The Fractal Nature Of Goal-Setting

This post is not specifically math-related, beyond the above-average appreciation for the phrase "fractal nature" that a mathematician might have.  Rather, this post is about getting good at stuff in general — math being just one possible thing you might want to get good at.

The following is a brief summary of stuff we've learned from various books that are not designed for mathematicians (many, in fact, were aimed at aspiring musicians) and have adapted to our desire to become better mathematicians.  The underlying idea is simple and closely related to the idea of "having something to do when you get out of bed in the morning" that we talked about last week, but, as always, the devil is in the details.

The TLDR is: set concrete goals, then break them up into smaller concrete goals (then break those up into smaller concrete goals (then break those up into smaller concrete goals (then ...)...)).  Now let's look at some of the details.

Set Concrete Goals

There are two important aspects of the word "concrete" in this context.  The first is a bit more obvious: make your goals specific, such that it's obvious when they're done.  So, a goal like "get good at math" is probably not great, but goals like "do every odd exercise in Graduate Textbook X" or "obtain a PhD in math" are things that have a well-defined end point.  That's a good start.

The second aspect of "concrete" that is important to understand — one that is a little less obvious — is that it should be (at least very largely) in your control.  This is perhaps the most important aspect of any individual goal, and it's also the most subtle.  Sometimes things that seem like they're in your control, are actually less-so than you might think.

For example, a goal like "getting the best score in the class on the next topology exam" is not in your control.  You can control how hard you study, but you can't control how hard anyone else studies.

One somewhat reliable way to test if your goal is in your control or not is to ask if there's any way to "game" the system.  For example, suppose everyone in your topology class partied all semester and got zeroes on the exam.  A score of 1% would then put you "at the top of your class", but in this case, would that actually reflect what your goal was meant to reflect?  Probably not.

So, set specific goals with a well-defined "done" status, and do your best to make sure that they're in your control.  But now comes the really fun part.

Apply Recursion (Build Your "Goal Fractal")

Once you have done this, it's time to apply recursion.  Namely, once you've set your specific, controllable goal, it's time to break it up into smaller, specific, controllable goals. And then take those smaller, specific, controllable goals, and do the same again.

This is how we get to the point of "having something to do when we get out of bed in the morning", or even just having something to do for the next 10 minutes.

With this approach, you'll be able to set lofty, long-term (but still specific, controllable) goals, and then lay down a plan for actually obtaining them.  The underlying assumption that one must agree with and internalize — the assumption that we very firmly hold to be true (but about which we are always willing to hear dissenting opinions) — is that humans can only ever make small, incremental progress on things.

I truly believe that this is an inevitable consequence of the continuous, linear progression of time — every 10 minutes, only 10 minutes has gone by.  The natural consequence of this is that you can only ever do 10 minutes worth of stuff in 10 minutes.  Therefore, if you can't break down your lofty ambitions into a collection of things that can be done in 10 minutes (or maybe an hour), then you'll forever be stuck wishing for a thing but not being able to make any actual progress towards it.

An Example

Another strong opinion that we hold is that if "a picture is worth 1,000 words," then "a good example is worth 1,000 pictures" (as a side note, a good counter-example might be worth about 500 examples).  So, let's give an example.

Let's suppose we're a high school student who loves math and our big, lofty goal is to "get a paper published in a well-respected, peer-reviewed math journal": 

That has a well-defined "completion time" and is pretty largely within your control. It is, however, going to take a long time and a lot of work. We therefore need to break this goal up into smaller, more immediately attainable (yet still concrete and controllable) goals. Fortunately, we can do this step-wise...

You might realize that in order to publish a paper, it'll help you tremendously to be at some kind of university studying math. You'd also probably want to be doing so at least at a graduate-school level, since there probably aren't too many publishable results accessible before then (there very well might be, but we're trying to take the most realistic approach).

