Wednesday night Yinh came out of the boys’ room and in a hushed tone relayed the 6th grader was feeling down after a bad day. He woke up feeling a bit under the weather (but fine enough to go to school), had a subpar hoops practice, got into a fight with his younger bro (over Halloween masks), his friend was moving to India the next day, and…he was super-stoked about a topic he learned about in Math Academy (interquartile range) but when he told me about it I immediately asked him a question that he didn’t know the answer to deflating whatever good the day had salvaged.

I felt like a total ass. I was 1) out of tune with how the day was treating him and 2) my tone in asking him about the math was out of sync with my message. My intention was curiosity about what he learned but it came out as a challenge or test.

I fumbled. Sometimes you clench the ball too tight.

I have high expectations of effort in 2 categories for the kids:

a) Areas they earnestly want to improve in.

They play soccer for fun but care about getting better at basketball. They create those distinctions. I remind them that getting better doesn’t happen without practice so if you want X and your actions betray you, I’m going to question what you actually want. I try to balance sensibility given their ages without watering down the truth of what being competitive means. Overall, I let their own motivation dictate how hardcore to be about this. I’ll come back to what I mean by this.

b) Things where they are too young to understand the option value of.

I’ll just caveat this with both boys have obvious academic aptitude. If they didn’t, I’d adapt my thinking to what they need as individuals. But since they are capable, I expect them to crush school. You’ll have more choices down the line if you can get yourself onto the equivalent of the honors track. I’m not suggesting this is deterministic — but there’s option value and the easier school work is to a child the cheaper the option premium. Orienting your drive in the direction of your innate strengths is a generally good principle (and possibly underappreciated because the second derivative of that function is opaque but likely convex with respect to success and satisfaction in life).

The flipside of the high expectation is what kids are capable of. I don’t mean this in a purely literal sense. You can search YouTube and find children performing extraordinary feats. Those examples round down to “rare exceptions”. Generally speaking, fulfilling high expectations requires sustained commitment. And sustained commitment relies on motivation. Motivation is a puzzle. What makes some kids (or anyone for the matter) obsessed and others float in the wind? I don’t know those answers but I’ve recently pieced together a partial explanation that bears keeping in mind especially in moments of frustration.

I’ll start with the moments of frustration. Watching my kids waste their time on YouTube Shorts or watching Brawlstars influencers. Nails on chalkboard. Go collect some bugs or build a bike ramp, just turn this crap off. [My nieces and nephews call me Uncle “Nonsense” because I always ask “what nonsense are you watching now?” It’s past the point of meme — my sister’s kids sent me a photo of themselves eating bubble gum flavored ice cream because they know I call it a “nonsense” flavor.]

Sometimes I just wanna yell “go be productive”. But I don’t because that’s not what I really mean — that would be me projecting. That’s my demon, and I don’t want my kids to meet him.

I’ve thought about this a lot. This urge towards productivity. Consider this Paul Graham excerpt I quote in Impedance Mismatch:

What I’ve learned since I was a kid is how to work toward goals that are neither clearly defined nor externally imposed. You’ll probably have to learn both if you want to do really great things.

The most basic level of which is simply to feel you should be working without anyone telling you to. Now, when I’m not working hard, alarm bells go off. I can’t be sure I’m getting anywhere when I’m working hard, but I can be sure I’m getting nowhere when I’m not, and it feels awful.

There wasn’t a single point when I learned this. Like most little kids, I enjoyed the feeling of achievement when I learned or did something new. As I grew older, this morphed into a feeling of disgust when I wasn’t achieving anything. The one precisely dateable landmark I have is when I stopped watching TV, at age 13. Several people I’ve talked to remember getting serious about work around this age. When I asked Patrick Collison when he started to find idleness distasteful, he said:

“I think around age 13 or 14. I have a clear memory from around then of sitting in the sitting room, staring outside, and wondering why I was wasting my summer holiday.”

Perhaps something changes at adolescence. That would make sense.

This line resonates “a feeling of disgust when I’m not achieving anything”. But while that disgust is useful gasoline it has fossil fuel level byproducts and externalities. It’s a tradeoff not an unalloyed virtue. So “go be productive” misrepresents what I really want (which stated simply is to be active in mind or body not numbed into zombie mode. A rule that we enforce poorly is “if you are going to watch YouTube you must search for what you want. You are not allowed to let the algo create the menu or serve the next video”).

But the main reason I restrain the “be productive” message is that it’s inappropriate at their age. Even these psychos that Paul Graham refers to didn’t feel the urge to be productive until 13 or 14. He wonders:

Perhaps something changes at adolescence. That would make sense.

I was listening to Dr. David Yeager on the Huberman podcast (the first time I’ve ever listened to that show believe it or not. Thanks Justin for the heads up, I really enjoyed Yeager’s insights.) and he talks about how puberty coincides with the first time kids really onboard status anxiety. It’s when they become deeply aware of a pecking order or being “popular”. The social dynamics are complex and stressful. It is the first time they start to think of their value in terms of what they can do, or if they are pretty, etc.

It’s the age when kids start bands. We know why they do that. And it’s not to “be productive.” But achievement is a byproduct. Ambition is a natural solution to status anxiety.

Meanwhile elementary school kids are beautifully unaware. Some kids wanna score a hattrick and some are picking daisies. But there’s room for everyone. Their lives are pre-achievement. Open exploration. No judgement. It’s a small window.

How dare I shorten it?

For their whole lives, others are gonna size them up. These boys’ thoughts will whisper “what are you bringing to the table?”. They don’t need extra pressure from me. They don’t need another form of love that comes with strings attached. The world’s love comes with enough conditions.

They’ll need guidance. They’ll be things they can’t forsee because they are unforseeable. But there will also be things they can’t foresee because they’re inexperienced. We can help with that. There will be useful resources and opportunities they didn’t even know were possibilities. We can help with that. There will be questions of reality. Like what it takes to learn. What it means to have integrity. How to model decision-making. We can help with all that.

But we are not here to create pressure. Or fear. I want them to face fear. I don’t want to generate it. The world will do that for free.

If I killed the enthusiasm for what Zak learned in math, I’m doing something wrong. I’m grateful Yinh told me, because like him, I’m learning and we will get better. Puberty is gonna be a trip.

The day after the bad one was uplifting.

I was traveling. Yinh texted me that the younger one was struggling with Math Academy and extra frustrated because the older one who he idolizes crushed his quiz.

But…the older one coached and brought him around and excited again.

Like any set of brothers they can be 2 cats in a bag (they were the night before), but it makes it that much sweeter when they get each other’s backs. This particular instance made me especially proud because I’m relentlessly on their ass about being patient (go slow to go fast) in learning and with each other so any indication that they hear me is a win.

Yinh was blown away at how Zak was so calm and methodical breaking down Max’s difficulty into manageable steps. I’ve had his friends say the same thing about how he helps them. Being patient and helpful because everyone learns at a different pace is an explicit and modeled value at home. But a lot of what you want to instill doesn’t transmit so it’s worth celebrating the small win.

Yinh captured the moment when cooler heads prevailed:

If you know about education and want to write about it, I’m just letting you know you should go for it. There’s an audience. That Principles of Learning jam I put out Wednesday got me a lot of inbound. Which I never would have predicted. It was an exercise in organizing my thoughts on a topic I find inexhaustibly interesting — learning faster and more efficiently. If I wasn’t loving building moontower.ai, I’d be pulling on that thread harder.

