselves, it is an aid to reasoning. Learning the language of mathematics is necessary to its application. As in any language, rules apply to its application and construct, and rules that follow consistent logic are necessary to this language. "...it allows us to describe the world around us in consistent terms." << This also follows the definition of language. A language is tool of communication, and must have agreed upon characteristics so the ideas may be effectively communicated. The words, sounds, shapes must have a common meaning to those using the language. ETA: Languages are extensible and change over time. The language of mathematics has followed this pattern. New symbols have been added to describe new concepts. Calculus adds significant concepts that are very difficult (cumbersome) to communicate with addition/subtraction/multiplication/division, so symbols were added, and continue to be added so more complex ideas can be more easily communicated. The foundations of the language tend to remain static over long periods of time, and some may see this as lack of change. |
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Language.
Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Language is used to communicate ideas/thoughts/concepts between people. Language also helps us put thoughts into coherent form for ourselves, it is an aid to reasoning. Learning the language of mathematics is necessary to its application. As in any language, rules apply to its application and construct, and rules that follow consistent logic are necessary to this language. "...it allows us to describe the world around us in consistent terms." << This also follows the definition of language. A language is tool of communication, and must have agreed upon characteristics so the ideas may be effectively communicated. The words, sounds, shapes must have a common meaning to those using the language. ETA: Languages are extensible and change over time. The language of mathematics has followed this pattern. New symbols have been added to describe new concepts. Calculus adds significant concepts that are very difficult (cumbersome) to communicate with addition/subtraction/multiplication/division, so symbols were added, and continue to be added so more complex ideas can be more easily communicated. The foundations of the language tend to remain static over long periods of time, and some may see this as lack of change. Correct, however, even in Calculus, once a problem is presented, and the rule is learned, you are right back to rule application. People have spent a lot of time mischaracterizing mathematics. There is a perception that it is problem solving. An example of problem solving goes something like this: Using the following materials, create a device, or devices, that can hold fifteen Calculus books 1/2" off a school desk for one minute: Materials: Three sheets of college ruled notebook paper. 1" of standard masking tape. No other materials are permitted. FWIW, this was an actual problem given to my HS Comp Sci class. The winner of this design actually held every textbook in the room. When the textbooks failed to collapse it, we asked the Center from the football team to stand on the device. He weighed @380lbs. It held him. Three sheets of notebook paper, and 1" of tape. Now, though, if I was given this problem, my brain 'knows' the solution and will revert to the re-application of that rule as well. There may be 'other' solutions (and there were) but this solution was 'the' correct answer. And it should be noted, I did not know if there was a solution when I gave the problem to the students. TRG |
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Correct, however, even in Calculus, once a problem is presented, and the rule is learned, you are right back to rule application. So if a wrench is used and the bolt is turned, the wrench is no longer a wrench? If language is used to write a book, the book is completed, then the language is no longer a language? No longer useful to write new and different books? ... There is a perception that it [math] is problem solving.
I think that impression is certainly incorrect. Language is used to communicate... to write a book... to convey an idea. Language is a tool used to to convey an idea, not the idea itself. Mathematics is a tool used to formulate solutions; it is not the solution itself. |
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Also, are mathematical rules, relations, and operations invented or discovered? I would say that the notations are invented (see Leibniz/Newton –– different notations for the same ideas), while the relations and ideas exist and are "discovered", though that's not exactly the right word for it. Perhaps "proven" or even "proposed" would be better? |
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Moving a discussion from another thread here, as it deserves its own thread. It has been contended that Math is, by definition, "rule application". I contend that it is absolutely *not* "rule application", but instead a vast science that deserves far more discussion. Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Math is a language –– in fact, some would contend that math is the one language all humans understand, even if at varying levels of comprehension. Discuss.... Math is rule application. /Thread TRG This is not GD. Argue the point, or troll elsewhere please. I'm not arguing, I am stating a fact. Math is not problem solving. Math is the application of memorized rules to determine a solution. I've stated my position. You have stated your position. You want links, instead of comments about how I *feel* about math? http://www.tutorfi.com/Math/rulesinmathematics http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx http://www.enotes.com/math/q-and-a/application-lagrange-rule-235653 http://www.sosmath.com/matrix/determ2/determ2.html http://mathworld.wolfram.com/LHospitalsRule.html http://betterlesson.org/lesson/7641/prime-factor-gcf-lcm-divisibility-rule-application Your move, Euler. TRG So, you posted links to some rules in math, and that proves that math is defined as "application of rules"? Application of L'Hopital's rule is not the definition of math. You've proved nothing with your links, other than that you can find some math rules, and some grade-school level tutoring sites that say to follow the rules to solve math problems. That's nice, if you're in grade school and learning how to count. It's useless when you attempt to derive the formula for the surface area of a sphere, or how a natural language works, or solve a computer science problem. Math is not rule application. Math is the study of, and the development of, rules that define natural phenomenon. What rule, for instance, did Sir Isaac Newton (and Leibniz, for that matter) use to develop the theories of calculus? What rule did Albert Einstein use to discover the principles of general relativity? Interestingly, you bring up Euler –– what rule did he apply to develop the Euler-Lagrange equation? You view math as an end product, and ignore the work that real mathematicians do to create the product you see. All they 'did' was develop the rule. Calculus did not need Newton to create it. Newton sussed out the rules, wrote them down, and taught others how to apply them. If math is a language, then your premise fails. Newton used existing rules to determine more complex rules in which to apply them. In Newton's world acceleration is based upon other, lower rules, reapplied to make a more complex rule. He did not invent a new dialect, he simply wrote a more complicated sentence. If math were an actual language, then parts of it would become outdated, disbanded, antiquated and inaccurate. Instead, it is a collection of agreed upon rules that we all use to transmit ideas using an agreed upon standard of rules. People refer to math as 'problem solving' and accept it as such. In reality, it is rule application, nothing more, nothing less. http://www.tutorfi.com/Math/rulesinmathematics To belabor the point, even the symbols used are agreed upon standards, along with base 10 numbering. When I give you a problem to 'solve' such as 7x7, we have already made several agreements. Additionally, although you may 'solve' this by quickly determining the answer to be 49, this is based upon the rule of addition. 7+7+7+7+7+7+7 So, did you really 'solve' this problem, or did you learn the rule of addition? Or did you simply memorize the multiplication table and apply a memorized 'solution' to the problem. You might see yourself as a 'problem solver' and you might see math as 'problem solving' in reality, it is not. It is memorization of rules, application of those rules in agreed upon structures, and determining the 'correct answer' from that agreement. I'm in Comp Sci, I used math ALOT when I was working for The Center for Cognitive Research back in the early 90s. I met alot of interesting psychology students, faculty and researchers. The brain is a weird thing. It likes rules. It craves them. It also loves problem solving. Rule application uses an entirely different portion of the brain from problem solving (with an unknown/novel solution). The brain, when confronted with a new stimulus will attempt to apply rules (from the memory portion of our brain). When this fails, other areas of the brain attempt to make new rules and connections. I saw it first hand in multiple research studies using the program that I wrote to present the stimulus. Math problems stimulated memory areas. When that failed, the brain activated other areas to make connections. As an aside, sign language is stored in a deaf person's brain differently than in a person who learned it later in life. Along with the associated memory/language responses. Auditory areas were active in the hearing person, since they also associated a 'sound it out' component in their memory. It glowed on the brain maps. "You only truly solve a problem once, after that it is recalled rule application." Dr. Holmes circa 1992, loosely paraphased, by me, from a lecture in the course on Human/Computer Interaction I was taking from him. So, unless YOU are inventing a new connection in math, you are applying rules. TRG First, you've really got to stop using grade school tutoring sites to backup your theory. Math relates to what's taught in grade school about how advanced CQB training relates to the local CCW class where you shoot 30 rounds at 3 yards to qualify. What rule, specifically, do you contend that Newton used to build the theories of differentials and integrals? Figuring out how to describe natural phenomenon is math –– not mindless rule following. Hell, you get into even Calc II and my professors were talking about creativity in solving complex problems –– not just blindly applying rules. You're looking at the rules that mathematicians created and believing they're the sum total of math. They're not. You say you're "in Comp Sci", what field and what level? I'm curious, because computer science *is* math, and computer science research has a lot to do with problem solving. You define math as rule application vs. problem solving, you're in a distinct minority of people in that field. |
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Correct, however, even in Calculus, once a problem is presented, and the rule is learned, you are right back to rule application. So if a wrench is used and the bolt is turned, the wrench is no longer a wrench? If language is used to write a book, the book is completed, then the language is no longer a language? No longer useful to write new and different books? ... There is a perception that it [math] is problem solving.
