quark Posted January 15, 2015 Share Posted January 15, 2015 I know you guys will always offer a thoughtful response and this question has been bugging me lately. Right now, I'm starting to believe that not all math-hungry kids might be both deep thinkers AND problem solvers. Do you think that just the process of thinking alone could drive someone to love and want to do math...but that that person might not actually want to problem solve all the time? I hope I am communicating my thoughts accurately. I know that problem solving definitely involves thinking...but thinking doesn't necessarily lead to solving. Can one still be considered a mathematician if s/he is not interested in the actual solving aspect? Only the process that leads to the solving? Which could lead to tons of unsolved problems I guess? 2 Quote Link to comment Share on other sites More sharing options...
1053 Posted January 15, 2015 Share Posted January 15, 2015 . Quote Link to comment Share on other sites More sharing options...
lewelma Posted January 15, 2015 Share Posted January 15, 2015 Quark, I'm not clear on the difference you are making. Are you saying that some people like to just think about mathematical concepts but not actually solve problems? Like appreciating complicated origami or considering the implications of chaos theory to many different fields? 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Quark, I'm not clear on the difference you are making. Are you saying that some people like to just think about mathematical concepts but not actually solve problems? Like appreciating complicated origami or considering the implications of chaos theory to many different fields? Yes! Thanks for putting that clearly for me! They just like to gather multiple concepts in their heads and ruminate and ruminate and ruminate...but hitting upon the answer itself is not thrilling so there might just be lots of unanswered problems left behind kwim? I'm just wondering...is this something to encourage or discourage? Quote Link to comment Share on other sites More sharing options...
lewelma Posted January 15, 2015 Share Posted January 15, 2015 I have a friend who is like this. At first I thought that he was a problem solver, but he is not. He is instead a visionary. He uses his knowledge of math (he has a BS in Computer Science) to think about complex ideas and the implications of these complex ideas to society. I'm sure there is a job out there for people with this type of skill set! He is fascinating to talk to. 3 Quote Link to comment Share on other sites More sharing options...
Barb_ Posted January 15, 2015 Share Posted January 15, 2015 Yes! Thanks for putting that clearly for me! They just like to gather multiple concepts in their heads and ruminate and ruminate and ruminate...but hitting upon the answer itself is not thrilling so there might just be lots of unanswered problems left behind kwim? I'm just wondering...is this something to encourage or discourage? Sounds like a philosopher. I have a math student who loved learning math, playing with numbers, proofs, concepts...then decided she wanted nothing to do with Math after Calculus. She loves to explore, discover and contemplate but drilling down is not her thing. Btw, she's the neuroscience/comp sci student at the university now. 3 Quote Link to comment Share on other sites More sharing options...
Barb_ Posted January 15, 2015 Share Posted January 15, 2015 I have a friend who is like this. At first I thought that he was a problem solver, but he is not. He is instead a visionary. He uses his knowledge of math (he has a BS in Computer Science) to think about complex ideas and the implications of these complex ideas to society. I'm sure there is a job out there for people with this type of skill set! He is fascinating to talk to. Visionary is a better word choice. 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Sounds like a philosopher. I have a math student who loved learning math, playing with numbers, proofs, concepts...then decided she wanted nothing to do with Math after Calculus. She loves to explore, discover and contemplate but drilling down is not her thing. Btw, she's the neuroscience/comp sci student at the university now. Wow, this gave me goosebumps because the person who sparked this debate in my head is a neuroscience lover too. Hmm...interesting! Thanks so much Barb_! 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 When you say "deep thinker", what kind of thing are they thinking about? Gosh, just tons of things. Mathy things mostly...e.g. abstract algebra stuff, number theory stuff. Unfortunately, I don't remember any distinct examples. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 I'm just wondering...is this something to encourage or discourage? I guess to add to this...what the parent of this person is worried about is (and as a result started my whole thought process on this issue)...could it be laziness? But in my opinion, a lazy person wouldn't want to think of hard things all the time either right? Hee hee, this is probably the first time I've quoted myself. :coolgleamA: Quote Link to comment Share on other sites More sharing options...
Barb_ Posted January 15, 2015 Share Posted January 15, 2015 Wow, this gave me goosebumps because the person who sparked this debate in my head is a neuroscience lover too. Hmm...interesting! Thanks so much Barb_! Sure! Jen considered majoring in Math or Physics but ended up finding the problem solving part tedious. Loved the theory though! 3 Quote Link to comment Share on other sites More sharing options...
Barb_ Posted January 15, 2015 Share Posted January 15, 2015 I guess to add to this...what the parent of this person is worried about is (and as a result started my whole thought process on this issue)...could it be laziness? But in my opinion, a lazy person wouldn't want to think of hard things all the time either right? Hee hee, this is probably the first time I've quoted myself. :coolgleamA: No, I don't think so. How old is the student? Child or young adult? Quote Link to comment Share on other sites More sharing options...