[If you are operating at the intersection of machine learning and human learning and want to connect hit me up. I have a close friend at the tip of that spear who I’d be asking for a job if I wasn’t being feral.]

Anyway, parents wanna know about this stuff based on my email inbound. Just sayin.

In that vein, I’m going to share a response I sent to someone who recently enrolled their 7th grader in Math Academy. The parent is concerned that by going faster (even in a selective private school that the child is already in) that boredom could become an issue.

I’ll be honest. I hadn’t considered that angle. But it’s a totally legitimate one considering that MA’s Justin Skycak addresses it directly in:

The Greatest Educational Life Hack: Learning Math Ahead of Time (5 min read)

Justin frames it in terms of risk and reward. That’s a valid approach. But for some kids, it’s still too conservative. Pre-learning is the ultimate option on having doors open that simply won’t any other way because you are compressing time.

In one of Paul Graham’s best essays, How To Make Wealth, he talks about the decision to join a startup in such terms. In Startup = Growth it’s spelled out — “raising money lets you choose your growth rate”. If the kid enjoys going faster, let ‘er rip, they aren’t aware of the option your giving them but they might thank you later.

[I wish I could make copies of myself to do all the stuff I want to do but at the same time, I consciously don’t want to burn the candle at both ends right now. There are doors that are closed because I didn’t go faster when the cost of going faster from a family POV was lower. But I wasn’t inspired to go faster then. Anyway, I’m not writing for therapy here, but if I’m projecting my own illusions you should at least have the disclaimer. ]

I suspect the downside is not especially sensitive to pre-learning anyway. Even if a motivated or math-inclined kid didn’t pre-learn they’re gonna be bored. The teacher will introduce a topic, the kid will get it immediately and still need to wait for others to catch up.

Something I tell my kids, and I don’t express it any type of subversive tone but just as a matter of fact…you can’t let the pace of school dictate what you think is a normal pace. School is built for everyone, but if you are good at sports you wouldn’t expect to move at the pace of the average kid in your class. You’d have a coach and play “up” or on a club team.

The subconscious message we pick up everywhere, especially in school, is that there’s a correct pace. But the error bars around that pace are massive. We know our children so deferring to what’s best for the average when we have specific info (whether they need more help or more stimulation) is wasting info. As always, it’s sound decision-making hygiene to consider the outside view but adjust it for your circumstances.

One last caveat — if I found out my bored kid was working on his fantasy football model underneath this textbook at school because he was bored my reflex would be “Sweet, show me what you got so far. But also you better get an A+.”

Friends,

In September I wrote about signing my 6th grader and myself up for courses on mathacademy.com. It’s been a month and we’re addicted and competing with each other to level up in our respective leagues by gaining XP. A unit of XP approximates “one minute of focused effort by a serious but imperfect student”.

[I’ve turned a bunch of readers onto this site just as I was turned on to it by another reader and now I got peeps texting me questions about it or telling me about their kids progress. You love to see it. Random side note — I have a good friend who just moved from my neighborhood to Austin because he’s deep in the education/AI intersection and the weird city is the scenius for education experimentation. I mentioned it to him and let’s just say he knew all about it from different angles. What he told me only got me even more stoked about what mathacademy is doing fwiw.]

In that post, I pasted links to 30 articles that I planned to read by the site’s chief quant Justin Skycak after already doing a fair bit of reading on the blog. I’ve plowed thru the 30 articles and then some, which is still just a fraction of what’s on there.

I’m personally highly interested in the entire topic of using AI to develop talent and learn at rates that were previously unthinkable. I have a large unfinished document with years of insights that I’ve pulled together from various sources that probably won’t see the light of day. For education I’m a big fan of writers like Scott H. Young, Cedric Chin, Matt Bateman, and Freddie deBoer. You can search the substack for all the times I’ve referenced their work and I have plenty more in backlog. I’ve also harped on the degree to which SIG’s education was extremely well-mapped out from a pedagogical point of view. It wasn’t until I heard Todd Simkin explain the educational influences that informed how they taught did I appreciate the extent to which education theory underpinned their methods.

See:

🔗Educational Ideas Inspired By Seymour Papert’s Constructionism (Moontower)
🔗Notes From Todd Simkin On The Knowledge Project (Moontower)
🔗General & Childhood Education Articles (Moontower)

I’m adding Justin to my list of must-reads. After spending most of Sunday with the blog, I’ve synthesized a much more condensed version of Principles of Learning except it’s fully based on Justin’s insights.

I reached out to him when I first discovered the site and made my interest in what he’s doing as plain as possible. I told him:

I think being born on 3rd is to get exposure to someone when you are young who shows just how self imposed our speed limits are.

He hadn’t heard it put that way before.

The wealth you give a youth is self-efficacy. A chance to match their abilities to the needs of communities they find themselves in as they get older. Autonomy and confidence through competence.

When I say “speed limit” I’m not referring to speed only, or even necessarily. It’s more about limits in general. In athletics, you can’t be Lebron no matter what you do. But whatever your limit is, it’s further than you think. It goes without saying that finding your limit requires brutal effort and commitment…but however far that gets you, personalized instruction will get you even further.

If a great teacher/mentor/coach will get you further than the frontier that caps out at a given level of effort, then that role has insane leverage. The very act of pushing through a previously-conceived frontier will increase your motivation and effort as you see what’s possible.

There was a Washington Post article several years ago referring to “America’s most advanced math program” in Pasadena. The kids were crushing the AP Calc BC exam in 8th grade.

Who were the teachers?

The founders of mathacademy.com

The Math Academy began as a tutoring program run by husband-and-wife duo Jason and Sandy Roberts before being formally adopted into the PUSD curriculum in 2017.

Seen narrowly, mathacademy is an AI program that helps you learn math faster.

I think this is to miss what’s coming.

The instruction portion of the personalized coach is being automated.

I’m fairly convinced that we aren’t too far from knowledge not just being democratized (I mean Wikipedia already exists) but structured for delivery on incredibly effective, personalized rails.

Before someone’s reactance reflex gets all buzzy, I don’t mean “education” will be solved by a robot. Instruction is simply one component of education. Motivation, support, guidance, as well the type of story-telling and conversation that relates classroom learning to the world and others is as human-based an activity as a warm hug. But if the price for personalized instruction craters, the secondary effects are going to be large and visible.

At scale, we are going to find out just how many kids are capable of finishing Calc BC by grade 8 or publishing a novel in middle school. We hear those stories now and we dismiss them as “genius” or “privileged”.

But what if a low price for personalized instruction tells us we’re wrong about this? There will always be examples of genius or privilege. But if stories of insane achievement start multiplying amongst broken-English immigrants or other groups who are not advantaged in any way EXCEPT in motivation than you’ll know that the things Justin is writing about turned out to be true.

The price of personalized instruction falling is not a panacea. The cost is only a bottleneck after basic needs like stability and safety are met. But the cost is an active bottleneck for all but the rich once those needs are met. Even expensive schools are only incrementally better on truly personalized instruction (their primary advantage might be the compression of the classroom range to a higher functioning average but that’s not the same as personalized instruction so much as a release from tolerating a small number of disproportionally disruptive students).

I’m fascinated by mathacademy because of what it telegraphs — a future of cheap personalized instruction. I’m not picturing slicker edtech apps here. This is a glimpse of something different.