I think that impression is certainly incorrect. Language is used to communicate... to write a book... to convey an idea. Language is a tool used to to convey an idea, not the idea itself. Mathematics is a tool used to formulate solutions; it is not the solution itself. Language is not problem solving. Is Pi the formula or the solution itself? If math was problem solving, and not rule application, then each time you encountered a circle, in which you wanted to calculate it's area, you would need to rediscover 3.1416 See? You may solve the problem of "what is the area of this circle, but, you do that through the application of established rules. You may have memorized the formula, or you might recall parts of the equation, but you are not solving a problem, you are applying rules. Now, if you resolved a new way to determine the area that does not follow established rules, then you may have something. TRG |
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First, you've really got to stop using grade school tutoring sites to backup your theory. Math relates to what's taught in grade school about how advanced CQB training relates to the local CCW class where you shoot 30 rounds at 3 yards to qualify. Why? which basic rule of math does not apply at the higher levels? Is there some principle of math that is different? Is 2+2 no longer 4 in Calc II? What rule, specifically, do you contend that Newton used to build the theories of differentials and integrals? If I was an expert on Newton and his path of discovery I would speak about. I have not read a book that explained his thinking process. I will speak to the way the human brain works to say this: Newton applied the rules of math in new ways. He developed ways to express the new rules, and since that point others have re-applied his rules. Figuring out how to describe natural phenomenon is math –– not mindless rule following. Nobody ever said rule following was mindless. I think you are mischaracterizing the argument. Hell, you get into even Calc II and my professors were talking about creativity in solving complex problems –– not just blindly applying rules. You're looking at the rules that mathematicians created and believing they're the sum total of math. They're not. Math professors, like most faculty, have an inflated sense of self-worth. Most feel that their course is critical, when in reality, most are not. They tend to be self-promoters. "rules that mathematicians created " Were the rules created, or were they discovered? My position would be that the rules were discovered, and then expressed in symbology that others agreed up as a standard. Take Pi for instance, again. You say you're "in Comp Sci", what field and what level? I'm curious, because computer science *is* math, and computer science research has a lot to do with problem solving. I 'say' ? You 'say' you are i the Navy. What field and what level? Unless I agree that your credentials are worthy, then your statements about the Navy are specious and I can ignore them. See how that works? 
Your phrasing is equivalent to stamping your feet and putting your fingers in your ears. If you step back further, and don't hold math in a bubble, you will see that I am right. You define math as rule application vs. problem solving, you're in a distinct minority of people in that field. No doubt, but Newton also held a minority position. TRG |
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Mathematics is a tool used to formulate solutions; it is not the solution itself. good point. True, and as a tool, consider the following. "You need to remove a 9/16" bolt. Which of the following tools will you use? Crescent wrench, Screw driver, Hammer. 9/16" wrench?" Now, is removing that bolt 'problem solving' or rule application? If you had never seen a bolt, a wrench, screw driver, or hammer, then you would be confronted with a real problem. However, if you have ever removed a bolt before you already know the answer and apply the appropriate tool. Just as you can add 7+7+7+7+7, on a piece of paper to arrive at 35 (crescent wrench), you could also multiply 7x5 (9/16" wrench) and use the answer you have memorized. Or, you could write the number 1, in groups of 7, 5 times, and then add up the ones (hammer). Same works in math. You see the 'problem' and you select the rules (tools) that you have already determined will lead to the solution. If someone shows you which tool to select, or allows you to experiment, then shows you the steps needed, how the bolt works, the way each tool is applied, then, are 'you' solving the problem, or are you simply learning the rules, from them, and reapplying the rules? TRG |
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First, a discussion of whether 2+2=4 based on some kindergarten tutor's discussion is irrelevant, and in fact useless in this discussion. We're not talking about 2+2=4.
