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IME the #1 problem is the failure to teach anything WELL. I took AP Calculus in high school, and today, as a 35-year-old, I'm still continually surprising myself with the problems I realize must be related to Calculus-- but nobody bothered to tell me. Things like the 'ol speed vs acceleration vs jerk kind of problems.

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.

Dude, the "with respect to" clause is a pretty deep idea in calculus that really matters in multi-variate but not so much in uni-variate.

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).

See my other post, but I have to agree stats is incredibly hard to teach at the high school level. I also see the problem with grandparent's failures at applying math.

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.

Looking at Common Core anything besides old arcane way of math will never be accepted :(

When calculus is taught badly no one knows how to apply it even when the problems are staring them in the face.

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.

> 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 :)

>BUT people are always afraid of anything past 4th grade math!

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.

In the US, problem #1 is that it is very hard to find people who understand Mathematics sufficiently well to teach it effectively, who are willing to work for the salary typically paid to a teacher.

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.

Spot on. Both of my parents were high school math teachers (for most of their career). They essentially forbade me from going into public education, which is what I thought I wanted to do when I was in high school.

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...

the other criteria that kicks some people out are the requirements for a K-12 "teaching certificate". if you don't get a degree in education, it can be difficult and time consuming to get a teaching certificate. a friend of mine some years back desperately wanted to teach high school math, and had a bachelor's in pure math, tutored for years and was well loved, but couldn't surmount the teaching certificate hurdles. so now he works in software QA.

Yeah, the line in the article about demand for calc teachers outstripping the supply suggests an obvious remedy...

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...

> Unfortunately, the fragmented (ie, redlined) us public school system means that higher teacher play is impossible in lower income areas.

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.

I think that a student not well versed in how to perform calculations will not do well with the theory. Last night in the calculus class I am currently teaching a student could not correctly translate the fact that f is increasing where f' > 0. This on a problem where f' was linear. He just stared blankly at the inequality 2x-3>0. Anyone well versed in this sort of calculation is not going to fumble around at this problem. He thinks calculus is the problem when it's his knowledge of algebra that hinders him.

I'd be very interested to hear your opinion about a book I've been working on. It's a 3-in-1 combo, math+mech+calc, I wanted to teach mechanics and calculus in an integrated manner and I realize, from 15 years of private tutoring, I need to review all the high school math material because many students don't know the basic.

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

"“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories."

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.

> BUT people are always afraid of anything past 4th grade math!

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.

This pattern of patterns and structures is the structure of structures

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

I would have done much better in math if the class after Geometry was something other than pre-trig.

As an alien to the US education system, I am always completely confused by these 'pre-' course designations. What on earth is 'pre-trig' or 'pre-calc'? I mean, I got all the way up to undergraduate level in mathematics and I never needed to learn 'pre-' anything.

The idea of a "pre-" course is that you cover ideas needed for the later course. For example, a "pre-calc" course typically covers the sort of algebra that you would need in calculus but not actually integrating or differentiating anything. It's a matter of naming. You could call it "Algebra II" or "pre-calc".

They're not always required, but they try to bridge what you already know (or are supposed to know) with what is to come and move at a slower pace. It's also sometimes like an extra option for the less mathematically inclined who are still required to take such a number of math courses. On the other hand the only 'pre-' course I ever took (I tested out of pre-algebra somehow) was 'pre-calculus with honors'. I don't remember it much but I think it was mostly reviewing the harder topics from intermediate algebra aka 'algebra 2', which included trig (no separate trig course in my system), introduction to limits, and toward the end differentiation and maybe discrete summation. The 'honors' variant was also geared toward students who were planning on taking the more comprehensive 'AP Calculus BC' variant rather than the 'AB' variant.

There's a difference between AP calculus in Alberta and British Columbia? :)

Yes. In Alberta, all you care about is the rate of change of the the price of oil and computing the volume of manure, so only the basics about derivatives and integrals are needed. In British Columbia, you have to worry about how to fit your oddly-shaped modern furniture into your tiny Vancouver apartment and find the region of convergence for your mortgage, so more advanced techniques are necessary.

Just in case you didn't know calculus BC is considered the harder AP calculus while AB is considered the easier.