1053 Posted January 15, 2015 Share Posted January 15, 2015 . 1 Quote Link to comment Share on other sites More sharing options...
EndOfOrdinary Posted January 15, 2015 Share Posted January 15, 2015 My son is totally the thinker, not the solver. He is not jazzed about math, but greatly enjoys AoPS because of how it delves into concepts and allows him to solve however he wants. He likes pondering the inner workings of the numbers, drawing and configuring these crazy artsy pictures of various principles, and enjoys the puzzle aspects for just thinking about it. His perfectionism anxiety kicks in at the idea of solving. Not such a fan there. Example: Ds got so very excited yesterday when we were doing mole conversations with chemical equations in chemistry. To solve the problem he had to balance the formation equation and then explain how much of an element was going to be present from the decomposition of a compound if the given sample was however many moles. It was basically finding a ratio and then multiplying by the size of the sample. Ds goes off on this ramble about how everything in life can be expressed as a line with a given slope. So really all of the world is really an elaborate geometry pattern. If we can see the patterns, then we can quantify life in visual representations. If we can create visual patterns, then we can create rhythms. So all of life is just music if you know how to hear it. I just stared at him. He had no desire to tell you how many moles of carbon were going to be released - that was long gone, and in reality only half started to begin with - but somehow it created a philosophy of the world and the fabric of life. 8 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 No, I don't think so. How old is the student? Child or young adult? Young adult in age. Childlike in some ways. Asynchronous like crazy. 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Ds goes off on this ramble about how everything in life can be expressed as a line with a given slope. So really all of the world is really an elaborate geometry pattern. If we can see the patterns, then we can quantify life in visual representations. If we can create visual patterns, then we can create rhythms. So all of life is just music if you know how to hear it. I just stared at him. He had no desire to tell you how many moles of carbon were going to be released - that was long gone, and in reality only half started to begin with - but somehow it created a philosophy of the world and the fabric of life. I think he is such a delightful young man! :) Hmmm, now, are you telling me, everything in life ISN'T just elaborate geometry? (I'm only half kidding!). 2 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 It's late here...just wanted to ask you guys one more thing before coming back to this tomorrow... So if this is your child, what would you do? I know that on this board especially, problem solving is greatly encouraged. So in this sense, would you say this person who just wants to think all the time, is this person still problem solving? If your child only wanted to think about problems but not actually solve most of them, what would you feel and do? The thinking is at a pretty high level...conceptually, no worries. It's just that there's little output (in the form of written, solved problems). Personally, I think I would be worried. I wouldn't overly worry, but it would still concern me. Quote Link to comment Share on other sites More sharing options...
Pegs Posted January 15, 2015 Share Posted January 15, 2015 I think a person can definitely be (genuinely and passionately) math hungry without necessarily being good at, or very interested in, problem solving. Such people are deep thinkers who make amazing, abstract connections, IME. I'm not at all surprised to see neuroscience and philosophy mentioned above. :) 1 Quote Link to comment Share on other sites More sharing options...
1053 Posted January 15, 2015 Share Posted January 15, 2015 . 6 Quote Link to comment Share on other sites More sharing options...
Pegs Posted January 15, 2015 Share Posted January 15, 2015 I'm trying to think of a funny pun that combines "line with a given slope" with "going of on a tangent". :rolleyes: Get to the point, already! Badala doom tish 8 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 I guess to add to this...what the parent of this person is worried about is (and as a result started my whole thought process on this issue)...could it be laziness? But in my opinion, a lazy person wouldn't want to think of hard things all the time either right? Thinking deep can be more mentally draining than problem solving. My former CTO thinks deep, holds four patents in high performance computing/parallel computing and happen to be a Cambridge scholar. The concept of one of his patent was used in the movie Contact. Thinking is part of the intellectual overexcitability too. Whether it's your child or your friends child, not everything can or need to be solved, let them enjoy thinking. Rodin's sculpture of the thinker at Cantor Arts Museum is lovely by the way. 2 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 I'm trying to think of a funny pun that combines "line with a given slope" with "going off on a tangent". :rolleyes: Chemical reactions :lol: Quote Link to comment Share on other sites More sharing options...
