
The Calculus Trap (2005) - adenadel
https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap
======
nextos
The way calculus is usually taught is a mess. A mix of epsilon-delta
formalism, without adequate motivation, differentials and excessive focus on
computations.

For young students, a great introductory textbook is _Calculus Made Easy_. It
is around 100 years old, and develops all the material using infinitesimals.
Which is essentially modern non-standard analysis, minus rigor. It is also the
way Newton and Leibniz thought about calculus, and the way most physicists
intuitively think about problems.

For a more mature audience, I like _Infinitesimal Calculus_ by Henle &
Kleinberg.

~~~
Hasz
No way high school or college level classes are teaching intro calc with
delta-epsilon proofs, it just won't make any sense. It's generally introduced
in Real Analysis, which is generally a sophomore level class.

Develop the intuition, then crystallize and formalize the idea with a proof.
Otherwise, it's just not going to make any sense.

~~~
sevensor
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It
was pretty standard at my public high school. I'm sure I didn't understand it
very well at the time, but it was good to have seen it when I took real
analysis.

~~~
Izkata
Had to look it up to be sure, since I didn't recognize the name, but it looks
like this refers to using limits to "invent" differentiation?

If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same
teacher both years, can't remember when exactly Calc was)

I remember it involving a lot of drawings of graphs so we could understand
what each term referred to, and can't really imagine an easier way to learn
it.

~~~
nextos
Epsilon-delta is the formalization introduced by Cauchy, Bolzano and
Weirstrass during the 19th Century.

It is _not_ the way calculus was invented by Newton and Leibniz. They thought
in terms of infinitesimals. However, this approach is relatively hard to
formalize. It was only done by Robinson in the 1960s.

To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an
excellent book.

[1] [https://www.macalester.edu/aratra/](https://www.macalester.edu/aratra/)

------
gibba999
I don't buy it.

AoPS is a great organization, but the focus is on pure, theoretical
mathematics.

Understanding calculus is key to understanding many beautiful areas of applied
mathematics: image processing, signal processing, control systems,
electronics, etc. I consider them more elegant than theoretical mathematics.

Now, for that, you don't need all the messy manipulation (integration-by-parts
and similar), but you do need the basics of area-under-the-curve, of
derivative-as-slope, and similar, as well as some of the theory.

But that's not too hard to learn.

My own opinion is that the basics of calculus should be taught alongside the
basics of algebra in elementary school. Plenty of people have had success
doing both.

------
dang
A couple of good past discussions:

2014:
[https://news.ycombinator.com/item?id=7207495](https://news.ycombinator.com/item?id=7207495)

2009:
[https://news.ycombinator.com/item?id=717982](https://news.ycombinator.com/item?id=717982)

~~~
svachalek
The 2009 discussion at least partly understood that the article is not about
calculus, while it seems the 2014 and 2019 discussions missed the point. I
don't know if that really says something about the culture of the times but
it's interesting.

~~~
dang
Discussions are sensitive to random initial conditions like who happened to be
online when an article was posted, so I doubt it says much about 2009.

------
hyperpallium
What does HN think of Khan Academy's treatmemt of calculus?
[https://www.khanacademy.org/math/calculus-all-
old](https://www.khanacademy.org/math/calculus-all-old)

Ideally, by an expert in calculus, who has done the whole KA course on it
(though why an expert would do that, I don't know...)

~~~
HiroshiSan
As someone who tried teaching themselves math with Khan Academy, I feel like
Khan Academy suffers from the same problem most sites do, in that they tell
you the information before you can discover it for yourself. This is different
than how AoPS teaches math, where they ask you questions which then guide you
towards self discovery, and as a result, you start asking your own questions
which create paths for you to go on and explore.

~~~
barry-cotter
You, and AoPS’ target audience of current and potential math nerds are very
far from the norm. Inquiry based learning is actively harmful to learning for
many students, and much less efficient for more or less everyone.

> Why minimal guidance during instruction does not work: An analysis of the
> failure of constructivist, discovery, problem-based, experiential, and
> inquiry-based teaching

> Evidence for the superiority of guided instruction is explained in the
> context of our knowledge of human cognitive architecture, expert–novice
> differences, and cognitive load. Although unguided or minimally guided
> instructional approaches are very popular and intuitively appealing, the
> point is made that these approaches ignore both the structures that
> constitute human cognitive architecture and evidence from empirical studies
> over the past half-century that consistently indicate that minimally guided
> instruction is less effective and less efficient than instructional
> approaches that place a strong emphasis on guidance of the student learning
> process. The advantage of guidance begins to recede only when learners have
> sufficiently high prior knowledge to provide "internal" guidance. Recent
> developments in instructional research and instructional design models that
> support guidance during instruction are briefly described.

