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Geometry ...proof problems!

 
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ChristiT
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PostPosted: Thu Dec 15, 2005 11:09 pm    Post subject: Geometry ...proof problems! Reply with quote

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
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babaika
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PostPosted: Thu Jun 01, 2006 4:36 pm    Post subject: Reply with quote

I wish I was a registered user when this question was posted. This is both superb and paramount in its importance issue. If we are talking about child whose only objective is to score relatively high on some standardized test and then to forget about all this "math stuff"--the issue of geometric (as well as algebraic) proofs is absolutely irrelevant. Although.......enough to mention the fact that ANY child subjected to even plane geometry with proofs of its fundamental theorems will fare much better in such field as......logic (not to mistake for reasoning) and will have more profound organizational skills, be it on mental level or in everyday life than those who are not. I, personally, consider proof apparatus of geometry and algebra to be one of the most significant teaching instruments as well as a true gift to a students. My favorite bet (which I always win Cool is to bet people, who consider themselves "mathematically impaired" on Gauss' derivation of fomula of the Sum of N terms of Arithmetic Sequence--the very title creates horror in those people's eyes. Well, usually in 3-4 minutes they LOSE and then admit--that it is not that difficult. As for the solution of practical problems on ANYTHING math-related, and especially for those who see themselves in technological or engineering field--proofs are absolutely essential.
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Theodore
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PostPosted: Thu Jun 01, 2006 10:12 pm    Post subject: I prefer math by doing, not math by proving: Reply with quote

I don't know - I would take the opposing view, that proofs are only useful as they help you understand the basic formulas themselves, and that a proof you can't understand fairly quickly isn't going to be much use to you. The important thing is being able to do the math, not being able to prove that your method works for every problem of its type (since other people already did that).

As for the Gauss proof (just for you), any set of evenly spaced numbers can be divided into N/2 pairs, counting from the outside, that equal the same number. All you have to do then is multiply N/2 by the sum, which is the first and last number added together. So 1 2 3 4 5 6 goes to (1+6) + (2+5) + (3+4) or 6/2 * (1+6) or 21.

You can prove that this works by checking the formula for a sequence of numbers, let's say 1 2 3 4:

N(N+1)/2 = 4(4+1)/2 = 10 (it works)

Then you add another number, giving you the following in terms of the original N:

(N+1)(N+2)/2
- N(N+1)/2
= (2N + 2) / 2 = N + 1 added to original (it works)

Incidently, I never learned the proof.
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Moti
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PostPosted: Fri Jun 02, 2006 7:43 am    Post subject: In between Reply with quote

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 Wink), 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 Smile [though the proof is trivial in this case, basically boils down to an observation]
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babaika
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PostPosted: Fri Jun 02, 2006 9:39 am    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

Theodore wrote:
I don't know - I would take the opposing view, that proofs are only useful as they help you understand the basic formulas themselves, and that a proof you can't understand fairly quickly isn't going to be much use to you. The important thing is being able to do the math, not being able to prove that your method works for every problem of its type (since other people already did that).

I never learned the proof.


Evidently You did Very Happy Actually Gauss' proof is, indeed, original and smart, which is true--finding TWO vertical sums of two same converging progressions and thtn multiplying them by the half of numbers of terms. But this is tiny example. I, however, will disagree with Your disagreement with me Wink But this will require slightly longer elaboration on WHY so, which could be discussed later.
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babaika
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PostPosted: Fri Jun 02, 2006 9:44 am    Post subject: Re: In between Reply with quote

Moti wrote:
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 Wink), then one should try.



Not necesserily Russian but rather European and, of course, pretty much classic American view on math (one of the reasons why I am a collector of classic American math books). As for proofs--yes, no pain--no gain--that is absolutely true. This conclusion has been confirmed many times, especially for the HS students who go into engineering and technological field. Objective, of course, does matter--big time. Or maybe a child's dream?? Wink
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Theodore
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PostPosted: Fri Jun 02, 2006 12:54 pm    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

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.
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babaika
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PostPosted: Fri Jun 02, 2006 1:45 pm    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

Theodore wrote:
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)?



Excellent point!!! First and foremost, children who target technological field in universities and colleges. Dealing with proofs for them becomes much more than dealing with some abstract concepts but most importantly--the matter of the method (approach) to the problem (task). Solution of practical enginnering and design problems VERY often requires their solution in general form (the same goes to actual problems on Physics). Without well developed skills (which primarily are developed through process of proof) it is very difficult to do. E.g. I do not even touch complex numbers before students are given trigonometry (and have a very good concept of projection and resolution) and vectors. Once that is out of the way--frankly, 90% of a time I do not even have to explain much what are the complex numbers and what are operations with them. Why?? Once they know what is the sum of the vectors (and their projections) it is enough to point out to the fact that complex numbers ARE vectors. How is it done--by proofs, of course. In this particular case by comparing given pattern with the result of their own resolution (addition) of vectors in Cartesian System. Does it develop their minds?? Of course it does but it gives them a little bit more than that--the wholeness of the picture and an ability to extrapolate. Of course, this is not for everybody--but children who plan their careers in hi-tech field, they should have this absolutely. Forgive me for such an ultimate statement. But in this field the REAL life starts not before but AFTER the SAT (or ACT).
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Theodore
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PostPosted: Fri Jun 02, 2006 3:42 pm    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

I guess the basic question is, is it enough to just have the correct answer, or do you have to prove the answer as well? Also, do you have to prove the answer mathematically, or can you prove it empirically (cycling through all possible inputs or a reasonable sampling and checking output for correctness)? If I have to prove something, I'm probably going to use the latter method, or just point to a proof someone else already did (everything's on the Internet), rather than doing things the hard way.

