kitchen table math, the sequel

This weekend's events:

I just got the answer to this problem right:

Ten out of every 1,000 women have breast cancer. Of these 10 women with breast cancer, 9 test positive. Of the 990 women without cancer, about 89 nevertheless test positive. A woman tests positive and wants to know whether she has breast cancer for sure, or at least what the chances are. What is the best answer?

I realize most ktm readers can do this in their sleep, but I had to reason it through ... and I did!

That really makes me happy.

Which brings me to yesterday's challenge: I wrote a proof!

On purpose!

I had been talking to friends about index funds, the stock market, and the ever-terrifying Federal Reserve...and pretty soon I found myself utterly confused.

My question was whether Ed and I needed to get the 401(k) out of an index fund and into cash while the stock market is losing its mind.

Both of my friends seemed to think that something called "dollar cost averaging," which I had never heard of, meant that you don't actually lose more money when the stock market declines because your dollar cost average now goes down as you buy more shares with each new deposit into the 401(k).

One said she sees stock market declines as opportunities to buy more stock at a discount. She doesn't seem to worry about stock market "corrections" at all.

Neither of my friends was making an argument about timing the market. Both seemed to be saying that when the market falls, your dollar cost average falls, too, so you come out OK regardless of whether the market has hit bottom or not.

I was so flummoxed that I pulled up an Excel sheet and did three hypotheticals comparing two investors starting with the same amount of money distributed between the market and savings. In each one the investor who stayed in the index fund as it sank, or who put more money into the fund from savings as it sank, ended up worse off.

Then I wrote a proof!

That was a moment.

Regardless of whether my proof is correct or incorrect, I had just had a real-life experience of the incredible power of mathematical proof. I was thinking about all those people asking when students will ever use algebra in 'real life' -- I just used mathematical proof in real life.

Several years ago, I read a book which explained that a loss is a loss is a loss: money you've lost in the stock market today doesn't come back tomorrow. That made sense to me, but now I find that, apparently, few people believe it.

And it is a hard idea to hold on to, because intuitively it feels like money "comes back" when the stock market rises again.

Intuitively, it feels like you lose the money only when you sell after the loss, not when you hold.

Now I can look at my proof and know that money lost is money lost. You can sell, you can hold. Either way, that money is gone.

What to do about it--if anything--is another question, of course.

Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials) (v. 1)

Thoughts?

re: TERC on Establishing Truth in Geometry

Oh really .... I don't know where to begin!!!!

"establishing the validity of ideas is critical to mathematics"

I know straight away that my blood pressure will need to be checked by the time I get to the end of this!!!

But wait, there's more!!!

"Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms."

Of course, that should read "Most mathematics instruction and textbooks AND ALL MATHEMATICIANS ..."

"evidence for its validity in the form of a proof"

By this stage, it's pretty obvious to me that the author isn't a mathematician. "Validity in the form of a proof" ..... what other type of validity is there??

Of course, the trained mathematician should have their 'proof by obfuscation' alarm bells ringing by now.

"For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument."

Is this even English? [ed.: I've been asking myself the same question. When I finally learn how to diagram sentences, I'll be able to answer it.]

"However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?"

They'd better see it as (ahem) "a way to establish the validity of their ideas" or their teacher hasn't really communicated the difference between Science and Mathematics too clearly.

Worthy of mention is the desire to "convince students" .... you may accuse me of semantic nitpicking here, but it's important!!

"Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work."

Which roughly translates as "In order to save the village we had to destroy it!"

---------------------------------

I think Melanie Philips (British author) summed it up best in her book 'All Must Have Prizes':

"A fundamental shift in emphasis from knowledge transmitted by the teacher to skills and process 'discovered' by the child has undermined the fundamental premises of mathematics itself. The absolutes of exactness and proof on which the subject is based have been replaced by approximation, guesswork and context."

Melanie Phillips, All Must Have Prizes

No one would deny that establishing the validity of ideas is critical to mathematics, both for professional mathematicians and for students. But how do people establish "truth"; how can they prove things? According to Martin and Harel (1989), in everyday life, people consider "proof" to be "what convinces me." Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms.