[https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_...](https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1?needAccess=true&)

~~~
jacobolus
Your link is to a very controversial polemic which IMO sets up a straw man,
and then makes its own argument in a (both theoretically and empirically)
questionable way, based on the authors’ pet “cognitive load” theory (which I
personally think is bunk, but YMMV). It should be read in the context of its
critics, and taken with a heap of salt.

To understand the problem with US-style mathematics pedagogy, I would
recommend reading [http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-
NEW.pd...](http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf)

For some advice and materials based on an alternative theoretical framework,
let me recommend
[https://www.map.mathshell.org/trumath.php](https://www.map.mathshell.org/trumath.php)

* * *

I think Khan Academy should be thought of as a consistently average-quality
US-high-school-style lecture, combined with US-style trivial exercises. It is
a slow, unimaginative, pedantic curriculum.

But it has the advantages of being free, always available, and self-paced (in
the sense that students can keep going through as much of it as they want
without needing to wait, and can return to previous sections any time). I’m
glad it exists, because it sets a quality floor; live teachers have variable
quality, and while many are better than KA lectures, some are certainly worse.

~~~
barry-cotter
If you can refer me to three relevant articles that critique or falsify
Kirschner et al. I’ll be happy to read them. I’ve read _Problem-Based Learning
is Compatible with Human Cognitive Architecture: Commentary on Kirschner,
Sweller, and Clark (2006)_ already.

------
RandomInteger4
> "For an avid student with great skill in mathematics, rushing through the
> standard curriculum is not the best answer. That student who breezed
> unchallenged through algebra, geometry, and trigonometry, will breeze
> through calculus, too."

That was me. I was great at calculus type things, but Matrix Theory hit me
like a ton of bricks. I still have that text book, sitting on my other desk,
staring menacingly at me from across the room; Matrix Analysis, Horn and
Johnson. Geometry in High School gave me a taste, but would have been nice had
we had available another proof based class in the math curriculum; Formal
Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?

~~~
msla
> That was me. I was great at calculus type things, but Matrix Theory hit me
> like a ton of bricks. I still have that text book, sitting on my other desk,
> staring menacingly at me from across the room; Matrix Analysis, Horn and
> Johnson. Geometry in High School gave me a taste, but would have been nice
> had we had available another proof based class in the math curriculum;
> Formal Logic or Discrete Maths at a high school level. Maybe even Linear
> Algebra?

I am deeply confused by a curriculum which separates Matrix Theory from Linear
Algebra. The description in the Wikipedia category just barely helps:

[https://en.wikipedia.org/wiki/Category:Matrix_theory](https://en.wikipedia.org/wiki/Category:Matrix_theory)

> Matrix theory is a branch of mathematics which is focused on study of
> matrices. Initially, it was a sub-branch of linear algebra, but soon it grew
> to cover subjects related to graph theory, algebra, combinatorics and
> statistics as well.

The University of Missouri has a Matrix Theory course:

[https://www.math.missouri.edu/class/matrix-
theory](https://www.math.missouri.edu/class/matrix-theory)

> Basic properties of matrices, determinants, vector spaces, linear
> transformations, eigenvalues, eigenvectors, and Jordan normal forms.
> Introduction to writing proofs.

... which specifies a textbook:

> _Linear Algebra with Applications (7th edition)_ by Steven J. Leon

... which deepens my confusion. If you're taking that course, how is it _not_
an introductory Linear Algebra course?

And this MathOverflow answer obfuscates again:

[https://mathoverflow.net/questions/11669/what-is-the-
differe...](https://mathoverflow.net/questions/11669/what-is-the-difference-
between-matrix-theory-and-linear-algebra)

> Let me elaborate a little on what Steve Huntsman is talking about. A matrix
> is just a list of numbers, and you're allowed to add and multiply matrices
> by combining those numbers in a certain way. When you talk about matrices,
> you're allowed to talk about things like the entry in the 3rd row and 4th
> column, and so forth. In this setting, matrices are useful for representing
> things like transition probabilities in a Markov chain, where each entry
> indicates the probability of transitioning from one state to another. You
> can do lots of interesting numerical things with matrices, and these
> interesting numerical things are very important because matrices show up a
> lot in engineering and the sciences.

> In linear algebra, however, you instead talk about linear transformations,
> which are not (I cannot emphasize this enough) a list of numbers, although
> sometimes it is convenient to use a particular matrix to write down a linear
> transformation. The difference between a linear transformation and a matrix
> is not easy to grasp the first time you see it, and most people would be
> fine with conflating the two points of view. However, when you're given a
> linear transformation, you're not allowed to ask for things like the entry
> in its 3rd row and 4th column because questions like these depend on a
> choice of basis. Instead, you're only allowed to ask for things that don't
> depend on the basis, such as the rank, the trace, the determinant, or the
> set of eigenvalues. This point of view may seem unnecessarily restrictive,
> but it is fundamental to a deeper understanding of pure mathematics.

If I try to parse charitably, I come away with the idea that Matrix Theory is
about matrices as a data structure, usable for many things outside the scope
of Linear Algebra, where they're all about using matrices to represent linear
transformations. It's the difference between a column of numbers on a shopping
bill and a column of numbers which represents a vector in a space with a
specified basis. Gotcha.

However, this answer directly contradicts what the University of Missouri
calls Matrix Theory, which is so Linear Algebra they even use a Linear Algebra
textbook. It also... I don't know, trivializes the field of Matrix Theory. So
you can manipulate matrices. So what? They show up a lot because they're used
to represent specific things. Is the course going to barely introduce a lot of
specific things and then focus on the matrix representation? What a waste!

