002: doing before reading, part I

first published: april 7, 2025
last updated: april 7, 2025

mathematics project laboratory
minds unleashed
doing before reading
teaser for part ii
some nice quotes about the class

mathematics project laboratory

During undergrad, in the spring semester of my sophomore year, I took an unassuming-sounding course called Mathematics Project Laboratory,^[1] taught by Professor Mike Artin. I really liked Prof. Artin. (He reminded me a little bit of a calmed-down version of Doc from Back to the Future, both in looks and attitude.)

In this class, we got into groups of 3 students, and each group would pick an open-ended problem to work on. There was a list of about a dozen problems, as I recall, and each problem was described in a couple of paragraphs. None of them were "standard" math problems, but they all related in some way to mathematically interesting topics.

For example, the first problem my group chose had to do with fleas hopping around on a chessboard, and any square they landed on would change color. This problem was a variant of cellualar automata, an area of math and computer science that examines how simple rules of behavior can lead to complex patterns and algorithm-like processes.

We had one month to work on our chosen fleas problem. Now, what do you suppose we were supposed to do first? Go study up on cellular automata? Get familiar with the general background and terminology? Read a research paper describing some simulation or experiment that we could implement as a starting point? Work through examples from a textbook?

NO! We were not supposed to do any of these things!

Most unusually, as we worked on this project, we were not supposed to look up anything specifically related to the project. So we had one month, three people, pencils and paper, and computers, for writing whatever code and simulations we might be able to dream up on our own. (I was about to add "coffee" to this list, but this was in my pre-coffee era! Ah, youth.)

We could not go read textbooks about cellular automata; we could not look up what research other people had done; we could not go download other people's cellular automata code packages; Stack Exchange, Github, and ChatGPT didn't even exist, but even if they had, we could not have used them, or any other Internet resources.^[2]

Each group did have weekly discussions with the TAs, but that was our only source of external information/feedback. Everything else that we generated during that month was the product of our own minds.

I remember having meetings with my teammates with just the three of us, sitting around a table with sheets of paper in front of us, drawing little grids and fleas and arrows, trying to work things out essentially from first principles. I remember inventing ridiculous equations while trying to calculate probabilities and permutations/combinations of various outcomes. I remember writing code to run little flea simulations.

We worked on this project a lot! As a math major, it was typical for us to spend 10-15 hours per week on homework for a single class, and so with this project, we probably put in 3 people x 15 hours/week x 4 weeks = ~180 person-hours of work.

It was a long time ago, so I don't remember what we actually ended up learning about the fleas, but by the end, I do remember feeling like my brain had just run a marathon. I was intellectually exhausted, but also strangely energized. It was a mental workout like no other I had experienced.

And then... we had to immediately turn around and do it again---our second project,^[3] for the second month. And then... ONE more time---a third project,^[4] for the third month. A triathlon.

(Culminating in a final project presentation that apparently had our audience's hair on end, because of the sheer physical danger involved... like when you watch trapeze artists at the circus performing without a net. Not what you'd expect from a math class presentation, is it?? <cue suspenseful music>^[5])

minds unleashed

In the end, Mathematics Project Laboratory was one of my favorite classes of all time. I remember very little about the specific topics that we worked on or what exactly we discovered.

But I do remember the fun of brainstorming, batting around ideas to see which ones felt exciting to us. Inspecting the problem from all sides, forwards, backwards, upside down---like it was a vault that we were trying figure out how to break into. I remember the frustration of finding mistakes in what we had done before, and having to scratch out days worth of work and start over. I remember that it was hard work, and a LOT of work, and the feeling of exultation when the work paid off and we uncovered some cool result.

Most of all, though, I remember how satisfying it was to fully unleash our minds against a problem, for hours upon hours, weeks upon weeks.

Part of this satisfaction came from not having a set project goal or outcome... but I've done lots of course projects that were similarly open-ended, and they didn't engender in me the same sorts of feelings of running a mental marathon.

I think the bigger part of the satisfaction came from being able to work on something, for an extended period of time, without having to follow what other people have done.

It was a bit like, imagine if you have been playing with legos your whole life, but you were always building something definite using those little instruction booklets and lots of specialized pieces. It can still be creative: maybe you are given a cool lego castle to start with, and your job is to figure out how to add on an extra tower.

Now, though, you are handed a big box of undifferentiated legos, and you get to just... build! And nobody is standing over your shoulder telling you that your castle should have these specific shapes of walls, or, oh look, here's how you can build a little dungeon add on. And so, you get to dream up what your castle looks like, you let the pieces run through your fingers as you contemplate how to start. You figure out a wall shape that pleases you, and the castle starts to rise up from a combination of your imagination, effort, and sheer will. Sometimes you build things that fail; your cantilevered drawbridge collapses, and you learn something that you didn't know before about the structural limits of cantilevered chains of legos. And then you build the next bridge a different way, and it stays up.

