Is there a difference between math thinkers and math problem solvers?

[Arcadia]

 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.

 

Don't students come up with problems/topics for their PhD dissertation and then find a supervisor who is willing to mentor/sponsor the student's PhD project?

Maybe its difference in educational systems as I don't know how things are done here.  Hubby's PhD dissertation project in nanotech was open-ended. Another PhD candidate can expand on it.

[Barb_]

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.

Interesting. My daughter, the one who dropped out of math after Calc, is also an ENFP. She loved her Euclidian proof-based geometry and rudimentary number theory, but Physics II is her only C. She scored 100s on the labs and high grades on the homework but blew the exams...like with scores in the 30s. There were 2-3 problems per exam and it was all calculus. She knew her calculus...got an a- in the weeder class...but she said her brain just shut down.

 

Agreed that even non-problem solvers must become accustomed to output, albeit maybe at a lower level than they can process theory. It's a matter of discipline and maturity.

[mathwonk]

The more imaginative and insightful students do generate their own problems, but in math this is not at all always the case.  It is very easy in math to pose a problem that one will not be able to solve as a student.  so the advisor can help focus and mitigate the difficulty of the project proposed by a naive student.

 

Experimental sciences are also different since one can do experiments and explore, generating both positive and negative results, and then comment on them and write them up.  In pure math. negative results cannot be published.  you either find a new truth or you wind up after months of work with nothing at all.

 

E.g. one may be interested to know whether every even integer > 2 is or is not the sum of exactly two prime numbers.  Since however this has been studied for many decades without success, it is unsuitable for a thesis.  The advisor might however suggest a modification, such as trying to prove that the ratio of the number of even numbers less than X, divided by the number of positive integers less than X which are sums of 2 primes, approaches 1 as X --> infinity.  this also may or may not be trivial, as i have not thought about it, but i would guess one could attack it with the famous "prime number theorem".

 

Or, a perfect number is one whose smaller factors add up to the number, such as 6 = 1+2+3, or 28 = 1+2+4+7+14.  It is widely believed that no odd perfect numbers exist, but if a student proposed to work on this, his advisor might say that a suitable PhD thesis might be just to prove that if one did exist, it would have a large number of prime factors.  someone of my acquaintance generated such a problem for himself as a student.

 

Often a student just knows he/she wants to work on a topic say in homotopy theory, or singularity theory, or moduli, and the advisor may then suggest several more narrow areas, and try to help the student discern those which interest her/him especially, finally arriving together at a specific problem.  

 

But many students just ask for a problem to solve and the advisor provides one.  If the student solves it too easily, the advisor may provide a more challenging one until he/she feels he has the one that teaches the student as much as possible in the given amount of time.  One friend of mine actually entered grad school with a "thesis worthy" problem already identified and already solved, and merely wrote it up after a requisite amount of time, along with other discoveries made in the meantime.  

 

I myself needed a lot of help identifying a suitable problem, but once I became familiar with it, made significant progress, and helped generate new methods that, in the hands of more talented workers, led to resolving previously unsolved questions of wider interest.

[mathwonk]

here is a nice example of problem finding by one of my 10 year old scholars.  he was interested in fibonacci numbers (1,1,2,3,5,8,13,21,.....) where each number after the second is the sum of the two previous ones.

 

He observed that it seemed as if, denoting by f(n) the n th fibonacci number, then f(k) divides f(n) whenever k divides n.

 

I had never noticed this and it did not seem obvious, but he had checked it for  maybe all k up to 6.  Together we proved it, using an idea of his.

 

It turns out to be a well known property of fibonacci numbers, proved early in maybe a grad class on them, or maybe any number theory class, but i was very impressed at his perspicacity.

 

After reading up a bit on these numbers, we were also able to show that the the gcd of f(n) and f(k) equals f(gcd(k,n)).

 

that was our best day in calculus class, i.e. the day when he asked "can we discuss something besides calculus today?".

 

To me this shows more clearly than anything else the fact that this boy can be, and really already is, a mathematician.

