In the last month or two, there's been a measurable
increase in the attention to a wide variety of smaller math channels on YouTube. My friend James and I ran a second iteration
of a contest that we did last year, the Summer of Math Exposition which invites people to
put up lessons about math online. Could be a video, could be an article – any
medium you dream up, whatever topic you dream up and we have some prizes available for the
ones that we deem, in some sense best, whatever that could mean.
The deadline for submissions was a little
over a month ago and if we just focus on the video entries, they've collectively accumulated
over 7 million views since that time. Considering that the vast majority of these
are uploaded to very young channels where the video is often just the first or the second
upload, this was really exciting for me to see. I suspect a big part of the reason for this
rising tide is that after the submission deadline, we ran a peer review process where an algorithm
would feed participants two different videos to compare and they'd be asked to vote which
one of these is "better" according to a few criteria. Now, that process generated over 10,000 comparisons
which, on the one hand helps to provide an initial rough rank ordering of all the videos
but more important than that, any judgments or rankings, it gave an excuse for many hundreds
of people to upload around a similar time and then collectively view each other's work
helping to jump start a cluster of videos with a shared viewer base.
And that 7 million number doesn't account
for other videos on these channels, for instance the many submissions people made to last year's
contest which since this year's deadline collectively jumped up by about 2 million views. One group who told me that last year's contest
was what inspired them to put up their first video mentioned to me that a video they had
made in between the two contests managed to suddenly jump from 1000 to 600,000 views during
the peer review process for this year's contest. Despite not being among those videos reviewed. This is all to say, there can be a surprising
value in the seemingly simple presence of a shared goal and a shared deadline. And again, these are just the video entries
where it's easy for us to run these analytics to quickly get a sense of the reach. Many of my favorite entries were the written
ones.
The spirit of the contest, as you can no doubt
tell, is getting more people to put out math lessons and on that front, mission accomplished. But I did promise to select five winners – lessons
that stood out as especially valuable for one reason or another and that brings us to
this video here. To choose winners, James and I both spent
a couple weeks giving a pretty thorough look at over 100 of the top entries as determined
by the peer review process and we also recruited a few guest judges from the community to help
look at a subset of these and make sure that our own biases and blind spots aren't playing
too heavy a role. Many, many things to them in the time they
offered. I won't tell you the final decision until
the end of the video. What I thought might be more fun is to lead
up to it by talking through the criteria that I had in mind when making this election, highlighting
as many exemplary submissions as I can along the way.
Hopefully giving any of you who are looking
to put out your own math lessons online at some point a few concrete things to focus
on. At a high level, the four criteria I told
people I'd look out for were motivation, clarity, novelty and memorability. The first two are probably the most important
and let's start with motivation. This actually has two meanings I can think
of – one on the macro scale and one on the micro scale.
By macro scale motivation, I mean how well
do you hook someone into the lesson as a whole. This video by Aleksandr Berdnikov opens by
asking why it is that when you hear a plane approaching, the pitch of its sound seems
to slowly fall. He points out how a lot of people assumed
this is the Doppler Effect, but that this doesn't actually hold up to scrutiny since
for example, that pitch actually rises as the plane is going away from you.
It's a good point and an interesting question,
you have my attention. One of my favorite articles in the batch by
Adi Mittal prompts you to wonder about an algorithm behind the panorama feature on your
phone and proceeds to explain one – why that's not trivial and then two, the linear algebra
and projective geometry involved in a DIY style project to stitch together two overlapping
images taken a different angles. Application and tangible problems can make
for great motivation, but that's not the only source of motivation. Depending on the target audience, a good nerd
sniping question can also do the trick. This video on the channel Going Null opens
with a seemingly impossible puzzle. Ten prisoners are each given a hat chosen
arbitrarily from ten total hat types available. It is possible for some of the prisoners to
have the same hat type as others and everyone can see all of the other hats but not their
own. After given a little time to look everyone
else's hat and think about it all, the prisoners are to simultaneously shout out a guest for
their own hat type.
The question is, can you find a method that
guarantees at least one of the prisoners will make a correct guess? This video by Eric Rowland motivates the idea
of p-adic numbers than the p-adic metric by showing how if you take 2^10th and 2^100th,
2^1000th, so on and so on and you assign distinct colors to each digit lining them all up on
the right. You can see that their final digits line up
more and more with larger powers. Then he asks the question of whether it's
reasonable to interpret this as a kind of convergence, despite the fact that these numbers
are clearly diverging to infinity in the usual sense. A completely different form of motivation
can come from showing the historical significance of a problem or field in a sense giving the
viewer a feeling that they are part of something bigger.