Your answer regarding Newton tells me you have essentially no math education. Read what he wrote, it's not hard. Math professors tend to be esoteric geniuses that are very hard to understand. It is what it is, there's no shame in admitting you don't get it. I've had more than one professor spend fifteen minutes drawing on a chalk board and finally turn around and see the blank stares and say "oh wait, wrong course". I've clearly articulated what I am and what I do in the Navy, many times. Your statement "in Comp Sci" does not square with your view of mathematics, and appears to be artfully constructed to give the appearance of expertise where there is none. So I ask, because I want to understand who I'm dealing with. That's all. If someone asks me what I do in the Navy, my answer wouldn't be to attempt to deconstruct their question into some kind of lawyerspeak for "I'm too cool to tell you", I'd simply tell them I'm a Chief Electronics Technician, BS in Computer Science, and 20+ years in Computer Science, Math, and IT related fields. If you honestly think that Newton just followed rules, see my comment above. |
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Mathematics is a tool used to formulate solutions; it is not the solution itself. good point. True, and as a tool, consider the following. "You need to remove a 9/16" bolt. Which of the following tools will you use? Crescent wrench, Screw driver, Hammer. 9/16" wrench?" Now, is removing that bolt 'problem solving' or rule application? If you had never seen a bolt, a wrench, screw driver, or hammer, then you would be confronted with a real problem. However, if you have ever removed a bolt before you already know the answer and apply the appropriate tool. Just as you can add 7+7+7+7+7, on a piece of paper to arrive at 35 (crescent wrench), you could also multiply 7x5 (9/16" wrench) and use the answer you have memorized. Or, you could write the number 1, in groups of 7, 5 times, and then add up the ones (hammer). Same works in math. You see the 'problem' and you select the rules (tools) that you have already determined will lead to the solution. If someone shows you which tool to select, or allows you to experiment, then shows you the steps needed, how the bolt works, the way each tool is applied, then, are 'you' solving the problem, or are you simply learning the rules, from them, and reapplying the rules? TRG And mathematics is the art of finding ways to solve those problems. Math research is not "rule following", it's developing solutions. |
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Mathematics is a tool used to formulate solutions; it is not the solution itself. good point. True, and as a tool, consider the following. "You need to remove a 9/16" bolt. Which of the following tools will you use? Crescent wrench, Screw driver, Hammer. 9/16" wrench?" Now, is removing that bolt 'problem solving' or rule application? If you had never seen a bolt, a wrench, screw driver, or hammer, then you would be confronted with a real problem. However, if you have ever removed a bolt before you already know the answer and apply the appropriate tool. Just as you can add 7+7+7+7+7, on a piece of paper to arrive at 35 (crescent wrench), you could also multiply 7x5 (9/16" wrench) and use the answer you have memorized. Or, you could write the number 1, in groups of 7, 5 times, and then add up the ones (hammer). Same works in math. You see the 'problem' and you select the rules (tools) that you have already determined will lead to the solution. If someone shows you which tool to select, or allows you to experiment, then shows you the steps needed, how the bolt works, the way each tool is applied, then, are 'you' solving the problem, or are you simply learning the rules, from them, and reapplying the rules? TRG And mathematics is the art of finding ways to solve those problems. Math research is not "rule following", it's developing solutions. You're wrong. The art is in determining which rules can be used to arrive at an established, and commonly 'accepted' solution to a problem. You are taking this personally, when it is not meant to be personal. I am relaying to your my direct experience, from men more educated and studied than myself, along with direct evidence from research I personally witnessed, that Math uses memorized rules to determine solutions. It does not artfully create new constructs to determine outcomes. Math is rule application. You may 'determine' a solution using a rule, but, unless you are creating brand new, never seen before, mathematical principles, then you are simply applying tools to arrive at predetermined solutions. Whether that solution was predetermined, or not, is not part of this equation. Twice now you have questioned my ability to speak on something because I have not 'proven' my credentials. Most people would see that as attempting to discredit my viewpoint. Show me one thing, one thing, in mathematics that is not based upon a previous rule. Dazzle me. Let me see how you are doing something that uses no predetermined, pre-established, rule to arrive at a solution to a problem. If mathematics is not rule application, then show me something that uses no rules to arrive at a solution. There are trillions of possible math problems at your fingertips, pick a single one and show me how there are no rules that determine your 'solution' to the result that you achieve. I'll go finish hot packing some vegetables from the garden while you work it out. P.S. Be sure to show your work.
To add, and to help you understand the issue, chess is also NOT problem solving. It is rule application as well. Show a chessboard to someone who has never seen the game, or heard the rules. Show them the board with pieces. tell them that their 'problem' is to take your King. Tell them to solve this 'problem.' They will reach across the board and take your King. To teach a computer to play chess, chess was translated in to mathematical rules. TRG |
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So you're just quoting things you heard from people who don't understand math.
I'm still waiting for you to explain what gives you the ability to think you know more than everybody else in the field (well, with the exception of the kindergarten tutoring site you keep trumpeting). I've asked for your credentials twice, because I don't think you have any. If I go into a thread on infantry tactics and start arguing with Sylvan, what do you think the first question is going to be? "What relevant experience do you have?" You keep trumpeting the fact you worked with some scientists once, but never will say exactly what your qualifications are. "in Comp Sci" doesn't mean anything at all. Calculus was not "based on a previous rule". Counting was not "based on a previous rule". I suppose there are two groups of people –– mathematicians and rule followers. You've clearly explained how you can only see one, but that does not preclude the existance of the other. Mathematics is form of reasoning –– not a form of rule following. I leave you with this, from your field. http://education.stateuniversity.com/pages/2203/Mathematics-Learning-MYTHS-MYSTERIES-REALITIES.html "Mathematics is first and foremost a form of reasoning. In the context of analytically reasoning about particular types of quantitative and spatial phenomena, mathematics consists of thinking in a logical manner, making sense of ideas, formulating and testing conjectures, and justifying claims. One does mathematics when one recognizes and describes patterns; constructs physical or conceptual models of phenomena; creates and uses symbol systems to represent, manipulate, and reflect on ideas; and invents procedures to solve problems. Unfortunately, most students see mathematics as memorizing and following little-understood rules for manipulating symbols." emphasis mine, and that would suggest to me that your views are the problem with math education today, and why I have to spend hundreds of hours dragging people through courses they can't understand because they were taught to follow rules and not to think. |
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So you're just quoting things you heard from people who don't understand math. I'm still waiting for you to explain what gives you the ability to think you know more than everybody else in the field (well, with the exception of the kindergarten tutoring site you keep trumpeting). Really? I kind of expect you to take this up at least one notch on the scale of discussion. Posting a link is touting? Or do you think, because it is kindergarten that it is somehow beneath you, and makes me to also appear beneath you? I've asked for your credentials twice, because I don't think you have any. If I go into a thread on infantry tactics and start arguing with Sylvan, what do you think the first question is going to be? "What relevant experience do you have?" Well, then I hope you don't work in Military Intelligence. Who I am, my full name, even links to my school's website, with pictures, have been on this website for ten years. Honestly, you are making too much out of this. I have a Master's Degree in Computer Education and Cognitive Systems. I have been teaching Computer Science since 1990. I worked for the Center for Research on Learning and Cognition as a Research Programmer. I personally wrote programs at the behest of professors who were studying how the brain functioned. They mapped the individual centers of the brain in hearing, non-hearing, subjects. We also studied I/O devices back when a 'lively debate' was whether the trackball, mouse, or full immersion body suits would be the I/O device of the future. You keep trumpeting the fact you worked with some scientists once, but never will say exactly what your qualifications are. "in Comp Sci" doesn't mean anything at all. You keep trumpeting your experience in the Navy, but you have not shown any credentials to speak on how a human mind works, experience in the field of neurological studies, or any experience as an educator directly observing the manner in which concepts are explored, explained and comprehended by both adult and minor students for more than 20 years. I'm sure you have some experience in direct observation of research in to the learning centers of the mind, memory, perception, auditory and visual stimulus, of course. You have participated in debates