Libraries were free. The internet is free, convenient, and wider reaching. Sal Khan is a prophet who built on its rails. Well, the tracks are being upgraded.

The trains are going to go faster.

The full document can be found as a moontower guide:

🎓Principles of Learning Fast

It’s a living document that I’ll add to over time.

This condensed version hits most of the highlights based on what I’ve read so far. I pontificate like a blowhard at the end a bit more.

Maximizing the Learning Rate: A Neuroscience-Informed Approach to Education

The objective function of educational strategy outlined below is to maximize the learning rate—helping students acquire and retain knowledge more effectively. There are certainly great programs for independent learning out there but the objective in this discussion is to leverage technology and cog sci to progress through levels of mastery faster.

What Neuroscience Has Taught Us About the Brain

These are some of the most durable findings in cognitive science.

• Neuroplasticity: The brain’s ability to rewire itself through new experiences is one of the most significant findings in neuroscience. Neuroplasticity means that the brain continually adjusts its neural connections in response to new learning. This allows learners to develop new skills and adapt to challenges. Methods like deliberate practice particularly the “effortful repetition” and “successive refinement” aspects repeatedly strengthen neural pathways until tasks become second nature (Talent Development vs Traditional Schooling).
• Dopamine and Motivation: Neuroscience has shown that dopamine, a neurotransmitter, plays a critical role in motivation and reward-based learning. When learners experience success, dopamine is released, reinforcing the behavior and encouraging continued effort. This makes motivation a crucial component of the learning process, as it directly influences how willing a learner is to persevere through challenges.
• Working Memory and Its Limitations: The brain’s working memory, or the ability to hold and manipulate information temporarily, is limited. Overloading this system can impede learning, as the brain can only focus on a few pieces of information at once. Techniques like chunking—breaking down complex tasks into smaller, more manageable units—can help mitigate this overload (When Should You Do Math in Your Head vs Writing It Out on Paper?).
• The Science of Forgetting: One of the most critical insights from cognitive psychology is the concept of forgetting curves. The theory, which dates back to Hermann Ebbinghaus’s pioneering research, shows that learners forget newly acquired information rapidly unless there is some form of reinforcement. The brain’s natural tendency to forget is often visualized in a forgetting curve, which steeply declines in the hours or days after learning.

  
  Forgetting Curves and Memory Decay: Ebbinghaus’s forgetting curve demonstrates that without review or rehearsal, retention of new knowledge drops quickly over time. However, the rate of forgetting slows down when learners engage in retrieval practice and spaced repetition—both of which can flatten the curve, leading to more durable retention.

  Spaced Repetition Leads to Automaticity: Over time, repeated retrieval practice pushes learners toward automaticity—the ability to recall information effortlessly. Once information is retrieved enough times across spaced intervals, it becomes deeply embedded in long-term memory. Efficiency is achieved through repeated activation and myelination – a process where neural pathways are coated with a substance called myelin, increasing the speed and efficiency of signal transmission.

  This is the key loop:

  Retrieval practice > Automaticity > Reduced demand on working memory >

  The learner frees up cognitive resources for more complex tasks, facilitating better problem-solving and higher-order thinking.

  “Automaticity frees up cognitive resources that would otherwise be consumed by basic recall tasks, allowing for higher-order cognitive tasks to take place in the working memory.”

Implications for Implementation

The implications of neuroscience and research on forgetting curves for learning are vast. Here’s how these insights translate into effective learning strategies:

• Retrieval Practice and Minimizing Forgetting: The act of retrieving information from memory, rather than passively reviewing material, significantly boosts retention. Each successful retrieval attempt strengthens neural pathways and makes the knowledge more durable. As learners engage in retrieval, they disrupt the forgetting curve and prolong the retention of knowledge (Which Cognitive Psychology Findings Are Solid, That Can Be Used to Help Students Learn Better?).
• Spaced Repetition for Long-Term Retention: By structuring review sessions at increasingly spaced intervals, learners allow time for memory consolidation. This reduces the steep decline of the forgetting curve, especially in the early stages of learning. Over time, the intervals between repetitions can be extended without significant loss in retention, enabling efficient long-term learning. The use of spaced repetition systems (SRS) has demonstrated significant improvements in student performance (Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures).

Common Misconceptions in Learning

It’s easy to fall into widely accepted beliefs about how people learn, but research has debunked many of these ideas. Here are a few myths that might surprise you:

• Learning Styles: Contrary to popular belief, the idea that individuals have specific “learning styles” (e.g., visual, auditory, kinesthetic) and that teaching should be tailored to these styles is unsupported by research. While students may have preferences, these preferences do not significantly improve learning outcomes. Instead, using varied teaching methods that engage multiple senses enhances learning for all students (Why is the EdTech Industry So Damn Soft?). Veritasium has also called this the “biggest myth in education”.
• The Myth of Productive Struggle: While allowing learners to struggle through difficult problems might seem beneficial, research has shown that this is often counterproductive, particularly for novices. Without proper guidance, prolonged struggle leads to frustration and disengagement. Scaffolding and explicit instruction provide the necessary support to avoid cognitive overload and enable meaningful progress (What’s the Best Way to Teach Math: Explicit Instruction or Less Guided Learning?).
• Discovery Learning vs. Direct Instruction: The idea that students should learn concepts through self-discovery has been largely debunked, especially for beginners. Direct instruction, which provides clear guidance and support, has proven far more effective in most learning scenarios. Discovery learning works well for experts but can leave novices overwhelmed and unproductive, a paradoxical finding known as the “expertise reversal effect”. (The Pedagogically Optimal Way to Learn Math).
• The Illusion of Comprehension: Learners often mistake familiarity with material for true understanding—a phenomenon known as the illusion of comprehension. Just because something feels familiar doesn’t mean the learner can apply it effectively. Combatting this requires practices like retrieval practice and interleaving, which force deeper engagement with the material (Which Cognitive Psychology Findings Are Solid, That Can Be Used to Help Students Learn Better?).

What Pedagogy Research Has Taught Us

Pedagogy research provides practical strategies that align with neuroscience insights, helping us understand how to optimize learning environments:

• Deliberate Practice: One of the most well-established findings in educational research is the importance of deliberate practice. Unlike passive or rote learning, deliberate practice focuses on honing specific skills through effortful repetition and immediate feedback. This approach helps students achieve automaticity, where foundational skills become second nature and free up cognitive resources for more complex problem-solving. This is why “deliberate practice” is regarded as the most effective training technique across talent domains (The Pedagogically Optimal Way to Learn Math).
• Worked Examples to Reduce Cognitive Load: Especially in subjects like mathematics, worked examples are invaluable for novice learners. By showing step-by-step problem-solving processes, worked examples reduce cognitive load, allowing learners to focus on understanding the process rather than inventing solutions. This strategy is effective in reducing overwhelm, a key barrier to learning (Which Cognitive Psychology Findings Are Solid, That Can Be Used to Help Students Learn Better?).
• Active Learning for Deeper Understanding: Research consistently shows that active learning—engaging students in activities like problem-solving, discussion, and teaching others—leads to better retention and understanding than passive learning methods like lectures. However, this active engagement must be paired with direct instruction, especially for novices, to prevent cognitive overload (Why is the EdTech Industry So Damn Soft?).
• Interleaving Practice: Interleaving, or mixing different topics or skills within a study session, forces the brain to continually retrieve and apply information, strengthening neural connections. While it may feel harder for learners, this desirable difficulty improves long-term retention and the ability to transfer knowledge to new contexts (Which Cognitive Psychology Findings Are Solid, That Can Be Used to Help Students Learn Better?).

