It’s confusing when someone changes their profile picture. I made a joke about how, to reduce confusion, there should be a feature that gradually morphs your old profile picture into your new one.

And then I figured, why not make it real?

I also added horse mode, where instead of directly morphing from your starting picture to your ending one, you turn into a horse in between.

For the last few weeks, as I’ve been building this out, my DMs have looked like this:

The majority of my friends didn’t even question why I was sending pictures of them as a horse or morphed with others, they just accepted this was a thing I was really into at the time for some reason.

And so they sentenced other people to be horseified.

Why did you do this?

AnimorPFP is a cultural critique of the fragility of one’s online identity, the inherent impermanence in a digital ecosystem in which the inhabitants are intangible.

We increasingly live our lives online, where our presence is represented by a static image, carefully curated to control how we are perceived. Making it jarring for that identity, that single image that represents the totality of an individual, to switch unexpectedly. It’s striking, the lack of object permanence in the digital sphere, as if we are once again infants, confounded by peek-a-boo.

So we must ask: In this era of tech, in a life lived online, what does identity entail? What does it mane to exist online? If one’s digital persona is masked with horse, if one’s visage is more strongly and more often associated with horse than one’s own face, then what separates mankind from equine?

My thoughts on this matter are too profound to be illustrated by prose alone, so I have prepared a poem:

There once was a fella named Harold
He planned to change his PFP but was imperiled
For it defined his identity
Without it, he felt a non-entity
A neighsayer might even claim he was Gerald!

Although to be serious for a paragraph, there is a genuine point to be had about this era of social media, where instead of following friends, the majority of people we follow are strangers. Even more so, our front pages are filled with people we don’t even follow, the algorithm decides who we stay updated on. The “problem” that AnimporPFP solves didn’t meaningfully exist ten years ago, before the rise of the influencer and algorithmic feeds. Back when social media facilitated mutual engagement with friends more than parasocial relations.

(Albeit there were always anonymous or follower-based forums like Tumblr, Reddit, and Twitter, but the quintessential social media was friends-based Facebook and old Instagram. Nowadays, essentially every major platform is grounded in serving you content from people you don’t know, and often don’t even follow.)

How does it work?

Why, with the beauty of mathematics, of course!

First, we need to understand what the faces in the start and end images look like, so we map facial features across 468 points (using MediaPipe Face Landmarker). Then, these points are used as references for dividing the face into a series of triangles (Delaunay triangulation). We compare the positions of the triangles in the start image to those in the end image, and as the slider moves along, the triangles warp to the end positions, while fading in the colors of the end image (via WebGL).

Thanks to existing software, creating this was actually very easy. Except for horse mode, which was very difficult and 99% of the work. I know what you’re thinking, that is an incredibly useless feature in an already pointless project. And to that I say fuck you, I’m turning you into a horse.

I also asked Chat for some technical advice, and this is where you can really tell it was heavily trained on Reddit content, as it first explained to me why my own joke was funny. And lowkey roasted me in doing so. I’m not sure what it means by “overengineered,” I think this is a reasonable solution to a serious problem.

I’m just serving AnimorPFP on one of my personal sites, instead of buying a new domain, because to be frank guys, I have spent hundreds of dollars on domains and it’s getting quite ridiculous at this point.

You can try this super helpful tool out for yourself at animorpfp.rawandferal.com.

See you next time, whenever that may be.

Love,

danielle (𝓇𝒶𝓌 & 𝒻𝑒𝓇𝒶𝓁)

Here’s a new preprint from Anthropic: “How AI Impacts Skill Formation”. AI coding bots make you bad at learning, and don’t even speed you up. [arXiv]

The researchers ran 50 test subjects through five basic coding tasks using the Trio library in Python. Some subjects were given an AI assistant, some were not.

The subjects coded in an online interview platform, and the AI users also had the AI assistant.

The researchers used screen and keystroke recording to see what the test subjects did — including those no-AI test subjects who tried using an AI bot anyway.

Afterwards, the researchers tested the subjects on coding skills — debugging, code reading, code writing, and the concepts of Trio.

