Hi Bob,
I have a son currently taking high school Geometry, and we are having problems with the proofs....my question is, to what extent should there be mastery of this, and is this a huge part of the SAT and ACT test...I am not shrugging this off, but want to prevent burn-out or possibly him losing interest and getting a mental block where this is concerned. Any and all help would be greatly appreciated. Thanks in advance and God bless you.
Christi
Geometry ...proof problems!
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In between
I think Babaika is approaching that from the classical [and if not mistaken Russian?] view of mathematics as a way to train the mind. The problem of course is that many get stuck and only frustration, not training, ensues. So what matters depends on the objective.
If the objective is to learn math theory and understand it, or to use it to train one's mind (and it is not necessarily transferable to other domains as many mathematicians would confess to
), then one should try.
If the objective is to have applied knowledge, then one should learn to use it, and not worry as much about fully understanding or knowing all the proofs.
As for the example, as you well know, you didn't prove it, only showed it works on this one example. But then, others have already taken care of that part
[though the proof is trivial in this case, basically boils down to an observation]
If the objective is to learn math theory and understand it, or to use it to train one's mind (and it is not necessarily transferable to other domains as many mathematicians would confess to
), then one should try.
If the objective is to have applied knowledge, then one should learn to use it, and not worry as much about fully understanding or knowing all the proofs.
As for the example, as you well know, you didn't prove it, only showed it works on this one example. But then, others have already taken care of that part
[though the proof is trivial in this case, basically boils down to an observation]Moti Levi
www.LearningByYourself.com
www.LearningByYourself.com
Re: I prefer math by doing, not math by proving:
Warmest Regards
Re: In between
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Re: I prefer math by doing, not math by proving:
Thing is, there are many other equally good ways of training the mind, that are incidently a good deal more fun. Chess or checkers, for instance, or any sort of large-scale strategic simulation where you have limited time to make your decisions. Why train your mind by doing mathematical proofs if you're not going into a field where they're needed (basically, professor)?
Moti: Actually I did just prove it for all sequences of consecutive integers. If you assume that the formula works for N consecutive integers (and I showed that it did), then prove that it works for N+1 consecutive integers if it works for N consecutive integers, then by mathematical induction, it works for all sequences of consecutive integers. The proof is easily modified for any spacing and starting point.
Moti: Actually I did just prove it for all sequences of consecutive integers. If you assume that the formula works for N consecutive integers (and I showed that it did), then prove that it works for N+1 consecutive integers if it works for N consecutive integers, then by mathematical induction, it works for all sequences of consecutive integers. The proof is easily modified for any spacing and starting point.
Re: I prefer math by doing, not math by proving:
Warmest Regards
It's all valid :-)
Basically, all points made here are valid. It depends on the person and his/her objectives, strenghes and weaknesses, likes/dislikes etc. For some, theoretical math is useful in many ways [theory, mind development, application] for others, it is not and even not needed to use it.
There are levels of understanding as well. If I start from low to higher (and the separation is not so clear cut since they overlap):
1. Technical skills
2. Applying skills
3. Understanding concepts [theory]
4. Applying concepts
5. Proving concepts
6. Creating new concepts [theory]
Not everyone needs even 3, let alone#5. There is no "right"or "wrong" on that question [unlike math]. It depends on what I said above.[/list][/list]
There are levels of understanding as well. If I start from low to higher (and the separation is not so clear cut since they overlap):
1. Technical skills
2. Applying skills
3. Understanding concepts [theory]
4. Applying concepts
5. Proving concepts
6. Creating new concepts [theory]
Not everyone needs even 3, let alone#5. There is no "right"or "wrong" on that question [unlike math
]. It depends on what I said above.[/list][/list]Re: It's all valid :-)
Warmest Regards
Re: I prefer math by doing, not math by proving:
Warmest Regards
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