Connecting It All: The Flywheel of Competence, Confidence, and Motivation

When neuroscience and pedagogy principles are applied in tandem, they create a reinforcing cycle that propels students toward continuous growth and mastery:

• Competence: Effective learning techniques, such as deliberate practice and retrieval practice, build competence. As learners master fundamental skills, they achieve automaticity, allowing them to perform basic tasks effortlessly, freeing up mental resources for tackling more advanced problems (Automaticity for Cognitive Efficiency).
• Confidence: With growing competence comes confidence. When learners see themselves succeeding—whether it’s mastering a math concept or improving in a skill—they are more likely to tackle new challenges with a positive mindset. This confidence feeds into their willingness to engage with difficult tasks (Recreational Mathematics: Why Focus on Projects Over Puzzles).
• Motivation: Confidence breeds motivation. As students become more confident in their abilities, they are more driven to continue learning. This motivation reinforces their engagement in deliberate practice, completing the flywheel and leading to greater competence over time. Accountability, whether through structured learning programs or paid educational platforms, also plays a role in keeping learners committed to their goals.

 

Key points and clarifications from select posts

 

• Recreational Mathematics: Why Focus on Projects Over Puzzles (2 min read)

  There’s only so much fun you can have trying to follow another person’s footsteps to arrive at a known solution. There’s only so much confidence you can build from fighting against a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way.
  

• The Situation with AI in STEM Education (11 min read)

  The major limitation of LLMs in education is their reliance on student-initiated questions. Effective teachers don’t simply answer questions; they guide students through a structured learning process, scaffolding information and addressing knowledge gaps. LLMs, like ChatGPT, primarily respond to prompts, lacking the pedagogical ability to anticipate a student’s needs or direct their learning path.
  

  The promise of AI in education overemphasizes the role of “explanation”. Scaffolding and learning management are equally important. He cautions against prioritizing AI’s ability to engage in conversational dialogue over its capacity to deliver well-structured, personalized learning experiences.

• Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures (22 min read)

  Theoretical Maximum Learning Efficiency In physics, nothing can travel faster than the speed of light. It is the theoretical maximum speed that any physical object can attain. A universal constant. In the context of spaced repetition, there is an analogous concept: theoretical maximum learning efficiency which posits that in a perfectly encompassed body of knowledge, it’s theoretically possible to achieve mastery through continuously learning new, progressively advanced topics without ever explicitly reviewing old material. This idea, while theoretical, underscores the power of leveraging knowledge interconnectedness.

  Importance of Encompassing Graphs (as opposed to prerequisite graphs) : Unlike prerequisite graphs which show learning dependencies, encompassing graphs map how practicing advanced topics reinforces prior knowledge. Constructing these graphs is a laborious, manual process requiring significant domain expertise, highlighting the importance of expert-designed learning pathways.

• Talent Development vs Traditional Schooling (12 min read)

  
  Orthogonality of Talent Development and Schooling: Traditional schooling, with its age-based grouping and standardized curricula, often fails to effectively nurture talent. This stark contrast emphasizes the need for specialized approaches outside the traditional classroom setting. Talent development is not only different from schooling, but in many cases completely orthogonal to schooling: “For one portion of our sample, talent development and schooling were almost two separate spheres of their life. … Usually the student made the adjustments, resolving the conflict by doing all that was a part of schooling and then finding the additional time, energy, and resources for talent development. … Mathematicians found and worked through special books and engaged in special projects and programs outside of school. Sometimes the schools or particular teachers made minor adjustments to dissipate the conflict. Mathematicians were sometimes excused from a class they were too advanced for and allowed to work on their own in the library. Sometimes they were accelerated one grade as a concession to their outside learning. … Whether the individual or the school made these adjustments, it was clear that these adjustments minimized conflict but did little to assist in talent development. The individual was able to work at both schooling and talent development, although with minimum interaction. … Talent development and schooling were isolated from one another. Schooling did not assist in talent development, but in these instances it did not interfere with talent development.”

  Individualized Instruction in Talent Development: Unlike the group-focused approach of schools, talent development thrives on personalized instruction, tailoring learning tasks to individual needs and ensuring mastery before moving on. This distinction underscores the importance of personalized learning pathways in maximizing potential.

 

A useful reminder from You Will Never Achieve Your Goals Unless You Transform Yourself Into a Person Who is Capable of Achieving Them:

You want to do something that sets you apart? You’re going to have to work harder than most.

Actually, let’s re-print the entire post:

The #1 confusion that I hear when people ask me about math, ML/AI, startups, etc., is they think there’s a way to achieve outsized success without putting in an outsized amount of work.

You want to do something that sets you apart? You’re going to have to work harder than most. There is no way around it.

You think you can get good at math by watching YouTube videos?

Develop cutting-edge ML/AI by asking ChatGPT to code it up for you?

Put a dent in the universe working 40 hours per week?

If you think any of those things, then you will never achieve your goals because you will never transform yourself into a person who is capable of achieving them.

And guess what? It’s not enough to simply work hard.

To achieve outsized success, it’s critical to not only put in enough time/effort, but also to work productively.

You have to work hard AND work smart.

And furthermore, work in a direction where you have some competitive advantage (or, at least, you’re not at a disadvantage).

Part of this work involves engaging in activities that maximize the likelihood of you getting some lucky breaks.

You have to work to maximize your luck surface area.

I have friend from my college days that always used to say ridiculous catch-phrases with his personal mix of cheekiness and seriousness. Like if you didn’t for the 5th set of squats he’d just dismiss you with “I guess I’ll just take you off my list of successful people for today”.

It was a phrase a bunch of us still parrot to this day in a joking way. Oh you didn’t moisturize after your shower? Off the list. Didn’t drink your coffee black? Just go to bed now and try again tomorrow.

The grindset earns its parody. But we don’t mock mediocrity because it’s suffered enough. We apologize for it readily but rarely our own. But we might bend over backward to apologize for others’ mediocrity. The reasons can range from genuine concern to signaling to grift. I don’t want to paint the motivation with a broad brush. Regardless of the motivation, excessive sympathizing without actually getting your hands dirty is just patronizing. If you care, then help.

A lot of education problems are not education problems so much as just family or stability problems. That can range from abuse to just having parents that are consistently crap decision-makers. Public school, as maligned as it often is, can be a refuge. A chance to get inspired by a great teacher or an authority figure whose influence counteracts the brainworms that might come from home life.

If you come from stability, this is hard to see and might even sound offensive. But kids are not possessions. It’s why we have laws to protect them but you’re free to stick your silverware in the microwave if you want. Where the lines of interference lie are legal matters (and by extension political — it comes with living in a representative government…shrug).

My belief is that education experimentation is good because making progress on instruction is a high-leverage activity. The fact that it is not evenly distributed because the spread is rate-limited by non-educational obstacles is not a reasonable objection to innovation. Of course, you can be skeptical or bearish. Hell, the education world is mass of twisted hot metal. But resignation to extending an uninspiring status quo or accepting low standards is anything but progressive.

[There’s probably some smart-sounding argument that goes “you can’t fix education because you can’t fix society” to which you can only wonder, then what are we even doing here?]