One excellent video on a channel A Well-Rested
Dog provides an overview of the history of calculus and the progression of how some of
the world's smartest minds grappled with the nuances of infinity and infinitesimals. It's the right mixture of entertaining and
detailed and he goes on to talk about how learning all of this made his own questions
and confusions in a calculus class feel validated. Which I think a of students can resonate with. This example is less about the intro of a
video motivating the lesson it teaches and more about the entire video motivating an
entire field. And another one like that would be this lecture
by the channel Thricery, laying out how Cantor's diagonalization argument and the Halting Problem
and a number of other paradoxes people might have heard of in math, computer science and
logic.
All actually follow the same basic pattern
and moreover, if you try to formalize the exact sense in which they follow the same
pattern, that ends up serving is a pretty nice motivation for the subject of Category
Theory. And the last flavour of motivation I'll mention
is if you can somehow make the learner feel like they're playing an active role in the
lesson. This is very hard to do with a video, may
be even impossible, and it's best suited for in-person lessons but one written entry that
I thought did this especially well was an inverse turing test, where you as the reader
are challenged to come up with a sequence of ones and zeros that appears random and
the article goes on to explain various statistical tests that you could apply to prove that the
sequence was actually human generated and not really random.
The content of the article is centered around
the particular sequence that you, the reader created and you're invited to change it along
the way to try to get it to pass more tests. It's a nice touch and I could easily see this
working really well as a classroom activity. Whatever approach you take, whatever flavor
of motivation is your favorite, it's hard to overstate just how important it is that
you do actually give viewers a reason to care. This is true for any piece of content but
I think it's especially true for educational content and even more so for math given the
amount of focus and thought that these topics sometimes require. I think this was articulated best by the author
of one of my favorite podcasts, An Opinionated History of Mathematics, who has a manifesto
on his website that lays out what he calls "the axioms of learning" beginning with the
first axiom, "In a perfect world, students pursue learning not because it is prescribed
to them but rather out of a genuine desire to figure things out.
It follows that we must not introduce any
topic for which we cannot first convince the students that they should want to pursue it." That said, one mistake that I think I've made
in past videos is to over philosophize in the videos introduction. Motivation is critical but it doesn't have
to take long, and often what actually keeps the viewer engaged is to get right to the
point and leave any commentary about broader themes and connections to the end. If you can, motivate using clear examples,
not sweeping statements or promises of what is to come. By micro scale motivation, what I mean is
whether each new idea that's introduced in the lesson itself feels to the learner like
it has a good reason to be there. For instance, this video by Joshua Maros gives
a fairly detailed overview of Ray Tracing and what I love about it is that before he
introduces any new technical topic, like the rendering equation, important sampling or
the ReSTIR algorithm, he's already outlined the main idea and intuition for that topic
with really well visualized examples.
It makes it so that once the equation on the
screen or the algorithm is described, it doesn't feel like an expression handed down with nothing
to hold onto. Instead, it arrives only once it's articulating
something that already exists at least loosely in the viewer’s mind, making it much easier
to parse. This video by Michael DiFranco about extending
the factorial offers another great example of a lesson with good motivation along the
way. You may have heard that there is a function
generalizing the factorial to real and even complex inputs, the gamma function. The usual definition is written down as a
certain integral expression sort of handed down from on high and the justification for
why this generalizes factorials is certain properties that you can prove about it but
a lot of students find this unsatisfying. Where does it come from? By contrast in Michael's explanation, he starts
by observing the properties that are true of the normal factorial function that you
would want to be true of a general version and uses those desired properties to motivate
various different alternate expressions massaged here and there to be more amenable to whole
number inputs ultimately leading to a pretty satisfying answer.
Another good template for this micro scale
motivation when introducing a pretty complicated solution to a problem is to start with a naive
but flawed solution and then progressively refine it. This article by Max Slater on differential
programming does this particularly well. The basic question is how you get computers
to evaluate derivatives – a ubiquitous task for machine learning. He starts by describing the most obvious approach
and then what flaws it has and uses that as for another approach. But that one has its own flaws and fixing
those motivates yet another approach and so on. The ideas he builds up to, dual numbers and
backward mode automatic differentiation, both could feel a bit confusing if presented out
of the blue. But in context, having motivated each new
idea by pointing out flaws with the previous ones, it all ends up feeling utterly reasonable. Turning back to that same prisoner hat puzzle
I referenced earlier, one of the other things I liked about it is how the author doesn't
just present the solution.
There are plenty of puzzle videos out there
which do that. Instead, he gives a pretty authentic look
at the wrong turns and tangents that are involved in the problem solving process, not even eating
up too much time to do so, and he justifies each new step with a general problem solving
principle. All of this micro scale motivation could just
as well be categorized as a subset of clarity. If motivating a lesson determines how much
attention and focus the viewer is willing to give you, clarity determines how quickly
you burn through that focus.