EndOfOrdinary Posted January 15, 2015 Share Posted January 15, 2015 I guess to add to this...what the parent of this person is worried about is (and as a result started my whole thought process on this issue)...could it be laziness? But in my opinion, a lazy person wouldn't want to think of hard things all the time either right? Hee hee, this is probably the first time I've quoted myself. :coolgleamA: I think it might make them the exact opposite of lazy. A lot of the anxiety, OCD-ish, neurotic stuff my son has comes from his anxiety about all the big thoughts. He is pondering the foundations of the universe and has been since he was tiny. Talk about weight to carry around. Moreover, he is always coming back to the idea of how very powerless that makes him, how very small, how he has to continue to work even though he is an ant trying to move a mountain. It is constant perseverance. Not lazy, more just not concerned with what he considers trifles....like wearing socks or combing his hair. The details oriented problem solvers (that is me) need the big idea thinkers. The big idea thinkers need the problem solvers. We both balance each other...and make sure the other person doesn't appear homeless. Most fields have room for both and this is really why think tanks and university projects have teams and cohorts work together. Most of the really amazing thinkers have had their group of great minds they hung out with. As a parent, I try to talk with Ds about how to either locate people who will work on the development of his big ideas or what small steps he needs to go about to move the ideas forward. The output is small comparatively. Ds doesn't want to show the pattern in mole samples. I might force him to, but probably not. He does have to write up a blurb and put it in the Idea Bin (the digital file of such great ideas, though some are not so great, but merely just put in the idea bin for later processing). I use the Idea Bin as output. He will also frequently come back to thoughts and ponder more. The blurbs are dated, so each time he returns is a new entry listed below the first blurb. It is a journal basically, but I know better than to call it that. When he was younger we would draw elaborate mind maps that showed how he got from some seemingly insignificant idea to a completely different revelation. Attending a hippie church (UCC, Unitarians, that kind of thing) has helped. Lots of opportunities to express his ideas in theological/philosophical ways to people wanting to listen. When other people ask him, he is far more willing to create a picture, comment on MLK for a service, discuss himself, put something in the newsletter. It is like pop culture math. People seem to dig it. It also has forced him to communicate himself through the eyes of non-math people. He has to ask himself, "what does this mean for the world?" It was a stretch that was initially hard, but has really taken him to some interesting places. Blogs and YouTube were made for big ideas to be expressed. Who knows, he might become the next Pioneer Woman! :) 5 Quote Link to comment Share on other sites More sharing options...
IsabelC Posted January 15, 2015 Share Posted January 15, 2015 I know you guys will always offer a thoughtful response and this question has been bugging me lately. Right now, I'm starting to believe that not all math-hungry kids might be both deep thinkers AND problem solvers. Do you think that just the process of thinking alone could drive someone to love and want to do math...but that that person might not actually want to problem solve all the time? I hope I am communicating my thoughts accurately. I know that problem solving definitely involves thinking...but thinking doesn't necessarily lead to solving. Can one still be considered a mathematician if s/he is not interested in the actual solving aspect? Only the process that leads to the solving? Which could lead to tons of unsolved problems I guess? Probably I'm missing something here (it's been a long day) but I'd say it's obvious that some people view math as a means to an end while others (fewer) view math as an end in itself. And of course some kids might be both 'thinkers' and 'problem solvers'. Your last line is confusing to me. It sounds as though you're suggesting that lack of interest in problem solving might indicate someone isn't a Real Mathematician. I would have thought that, if anything, it would be the opposite: the person who simply loves thinking about math would be considered the 'more mathematical'. That's how it would be, I think, in any other field, for example a person who makes pictures to express her ideas about the world (true artist) vs a person who makes pictures for advertising ('less artisty artist' AKA graphic designer). Or a person who drives cars in order to win the F1 championship (a driver) vs a person who drives in order to cart the kids here, there and everywhere (a mom!). 2 Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 15, 2015 Share Posted January 15, 2015 Honestly, it sounds like you are just describing the difference between common personality traits, both of which could lead to success in any number of fields. I think it's dangerous to say "type X belongs in field X," and "type Y belongs in field Y." Every field needs a balance. In math, there are theorists, and there are experimentalists. Mathematics started as a branch of philosophy. Computer science is applied discrete math. In every case, though, the branch needs both theorists and experimentalists. Very few mathematicians are equally adept at both. Quitting at calculus as described earlier (just using the example, and not meaning to pick on any case here) may be a sign that the child doesn't really like application, and would prefer pure math. Very few schools offer a pure math option, though. AoPS has number theory and combinatorics, but beyond that, most kids' exposure is limited to geometry. Who here has taken their kids into abstract algebra, real analysis, symbolic logic, or topology? Analysis requires some advanced studies, but the other three really don't. They are hard to teach, but even young kids can get the concepts. OP, the answer is yes, both are real, and both are needed in mathematics. 8 Quote Link to comment Share on other sites More sharing options...
wapiti Posted January 15, 2015 Share Posted January 15, 2015 . 2 Quote Link to comment Share on other sites More sharing options...