When it comes to teaching and coaching there’s a delicate balance of toughness and love. It’s like parenting to be honest. It’s hard because it often hurts plus its mired in bureaucracy.

On the supply side, great teachers might be scarce because the right mix of tough but fair + smart is just scarce in the population and now we have to choose a subset of those people. If you want great teachers you’re asking for a legion of special individuals. Attracting special individuals requires a special effort to recruit, train, support, and enable. We get what we are willing to pay for. I don’t have a full understanding of the frictions that make our spend inefficient, but addressing them is independent of trying to make inroads with technologies that improve instructional efficiency.

🍰Justin has plenty of criticism on technology by the way: Why is the EdTech Industry So Damn Soft? (11 min read)

If you don’t see technology as being more than incrementally useful (at least on a longer-time scale) then you’ve given up.

Because we aren’t going to get much better without it.

(Collectively at least. The resourceful are going to have robot tutors if they can’t afford human ones. When you get down to it, it’s your move either way.)

I hate everything about this article:

How Rich (or Not) Do You Have to Be to Get Into the Ivy League? (NYMag)

Just to pick one example — the “recruit to reject” tactic where elite colleges encourage applicants unlikely to be admitted to apply to pad selectivity numbers. This practice is part of broader slate of strategies that create a more competitive admissions environment.

It just feels like colleges are overplaying their hands in a world where gate-keeping is becoming harder. Their cocktail of deadweight loss and hubris goes down harsh and burps up schadenfreude.

I enjoyed Byrne Hobart’s discussion of education in this podcast interview:

Antitrust, the Future of Competition, and Harvard (The Riff)

2 delightful excerpts:

Say more about what you expect your kids to do. What do you predict for their higher education, “your kids” being a stand-in for high ambition smart, parents of kids.

College might not be an option. However, I would like it to be more of a backup plan for them. If they haven’t found a worthwhile use of their time or developed meaningful talents, despite having nearly two decades to do so, then yes, they can and should consider college. They should aim for a good school, although if you extrapolate, they’ll be borrowing significant amounts of money, it’ll be incredibly expensive, and who knows when the school will all burn down during a protest or something.

Assuming things revert a little bit, going to a good school is still a good way to meet a high concentration of very smart people and those who have dedicated their lives to understanding something that you also find interesting. You can have information-dense conversations with world-leading experts on a topic you find interesting. They can provide insights on ideas that you suggest, tell you who came up with an idea 20 years ago, what the rebuttals were, who tried it, how it worked, how it didn’t, etc. This can be very valuable.

But these people also have email addresses. If you find an academic paper on a topic you like, read it carefully, and provide constructive criticism, you are probably at the 99th percentile for a young person engaging with their work, inclusive of the people who are getting graded in their class. You don’t want to abuse their time, but in many cases, these people are either part of the public sector or their existence is government-subsidized. They are public servants, and their job is to be helpful. And if you can actually engage with their material in a meaningful way, then it’s valuable to them too. This is what they wanted to do with their lives.

You can 80/20 a lot of the benefits of college without paying huge amounts in tuition by reading things by people you like, writing them thoughtful emails about what you read, and sending them the thoughtful thing that you wrote in response to their work. You can get some fraction of the college experience without having to live in a dorm or something similar.

I think it’s a bad idea to over-optimize for some specific hypothetical future plan. I do emphasize to them that schooling is optional, and the better they’re doing in life, the more it is just a purely optional thing. Some of them will likely go to school, hopefully not all of them will finish, and hopefully, it will be because the opportunity cost of getting that degree from Stanford, MIT, Harvard, or whatever, is simply too high compared to their other slightly better options.

I understand the YC analogy [as a substitute for college] as it relates to startups. But what do you think is the equivalent for lawyers and doctors? Another way to put it is, imagine your kids going to college in 10 years. Are they going to go to college? If they are, what’s an alternative look like? How do you think the ecosystem is going to look in a decade? When is it going to materially change? What could that look like?

There’s not going to be a YC for doctors or lawyers. Part of the reason for that is those are both cartel businesses where they want to limit entry into the business and they can. Especially the lawyers, their whole job is parsing rules and writing rules, so they’re really good at coming up with rules that are advantageous to them.

It’s hard to replace those, but there’s room to try to route around them. In general, it’s good life advice to push back on 18-year-olds who have an entire plan for how they will get their dream job and it can’t happen until they’re 25. You have a lot of life ahead of you at that point and a lot of changes you will go through. It’s hard to reverse course. Getting half of an MD or JD is very expensive and not incredibly valuable. [Kris: I’ve had the same thought. Premature optimization is a recipe for waste and sadness for humans. It also discards the timeless, nature-tested decision-making algorithm of evolution —> optionality]

A lot of those jobs are easy to glamorize from the outside, and they’re less fun on the inside. Doctors and lawyers have high divorce rates, high substance abuse rates, and a lot of people try to get out of those professions. They’re great for some people, but it’s easy to see a cultural narrative that is out of date when you saw it. By the time you’re actually doing that thing, it’s even more out of date.

[Kris: notice the market principle embedded in Byrne’s comment — the bullwhip effect. There’s a lag between heightened job demand and the skill to supply it. I don’t want anyone to get bent out of shape, but here’s a generalization (that means exactly that — it may not apply to any specific instance) — tiger parents are risk averse. They project their risk aversion on their kids. Either by command-and-control or Ouija board nudges, they direct their children towards legible paths that are currently in vogue. It’s probably software engineering more so than lawyers these days but you know what I mean. Legible paths get overbid. And competition is a form of hormesis — it’s all about the dose. If you take too large a dose relative to the reward, especially if the reward is only externally motivating, you will find yourself in the latest new-age rehab — slurping health gurus, chasing real estate syndications for passive income or heaven forbid writing a substack. Also these are best outcomes. The dark side is having the narrow path steal whatever’s left of your vitality. You know, that youthful perspective that still couldn't read a line chart.]

In the piece that I wrote on education financing, I have a footnote on this. Sometimes the cultural norms for a profession get established because there was some group that was very idiosyncratic and someone wrote a book about them or made a movie about them. That solidified that particular group’s norms, and they became the norms for anyone with that job.

A lot of people who are law students or aspiring law students watch the movie, The Paper Chase, where it’s all about a guy’s complicated relationship with his brutally demanding contracts professor. The movie is about 50 years old, so a lot of the professors have also seen it. They sort of know what the great law school professor is supposed to be like, and what the great student is supposed to be like. Everyone sort of plays these roles.

This was also revealed by law enforcement wiretaps of mafiosos that they actually started using cliches from The Godfather after the movie came out, and they all loved the movie. So the movie’s stereotypes became more accurate because it got more widely accepted.

But if you’re entering some career because you’ve heard about it in some context where you know exactly what it’s going to be like, and it sounds incredibly glamorous, you just have to know that to the extent that it actually meets your expectations, it’s because everyone is LARPing. They read the same thing you read, or they watched the same thing you watched, and they’re just trying to bring it back.

This is probably a good reason to not work in the White House. Everyone just wants to be pretending it’s a West Wing episode, and it’s just going to feel so weird and dated and retro and 90s. If you see someone in a meeting reading the Financial Times in the West Wing, you’re just obligated to give them a swirly or something, tell them to knock it off. It was just a TV show. The writing was good, but it shouldn’t have changed your life.

