And that will get you a real Feeling of Power
I can see some elements of set theory being taught to a kid, after all, the ideas are fairly basic and seem to support computer science pretty well, which is another topic being taught at an earlier age these days. I think I saw the basics of set theory as early as high school.
You may find it strange, or funny (depending on your point of view) that the math-intensive Aerospace Engineering PhD qualifying process at my university does not require set theory. Sure, we need to know linear algebra, ODE, PDE, complex analysis, statistics and a couple other topics…just not set theory. Which is the first thing I had to go learn on my own after my research got started, because I needed it. (Incidentally, the Industrial Engineers were well grounded in set theory, so I took an IE optimization class to get my exposure to sets.)
No, every learning starts with memorizing a few basic facts; then learn rules to apply them. Then the more intricate details. reading requires memorizing the alphabet to start, and the sounds associated with it.
But if you haven’t memorized addition, subtraction, and multiplication tables, you will be more handicapped than if you do know them. Boolean logic does not require set theory, but it does require an understanding of “and” vs “or”.
We learned about whole numbers, integers, rational and irrational before the curriculum demanded they be labelled as sets. Telling a child about “set is closed under this operation” makes more sense (I am guessing) if he has such real-world examples to relate it to.
I did watch it. Did you read the Common Core standards? They’re right here:
http://www.corestandards.org/Math/Content/
That sounds an awful lot like what most people were taught in school - decomposing tens of hundreds, paying attention to place value, etc.
The Common Core standards are public information; rather than reading a short video or relying on anecdotal evidence, check them out. They’re pretty clear and rarely define an actual pedagogy, as much as defining what students should know how to do.
We had New Math, but I think what we had was not “new” enough. There was a bit of set theory, but everything else was pretty traditional. By third grade I had about ten pages of three and four figure multiplication and division problems in my daily homework. There was also perhaps a week of geometry in any given semester, and we were certainly never taught or asked to prove anything about triangles or anything else. If I’d been given full-on New Math I might have done a lot better.
From what I’ve read in recent years, in its fullest expression the New Math was supposed to present set theory, arithmetic, and geometry as a cohesive whole, so that among other things we would intuitively understand much of elementary algebra by the time we got there.
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From an early 1960s Time Life book, here is an appendix on the new math.
I hope it’s all right to link to these excerpts; I placed it in a separate post to make it easier to moderate if needed.
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This actually works the same whether its exterior or interior, doesn’t it? Whether an ant perambulates the inside or the outside of an n-gonal box, it will have to make the same turns in either case.
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It works the same way, but the word “interior” and “exterior” mean something different.
If you have a right angle (a corner of a square), the “interior” angle is 90. The “exterior” angle is measuring the same angle the other way around, on the outside: 270 degrees. One way you turn to your right: the other way you turn to your left, your left again, and your left again.
I’m pretty sure that is not what it means. Forget about turning around; say you keep turning counter-clockwise. If the internal angle is 90 degrees, the exterior angle must be exactly 90 degrees, since that is by how much you must turn left at each corner.
ETA it is easy to see that the internal and the external angles sum to 180 degrees.
It’s a matter of perspective. If you regard the angles as just the corners of the polygon, then for a pentagon or anything higher the angles will total more than 360. But if you regard them as the amount by which you have to change your heading as you walk around the polygon (inside or outside), then the proposition holds. The more vertices you have, the smaller each angle–i.e., change of direction is–so when you have infinitely many vertices, or a circle, the changes in direction are infinitesimally smll.
Picture yourself walking around the perimeter of a polygon. The interior angle is the angle between where you came from, and where you’re now going to. The exterior angle is the angle between where you were going to, and where you’re going to now.
This is the way we had it. The books were mostly old math but the teachers would go over how numbers worked too, pretty much the techniques of new math and core math without so much designated vocabulary.
Yes, 270 degrees of a circle is 90 degrees of a circle. We use the word “interior” to mean the 90 degree angle, and “exterior” to mean the 270 degree angle. Sometimes it’s important to some people: sometimes it’s not important.
