
Do the math: too much calculus? (2012) - carlosgg
https://www.washingtonpost.com/blogs/class-struggle/post/do-the-math-too-much-calculus/2012/08/16/9a35034e-e4fd-11e1-8f62-58260e3940a0_blog.html
======
krschultz
I graduated with a mechanical engineering degree, meaning I took 3 semesters
worth of Calc and 3 more semesters worth of differential equations. So I'm not
one to hate on math or calculus specifically. But I think high schools should
focus more on statistics.

Statistics is relevant to more fields than calc. It matters a lot more in
business and in all the jobs where you use calc you will also have to use
statistics. It's also useful for citizens to understand statistics better -
and educating citizens is truly one of the purposes of public education.

~~~
pps43
The problem is, it's hard to understand statistics without knowing at least
basic calculus. As soon as you move from coin flips to probability density,
you need derivatives and integrals.

These days you can solve many practical statistical problems with Monte Carlo,
but understanding why those curves look the way they do is still very useful.

~~~
analog31
I had kind of a strange experience. While in college, I worked as a math
tutor. Since I had taken one semester of stats, the tutoring center decided I
was qualified to run a tutoring session for the non-calculus stats course. I
was happy for the money, and had been reasonably successful with other
tutoring assignments.

When we got to continuous distributions, we kind of hit a wall. Here's the
puzzle. Suppose the data are recorded to two decimal places, and are
reasonably bounded, for instance they are between 0 and 100. That's a discrete
distribution. All of the data in the examples and problem sets were fixed-
point.

I literally couldn't justify to someone who had never taken calculus, why a
wholly different kind of distributions with its own collection of formulas,
was necessary or useful. The teacher had never explained it either.

~~~
rocqua
The chance of choosing 0.5 from the interval 0 to 1 is 0. So if you want the
change of choosing between 0.4 and 0.6 in that interval, you can't just sum
all chances between 0.4 and 0.6 you'd have infinity x 0 which isn't 0.2 .

Instead, to deal with this 'a lot of really small things' we need a slightly
different approach. Then draw a pdf, and talk about area under the curve.
Perhaps relate this to the discrete case where the curve is just really
blocky.

------
baldfat
I think we teach Calculus wrong in a deep fundamental way. We should
incorporate more high math concepts earlier into education BUT people are
always afraid of anything past 4th grade math!

"A minority of students then wend their way through geometry, trigonometry
and, finally, calculus, which is considered the pinnacle of high-school-level
math.

But this progression actually “has nothing to do with how people think, how
children grow and learn, or how mathematics is built,” says pioneering math
educator and curriculum designer Maria Droujkova."

...

"“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."

[http://www.theatlantic.com/education/archive/2014/03/5-year-...](http://www.theatlantic.com/education/archive/2014/03/5-year-
olds-can-learn-calculus/284124/)

Why is calculus the pinnacle when I actually feel that pre-calculus would be
better students even make it into middle school? These concepts of _patterns
and structures_ is the foundation of not only math but also of language and
reading.

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

~~~
baldfat
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."

~~~
fao_
... _Ohhhhhh._

I wish you would teach me calculus =)

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

------
japhyr
I have a 5 year old, and I was talking to some acquaintances who also have a
kindergartener. The kids are learning math, by playing with manipulatives.
It's great; they're learning mathematical concepts, and they have no idea
anyone in the world doesn't like "math". They don't know what "math" is, they
are just starting to discover that the world is made up of interesting but
identifiable shapes, and that numbers have some really interesting patterns as
well.

These parents, however, don't see anything they recognize as "math". They're
afraid their kid is going to fall behind, so they're giving their kid
worksheets at home focusing on two-digit addition and subtraction. Their kid
doesn't like math already, because at home math is just writing things on a
piece of paper with little meaning. This is sad to me, because there are so
many interesting things you can do with your kids at home to help them develop
their mathematical understanding.

Our educational issues with math start young, and they come from adults. I'm
fortunate to work in a small high school where I get to treat each kid
individually. It's wonderful to meet kids where they're at, and help them move
forward and start to enjoy math again.

