
Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf] - capocannoniere
https://web.williams.edu/Mathematics/lg5/Rota.pdf
======
woopwoop
I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a
public university with very good undergrad engineering students. Nonetheless,
with the course I taught, at least, the main problem was that WAY too much was
packed into a single semester. I suspect that this is the same everywhere. For
example, about a month and a half of the course was devoted to systems of
linear equations. These were students who, a priori, knew basically no linear
algebra, being asked to understand an entire linear algebra course, with some
extra stuff thrown in (you know, the differential equations), in six weeks. In
order to solve a general system of constant coefficient linear odes, you have
to take a matrix and compute its Jordan canonical form. The students I taught
were performing this calculation by the end of the unit, but it was of course
a joke. On their exam, they were asked to do this calculation (in differential
equations language), and most of them were able to do it because they were
good students and had memorized the procedure. Then, in another question, they
were presented a 5x5 matrix, told that its only eigenvalues were 2 and -2, and
asked if it was invertable. I don't think anyone gave a reasonable answer.

What is the point of that?

Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the
entirety of a book review on contemporary philosophers: "When pygmies cast
such long shadows, it must be very late in the day."

~~~
tomxor
>The students I taught were performing this calculation by the end of the
unit, but it was of course a joke. On their exam, they were asked to do this
calculation (in differential equations language), and most of them were able
to do it because they were good students and had memorized the procedure.

This is a prime example of what I believe is wrong with traditional education
(i'm not judging your teaching ability of course), as you are obviously aware,
this is more of a box ticking activity than learning in the sense of having
any actual understanding.

This has always personally caused me a great deal of trouble, for whatever
reason I can only learn substantially through understanding - attempting to
memorise procedures when I don't have time to understand the underlying
workings is like trying to eat cardboard for me, in that sense I am a bad
student and am exiled to purely self taught methods.

Although I am quite content learning on my own, it makes me feel like much of
my formal education was a big waste of time, If I could go back in time
knowing this I would avoid it altogether.

My experience is hopefully a more dated view by now, I also hold out hope that
the new exposure the internet has given courses through MOCs who provide no
kind of certification will by proxy refocus traditional education more on the
value of learning than certification.

Also in the UK at least there are some new ranking systems emerging for
universities that are more focused on learning like TEF, although indirect
that is at least a step in the right direction.

~~~
sordidasset
I'm in the same boat as you. 25 and starting my career, severely limited by my
poor performance in school.

Sundays are study days from now on. Hopefully I can catch up.

~~~
tomxor
I think it's widely accepted that if you are in tech, your success is far more
dependent on your actual ability, and schools haven't ever exactly been able
to help with tech careers so far.

It's only University level certifications that make any difference for your
prospective employers, but that does not necessarily guarantee the skills you
need as an engineer (if that's what you are aiming for), and many businesses
recognise this.

I got into full time programming job at around your age after a variety of
education routes in sciences and arts (no comp-sci) and variety of crappy
jobs.

No one seemed to be particularly interested in my lack of certifications when
it came to interview time (even though they all have them on the job app).
Once I could talk shop and show some small pieces of work and my potential
could be recognised I was hireable.

I'm not saying it's ever easy to get a foot in the door, but if you focus on
the work, your ability will show through in the interview process (if they are
not simply a large corporation hiring through box ticking criteria, probably a
bad place to start out when you are a junior anyway).

~~~
tomxor
[Edit] I'm presuming your path is programming in some form... of course there
are other areas of tech where you can't really get away from formal education
and certification, e.g no company is going to let you near their ASIC design
until you have some sort of qualification in electrical engineering + basic
comp sci. If you want to just to coding generally however then my previous
post applies.

------
stephengillie
Please teach calculus before teaching trigonometry. There's no prerequisite to
learn trig first, and forcing people to learn trig-calc excites many
mathophiles but is a major turn off to other students. Calculus can be taught
using just basic algebra, and most students will benefit from already
understanding calculus, when they are learning trigonometry.

~~~
uoaei
Interestingly, children as young as 5 show an aptitude for understanding
overarching concepts of calculus.[1]

This makes sense: it is much easier to talk about "rates of change" and
"accumulation" in simple terms and show how they are related using models that
appeal to children. We don't need to dive right in to the notation and
algebraic manipulations to get across the basic idea. That can come later when
children can handle the rigor. For now, let them play with it. That makes math
a lot more fun and less draining, esoteric, impenetrable.

[1]
[https://www.theatlantic.com/education/archive/2014/03/5-year...](https://www.theatlantic.com/education/archive/2014/03/5-year-
olds-can-learn-calculus/284124/)