But mathematicians most often "find" truth by methods that are intuitive or empirical in nature (Eves 1972). In fact, the process by which new mathematics is created is belied by the deductive format in which it is recorded (Lakatos 1976). In creating mathematics, problems are posed, examples analyzed, conjectures made, counterexamples offered, and conjectures revised; a theorem results when this refinement and validation of ideas answers a significant question. Hanna (1989) argues that because mathematical results are presented formally by mathematicians in the form of theorems and proofs, this rigorous practice is mistakenly seen by many as the core of mathematical practice. It is then assumed that "learning mathematics must involve training in the ability to create this form" (pp.22-23). The presentation obscures the mental activity that produced the results.

In fact, according to Bell (1976), personal conviction grows out of internal testing and forming a judgment about whether to accept or reject a conjecture. Later, one subjects this judgment to criticism by others, presenting not only the generalization formed but evidence for its validity in the form of a proof. For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument.

In sum, formally presenting the results of mathematical thought in terms of proofs is meaningful to mathematicians as a method for establishing the validity of ideas. However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?

Let me guess.

No?

No, students do not see proof as a way to establish the validity of their ideas?

Is that it?

Conclusion

Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work.

I had a feeling.

By focusing instead on justifying ideas while helping students build the visual and empirical foundations for higher levels of geometric thought, we can lead students to appreciate the need for formal proof. Only then will they be able to use it meaningfully as a mechanism for justifying ideas.
Geometry and Truth
by Michael T. Battista and Douglas Clements

Only then, after sophomore year has come to an end and so has geometry.

Here's a question.

How many sophomores in high school have mathematical ideas?

With Catherine's encouragement, I've been blogging about my nine-year-old mathematically gifted kid. I call him GK for short. Here's our math lesson from this morning:

GK proves negative exponents.

It is Catherine's theory that how our schools treat our gifted can be a pretty good measure of academic rigor in general.

I would add that we can learn a good deal about mathematical thinking, and all its various forms, from observing the gifted kids. The problem is keeping up with them. That's why my blog is called "Clueless Mom of Gifted Kid."



update from Catherine:

I asked Barry whether this is a proof - it is!

Yes, that would constitute a proof, and even though it proves it for 3 ^(-2) one can see that it extends to all numbers. A more general proof would be that since a^m/a^n = a^(m-n). If n > m, then m-n is negative. Since it's the same as dividing a^m by a^n, one can see that there are m a's in the numerator and n a's in the denominator. Through cancellation one is left with 1/a^(m-n).

There are more rigorous and formal proofs but the above is suitable for an algebra 1 course.

Steve H left this demonstration (proof?) of why, when calculating slope, it's OK to use either point as "Point 1":

(y1-y2)/(x1-x2)

= -(y2-y1)/[-(x2-x1)]

= -1*(y2-y1)/[-1*(x2-x1)]

= -1/-1 * (y2-y1)/(x2-x1)

= (y2-y1)/(x2-x1)

Here's my question.

What are the justifications for each step?

(y1-y2)/(x1-x2)

= -(y2-y1)/[-(x2-x1)] mutiplicative identity ??

= -1*(y2-y1)/[-1*(x2-x1)] mutiplicative identity again??

= -1/-1 * (y2-y1)/(x2-x1) commutative property of multiplication??

= (y2-y1)/(x2-x1)

I'm definitely ready for something more formal...although I'm not sure where I'm going to find it.

Think I'll check Dolciani's & Foerster's books.


update from Steve:

I think schools should spend much more time on these identities. They should show how they are used in all sorts of interesting ways, forwards and backwards. I don't think I really learned algebra until this happened. [Catherine here: I agree absolutely. I'm really feeling a need for this, and it's not really something I can "provide" for myself - at least, not without a HUGE amount of effort.]

(y1-y2)/(x1-x2)

= -(y2-y1)/[-(x2-x1)]

This is the Distributive Law, in reverse, if you will.

[Catherine: I didn't see this! Now I do!]

= -1*(y2-y1)/[-1*(x2-x1)]

I suppose you could call this the Multiplicative Identity. I mentioned before that the sign "belongs" to the term that comes after it. I always like to think of a sign as a +1 or a -1. [Catherine: me, too]

= -1/-1 * (y2-y1)/(x2-x1)

I suppose you could call this the Commutative Property of Multiplication law. In other words, it doesn't matter which way you multiply fators. That's why I liked to have students circle the factors in a rational term. Then they know that they can move the factors to any position.


= (y2-y1)/(x2-x1)

I got rid of the -1/-1 using the Multiplicative Inverse law.


It's interesting that in the list of basic identities I found online that there wasn't an identity for

a/1 = a

I suppose this would come from the Definition of Division