~~~
ivan_ah
I don't have kids, but if I had a 5 year old, I'd sure get them some of these:
[https://en.wikipedia.org/wiki/Cuisenaire_rods](https://en.wikipedia.org/wiki/Cuisenaire_rods)

I've often thought about inventing math toys, e.g., group theory for toddler:
S_3 = triangle toy, S_4 square toy, etc. I'm sure there are many advanced math
topics that could be turned into toys. No symbols, no equations, just toys,
but subtly you're getting kids familiar with important math structures.

~~~
kol
Rubik's cube can be used to teach group theory, see
[http://geometer.org/rubik/group.pdf](http://geometer.org/rubik/group.pdf) or
[http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20an...](http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf)

------
castratikron
AP Calculus was the most useful course I took in high school. It was rigorous,
and I scored high enough on the exam to get two free college classes and start
freshman year at the sophomore level, which left room at the end of my college
career for more specialized courses.

There should be some vetting process for the students who want to take AP
classes. A lot of the students in my AP Calculus class had no business being
there. One guy got a 4 out of 50 on a test once, another guy didn't even take
the AP exam, and another guy just put his head down on the desk during the AP
exam. But this is really a problem for individual schools and has nothing to
do with College Board.

~~~
mhb
Isn't it kind of weird that a college won't let you take whatever math class
you want?

~~~
gnarbarian
No. Tuition only covers a portion of a student's cost to a university. I think
it's around 30% at mine.

1) If a class is way over your head the university is wasting money having you
there.

2) Having stragglers in the class that are far behind slows down the whole
class.

3) Every seat that is filled with a student who is not ready for the class is
a seat that can't be filled with a student who could actually pass it.

~~~
jessriedel
> Tuition only covers a portion of a student's cost to a university. I think
> it's around 30% at mine.

Where are you getting that from? Are you just dividing the tuition revenue by
the university's budget? Because the university is there for a lot more than
educating undergrads.

~~~
gnarbarian
Most universities aren't primarily focused on research.[1] They are there to
educate students. Research does occur at every university, but looking at
their research budget as a fraction of their whole budget will show what sort
of institution it is.

Even if you subtract the research budget out first you will still arrive at a
figure less than 50% for most universities.

[1]
[https://en.wikipedia.org/wiki/Carnegie_Classification_of_Ins...](https://en.wikipedia.org/wiki/Carnegie_Classification_of_Institutions_of_Higher_Education)

edit:

[https://nces.ed.gov/fastfacts/display.asp?id=75](https://nces.ed.gov/fastfacts/display.asp?id=75)

~~~
jessriedel
> Most universities aren't primarily focused on research.

Where are you getting "many" (rather than "some")? The page you link to
discusses _colleges_ in addition to universities, and it's well known that
colleges exist primarily to educate students.

> Even if you subtract the research budget out first you will still arrive at
> a figure less than 50% for most universities.

Where is that numbers coming form?

~~~
drakonandor
Universities are composed of colleges in USA. You seem to have more of a
'Canadian' definition, where college means something like "community college"
in USA.

~~~
jessriedel
I am not using such a definition. I am an American, and I am thinking of
American places like Kenyon College, Hamilton College, Harvey Mudd College,
Union College, Claremont College, etc., etc.

> Universities are composed of colleges in USA.

This is pretty rare terminology in the US, being mostly found in Ivy league or
other old schools. Most "colleges" in the US are independent institutions.

[https://studyusa.com/en/a/107/what-is-the-difference-
between...](https://studyusa.com/en/a/107/what-is-the-difference-between-a-
school-college-and-university-in-the-usa)

------
ivan_ah
I think linear algebra is a much more valuable and general-purpose math tool.
I'm not sure how feasible it would be to teach linear algebra topics
"properly" to high schoolers, but even a half-assed linear algebra would be
more useful than calculus.

The connections to geometry, general systems thinking, formal math methods,
and countless applications of linear algebra are just the kind of thing
students need to get them interested in learning more math.

Anyone interested in LA an its applications should check out my upcoming book,
the _No bullshit guide to linear algebra_ availale on pre-order here
[https://gum.co/noBSLA](https://gum.co/noBSLA) (it's almost finished; just
beefing up the problem sections).

~~~
lqdc13
They taught me about matrices and vectors and common matrix/vector operatoins
in pre-calculus and even earlier in a shitty public school.

Calculus needs to happen in high school so that people are more prepared to
take it again in college. For me, calculus 2 was the hardest class in college.
This is coming from someone who went to a math grad school.

Some people need to study way more for calculus than for any other class in
high school or college.

~~~
sidusknight
I'm guessing you did applied math in grad school? I think very few (pure) math
majors had any issue with calc.