The coders in the AI group were slightly faster, but it was not statistically significant. The main thing was that the AI group were 17% worse in their understanding:

The erosion of conceptual understanding, code reading, and debugging skills that we measured among participants using AI assistance suggests that workers acquiring new skills should be mindful of their reliance on AI during the learning process.

It’s just a single study and quite limited. You should expect to see AI bros dismiss the study saying it’s one library, it’s not enough coders, it’s an old model — and not to do better studies addressing their own objections.

If you don’t do the work, you don’t learn, and you don’t remember. Watching a bot do your job teaches you nothing. You end up incompetent. And you won’t work faster anyway.

• Video — Podcast

I. From “Show Me” to “Let Me Try”

Early math is very example-heavy. Both in classes, textbook explanations, and in worksheets. In the worst cases, you run into what is commonly called “Drill and Kill”, which means providing students with an enormous number of repetitions on a particular skill until their intellectual curiosity is crushed. That is, drill the topic until their spirit is killed.

That said, it does get a bad rap because when kids are starting to learn math, they do not have enough maturity to instantly get what is being shown to them, so they need to be shown *what* to do so that they can do it themselves.

Ideally, it’s a drill with decreasing support until the student masters the skill with no support given.

As kids get older and math gets complicated, the support and examples become less and less, until you get into higher-level undergraduate and graduate textbooks, where the student is supposed to come up with all of their own examples. Ultimately, the books at that level present the material in the form of axioms, definitions, theorems, and proofs. Two undergraduate math books famous for this are Landau’s Foundations of Analysis and Rudin’s Principles of Mathematical Analysis (a.k.a. Baby Rudin).

However, parents and kids are rarely told that this transition is coming or how to even deal with it.

One way to start very early in preparing for this transition is to have your kid start making their own example problems for the math material they are learning.

As they get older, their examples will improve, and by the time they reach the definition-theorem-proof format, they’ll be rock-solid at creating examples.

Not only that, but if your kid is very mathematically advanced, it gives them something to do when they are bored in math class, since they’ve already seen the material being presented ages ago.

II. The Cognitive Shift From Examples Provided to Examples Created

You can start with this cognitive shift regardless of where your kid is mathematically and what age they are currently celebrating. The earlier you start, the longer your kid will have to train their example-creation muscle, but you can start at any time.

The cognitive shift is from being a consumer of examples to a creator of examples.

Instead of thinking, “doing these exercises will teach me the technique”, it’s more of a “doing these exercises will show me how the technique works, so I can teach and test my understanding of how the technique works.”

It’s a bit of a small, subtle shift, but it does two things: a) creates a sense of ownership about learning the material, and b) enhances doing and reading the exercises. Once learned, your kid will constantly be on the lookout for how they could potentially create a similar problem.

Even at the earliest level, if a kid is doing a bunch of exercises to study multiplicative identity, for example:

• 1 x 1 =

• 2 x 1 =

• 3 x 1 =

• 4 x 1 =

They can start thinking about things like: What if we multiply by 1 to the 1s we’ve already multiplied?

• 1 x 1 x 1 =

• 2 x 1 x 1 =

• 3 x 1 x 1 =

• 4 x 1 x 1 =

Or maybe they think about whether they can substitute a fraction for the first number

• 1/2 x 1 =

• 2/2 x 1 =

• 3/2 x 1 =

• 4/2 x 1 =

Notice that this isn’t about making their math skills “faster” or solving more “difficult” problems; it’s about making sure they are exploring the technique even more deeply.

III. Why This Helps Kids Who Are “Bored”

Boredom often creeps into your math kid’s life in school math classes. They enjoy working on math, so it comes easier to them the more they do it. Suddenly, they are “ahead of the class” and bored with the material being presented.

One approach is to start petitioning the school for advanced work, grade-skipping, pull-in instruction, pull-out instruction, grade-telescoping, etc. However, it’s not clear whether this will alleviate boredom in the long term, let alone whether the teacher, school, or district will even consider it.