regentrude Posted January 15, 2015 Share Posted January 15, 2015 It's late here...just wanted to ask you guys one more thing before coming back to this tomorrow... So if this is your child, what would you do? I know that on this board especially, problem solving is greatly encouraged. So in this sense, would you say this person who just wants to think all the time, is this person still problem solving? If your child only wanted to think about problems but not actually solve most of them, what would you feel and do? The thinking is at a pretty high level...conceptually, no worries. It's just that there's little output (in the form of written, solved problems). I consider the problem solving aspect important because the "just thinking" about something without ever bring a project to an actual conclusion would have me concerned: I know people who have their head full of brilliant ideas and start a bunch of stuff, but lack the tenacity to bring a tedious project to its conclusion. So this is an aspect I would work on with my student - BUT on an age appropriate level. It may well be that what you are seeing could be a sign of asynchronous development: thinking way above age level, but executing/creating output closer to age level. So, when pushing the problem solving (not just mathematically, but in all life aspects), I would make sure my expectations are more age appropriate, which may not necessarily reflect his cognitive maturity. Does that make any sense? 8 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 So in this sense, would you say this person who just wants to think all the time, is this person still problem solving? If your child only wanted to think about problems but not actually solve most of them, what would you feel and do? The thinking is at a pretty high level...conceptually, no worries. It's just that there's little output (in the form of written, solved problems). If it is work that is assigned then it becomes a discipline issue. Their assigned work isn't much so no or little output would require me to have a one to one discussion with my kid. If it is not assigned work then the child is still making connections in his head. It is still useful work. I don't require or worry about verbal or written output. My older is a silent thinker and his writing stamina hasn't caught up. Requiring him to talk or write would be a killjoy to his extracurricular thinking. ETA: Tangentially older has been thinking about Big Bang, wormholes and time travel for years. No output there :lol: He was also interested in the four color theorem and the konisburg bridge problem/travelling salesman problem. That has plenty of output in the form of drawings but no computation. 3 Quote Link to comment Share on other sites More sharing options...
Dmmetler Posted January 15, 2015 Share Posted January 15, 2015 I'm thinking that it's kind of like musicologists vs composers. Composers want to create music. Musicologists want to really think and focus on the impact music has, through time, through space, through history, etc. Similar paths, but fundamentally just a different psychology. I've also noticed that a lot more musicologists/thinkers are really, really happy teaching, while the composers/creators often aren't. My DD is a problem solver, but the problems have to be practical and real, not theoretical. I can see her eyes light up when we get to a section in AOPS that she can see the applications of immediately. DH is the same way-he's not a pure mathematician, but is an amazing applied problem solver. 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Thinking deep can be more mentally draining than problem solving. My former CTO thinks deep, holds four patents in high performance computing/parallel computing and happen to be a Cambridge scholar. The concept of one of his patent was used in the movie Contact. Thinking is part of the intellectual overexcitability too. Whether it's your child or your friends child, not everything can or need to be solved, let them enjoy thinking. I find the bolded an interesting idea. I always think of problem solving as including thinking as well. I think (lol) of thinking and thinking as leading to solving but not yet at the solving level if that makes sense? So some problem solving can be quick and perhaps not as draining but some problem solving can take days and even months and years, done in a slow, consistent way (kiddo has done that and I'm sure all you guys have seen it with your kids). Wouldn't both be draining? I do agree thinking is draining. Just thinking about this whole idea of thinking vs problem solving (and should it even be a vs situation) has been both invigorating and a bit exhausting for me to process! I'd like to know what that CTO was like as a child and then a teen. :) I think the parent is especially worried because this is a teen. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 I think it might make them the exact opposite of lazy. A lot of the anxiety, OCD-ish, neurotic stuff my son has comes from his anxiety about all the big thoughts. He is pondering the foundations of the universe and has been since he was tiny. Talk about weight to carry around. Moreover, he is always coming back to the idea of how very powerless that makes him, how very small, how he has to continue to work even though he is an ant trying to move a mountain. It is constant perseverance. Not lazy, more just not concerned with what he considers trifles....like wearing socks or combing his hair. The details oriented problem solvers (that is me) need the big idea thinkers. The big idea thinkers need the problem solvers. We both balance each other...and make sure the other person doesn't appear homeless. Most fields have room for both and this is really why think tanks and university projects have teams and cohorts work together. Most of the really amazing thinkers have had their group of great minds they hung out with. As a parent, I try to talk with Ds about how to either locate people who will work on the development of his big ideas or what small steps he needs to go about to move the ideas forward. The output is small comparatively. Ds doesn't want to show the pattern in mole samples. I might force him to, but probably not. He does have to write up a blurb and put it in the Idea Bin (the digital file of such great ideas, though some are not so great, but merely just put in the idea bin for later processing). I use the Idea Bin as output. He will also frequently come back to thoughts and ponder more. The