
Learn Difficult Concepts with the ADEPT Method - jhund
http://betterexplained.com/articles/adept-method/
======
saeranv
When I was taking linear algebra and calculus in university, I found that
there was a lot of focus on deriving formulas from underlying principles, with
the notion that this constituted a "fundamental" understanding of the
mathematic concept. I got quite good at being able to derive formulas for
anything, and did well enough to scrape by on my exams. However the concepts
didn't really stick with me. Going through Khalid's site I quickly discovered
I had a terrible intuitive understanding of mathematical concepts, almost
embarrassingly so. Somehow, derivation from first principles doesn't quite
capture intuitive insights for me, especially once I start worked at higher
levels of abstraction removed from easily understood foundations (i.e. multi-
dimensional vectors).

The two things that I found most helpful in relearning math is (1) building up
a foundation for mathematical concepts through betterexplained's intuitive
method and (2) turning it into code as soon as possible. For the latter, I
have a side project that is a sort of platform to test all my various ideas,
from city performance modeling, to procedural form generation, where I am
constantly trying to rework or tweak with new math formulas. It's amazing how
much more efficient and useful this is as a learning method.

~~~
kalid
Kalid from BetterExplained here, it's awesome to hear the approach is working
for you. I had very similar reservations, I could derive many results (like a
robot) but had little intuition behind e, i, pi, radians, etc. (let alone how
they all came together in something like Euler's Formula).

My litmus test became: If I can't intuitively describe i^i (an imaginary
number to an imaginary power) I don't understand it. I don't care if I can
derive the equation 15 different ways. If I couldn't spit out some properties
of i^i (positive or negative? Real or imaginary? Big or small?) after a glance
then I knew I didn't know it. (Why can I spit out properties of 2^3 or 3^(-4)
in a few seconds, but not i^i?)

Code is an excellent way to practice these ideas; the bugs in your logic
correspond to the bugs in your thinking, and you see (very explicitly) where
to correct them.

~~~
jacobolus
Kalid: you should really check out geometric algebra (a.k.a. Clifford
algebra). It will give you a much deeper understanding of what _i_ is and what
the exponential function is, and how they generalize to higher dimensions and
more complicated models, and it will help stitch together the weird
inconsistent little fragments of understanding provided by imaginary numbers,
quaternions, matrix algebra, Lie theory, differential forms, etc. into a more
unified/cohesive model.

This is the model all high school and college students will be taught in 100
years, or perhaps even in 50 years, and it will prevent an enormous amount of
confusion and misunderstanding. It’s already becoming the practical tool of
choice in many geometric computing problems, and among niche groups of
physicists.

”Reforming the Mathematical Language of Physics”,
[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf](http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)

“Grassmann’s Vision”
[http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf](http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf)

“Imaginary Numbers are not Real” [http://geometry.mrao.cam.ac.uk/wp-
content/uploads/2015/02/Im...](http://geometry.mrao.cam.ac.uk/wp-
content/uploads/2015/02/ImagNumbersArentReal.pdf)

“Geometric Algebra”
[http://arxiv.org/pdf/1205.5935v1.pdf](http://arxiv.org/pdf/1205.5935v1.pdf)
(this one is a good place to go if you get stuck in another source).

(Or books might be better sources for going in depth. Search for _New
Foundations for Classical Mechanics_ , _Geometric Algebra for Computer
Science_ , _Geometric Algebra for Physicists_ )

You’d also probably enjoy Hestenes’s work on modeling in physics teaching,
e.g.
[http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf](http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf)
[http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%2...](http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%20in%20the%20Newtonian%20World.pdf)
[http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf](http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf)
[http://modeling.asu.edu/R&E/Hestenes-
ModelingTheory2007.pdf](http://modeling.asu.edu/R&E/Hestenes-
ModelingTheory2007.pdf)

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theobon
I'd love to seen an ADEPT method explanation of Laplace Transforms.

I got through most of math fairly easily by having a mental model of what was
going on and could always check that I was on the right track as it made sense
in my mental model. However, when I got to Laplace transforms I never figured
out how to visual what that meant. Everything collapsed into transform into
the magical space where you can do some things easier and then you can
transform out into a new place. I could never be sure of how I got from a to b
without a tedious examination of every step to ensure I applied the rules
correctly.

I'd love to have a mental model for Laplace Transforms.