~~~
dotancohen
My oldest, while in second grade, learned enough calculus to determine the
_location_ of a train given it's acceleration and since it started. That's
because she was interested, and asking questions, and luckily I had explained
to her the slope of a graph just a few days before the train ride. She didn't
learn the formula to memorize, but rather the concepts. Only then did we do
the calculations the long way, on paper, pen in _her_ hand.

~~~
g10r
why?

The question my three-year-old son asks over and over each day. It's
exhausting, and I love it. I do my best to provide the answer instead of
simply stating "because" or "just do it" as one of my greatest fears is to
suppress his natural desire to understand as much as possible about the world.
Also, as a child I hated memorization, yet loved delving into a subject that
intrigued me.

------
fmap
And 20 years later, this essay is still as relevant as the day it was written.
I agree with pretty much everything that's in the essay, except for a few
small points.

> There is nothing wrong with keeping the functional notation for density
> functions – as physicists and engineers always did – as long as one bears in
> mind that density functions cannot be evaluated, but only integrated.

This always bothered me, since, as noted in the very next section,
distributions don't have an analogue to pointwise multiplication. Even worse,
there is a perfectly servicable notation for such "dual functions/vectors"
that physicists have been using throughout the second half of the 20th
century. We could just use a consistent notation and not confuse new students,
but no. "It's always been done this way" is a terrible argument and leads us
to the confusing mess of notations that people still use for integrals and
integral transforms...

\---

Apart from that I would teach people recurrence equations/stream calculus
before going into the limiting case of differential equations. It's true that
differential equations are sometimes easier to handle analytically, but this
is neither relevant (as the article notes) nor a great point in their favor,
since we just end up teaching students a bag of tricks instead of explaining
why something works...

------
jessaustin
_Why is it that no one has undertaken the task of cleaning the Augean stables
of elementary differential equations? I will hazard an answer: for the same
reason why we see so little change anywhere today, whether in society, in
politics, or in science. Vested interests dominate every nook and cranny of
our society, even the society of mathematicians._

Truly we live in a decadent age. With this much fuel piled up, who will be
surprised by the conflagration?

~~~
mcnamaratw
Haha!

------
inbox
I wish I came across the so called proof-based math before calculus and
trigonometry. It would have grabbed me instantly. High school math (and
especially physics) classes would leave me with very uneasy feeling that
something crucial is left unmentioned, something important is swept under the
rug and something important is hidden for whatever reason. Turns out I was
wanting for proofs, but couldn't articulate it - I just sensed that something
was off :)

~~~
jessaustin
Your high school didn't teach geometry?

~~~
adamnemecek
Geometry proofs aren't real proofs.

~~~
TheRealPomax
They most certainly are real proofs. However, they are a different _kind_ of
proof if you mistakenly believe that only algebra can yield proofs. Provided
you have the algebraic rules for the type of geometry you are working with,
any geometric proof can be expressed as algebraic proof and vice versa, but
the trick is to appreciate that depending on what needs to be proven, one can
represent in a single step what the other takes many tedious pages of step
upon step upon step. And that goes both ways of course.

~~~
adamnemecek
Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type
theory) but that's neither here nor there.

~~~
TheRealPomax
You don't get the luxury of hand waiving: the two are equivalent, and thus any
rigorous proof in one has an equivalent proof in the other. In acknowledging
this, you accepted your original claim was false.

So what you really meant here was:

"Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy
type theory). I must have had a brain fart when I implied that one could be
more, or less real than the other. That made no sense."

------
madengr
Regarding item 10 in that list, I learned plenty of Laplace transforms,
partial fraction expansions, stability, phase planes, etc. It's the core of
control systems theory. It was just not taught in differential equations
class. The DE class was a bag of useless tricks. All the other EE classes were
very useful tricks of how not to directly solving DEs.

Going to drink a beer now for Oliver Heaviside. The invention of the Laplace
transform is one of the greatest contributions to engineering.

------
mathattack
Apparently he passed 2 years after this was written. RIP. A great mind.

[https://en.wikipedia.org/wiki/Gian-
Carlo_Rota](https://en.wikipedia.org/wiki/Gian-Carlo_Rota)

~~~
GeorgeTirebiter
I clearly remember his class on, basically, Mathematics for Philosophers. As
an engineering student, I thought my pal from Urban Studies who suggested
taking it was a way to pad one's GPA, which wasn't my style. Nonetheless, I
agreed, and, now, I regard it as one of the most important courses I've ever
taken. I have never met anyone with that level of enthusiasm, wonder, and love
of Mathematics - who also knew how to partially impart it to his students!
Mainly, it's the Wonder of it all that remains with me. What a privilege.

------
tempw
It's sad one needs to reinforce the teaching of concepts rather tricks.