------
Yhippa
> Too many students experience a secondary-school calculus course that drills
> on the techniques and procedure that will enable them to successfully answer
> standard problems but are never challenged to encounter and understand the
> conceptual foundations of calculus,” he said.

As I get older I've realized that I learn better when I understand the
concepts about things. I then get excited to apply what I've learned into
practice.

I think that if I learned about "why calculus", what problems did it solved at
the time it came around, and how we use it in practice I would have been able
to grok it quicker and deeper. I didn't end up getting above a 3 on the AP
calculus exam.

I did however get a 5 on the US government exam. I had a teacher who did a
great job throughout the course having us work out essays and testing
arguments in class to help solidify our knowledge as opposed to memorizing
facts.

~~~
hood_syntax
Conceptual background is so important for understanding something. I was lucky
to have a great teacher for BC calc, and I got an easy 5 on the exam. The
concepts make sense, they are consistent, and they build on each other, so
with the right progression of ideas students can get a lot farther than they
would otherwise.

------
hanoz
I remember once when when driving I was pleased when it occurred to me that I
should be seeking to minimise the third derivative of distance with respect to
time for the comfort of my passengers. That was the sum total of calculus's
contribution to my adult life. Not sure it was worth it.

~~~
ThrustVectoring
Curves in freeways are designed that way as well - they don't just start
turning suddenly, but gradually transition into and out of the curvature.

~~~
0xCMP
Good ones at least.

~~~
ThrustVectoring
Don't quote me on this since I don't remember why I believe it, but I think
it's part of the legally mandated design requirements for the US interstate
system.

~~~
0xCMP
No I'm sure there are requirements, but sometimes there are also space or
budget constraints (I assume) since some curves are definitely safer and
easier than others for the same highway (I-95 for me).

I wished they were more consistent

------
jimmydddd
For lower level math, my kids seemed to revisit the same subjects over and
over from k-9th grades. For example, mean, median and mode, year after year.
The first few years it was just robotic problem solving. After a few years,
they started to get a broader understanding. Maybe some people need to take
several passes through the higher level stuff too.I guess the problem is that
there is not enough time.

~~~
douche
What infuriates me, looking back on it, was simply how much of my _time_ the
educational system was allowed to waste. Thousands and thousands of lost
hours, some of them filled with pointless busy-work, but many more just
squandered on inanity. At least some of the teachers would let you finish the
daily penance quickly, and sit quietly in the corner reading something
interesting. Too few, though.

~~~
pdm55
I look at this way: early education focuses on learning to read and write and
do arithmetic. The basic purpose is to give you tools whereby they can
continue to educate yourself throughout your life. So basically, our early
education aims to put our future education into our own hands.

Let me qualify the above a little. You might ask, well what are the purpose of
high-school teachers or university lecturers? Once we are given these tools
shouldn't we just be given the books to progress on our own? Hopefully,
teachers are people who can take a complex subject and guide us towards an
understanding of that subject. To me teachers are like guides through a jungle
of knowledge. We need a guide so that we don't get lost. For me, a teacher is
part guide and part "composer of problems". I consider teaching to be guided
problem-solving. When I am teaching a course, I pose problems, whether it be
in Math or Chemistry, and guide the students as they try to solve those
problems.

And as for your comment about "wasted time", I am not sure what to think about
this. The difficulty with learning is that we need repetition and
reinforcement. Yes, I see a lot of pointless repetition, but we do need some
repetition. I am presently struggling with improving my chess skills, as well
as trying to learn to program an Arduino. Mastering these skills require a lot
of repetition in order to cement that knowledge. Wish me luck.

------
euroclydon
I have a 5th and a 6th grader. I'm going to buy them an educational copy of
Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs,
solve equations, do derivatives and integrals, and factor algebraic
expressions.

I want them to do all this in high school, whether or not they take Calculus,
so they can get an intuition for it. Then they can memorize the trig rules in
college.

~~~
Oletros
AS an Spanish not knowing the US education system, at what age one kid goes to
5th grade?

~~~
euroclydon
My 5th grader is currently 10 years old.

------
abakker
Sample size of one: I think I understand calculus, the needs of calculus, and
how it is profoundly meaning to understanding things around me. Can I _do_
calculus? no. I couldn't do calculus because I really couldn't do algebra. I
think that if I had been taught the concept and relevance of calculus earlier,
and only had to worry about the execution of calculus later, I would have been
much better off.

understanding what derivatives and integrals and trig functions _mean_ should
come before we are forced to execute their use, or learn proofs, or even
really go beyond basic notation.