and discussion on how to best isolate specific stimuli to determine the neural pathways in a subject with the exclusion of other inputs? You also, then, wrote the software that met these standards so that the research study could be conducted? You have then participated in discussion and debates on methods to further isolate regional centers of the brain to separate types of stored memory from other stimulus to determine possible outcomes. So, that is why you know, first hand, about how the human mind perceives a mathematical problem, versus, let's say an apple, or a word, a sound, a smell? This is how you know that mathematical problems stimulate the creative centers of the brain, instead of directly being observe to stimulate memory and recall centers? Calculus was not "based on a previous rule". Counting was not "based on a previous rule". You are missing the point here. I suppose there are two groups of people –– mathematicians and rule followers. You've clearly explained how you can only see one, but that does not preclude the existance of the other. Mathematics is form of reasoning –– not a form of rule following. You are almost there, but you still can't quite let go of your preconceived notion about mathematics. emphasis mine, and that would suggest to me that your views are the problem with math education today, and why I have to spend hundreds of hours dragging people through courses they can't understand because they were taught to follow rules and not to think. No doubt there is a problem. I spend very little time in my courses telling students 'how' to solve a problem on their computer. Their pathway to understanding is in their hands. I may point them down the path, but the course they take is their own. It pisses off some students, frustrates others, and from a distance makes me seem to be less involved in their education. In reality, it makes them better students. I am not part of their solution chain. BTW, I read the website, it is concept that goes back to Aristotle and Socrates. The Socratic method of math education does work, but not in a public school setting with standardized test, predetermined learning objectives, and goals determined by outside forces. It's a nice idea to think of students sitting around 'discovering' the solution to a question about lunar orbits, with no preconceived notion about Newton's laws. It looks great on the big screen, but it is not the way schools operate. It is also extremely frustrating for the students when every question from the student is 'answered' by another 'question' from the teacher. "How far away is the moon, teacher?" "How big is the moon, my student?" The Socratic method breaks down when the student does not 'own' the problem. The website spends alot of time talking about pie in the sky concepts that are, really, old news. It works, we know it works, but not unless the student is interested in the subject and has an interest in achieving the result. It also, when it does work, only generate results in small groups. Not many people are willing, or financially able, to have a research physicist, with Socratic training, sit with their kids to 'explore' gravity as a concept. Try that in a HS physics class some time, let me know how it goes. BTW, you still haven't posted a problem that is exclusionary of rules...? TRG |
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True, and as a tool, consider the following. "You need to remove a 9/16" bolt. Which of the following tools will you use? Crescent wrench, Screw driver, Hammer. 9/16" wrench?" Now, is removing that bolt 'problem solving' or rule application? If you had never seen a bolt, a wrench, screw driver, or hammer, then you would be confronted with a real problem. However, if you have ever removed a bolt before you already know the answer and apply the appropriate tool. Just as you can add 7+7+7+7+7, on a piece of paper to arrive at 35 (crescent wrench), you could also multiply 7x5 (9/16" wrench) and use the answer you have memorized. Or, you could write the number 1, in groups of 7, 5 times, and then add up the ones (hammer). Same works in math. You see the 'problem' and you select the rules (tools) that you have already determined will lead to the solution. If someone shows you which tool to select, or allows you to experiment, then shows you the steps needed, how the bolt works, the way each tool is applied, then, are 'you' solving the problem, or are you simply learning the rules, from them, and reapplying the rules? TRG It sounds like you are trying to argue against Math being a language and toward it an application of rules, yet studying your arguments supports math being a language. Math involves application of rules, and so much more. If one uses only the application of rules in math, that would be like using English and Shakespeare never writing a play. Now, is removing that bolt 'problem solving' or rule application? Both, and more you don't mention.
When the bolt is removed, the problem is solved. Therefor removing the bolt is problem solving. The bolt is removed by way of applying rules to the tools at hand. For instance, one could use the crescent wrench as a hammer and the screwdriver as a punch and successfully remove the bolt. Or one could use the wrench as designed to make the solution easier to achieve. You could similarly abuse words by forming a poorly crafted sentence and still successfully communicate an idea (or fail to, as is all too common) How well one uses that tool (or abuses it) is up to the user. Your example of different ways to use math and still successfully arrive at the solution follows. Therefor your statements above support my position that math is a language... and language is a tool (or set of tools) used to solve problems. The problems any generic language solves is to organize thoughts/ideas/concepts and hopefully to communicate them. <<< This is what math does. A common task in math class is to SIMPLIFY. If one uses the rules of the language successfully, the same concept (or value) is expressed in a different way; one which is easier to understand or to communicate. All languages use rule application. To be stuck on just the application of rules misses most of the capabilities of whatever language you are referring to. ...though that would address the most common usage of the language. |
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I have to spend hundreds of hours dragging people through courses they can't understand because they were taught to follow rules and not to think. Why drag them? Use the Socratic method, as you posted, and let them discover mathematical concepts for themselves. You can be the one to show that these methods can be used, instead of traditional methods. BTW, I am glad you are a teacher as well. I am curious to know if you have seen a drop in overall performance in your students in the past two years? I think there is a global, 'Facebook Effect' on education. when I was in Russia, the faculty there cited the same drop, two years ago, in overall student effort and outcomes. Also, noted, was the leap in interest in Facebook, and vKontake (russia's version of Facebook) TRG |
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Therefor your statements above support my position that math is a language... and language is a tool (or set of tools) used to solve problems. The problems any generic language solves is to organize thoughts/ideas/concepts and hopefully to communicate them. It is definitely used as a language. Although it does break down, at some point, in that model. I don't think I meant to imply that it was not used as a language. There are agreed upon symbol and constructs, just as you and I agree an 'apple' has a sound that we associate with it. Others associate a different 'sound' for apple. In math, that also occurrs, and cause its own problems. For instance an X is known, mostly to be multiplication, unless you are using a computer, then it is an *. I still see students who do not know that the * is a symbol for multiplying. However, almost everyone in the world nods their head for "yes", and shakes it for "no". Although there are exceptions to this as well. TRG |
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Now, is removing that bolt 'problem solving' or rule application? Both, and more you don't mention.
When the bolt is removed, the problem is solved. Therefor removing the bolt is problem solving. The bolt is removed by way of applying rules to the tools at hand. For instance, one could use the crescent wrench as a hammer and the screwdriver as a punch and successfully remove the bolt. Or one could use the wrench as designed to make the solution easier to achieve. You could similarly abuse words by forming a poorly crafted sentence and still successfully communicate an idea (or fail to, as is all too common) How well one uses that tool (or abuses it) is up to the user. Your example of different ways to use math and still successfully arrive at the solution follows. Therefor your statements above support my position that math is a language... and language is a tool (or set of tools) used to solve problems. The problems any generic language solves is to organize thoughts/ideas/concepts and hopefully to communicate them. <<< This is what math does. A common task in math class is to SIMPLIFY. If one uses the rules of the language successfully, the same concept (or value) is expressed in a different way; one which is easier to understand or to communicate. All languages use rule application. To be stuck on just the application of rules misses most of the capabilities of whatever language you are referring to. ...though that would address the most common usage of the language. You are correct, IMHO, the analogies, mostly, about math and its ability to communicate, even crudely, as a language. <3
You also forget that you have preconceived information about how a bolt works.
BTW, have you tried the problem I posted with three sheets of notebook paper to support textbooks? TRG |
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I kept a hard copy of the research that we completed in one year. Although these are just the names of the individual programs it should show some 'evidence' of my background experience in how people learn and the cognitive properties that we studied. I was the programmer and a Graduate Student in the department of CECS.
