TITLE: Are We Teaching Pre-Calc Wrong?
QUESTION [39 upvotes]: It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial.  When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that went into its creation, we risk dragging our students past that conceptual understanding. Source.
By the time Newton and Leibniz were developing Calculus mathematicians already knew how to solve particular problems of derivatives and integration. There was a good understanding on the way to resolve problems, too. Newton's teacher, Isaac Barrow, already had some understandings of the topics Newton would push to the perfection. Source.
For me, pre-calc was plugging and chugging trigonometric, geometric, and algebraic equations. Math books follow a rather narrow and strait path across some imaginary border from the province of pre-calc into calculus. As was mentioned already, that historic transition was anything but discreet. Should Pre-Calc teach the foundations of the fundamental theorem of calculus that were known Pre-Lebniz/Newton? I think that would energize what is otherwise a boring exercise in tedious calculation. 
A beautiful example of a Pre-Leibniz integration: Who realized $\int \frac 1x dx =\ln(x)+c$?

REPLY [15 votes]: I like @Arkamis' answer a lot and I'd like to add to it by expanding the ending part of that answer.
Primary and high school mathematics is often seen as a bunch rules and formulas to solve certain problems, and very few teachers give any explanation for those.
For example, a colleague of mine asked her student something along the lines $2^3 \cdot 2^4 = ?$. He was trying hard to remember the formula from his high school, but couldn't. Then she went on with $2^3 = 2 \cdot 2 \cdot 2$, $2^4 = 2 \cdot 2 \cdot 2 \cdot 2$, so $2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2)$, so we have $7$ twos, so that's $2^7 = 2^{3+4}$. The student's response was, in complete astonishment: "Wow! So that formula actually makes sense!?!".
When P.E. is "taught", it is often customary to learn basics of few sports, do various gymnastic stuff, run around a bit, train a body,... why is mathematics not seen as a polygon to train brain to think, but instead it's a place where they try to pour lots of cryptic stuff in that same brain?
"You'll need it later in life" is a lie for most of the mathematics, unless the student is to become a mathematician, physicist, engineer,... But, thinking in order to solve problems, even if they're not real-life, would be priceless "in life".
And just like the aim of P.E. is not in those results you make there (none of the measured running times or climbing heights will ever be used anywhere), a bunch of theory and formulas should not be the aim of mathematics. We make kids run on P.E. to train their bodies in a certain way; the process of understanding how and why, to train brains, should be the aim of mathematics, not encyclopedic knowledge that can be found on Wikipedia and bunch of other sites.
Students don't need $a^b \cdot a^c = a^{b+c}$, but they do need to understand why it works like that. They need to see that the above "proof" works for the natural numbers, it is good for them to think how would they expand it to integers, maybe even rationals, and then be explained how it is actually done. Sure, it easier to just spit out the formula, but such approach misses the point entirely.
It is worth noting that this is the problem with most schooling. History, for example, is often taught as a combination of dates and names. Lots of boring facts, without concept. In my opinion, it is much more important to explain how Hitler could've happened, than what is the exact date he attacked Poland. It is the understanding of the phenomena that lead to a completely different approach towards Germany than after the WWI. That explanation is important and, more importantly for the scope of this discussion, far more interesting to students.
Geography... I was taught how much coal does Russia export (?!?), but I learned more about countries from "Where in the World is Carmen Sandiego" than I did in school, just like I learned more history in 2-3 years reading of Cracked (a comedy site!) than I did in 12 years of primary+high school. In the end, all I was left with was contempt for that stuff, just like many students are left with contempt for mathematics.
"Why?" is the most important question in the World, and most schools seem to avoid it. Try asking a student "Why?" for an answer (s)he wrote on the blackboard, and they'll immediately try to erase the answer, thinking that "Why?" equals "This is wrong". And that is the true failure of the schooling system.
A bit of context for this answer: I was born and raised in ex-Yugoslavia and Croatia; I've read about schooling systems around the World, but I work at the university, so I'm not part of school education, but I do see the consequences.
Edit:
Today's SMBC is very appropriate for this discussion, and it'd be a pity not to include it here:

-- Source: SMBC, 23.6.2016. (extra panel, off-topic for this discussion, here)
There is also a quote I recently read:

Every maker of video games knows something that the makers of curriculum don't seem to understand. You'll never see a video game being advertised as being easy. Kids who do not like school will tell you it's not because it's too hard. It's because it's boring.
  --  Seymour Papert [source]