 

It is often true that a senior mathematician, equipped with more technical tools, can prove something he has his attention drawn to.  But we all admire, even envy, the imagination and insight of the young person with a fresh eye, who sees the truth in nature even before having the power to prove it.  Without this ongoing process of new discovery, mathematical research would stop.

 

Of course many senior mathematicians also have fine insight and can look into the future past what their skills can yet reveal, to suggest avenues of research for the next generation.

[mathwonk]

here is another problem that interested my student, from a contest, that he asked me how to solve:  start from a circle of radius 1, and inscribe a square around it.  Then another circle around that square, then a regular octagon around that circle, another circle, then a regular 16 sided polygon,... and continue, alternating circles with regular polygons of 2^n sides.

 

what is the limiting value of the radii of the circles?  (answer Ï€/2).  After I solved this, I was asked by the contest maker if my method would also solve the case where each succeeding polygon has, not twice as many sides, but only one more side than the previous one.  I could not see how.

 

Then my student suggested doing the case where each succeeding polygon has 3 times the number of sides as the previous one, a much better next case scenario.  I made progress on this one but did not settle it.  Still his quick insight into finding the next natural question impressed me.  Can anyone solve this?

 

Forgive me if I am just indulging myself here, outside the realm of interest of the thread.  But it is so interesting it sparks these comments.

[8filltheheart]

Quark,

 

I don't have time to read through this thread, but I did want to share that Ds has STACKS of thought experiment notebooks (he used the black and white marble composition books.). He would write down ideas that came to him and the various ways considering different angles of those ideas might impact them. He has all sorts of theories regarding dark matter, palnet formation.....all very theoretical stuff that he would research and play around with but none of them were actually answerable by someone without much greater academic levels and a pretty massive lab (most of them are areas that still don't have real answers.) ;)

 

I think they were incredibly beneficial for him and his ability to puzzle out solutions to problems he did have to solve. His free time was his to do whatever he wanted. Thought experiments were a huge part of that. The only problems he ever had to solve were the ones assigned for school. :) he was not interested in math competitions,etc.

 

He is also a philosopher at heart. Several centuries ago, I am sure that what he would be.

[lewelma]

Quark, 8's ds's notebooks are what I was talking about upstream!  I think deep thinking with notebooks is going to get you further than deep thinking without them.

 

DS and I had a long conversation about this, and he concluded that discussion makes deep thinking deeper, and writing makes it clearer. He is uncertain if you can actually have deep thinking without trying to express yourself either verbally or in writing.  Apparently, he does his math thinking without language, but does not feel the thoughts are complete until he tries to express them.

[quark]

Thanks 8 and Ruth! I will pass this on. We have stacks and stacks here too but I don't want to overwhelm that other parent (in my OP) about what my DS is doing so I haven't showed them our stacks of stuff. I think writing it all down is fabulous...but there is reluctance (or maybe lack of time) involved. Thanks again!

 

On another note...a PSA to others with young ones...save all the written down stuff you can. I just encountered someone doubting that my DS was really doing what I say he is doing!! :confused1: Actually DS is doing more but I don't want to tell this person that or sound like I want to brag kwim? But having notebooks, screenshots validates what my DS needs in terms of higher level work. He doesn't shine in other ways until you start talking math to him so people don't realize that this goofy, scruffy-looking kid is doing all this stuff and they look askance at you when you ask if they think your DS can audit a class their (older) kid is taking!

[ebunny]

On another note...a PSA to others with young ones...save all the written down stuff you can. I just encountered someone doubting that my DS was really doing what I say he is doing!! :confused1: Actually DS is doing more but I don't want to tell this person that or sound like I want to brag kwim? But having notebooks, screenshots validates what my DS needs in terms of higher level work. He doesn't shine in other ways until you start talking math to him so people don't realize that this goofy, scruffy-looking kid is doing all this stuff and they look askance at you when you ask if they think your DS can audit a class their (older) kid is taking!

 

Q- I began saving my DD's work as per your suggestion a while ago. That was/is valuable advice and helped us advocate for her needs in some places.