From My Actual Life

The bullwhip effect in Byrne’s take is humbling because there is no solution to “safety” if safety requires sacrificing your life for it. Instead of checking a bunch of boxes, flourishing requires a fuzzy mix of imagination, persistence, self-awareness, and perhaps in no small way, savvy. Sheep get shorn eventually.

Once of the glaring examples of this bullwhip thing has been my read on how friends’ feel about their occupations. Doctors and lawyers, what we were told to chase as 80s kids, don’t exactly seem thrilled about their careers. The happiest ones feel like they make a difference. And they do. That matters and it’s awesome. But they don’t think it’s a good deal.

On the other hand, my contemporaries in tech and finance, while having their share of complaints, as a cohort think it’s a pretty damn good deal. Sure it can be dystopian and soul-sucking. But nobody wants to switch spots with a doctor.

I do have a HS friend who was an I-banking analysts for a year and decided the job was full of the worst people. The bosses are miserable, but…rich? The middle-class mind could not comprehend this. He went back to school and became an oncologist.

There’s always a couple older dogs in med school so these stories happen.

In contrast, the list of doctors and dentists I am at least one degree separated from who pulled the chute and went into RE, business, or franchises is long. Nobody leaves trading to buy a McDonalds. It’s hard to think of a crappier job to buy which makes me think the bar of “what’s a satisfactory way to spend my day” must get real low in some branches of medicine.

Then the debt, extra years of schooling…I don’t need to explain, you have friends and family too. You see it.

I don’t understand the underlying forces that led to this but I can see where we are now. How much of it is bullwhip? Growing up, nobody said become a plumber or electrician. Today, you called in a favor to get one to swing by the house for less than $150/hr.

Prestige has a benefit. And a cost. The market is looking for a clearing price every time someone fills out an application for trade school or a school school.

One day in the future, a chip cluster will re-write this essay and “software developer” will replace “doctor” and maybe “doctor” will replace “plumber”. Maybe deciding what we do should have less to do with how far in the future these replacements will happen but on something else. A few months ago I gave a talk at Cal to the quant trading club. It had some broader relevance on the topic of careers. (See Make Yourself Future Proof)

Parents push smart but innocently naive teens towards highly competitive fields because they think the path is safer.

Safer for the parents’ egos maybe.

Further learning:

Marc Andreesen ranted on education almost 5 years ago. I published my notes on it back then.

Notes from Marc Andreessen on Education (Moontower)

It remains highly relevant. Here’s a few excerpts:

The value proposition of university for people in “show your work” fields is changing.

One of the most basic revelations the internet has surfaced is the different nature of professions.

Internet has made the largest difference in “show your work” professions: occupations where it is valid and easy to demonstrate your value online. For example, coding, design, music, art, game dev, animation. Open source projects and writing, democratized, pure examples of “show your work” fields.

From an employer’s point of view conscientiousness is a proxy for being a good employee. But this can be circumvented by just showing your work online. This erases the value of a degree that derives from employer demand.

GitHub has like an internal ranking and rating system for software code, and for programmers. So you can actually build an actual professional reputation as a software developer on GitHub without ever actually being face to face with another human being. People all over the world today who were basically taken advantage of this to be able to basically build these incredible track records as a software developer and make themselves more employable. Employers like my venture firm. We recommend that our employers spend as much time on GitHub looking for good programmers as they do on LinkedIn, or going to college fairs.

YouTube, blogs, Figma for design all play a similar role as GitHub does for software developers. He tells the story of South Park as an early example of a viral video that was able to spread organically through a distributed technology. The show born from Matt Parker and Trey Stone’s irreverent holiday card which made its rounds as a downloadable Quicktime vid!

“If you can go to college, go to college”

• Even jobs that probably shouldn’t require degrees require them.

I think it’s actually quite dangerous to give somebody, somebody as an individual the advice, don’t go to college, like in the current system that we have that’s basically saying don’t prove that you’re smart don’t prove that you’re industrious, and conscientious and then basically be prepared to settle for fundamentally lower income for the rest of your life.

• Understand the proposition

Gates and Zuckerberg notwithstanding, if you go to college finish college. Get the piece of paper.

• The 2×2 matrix of what to study and where to study.

The spread of outcomes for technical degrees is not that wide. If you have a technical degree your choice of school matters less. This is exactly the opposite of what you find with liberal arts degrees. Since the output of a liberal arts degree are more subjective or uneven the school issuing the diploma carries more weight.

Possible explanation: in absence of concrete skills, the network from a top school is valuable.

Tips for those in college or considering college

Execute on the opportunity — take the hardest course load you can. Get the skills (obviously get good grades but focus more on getting the skills).

If you are at a sub-tier college taking liberal arts, de-risk by acquiring marketable technical skills.

For those considering alt paths

At this point Marc, still recommends college and acquiring technical skills but if you choose an alt path be aware of the trade-offs. For example, if you choose to do open source work recognize it’s better to make major contributions to one project (as opposed to minor contributions to multiple projects) because that really demonstrates what employers are looking for. Put yourself in the mind those who will be evaluating you years down the road.

Consistent work demonstrates conscientiousness and the nature of the work is an embedded intelligence test.

What should a software developer do? Unquestionably the answer is create an open source project or go become a member of an existing open source project and make successful high quality sustained contributions to that project over time. At this point I think that’s clearly a better credential than getting a computer science degree. I’d hire people like that myself and the great thing now is you can do that from all over the world.

So what matters to Andreesen when they hire or fund someone?

The good news:

They do not care about a degree or GPA or test scores and in fact question if too much conscientiousness means you are too much of a rule-follower.

The tough news:

They measure you by what you have actually done. Building companies requires being able to do things so that is the capacity they are looking for. List of things a founder will need to be able to do:

• Building an actual product that somebody will actually pay for.
• Figuring out a way to actually sell it to them
• Actually collect the money
• Actually service the customer so they actually have a good experience
• Actually tell their story so that anybody will even know that they exist
• Run a finance function so that they don’t lose all the money
• Run a legal function so they don’t get sued all the time
• Actually get others to work with them.

There are many talented people so the way to stand out is to actually demonstrate the ability to build or create.

Steve Martin best career advice ever: Be so good they can’t ignore you.

A friend in one of the chats I’m in revived the practice of sharing wordle scores. I joined in because I’m playing the NYT games again thanks to my kids. The 7-year-old comes into my bed every morning to do the cycle: wordle, mini, connections, letterboxed and tiles. In that order. Kids + routine are like pb & j.

Anyway, one of the younger members in the chat goes “sometimes i feel like it would be nice to have a kid just so i can teach them everything i know.”

Very cute. But then a dad injected reality:

nah, they don’t want to learn anything from their parents

Ding, ding ding.

Here’s my impression. Kids learn from you by osmosis. Modeling, not direct instruction.

I think it has something to do with your lessons being a “you” thing. They don’t feel like the knowledge is theirs. Like they can smell your desperation in wanting them to learn no matter how chill you are about it.

It seems like a brand of what David McRaney calls reactance.

This is a concept that has been studied extensively in the context of clinical therapy.

They would come to the therapist and the therapist would say, ‘Well, you know what’s your real problem is. You should be doing this.’ Or, ‘I don’t know if you’ve noticed, but you don’t do this very much. You should do this.’..All that feels pretty good. They now call that in psychology the “writing reflex”. And, we’ve all felt that whereas someone is saying something and you’re like, ‘Oh, I have the advice for them. I know what to tell that person.’