blurbs are dated, so each time he returns is a new entry listed below the first blurb. It is a journal basically, but I know better than to call it that. When he was younger we would draw elaborate mind maps that showed how he got from some seemingly insignificant idea to a completely different revelation. Attending a hippie church (UCC, Unitarians, that kind of thing) has helped. Lots of opportunities to express his ideas in theological/philosophical ways to people wanting to listen. When other people ask him, he is far more willing to create a picture, comment on MLK for a service, discuss himself, put something in the newsletter. It is like pop culture math. People seem to dig it. It also has forced him to communicate himself through the eyes of non-math people. He has to ask himself, "what does this mean for the world?" It was a stretch that was initially hard, but has really taken him to some interesting places. Blogs and YouTube were made for big ideas to be expressed. Who knows, he might become the next Pioneer Woman! :) Thank you EoO...you gave me the words to use for clarity. Big idea thinkers...that was what I had wanted to say but couldn't think of it last night in my sleep befuddled state. I like the Idea Bin idea very much and might use that myself! :) The concern is there is no output or interest in output of *any* sort. With a younger child, a parent might not be worried but one who is already an adolescent...I think there is worry about not performing to potential and things like that...but you guys have given me lots of food for thought to pass on. Thank you! 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Probably I'm missing something here (it's been a long day) but I'd say it's obvious that some people view math as a means to an end while others (fewer) view math as an end in itself. And of course some kids might be both 'thinkers' and 'problem solvers'. Your last line is confusing to me. It sounds as though you're suggesting that lack of interest in problem solving might indicate someone isn't a Real Mathematician. I would have thought that, if anything, it would be the opposite: the person who simply loves thinking about math would be considered the 'more mathematical'. That's how it would be, I think, in any other field, for example a person who makes pictures to express her ideas about the world (true artist) vs a person who makes pictures for advertising ('less artisty artist' AKA graphic designer). Or a person who drives cars in order to win the F1 championship (a driver) vs a person who drives in order to cart the kids here, there and everywhere (a mom!). No, not suggesting...just wanting to clarify the distinction in my own head and also help me to explain it to others! :001_smile: Thank you, I like your examples about artists and ad creatives very much because it is a personal experience for me and helps me process this issue more thoroughly. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 In math, there are theorists, and there are experimentalists. Mathematics started as a branch of philosophy. Computer science is applied discrete math. In every case, though, the branch needs both theorists and experimentalists. Very few mathematicians are equally adept at both. Quitting at calculus as described earlier (just using the example, and not meaning to pick on any case here) may be a sign that the child doesn't really like application, and would prefer pure math. Very few schools offer a pure math option, though. AoPS has number theory and combinatorics, but beyond that, most kids' exposure is limited to geometry. Who here has taken their kids into abstract algebra, real analysis, symbolic logic, or topology? Analysis requires some advanced studies, but the other three really don't. They are hard to teach, but even young kids can get the concepts. OP, the answer is yes, both are real, and both are needed in mathematics. I loved this! See, I'm so glad I asked. You guys think of things I might never have. Thank you Mike! Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Last night I was struggling to get ds11 off the computer to get ready for bed. He was trying to debug his new game on Scratch and of course I was interrupting his train of thought. Somehow I ended up suggesting that debugging was problem solving and he insisted that no, he was just thinking. He was under the impression that problem solving is work but thinking isn't. (dude, people get paid to do that!) This may not be what y'all are talking about but it just came to mind... (EoO, FWIW, he has a lot of big-thought anxiety and unfortunately has actual OCD, but occupationally I'm quite sure he'd prefer problem solving to big thoughts.) I admit that sometimes I am under this (bolded) impression too. :leaving: It's easy to not consider it work when you don't see something materialize from it...but there is an output of sorts...it's just not manifested on paper or through a video etc. The output is within the person somewhere, ready to be used another day or perhaps just stored within for the beauty of it. It is not expressed outside so people might think it is not output, but like 1053 said, it could be verbal output when the person verbalizes their thoughts. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 I consider the problem solving aspect important because the "just thinking" about something without ever bring a project to an actual conclusion would have me concerned: I know people who have their head full of brilliant ideas and start a bunch of stuff, but lack the tenacity to bring a tedious project to its conclusion. So this is an aspect I would work on with my student - BUT on an age appropriate level. It may well be that what you are seeing could be a sign of asynchronous development: thinking way above age level, but executing/creating output closer to age level. So, when pushing the problem solving (not just mathematically, but in all life aspects), I would make sure my expectations are more age appropriate, which may not necessarily reflect his cognitive maturity. Does that make any sense? Makes perfect sense. This is an older child who produces output for purposes deemed necessary but not interesting (like school work). But the cognitive stuff that is at very high level is not expressed on paper or other media (other than verbally processing it) and makes this person seem like an under-performer. It could also lead to disillusionment with learning at some point. So now I think I should suggest a mentor to discuss it with but I know they don't have the means to look for a mentor right now. 1 Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 I'm thinking that it's kind of like musicologists vs composers. Composers want to create music. Musicologists want to really think and focus on the impact music has, through time, through space, through history, etc. Similar paths, but fundamentally just a different psychology. I've also noticed that a lot more musicologists/thinkers are really, really happy teaching, while the composers/creators often aren't. My DD is a problem solver, but the problems have to be practical and real, not theoretical. I can see her eyes light up when we get to a section in AOPS that she can see the applications of immediately. DH is the same way-he's not a pure mathematician, but is an amazing applied problem solver. Yes! More of those "spot on" words for me to use to help them! Thanks so much! I really appreciate it. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 If it is work that is assigned then it becomes a discipline issue. Their assigned work isn't much so no or little output would require me to have a one to one discussion with my kid. If it is not assigned work then the child is still making connections in his head. It is still useful work. I don't require or worry about verbal or written output. My older is a silent thinker and his writing stamina hasn't caught up. Requiring him to talk or write would be a killjoy to his extracurricular thinking. ETA: Tangentially older has been thinking about Big Bang, wormholes and time travel for years. No output there :lol: He was also interested in the four color theorem and the konisburg bridge problem/travelling salesman problem. That has plenty of output in the form of drawings but no computation. My own guy prefers thinking any day to producing. But he will produce when something is assigned, do it willingly, mostly correctly, and sometimes discover that he does enjoy producing output after all (in the case of geometry especially, there were so many proofs to write but he loved it despite being a reluctant writer in other areas). In the case I brought up, the producing is an unwilling act, just churned out without enthusiasm, and possibly killing the joy of learning. But this is not an age where most of us will say let maturity catch up, or adjust our output requirements to age. This is already a much older adolescent. ETA: I wanted to add...I don't believe there are any learning disabilities involved. I think they've already investigated that quite thoroughly. Quote Link to comment Share on other sites More sharing options...
quark Posted January 15, 2015 Author Share Posted January 15, 2015 Thank you everyone. I think I have the words and clarity to pass on to the family now. This has been a very helpful discussion for me personally too and I truly appreciate each and every post! Thanks so much again. 1 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 I'd like to know what that CTO was like as a child and then a teen. :) I think the parent is especially worried because this is a teen. Deleted TMI portion. I think producing something tangible be it written or verbal output is so expected that we worry if our kids produce "too little". It's hard to show the thinking going on in our kids head other than putting all those brain activity probes to measure synapses. Output has kind of become a societal expectation. Understandable for assessment/exam situations of course. As for deep thinking being more draining, I was thinking of the open ended problems or issues that may take more than a lifetime to solve. 1 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 In the case I brought up, the producing is an unwilling act, just churned out without enthusiasm, and possibly killing the joy of learning. But this is not an age where most of us will say let maturity catch up, or adjust our output requirements to age. This is already a much older adolescent. Hubby read our kids the riot act about some things you do just to bring food to the table :lol: Teens can be a very rebelling, forge your own way stage. It's tough though on the parents and for an older teen it is more worrisome. How long has the unwillingness been? Some kids just drag their feet through unpleasant tasks all their life. Some do that only at puberty stage. It's hard to go ahead without knowing the root cause. All the best in supporting your friend. Quote Link to comment Share on other sites More sharing options...
sunnyday Posted January 15, 2015 Share Posted January 15, 2015 I think it would be important to make sure the student knew that there *is* such a thing as a process for solving big, incomprehensible questions. I didn't know that as a student. Right up through my undergraduate degree in physics, I thought there existed only problems you were given that already had solutions and "right answers", or big ideas you simply thought about but didn't act on or express. Lab science and experimental design were pretty incomprehensible to me, and I thought I wanted to be a theoretical physicist, imagining that would look somewhat like taking existing mathematical equations and just twiddling them until I came up with something new. But my lack of persistence for problem-solving, proof-writing, and project-planning were my downfall; I "burned out", failed to develop a plan for my honors research project, and limped through the last courses for my undergraduate degree, never again to return to the sciences. As an adult I'm starting to grapple with these flaws and trying to prevent them in my children. One thing I have found helpful is to learn more about the writing process. That's one very concrete example of how you move in iterations from nebulous ideas to logical, connected thoughts that you can share with others. I've found Peter Elbow's writing pretty inspirational in that way. Zen and the Art of Motorcycle Maintenance was also a fun read but a lovely exposition on taking big philosophical ideas and connecting them and creating a framework for explaining them; the actual philosophy discussed, the pursuit of Quality, is also kind of a nice concept to focus a teenaged dreamer of big ideas and a reluctant follower of others' paths. I believe one of my fellow physics majors mentioned it as an inspiration for him when he was in high school (he's now a neuroscience PhD and a medical intern about to move on to his ophthalmology residency.) 2 Quote Link to comment Share on other sites More sharing options...