Another basic question- are there better things you could be spending your time on? For instance, if you're a programmer, you could take several advanced theory courses, but it's probably going to be more useful to take several programming courses, as all the theory in the world won't help if you can't write your brilliant solution in the programming language your client needs. I imagine there are similar parallels for engineers.

I'd rather build something than explain how it can be built in theory.
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Moti
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PostPosted: Sat Jun 03, 2006 8:09 am    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

Theodore wrote:

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.


Yep. My oversight when I was writing my reply I forgot you did the inductive step and remembered only the example Smile So, you should trust yourself more about being able to replicate Gauss Wink
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Moti
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PostPosted: Sat Jun 03, 2006 10:01 am    Post subject: It's all valid :-) Reply with quote

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 Wink]. It depends on what I said above.[/list][/list]
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babaika
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PostPosted: Sat Jun 03, 2006 11:25 am    Post subject: Re: It's all valid :-) Reply with quote

Moti wrote:
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 Wink]. It depends on what I said above.[/list][/list]


Couldn't have said better myself!!! Very Happy Although--there are still some details in all of that, which should be pointed out. One of them: professional (occupational) aspirations of children, which in many (very many) cases begin to manifest themselves at the age of 16+ . The most important shift being switch from humanitarian (or none at all) to technological (or otherwise) preferences.
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babaika
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PostPosted: Sat Jun 03, 2006 12:01 pm    Post subject: Re: I prefer math by doing, not math by proving: Reply with quote

Theodore wrote:


I'd rather build something than explain how it can be built in theory.


I clearly understand both conviction and degree of healthy sarcasm behind this answer and I do not argue in this case in general. I am talking about children (and this is a very large audience) who plan and try to go into technological field and I do not MEAN computers. I am talking more about aeronautical or mechanical engineering, naval engineering, civil engineering etc. Presently I have couple of dozen HS students, whom I literally pushed through into such schools as ranging from Washington University in St.Louis to Naval Academy. The fields, which require an extremely strong conceptual thinking and, of course, a very profound KNOWLEDGE of physics and math. And here is the Mother of all Problems, which I found to be absolutely true--one facet of problem discussed in USA Today (sadly have no date--but have copy of an article) High Schools Skip Over Basics In Rush To College Classes and later in April 13, 2006 of Wall Street Journal in large article Top High Schools Fight New Science As Overly Simple. Both articles are Bingos, albeit are only small portion of the much larger stream of the opinions in the last years on how to teach natural and precise sciences be it in PS or in HS environment. And the problem here lies (in my opinion--well, not only in mine) in this astounding gap, which exists between whatever PS-HS program in precise scienses and actual volume and intensity of programs already in colleges and universities. Sad examples are plenty, sadly. Of well-rounded kids with very high scores on.....whatever being steam-rolled in the hi-tech faculties of serious colleges once they get there. My latest personal experience accounts for at least 5-6 of those students (most of them HS). And all this is not some abstract problem--this is as real as anything around us and it does affect lives of VERY many kids, who very often see their benign and ambitious dreams being ruined in a front of their eyes.
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babaika
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PostPosted: Sat Jun 03, 2006 12:24 pm    Post subject: Reply with quote

Just a bit in continuation: we very often forget that in the end we are dealing with children. But life does not limit itself to this wonderful season of juvenile foolheartidness--they grow up and they go out there as already grown ups with a single objective--to realise their dreams. At least, the majority of them does. Reality of the modern life, however, is such that I personally have to go with abusive ITT Institute TV slogan: You Are What You KNOW. With emphasis on KNOW, not informed or being aware of. They (children), not US will have to make their choices, make up their minds and in the modern world--this is a very tall order. And one have to be prepared and have actual cognitive weapons to deal with this reality, especially if the big dream is in plans. While this point might seem as almost truism, there are many layers to it and one of them is quite simple--only very educated (and not pro forma!!!) people novadays make it far, even considering epic examples of "self-made men (or women)" or even winning the jackpot in state lottery. The Bottom Line is simple: One is, indeed, what he or she Knows.

PS. And of course--both: constructing a backyard shed or calculating the stresses on the tip and roots of the wings in aeroplane could be FUN, and actually they are and related--the same way as going to see clowns on County Fair or going into Meropolitan Opera to listen to Verdi's masterpiece related too. Both are FUN and both have the right to exist.
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