But, you also have also experienced this other thing that happens, and this seems to be something that’s universal to human beings across all cultures. It’s just something that the brain that we’re issued at birth, it’s something that’s a feature of human thinking, rationality, psychology. Human brains do this. It’s called reactance. In the psychological parlance, they’ll say something along the lines of, you feel motivationally aroused to remove the influence of the attitude object, which just means: ‘You made me feel a feeling I don’t like and I want it to go away. So, I’m going to push you away,’ or ‘I’m going to disengage.’

What is the feeling that’s causing the motivational arousal? It’s the sense that your agency is under threat–your autonomy is under threat. It’s the ‘Unhand me, you fools,’ feeling. You’ve all felt this. If you’ve ever been a teenager or you’ve ever spoken to a teenager, you know what I’m talking about…”You shouldn’t do this. You should study more.”‘ This is good advice that the person when they’re 35 will go, ‘Man, my parents were right about that.’

But, in that moment it’s just the fact that you’re saying, ‘I have a thing in my head that should be in your head and I want it to be in your head.’

And, oddly enough, it’s the want that creates the reactance. The person’s feeling that you have approached them in some way and said, ‘I want you to think, feel, or believe, or act in a certain way that you’re not doing right now,’ and it feels coercive. It feels like they’d come at you and they’re threatening you. They’ve got a knife in their hand, and they’re saying, ‘Walk this way.’ That’s what it feels like.

We just, at a visceral level, will react by saying ‘no thanks’ to that, and we’ll push against it.

Back to the wise dad in the chat. He goes on to say:

you’re better off teaching someone else’s kid

Despite the reactance thing, this also seems true. My kids’ friends are around a lot or I’m shuttling them somewhere so there’s always an interesting spot to drop a fun riddle or word problem into a dull moment especially if the kid is, I don’t know, a “solver” for lack of a better term. (Some kids have the “ugh, I have to use my brain” reflex. Tread lightly…most kids are solvers at some level but you might need to hide the pill in the peanut butter.)

[My 10-year-old likes to needle me by announcing to his friends “Imagine having to live with this all the time” but I’m getting better at dialing in the dose. I think.]

Meanwhile, some kids are just thirsting for enrichment. I have one friend who asked me to point them to resources for their precocious son (the kid is an alien — genius, super athlete, and socially adept…I wish I could share some of the stories). I figured I could teach them about something I know about or just point them to things that might align with their interests. That might be a more productive frame. Just shepherd them towards sources of intellectual nourishment.

I sent this particular kid Ed Thorp’s bio not just because the action arc will provoke but because Ed’s own childhood will be relatable to a boy who sees through his or her surroundings like an x-ray. The Founders’ podcast is also an effective way to introduce a powerful mind to its greatest lever — the belief that the rules are all just made up. The world is malleable. See The Podcast I Listen To With My 9-Year-Old.

While today’s musing is mostly an off-the-cuff reaction to the wordle chat, I also had a related exchange this week with a college student who DM’d me on Twitter. I got permission to share it here. The first part conveniently flatters Moontower so you’ll have to sit through that for a scrubby second. (I added the hyperlinks for this reprint).

Original DM: Hey Kris, Hope you’re well. I’m a junior at [Ivy] and will be interning at SIG (Bala Cynwyd) this summer. I’ve been reading your posts for over a year now, and they really helped me gain intuition around options (and also other math concepts – like thinking in log space and Jensen’s inequality). I recently joined this mentoring program called Big Brothers Big Sisters, and I got matched with a Little from a middle school in [city]. He really enjoys sports and board games. Today, we played some games from the book Math Games with Bad Drawings (which I found from your blog) – it was really fun! I was wondering if you had any recommendations for fun 2-person activities, which could also teach useful concepts like decision-making and strategic thinking

My response: First glad the site has been useful for you. Cool that you are doing that program. My general recommendation is to just find interesting scenarios to analyze in whatever the kid is interested in. Tree (ie binomial) thinking can be applied anywhere in decision land. For example, whether to punt or go for it if it’s a football fan.

I recently taught some kids about the idea of volatility but in a light way. We were talking about daylight savings time and I told them it actually causes accidents. And I just socratically asked them why that might be true…(people rushing to work, sleeping in, not sleeping enough, etc). But I asked them how they would know that DST was the culprit? And where I’m leading them to is “What does it mean for there to be a number of accidents that exceeds the norm by an amount that isn’t noise”…ie statistical significance.

But I don’t burden them with these words. I just have them propose a guess for the number of typical accidents in a day in say their county or state. Then I make up a fake data set and have them compute the mean and the mean absolute deviation. This is basically what our minds do automatically when deciding if something is abnormal but here we just lay the thinking out step by step. And then maybe on our fake data, they compute some MAD of like 50 accidents a day and then we see that there’s 150 on DST day. That’s how you know DST is the culprit.

Generally, I look for learning moments in everyday stuff or their interests. Better to be socratic not pedantic so they can discover the lessons themselves. Then they “own” them.

And then I attached some resources.

This is one I just published and I will add to over time:

A Collection of Math Riddles/Puzzles (Moontower)

which can be found in this wider portal:

🧠Moontower Brain Plug-In

And here’s a nice thing about helping kids that aren’t your own. They appreciate it. Got this in the mail:

Just a reminder. Do stuff. If you can make it cool that’s a bonus. But even corny with enough persistence eventually endears because it’s authentic. It’s all better than apathy.

Try. The world vibrates back.

Some fun math stuff today all inspired by discussions I’ve seen online in the past week.

Another Monty Hall Explanation

I saw a discussion online recently about the Monty Hall Problem. It was a debate about the best way to explain the solution. If you need a refresher on the problem and the solutions I wrote this a while back: The Monty Hall Problem Is More Than A Game.

Many people are familiar with 99 or 1,000,000 doors explanation. I think this explanation is easier to grok after you’ve been given a more probability based explanation. The value of the 99 doors version is it highlights the beating heart of the problem — the host knows where the goats are!

Anyway, the discussion prompted me to draw out a solution I haven’t seen laid out this way before. It uses tree thinking (big surprise right?).

X² and its neighbors

What’s bigger:

20 x 20 or 19 x 21?

And is your answer generally true.

This is the same as asking, what’s bigger:

X² or (x-1)(x+1)?

Somewhere in your mental library, there’s a dusty text probably wedged between the book called “Polynomial” and the volume named “Completing the Square” that will remind you that (x-1)(x+1) = X² – 1

Which means your answer generalizes:

400 > 399

X² > X² – 1

So just knowing what 20² is means you immediately know what 18 x 22 is 396, X²-4.

The sandwiches around “20” continue down the squares:

17x 23 = 400 – 9

16 x 24 = 400 – 16

15 x 25 = 400 – 25

Et cetera

Clever Polling

This is neat:

I posted that I remembered reading about a sampling technique a long time ago for how to poll for things that people wouldn’t answer truthfully (ie “Have you cheated on a test/spouse/taxes?”).

@LoneSands recognized what I was referring to:

The version I saw had you flip a coin twice before answering an agree/disagree question. If first flip was heads then answer honestly; on tails use the 2nd coin flip to decide.

Ah yes, that was it. So how do you use this tree math to infer the “true” response rate assuming people follow the rules?

This is how I solved it….I just used some real numbers.

Suppose n = 1000 and I had 550 agree.

I expect that 250 are “agree” and 250 are “disagree” because of the second coin flip.