Not sure what you are alluding to, but when I typed “interior” and “exterior” I meant exactly what Euclid meant by those words in Proposition 16, nothing tricky. It is true that a corner apparently makes two, complementary, angles, but in a triangle or a polygon only one of them is inside the figure. As for the external angle, if you extend side BC of triangle ABC to D, then it is angle ACD (it’s easier to look at the diagram than to discuss it).
I do not agree with the statement that 270 degrees is the same as 90 degrees, though, since one is bigger than the other (it fits inside). You might say that 270 is equal to -90 modulo 360 degrees, but we said we were not being tricky and I’m not sure even the New Math deviates from straight Euclid.
The real problem with the so-called new math was they expected teachers to learn it and teach it on their own. I have two stories about my own kids.
My daughter was given the question in a textbook assignment, “How many subsets does a set of three elements have? Express your answer as a power of 2.” Obviously the answer they wanted was 2^3. The answer the teacher gave was 256, which is 2^8. She had not only misunderstood the question, but more seriously she utterly misunderstood the reason why the question was asked. When my daughter objected, the teacher could only ridicule her: “See you don’t get all the answers right.” Or words to that effect.
My son had third grade in Switzerland where he learned about base 6 arithmetic (presumably among other bases). Back in Montreal for 4th grade, they were learning about number bases and had a bunch of base 10 numbers to convert to base 6. Except for the last problem of the set, all the numbers were less than 36 and the teacher had inductively concluded that you do the conversion by dividing by 6 and than the answer is qr where q is quotient and r the remainder. So 25_10 = 41_6 since 25/6 is 4 with a remainder of 1. Had she realized that when the quotient is > 5, then you have to rinse and repeat, she would have been fine. But the last problem was 61 and she insisted the answer was 101. My son insisted it was 141. She said, “How do you know?” “I learned it last year in Switzerland.” “Maybe that’s how they do it in Switzerland, but this is how we do it here.”
ETA: We also had expanded notation, e.g. 5824=51000×8100+210+41.
However neither in grade school arithmetic nor in high school algebra was the obvious connection made with single variable polynomials in algebra. At both times, math seemed to me like a random bag of tricks that I was meant to use without being told why they worked. For instance, I was taught that of the digits of a number in base 10 add up to a multiple of 9, then the entire number is also divisible by 9. But I’m pretty sure I was never taught why that works.
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Euclid said that?
And that?
You must have a different Euclid than I did.
I am willing to have my ignorance fought, but it says “μιᾶς τῶν πλευρῶν προσεκβληθείσης ἡ ἐκτὸς γωνία”. Ie the exterior angle when one of the sides is extended. (Better to use the full phrase than to talk about an unspecified “exterior angle”.) When you do that __ there are two angles meeting at that vertex and only one of them is inside the triangle, so it’s pretty clear which is the external angle.
Where else and in what other context does Euclid talk about angles exterior to a polygon?
Melbourne, simply Googling “exterior angles” will tell you that you are wrong about what that term means. Google itself says “the angle between a side of a rectilinear figure and an adjacent side extended outward.” Wikipedia has an article showing the concept. Multiple math articles are shown where they discuss it. All agree with this definition.
The fact that the exterior angles add up to 360 on any convex polygon is something that is part of the basic geometry curriculum. The fact that you get a different answer shows that you are not using the correct definition.
The internal and external angles add up to 180 degrees. That is how the terms are defined. You appear to be mixing up that, for instance, a 90 degree angle is also a -270 degree angle. You use the negative sign when going backwards.
I don’t know that he is necessarily wrong, but mea culpa for not specifying a priori which exterior angles we were talking about. Euclid certainly does. It’s not like anyone reading this textbook, in which things like “angles” are defined, is expected to guess what an “exterior angle” might be, but Euclid only refers to the exterior angle when a side of a triangle is extended. (Or however you want to literally translate it.) at least when proving it is equal to the sum of the opposite interior angles.
I did note that a wedge-shaped corner forms two angles, in which case I suppose one would be the interior angle and the other the exterior angle, but I ask again, does that actually occur in The Elements?