All calculus ever taught me was to apply these formulas to those problems (for some reason) and that dx/dy meant derivative of x with respect to derivative of y.

It boggles my mind that nobody simply bothered to tell us what a goddamned derivative was; what the word meant, and what real-life examples were. The "with respect to" was the most confusing and least useful concept. If we knew what the things were in the first place, the fact that one was "with respect to" the other would have been obvious. And "with respect to" could have been replaced with "relative to" in our minds, because we'd know what the things were.

Calculus seemed to be entirely repeating nonsensical mantras and applying impenetrable methods to problem sets and receiving an answer out the other end.

I'll also echo calls for more time on algebra. Surely the problem with algebra was that I wasn't a devoted enough student to repeating problem sets ad nauseum, but I just never got the 'knack' for it. I understood what I was supposed to be doing, but I was never satisfied with the answer "you'll start getting a feel for what to simplify to what after you do enough problems" as an explanation. It was frustrating to not feel like there was a concrete process to it all.

This made later Calculus and Statistics hard for me, because many exams would simply be 2 enormous problems ; the calculus I could get through but I couldn't simplify the problems quickly enough to be able to apply calculus or statistics at all.

I TA'ed physics at one of the top Ivy League universities in the US, and what I found amazing was that we didn't require calculus for our introductory mechanics course. You don't get into these universities without taking all the highest level courses in high-school, which means that almost everyone in the class must have taken at least one semester of calculus. But we still avoided the simple v = dx/dt calculations, the logic being that physics was too difficult to combine with the math they'd all learned years before.

My friends in economics told similar stories: "Don't worry, we don't have to use calculus to calculate marginal cost, we can use 'the midpoint method'", followed by 20 minutes of explaining an arcane nondeterministic procedure to approximate a derivative.

Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.

I like a lot of what you said, but I do disagree with that point. Immediate application of mathematical concepts may be gratifying, but making it the _only_ motivating factor behind maths can result in students who conflate the underlying machinery with its application [1].

Taking a pure maths course in undergrad with related subject matter prior to one of my Controls classes made the class substantially easier to understand. Learning the abstract concepts beforehand let me see the common applications more quickly than others.

I think the biggest difference is that when I learned some Calculus through Physics, it was a little more difficult to go from a concrete basis to a more general one. My understanding ended up being based on analogies to other concepts until I went back and covered the theory again.

[1] This is all my own opinion, I'm not an educator so the most I can do is pull from my own pedagogical experience :)

I keep seeing how society needs better math education, whether it be statistics or calculus. Well no shit. Unfortunately those are extremely hard subjects to teach in high school. I'd argue statistics is harder to teach than calculus. In calculus you usually derive things by following the formulas. In statistics there isn't always a formula, logic, or path you can follow.

I took both AP Calc BC and AP Stats in HS and I'll tell you I learned a lot of calculus then and am learning a lot of statistics now (I'm 32).

The huge problem as I mentioned elsewhere is that stats is really applied math. You do learn a lot of regurgitation-style stuff - formulas, proofs, blah blah, but it's all useless without applying it. Moreover, stats draws from so many different areas in math and requires analytical skills that are closer to many of the things scientists emphasize in learning to conduct studies and experiments. This makes it almost unsuitable to teach the more valuable parts of stats at lower levels because the students can't possibly be prepared.

At best, I think you can teach some general things about stats and the mentality I described of being analytical and skeptical. Unfortunately, stats is just really hard for anyone to properly and comprehensively learn who isn't going to be able to invest a lot of time both learning pre-requisites and then all the different areas of stats. It's like learning to be a carpenter and only understanding how to work a hammer, but not a saw, measuring tape, or anything else.

Then the solution is simple! We teach calculus to 3rd graders, then no one will be afraid of it!

But seriously, I agree that calculus is taught way to late. I wasn't introduce to calculus until college, and if I had been even by senior year of high school, I would probably be a mathematician right now.

For example, my 8th grader is being taught "Higher School Algebra" by a teacher who last year taught 5th grade and by all accounts has to leave the classroom often to get tips on the material from the 7th grade math teacher.

Boy, am I ever glad!

I am much more financially comfortable, enjoy the hell out of my job, and don't have to deal with the annoyances and politics of being a high school teacher. Would I be technically capable? Absolutely. Would I be good [enough] at it? Probably. Would I do it for a mid (or even high) five-figure sum, even with the good retirement package? Not on your life...

Unfortunately, the fragmented (ie, redlined) us public school system means that higher teacher play is impossible in lower income areas. So private schools will pay more for the calculus teachers, and the low income public schools will continue multi-classing pe teachers for the task. And in another twenty years we will continue to lament the increasing lack of social mobility in this country...