That means 300 people actually agreed.

The true agree rate then is 300/(200 disagrees + 300 agrees) = 60%

LoneSands gave the general formula:

True agree rate = 2*(observed agree rate) – .5

That’s friggin’ tight. I like that.

I wanted to bridge the general solution to mine for validation. This is a great opportunity to demonstrate how to use GPT for self-tutoring.

I gave GPT the general formula and then prompted:

I needed to push a bit more because the bridge wasn’t complete. Luckily, GPT never gets tired of my tediousness.

I’ll tell you a secret.

I’m taking Algebra 2 on Khan Academy right now. These notes remind me what has been refreshed or reinforced (I’m about 2/3 finished with the course):

My plan is to re-do all my HS math through Calc BC and then assess what math I want to learn for the first time after that. It’s really different to take math as an adult because:

a) There’s no deadlines so the stress is gone

b) I actually care beyond a test score

c) Re-learning math after you’ve spent 20 years doing math at work (usually through heuristics and shortcuts passed down), you naturally try to draw connections between the math class examples and the problems in your field. For example, deriving the geometric series in Algebra 2 is synergistic with thinking about compounding problems. But the first time I took Algebra 2 I surely could care less about how much total distance a monkey swinging from a branch with a starting arc length of 50 ft who loses 15% per swing covers after 5 swings.

Math scratches the same itch as NY Times puzzle games for me, which is to say I don’t need an instrumental reason to do it. But it’s also a convenient discovery to hear one of the the late Charlie Munger’s big pieces of advice (amongst many amazing bits of advice) to be “pay attention in HS math”.

☮️

Stay Groovy

One of the Substacks I never fail to read is Range Widely by David Epstein, author of Range (my notes on his interview about the book) and The Sports Gene: Inside the Science of Extraordinary Athletic Performance.

David is a journalist by trade. His writing is well-researched. Social science research, especially the kind of pop-sci stuff that climbs the heap to find itself in airport bookstores, should require a “grain of salt” rating (G: “germane”, PG: “possibly garbage”, R: “rumored at best“). David’s process and intellectual demeanor indicate care — he resists the temptation to oversell conclusions.

Personally, I rarely read social science books — I’ll just listen to a podcast with the author if I care. The insights in such books feel like they have an asymmetrical yield — if they confirm what you already thought then the opportunity cost of reading that book is high (I’ll be lucky if I read 500 more books before I’m dead) and if the book has a ground-breaking insight it’ll almost certainly be out of fashion within a decade (“the game theory of getting published in social science” is a comically fractal idea. If you google that phrase, you’ll see why).

Anyway, David’s history of intellectual care makes him an ideal candidate to interview other social science authors about their books — critical enough to ask good questions but friendly enough that he can get the interviews in the first place.

Enough preamble…some excerpts I enjoyed from David’s Q&A with psychologist Adam Grant on his new book Hidden Potential: The Science of Achieving Greater Thing (emphasis mine):

• Many people believe that if you’re not precocious, it’s a sign that you lack potential. But potential is not about where you start — it’s a matter of how far you’ll travel. And the latest science reveals that we shouldn’t mistake speed for aptitude. Our rate of learning is driven by motivation and opportunity, not just ability. Think of all the late bloomers who weren’t lucky enough to stumble on a passion, or to have a parent, teacher, or coach early on who recognized and developed their hidden potential.
  This doesn’t mean we should ignore “gifted” students. We need to think differently about how we nurture their potential too. Empirically, the rate of child prodigies becoming adult geniuses is surprisingly low. I suspect one of the reasons is that they learn to excel at other people’s crafts but not to develop their own. Mastering Mozart’s melodies doesn’t prepare you to write your own original symphonies. [Kris: this is exactly the point Trent Reznor made to Rick Rubin as he wrestled with his own potential]. Memorizing thousands of digits of pi does little to train your mind to come up with your own Pythagorean theorem. And the easier a new skill comes to you, the less experience you have with facing failure. This is a lesson that chess grandmaster Maurice Ashley drove home for me: the people who struggle early often build the character skills to excel later. We need to start investing in character skills sooner.
• Because Glennie is deaf, she had to find nontraditional ways to learn, like using different parts of her body to feel vibrations that correspond to different pitches. She and her teacher were constantly trying different ways to do that, and different ways to do everything, really. As you write: “Continually varying the task and raising the bar made learning a joy.” I’ve long been fascinated by this issue of variable practice. Mixing things up constantly might seem counterintuitive, but it turns out to be better for learning.
• You note that concert pianists who reach international acclaim by age 40 typically were not obsessed early on, and that they usually had a slow but steady increase in their commitment to music. It just made me think of the first page of Battle Hymn — in which the author promises the secrets to raising stereotypically successful children, and recounts assigning her daughter violin and soon she’s supervising five hours of deliberate practice a day. That part was excerpted in the Wall St. Journal, and it was the Journal’s most commented upon article ever! It really seeped into the public consciousness, I think. What didn’t make as much of an impression was the part later in the book where the author (to her credit) recounts her daughter turning to her and saying: “You picked it, not me,” and more or less quits. [Kris: I’m very careful riding our kids in areas that they are naturally drawn to because of such “reactance”. I don’t want to turn “their thing” into “my thing”. You have an extra gear to give for those things that you discover independently.]
• The issue of “learning styles.” This is the very popular idea that some people learn best by listening, others by reading, others by looking, etc. Maybe someone prefers podcasts to books because they style themself an “auditory learner.” Trouble is, a mountain of research has failed to back this idea up [Kris: Veritasium calls this “the biggest myth in education”. Although I suspect the testing design for experiments that dismiss the idea might be strawmanning the contention or interpreting it too narrowly].People may indeed have a style of learning that feels most comfortable, but that doesn’t mean they’re actually learning more that way. In fact, to use a line from Range, in many cases, difficulty is not a sign that you aren’t learning, but ease is [Kris: I’ve found that many teachers I respect agree with this so it’s not as bold a statement as it might appear even if this is the first time you’ve heard that. I remind my kids — if it’s easy it’s just review, not learning. Non-superficial learning hurts. I might even go as far to say that learning and pain are nearly synonyms. To be clear, such a statement is more useful as a reminder than a universal truth. Experiential learning is an easy counterexample]. As you write: “Sometimes you even learn better in the mode that makes you the most uncomfortable, because you have to work harder at it.” I was just reading a study (“Measuring actual learning versus feeling of learning”) which showed that Harvard physics students preferred lectures from highly-rated instructors to active learning exercises. But they learned more from the latter. The main difference in the active group was that students had to try to solve problems in groups before they really knew what they were doing, and so they would discuss, generate questions, and hit dead-ends, all before seeing correct solutions. We know that forcing learners to try to generate solutions before seeing them enhances learning (the so-called “generation effect”), but it doesn’t feel great, so we may avoid it.
• Back in December, you helped me get in touch with RA Dickey, and he was every bit as stellar of an interview as you promised. His story helps to illuminate why so many people fail to try new methods when we get stuck. It’s not so much that we’re stubborn or resistant to change. We hate the thought of giving up the gains we’ve already made. We forget that sometimes, the best way to move forward is to go back to the drawing board. [Kris: Feeling seen] If your fastball is slowing down and your career is stalling, you have nothing to lose by tinkering with the knuckleball. We shouldn’t be so afraid of failing that we fail to try.