What would you still try, if you knew you would fail?

June 30, 2024

Despite having no plans to write a novel, I love reading advice for authors—maybe because it helps me understand what I love about my favorite books. Recently I was recommended Story Genius by Lisa Cron, which walks you through the stages of building a compelling novel around a compelling character arc, and I got to this character-building prompt:

[A]ll protagonists stand on the threshold of the novel they’re about to be flung into with two things about to burn a hole in their pocket:

  1. A deep-seated desire—something they’ve wanted for a very long time.
  2. A defining misbelief that stands in the way of achieving the desire. This is where the fear that’s holding them back comes from.
Story Genius, p. 74

Since I also love any chance for a bit of self-therapy, I immediately wondered, “What misbelief do I have that’s keeping me from achieving my goals?”

The answer came immediately, as if a corner of my mind had been patiently waiting for me to ask: “You think that it’s worse to try and then fail at something than not to try at all.” When I imagine writing to my representatives about an issue that’s important to me, I think about how crushing it will be to have gone to all that effort when they vote the other way. In my math research, I tend to shy away from trying to prove big, important conjectures, because the odds are low I’d be the one to crack them. I read book-writing advice because that feels safer than trying and failing to actually write one.

But I can also recognize that these choices are not in line with my values. So, to riff off cringey self-help gurus, I asked myself the title question of this blog post: What sorts of things would it be worth doing anyway, even if I knew that I would *fail*?

After getting past the gut reaction of “nothing,” I thought of three kinds of situations where I do feel like it’s worth trying even if assured of failure:

  • When there’s important information to be learned from the attempt. Gardening is like this: it’s hard to know what will grow well in our microclimate, and there’s no substitute for trying and seeing what works.
  • When initial failure paves the way for eventual success. I ask this of my students all the time: Study in such a way that you can make mistakes, notice them, and then learn to do it correctly. And the reality of how big math conjectures are eventually proven is from decades of community efforts, most of which are in directions that don’t pan out — you just can’t tell in advance what will.
  • When human dignity demands that we try anyway. When I write to my representatives and they don’t listen, I don’t do it because I’ll learn something valuable from the process, or because I had to fail so that others could succeed later. But I do believe that some battles are worth fighting anyway.

And one bonus reason that’s often in the background as well:

  • When we could be wrong about the likelihood of failure. In reality, we’re never actually certain that we’ll fail. But we can’t let an incorrect belief that we’ll fail be the only reason not to try.

I still allow myself not to try more often than I’d like, and that’s partially because I really do have limited time and energy and have to decide where it’s most worth spending. But if I sense that I’m reluctant to try something that falls into one of these categories, I can remind myself why sometimes failure is the right choice.

What do you want to start noticing?

January 29, 2023

Like many people at the start of a new year, I think about what habits I want to build, but I also like to spend some time thinking about habits in general and how I might structure them differently.

I’ve written before about the three kinds of habit framework:

  • Automatic habits, behaviors you do without even thinking about them
  • Intentional habits, choices you make deliberately over and over again
  • Mindset habits, your default ways of thinking about the world

Over the past decade, many of my personal habits have rotated in and out between automatic, intentional, and just plain nonexistent. Mindsets seem longer-lasting, but more mysterious in their formation: often I’ve only seen progress in retrospect, when I suddenly notice I’ve been thinking and feeling differently. Getting my mindsets in line with my values also feels more satisfying of my desire to be the person I want myself to be (not just to do the things I want myself to do).

My current mental model for how to change mindset habits is the following process:

more “What do you want to start noticing?” …

Moving to Connecticut!

May 31, 2022

Things have been a little slow here on the blog lately, with fewer and shorter posts than usual — the reason, I’m happy to announce, is that I’m finishing up my time as a visitor here in Minnesota and this summer we’re moving to Connecticut! I’m starting a tenure-track position at Southern Connecticut State University. I’m thrilled to get to participate in the university’s mission, and I can’t wait to put down roots* not far from where I grew up — I still have a lot of extended family in the area, and I’m so glad my daughter will be able to know them as she grows up too.

