I'll be teaching my oldest son to do proofs from number theory

[NNadir]

this weekend.

Anyone have experience with how these things work with kids? He's 11.

I'm teaching him the proofs for showing that the sum of a sequence of integers from 1 to n = (n)(n+1)/2 and the proof that any number whose digits sums to 3 or 9 is also divisible by 3 or 9.

So far this week we've covered summation notation, reviewed the distributive law, and communitivity and the use of indexed variables. He's been really into it (he also has gotten into teaching himself the greek alphabet after I showed him some introductory texts on set theory) but I haven't shown him the big one, proof by induction.

I hope he can get it.

They do a pretty good job with his math education in his school, but I'm hoping that he's the right age to go independent. I also don't want to overwhelm him. Anybody have any experience with this outside the school hobby home schooling?

[phantom power]

1. I used to tutor number theory.

however, that was tutoring other college students. The proof is inductive. Does he understand induction?

Oh, I see that's the topic. Well, I don't think the principle of induction is too hard to grasp. The mental image of dominos falling is a good one. After that, the proof itself is just a bit of algebra.

[NNadir]

2. Not yet. But I think he can get it.

I know the proofs, and that's the point of trying to teach them to him, to show how proof by induction works. I think these two proofs are simple enough, that they should be easy for him to grasp the concept from them.

But if you can suggest an even simpler proof, I'd love to hear it.

[phantom power]

3. Nothing simpler comes to mind.

Dominos. It's all about the dominos.

[mcg]

24. Sum of 1 to n = (n)(n+1)/2 proved visually

A way to prove that s, the sum of 1 to n, is n*(n+1)/2 without induction is visually, by considering an n x n grid. The sum of 1 to n is the number of cells in the lower triangle including the diagonal, ditto for the upper triangle and the diagonal, these two triangles together fill the n x n grid plus the diagonal (which is n blocks).
So 2*s = n^2 + n.

[The_Casual_Observer]

4. 1-1+1-1+1-1+1-1+1-1+1-1....... = 1/2

or so they thought.

[phantom power]

10. My recollection is that sum is actually ill-defined.

You can "prove" it sums to any number, by regrouping.

[Salviati]

12. I'm not sure about that...

I know that you can get any conditionally convergant series to converge to any number you want by regrouping the summation, but I think that for it to be conditionally convergant, the terms in the sequence must approach 0 as n goes to infinity. I think that with a sequence like this, there's no way to get it to converge to anything, since the addition of each term will change the value by +/- 1.

[phantom power]

13. Yes, it's non-convergent.

[struggle4progress]

17. It depends on what you mean by convergence. There are some ..

.. generalizations that agree with the standard notion, whenever the sequence converges in the standard sense, but yield numerical answers for some sequences that do not converge in the usual sense.

For example, averaging the first n terms of a sequence may yield a convergent sequence:

in the case 1 -1 1 -1 1 -1 ... this yields 1 0 1/3 0 1/5 0 ... with limit 0, so in some sense the sequence converges to 0.

IIRC, various different generalizations of the convergence notion do not necessarily give the same answers ...

[mcg]

23. e.g. it is Abel summable to 1/2

http://planetmath.org/encyclopedia/AbelSummability.html

or if a sequence has more than 1 "limit point" (in the sense of being a limit point of a subsequence) than the "limit points" could be averaged to get a "sum" which in this case would be 0.

A much more complicated example of averaging more than one "limit point" can be seen on
http://mathworld.wolfram.com/FeigenbaumConstant.html

[applegrove]

5. After reading the title of your thread - there is coffee all over my

computer screen. I just wanted you to know that.

[s-cubed]

6. I tutor junior high and high scholol students in math.

First word of advice is to keep it fun. As he gets older, he may very well resist working with you, just because it is the job of a teenager to become independent of his parents. I could never work with my boy. Don't take it personally, and don't insist that he work with you,. Enjoy the NOW.

I have seen a number of enrichment programs for kids at different schools. The best ones broaden the kids, rather than accelerate upward. (That's debatable, but it's my opinion).

I think probability and statistics are excellent, especially the former. Also, there are books on computational tricks: how to compute in your head. These are especially good if you discuss the number theory behind the computation. Exploring why divisibility rules work is good. Different bases, especially base 2 and how it relates to computers and logic. Computing in a different base. Also, modular arithmetic.

I an very fond of the books put out by Mathcounts, because the problems vary in difficulty and are generally just interesting. They are geared to 7th & 8th grade kids. There are workbooks for each year the contest has been run, and the problems are organized as school level (easiest)), district, state, and national (hardest). Depending on your confidence level, buy the solution book as well.