~~~
yequalsx
I'm skeptical that a person can really understand why x^(2/3) is not
differentiable at x=0 but has an absolute minimum at x=0 without understanding
the algebra. I think one can get an idea of the concepts without knowing
algebra but I doubt one can understand calculus without knowing algebra.

We always teach the meaning of a given mathematical concept before doing using
them or doing calculations. It's just that true understanding comes from the
doing the calculation part.

~~~
rconti
Not the parent, but I had the same problem. I understood that kind of algebra
but could never figure out how to simplify things well, which hamstrung me in
Calculus. Also, they never bothered to explain the purpose of any of the
equations in calculus.

------
AstralStorm
Personally, I have always wondered about the prominent position trigonometry
and geometry get.

As it is typically taught, it is the most useless thing, the time spent there
would be much better spent on algebra, calculus. From basic theorem of algebra
flows trigonometry and a lot of number theory, while from calculus with some
linear algebra flows geometry.

The rationale is probably history.

~~~
elihu
For a lot of people, geometry is their primary or perhaps only exposure to
proofs. Not that everyone needs to understand proofs in their day-to-day
lives, but it's a good mental discipline to at least be exposed to, and
geometry proofs are fairly straightforward and relate to things that many
people will have a good intuition to convince them that the thing proved by
each correctly-constructed proof is really true.

Also, geometry gives students something concrete to visualize, and so might be
easier for some to understand than more abstract kinds of math.

Maybe trig is over-emphasized at the high school level, but it is pretty
useful in a lot of real-world situations, and it's easy to apply without
having to know a lot of other stuff.

~~~
analog31
From what I can tell, proofs are gone. I got my daughter all excited about
geometry, by promising her that she could do proofs. The curriculum was almost
exclusively focused on problem solving based on remembering formulas. She went
ahead and did all of the proofs anyway.

~~~
elihu
:(

I hope that's not universally true, and that proofs are still actively taught
in some high schools.

------
hprotagonist
>This is not true for students who take Advanced Placement calculus

I __hated __BC calc. Bad teacher. I learned the same material at the same time
in AP Physics, and loved it.

I think a related issue to the author's point, though, is that students are
taught in a way that implicitly suggests that calculus is the pinnacle of
mathematics. Which is blatantly wrong.

~~~
baldfat
I have had the pleasure of pushing many female students into taking calculus
but my success rate was about 50%. Each one of my children's friends LOVED
STEM and were thinking about going into engineering. All of them said that
they were the only girl in the class and were intimidated and the other said
it was REALLY boring. Calculus should be easily made fun just exploring the
actual concepts in real life similar to a good physical science teacher.

~~~
hprotagonist
As it happens, my teacher was female.

She was a firm believer in "teach to the test", "do these exercises and follow
the magic rules of differentiation, shut up and just do it don't ask why it
works".

That's not my jam.

~~~
baldfat
I think many "Math" people get the mechanics and like the functions and miss
out on the application.

I remember trying to figure out how the Cha Cha Slide could be a decent
Calculus problem. How does the song change the pattern?

------
spike021
I think this is just another example of standardized testing (specifically AP
course exams in this story) almost creating a niche-like education for
students.

Maybe that's an odd way to put it, but some students wind up spending hours
and hours studying AP/SAT/ACT/etc.-specific materials (literally 500+ page
textbooks for each type of standardized test). Sure this method of studying
helps them achieve great scores on the standardized tests, but my impression
as a recent student in academia (finishing college now) is that this creates
an issue where students are almost "overly" dependent on more specialized
areas.

I don't know if it's an issue of not having time to learn "street smarts" (so
to speak) because standardized studying takes up all their time, or if it's
something else. But there's definitely some improvement that can be made.

~~~
smallnamespace
A pattern that you see almost everywhere is: imperfect incentives/objectives +
strong competition = perverse outcomes

Over the last decades, we're increasingly seeing this in academia (publish or
perish), the corporate world (boost the bottom-line at the expense of
everything else), politics (win the election even if you have to burn your own
party to the ground), and many other walks of life, as the world has gotten
more competitive.

