Harvard 1869 entrance exam • Hacker News Board

Re: Harvard 1869 entrance exam

#61

Post by twelvechairs » Mon, Dec 05, 2011, 4:28 AM UTC

derleth wrote: post #9

Where is the modern history? Where are the questions on governments? Worse, this test doesn't call for much analysis; it's mainly regurgitation of memorized trivia and the most mechanical arithmetical techniques.

I agree, but sadly I think a lot of testing is the same now. We are still strongly skewed towards giving people who have rote learned the right things a 'leg up', rather than those with true 'ability'.

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....

Re: Harvard 1869 entrance exam

#62

Post by mmaunder » Mon, Dec 05, 2011, 4:29 AM UTC

The source of the Amazon: Seattle?

Re: Harvard 1869 entrance exam

#63

Post by impendia » Mon, Dec 05, 2011, 4:29 AM UTC

twelvechairs wrote: post #50

hammock wrote: post #34

Earlier quoted context omitted.

Who here can come up with the most concise explanation?

The process of explanation by example (though I agree with others that it is really intrinsic in the meaning of a power - id like to hear impendia's explanation): 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

> id like to hear impendia's explanation

I personally like cousin_id's the best, as it is (IMHO) the simplest.

Re: Harvard 1869 entrance exam

#64

Post by jerf » Mon, Dec 05, 2011, 4:29 AM UTC

boredguy8 wrote: post #28

mortenjorck wrote: post #22

In a way, this reinforces my hypothesis that Latin in traditional Western higher education was never quite so much about Latin itself as it was about gaining a deeper understanding and greater praxis of your native tongue by reading its source code.

Uhm, what? You couldn't be more off. Traditionally, any liberal arts education would include extensive familiarity with the classics. That means Plato & Dante, at the minimum, whether a BA or BS. You were expected to know Latin and Greek because you were expected to read Latin and Greek. If you were pursuing a BS, then perhaps you would read Euclid's "Elements" instead of Thucydides' "The History of the Peloponnesian…

It isn't "chronological hubris" that the vast bulk of human knowledge has been generated since 1869. Precise measurement is difficult to even define but it's difficult to imagine a non-pathological definition for which that would not be true.

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.

Re: Harvard 1869 entrance exam

#65

Post by dangrossman » Mon, Dec 05, 2011, 4:36 AM UTC

baby wrote: post #55

BrandonM wrote: post #53

Earlier quoted context omitted.

> It's exactly like learning scheme. Seriously who fucking cares about scheme? 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 think this website has a pretty poor UI. And it's PHP also.

> And it's PHP also.

I have never heard that before. What part is PHP?

Re: Harvard 1869 entrance exam

#66

Post by dlss » Mon, Dec 05, 2011, 4:37 AM UTC

barrkel wrote: post #58

michaeldhopkins wrote: post #47

Earlier quoted context omitted.

That's the point -- you don't understand it well enough to explain it.

I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason". Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.

Being able to explain proofs intuitively is a valuable way to check how deeply you know them.

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.

Re: Harvard 1869 entrance exam

#67

Post by jerf » Mon, Dec 05, 2011, 4:37 AM UTC

CountHackulus wrote: post #15

I'm surprised there's no calculus. Not even a basic derivative. The rest of the exam is certainly difficult to warrant a calculus question.

While Calculus may nominally have been invented a couple of centuries earlier, even by 1869 it still had a lot of development to go through before it would approach the relative ease of modern calculus. Limits were still only a few decades old, with the modern notation for it not developed until 1908[1]. It takes a long time for things to go from the cutting edge of math down to pre-college curricula. Perhaps some students had been exposed to it, but testing for it on the entrance exam may not have provided much information for the examiners.

In fact, reading that sort of makes me curious about what pedagogical approach was taken to teach pre-limit calculus.

[1]: http://en.wikipedia.org/wiki/Limit_of_a_function

Re: Harvard 1869 entrance exam

#68

Post by impendia » Mon, Dec 05, 2011, 4:38 AM UTC

baddox wrote: post #38

cousin_it wrote: post #35

Earlier quoted context omitted.

Me, me! Let me try! X^N * X^M = N copies of X, multiplied by M copies of X = N+M copies of X multiplied together = X^(N+M)

That's not very rigorous, especially with fractional (or God forbid, irrational) exponents.

Indeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.

Re: Harvard 1869 entrance exam

#69

Post by impendia » Mon, Dec 05, 2011, 4:40 AM UTC

fuzionmonkey wrote: post #36

impendia wrote: post #31

> What is the reason that when different powers of the same quantity are multiplied together their exponents are added? 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.

Do you actually find college students that don't know what an exponent is? I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.

They have a working knowledge of it, with very rare exceptions -- but I find that students are taught math as a bunch of rules and not all of them are comfortable giving explanations.

Re: Harvard 1869 entrance exam

#70

Post by baddox » Mon, Dec 05, 2011, 4:46 AM UTC

michaeldhopkins wrote: post #47

baddox wrote: post #37

Earlier quoted context omitted.

That question stood out to me as a particularly bad question. What is the answer supposed to be? I completely understand how multiplication of exponents works, but I have no idea how to describe the "reason." You can give a simple algebraic proof quite easily (especially if we're just dealing with integer exponents), but unless "reason" had a more specific mathematical meaning in that time, it seems like a very vague…

That's the point -- you don't understand it well enough to explain it.

I would have to brush up a bit, but when I was in school I wouldn't have had much trouble proving it. But that's not the real issue.

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.