Harvard 1869 entrance exam
spectrum.columbiaspectator.com
spectrum.columbiaspectator.com
Hell, this makes the SATs, with its ragged edges and sloppy Times New Roman straight out of MS Word, look like carelessly produced junk.
Which makes it all the more impressive--each page took considerable craft and attention to detail.
For more information on this, you'd have to get a book like this one--I can't find anything more online: http://www.worldcat.org/title/printing-of-mathematics-aids-f...
As a math professor, I think this is a great question. Students learn that math is about manipulating formulas and equations, or about excessive formalities. But being able to explain simple arithmetic facts in clear and plain English is often neglected, and is of the utmost value.
X^N * X^M
= N copies of X, multiplied by M copies of X
= N+M copies of X multiplied together
= X^(N+M) =
(a*a*a*a*a) * (a*a*a*a*a*a*a)
^ ^
n times m times
a^(n+1)=a^n * a
a^(n+2)=a^n * a^2
a^(n+3)=a^n * a^3
a^(n+0)=a^n * a^0
therefore
a^(n+m)=a^n * a^m
I personally like cousin_id's the best, as it is (IMHO) the simplest.
x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED.
By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.
I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.
It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it.
[1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.
I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon.
I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications:
2^4 == 2 * 2 * 2 * 2
When you multiply 2^4 * 2^4, that is short hand for:
2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8
Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
So why e^pi * e^e = e^(pi + e)? Yes, it follows from the fact that it works for rational numbers, but in order infer this, you'd need to prove the continuity of exponential function, which is nontrivial at best.
Of course, if you define a^x to be the unique continuous function f: R -> R, such that f(1) = a and f(a)f(b) = f(a+b), as soon as you proved the existence and uniqueness of this function, this follows straight from definition.
There are also different definitions of exponential functions, like exp(x) = lim n->inf (1+x/n)^n, or exp(x) = sum_{n=0}^inf x^n/n! . How easy it is to prove now that exp(pi)exp(e) = exp(pi+e) ?
2^2 == 2 * 2
2^3 * 2^2 == 2 * 2 * 2 * 2 * 2 == 2^5
In fact, I was impressed at how open-ended these questions were. For instance: "Leonidas, Pausanias, Lysander" is about as open-ended as you can get.
I for one, would just create a list of interesting anagrams. (eg: Paranoia Saleslady Snide Sun) I figure it would show my moxie.
In this particular case, the property a^x a^y = a^(x+y) (plus some very weak technical condition, like Lebesgue measurability) uniquely defines exponential functions.
So, in hindsight, you can think of exponentials as arising in the classification of homomorphisms from the additive group to the multiplicative group of reals.
It actually goes deeper than that. You can extend the reasoning to complex numbers (as everyone knows), to matrices, to Lie algebras, and probably beyond.
My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (things get much more complicated with rational and irrational exponents)? If you're allowed to assume the definition of exponentiation, then the behavior of multiplied exponents probably follows almost trivially.
To me, this question is equivalent to asking the "reason" that 2 plus 2 equals 4. Everyone "knows why," and understands it pretty well (and could even give an intuitive "proof" by counting), but the question is poorly specified.
Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.
The generalization of exponentiation makes the "multiply N times" explanation fail.
How do you explain irrational exponents this way, for example? What about complex exponents?
Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well.
I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.
http://dictionary.reference.com/browse/reason
1. a basis or cause, as for some belief, action, fact, event, etc. ...
3. the mental powers concerned with forming conclusions, judgments, or inferences.
So, its the difference between "reasoning", which we do in math and logic all the time, and "belief" or "motive". That is "reason" did indeed have a specific mathematical meaning at that time, and it still does today.
One might also google "mathematical reasoning"...
Besides the river questions, the history and geography questions were about ancient Greece and Rome.
There was nothing, for example, on medieval or renaissance history. I suspect that a comparable modern student would have a broader understanding of ancient history than what was shown on this test.
Relevant xkcd: http://xkcd.com/903/
I can't even comprehend what is being asked on most of these questions let alone what it's about, at least if I knew what they were asking I'd be halfway there.
Charles W. Eliot, president 1869–1909, eliminated the favored position of Christianity from the curriculum while opening it to student self-direction.
I really want to know how you concluded that this has something to do with religion...
I studied ancient Greek too for several years. I hadn't heard of this 'General Supposition' as a way to describe conditional clauses. Apparently -- just looked it up some -- it was an older way of describing the categories we use now (which are mostly temporal based -- future less-vivid, etc.). So there are two spheres for classifying conditional clauses.
If you look up the Goodwin grammar reference for 'General Suppositions', the vast majority are from NT Greek. So that's what made think it was mostly used in NT Greek.
So in a way, thinking about conditionals as 'General Suppositions' is sort of biased in that direction.
But you're right, could apply to Classical Greek too. I should've remembered that Classical Greek is mostly a superset of NT Greek, so there would be examples there too, etc.
But it's actually a rather deep question. Classifying conditions as 'general' asks for a kind of aphoristic understanding of things (talking only about the very general case). I would hazard a guess that the majority of this is in NT Greek. But I suppose some historians and philosophers also generalized (Thucydides, Plato) in this way and could have their protases classified as such. But it is sort of a different way of thinking. Bad to generalize ;-) -- but it may go quite to the heart of certain differences between aspects of NT and ancient Greek.
In fact, reading that sort of makes me curious about what pedagogical approach was taken to teach pre-limit calculus.