What we can do as adults who work with kids who love math is explain that the surface is easy, but the idea may not be. Boredom often means the *presentation* is easy—not the concept. That is, the work they are doing in their classroom is easy, but that doesn’t mean that the idea is simple.

For example, consider multiplication as repeated addition. A mathy kid will soon have most of the multiplication tables memorized, so they can sail through any work in this area easily and quickly. If they talk about being “bored” with this, ask them to come up with example questions where repeated addition may or may not work, or how it might work differently.

For instance, does multiplication as repeated addition work with negative numbers? Have them come up with some simple examples where either the first number or the second number is negative. Does it work? What does it even mean to do a repeated addition of a negative number?

Creating more complex versions of easy problems restores challenge and because the kids themselves are coming up with the questions, not you, the problems will be right at the edge of their mathematical maturity. Ask them to change numbers or constraints, or even context.

This can be done at home and, probably more helpfully, at school. At school, they can do it silently in their head or on the side of a piece of paper. They still do the school work, but they also build their problem-creation muscle at the same time.

What we really like about this method is that it doesn’t require the teacher’s permission or the creation of new material. Additionally, if they finish an assignment/test/work early, they can turn the waiting time into time for mathematical thinking.

IV. A Practical, Research-Aligned Ladder for Problem-Posing

Once you’re bought in and your kid, maybe with your help, starts creating their own problems, a question that comes up is - well, is it a well-posed problem? That is, is it really using and testing the technique correctly?

There is a huge mountain of research and practical experience that goes into problem creation for helping cement learning. I’m not going to go into that here.

I think it would be more helpful to give you the synthesized version.

There are 5 general levels that the problem-posing can go through:

Level 0 — Solve (baseline)

• Solve a standard example that’s an exact copy from the book/lecture

• This is to make sure the student understands the method

• Example: The lecture/book showed an example of 3 * 4 = 3 added four times = 12, so the student poses the problem 3 * 4.

Level 1 — Near-Transfer Problem Posing

• Change one element (numbers, context)

• Same structure, same method

• Strongly supported by research for young learners

• Example: The lecture/book showed an example of 3 * 4 = 3 added four times = 12, the student poses the problem 2 * 4.

Level 2 — Method-Preserving Problem Posing

• Change many elements, same underlying technique

• Structure preserved, surface changed

• Student explains *why* the same method applies

• Example: The lecture/book showed an example of 3 * 4 = 3 added four times = 12, the student poses the problem 7 * 5.

Level 3 — Constraint-Driven Problem Posing

• Given constraints, not a template

• Requires planning, verification, and reasoning

• Example: The lecture/book showed an example of 3 * 4 = 3 added four times = 12, the student poses the problem 3 * some number = a number that must be less than 10. As a bonus, they could say what the minimum and maximum numbers are that would work to be less than 10.

Level 4 — Error-Sensitive Problem Posing

• Design a problem that exposes a misconception that they or they think another student may trip up on

• Shows near-teaching-level understanding

• Example: The lecture/book showed an example of 3 * 4 = 3 added four times = 12, the student poses the problem 3 * 0 because what does it even mean to add 3 zero times? Do you end up with 3 or 0? That may trip some students up. Being able to spot potential sources of confusion indicates the edges where the mathematical technique might not work or require a different approach.

As with most aspects of parenting, it’s helpful to work with your kid through examples of different types of problems they might encounter. You and they don’t always need to reach Level 4 for each topic/technique, as higher levels are unlocked only when a deeper understanding of the mathematical technique develops. So when they first encounter a technique, ideally, you start at level 0 and try to replicate examples they saw in class or in the book. Then, as they work on homework, they may move up one or two levels as they see other worked examples and solve problems on their own. Eventually, maybe a few sections or even chapters, they (and you) might arrive at level 4 problem posing.

V. Problem Posing Builds Mathematical Maturity

As in the last paragraph, as the student becomes more comfortable with the material, their ability to delve deeper into questions that test the actual technique will improve. They’ll be able to identify the structure and formulate questions that probe it. They’ll come up with questions that test the boundaries of a technique, when it works, and when it doesn’t. They may reach the point where they can ask questions about when this technique breaks and what to expect when it does.