With these realizations, one of your smaller goals could be "get admitted to math grad school." This is specific with a well-defined completion date, but it's not totally in your control.  For example, maybe your graduating class (globally) is one of the most mathematically gifted classes in the history of the world.  That said, I think this is "in your control" enough where we can add it:

That obviously won't be enough, though.  There's also a certain amount, and type, of work that you just have to do.  Maybe you read our post last week about reproducing papers, so you also set the smaller goal of "reading and reproducing 10 papers in a specific field".  That's specific, in your control, and an easier-but-probably-necessary-step towards the ultimate goal, so let's add it:

Let's now hone in on the grad school bubble.  To do this, you'll probably need to "get a bachelors in math".  Specific and controllable.  You'll also probably want to stand out a bit as an undergrad, so maybe we should add "Complete 2 graduate-level courses as an undergrad".  Let's also add "perform well on the math GRE":

Let's now hone in on the "take grad courses while an undergrad" bubble.  One necessary step might be to "complete the normal math major coursework by the end of my junior year to free up time in my senior-year schedule for the grad courses," and another is probably to "identify which subfield I want to take these grad courses in" (which itself can then be broken down into "talking to professors," "going to seminars," etc.).  We then have:

You hopefully see where we're going with all this.  It will take some time — maybe about an hour — but if you keep breaking down each bubble into smaller, specific, controllable goals, you'll eventually get to a bunch of bubbles that say things like "complete tonight's math homework with no mistakes", which itself can be broken down into "complete the next problem and check its validity".

It is vitally important, in fact, to get down to this level of specificity.  Namely, the whole point of this exercise — and of viewing goals as fractals — is to make it so that you know what you need to do today, or in the next hour, or in the next 10 minutes, or right now, to take one small step in the direction of your goals.

Once you have your bubbles laid out, the satisfaction of checking them off and seeing a visual representation of your journey towards your goal often provides the very motivation that you need to attack the next bubble.

Personalized Goals

The example we gave above is still more generic than I would have liked it to be, primarily because goals should be more personal to your current needs.  For example, maybe you have test anxiety.  That's fine, plenty of people do.  It can, however, hold you back from your goals, depending on what your goals are.  So, if this is you, put a big bubble on your chart that says something like "conquer my test anxiety" (though, maybe something more specific).

Then break this up into smaller goals (this process might help make the bigger goal more specific as well).  This could be "do my homework with a stopwatch for a week", to try applying artificial time pressure. This could be "read 3 books about test anxiety". This could be "before each exam, do every problem in each section that is being tested" to maximize your confidence going in to the exam.  Then, of course, break all of these up into smaller goals.

Importantly, though, make sure your "Goal Fractal" is unique to youyour hopes and dreams (not society's, not your parents', etc), and your particular circumstances.  If, for example, you're ex-military who's getting a later start on your academic career than most, that's perfectly okay (and can, in fact, give you a leg up in terms of maturity, work habits, etc., compared to your younger peers). But you need to make your Goal Fractal reflect that.

Goals Are Not Static

There is one final vitally important aspect to this type of goal setting — moving from lofty, distant goals to concrete, immediate goals — that we haven't yet talked about.  This final ingredient is the willingness to change your goals.  After all, if you're a high school student who wants to write papers on algebraic topology, you might be 10 years away from achieving your goal.  One must understand that a lot can happen in 10 years — a lot can change (about you, about the world, etc.).

Having your goal written down in front of you every day is a fantastic way to stay focused, and breaking it down "fractally" is a great way to make consistent progress towards it, but don't become a slave to your own Goal Fractal.  The whole point of a Goal Fractal is to make you feel good watching your progress pile up, day after day.  "Feeling good" about your goals and the progress you're making towards them is key, so be willing (and happy) to change them if circumstances call for that.  There's no shame in that.

Not Actually A Fractal

A Goal Fractal is, of course, not an actual mathematical fractal, for one big reason — there is a minimum resolution to the Goal Fractal. That minimum resolution is "the time it takes to write down and/or check off the goal".

Once I get my Goal Fractal down to the granularity of "do the next problem on my homework correctly," I probably shouldn't break that down into "start the next problem" or "open the textbook" or "think about opening the textbook." Just imagine thinking about opening the textbook and then checking off that bubble and then going back and actually opening the textbook and then checking off that bubble, etc.  At some point, you just actually have to get to work :)

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