 

I'm sorry you had that experience. :grouphug:

 

[Barb_]

Quark, 8's ds's notebooks are what I was talking about upstream! I think deep thinking with notebooks is going to get you further than deep thinking without them.

 

DS and I had a long conversation about this, and he concluded that discussion makes deep thinking deeper, and writing makes it clearer. He is uncertain if you can actually have deep thinking without trying to express yourself either verbally or in writing. Apparently, he does his math thinking without language, but does not feel the thoughts are complete until he tries to express them.

Thank you for posting this. I'm going to have to give it some more thought, but I suspect this comment may fundamentally change the way I do things with the deep thinker or two I may still have around my house.

 

Do you all think this applies to introverts as well as extroverts? I know for a fact, my extroverts benefit greatly from verbal expression...I've often said that we don't know what we think until it pops out of our mouths. But the introvert who I suspect is a deeper thinker than she lets on has trouble expressing herself without tripping over tongue.

[quark]

Q- I began saving my DD's work as per your suggestion a while ago. That was/is valuable advice and helped us advocate for her needs in some places.

 

I'm sorry you had that experience. :grouphug:

 

 

:001_wub: Thank you e! It was painful for a day or two. But I don't talk about DS IRL much so I also can't blame this person (I guess).

[quark]

Thank you for posting this. I'm going to have to give it some more thought, but I suspect this comment may fundamentally change the way I do things with the deep thinker or two I may still have around my house.

 

Do you all think this applies to introverts as well as extroverts? I know for a fact, my extroverts benefit greatly from verbal expression...I've often said that we don't know what we think until it pops out of our mouths. But the introvert who I suspect is a deeper thinker than she lets on has trouble expressing herself without tripping over tongue.

 

Personally, for me yes. I trip over my tongue a lot. I struggle to find words that are truly accurate to what I mean. DS verbalizes his ideas much more freely. The teen in question seems asynchronous (to me) in this way...introverted with ideas, extroverted when engaged in play/board game activities.

 

[Arcadia]

Quark,

 

Kiddo was doing his hyperbola section (int algebra) at the library and the middle school kids just look out of curiosity and walk away :)

 

But the introvert who I suspect is a deeper thinker than she lets on has trouble expressing herself without tripping over tongue.

My perfectionist neither introvert nor extrovert older boy does not like to present his ideas until he thinks they are good enough. He is the paper killer in my house.

 

He also prefers written to oral tests. When he has to do a presentation for outside classes, he would tape himself on the iPad and rehearse until he thinks it is okay.

 

ETA:

Once older thinks it is up to his standard, he can be very chatty. My younger boy's love for talking overwrite his perfectionist tendencies.

[Barb_]

Quark,

 

Kiddo was doing his hyperbola section (int algebra) at the library and the middle school kids just look out of curiosity and walk away :)

 

My perfectionist neither introvert nor extrovert older boy does not like to present his ideas until he thinks they are good enough. He is the paper killer in my house.

 

He also prefers written to oral tests. When he has to do a presentation for outside classes, he would tape himself on the iPad and rehearse until he thinks it is okay.

I'll have to think about how to get her to do more free writing. She's an artist, but rarely shares her creations. I'll have to convince her that no one will force her to share.

[Arcadia]

The teen in question seems asynchronous (to me) in this way...introverted with ideas, extroverted when engaged in play/board game activities.

 

Could it be perfectionism at work? Ideas are more personal, playing and board game are more social.

[lewelma]

Thank you for posting this. I'm going to have to give it some more thought, but I suspect this comment may fundamentally change the way I do things with the deep thinker or two I may still have around my house.

 

Do you all think this applies to introverts as well as extroverts? I know for a fact, my extroverts benefit greatly from verbal expression...I've often said that we don't know what we think until it pops out of our mouths. But the introvert who I suspect is a deeper thinker than she lets on has trouble expressing herself without tripping over tongue.

 

 

From our discussion, it is not an introvert/extrovert thing.  It is simply the complexity of deep thinking that requires language in his eyes.