Even worse is when you have all the proof based and concept oriented course
and are tested on trickeries on exams.

~~~
analog31
I believe this is a huge shortcoming of how math is taught. You can bet your
last dollar that the _teacher_ doesn't think it's about tricks. But the
students are convinced that it is. Students and teachers are both exposed to
the exact same material but end up with diametrically opposing conclusions.

Disclosure: I taught college freshman math for one semester, long ago. It was
a course where I was supplied with a syllabus and exams, and the students
could buy a packet of exams from previous years.

The tricks are what you remember from doing problems over and over, and
recognizing patterns. There is also a higher level pattern that isn't
mentioned in class, but is vital to solving problems: You learn to identify
each problem with a particular chapter or section in the textbook, and then
solve the problem by recalling the methods in that section. This is of course
a grotesque distortion of what math is, but will get you through the lower
level college math courses with good grades.

The other skill is being able to perform the manipulations quickly enough that
you can try one or two before hitting on one that works.

Disclosure: I taught college freshman math for one semester.

------
hprotagonist
you can fake knowledge of undergraduate differential equations if you know
about 3 hours worth of linear algebra.

More profound understanding is difficult but not terribly impossible.

The best diffeq text i've found is by Blanchard, DeVaney and Hall, and it
remains the only math textbook i've ever been able to read.

------
abhirag
I have realized that Difference Equations i.e. the discrete variant of
Differential Equations are much more common in Computer Science/Data
Science(recurrence relations etc.) and it would have been great if they were
atleast given half as much attention as Differential Equations get in
universities.

------
falcor84
I'm not sure how well it would fit as a first course, but the thing that
really helped me understand what's behind differential equations is Steven
Strogatz's wonderful book "Nonlinear Dynamics And Chaos"[0].

[0]
[https://www.amazon.com/gp/aw/d/0813349109/](https://www.amazon.com/gp/aw/d/0813349109/)

------
burnerOh2125
Interesting, I'm a phd-drop out in computational biology, working as a data
science consultant: I use mathematics including multi-dimensional statistics,
linear algebra, and calculus everyday. Being self-taught, I'm very self
conscience about the math I don't know, but so far, not knowing differential
equations doesn't seem to have hurt me. I actually just ran into a problem
that uses Hamiltonian dynamics, so maybe I will end up learning differential
equations, but it does seem like the course, as taught many places and in the
Dover books I own, presents either no useful conceptual insight, like why I
learned geometric algebra, or a powerful toolset, like some multi-dimensional
statistics.

~~~
codewithcheese
Hi I am interesting your advice on which math to learn if you have a spare
moment to provide it. I too am starting to teach my self the requirements of
data science and am also self conscious of the math I don't know. I am very
excited about what is now possible with machine learning and deep learning as
I believe it will become increasingly necessary for developers to stay
relevant.

Could you comment in more detail on which mathematical skill you have found
useful as data scientist? Which resources did you use to teach yourself? Very
appreciative of any help.

~~~
ragulpr
* _Solid_ linear algebra

* Probability theory

* Statistics (bayesian if possible)

* Optimization (this is less important but extremely useful)

If you know the mechanics of multivariate calculus you'll be fine learning the
above. The course that personally have had most payoff was functional
analysis. Purely theoretical course that will give you no practical skills and
at first glance seems unrelated to ML but it (subtly) gave me a much deeper
understanding of what ML is all about.

------
ben_jones
My differential equations class was the only course I ever took with over 30%
of the final grade being derived from homework assignments. Unsubstantiated -
but my peers and I all believed this was because everyone was failing and
dropping the course.

------
gumby
Due to an offhand comment from a friend, I started my kid on continuous math,
not discrete like they do in school. Went great, until about grade 4 when they
actually started to care whether he was aligned with his classmates. Germans
are actually pretty intolerant of non-professional pedagogues teaching kids.

But it greatly improved his grasp of elementary math as it matched what he saw
in the real world.

------
colinator
_My colleague’s error consisted of believing that the more testable the
material, the more teachable it is. A wider spread of performance in the
problem sets and in the quizzes makes the assignment of grades “more
objective.” The course is turned into a game of skill, where manipulative
ability outweighs understanding._

Sounds familiar.

------
Chris2048
Previous Discussion:
[https://news.ycombinator.com/item?id=11207183](https://news.ycombinator.com/item?id=11207183)

------
katastic
I took two courses in ODE. I got an A in the first semester. I still have no
idea what they are or what's going on. It's all just relationships and
patterns to me--but with no intuition or understanding behind them. It never
"sunk" in like Calculus did where "Area under the curve" and "tangent line"
are super obvious in retrospect and immediately applicable to your daily life.
"What's velocity? The change in distance over a unit of time." Done. Presto.

------
johan_larson
When engineers and scientists have to get some real work done with DEs, what
do they do? Is it all solved numerically these days?

~~~
eiji
Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list
goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical
software around it to approximate solutions for real world applications.

High and low electro magnetics (Maxwell), Navir-Stokes, ...

~~~
chillingeffect
What tools are used for parameter estimation? E.g. I made a model that
describes my system behavior, then collected a lot of data and next need to
find the constants in the model that make the best fit. Thank you.

~~~
eiji
The search terms there would be "Model Calibration" and/or "System
Identification" Software. I think those are again very domain specific. But
there are a few. PS: While AI and machine learning are catching on in many
different fields, the software packages in those domains make heavy use of
fairly complicated fitting and approximations methods already. Lots or
correlation and effect estimation, which builds many of the underlying guts of
machine learning.

------
eli_gottlieb
>A teacher of undergraduate courses belongs in a class with P.R. men, with
entertainers, with propagandists, with preachers, with magicians, with gurus.

Well _that 's_ depressing. If not undergraduate courses, where do people
expect graduate students to come from?