Mathematical Maturity means being able to do the above with skill and efficiency, unprompted. This is how mathematicians actually work, which is why, as students progress further along their mathematical journey, the books and courses they study have fewer and fewer worked examples. And as they explore the four levels of questions for each new mathematical object/technique/idea, it reinforces what they’ve learned about previous mathematical ideas. Which further entrenches and enhances their understanding of the subject.

Lastly, as they keep posing these problems, they will start to build intuition about how things might work, where they may break down, and how they can apply them in multiple places such that when they hit the abstraction of high school algebra or college mathematics or Ph.D. level mathematics, it’ll feel natural rather than a step up that may be too big to cross.

Down the line, kids will start seeing math as mathematicians see math: objects to explore, not rules to obey. Mathematicians treat concepts like numbers, shapes, sets, and many other things as real abstract objects that can be investigated, much like a biologist would investigate a new creature. Instead of applying memorized formulas, mathematicians “poke and prod” mathematical objects to see what happens, looking for patterns and relationships.

VI. The Parent–Child Dynamic Shift

One fun aspect of this question-posing technique is that it inverts the power dynamic between the adult and the child. Since the child is the one asking the questions, it’s now the adult who has to do the work and come up with the answer. Which the child then checks. In effect, the kid is now the teacher, and you are now the students.

As an adult, it can be helpful to make silly mistakes or make a big show of how hard the problems they made are. Obviously, don’t overdo it. Kids are very smart, and you don’t want them to think you’re making fun of them. But gentle teasing can be fun to work with and will give you some insight into what they understand.

Plus, after a day of being in school where they have the lower hand in the power dynamics with the teacher, it can be wonderful for the kid to feel like they have the upper hand. Especially when the rest of their day involves being corrected, evaluated, or rushed.

You can even ask them to assign you homework for the next day or week. It’s a fun way to make math time feel less “let’s do more worksheets” and more “let’s figure out how to trick my parent.”

VII. What Happens When Kids Create Examples They Can’t Solve

They got you. They asked a problem you couldn’t solve. And when you ask them to solve it, they can’t solve it either! Oops! And maybe a hooray as well!

This will happen all the time because there’s a lot of math that’s either not well taught at the elementary school or not well specified. This is a feature, not a bug. Many well-known problems in professional mathematics are easily stated but remain unsolved.

Going back to our multiplication as a repeated addition example. What if the kid asks you to multiply 2 and 1/2, that is, 2 * 1/2? If you subscribe to the first number being the thing you’re going to add repeatedly, what does it mean to do “1/2” of an addition? It leads to either trying to work around it by introducing new techniques (multiplication is commutative: a * b = b * a; 2 * 1/2 = 1/2 * 2, so it’s two repeated additions of 1/2), or trying to explain multiplication another way (multiplication is later taught as “scaling” up and down, rather than repeated addition”).

When it happens, it leads to great questions like:

• Why does this one work?

• Why doesn’t this one?

• What changed?

These questions help you discuss each of your understandings, and you may even revisit definitions to better understand the mathematical idea. These questions lead to recognizing patterns that stick because they were studied and prodded, not memorized.

VIII. The Bigger Picture: Play as Serious Work

For kids who love math, this form of building understanding almost feels like “play”. The goal isn’t to complete 100 problems before bed; it’s to poke and prod this “thing” to figure out how it actually works.

Playing around through exploration then opens up many fun questions and possibilities.

One exploration that I loved as a kid and helped me memorize the 9’s multiplication table row was *that the “tens” place goes from 0 to 9, and the “ones” place goes from 9 to 0.*

09
18
27
36
45
54
63
72
81
90

What is it about multiplying by 9s, that is, doing repeated additions of 9, that makes this pattern form?

Why is it that multiplying by 8s, that is, doing repeated additions of 8, doesn’t make this pattern form?