To fix this, you can either fix the objective (which may be very difficult),
or you can reduce the level of competition.

~~~
Retric
This also causes some odd results. If the MIT/Caltech/Harvard/etc student
bodies are generally less intelligent than you might expect and often boring
people.

You see the same thing at most national level high school competitions. The
surface details may seem very diverse, but underneath that they tend to have a
lot in common.

------
CN7R
Schools should adopt more dynamic curriculum, instead of a one size fits all,
for individual students. Not everyone will grow up to be an engineer, but that
doesn't mean those who have an interest in advanced mathematics should be held
back. Encourage students to pursue their academic passions, and don't limit
certain subjects depending on grade level.

A personal anecdote: I didn't start liking math until Calc AB, as subjects
beforehand were presented as dry and were relatively easily so I didn't
individually pursue mathematics outside of school. But by that time, I was
already a junior and as a result, I often wonder what better math teachers and
a non static curriculum would've done for my education.

------
angry_octet
So let me get this straight: kids who didn't study enough math in middle
school have difficulty doing harder math later on, so they rote learn it to
get into college, then drop out of math. Okay?

And the kids that studied math, e.g. because they enjoy it and/or have good
teachers early on, take AP math and go on to do well in math in college.

Is this a Captain Obvious moment? When will people learn that there is no
shortcut and they actually have to do the work, and pay a living wage to
teachers?

It reminds me of the 'coding in schools' thing. Who exactly is going to take a
massive pay cut to teach computing? And do we just give them a walled garden
to code in?

------
gnarbarian
It sounds like the issue is there are too many high schools offering classes
in AP calculus which don't do a decent job of actually teaching the concepts
to the students. Having a good teacher for these classes makes all the
difference. Additional testing at the college level would help assess a
student's ability level but it does not solve the problem of time wasted when
the course must be retaken.

Another issue this brings up is it causes a larger gap to form between
trigonometry and calculus 2. Where memorization of hundreds of trig identities
builds on top of the identities learned in trig. If a student takes Trig ->
pre-calc->AP calc -> Calc 1 -> Calc 2 They will have a much harder time in
Calc 2. That is what happened to me. I had to take Calc 2 a couple times
before I got it sorted out.

[http://www.ansep.net/programmatic-outcomes/statistical-
data](http://www.ansep.net/programmatic-outcomes/statistical-data)

I think the answer is to get more qualified teachers in the schools teaching
the AP level classes. There is a program in Alaska called ANSEP (Alaska Native
Science and Engineering Program) who have this figured out. Middle school
students who enter the program come to the university during the summer time
and take math classes from university professors. Many of the students are
completely finished with their math track (for engineering programs) before
they start university their freshmen year. These kids are all the way through
partial differential equations at 17.

The graduation rate for ANSEP students who enter engineering programs is
around double the national average. There are many other important aspects to
it. Including mandatory study sessions and living with other students who are
also taking the same classes. Students are also given well paid internships in
the industry during the summer time. When looked at as a whole, there is no
doubt that they have figured out a far more effective means for getting
students through STEM programs and jobs once they graduate.

Full Disclosure: I am good friends with a few of the people running this
program.

------
noobiemcfoob
I feared calculus, right up until circumstances forced me to not only take
calc in high school but to take the hardest level of calc my school offered
(AP BC Calculus, counted as 2 semesters of college calculus). I was greatly
benefited by an amazing teacher.

I've always felt that calculus is one of the most important branches of
mathematics for the simple insight it should hammer home time and again: An
infinite amount of infinitely small things can and _will_ add to infinity. All
too often, humans fail to see the cumulative effect of an integral.

Every branch of math has these types of insights built into them. Calc the
power of instantaneous change. Geometry shows the beauty of a proof - and if
you scratch a little bit, the shattering epiphany that triangles are literally
the same everywhere, always and forever. The reality is that many of these
concepts are simply absorbed into our culture, and we fail to realize it until
someone explains that it took centuries of work to formulate the concept of 0.

When it comes to education, I believe the struggle has far more to do with
testing standards than the material. It's easy to teach to process, much
harder to teach insight and further harder to test. In reality, none of this
matters until your institution decides and commits to truly educating its
students.

tldr - None of this matters until your institution decides to truly educate
its students.

------
hal9000xp
I do recommend to read these articles:

[http://www.artofproblemsolving.com/articles/calculus-
trap](http://www.artofproblemsolving.com/articles/calculus-trap)

[http://www.artofproblemsolving.com/articles/discrete-
math](http://www.artofproblemsolving.com/articles/discrete-math)

[http://www.artofproblemsolving.com/articles/what-is-
problem-...](http://www.artofproblemsolving.com/articles/what-is-problem-
solving)

Short recap: Mathematics is not bunch of dull rules. Education system puts way
too much emphasis on memorization of dull rules instead of problem solving and
developing strong intuition. Calculus overrepresented and discrete math
underrepresented in education system.

As a person who currently is rediscovering math from scratch, I find these
articles very insightful. I rediscover math from problem solving/intuitive
point of view rather than beating my head against the wall of formal
definitions.

------
Shorel
We can teach intuitive integral calculus in kindergarden:

Have lots and lots of ping pong balls handy, and try to measure the volume of
everyday things in ping pong balls.

Exact results are not a requirement and in fact they don't matter at all for
this early age, only the intuitive visualization.

We can teach the concepts of squares, cubes and prime numbers using the ping
pong balls as well: Get for example nine balls and form a square shape.
Squares are numbers that can form that shape. Similar for cubes. Factorization
in two factors is arranging the balls in a rectangle. A prime number is any
number that can't be arranged in a rectangle.

All this is intuitive, children can understand it easily, and it will help
them in the future.

And all this is before they learn about equations and algebra, it's even
before fractional numbers.

------
tnecniv
I have a specific problems with the HS math curriculum I experienced (as
opposed to the general problems with the whole math curriculum I had before my
junior year of college). Mainly, there was just so much wasted time it was
unbelievable.

I actually cannot remember a single thing that I learned in Algebra II because