It's creative, it's fun, and you learn a heckuva lot not just about castles <cough: the topic you are working on> but also about how to build anything with legos <cough: how to think>.

doing before reading

Don't get me wrong; I do agree that reading is also important for learning. But I often feel that the common approach of having students read a bunch of stuff about a topic before starting to work on it themselves is not the best way to learn about something new.

I think that doing something first, on your own, using only what is inside your own head, will teach you an incredible amount about a topic, even though it may seem counterintuitive because you aren't getting any "new" outside information about it. (In fact, you are; it just happens to be information that you yourself are generating!^[6])

After you have tried the "doing" for awhile, then comes the perfect time to go start reading. And, if you do it this way, the reading will be far faster, easier, more intuitive, and more useful than if you had taken the approach of reading before doing.

For our Mathematics Project Laboratory class, we didn't do any follow up reading, but after spending a month on each project, I feel like I could have sat down and just inhaled scientific papers about cellular automata, random walks, and floating bodies.

I knew each problem so thoroughly, inside and out. I knew the edge cases, the loopholes, the weird tricks you needed to use to implement a computational simulation that would work properly.

And the equations! Every equation was not just squiggly symbols on a page; I knew what it meant. I knew what each variable represented, and I could easily describe all kinds of relationships each variable had with everything else. I could rewrite each equation in a dozen different forms, because I knew how the whole problem functioned, as a coherent system of concepts and behaviors, and not just as isolated bits of mathematical notation.

In cognitive science terms, knowing about something means having a "mental model" of it---having an idea not just about various facts, but also how they fit together, which parts are easy or hard, how they fit in with other things that you know, etc.

After each project in my class, I had an extremely good mental model of each problem. I didn't know the official terminology for anything; I think we probably invented a lot of our own terms to describe things. But once you have a good mental model of something, it is extremely easy to go read about what other people have done, even if they are using different terminology.

You recognize what people are talking about, instead of having to learn what they are talking about.

teaser for part ii

Briefly (because I need to go write a bunch of emails now 🙄), there are three things you get from doing before reading that I believe are much harder to get if you take the approach of reading before doing:

1. Doing before reading helps you build up very good mental models about a topic.
2. Doing before reading helps you be more creative.
3. Doing before reading helps you exercise your brain so you can think better in the future.

I have a lot more to say about each of these points, including cameos from Richard Feynman, Richard Hamming, Ivan Sutherland, and a Rutgers undergrad who recently made an incredibly unexpected algorithmic discovery... so stay tuned for Part II in the near future.

some nice quotes about the class

In the course of writing this blog post, I discovered two papers^[1:1] written about the Math Project Lab course, how it came to be, and some of the outcomes observed by instructors and students. So, to finish up, here are some nice quotes from the Greenwald & Miller paper:

Students at MIT are used to frequent, directive problem sets where plugging and chugging sometimes takes the place of deeper, riskier, experimental engagement. The Math Lab offers students “license to wonder.” Students who when given the license to navigate the far reaches of these projects, say Math Lab Instructors, “can sometimes surprise you with how far they’ll go.” (p. 7-8)

Because of this high ‘market value’ placed on discovery, Artin sees CATAM, the progenitor [a previous course taught at Cambridge University], as overly directive and therefore under-nourishing. The open-ended approach, as Artin sees it, does not dictate to students what to notice, what to do, or even what a certain algorithm is named (so they can find clues on Google, for example). In this process, says Artin, students sometimes “flounder,” sometimes go down “dead ends,” but ultimately, on their own doing, they find a path. On this path, students do their own thought experiments, get somewhere, make their own conjectures, sometimes get creative, but ultimately, they “learn to go deeply into some particular area.” This, explains Artin, is the essence of “doing mathematics.” (p. 8)

In addition to discovery, independence and choice, students also valued the opportunity to gain depth of understanding in Math Lab. Clarification through explanation dominated students’ records of their experiences with this course. Theirs was a powerful description of active as well as deeper exploration of topics. Relevant activities noted included ‘explaining,’ ‘starting from scratch,’ ‘clarifying,’ ‘backtracking,’ ‘thinking,’ ‘correcting,’ and, last but not least, ‘throwing everything you know at it.’ One student even commented on the effect this course had on his “confiden(ce)…to do real research.” And finally, another student sarcastically noted that he “liked coming up with new ideas and watching them fail.” (p. 9)

mkfootnotes™