As to proof by induction, that's a touch one for a kid to grasp. Howver, you can look for a book on finite difference analysis. There's a large format book that is geared to kids that is very good. Unfortunately, all I have are copies of some pages - no titile, or reference.

Another good book is "1000 Play Thinks" by Ivan Moscovich, Workman Publishing Co, NYC. Lots of puzzles to help kids and parents think outside the box.

hope this helps
sss

[NNadir]

7. Thanks for your advice. We are keeping it fun.

I think he just gets off on having the time with me.

Last year we did quite well on some of the trigonometric functions, and graphing them in Excel. Most of that seems to have slipped away now, but he still understands radian measurement. As part of his interest in the Greek alphabet, he wanted to know how scientists use omega and I explained radial velocity. He grasped that quite well.

He actually has a fairly difficult time in school, since he's dyslexic. New Jersey educational policy pretends that dyslexia is non existent, and we've not been able to get any formal help with that.

I'll look into your suggestions.

Thanks again. Excellent advice, especially about the broadening part.

[s-cubed]

15. My son was not dyslexic,

but had terrible handwriting, due to a problem with small motor control. I had him tested privately. The resulting 4 page report did wonders in getting the teachers to understand that this was a real problem, not just him being lazy. I'm surprised about the school not recognizing dyslexia: you can ask for an evaluation as to whether or not he should be classified as having a handicap. If so, they are required by law to produce a plan for helping him. (I'm not an expert here.) I do know that if he is classified, any money you spend on tutoring is deductible - for as long as we still have any deductions.

[NNadir]

16. Believe me, I have been through this.

For them to recognize a "handicap" they have to recognize its existence.

New Jersey is horrible with dyslexia this in spite of the fact that one of the former Governors of New Jersey, the Repuke Tom Kean, he of the 9/11 commission fame, is a dyslexic himself.

My boy also has apraxia, the handwriting problem, and they've recognized that and are giving him OT for it. The funny thing is that he can draw quite well, but for some reason he can't write all that well.

We - he, his mother and I - know what's going on, and we're just making accommodations. If you read about dyslexia on the web, you understand that elementary school will sometimes border on the horrible and it has. Some of his teachers recognized his dyslexia as well - and admitted they were powerless to do anything within the system.

Interestingly, dyslexia is a genetic syndrome, connected with the 18th chromosome pair (other chromosomes may be involved), that only manifests itself as a "disability" in the speakers of certain languages, notably English and French. Although the gene for dyslexia is present in Asian populations it's phenotypical expression has no effect whatsoever on speakers of Chinese. It is believed that the gene afforded certain advantages to its possessors in preliterate times.

But my boy has a lot of strengths as well as weaknesses and he's coming to an age where we can focus on the strengths. Many dyslexics have gone on to do great things and I harbor high hopes for him, should humanity survive.

[s-cubed]

19. good luck

Focus on strengths is great idea. The mental calculations I memtioned never fail to impress people. Once my boy realized how proficient he was on both PCs and MACs - a rare combination - his self confidence soared.

by the way, scubed is for my initials: sss = s cubed

edit to add: some kids really like learning about groups, and that different groups have different properties. The rubik cube operations, for example, can be described this way.

[longship]

8. Mathematical induction

I am a former HS math teacher.

First, prove for n = 1 (or the first case, as appropriate)

Then, prove that if the equation is true for a particular value n=k (where k is an arbitrary number) then it is also true for the case of n=k+1. You do this by making a substitution of k+1 for n in the original expression and simplifying. The result will be a true statement in the form of the original expression.

There are any number of web sites on the technique. Google mathematical induction.

Here are some that seem to be at a middle school level:

www.themathpage.com/aPreCalc/mathematical-induction.htm
www.acts.tinet.ie/induction_645.html

The following is a very good description:
www.purplemath.com/modules/inductn.htm

Finally, another very good description at Wikipedia:
Wikipedia - Mathematical Induction

Good luck.

[htuttle]

9. I tutored both my sons in algebra and trig

They're bright, but fairly practical minded, and did not want to learn all the terminology and proofs involved until they could figure out what it was good for. If I would have started with formal proofs, they would have been turned right off (as they had been by their classes in school).

What I ended up doing was helping them to write a computer 'game'. It was a simple little thing in Hypercard, but was able to show them the basics of a coordinate system, how to use trig to 'aim the gun' at the right angle, some basics about logic (like 'and', 'or' and 'not', and how they work together), and elementary Newtonian physics (like how 'acceleration' is different than 'speed', etc...). It was also extremely helpful in getting them to understand what a 'variable' was (they had problems with that concept at first).

We ended up talking as much about computer programming (like code reuse, abstraction and extraction of common functionality, encapsulation, etc...) as math, probably because that's what I do for a living, but I think to this day what I taught them is all they remember about algebra and trig. I just wish that I'd had the tools I have now -- we could have gotten into far more complicated concepts if we'd had a 3D engine to work with.