Raven's Matrixes Basic Cognitive tasks Visual Memory Planned Connections Trackball Mouse Trainer Scotopic Sensitivity Semantic vs Episodic (memory) Sign language Sign Language - P300 Sign Language - Hemifields Letter Pairs Number Finding. Some of the research projects have this guy: http://www.twu.edu/psychology-philosophy/miller.asp as the researcher listed for Copyright on the coding I did. This guy is the one that explained how math was rule application, and the tenets of problem solving :https://faculty.coe.unt.edu/jon-young TRG |
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Learning any language in greater depth and breadth allows better and more elegant communication.
... You are correct, IMHO, the analogies, mostly, about math and its ability to communicate, even crudely, as a language. <3
You also forget that you have preconceived information about how a bolt works.
BTW, have you tried the problem I posted with three sheets of notebook paper to support textbooks? TRG Math is the only language that allows us to communicate some things. Listening to mathematicians talk about a problem has been both frustrating and illuminating. I didn't forget about my pre-conceived notions about how a bolt works. My brutish example above was chosen in part because of what I already knew. No, I have not tried the 3-sheets problem... I'm too fookin' busy posting on ARFCOM 
ETA: Dr. Jon Young
.... Courses I actively teach are Statistics, Learning, and Human Development Ah.... stats. That sheds light on your "rule application" stance. "Lies, damn lies, and statistics" Fook up the rules (easy to do) and the results will tell you lies. Playing fast & loose with the rules will allow you to produce most any solution you desire. Catching the falsehoods is as complicated as formulating them. ...like other languages! Politicians parse words carefully to say nothing while talking much, or to lie without sounding like they have shit coming out of their mouths. ETA2 - Short version of above: Very careful application of the rules in stats is absolutely critical. |
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Learning any language in greater depth and breadth allows better and more elegant communication.
... You are correct, IMHO, the analogies, mostly, about math and its ability to communicate, even crudely, as a language. <3
You also forget that you have preconceived information about how a bolt works.
BTW, have you tried the problem I posted with three sheets of notebook paper to support textbooks? TRG Math is the only language that allows us to communicate some things. Listening to mathematicians talk about a problem has been both frustrating and illuminating. I didn't forget about my pre-conceived notions about how a bolt works. My brutish example above was chosen in part because of what I already knew. No, I have not tried the 3-sheets problem... I'm too fookin' busy posting on ARFCOM
I found, through my interactions with the Russians, that we know more common sign language than you might think. I would lay a bet that you could walk in to any grocery store or market, in any country, using sign language and gestures, and get a coke, a sammich and a pack of smokes. Let's see you try that with trigonometry...
And I caught your heavy brow ridge in your example of using the tools to bang out the bolt. BTW, you comments amount simplifying are correct. I saw a Star Wars/Trek movie once about the 'binars'. A race of aliens who talked only in binary. They claimed it was for speed. I laughed out loud. Try saying hello in binary... 1010101010101010101010001010110110010111010101011010, instead of just 'hi!' TRG |
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Kindergarten tutoring isn't what's being discussed, and it's silly to keep bringing it up as though kindergarten methods of teaching are somehow the arbiter of how math is defined.
I see that you are a career educator. I suspected as much. I don't spend my time hunting people on this website down to find out who they are. Simply not something I have time for. I asked, because I was curious, and I had a suspicion that you've confirmed. I actually have not "trumpeted" anything about anything I've done in the Navy. I haven't done cognitive research. What I have done is install a modern combat direction system on a CVN, troubleshoot it, fix its install problems, and supervise its maintenance for 5 years, training about 50 technicians in complex systems analysis, maintenance, and operations during that time –– including a deployment to the Persian Gulf and nearly round-the-clock flight operations. What I have done is install and maintain computer labs, again teaching and training my subordinates how to troubleshoot, how to think, and how to become effective technicians. What I have done is run RDT&E programs for NSW, analyzing and solving complex technical problems for which there aren't any "rules", in a no-fail environment including multiple deployments overseas to implement those solutions. I am a Navy instructor –– have never been billeted in that role, but I'm trained in it and I've taught almost every single day of my life as part of my real job and in school before that. I see the products of the education system every day. I do everything I can to re-teach them how to think –– but after 15+ years of indoctrination into "which rule do I apply", many of them cannot be salvaged. I don't really have any "preconceived notions" about mathematics. I learned how to think mathematically at such a young age I don't remember not being able to. I remember sitting down to take the ACT at the age of 12 and realizing I didn't know a formula for something, so I derived it. Most kids these days don't have that capability –– not because they can't, because they've been taught to plug the numbers into a formula and ignore the how and why. The Socratic method works well. It's how I educate my child. I use a variation of it when I instruct, and even in a room of 50 or more people, it can still work, if used by someone with experience in drawing the students into the discussion and allowing them to build on the knowledge they have. I don't teach high school –– I never attended high school. I was too busy working full time and going to college. So perhaps you're right –– or perhaps our education system has so badly failed that we're not even attempting to teach anything but how to pass a bubble test anymore. That's a sad state of affairs if it's the case. To answer your facebook question, I have not seen any specific drop in interest/education/level of effort in the last two years specifically. It has not appeared to change over the last 16 years to me –– but again, I don't teach high school, the training and instruction I do is almost all military or technical in nature, to college age junior Sailors, most of whom are significantly above the norm in intelligence and motivation. My Grandmother was a professor at TWU for 30 years, btw. Ph.D in Education from KU. |
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Kindergarten tutoring isn't what's being discussed, and it's silly to keep bringing it up as though kindergarten methods of teaching are somehow the arbiter of how math is defined. There is only one person who keeps bringing up kindergarten around here, and it ain't me. I see that you are a career educator. I suspected as much. So, you aren't a Flag Officer, I suspected as much. I don't spend my time hunting people on this website down to find out who they are. Simply not something I have time for. I asked, because I was curious, and I had a suspicion that you've confirmed. Really, you 'confirmed' a well known fact? Something I have posted about for ten years? You suspected I was a teacher? As if that is some sort of 'lesser' form of career? I actually have not "trumpeted" anything about anything I've done in the Navy. [span style='color: red;'][span style='font-weight: bold;'] I haven't done cognitive research. So, we can just assume, now, that anything you claimed about how people learn was, in fact, talking our your ass, right? The same as if you told Sylvan about infantry tactics. You have no basis in study, training, or experience in cognitive research, learning theory, or even basic human psychology, and you still feel qualified to state that math is not rule application and memorization? Even in the face of a link, which you posted, that decries the fact that math is... wait for it... taught as rule application and memorization. I am a Navy instructor –– have never been billeted in that role, but I'm trained in it and I've taught almost every single day of my life as part of my real job and in school before that. It's nice to meet a colleague, I hope you don't feel that being an instructor somehow implies that your role in the Navy, or society, is diminished because you are an Instructor, instead of a Navy Pilot, or Admiral. It's ok to be an Instructor. I've learned to live with the shame of helping people learn new skills, pursue career goals, and I have even learned to accept the 'thank you for teaching me' as something of a compliment. I hope you can too, one day. I see the products of the education system every day. I do everything I can to re-teach them how to think –– but after 15+ years of indoctrination into "which rule do I apply", many of them cannot be salvaged. No, actually, you don't see the 'real' products of the educational system. I don't either. The 'real' products will never be productive members of society, they still cannot read, they cannot write a simple document, or even save a file to a USB drive. You, more so than I, work with people that at least hae some basic interest in bettering themselves. Most leave public school with no initiative whatsoever. Be thankful that the ones you are working with made the choice to serve the nation, and make something of themselves. you have it better than you might think. I don't really have any "preconceived notions" about mathematics. If that statement were true, you would not be having this conversation, or be taking it so personally.
