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i could remember how to complete the square but i couldnt remember the quadratic formula so i would derive it at the top of each test before i started . . i can remember processes but i cant memorize things like dates, names, formula . . .

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to me this derivation makes the formula more memorable, i.e. if you can remember

 

(a-b)^2 = (a+b)^2 - 4ab

 

and 2x = (a+b) ± (a-b)

 

then you get 2x = p ± sqrt(p^2 - 4q).

 

but re deriving it is definitely the best way to be sure it's right!

Edited by mathwonk
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i could remember how to complete the square but i couldnt remember the quadratic formula so i would derive it at the top of each test before i started . . i can remember processes but i cant memorize things like dates, names, formula . . .

 

Yes, me too!

 

I have so many tricks to make things more memorable or to reduce them to something I already have memorized precisely because I'm dreadful at memorizing.

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Well I suspect you're just better than me at understanding completing the square.

 

Gosh that's not at all what I meant, I hope it didn't come across as arrogant. :)

 

I'm really interested in everyone else's 'aha' moments. It's always been something that just makes me wonder, why something is so easy for person A, they see it immediately, and person B struggles and struggles and has multiple explanations and still doesn't grok it.

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Please forgive me klana, i did not mean to suggest that was your opinion, that is my opinion.

 

i am also aware that everyone's aha moment is different, which is why teaching math is so hard. there is no size that fits all. that's why home schoolers and other flexible teachers, have an advantage, in my opinion.

 

by the way, maybe the reason i had trouble with completing the square is i had trouble with fractions, like 99% of my students. finally in graduate school at brandeis, one of the most brilliant mathematicians i ever met told us: "to deal with fractions, the first step is always to get rid of the fractions".

 

no one had ever told me that before.

 

so lets see if i can derive the quadratic formula this time by getting rid of the fractions first.

 

ok we have x^2 + bX + c= 0, so then we have x^2 + bx + (b/2)^2 - (b/2)^2 +c = 0,

 

so [x+ (b/2)]^2 = (b/2)^2 - c. now what? i thought this would be trivial.

 

well let me take my teacher's advice and get rid of the fractions by multiplying through by 4 i guess.

 

that seems to give 4[x+ (b/2)]^2 = b^2 - 4c.

 

then taking square roots gives 2(x+b/2) = ± sqrt(b^2 - 4c.)..... and so on.....

 

well ok, but i was not that skillful at algebra in high school. so how did i win the state algebra contest? ( answer: i lived in tennessee, had a good teacher, and practiced test taking strategies.)

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by the way we all also have our own personal styles of seeing. i am a visual geometric visualization guy and many others see what equations say and mean better. i have often sat in seminars and had friends point out errors in equations on the board which were still opaque to me. i felt very stupid but tried to keep my composure as well as possible, knowing that i too had my strengths. If you want to survive in a subject you cannot allow your weaknesses and failures to discourage you.

 

and in fact i chose my specialty of algebraic geometry because it combined my strength of geometric visualization with my weakness of algebraic manipulation. i wanted to challenge myself and simultaneously take advantage of my strong points. it worked out well, but i was greatly helped by other colleagues with complementary strengths. math is often a collaborative effort.

 

some of you and your children definitely have more math talent than me. If you wonder how i received a PhD when other more talented people may not have, my secret was I just kept at it, maintained the love of the subject, and received a lot of help. If you love the subject you will get a lot of help. And if you or your child want it enough, you will find the same support.

 

We often think tuition is very expensive, but that is just a scam for selling diplomas, not knowledge. A wise man told me once "attention will get you teachers". That means of course that a dedicated student gets a break on tuition. this is still true. If you seek a degree and have limited funds, I recommend keeping this in mind.

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A remark on recommended books. It occurred to me that if e are going to teach elementary age children, we might benefit from giving ourselves at least as much preparation as is recommended for prospective elementary teachers. Of course this is obvious to those here with more experience than me, but it made me think to recommend a book that had not occurred to me before. Rather than a book aimed at the child, this is a book aimed at college educated adults who intend to explain elementary school mathematics.

 

It is Mathematics for Elementary school teachers, by Sybilla Beckmann. There is also an excellent Activities Manual to go with it. This book was written by UGA colleague Dr. Beckmann, formerly a professional algebraic geometer like me, over the last 20 years or so, in a dedicated attempt to improve the training of elementary math teachers. This book has been called the best one of its kind in America by an agency evaluating materials for mathematics education.

 

Best of all for us, it has been required for so many students for so long that it too is available used for as little as $1. I would hazard an opinion that Dr. Beckmann's book (with the activities manual) is better for developing and fostering understanding of mathematics than Michael Serra's book. And Dr. Beckmann's book covers everything, arithmetic, number theory, algebra, geometry, statistics, and probability.

 

 

If we read a book like this, we can more easily do the kind of teaching that was mentioned above (by kohlby, Beth, happycc.....), passing from one resource to another as best suits the child, independently of any fixed curriculum. (I started to say ciriculoum!) Here is the link for cheap copies:

 

http://www.abebooks.com/servlet/SearchResults?an=sybilla+beckmann&sts=t

Edited by mathwonk
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