The lowest-income school districts around here pay higher salaries than the nicest districts by a solid 20-30%. Most teachers with options still won't work there because the environment is frustrating, unrewarding, dangerous, and toxic.

Please drop me a line if this sounds interesting. Email in profile.

I used to think this until I had kids and started teaching them math. When you know math you can abstract it away and really focus on the concepts. But you don't realize how many simple calculations are done when abstracting these concepts.

The problem is that elementary and middle school math teachers can't teach except out of a book. Your child's math teacher isn't teaching your child math, she is just the proctor for their math textbook, which isn't teaching them anything at all really.

This is not a dig on (all) teachers. These workbook-driven curriculum are a band-aid for bad teachers, and all they really do is harm good teachers and students.

While not wrong, that sentence is devoid of meaning, with or without the synonyms swapped out.

Scottish first degrees are 4 years rather than 3 in England - presumably to allow for this.

Where I live, it was just called 'math'. The first time I encountered algebra was in university, and that started with groups (a set and an operation on pairs of items in the set)

As to the 'pre': I have a masters in math, and lots of what we did was in some sense pre.

For example, calculus on real variables showed you how to define differentiation and integration and how to rigorously prove lost of theorems (pro tip: if you can't make heads or tails of an exam question, at least write down "let epsilon > 0")

Next, we basically went through the same set of theorems but in multiple dimensions, but showed that those rigorous proofs had glaring holes, but hey, now you know better. Also: infinities are weird (pro tip: the area under a 2D curve can depend on the order in which you add up its parts).

Next, calculus on complex numbers extended that and showed that those rigorous proofs had glaring holes, but hey, now you know better (pro tip: every contour integral you compute at an exam has a value of plus of minus 2n pi, with n typically being plus or minus one)

For math majors, the next step was measure theory: that definition of integration we started with? Forget it. Lebesgue came up with something that's better. And by the way, those proofs we told you about have glaring holes (let's spend a few hours proving that cutting a rectangle across a diagonal gives you two parts that each have half the rectangle's area). A few of the functions that we proved cannot not be integrated can be, if you use this new definition of how integration works.

That seems a waste of time, but I don't think it is. Very few people can jump in at the deep end and survive; the first iterations are necessary to prepare one's mind for the next steps.

Given the above, I think you will have had various 'pre-' things, even though they weren't named so.

Makes no sense to me. All those things are related. There's a picture that shows you why Pythagoras' Theorem is true. And why the difference of two squares is never prime.

You'll have had a whole summer holiday between trig and algebra if they're separate things. Surely they are taught as a kind of whole?

Got a bit more separate at university, but not really.

"pre-calculus" is whatever isn't covered in intro to algebra and geometry, but needed for differentiation and integration:

https://www.khanacademy.org/math/precalculus

* Trigonometric equations and identities

* Conic sections

* Vectors

* Matrices

* Imaginary and complex numbers

* Sequences, series and induction

* Probability and combinatorics <-- more of a side topic, usually.

After Geometry we had Algebra 2 (quadratics, etc.), then Advanced Math (trig, pre-calc). To my detriment I made up "pre-trig" to illustrate being thrust back into a cold curriculum after (what I felt were) warmer concepts in Geometry.

Like you said, I didn't have to bother memorizing anything, because everything could pretty much be done with simple integration or differentiation.

I agree it's nice to know the logic and background that is the foundation of how we got _here_, but sometimes the older details aren't really as important and take away time from students learning the modern, more useful techniques and concepts.

One of the big things about kids and math these days is that you have less and less of their attention. If they see it's getting too hard, most don't take the challenge and they just put less effort in.

I don't see how students can truly learn if they don't understand the formulas they're applying and their relevance to the real world.

That was the point of the class, that there area two kinds of problems, linear problems and problems you can't solve.

I'm not in the least suggesting that an engineer shouldn't go through three semesters of Calculus, Linear Algebra+DiffEq and a Discrete Math course.

For example I taught my daughter factions this way.

Me: "These factions you see them? Now see these decimal points?"

Her: "Yeah I hate factions they are stupid!"

Me: "Factions are the only consistently real numbers below 1 and decimal points for the most part are made up numbers. 1/3 is 1/3 and never .3333."

Her: "I always thought it was the otehr way around."

You can construct a complete ordered field from infinite decimals

This has nothing to do with calculus, though.

I wish you would teach me calculus =)

FYI: infinite sums are calculus.