Lots of thoughts about math and life still to come, so stay tuned! Thanks as always for reading.

*Figuratively and literally: outdoor gardening here we come!

Brokenness is inaccurate emergence

April 30, 2022

One of the weirder tech issues I’ve had is that my laptop refuses to connect to wifi while its backup hard-drive is plugged in — it took me a while to figure out that’s what was even going on! I want my things to “just work,” and it’s so frustrating when they don’t. But it got me thinking about what “just working” means, and what it means for something to be broken — my conclusion is there aren’t really broken things, just places where our abstracted descriptions of reality are inaccurate.

more “Brokenness is inaccurate emergence” …

Memorize with Acronyms, not Alliteration

February 28, 2022

To memorize a collection of words or concepts, it can help to form a familiar word out of the first letters, making an acronym like HOMES for the five Great Lakes of North America:

  • Huron
  • Ontario
  • Michigan
  • Erie
  • Superior

For another example, I still remember the order of the four stages of mitosis from high school biology (“Prophase, Metaphase, Anaphase, and Telophase”) by remembering that their first letters in reverse order spell “TAMP.”

Another popular choice of mnemonic, if you have wiggle room in your word choice, is to express the concepts using alliteration: words that all start with the same letter or sound. Hearing three elementary school subjects referred to as “Reading, Writing, and ‘Rithmatic” was probably the first time I encountered this idea, but once you notice them, they’re everywhere. (“The 5 M’s of advertising” are Mission, Money, Message, Media, and Measurement. “The 4 D’s of bystander intervention” are Direct, Distract, Delegate, and Delay. “The 3 C’s to avoid in the covid-19 pandemic” are Crowded places, Closed spaces, and Close-contact settings, and so on.)

I find that it’s easier to make a mnemonic using alliteration than using an acronym, but acronyms are easier to use once you have them. Here’s why:

more “Memorize with Acronyms, not Alliteration” …

Every vicious cycle is also a virtuous cycle

January 31, 2022

What I have to share today is a simple idea that I’ve found really motivating. When I feel like I’m stuck in a loop of choices I regret, each leading to the next in a vicious cycle, I remind myself that every vicious cycle can also be a virtuous cycle.

For example, if I don’t make time to regularly go on long walks, I find that the joints in my toes can sometimes get painfully stiff. That makes it a little more painful to walk on them, so I’m less likely to go for a long walk at all, and so on: once this cycle gets going it’s hard to stop. But viewed another way, when I do go on long walks regularly, my feet feel better, and I’m more likely to want to go for a walk — a virtuous cycle. Remembering the possibility of this virtuous cycle makes it easier to choose to go for a walk despite the pain in my feet.

In fact, probability theory tells us that every vicious cycle hides a virtuous cycle in this way, if we define those terms as follows. Let’s say we have a sequence of choices to make between a good option and a bad option, and we’ll encode these choices by a sequence $latex C_n$ of 1’s (for the good options) and 0’s (for the bad options). We’ll call this scenario “a vicious cycle” if making a bad choice makes the next choice more likely to be a bad choice too, i.e. if

$latex P(C_{n+1} = 0 | C_n = 0) > P(C_{n+1} = 0)$.

And we’ll call the scenario a “virtuous cycle” if making a good choice makes the next choice more likely to be good:

$latex P(C_{n+1} = 1 | C_n = 1) > P(C_{n+1} = 1)$.

It turns out that these two conditions are equivalent! In general, for two events A and B we have $latex P(A | B) > P(A)$ (B is evidence for A) if and only if $latex P(A | B^c) < P(A)$ (not-B is evidence against A) if and only if $latex P(A^c | B^c) > P(A^c)$ (not-B is evidence for not-A). So if a bad choice today makes a bad choice tomorrow more likely, it must also be the case that a good choice today makes a good choice tomorrow more likely too.

