I was somewhat surprised to find out that geometry wasn't as popular in the 19th and 18th centuries in the US. I was under the impression that synthetic geometry was the cornerstone of high school math education in the past but it turns out that isn't quite true. This table shows a survey of offerings in various towns in Massachusetts during 1834 - 1840. At that time Harvard didn't even have geometry as an entrance requirement.
The Rise of the High School in Massachusetts By Alexander James Inglis
To give you an idea of small enrollment of geometry compare it to the more than ten thousand students taking US history. Geometry is listed second from the bottom.
The Rise of the High School in Massachusetts By Alexander James Inglis
And all this isn't to say that it ought to be done this way in general, or that it "worked for us." Marcus de Satoy describes America up until WWII as a backwater of mathematics compared to Europe. I was able to glean from Florian Cajori's very detailed and amusing history of math education in the United States that even at the university level math was a matter of an engineer teaching students what they needed to know to become engineers, and wasn't anything like I imagine a modern mathematics department to be. While Cauchy's students in France were rioting against him teaching them the newly formalized definition of a limit, VMI was struggling to find even decent math texts for its students. Cajori did such a good job with math history that one of his books, the history of notation, is still in print. If you want to know the history of your math department at your university, assuming it existed in the 1800s, you can look up all the details.
At any rate, for a better idea of what classical education might have consisted of in a 19th century American high school check out these courses:
The Rise of the High School in Massachusetts By Alexander James Inglis
You can peruse the book linked above to see exactly what authors and texts were used. They are listed out in various places and I bet you can also find them online as well in Google Books. Did you notice that there was no English, just Latin and Greek? And, by the way, Euler's Algebra is listed as an algebra textbook used in some high schools. How cool is that? I was compelled to go looking through it and found some real gems. So, do you remember learning how to divide the polynomials using the long division algorithm? Euler's idea of a good long division problem is 1/(1-a):
Elements of Algebra By Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange
And then he says, "let a = 1" and our modern day algebra book keeps telling repeating the mantra that it's undefined, it's undefined, it's undefined, but look what neat shizzle you would have learned in high school algebra in the 1800s. See, he divides by zero and it did not cause a rip in the time-space continuum! Dividing by zero isn't just great, it's infinitely great.
Elements of Algebra By Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange
Finally, to respond to the idea of removing all proof from the curriculum I wanted to end with the words of Steven Krantz
If one were to remove "proof" from mathematics then all that would remain is a descriptive language. We could examine right triangles, and congruences, and parallel lines and attempt to learn something. We could look at pictures fractals and make descriptive remarks. We could generate computer printouts and offer witty observations. We could let the computer crank out reams of numerical data and attempt to evaluate those data. We could post beautiful computer graphics and endeavor to assess them. But we would not be doing mathematics. Mathematics is (i) coming up with new ideas and (ii) validating those ideas by way of proof. The timelessness and intrinsic value of the subject come from the methodology, and that methodology is proof." --History and Concept of Mathematical Proof
7 comments:
Well, if we get to do it that way, then consider 1/(a-1) = -1/(1-a) = -(1+a+aa+aaa+...) = -1-a-aa-aaa-...
And, so 1/0 = 1/(1-1) = -1-1-1-1-... = - infinity.
Or, if you don't like the way I distributed the negative through an infinite series like that (which I don't know why anyone should object here given what else we seem to be willing to accept), then let's do it just like Euler.
(a-1)*(-1-a/(1-a)) = -(a-1)+a = 1
(a-1)*(-1-a-a^2/(1-a)) =
-(a-1) - a(a-1) + a^2 =
1 - a + a - a^2 + a^2 = 1
(a-1)*(-1-a-a^2-a^3/(1-a)) =
-(a-1) - a(a-1) - (a^2)(a-1) + a^3 =
1 - a + a - a^2 + a^2 - a^3 + a^3 = 1
and so on any number of times you care to do it. (Isn't that what he says?) So, we do have by Euler's very same reasoning that
1/(a-1) = - 1 - a - aa - aaa - ...
And so, by the same reasoning he seems to employ, I'm getting that 1/0 is a number infinitely small (i.e. minus infinity not plus infinity).
Who is this "Euler" turkey?? He clearly doesn't know what the hell he is talking about! ;oP
Adrian,
Now when I told you last night that I found this in Euler's Algebra you just laughed. I thank you for responding to this because I was worried I might actually have to do something extreme like...figure it out on my own.
In fact, for that matter -0=0! So, infinity = 1/0 = 1/(-0) = -(1/0) = - infinity? Infinity equals minus infinity?
Well, I have no idea what either of you are talking about, but the part about dividing by zero Not causing a rip in the space/time continuum made me laugh!
I appreciated your high school schedules! I am struggling with the temptation to add Elementary Greek to our schedule next year. We'd keep going with the Latin, add the Greek in 5th grade, and then add Spanish (with Rosetta Stone or something) in 6th. Only I can't think of any Practical reasons to study Greek. Neither of my kids are likely to end up as priests. But it looks like Fun. But obviously people Used to do it, so there must be Some justification. Thoughts?
P.S. Did I thank you for the microscope recommendation? I meant to -- I have that one bookmarked now.
FYI in some (rigorous, but not particularly common) topological ways of looking at the number line/plane it makes sense to say infinity = -infinity. Just pointing that out in case you were starting to get too comfortable with thinking you knew what infinity was. Never underestimate the potential of a discussion about infinity to surprise you!
These times that it's a problem, nobosy wants to study math and it's very important than any other suject in high school, that's what I think because math teach you to think, use logic and do quick operations, I work with stadistics, logic, reason, porcentage and everything related.
Thanks for sharing, good luck men.
This is terrible that's the fact American people are bad in math, imagine neither Harvard got good results during those times, now what we can expect about future generations.
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