For the past few years, I've had the pleasure of helping my children with homework. I think the best times are when we study mathematics. It's not just that I like math but I find that in helping each of them I learn more about math that I didn't learn in high school.
For example, not long ago my son, who's in the eighth grade, was learning about functions. So, in his homework there was a function, say, g(x) = x+5. Pretty simple.
One problem asked him to evaluate g(3). Now, most of us would say g(3) = 8. Simple. He didn't.
He said, well, g is a function, the parentheses without any operator imply multiplication. So, g(3) is not, as we might say "g of 3" but is simply "g times 3." "It's 3g, Dad, that's it."
I had to think about that because on the one hand one response is "No, that's wrong." On the other hand, we do write terms like g(3) to mean "g times 3." It came to me that here is an example of ambiguity within mathematics and how one has to know what the statement means at a level above the statement itself.
Here's another observation. Two nights ago I was helping one of daughters with math; she's in the 11th grade. She had to solve quadratics and one problem involved a term such as sqrt(x-3). So she manipulated the equation, put the square root on one side, with other terms on the other side of the equation. She squared both sides and solved for x. Simple. Not really.
After she determined the answer, I told her, "Let's try these answers in the original problem." As quadratics we found two solutions. So, she substituted her solutions, and lo and behold, with each answer she had two possible paths. For x=12, one of the solutions, the term sqrt(12-3) is 3 but it's also -3. As she explored each possibility, only the solution of 3 worked. Another solution was, I believe, 7. So the term sqrt(7-3) is 4 or -4. Well, x = 4 did not work but when she selected x = -4 this worked!
I told her, you're on to something deep here. You found two solutions but when you substitute them back in the original problem you then have to make choices as to how to use them to get the solution to work.
I asked her to talk with her teacher about this and I haven't heard if she did.
My takeaway message is this: When you have a chance to go back to elementary levels of work, you may find new insights worth your time. I know I have.
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Welcome back to posting! The blog world missed your comments and insights. And as always, an insightful one here. One of the reasons that the cliche "look at the world as children" is often true (though it needs an adult "observer";-) is the wonderfully unfiltered/unconstrained views they take we adults have long ago started taking for granted and seldom if ever question anymore. Of course, it is precisely in the questioning of absolutely "trivial" (i.e., foundational) matters/concepts, that some of the most important scientific and mathematical breakthroughs occur. Chaos theory, as you know, was born of the genius of Feigenbaum "questioning" the conventional wisdom regarding a lowly parabola!
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