 

So for my ds he will think about a problem for 10 hours on his own and then require that I sit there an listen to his ideas.  The thinking part is nonverbal, apparently even nonverbal in his head.  But what he was sure of was that if he only works nonverbally (including no writing) that the ideas are still fuzzy, not fuzzy like I would describe them because they are very well thought out, but fuzzy by his definition.  Until he tries to take a web like mash of ideas and sort them into a linear argument, the ideas lack clarity.  I asked him if he requires feedback or interaction to make these deep thoughts better, and he said although that would be preferred, that just having to order them enough to vocalize them is the last required step.  Next, I asked him if he could just explain them to a wall instead of a person, and he said that no, that would not do it.  Because it is the effort to *explain* the thought so the other person understands is what is required.  Finally, I asked if he had to have an educated person in his field, and he said no.  He did not need detailed feedback although this is the best for deepening any ideas, but he would need someone who had enough background to at least be able to understand.  He also said that writing it down does clarify it more than verbally, however, the written step would just be icing on the cake.  He felt that if he could verbally explain his idea it would basically be complete, just not polished.

 

He also said that you could skip the verbal step and just write it down if you would rather, but that takes quite a bit more time in his eyes and is less fun.

 

It was an awesome conversation that took more than an hour!

[quark]

Could it be perfectionism at work? Ideas are more personal, playing and board game are more social.

Yes, very likely. And perhaps also not having anyone to talk math with on a regular basis (at the level he wants to). Sometimes confidence also comes from having your ideas validated, approved by someone experienced etc. Anyway, I hope they figure it out. He is a young man with amazing potential.

[quark]

From our discussion, it is not an introvert/extrovert thing.  It is simply the complexity of deep thinking that requires language in his eyes.

 

So for my ds he will think about a problem for 10 hours on his own and then require that I sit there an listen to his ideas.  The thinking part is nonverbal, apparently even nonverbal in his head.  But what he was sure of was that if he only works nonverbally (including no writing) that the ideas are still fuzzy, not fuzzy like I would describe them because they are very well thought out, but fuzzy by his definition.  Until he tries to take a web like mash of ideas and sort them into a linear argument, the ideas lack clarity.  I asked him if he requires feedback or interaction to make these deep thoughts better, and he said although that would be preferred, that just having to order them enough to vocalize them is the last required step.  Next, I asked him if he could just explain them to a wall instead of a person, and he said that no, that would not do it.  Because it is the effort to *explain* the thought so the other person understands is what is required.  Finally, I asked if he had to have an educated person in his field, and he said no.  He did not need detailed feedback although this is the best for deepening any ideas, but he would need someone who had enough background to at least be able to understand.  He also said that writing it down does clarify it more than verbally, however, the written step would just be icing on the cake.  He felt that if he could verbally explain his idea it would basically be complete, just not polished.

 

He also said that you could skip the verbal step and just write it down if you would rather, but that takes quite a bit more time in his eyes and is less fun.

 

It was an awesome conversation that took more than an hour!

 

Thank you for sharing this Ruth! Although he processes things a little differently from your DS, my DS would definitely agree with the bolded. He had an amazing journey when he was working with his mentor, especially when they did abstract algebra and group theory fundamentals together. Their meetings were filled with some lines of proofs but also doodles of every color which sadly, didn't make much sense to me although I loved the visual aspect of it. Putting it down on paper (or in this case the computer screen/ online whiteboard) was probably about as fun for them as trading ideas verbally (the mentor has a lovely way of treating DS as a peer when talking math). It's sad that DS is not able to use the mentor now (schedule is just too busy). Such lovely days. I am so glad I saved a video of them discussing geometry together. This is something the CC is not able to give him at all (although there are other benefits to the CC for DS).

 

[Arcadia]

And perhaps also not having anyone to talk math with on a regular basis (at the level he wants to). Sometimes confidence also comes from having your ideas validated, approved by someone experienced etc.

I agree. So far we haven't hit that problem yet.