The best hook in the world is wasted if the
lesson which follows is confusing. This presentation by Ex Planaria talks about
how to describe various crystal structures using group theory, which considering the
complex 3D forms involved and the fact that most people don't know Group Theory, has the
potential to be very confusing. But they do a really effective job at keeping
concrete examples front and centre guiding the reader to focus on one relevant pattern
at a time and distilling down to a simple version of an idea before seeing how that
fits into a broader more general setting. In general, entries that struck me is especially
clear would often keep one or two examples front and center and that often give a feeling
of playing with those examples – may be running simulations or tweaking them to run up against
edge cases and all around giving the viewer a chance to build their own intuitions before
general rules are presented.
The example doesn't even have to be explicit. In a visually driven lesson, the choice of
what to show on screen when making general points is often a great opportunity to offer
the viewer a concrete example to hold on but without wasting too much time explicitly talking
about that example or over emphasizing its importance. This I think is part of what gives visually
driven lessons the opportunity to be clearer. As a brief side comment by the way loosely
related to clarity – for any of you who want to use music in the videos, while music can
enrich the story telling aspect of a lesson – setting the desired tone and the momentum,
once you're getting into the meat of a technical explanation, it's very easy for the music
to do more harm than good. If it's there at all, you want it to be decidedly
in the background and not calling attention to itself. I recognize some hypocrisy here, it's definitely
something I know I've messed up with past videos and it's just worth thinking about
whatever benefit you see from the music.
You don't want to incur a needless cost on
clarity that outweighs that benefit. Moving on to novelty, this is another category
that has two distinct interpretations. One would be stylistic originality. Back when I created this channel, part of
the reason I wrote my own animation tool behind it was to ensure a kind of stylistic originality. Well, the main reason was it was a fun side
project and having my hands deep into the guts of some tool helped me to feel less constrained
in trying to visualize whatever came to mind.
But being a forcing function for originality
was at least a small part of my reasoning. This means there's at least a little hint
of irony in the fact that if we fast forward to today, so many of the entries in this contest
used that tool (Manim) to illustrate their lessons. I have nothing wrong with that. It actually delights me. It's why I made it open so and I'm very grateful
to the Manim community for everything they've done to make the tool more accessible.
But I would still encourage people to find
their own unique voice and aesthetic whatever tools they use and whoever they take inspiration
from. I don't want to over emphasize that point
because it's the much less important half of novelty. The much more important kind of novelty is
when the thing you present would have been very hard to find elsewhere on the internet. Either because it's a highly unique topic
or because it's a very unique perspective. For example, this video on percolation showed
a completely fascinating toy model for studying phase changes, a model where it's easier to
make exact proofs. And considering the level of depth and the
level of clarity that the authors provided, I think it's fair to say you wouldn't find
something like this on YouTube if this group hadn't made it.
As to memorability, I'll keep this one quick. Lessons take off this box when they ask a
question that's just so fun to think about or provide such a satisfying aha moment that
it stays with you long after watching it or reading it. Admittedly, this one is highly personal and
subjective. To my taste for example, this video by Daria
Ivanova discusses the question of when it's possible for a single track to have been left
by a bicycle – that is, the back wheel goes through the same path that the front wheel
does, which is just so fun to think about. This video by Gergely Bencsik about how involute
gears work had a really satisfying way of explaining why a certain gear design pattern
works so well that for me at least just stuck. So, with all of that, who are chosen winners? In the announcement, I promised that one winner
slot would go to an entry that was made as a collaboration and that one goes to the percolation
video.
The other four winners are, perhaps unsurprisingly,
also ones that I've already mentioned. They include the post about describing crystal
structures with group theory, the video covering the history of calculus, the one about ray
tracing and the algorithms to make it faster and the problem solving lesson centered around
the tricky hat riddle. These entries really do speak for themselves. So, rather than telling you too much more
here, I encourage to check them out. To be honest, after I got it down to about
25 entries that I wanted to at least be honorable mentions, it was exceedingly hard to actually
choose winners from that. Since for each of these, I could easily envision
a target audience for whom that entry would actually be the best recommendation. It was a game of comparing apples to oranges
but times twenty-five. Below the video, I've left links to the other
20 that I chose as honorable mentions and to a playlist that contains all the video
submissions and also to a blog post containing links to all of the non-video submissions.
Thanks to a sponsorship from Brilliant. Each winner will get a thousand dollars as
a cash prize and also – and much more importantly I think – a rare addition golden pie creature. Also, after the initial announcement, Risk
Zero and Google Fonts both generously reached out offering additional prize sponsorships. I'd also like to thank Protocol Labs for another
contribution to help us cover the cost of managing the whole event. Thanks to everybody who participated and to
everybody who helped in creating this rising tide for new math channels and new math blogs
that we've seen in the last month. It was genuinely inspiring to see just how
well this all went..