bakpak Posted January 15, 2015 Share Posted January 15, 2015 I agree with Regentrude....kids/anyone can get intimidated about physically solving problems, while thinking about advanced problems is potentially freeing. Thinking 'mistakes' are quickly resolved and don't involve huge amounts of time sorting through dense details. I agree that the child should be encouraged to solve problems at their maturity level, but thinking ahead at their comfort level be encouraged as well. My DD is small, but she definitely wants to think and learn about advanced concepts; but sitting down and doing them? No thanks! She gets to that stage in her own sweet time, and it's closer to the expected age than the thinking only level was. This makes sense to me, as having thought about the concepts for a long time beforehand helps her tackle the doing with ease when she's developmentally ready. In general this is my philosophy - to introduce advanced concepts early so when the real time comes it seems like a next logical step, not a step into a scary abyss. Also, I'm more of a thinker than a doer, which I didn't realize until AFTER I finished my PhD and got a real job. I would never have realized how much I love theoretical problem solving. I have always enjoyed tangible results, but I'm finding now that what really gets me pumping is thinking and talking through problems until the big picture resolves; then I'm ready to move on to the next big problem. It's an interesting (and very important!) thing to figure out about yourself many years later :) This realization and the above posts makes me laugh thinking about a professor my freshman year of college trying to get me to switch majors to philosophy instead of biology; maybe he was onto something!! 2 Quote Link to comment Share on other sites More sharing options...
Arcadia Posted January 15, 2015 Share Posted January 15, 2015 But the cognitive stuff that is at very high level is not expressed on paper or other media (other than verbally processing it) and makes this person seem like an under-performer. Just notice this. Is this child willing to be video taped or even tape himself doing a verbal "presentation". My perfectionist older can be verbose to his iPad so the videos come in handy when he refuse to speak. If by any chance the child is writing phobic, the iPad and windows speech to text loves my kids more than me, we haven't tried that on kindle but Amazon Echo understand my boys perfectly. ETA: Is this child labeled an underperformer in school/classes? ETA: My writing phobic kids love mind maps or any visual way of writing down their ideas. That is for outside of assigned work. 2 Quote Link to comment Share on other sites More sharing options...
1053 Posted January 15, 2015 Share Posted January 15, 2015 . 1 Quote Link to comment Share on other sites More sharing options...
Crimson Wife Posted January 15, 2015 Share Posted January 15, 2015 I guess to add to this...what the parent of this person is worried about is (and as a result started my whole thought process on this issue)...could it be laziness? But in my opinion, a lazy person wouldn't want to think of hard things all the time either right? My "I hate math" kid enjoys reading about math concepts and discussing them, but hates doing math problems. In her case, I do think it laziness, because she struggles with similar issues in learning the viola and other things that require disciplined practice. She resents it when she actually has to WORK at something. 2 Quote Link to comment Share on other sites More sharing options...
lewelma Posted January 15, 2015 Share Posted January 15, 2015 I was thinking a bit about this last night. Quark, if the ideas he is contemplating are really complex, he would be unable to think them through in his working memory alone. He would have to start writing stuff down and organising it in order to see connections. If he is not doing this, he is not thinking about complex ideas. Either the ideas are not complex, or he is just skimming the surface and not thinking in a deep manner. If he were my kid, I would be encouraging him to make graphic organisers of some sort and work to make connections. The end product does not need to be anything to show anyone, it could just be a notebook like Da Vinci's, but I would still want to see some stuff in writing. Ruth in NZ 4 Quote Link to comment Share on other sites More sharing options...
quark Posted January 16, 2015 Author Share Posted January 16, 2015 Aargh, I wish I could multiquote like I used to be able to do... Arcadia: It's tough though on the parents and for an older teen it is more worrisome. How long has the unwillingness been? 1053: How is OP's friend's "young adult/teenager" going with traditional academic goals such as going to a decent college? I don't have a lot of layered details atm, I'm afraid. I'm not sure how much to tease out from the parent without being prying. I don't have very frequent contact with this family. I just feel that it's such a waste to not think of something for them when the love of math is still obviously strong. Crimson Wife: My "I hate math" kid enjoys reading about math concepts and discussing them, but hates doing math problems. In her case, I do think it laziness, because she struggles with similar issues in learning the viola and other things that require disciplined practice. She resents it when she actually has to WORK at something. It's hard to say with this kid. I think the parent feels and fears it could be laziness but to me a lazy kid wouldn't be contemplating and enjoying higher level work that obviously takes a lot of effort to wrap his/ her head around. So I don't really know what it is. Maybe boredom, a feeling of being disillusioned with the boring math that does have to be written down so this teen doesn't want to also have to write down the other math...I don't know. Trying to think what it might be. Arcadia: Just notice this. Is this child willing to be video taped or even tape himself doing a verbal "presentation". My perfectionist older can be verbose to his iPad so the videos come in handy when he refuse to speak. If by any chance the child is writing phobic, the iPad and windows speech to text loves my kids more than me, we haven't tried that on kindle but Amazon