I learned how to think mathematically at such a young age I don't remember not being able to. I remember sitting down to take the ACT at the age of 12 and realizing I didn't know a formula for something, so I derived it. Most kids these days don't have that capability –– not because they can't, because they've been taught to plug the numbers into a formula and ignore the how and why. Some students are gifted. If you had read some of my previous rants about public school, the main problem is that we force kids who are gifted in math to also take English, history, Art, Humanities, Social Studies and P.E. Who would ever think to tell Tiger Woods, halfway around the course, that he must stop and study History for an hour? Who would tell Michael Jordan he cannot play a game of hoops until he finishes learning trigonometry. Public school does a piss poor job of dealing with outliers in the system. I was pretty lucky, as a HS Comp Sci teacher, I was able to attract student, like you, to Comp Sci. I still speak to two of them from more than 10 years ago. One became a US Marine, and is still one of my best friends. The other went in to the US Army,and still uses computers. A third, and his brother, I have not heard from in five years, but they were both working their way up through major IT companies. Another, last I heard, had moved to the East coast and was quite a crackerjack in IT. But, as a 'career educator' I know this means I am beneath contempt for my lowly career aspirations.
The Socratic method works well. It's how I educate my child. I use a variation of it when I instruct, and even in a room of 50 or more people, it can still work, if used by someone with experience in drawing the students into the discussion and allowing them to build on the knowledge they have. I don't think we disagree here. It is effective, in limited cases, with a competent instructor. However, even Socrates admitted when he did not know the answer. The difference between being good, and being great, in the classroom is to admit that you do not know an answer and watch the students as they teach it to you. I don't teach high school –– I never attended high school. I was too busy working full time and going to college. So perhaps you're right –– or perhaps our education system has so badly failed that we're not even attempting to teach anything but how to pass a bubble test anymore. That's a sad state of affairs if it's the case. I had a friend, when I taught HS, who was a retired US Army Captain. He taught Science. Lots of retired Army/Navy/Marine/Ari Force guys go back to the classroom. You should consider it when you retire. I'll give you some guidance on it, when you get to the retirement age, and help keep you out of some of the traps they (admin) set for new faculty. My HS Economic teacher was a big influence on me. He washed out of the Air force because he could not land a T-38(?) on one engine without panicking. I am glad he washed out. He made a damn fine teacher. I still quote him to this day. To answer your facebook question, I have not seen any specific drop in interest/education/level of effort in the last two years specifically. It has not appeared to change over the last 16 years to me –– but again, I don't teach high school, the training and instruction I do is almost all military or technical in nature, to college age junior Sailors, most of whom are significantly above the norm in intelligence and motivation. My Grandmother was a professor at TWU for 30 years, btw. Ph.D in Education from KU. My father was the first in his family to graduate from college, so was my mother. I grew up, around the kitchen table, listening to educational theory, readability issues, left/right brain dominance, and classroom management. It is in my blood. Everyone in my family is now a college graduate. My wife will graduate with her Master's Degree in May. She will *probably* pursue a PhD in International business. Her Master's is in Mathematics.
TRG |
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You're taking everything I write as a personal attack, and responding with absurd insults and condescension, so I'm done with you.
I will simply say that the hubris you continue to demonstrate is fairly typical. I don't follow you around the website reading your posts and trying to figure out who you are, because I don't care who you are, or didn't until you started attempting to condense thousands of years of thinking and the foundation of civilization into "following rules". It's impossible to have a civil conversation with someone who takes an untenable position, defends it with "everybody knows", and condescends continually to everyone. When I leave the Navy, I'll be teaching at the college level with a Ph.D. |
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You're taking everything I write as a personal attack, and responding with absurd insults, so I'm done with you. Awww, I was just beginning to think we were making headway. I knew I shouldn't have questioned your credentials, even though you kept questioning mine.
I proved that I had direct, hands-on, training and experience, not to mention a Master's Degree in the field of Learning and Cognition, so you quit playing the 'show me your credentials' argument and leave the playground? Weak sauce. Mathematics is rule application. A select few may have savant-like skills that supersede that premise. However, just because Beethoven could play the piano, does not mean every pianist is Beethoven. The same is true for math. Some, those few, rare, geniuses may be able to convey mathematical concepts in some new way, however for 99.99999% of the human population it is as I stated. TRG |
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You're taking everything I write as a personal attack, and responding with absurd insults, so I'm done with you. Awww, I was just beginning to think we were making headway. I knew I shouldn't have questioned your credentials, even though you kept questioning mine.
I proved that I had direct, hands-on, training and experience, not to mention a Master's Degree in the field of Learning and Cognition, so you quit playing the 'show me your credentials' argument and leave the playground? Weak sauce. Mathematics is rule application. A select few may have savant-like skills that supersede that premise. However, just because Beethoven could play the piano, does not mean every pianist is Beethoven. The same is true for math. Some, those few, rare, geniuses may be able to convey mathematical concepts in some new way, however for 99.99999% of the human population it is as I stated. TRG Your problem, as with most academics, is that you have a great deal of very focused knowledge in something completely irrelevant to the discussion, and believe it gives you perfect authority to pontificate about everything, related to your field of expertise or not. You know how some people process information –– but that doesn't change the nature of the discipline. The fact that many people only see math as "rule following" does not change what math is, and I see you now admit that. 99.9999999%? Is that a real number, from some study, or made up on the spot to bolster a weak argument? No, most people don't see math as more than "rule following". The Newtons and Leibniz and Eulers do, and they're the ones who advance the field of math –– not the people that simply know how to plug in and solve for x. |
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You're taking everything I write as a personal attack, and responding with absurd insults and condescension, so I'm done with you. I will simply say that the hubris you continue to demonstrate is fairly typical. Typical of what? I don't follow you around the website reading your posts and trying to figure out who you are, because I don't care who you are, or didn't until you started attempting to condense thousands of years of thinking and the foundation of civilization into "following rules".
FWIW, almost everything in life is defined by the human mind's attempt to derive rules. Some of the most basic things, that you assume are not rules based, actually are rules that you learned at such an early age that you cannot emote them. It's impossible to have a civil conversation with someone who takes an untenable position, defends it with "everybody knows", and condescends continually to everyone. . I think you should scroll back up, I don't believe I ever took that position of 'everybody knows' that math is rule application. In fact, I posted the opposite. Everyone says math is problem solving, in fact it is only problem solving if you solve it yourself, with no rules given, or taught. After you solve it once, however, you will reapply the previous rule. And, I know you got hte bit in your teeth, but I would interested to see your solution the the 'three sheets of notebook paper' problem I posted. (welcome back to the thread. I was disappointed you left) TRG |
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You're taking everything I write as a personal attack, and responding with absurd insults, so I'm done with you. Awww, I was just beginning to think we were making headway. I knew I shouldn't have questioned your credentials, even though you kept questioning mine.