So I went to school in a country that is notorious for forcing kids to specialize early - the UK. In my age cohort, at age sixteen we had to pick just three subjects to study for the next two years. I chose 'pure and applied mathematics' as one of those three subjects. Some people chose 'pure mathematics' and 'applied mathematics' as Two of their subjects. I guess, vaguely, that doing the expanded applied maths would have included more calculus (not sure what that might have entailed - more second order differential equation stuff maybe?) - but there was definitely no point where I was faced with the choice of 'should I do a bit of calculus, or a lot of calculus?'. 16 year olds are not good at knowing how much calculus they need to learn. I was pretty happy just choosing 'do I want to learn some mathematics, no mathematics, or a lot of mathematics?' - and leaving it up to the school to figure out how much calculus to include. Based on my career since, the amount of calculus included was, it turns out, 'enough'.

Worth noting that the Scottish education system is different - people do 5 or 6 highers at 16/17 and those are used for university entrance rather than 3 A-levels at 17/18.

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

Hadn't thought of it like that before. :) A bit more usefully than the "one is harder one is easier" distinction, you can kind of think of the letters corresponding to semesters in a college course. A would be an introductory level (and possibly the only exposure a student might get if they're not interested in math but have to take a course), B would be a semester of Calculus 1, and C would be a semester of Calculus 2. Since high school the class goes year round you're still at a slightly slower pace than college, but if you do well on the AP BC test at the end many colleges will waive Calc 1 and Calc 2 from your requirements, whereas the AB test would only let you waive Calc 1. (A few other AP tests have different letter levels -- for physics only the C level seems to count and that's just basic Newtonian physics with very little calculus, though there is an optional E&M portion I never got to that may approach the Maxwell equations by the end... For computer science they used to have an A and an "AB", but they've dropped the "AB" since I took it. The "AB" basically got into actual algorithms and data structures, the A portion was basic programming and OOP heavy. All in Java.)

Same experience here. I went to international school, so I often ended up with American friends who'd talk about Algebra 1/2/3, pre-calc, trig, geometry, etc.

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?

Did you learn all of K-12 math in one year? Which year?

You certainly can't learn it as separate things. I had classes called "Mathematics" every year, all the way to the end of high school.

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

I'm already baffled by the distinction between algebra, trig, and calculus.

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.

"pre-trig" doesn't exist most anywhere.

"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.

I tried to post this yesterday but was getting rate-limited.

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.

We should teach early calculus along with entry-level SR/GTR instead of rehashed Newtonian crap. Unite math and physics for a while, and kids might actually see the point of it.

Exactly. Physics becomes so much simpler with calculus. Once you realize e.g. the relationship between position, velocity and acceleration you don't need to memorize 20 different equations for all the different flavours of problems that are basically down to integrating or differentiating ...

But "you need to learn the hard way everybody else did it before X formula/equation was developed!"

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 couldn't disagree more. It gets too hard when you're asking students to memorize arbitrary equations that make no sense to them. It gets too hard when you give them 30 problems that look identical and they do 3 or 4 and figure that's good enough because they know how to run numbers through the meat grinder of a formula.

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.

I held off taking physics in college until my senior year because I dreaded the 3 hour labs. By the time I got there I had 3 levels of calculus (and linear algebra, and plenty of proof based CS classes) under my belt, and it turned out to be one of the easiest classes I've taken.

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

Berkeley EECS grad. I didn't go the device physics route but I used calculus about twice in 4 years except in physics classes.

Circuit analysis turns differential equations into algebra, so you might have been using calculus more than you realize.

Yeah, I might have been guilty of some hyperbole. I remember a nasty integral on the Linear Systems midterm. It probably would have taken an hour or so if I didn't make any mistakes; but you could transform it and solve it as a linear equation in 10 seconds.

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.

That really makes no sense, unless the "EE" part was optional in your case.

There are different "tracks", and some of them have very little EE (with way more CS focus). I think I only took two EE courses IIRC..

The beauty of calculus is that you actually understand the Real World.

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."

... Ohhhhhh.

I wish you would teach me calculus =)

This... might have come out wrong. What I meant was something along the lines of: You should teach, have you considered mentoring?

The first sentence in Your last statement in your conversation to your daughter is profoundly wrong. Your daughter was not in any way made to better understand fractions by this interaction. By the way, .3333 is a rational number and therefore equal to a fraction.

is there a reason you call them factions and not fractions?

Your idea did not translate into English. I think you are trying to say that most decimals we see in real life are approximations, not exact measurements, while fractions are exact?

This has nothing to do with calculus, though.

https://www.dpmms.cam.ac.uk/~wtg10/decimals.html

You can construct a complete ordered field from infinite decimals

Berkeley's a really good university.

lel

Berkeley lets you skip learning asymptotic runtime analysis of algorithms if you're doing EECS? Weird.

FYI: infinite sums are calculus.