08
16
24
32
40
48
56
64
72
80

In the 8’s example, the “tens” places go: 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, while the “ones” place goes “8, 6, 4, 2, 0, 8, 6, 4, 2, 0”.

I won’t ruin it for you by answering it here, but ask your kid to work with you to figure out why this happens. As a hint, think about how the “tens” place and the “ones” place are both used when counting up to 10.

IX. Closing: A Quiet Superpower

When your kid can invent their own examples, they will have a very useful tool for when they are stuck or bored. They will know how to test ideas, come up with examples that might help them solve the problem, and explore why something isn’t working on their own.

Later, they’ll be able to trust themselves when math (or other STEM subjects) textbooks stop holding their hands.

It’s a long process, and it’ll take years to get there, but if you start now, gently, playfully, and without pressure, they’ll get there. And over time, math stops feeling like something that happens *to* them and starts feeling like something they can explore. And as a bonus, your math will also get better!

X. Closing

That’s all for today :) For more Kids Who Love Math treats, check out our archives.

Stay Mathy!

Talk soon,
Sebastian

The PS1 was an unrivaled time for soundtracks that go incredibly hard. Games like Ridge Racer Type 4,  No One Can Stop Mr. Domino, Ape Escape (and basically anything that Soichi Terada ever did), Ace Combat 3 Electrosphere and Ghost in the Shell: Megatech Body all punch well above their weight. Hell, even the Animorphs game kinda goes. In that vein, if you have never listened to the soundtrack to Sheep, Dog 'n' Wolf AKA Looney Tunes Sheep Raider, you should correct that right now because it contains at least one incredible heater.

Sheep, Dog ‘n' Wolf was a bizarre celshaded stealth action puzzler that came out in 2001 starring Ralph Wolf (NOT Wile E. Coyote) and described at the time by Infogrames’ Director of Marketing as “E-Rated Metal Gear.” The soundtrack was done by a musician named Eric Caspar, and though a lot of it feels like boilerplate production music, it does have bangers. 

One track in particular stands out, an ambient DnB track that could easily fit into any racing game from the era. It is unclear what this track is called or if it has a name, although I have seen a few sites like KHInsider and Discogs cite it as “Fronzy Last/Fronzylast_v3.” The track itself appears in the Planet X stage later in the game, which you can see by watching this speedrun attempt.

Anyway, enjoy this oddly specific track that keeps appearing in my YouTube suggested feed. Also check out this extremely of-the-era full page print ad complete with Typewriter font.

Researchers at Anthropic published their findings around how AI assistance impacts the formation of coding skills:

We found that using AI assistance led to a statistically significant decrease in mastery […] Using AI sped up the task slightly, but this didn’t reach the threshold of statistical significance.

Wait, what? Let me read that again:

using AI assistance led to a statistically significant decrease in mastery

Ouch.

Honestly, the entire articles reads like those pieces you find on the internet with titles such as “Study Finds Exercise Is Good for Your Health” or “Being Kind to Others Makes People Happier”.

Here’s another headline for you: Study Finds Doing Hard Things Leads to Mastery.

Cognitive effort—and even getting painfully stuck—is likely important for fostering mastery.

We already know this. Do we really need a study for this?

So what are their recommendations? Here’s one:

Managers should think intentionally about how to deploy AI tools at scale

Lol, yeah that’s gonna happen. You know what’s gonna happen instead? What always happens when organizational pressures and incentives are aligned to deskill workers.

Oh wait, they already came to that conclusion in the article:

Given time constraints and organizational pressures, junior developers or other professionals may rely on AI to complete tasks as fast as possible at the cost of skill development

AI is like a creditor: they give you a bunch of money and don’t talk about the trade-offs, just the fact that you’ll be more “rich” after they get involved.

Or maybe a better analogy is Rumpelstilskin: the promise is gold, but beware the hidden cost might be your first-born child.

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Been thinking a lot about this Ted Chiang quote recently: “I tend to think that most fears about A.I. are best understood as fears about capitalism. And I think that this is actually true of most fears of technology, too.”

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