Three Good Decisions

November 30, 2021

I have a hard time remembering things that went well. It’s much easier for me to call to mind mistakes I’ve made: times I dropped the ball, or tried and failed to do something that matters to me. This is definitely a phenomenon many people experience (negativity bias), but at least in my case there’s something more going on. Somehow, my brain seems to naturally flag only the negative experiences as things worth remembering (I guess to avoid similar mistakes in the future?), and at the same time, the fact that I can’t fix things that went badly in the past is so painful, I’d rather avoid thinking about the past entirely. Talking to friends and relatives, I realized that there are whole trips and visits I’d completely forgotten about (but can, with difficulty, vaguely remember), either because:

  • I felt bad about the way something happened, and so tried to put it out of my mind, or
  • Everything went fine, so apparently there was no benefit to dwelling on it.

But I don’t want to live my life forgetting huge swaths of it as I go, and expecting things to go badly because that’s all I can remember — so what can I do?

more “Three Good Decisions” …

“A and B” is equivalent to “B and A”, and the order matters

October 31, 2021

In math, I’m used to making no distinction between saying “A and B” and saying “B and A” — they each assert that both of the component statements are true. In fancy terms, we say that the conjunction “and” is commutative, like addition (a+b=b+a) and multiplication (ab = ba). Many mathematical operations are not commutative in general, like subtraction (a–b ≠ b–a), or matrix multiplication (AB ≠ BA). I’ve noticed that the way we use “and” in regular speech, there’s a subtle difference between saying “A and B” and saying “B and A” — here, take a look at the difference between these two examples:

more ““A and B” is equivalent to “B and A”, and the order matters” …

Binary compression of continuous data

September 30, 2021

“Which am I supposed to use: mean or median?”

Early on in my stats classes, we talk about how to describe the distribution of a piece of numerical information — things like height, weight, age, income, and so on that can vary continuously. A good description should cover three aspects of the distribution:

  • A measure of center — this is a single number that somehow is supposed to be in the middle of the range of values. Measures of center include means (averages), medians (halfway points), and modes (most common values).
  • A measure of spread — this is a number that represents how widely the values range from your measure of center: a distribution with a small spread will be tightly clustered around a single value, while a distribution with a large spread will have a wide range of values. If you’re measuring the center with the mean, you’d often use standard deviation to measure the spread, while if you’re measuring the center with the median, you might use something like interquartile range instead.
  • A description of the shape of the distribution: is it symmetrical? bell-shaped? skewed to the right or left? Does it have one peak, multiple peaks, or no clear peak?

My students often want to know which is the best measure of center — mean, median, mode, or something else? I tell them, different measures of center have different strengths and weaknesses, but there’s no way to perfectly capture all of the rich information in the whole dataset with just one number. That’s why we look for multifaceted descriptions that also describe the spread and shape!

But the question got me thinking: asking for a measure of center is a little like asking for the best way of collapsing the whole distribution into one single representative value. That definitely loses a lot of information, but what if we try to lose just a little less — say, by collapsing the distribution into two representative values? The result would be a “binary” distribution: one that only takes on two distinct values with some proportion each.

For example, the people in my daughter’s daycare building have an age distribution that looks something like this:

There are a lot of young kids under 10 years old, and then a few adult staff members of various ages. I’d be hard pressed to use a single number to capture the “center” of this distribution — the mean might be around 15 (not a typical age for either kids or adults), the median would be about 5 (the ages of the oldest kids), and the mode would be 3 or so (completely ignoring the adult staff). But it’s not so hard to compress the distribution into just two ages:

It’s not such a stretch to say something like “Of the people at the daycare, 75% are around 3 years old, and the rest are around 35 years old.”

This makes especially good sense for the daycare, where there are already two groups and we can just look the averages within each group. But it turns out there’s a precise way to approximate any* continuous distribution with a binary version! Here’s how:

more “Binary compression of continuous data” …