As a generalization, how does one explain things where no good mental model
exists?

~~~
kalid
Kalid from BetterExplained here, my quick intuition:

The Fourier Transform breaks a signal into its "cycle recipe" (what circular
paths are present?). The LaPlace Transform breaks a signal into its "spiral
recipe".

Circles are made from a type of exponential (given by e^ix), and spirals are
the more general version, where the radius changes (if s=a+bi, then e^is = e^a
* e^bi, aka a circular path where the radius changes exponentially).

The LaPlace transform actually deals with decaying spirals (negative s) -- why
is this useful?

Well, perfect circles that never decay (the Fourier Transform) are nice for
analyzing audio samples, as in music. (Repeated drumbeat throughout the song.)

Decaying spirals model things in the real world, where friction, etc. dampen
the signal over time. The Laplace transform can cleanly represent this
scenario, whereas you need an infinite number of cancelling terms in the
Fourier Transform to represent the "decays over time" setup.

Engineering applications prefer Laplace, compression mechanisms may prefer
Fourier. (Separately, Laplace/Fourier make differential equations easier to
solve by writing functions in terms of exponentials, which are easy to
derive/integrate. The Laplace transform is more general and powerful in this
regard, since it can handle any rate of decay, including 0. The Fourier
Transform is embedded within the Laplace.)

Just some quick thoughts from an amateur on this :).

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bbcbasic
Hard to fully get the Haskell Monad with analogies. The way to understand them
is to use them.

Coming to think of it same for OO. All those silly Dog is an Animal examples
spring to mind.

~~~
a_imho
I agree, if you want to learn something, you have to spend your time and focus
on it. There is really no other way. I can accept it can be more digestible
for many if it is packaged nicely in method X. Helping people learn is a very
noble goal, but it should come with a fair warning there is no shortcut in
understanding.

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hodgesrm
Great article! This is the most intuitive description of imaginary numbers I
have ever read. I'm going to try the same technique with Software Defined
Networking (SDN), which has been on my "must figure it out" list for a while.
:)

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karimdag
Having skimmed the article I want to ask: How about the first principles
method ? How does this [ADEPT] compare to it [fp] ?

~~~
Silhouette
It seems the main point being made by the article is that even if you are
going to teach principles, it can be more effective to start by giving some
context and motivation before getting into the detailed technical
explanations. Deriving results from first principles may well result in better
understanding than simply learning results and proofs by rote, but it's still
useful to have some indication of why those principles might be useful and
relevant as well.

~~~
kalid
Exactly. Even if teaching or reasoning from first principles, you would
explain them via analogies, diagrams, examples, etc.

Teaching from first principles solely from the technical definition wouldn't
be very pleasant for most.

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vikeri
Excellent! I always think that people, like myself, that work with somewhat
abstract and/or complex concepts need to improve our ability to explain them.
Really happy to see a concrete method to do this.

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copperx
I see one thing missing from the ADEPT method, which is "what's the story?" in
the literal sense. Storytelling is one of the most powerful learning
techniques and even a self-learner can create a story about the concept to be
learned. The story doesn't need to be picturesque; even a dry story is much
more useful than just the facts and description.

Some of the examples they provide are stories ("Academic progress on imaginary
numbers took off only after the diagrams were made!"), but the notion of
teaching with a story isn't made explicit in the steps of the method.

~~~
kalid
Great point. I enjoy using stories, history, humor as effective tactics for
presentation. ADEPT is mainly about the raw ingredients I find helpful.

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joelg236
Fourier transformations clicked for me from your previous articles, this
outline really shows the skill of "sharpening" rather than "building"
knowledge. Helpful for young kids to adults alike.

~~~
thomasahle
I had the same experience. However later when I encountered new properties of
fourier transforms, such as the convolution theorem, fast fourier transform
and the fourier transform of boolean functions, I felt my acquired intuition
no longer fitted. Did you also experience this? If so, how did you get past it
to that "next level"?

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wingerlang
Cool. I am currently learning something new and something that have stuck with
me is the "if you can explain it, you know it". So while I am learning I am
writing a set of instructions about what I am learning and some of what I have
written down seems to resemble parts of this article.

Some analogy, visualisations and plain english.

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kitd
Interesting article.

I think I'm going to try explaining git branching/merging to newcomers via
interpretative dance in future.

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zump
Does anyone have a link that summarizes this new field of "learning methods?"

How does it compare to "How to Solve It"?

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kristianp
Anyone want to explain the Y combinator with this method? I find the wikipedia
article on it completely inscrutable.

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supergirl
Analogies are dangerous in math. They give you the illusion that you
understand, but you actually don't.

~~~
kalid
Only if you stop at the analogy :). Last step is Technical Definition.