I proved that I had direct, hands-on, training and experience, not to mention a Master's Degree in the field of Learning and Cognition, so you quit playing the 'show me your credentials' argument and leave the playground? Weak sauce. Mathematics is rule application. A select few may have savant-like skills that supersede that premise. However, just because Beethoven could play the piano, does not mean every pianist is Beethoven. The same is true for math. Some, those few, rare, geniuses may be able to convey mathematical concepts in some new way, however for 99.99999% of the human population it is as I stated. TRG Your problem, as with most academics, is that you have a great deal of very focused knowledge in something completely irrelevant to the discussion, and believe it gives you perfect authority to pontificate about everything, related to your field of expertise or not. The fact that many people only see math as "rule following" does not change what math is, and I see you now admit that. 99.9999999%? Is that a real number, from some study, or made up on the spot to bolster a weak argument? No, most people don't see math as more than "rule following". The Newtons and Leibniz and Eulers do, and they're the ones who advance the field of math –– not the people that simply know how to plug in and solve for x. Why do you keep going back to these kinds of statements? 'career educator, as I assumed' 'as with most academics'? Do you, really, not see them as specious and ad hominem? If I kept saying, "...like most career Navy men, you have no experience in the real world." Would you *not* see that as an attempt to draw a reaction from you? It really is annoying, and has nothing to do with the topic at hand. However, your other statement is downright bizarre " very focused knowledge in something completely irrelevant". I have focused knowledge on how the human brain stores, catalogs and develops learning pathway. I have done direct research that deals with how the brain synthesizes, and then recalls information. I made a statement about how the brain recalls mathematical information based upon that direct, first hand, training. How can you simply dismiss that as irrelevant? If anything, installing a system on a CVN has nothing to do with your ability to speak to how people learn. I still don't think questioning your credentials, or dismissing you out of hand because of your lack of them, is appropriate. I'm still shaking my head at some of the things you have claimed I wrote, or positions you feel I hold. TRG |
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Now we're getting somewhere.
You know how people learn. Your statements have been "math is following rules" or something to that effect. If you said, as you did before (minus the ridiculous "99.9999999%" thing), that most people don't think in a manner beyond learning to follow rules that have already been laid out by people who understand and practice the discipline of mathematics, I'd agree with you. Instead, you took the absurd position that because people you studied and people you've taught don't understand the discipline, it must be nothing more than following rules. We're talking past each other, and I don't believe that's going to change, unfortunately. I've lived and worked with academics all my life, it is what it is. The good ones know their lanes. The others don't. The only reason I brought up my experience on a CVN is that what I did there was both learn and teach troubleshooting and a whole plethora of technical disciplines. It doesn't require a lab and research grants to understand how people learn. |
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Quoted: Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. |
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Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. I would argue that math is a foundation of physics, as most of physics doesn't get described without math, and both describe the real world, but I don't disagree with the rest. |
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Now we're getting somewhere. You know how people learn. Thank you. I do, sincerely, appreciate the acknowledgment. Your statements have been "math is following rules" or something to that effect. I stated the position, as fact, based upon the study, conducted by others. As much as I can state the world is round, although I have never been in space. My position is clear, "Math is rule application." Nothing, 'to that effect'. It is rule application. If you said, as you did before (minus the ridiculous "99.9999999%" thing), that most people don't think in a manner beyond learning to follow rules that have already been laid out by people who understand and practice the discipline of mathematics, I'd agree with you. Fair enough. Instead, you took the absurd position that because people you studied and people you've taught don't understand the discipline, it must be nothing more than following rules. And, there is where you mischaracterize both the position and the people who made the statement. You do not need to be a Mathematician to understand how a Mathematician's brain stores information. Do you really think that to receive surgery, you must first be a surgeon? We're talking past each other, and I don't believe that's going to change, unfortunately. Actually, it has changed. I think my position offended you. Based on your experience, with academics and public school (the part about no HS I found intriguing), you found my position to be an affront to something you hold in high regard. I've lived and worked with academics all my life, it is what it is. The good ones know their lanes. The others don't. I would not go to a Mathematician if my dog was sick. I would not ask an Psychologist to do my taxes. Psychologists, especially those who focus on neural pathways, will tell you how the brain functions, how we learn. Irrespective of your field of study, as a human, you are constantly learning rules. These rules are something that define us. Step back from the issue a little and you will see rules all around you. Some you have taught yourself, others were taught to you. FWIW, there are only two instinctual fears. Loud noises and fear of falling, everything is a construct of rules of your creation, or others, but you absorbed these rules and now use them in your daily life. It is, somewhat, like seeing behind the curtain when you see that what 'we' do is make rules. The novel, White Fang, does a great job of describing this from a dog's point of view. it is worth a read also, if for nothing else than insight n to the way the brain learns to deal with the world. The only reason I brought up my experience on a CVN is that what I did there was both learn and teach troubleshooting and a whole plethora of technical disciplines. It doesn't require a lab and research grants to understand how people learn. No, it may not qualify you to speak about how the brain forms connections, but, it is damn sure something I wish I had a chance to do. Not many people get that opportunity. I taught a woman to save her work on to a USB the other day, doesn't hold a candle to teaching servicemen to pull cable and wire a CVN. But, the commonality is in the transmission of knowledge. BTW, in the 'walk a mile in their shoes' category, I work with faculty, every day, who never spent a day in a public school classroom. Before you head to the Naval Academy, take a swing at teaching in public school for a year or two. Hall duty. Lunch duty. Bus duty. Teacher meetings. Report Cards. Study Hall. Passing periods. Bathroom fights. .... ahh, memories. It will add an level of appreciation to your career in 'higher' academia. You've never really 'taught' until you have spent an hour in front of a seventh-eighth class, teaching them to use a Macintosh to create a school newspaper ... only to find that you had a 1/2" booger stuck on your forehead the entire time. No wonder they kept asking questions, that required me to walk to their workstation, and making eye contact. TRG |
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Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. What is a quark? Up, down, left, right, strange, normal? I have always heard about them, but never really understood what they were. TRG |
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"to that effect" simply meant I couldn't remember if it was an exact quote, nothing more.