Your son sounds like he takes to abstraction a lot easier than mine did, so you probably don't need to 'trick' him into learning things like I had to, but you still might find it helpful to find practical applications of a concept if he has problems with it.

[NNadir]

11. Good points. Kids differ.

My boy has had little trouble with the concept of a variable. It didn't take much explanation at all for him. They covered it briefly at his school beginning in third grade, and they touch it regularly. I do ask him to check his alegebra by substituting numbers. He's actually pretty poor at multiplication and division of long numbers, though, although he's better at it when - for reasons I don't get - he's dividing by "pi" which he loves to approximate. I'm not sure I really get their point with the way they're teaching him math, but it certainly has some positives.

He also gets the co-ordinate system in two dimensions, but not really in three. I think also he gets the basic idea of polar co-ordinates.

My son by the way is at best a marginal fifth grader in math, at least in school. Consistently he gets C's on his tests, which he reminds me, is passing. I don't really ask more of him in that department, grades. Most often the trouble is that he misses reading part of the problem and simply doesn't answer it.

[Geoff R. Casavant]

14. Geez, I used to be a whiz at calculus . . .

. . . but some aspects of number theory were always a bit beyond me. I didn't recognize half the terms you were using, so it appears your son has already surpassed my ability.

[struggle4progress]

18. I taught college number theory for some years. The number of possible ..

.. approaches is mind-boggling.

I used to start students off with games involving heaps of pebbles: this has the advantage of being historically accurate, visually immediate, and potentially quite tricky -- for example, you can define square numbers and triangular numbers, then you can ask if a square can be twice another square, if a square can differ from twice another square by one, and if a square number can be triangular. The Greeks knew the answer to the first question (no) and had the general formula for the second question; the third question can be attacked in a way that leads back to the second (but algebra helps and IIRC some algebraic number theory is useful to proving that the obvious formulae really do account for all the triangular squares).

Heaps of pebbles also lead naturally to so-called "unbounded register machine" models of computing: this is potentially fun, because you can write little programs and simulate them by hand. It also leads naturally to (say) Peano arithmetic, which is a reasonably simple model for number theory.

And -- thinking ahead somewhat -- Peano arithmetic, once absorbed, leads to the fact that Godel has proved some seventy years ago that anything, which is as strong as (or stronger than) Peano arithmetic, is necessarily logically incomplete. Godel's proof is accessible to a dedicated late adolescent, who has interest, some acquaintance with proofs in symbolic logic, and knows a bit of number theory (I think all that is required there is unique factorization plus the Chinese remainder theorem).

For now, visual approaches may help: try browsing Conway & Guy's Book of Numbers, which contains a lot of pictures and is intended for the general reader. How do you discover and prove the formula for the number of cannon balls in one of those pyramidal heaps.

The set theoretic approaches to the laws of arithmetic is instructive: some content of Kamke's old, little book may be accessible, if approached correctly. If you're interested in the philosophical question of what a number is, surrounded with some real mathematics, for your own edification, and potential tasty tidbits for later, you might examine the multi-author (Ebbinghaus et al) Numbers put out by Springer-Verlag in their GTM series.

Quite a lot can be done with prime factorization: e.g. compare different proofs of irrationality of the square root of two. Once the kid knows about prime factorization and about complex numbers, you can point out that the so-called Gaussian integers have unique prime factorization and that the ancient formulae for Pythagorean triples can be deduced from this fact.

A neglected topic, that I often wished more students knew about, was the Euclidean algorithm. It is, incidently, closely related to continued fractions, and there was probably a time when anyone with a scientific interest who wanted simple fractional approximations to numbers knew about continued fractions. The old classic by Dantzig (Number the language of science) contains a nice bit on the infinite continued fraction expression for the square root of two -- which, of course, can be regarded as the key to the question asked above, whether one square can differ from twice another by one ...

[nashville_brook]

20. i'm teaching my dogs to read flash cards

so far they get "sit" and "dance" and "toothbrush."

[NNadir]

21. How's it going?

There was some movie where they had the dog using the toilet bowl. I've always thought that would be useful.

I've got both my kids using the toilet bowl.

[NNadir]

22. Well we got through the weekend and we worked on it.

I think a good time was had by father and son, but I'm not certain he's mastered all the concepts.

We worked on the "sum of digits" proof, since that concept has been introduced in school (without proof.)

We had some trouble with the notation for representing the digits in the integers, but I think he's got that part of it now, having become more comfortable with how to work with subscripts.

I think we'll have lots of opportunities to talk math now, which is, of course, a good thing. I thank those who gave me good advice.