 

Are there any elderly chess players near him? A lot of elderly chess players we met are very good at math. If he can't find peers at his level for math, I'll be tempted to hang out at chess tournaments and chat people up just to find math peers.

If I am really desperate to get someone to listen to my kids talk math or science, I might just let them talk to the NASA Ames visitor center volunteers.

[lewelma]

Unfortunately, I'm the one who listens to ds talk math.  He needs me, so I do it.  Problem is I don't really understand what he is talking about, so I just fake it by asking leading questions to draw out more information or by challenging him in some vague way that makes him think differently even though I don't know what I have really said.  I try, but boy I wish my kid had a math parent!

[Barb_]

Unfortunately, I'm the one who listens to ds talk math. He needs me, so I do it. Problem is I don't really understand what he is talking about, so I just fake it by asking leading questions to draw out more information or by challenging him in some vague way that makes him think differently even though I don't know what I have really said. I try, but boy I wish my kid had a math parent!

This makes me laugh! Faking I can be a valuable skill for any homeschooling mom. I find they think we still know everything long after they begin to outgrow us.

[quark]

This makes me laugh! Faking I can be a valuable skill for any homeschooling mom. I find they think we still know everything long after they begin to outgrow us.

 

It doesn't work with my DS anymore! :crying:

[quark]

Are there any elderly chess players near him? A lot of elderly chess players we met are very good at math. If he can't find peers at his level for math, I'll be tempted to hang out at chess tournaments and chat people up just to find math peers.

 

A great idea, Arcadia, thanks. I'll pass this along too.

[Jackie]

I haven't been participating in this thread. I'm from the opposite side - I love the calculations, but I'm not a theorist. Doing Algebra was relaxing and soothing and Calculus was stressful. So I have to imagine there have to be people for whom the opposite is true. FWIW, I think there is value in both and am trying to lay a math foundation with my daughter that both solidifies basic math concepts while encouraging a wonder about mathematics, rather the same as I do for science.

[lewelma]

It doesn't work with my DS anymore! :crying:

 

Either I'm a better faker than you, or your ds is more perceptive than mine! :D

[Barb_]

I haven't been participating in this thread. I'm from the opposite side - I love the calculations, but I'm not a theorist. Doing Algebra was relaxing and soothing and Calculus was stressful. So I have to imagine there have to be people for whom the opposite is true. FWIW, I think there is value in both and am trying to lay a math foundation with my daughter that both solidifies basic math concepts while encouraging a wonder about mathematics, rather the same as I do for science.

I find it funny that neither theorists nor problem solvers seem to be fans of calculus.

[Mike in SA]

I find it funny that neither theorists nor problem solvers seem to be fans of calculus.

 

Probably because it's equal-opportunity -- lots of theory to bug the problem solvers, and way too may problems for the theorists.  :)

 

It's an important platform to higher math and science.  It just happens to be one of those "gotta make your way through it" subjects.  Thinking back, it seems that most students either like calculus I (more theory) and dislike calculus II (more application), or the other way around.

[8filltheheart]

Unfortunately, I'm the one who listens to ds talk math. He needs me, so I do it. Problem is I don't really understand what he is talking about, so I just fake it by asking leading questions to draw out more information or by challenging him in some vague way that makes him think differently even though I don't know what I have really said. I try, but boy I wish my kid had a math parent!

This is what I did with ds starting in 8th grade and still continue now that he is 19. I don't have to really understand to be a sounding board or ask questions that challenge him to consider an alternative perspective. In no way has he ever been under the illusion that I am even close to his equal on these topics! ;). But, he likes to discuss them and he appreciates being asked questions that make him think about he needs to explain details he hadn't thought about explaining. For sure, being able to articulate complex ideas in simple language forces deeper understanding BC they have to teach what the know to someone completely clueless. ;)

[8filltheheart]

Ds likes calculus even though he is definitely a theorist. He just enjoys all math in general.

