Recent content by Bearded Man

The concept of a "true interest" or "innate passion" is interesting. It's like we have an a priori passion, and the goal of life is to discover this passion and pursue it. I think it's more likely that we have affinities to certain areas and preferences, but that passion is largely created. You...

It can be taught. College level math classes are very different than high school level math classes. Although they can be taught, that doesn't mean you won't have to put in a lot of time studying them independently though. But to make it in academia you need to both very intelligent and very...

Would you describe the method you used to teach Spivak in 8 weeks?

I recently wrote a fairly lengthy article on the need for educational reform in mathematics for my high school newspaper. I am appalled that mathematics is approached as a mere collection of formulas, which students are told to use because they magically work when necessary. There is no emphasis...

You can self study BC, especially if you've already taken AB.

He could learn Calc 1 over the summer, but I don't know if he should. It might be best to wait until he's more "mathematically mature" and introduce him to calculus a la Spivak or Apostol instead of teaching him differentiation like "carry down the exponent and subtract one...". OP, what do...

You cannot make a meaningful contribution to physics as a high school student (usually). To do so you would first need to be proficient in the various areas of physics, all of which have advanced mathematics as a prerequisite.

What's the date of the Pre-Calc test? You could teach yourself Algebra II by the start of July, assuming 8+ hours of studying per day in the summer (in addition to around 20 hours a week from now until the end of the school year) then teach yourself precalc in the roughly 50 remaining days of...

Theoretical Physicists need to be incredibly talented in mathematics. Richard Feynman was a Putnam Fellow, making him one of the most gifted undergraduate mathematicians in the nation at the time. There is a difference between the math you are doing now, which puts emphasis on a rather dull...

When really low grades (like <40%) are considered passing, are students even learning anything?

Aight brah, check it - pure mathematicians are da bomb. Honeys be crawling up to you tryina get in yo pants, and u got to be all like "contain yourself, saucy wench, I have to go prove the Riemann Hypothesis." Algebraic Structures? More like big black booty, playa. One caveat of pure...

Recall that we can add two equations together. S_n = a + ar + ar^2 ... + ar^n rS_n = ar + ar^2 + ar^3 ... + ar^{n+1} S_n - rS_n = a - ar^{n+1} S_n (1 - r) = a(1-r^{n+1}) S_n = \dfrac{a(1-r^{n+1})}{1-r} That is the sum of a geometric sequence. You can derive the sum of an arithmetic sequence...

an is an expression for a given term in the sequence. so a1 is the first term in the sequence, 1, and a2 is the second term in the sequence, 4, etc. If you wanted to find a5, you would find 45-1.

A geometric sequence is simply a sequence of terms where each successive term is multiplied by some r. Take this, for example: a_n = 4^{n-1} 1, 4, 16, 64, ... It is clear the r is 4. The sum of a geometric sequence is given by S_n = \dfrac{a_1(1 - r^{n+1})}{1 - r} The authors note that the...

Are you familiar with geometric sequences and their sums? Looks like he jing ^ beat me to it!