Personally I'm a huge advocate for the fact that what one might call "wisdom" is not unique to our age and may indeed be getting a bit lost in the shuffle, but nevertheless, there's no way that we can go back to covering everything that took 12 years to learn in 1869 and cover all the things that take 12 years to learn today. When you push an hour into the curriculum to cover, say, the basic functioning of electricity, to name just one thing that I think one should not be able to escape from modern schooling without having gotten exposed to at some point, an hour has to come out of it somewhere else.
It's exactly like learning scheme. Seriously who fucking cares about scheme?
I see latin and greek in a Harvard test as a part of distinguishing highly educated kids from the others.
Err... This website is written in Arc (a cousin of Scheme), which itself is implemented in MzScheme. So by extension, you care about scheme, and so does everyone else here.
I have never heard that before. What part is PHP?
I'm a CS student at the University of Waterloo, where most early CS courses and many advanced courses here use scheme. There are thousands of students taking CS courses here so in fact there is a generation of grads being produced where a significant number (thousands at least) of them base much of their CS knowledge on scheme. That makes it very relevant.
I can see how Latin will probably not help with English much (and with French I couldn't see it either) but in comparison German has a LOT more grammatical possibilities and rules and they are very much like Latin except for maybe 2 cases and a few other Latin oddities. This probably makes German comparatively harder for a native English speaker..
The way we had to study Latin was very analytical, never like a spoken language but like dissecting word by word until you could finally understand the sentence. So through studying Latin vocabulary and grammar like that, you got a different and actually excellent insight into your own German mother tongue. And a lot of words, vocabulary and expressions (also in English) at least have Latin (if not Greek) roots.
Some university degrees used to require you to have had Latin in school, medicine was one of those.
The more interesting part of Latin, however, is reading all the great writers and learning about the times they lived in. In a modern Latin class, this should get much more emphasis, even if it happens at the expense of pure language analytical skills.
Rather than in the current education system where very very few people want to proceed working in these pure sciences and majority of them want to become leaders and thanks to Univs of US in which leaders are "annointed" by dishing out MBA's based on GMAT / CAT scores.
Heck, I learned Latin in high school in the late 90s. I hated it when I took it, but it certainly made learning other things a lot easier.
In practice, the requirement for the classics was no impediment to elites; one could be admitted on the condition that shortcomings would be remediated. Once admitted, you could easily purchase the services of a tutor or cheat your way through.
(The history of mathematics doesn't get so much as a mention, either, but I don't expect it to.)
Maybe a bit harder than some of the other stuff, but still regurgitation.
That would be me. 1869 might be a bit early for Civil War history to show up, but nothing on the War of 1812 or even the Revolution? Nothing on the history of Westward Expansion?
I understand going back to the sources ("ad fontes") and education for its own sake, but I've never seen something that implies an education that is so divorced from anything of the time the people receiving it are living in.
Still, I don't think this is necessarily a bad thing, since it's fair to make a test that selects for students that have been well-educated in that time period. I just don't think it's necessarily more difficult than a modern equivalent test would be.
[1] See today's links.
Nice, reminds me of
> Find 7th root of 0.9999 to four places of decimals
:)
Worse, this test doesn't call for much analysis; it's mainly regurgitation of memorized trivia and the most mechanical arithmetical techniques.
To be fair though, at least now you can rote learn the right things to pass the tests in any country, not just a few wealthy western schools that teach latin and greek....
I also have a very close friend who is now an elementary school teacher (4th grade), and says this as well (they don't just teach with rote memorization). Recently, she was telling me about the subjects she was teaching and the different learning tools they were employing in the class room, and we were having a discussion about how great it was that they are focusing on problem solving and knowledge discovery and not just on rote memorization.
Now the students who do math contests and are AIME/USAMO level could probably do the geometry section here without a lot of trouble, but they are the exception.
Check G-SRT.4 at http://www.corestandards.org/the-standards/mathematics/high-...
There's 2 trigonometric proofs, and a number of less than obvious geometric proofs, especially the latter ones pertaining to the circle. The rational equation in #8 on algebra going to involve solving a cubic. And although #7 in arithmetic wouldn't be too hard if you worked entirely in pence, it is still a trickier problem in the days before decimalization.
Granted, it's weird in this day of students taking 5+ AP classes to see no calculus, but the math isn't weak at all. Remember, there's no calculators here. Maybe a slide rule and a table of trig values/logarithms. But that's still a lot of work by hand.
Also, as you may know, many calculators today have a CAS, a Computer\Calculator Algebra System, that will gladly give responses as a surd or fraction (or even solve a differential equation). To my knowledge, College Board has yet to ban a calculator for a CAS.
In terms of proofs, they are pretty basic, and I learned that stuff in third grade. I mean, I went in and out of gifted programs, but I think the better question is how long does it take to get people to add integers correctly? Ten years?
I drove myself a lot as a child, and drove enough teachers crazy to have to switch schools about ten times before middle school, but I would be surprised if my experience is no longer strictly atypical for people seeking a world-class education.
Note: I didn't attend an Ivy-League, but I did end up skipping a few grades. All those were after elementary school, so it's likely I was ahead of students at the time -- ahead of average, not ahead of the expectations we should have for first class minds seeking a higher education in an age when this is exceptionally uncommon.
There are communities of people who would indeed consider this an easy test but they tend to delve even more deeply into classics than a typical contemporary of the test.