[quark]

In no way has he ever been under the illusion that I am even close to his equal on these topics! ;). But, he likes to discuss them and he appreciates being asked questions that make him think about he needs to explain details he hadn't thought about explaining. For sure, being able to articulate complex ideas in simple language forces deeper understanding BC they have to teach what the know to someone completely clueless. ;)

 

I think DS was under the illusion for a little while, then he started to say "Mom, you have no idea what I'm saying do you?" to which I would nod vigorously and say "but you can still tell me because I love listening to you". And so he would tell me. He really needs a sounding board and I am one, and I sometimes do ask questions if he isn't precise about the way he explains something, but generally, yeah...I have no clue but I like to listen. :laugh:

[Dmmetler]

Often I think DD would talk to a brick wall-and that I'm a brick wall that can give the illusion of understanding (not about math so much as science)

[daijobu]

I haven't been participating in this thread. I'm from the opposite side - I love the calculations, but I'm not a theorist. Doing Algebra was relaxing and soothing and Calculus was stressful. So I have to imagine there have to be people for whom the opposite is true. FWIW, I think there is value in both and am trying to lay a math foundation with my daughter that both solidifies basic math concepts while encouraging a wonder about mathematics, rather the same as I do for science.

 

I'm glad you spoke up.  I love solving problems.  I get a little rush ("yes!!!") when I check the answer sheet and find I got the right answer.  I'm not such a big fan of open-ended problems, but I do enjoy them sometimes.  

 

But I also like calculus.  I like rotating 2D shapes around a line to create some 3D object.  We were just doing a MathCounts problem last week that involved rotating a trapezoid that has two adjacent right angles around the rectangular side to form the frustrum of a cone.  You didn't need calculus to solve the problem but it reminded of all that cool stuff.  

[MarkT]

 

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.

 

 

Absolutely  -  still trying to get my smart but lazy DS to buy into this concept which is the whole point.

[dovrar]

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.

 

DS has expressed an interest in symbolic logic. Any recommendations for good curriculum?

[Mike in SA]

DS has expressed an interest in symbolic logic. Any recommendations for good curriculum?

 

Not as a curriculum, no.  You may well find something along the lines of "introduction to logic" offered by a philosophy department or on EdX, Coursera, et al.

 

[wapiti]

DS has expressed an interest in symbolic logic. Any recommendations for good curriculum?

How old is the student?  The only one I know of is Patrick Suppes' First Course in Mathematical Logic.  I haven't looked at it in a long time so I can't remember what age, but I'm thinking middle or high school for an accelerated kid.

 

Link to free solutions PDF.  If you search the forums here for Suppes, you'll find some discussions about it.  ETA, the Amazon link includes a preview.  It looks like a middle schooler could handle the beginning though the preview doesn't go very far.

 

Ah, a quick google turned up one of those fascinating old posts by Charon.

Edited May 31, 2017 by wapiti

[Matryoshka]

DS has expressed an interest in symbolic logic. Any recommendations for good curriculum?

 

One of my dds always said she 'wasn't good at math' and claims to have forgotten most of what she got to in Precalculus (which is as far as she got in high school).

 

Now she's wanting to major in Logic/Philosophy.  She just took Symbolic Logic last semester and loved it.  She reads logic books and theory in her spare time for fun.  She's all excited to get to Modal Logic and has been pre-studying.  Whaaaat?

 

Not as a curriculum, no.  You may well find something along the lines of "introduction to logic" offered by a philosophy department or on EdX, Coursera, et al.

 

I think Mike's suggestion is as good as you'll get - I don't think they design Symbolic Logic courses for high school students.  I don't think dd's prof even liked/used the text they bought much, so I'm not sure I should recommend it...

[carrierocha]

In our house we consistently talk about two major things we are working in during math. They are 1) math thinking and 2) math calculating. It sounds like you are referencing that distinction. Sometimes we enjoy the math thinking that is required to analyze a situation and turn it into a mathematical equation, but aren't so interested in actually calculating it out. Makes perfect sense to me.

FWIW - as the math teacher in my house I give specific feedback on both aspects of my kid's math work. I have one who prefers the calculating, but tolerates the thinking. My other is just the opposite and the thrill for her is thinking it through.