My highschool education did a decent job of this (albeit not quite as strict on explicitly writing out each step), but, as you mention, I was homeschooled and my dad took the time to find textbooks that took this approach. EDIT: I also spent a year studying classical and symbolic logic, which was frustrating at the time but incredibly beneficial in retrospect. On the other hand, my wife was homeschooled but with a different math curriculum (Saxon math, if you're curious) that heavily focused on memorizing the mechanics (i.e., "this is how you do long division", etc.) but wasn't good at promoting an overall understanding of the material. Instead, it shows math more as a "bag of tricks" and solving a problem means picking the right trick out of the bag and then applying it. At the beginning of this school year I taught "Algebra in a week"--a review of algebra for students that didn't do well on their math placement exam. I'd sneak in stuff like proofs that (a^b) * (a^c) = a ^(b+c) and comments like "we can rearrange the order of these terms because of the commutative property". At the end, several of my students thanked me for doing this! (Unfortunately, having TA'd for "Calc II for engineers", I can say that even at the university level not every math professor teaches this way. It's incredibly frustrating to watch students struggle to understand the material because they're being taught a process instead of why that process works. On the other hand, I estimate that most expect math class to be taught that way, and view it as a hurdle that they have to pass in order to get to the classes they want to take. Furthermore, teaching focused on the process does little to discourage the "I can just do this on a computer in the real world" mindset. I could go on for hours on this topic, since it's hardly specific to teaching math...)
Here, it's a little different: Calc I: - Limits - Derivatives, antiderivatives for single-variable functions - Riemann sums - Basic integration, up to u-substitution Calc II: - Trigonometric substitutions - Partial fraction decomposition; applications to integration - Natural logarithm/Euler's constant - Logarithmic differentiation - L'hopital's rule - Indefinite integrals - Sequences & series - Taylor series - First-order ODEs The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page. So many students give up on that section since it's not well connected to any other material in the class and some of the concepts (especially the various remainder theorems) cannot easily be rotely applied to a problem. And, yeah, "I can do this on a computer" does you no favors when you're trying to understand why some technique (even in another field of study) works, rather than just "oh I have this technique, let me apply it to some problems". Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.
First day of Calculus II happened Tuesday. Professor stated we are going through series and sums first. It's going to be a pain on top of all homeworks being online. This is my first time taking a math class in two years as well which is a bit of a pain having to go back and relearn (read: brush up on) many rules. Thanks for the article. I couldn't help but relate to this section of the comments. Thankfully, I'm also taking Proofs as a class, hired a tutor and plan to attend all the extra supplementary instruction sessions I can. As you can see, Devac, it really can be an odd system [from the outside].The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.
I think sequences and series are quite cool, and they tend to pop up all over the place in other fields, including probability, combinatorics, computer science, and communications. I'm interested: what all does your proofs class cover? Is it something like a foundations class, where you start off with a few definitions and axioms and prove a bunch of stuff, or does it focus on logic and different proof techniques, or a mix? Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.
According to the syllabus, we'll be going through Logic, Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, and Functions. Some Proofs in Calc. Bless my HS Calc teacher for getting our toes wet early on since some of this sounds familiar. Just a matter of dusting of memories. My tutor has gone through the class so he's my first line of defense in the comparison department. If not, I'd like to start a study group (especially so if some from my Calc class are in the Proofs as well).
This is a disturbingly apt comparison. I say leave it as-is. At least part of the problem I see is that bad math education tends to hand people properties that seem to have appeared from thin air. On the other hand, if you say, "why is X true?", then that gives an explicit clue to people that they should (if interested) sit down and try to work that out. Proofs are tricky things to write, but I do prefer a proof written to explain, rather than simply to derive the conclusion. A little exposition here and there can go a long way towards showing others the relationships you see. I am hardly innocent here, though! I do need to work out a good way of writing proofs in LaTeX. It feels like mine always get compressed into a paragraph of math symbols. Maybe I just need to focus on writing a sentence or two of exposition per step and make one paragraph per conceptual step.Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.
Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.