I either relearned it in Pre-Calc or never used it again. Similarly, we spent
the whole first semester of Calc BC redoing all of Calc AB. What a waste!

If I could, I would replace all that wasted time with Linear Algebra (with
proofs) and Probability (throw in statistics if you want). Calculus should
stay, but teachers should motivate it better (this goes back to the general
problems I alluded to). Those two classes are immensely useful and teach a
whole different modes of thought compared to the regular pre-college
curriculum.

------
thearn4
As someone currently teaching college calculus on the side this semester &
works in aerospace engineering, I agree. The two math subjects that I notice
most people are deficient in post-college are statistics, and linear algebra.
In my opinion, we teach way, way too much calculus.

------
jiaweihli
I grew up 3 years ahead of my classmates in math. I found everything up until
calculus to be easy but elegant - algebra, geometry, probability,
trigonometry. Much like programming, there were often multiple ways of the
solving the same problem, and deriving a formula was feasible and logical.

In contrast, I found calculus to be mind-numbingly boring. It required
memorizing and reciting formulas upon formulas. It was the first time I felt
that math was tedious, and pushed me away from majoring in math in college.

------
dver23
Terrible teachers making difficult subjects impossible for students.

It's the same experience for my kids that I recall from my time. Industry can
pay more, and outside the rare person who has come to teaching to give back,
the instructors are those who can't.

Even when I was at Cal Poly, the only instructors in eng/sci who I recall
being helpful were those from JPI that would come in and teach a class or two.
One taught us three classes worth of material in course when he realized how
much we had missed from our previous instructors.

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LouisSayers
We're building out an online solution over the next year to help prepare final
year high school students for exams - with an initial focus primarily on
Maths.

Would love to hear from people with thoughts around Math tools they feel would
be valuable - especially taking into account current resources such as
KhanAcademy, and how we could potentially provide something complementary.

Either comment here, or hit me up - louis@connecteducation.com.au

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waynecochran
I think a Physics course taught with "Just in Time" Calculus would be fruitful
for motivating the Calculus. Calculus was invented to do Physics.

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anu7df
I have no idea why calculus gets such exalted status in the annals of
highschool nightmares. No I am not bragging, humble or otherwise. I found
calculus to be simple neat and logical, and not so mind bending. Linear
algebra on the other hand was far less trivial; not the mechanics, the aha
moment bit. Geometry.. that for me was genuine mind bending pleasure pain.

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madengr
"Instead, students who took high school calculus often find they have to
retake college calculus, or even pre-calculus. Many flee to the humanities."

Well no sh!t. I find it funny that people who aced calc in high school get
their ass handed to them in college.

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Koshkin
The best thing about learning calculus is that you see how everything you have
learned so far - algebra, geometry, trigonometry - everything comes together
and becomes equally useful and important. This is, in fact, what mathematics
is all about.

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WhitneyLand
I accidentally took Combinatorics as a fill in and it has been surprisingly
useful.

The need to count things well comes up on a regular basis, not to mention
helps with learning poker.

Would recommend this over some caclulus for a lot of folks.

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maxerickson
There should a high school class on numeracy, followed by statistics.

Wrote learning of algebra does not necessarily lead to numeracy.

Sweating over the students that are (mostly) ready for calculus and college is
missing the forest.

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ThatGeoGuy
I am not sure how much of this is specific to the American education system,
as I am Canadian, and have not experienced it, but my own take after going
through high school to my current graduate studies is that

1) We teach algebra, geometry, and their relationship wrong in many ways

2) We don't teach enough calculus at lower levels of mathematics, instead
opting to prefer a "poor man's" algebra.

I'm sure there's too much to write on for one HN comment, but a lot of 1)
comes down to how we teach "solve for x" type problems which are more
interested in re-ordering and simplifying equations than we are about giving
people a fundamental understanding of what our relations and equations
actually mean. Often I notice that many students couldn't graph or draw or
even begin to reason about what they're trying to solve for. If you give
someone a graph and ask them to find the maximum point of a line on a graph,
they can very easily point it out. The next step is doing it numerically, but
a lot of times the relationship between what we can determine visually and
what we can determine computationally is lost.

Which is sort of what prompts 2). You can't learn Calculus without learning
limits, and Calculus places explicit meanings to relations that we can graph,
and how those relations change according to specific variables or unknowns.
Finding the maximum of a relation using Calculus is much easier than strictly
using algebra for this reason, because we can make the relationships between a
curve and it's derivative explicit. Most students cringe at the thought of
Calculus because we place it on a pedestal and students just assume Calculus
is the peak of mathematics, but we really need to introduce Calculus as more
"normal" math earlier on, IMO. I'm talking basic tasks like derivatives
finding the equation of a line tangent to a curve, or finding the limit as we
approach a point on a curve, or even just (re)factoring equations so that we
can plot them easier and / or take their derivative easier.

In almost every example I can think of where I learned Calculus, it was easier
than building intuition for Algebra because there is less robotic crunching
and more reasoning about what specific problems mean. What does it mean to
take the derivative? Why are volume and area and perimeter related? I know in
many ways it seems like I'm just advocating for thinking spatially or visually
about mathematical problems, but that's part of what (I think) makes Calculus
more approachable. You can get a lot farther with a basic understanding of
Algebra and a very small amount of Calculus than you can with just Algebra
alone. Should our curriculum be entirely Calculus? No, that's ridiculous;
however we should ease up on the systems-of-equations type problems and obtuse
word problems that require students to produce or remember all manner of
expressions and formulae, and instead focus on building that first intuition
of how we can use Calculus as a tool for problems that are much, much harder
without. I expect that it would give Calculus a more realistic reputation, and
would probably put off less students who are already giving up on math.