Echo understand my boys perfectly. ETA: Is this child labeled an underperformer in school/classes? ETA: My writing phobic kids love mind maps or any visual way of writing down their ideas. That is for outside of assigned work. Ruth: I was thinking a bit about this last night. Quark, if the ideas he is contemplating are really complex, he would be unable to think them through in his working memory alone. He would have to start writing stuff down and organising it in order to see connections. If he is not doing this, he is not thinking about complex ideas. Either the ideas are not complex, or he is just skimming the surface and not thinking in a deep manner. If he were my kid, I would be encouraging him to make graphic organisers of some sort and work to make connections. The end product does not need to be anything to show anyone, it could just be a notebook like Da Vinci's, but I would still want to see some stuff in writing. These -- and ideas from the rest of you that I am unable to take time right now to copy and paste because the darn multiquote thingy refuses to work -- are good thoughts and I'll pass them on. Thanks so much for taking the time to think about them! I think limited time might be an issue too...there's so much else required of this teen during the day...but I'll suggest all that I can and all that you guys have thoughtfully suggested. Thanks again! Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 16, 2015 Share Posted January 16, 2015 I was thinking a bit about this last night. Quark, if the ideas he is contemplating are really complex, he would be unable to think them through in his working memory alone. He would have to start writing stuff down and organising it in order to see connections. If he is not doing this, he is not thinking about complex ideas. Either the ideas are not complex, or he is just skimming the surface and not thinking in a deep manner. If he were my kid, I would be encouraging him to make graphic organisers of some sort and work to make connections. The end product does not need to be anything to show anyone, it could just be a notebook like Da Vinci's, but I would still want to see some stuff in writing. Ruth in NZ That's an interesting thought. I may be an odd one, but the more complex a problem is, the less likely I am to write it down. I'm a MBTI "ENFP," where complexity creates an environment pregnant with possibilities. I couldn't begin to write those thoughts down until I've processed them sufficiently, and often out loud. I write only if nobody is available to bounce ideas off of. My absolute favorite class was modern algebra. I disliked math until I reached calculus 3, because there was no beauty evident in rote calculation. The Kelvin-Stokes theorem changed my mind -- just trying to imagine what thought process would lead to such a beautiful abstraction, where multiple motions need to be visualized simultaneously in order to grasp the impact across a 3-dimensional manifold (ahh, college...). The proof itself was just a byproduct of the thought process, and wasn't nearly as impressive. Conversely, I most disliked statistical thermodynamics, not only because the prof made us learn partial diff eq in the first 2 weeks (it was a tough class), but because there was just SO MUCH crunching of formulas. I did well, but the process just didn't suit my tastes. I had a friend who absolutely loved it. Statistics is a very different beast. 3 Quote Link to comment Share on other sites More sharing options...
Mike in SA Posted January 16, 2015 Share Posted January 16, 2015 Can one still be considered a mathematician if s/he is not interested in the actual solving aspect? Only the process that leads to the solving? Which could lead to tons of unsolved problems I guess? By the way, Fermat was a pretty famous mathematician who showed this trait... :001_cool: 2 Quote Link to comment Share on other sites More sharing options...
mathwonk Posted January 16, 2015 Share Posted January 16, 2015 this reminds me of the importance i once had pointed out to me, of "problem finders" in mathematics. some of us just try to solve problems given to us by others, but who decides what problems are interesting? most PhD students depend upon their advisors to provide a problem, which is both interesting and solvable, but some students have noticed problems on their own with which they are fascinated. I suspect problem finders are generally held in even higher regard than solvers, but that regard is usually established by their first solving an important problem, and afterwards posing one that they have not solved. this latter contribution allows others to play in the game as well, and every mathematician would prefer to be given a problem to solve rather than just listen to someone explain how he solved one. Similarly, a good teacher is not one who just explains clearly how to solve a problem, but who presents an interesting problem to his students and inspires them to attack it. Learning math is not about just acquiring information and technique, but about acquiring a way of thinking, a habit of mind. Sometimes we meet students who do not like books that present math as a sequence of problems to be solved, or discoveries to be made, but merely want a simple statement of ":what do i do next?" These students are missing the main benefit of a mathematics education, and it is our challenge to initiate them gently into the wider world of thought somehow. In my own case, I think I shrank from such exploratory presentations for a long time , out of fear of not getting "the right answer". This is also why we seek books with such answers in the back. If we can somehow convey or at least suggest that the search, or journey, is part of the game, and try to remove the compulsion to get the answer, we may help. I.e. the benefit is in the struggle. It may help also to encourage students to pose questions in math whose answers they are merely curious about. As long as I only wanted to always know the answer, I was afraid to raise too many such questions. [edit: see Barb's following post and the end note: “I am not a teacher, but an awakener.†― Robert Frost] 7 Quote Link to comment Share on other sites More sharing options...
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