~~~
RandomInteger4
I didn't say that Linear Algebra and Matrix Theory were separated. I said that
Matrix theory hit me like a ton of bricks. I took Linear Algebra in college
prior to that, obviously.

I further stated that I think Linear algebra might benefit students if taught
earlier, in high school.

~~~
Gibbon1
We were taught a little linear algebra in high school. And more in college.

My impression of three semesters of calculus in college was that much was a
waste of time. It was probably useful for a mechanical/electrical engineer
circa 1950. But today no one solves problems that way.

I think more linear algebra and matrix theory would have been better.

------
morpheuskafka
> That student who breezed unchallenged through algebra, geometry, and
> trigonometry, will breeze through calculus, too.

I took calculus last year (AB Calc BC, 10th grade), and I can say that my
experience was certainly a counterexample. I did OK, but it was definitely a
marked difference from "breezing through" algebra.

~~~
tomrod
I expect it was more of a teacher issue then. A solid math teacher is worth
their weight in platinum.

~~~
dTal
I had an excellent Calculus teacher in high school, and I did very well on the
AP exam. It was still vastly more challenging than the trigonometry class that
preceded it. Even today, I use trigonometry all the time almost without
thinking, but I can't for the life of me remember how to long-divide
polynomials (or picture a scenario where I'd need to).

There's just so _friggin_ much deeply abstract symbol manipulation in calculus
class (which in school also covers essentially "advanced algebra"). It's a
different ball game.

~~~
jacobolus
Long division of polynomials is literally exactly the same algorithm as long
division of base-10 integers, except with no carrying.

------
sevensor
I think this is a side-effect of the way our educational system is structured.
We rely on big, standard tests to evaluate schools and their students. It
would be too expensive, and somewhat subjective, to evaluate students'
reasoning abilities, their conceptual knowledge. Can you imagine if these
tests were evaluated by rooms full of people reading proofs, rather than by
bubble-sheet scanners? So we test what we can test, which is procedural
knowledge, and we teach that at the expense of deeper understanding. And
because the best way of measuring procedural knowledge is to measure how many
procedures you know, we race students forward to as much calculus as they can
memorize.

------
hackermailman
This article is primarily a marketing piece for their $500 courses, which I'm
sure are good but if you don't have money or accessible local math student
clubs and want to like the article says 'explore math' try these math
foundations playlists for free
[https://www.youtube.com/user/njwildberger/playlists](https://www.youtube.com/user/njwildberger/playlists)
Wildberger starts from the very beginning, proving laws of each ring/field
with basic arithmetic. He also has an algebraic trig method that a primary
school kid could do and an interesting discrete algebraic calculus method,
plus plenty of abstract algebra content.

~~~
HiroshiSan
Not sure how you got this impression, the concluding paragraph is "However, we
are not the only other option. Other options students have are to become
involved in extracurricular programs, such as math teams. Math contests should
be selected with some care: those that encourage mass memorization or just
test standard curricular tools tend to exacerbate the ills of the calculus
trap rather than enhance problem-solving ability. Students can also pursue
independent study if they are able to find mentors. University professors are
occasionally willing to fill this role to some degree. There are also many
summer programs and good books for extracurricular study, and some communities
have developed grassroots programs to provide opportunities for eager
students. These options are usually not as easy as “enroll in the next
course,” but they will be far more rewarding than settling into the calculus
trap."

Also these articles are normally targeted to those already invested in the
AoPS ecosystem, whether it be books, courses, or their forums.

------
WillPostForFood
_If ever you are by far the best, or the most interested, student in a
classroom, then you should find another classroom._

It is unfortunate that the prevailing educational trends are to get rid of
tracks, lanes, and advanced classes, and dumping all the kids into mixed
ability classes.

~~~
threatofrain
I believe it's been roughly found that tracking helps smart kids moderately
and hurts everyone else with the brain drain. It's a question of to what
degree we are individuals, and to what degree we are a community.

I think it's a question that cuts right down the middle of many people's value
systems.

~~~
WillPostForFood
The data isn't very strong to support the idea that leaving advanced kids in
the room with those that need more attention is of any help, and there is now
growing evidence of achievement gaps widening, possible because it can be
demoralizing for the slower kids, possible because the more advanced kids are
now more likely to get outside supplementation which boosts them even further
ahead.