~~~
jpfed
>Finding the maximum of a relation using Calculus is much easier than strictly
using algebra for this reason

I recently-ish helped my stepdaughter with an algebra assignment where she was
asked to find the maximum of some polynomial. I totally blanked on how to do
this with only the tools that she was supposed to have. I didn't know how to
do it or explain it without calculus.

~~~
pdm55
Use a tool like Symbolab. It shows you the steps for different methods. Very
useful.

Further, we should all remember to draw a graph as our first step. Symbolab,
Desmos and Geogebra (my favourite) are all fantastic graphing software.

Also, for a quadratic if you know the zeroes (roots), it's half-way between
them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the
mid-point (maximum) is at x=3. Your graphing will confirm this.

Your stepdaughter might also have learnt that the mid-point of a quadratic
ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative
equal to zero, but is generally just given to the students.) Her teacher may
be expecting her to use that formula.

Alternatively the students might be required to determine and plot various
data points using, say, Excel, and find the maximum this way.

The other point is that solutions often do not spring to mind immediately.
That is why I ask students to send me their problems before our tutoring
sessions. I often need time to think about them.

The fun of math is this exploration to find an answer. So try to get problems
and give yourselves time (days) to explore them together.

~~~
jpfed
It was a cubic, but the point is that calculus so obviates the need for any
degree-specific tricks that I don't have any particular memory for them.

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danielmorozoff
Maybe logic should be the first mathematical discipline to be taught in middle
school. The notions of and / or and the representation of language in
mathematical form?

Before even starting on algebra

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strathmeyer
Why did we take two years of Calculus in high school when there weren't any
jobs when we came out of college?

~~~
Retra
Those things have no relationship, and the expectation of one is a fault in
your understanding of the world.