Again, I don't think most people, and I would argue almost no mathematicians, would agree with your premise. I'm not offended, I just think it's silly to say that Newton was just following rules when he built the concepts of Calculus. Creation of the rules (or even discovery of them), is what the people who don't need the rules to follow do. I understand your point better now –– but I still don't agree. BTW, I'll never teach at the Academy –– wouldn't want to. |
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Quoted: Quoted: Quoted: Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. I would argue that math is a foundation of physics, as most of physics doesn't get described without math, and both describe the real world, but I don't disagree with the rest. Math is an important language we use to communicate, and one major model we create through analysis in physics is a mathematical model. I would not refer to math as the foundation of physics. Both math and physics are more than what that would imply. We have discovered and invented much math as we have investigated the universe, but the foundation of physics is the universe, the concepts themselves, regardless of what language we use to describe them. The fact that we can create math models that come very close to describing the universe is an interesting thing. Is that true because the universe is fundamentally mathematical in nature, and we as part of this universe are naturally able to understand and build that mathematical framework? |
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Quoted: What is a quark? Up, down, left, right, strange, normal? I have always heard about them, but never really understood what they were. TRG You are in excellent company there. I do not think there is anyone who yet knows the answer to that question. String theorists kind of sort of think they might have an answer. |
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Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. I would argue that math is a foundation of physics, as most of physics doesn't get described without math, and both describe the real world, but I don't disagree with the rest. Math is an important language we use to communicate, and one major model we create through analysis in physics is a mathematical model. I would not refer to math as the foundation of physics. Both math and physics are more than what that would imply. We have discovered and invented much math as we have investigated the universe, but the foundation of physics is the universe, the concepts themselves, regardless of what language we use to describe them. The fact that we can create math models that come very close to describing the universe is an interesting thing. Is that true because the universe is fundamentally mathematical in nature, and we as part of this universe are naturally able to understand and build that mathematical framework? Note that I said a foundation, not the foundation... I agree they're intertwined in many ways. I think the universe runs on order, and math is one of the languages that describes that order. |
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There is a lot of research on how people learn the various disciplines. I am most intimately familiar with the research into the learning of science and specifically physics. Among the texts I have read that are very interesting are Arnold Arons "A Guide to Introductory Physics Teaching" and the NAP book How Students Learn Science and Math. The NAP book is available free in PDF form from the NAP site. There are a lot of derivative and related works on the NAP site. |
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Quoted: Note that I said a foundation, not the foundation... I agree they're intertwined in many ways. I think the universe runs on order, and math is one of the languages that describes that order. The universe also appears to run on randomness, which is much less satisfactorily treated with most mathematics. |
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Note that I said a foundation, not the foundation... I agree they're intertwined in many ways. I think the universe runs on order, and math is one of the languages that describes that order. The universe also appears to run on randomness, which is much less satisfactorily treated with most mathematics. There are entire fields of mathematical study in the world of randomness, but you're right –– it's certainly not well described. |
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"to that effect" simply meant I couldn't remember if it was an exact quote, nothing more. Again, I don't think most people, and I would argue almost no mathematicians, would agree with your premise. I'm not offended, I just think it's silly to say that Newton was just following rules when he built the concepts of Calculus. I understand your point better now –– but I still don't agree. BTW, I'll never teach at the Academy –– wouldn't want to. "When I leave the Navy, I'll be teaching at the college level with a Ph.D." My mistake, I read this too fast, and thought you meant you were going to the naval academy after you left the Navy with your PhD. I'm not concerned if most people don;t agree. People did not agree the world was flat, the Earth went around the Moon, or that irrational numbers are good decision makers (note, that's a joke.). Newton, unarguably, invented knew ways to explain the world around us. He got alot of things right, but, he still did not get it all right. He defined some pretty damn good rules, (laws are just a better way to say 'rules') that others have learned to follow. Creation of the rules (or even discovery of them), is what the people who don't need the rules to follow do. Not really. That sounds good, but, determine the rules, which others have not been able to define, is the path of those who live the deepest by rules.. Newton needed, wanted, craved, to know 'the rule' that made an apple fall. If he did not care if the apple fell, or floated, he would have never invested the time in learning the rule. The rules are the 'whys' and 'what happens if' that we live by. Newton, although a pioneer and free thinker in popular literature, was searching, deeply, for the rules that he knew existed, but nobody else had defined. Newton was not only following rules when he created Calculus, he was smart enough to realize that he was laying the stones for others to follow in his footsteps. He/they did not call it "Newton's third possibility of some shit that might happen is something else occurs, sorta, maybe, one time, I think..." It's Newton's third [span style='font-weight: bold;']law For fun, pour a glass of scotch sometime. Try counting all the 'rules' that you apply in that simple act. If it helps, mentally draw a flow chart, or decision tree, that determines the rules and outcomes for each step. You'll be surprised how many 'rules' you are living by, without even knowing it. If you stop, and self examine one day, you will probably find that you are probably, despite what you think, a rule follower. Our own description of ourselves is almost entirely opposite of what we think we are. You don't get to be in charge of IT installation on a CVN by being a free spirit that makes his own rules. You might think outside the box, and reach a solution in new ways, but... When you make a sammich, do you use white or wheat bread? Would you feel odd if the top was white and the bottom was wheat? What if the top was white, the bottom was a bagel half? What if the top is a tortilla and the bottom is toasted wheat? What if someone ate two bites, then handed it to you? Any 'rules' that you just saw? See my point? TRG |
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Math is the study of the world around us, and the development of rules to describe natural phenomenon in an orderly fashion. Your above definition is the definition of physics. Math is a method of reasoning. Math is the study of logic. Math is a discipline. It is the foundation of all science, as it allows us to describe the world around us in consistent terms. Applied math cares about the world around us. Just plain mathematics is, like philosophy, a study of logic and reason. It is an abstract study of quantity and space. That it can be applied (through science, engineering, technology) to the real world is a fantastic thing, but math is not specifically concerned with those applications. Frankly, it seems to annoy mathematicians, as they keep developing new math that seems to have no applications to physics, and then we go and find an application for it in physics. I would argue that math is a foundation of physics, as most of physics doesn't get described without math, and both describe the real world, but I don't disagree with the rest. Math is an important language we use to communicate, and one major model we create through analysis in physics is a mathematical model. I would not refer to math as the foundation of physics. Both math and physics are more than what that would imply. We have discovered and invented much math as we have investigated the universe, but the foundation of physics is the universe, the concepts themselves, regardless of what language we use to describe them. The fact that we can create math models that come very close to describing the universe is an interesting thing. Is that true because the universe is fundamentally mathematical in nature, and we as part of this universe are naturally able to understand and build that mathematical framework? Note that I said a foundation, not the foundation... I agree they're intertwined in many ways. I think the universe runs on order, and math is one of the languages that describes that order. FWIW, Newton and his contemporaries fail to describe that order. TRG |
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Note that I said a foundation, not the foundation... I agree they're intertwined in many ways. I think the universe runs on order, and math is one of the languages that describes that order. The universe also appears to run on randomness, which is much less satisfactorily treated with most mathematics. There are entire fields of mathematical study in the world of randomness, but you're right –– it's certainly not well described. As for randomness, Einstein is noted to have said, "God does not play dice!" Have you ever heard the theory that *maybe* the Universe, and the world around us, are actually a computer simulation? The principle is based upon pixelation. Meaning, that just as in a video image, the closer you look the more things pixelate. If you look even closer, the pixels blur. Look closer still and you get a larger blur with less definition. I don't buy the theory, but it is as fascinating concept that seems to fit the 'why' question of why we can't see what matter really is made of. The closer we look, and the harder we squint, the less we can define. TRG |