- Discrete growth: change happens at specific intervals
- Continuous growth: change happens at every instant

Here's the difference:

**The key question: When does growth happen?**

With discrete growth, we can see change happening after a specific event. We flip a coin and get new possibilities. We have a yearly interest payment. A mating season finishes and offspring are born.

With continuous growth, change is *always* happening. We can't point to an event and say "It changed *here*". The pattern is always in motion (radioactive decay, a bacteria colony, or perfectly compounded interest).

(Brush up on the number e and the natural logarithm.)

I visualize change as events along a timeline:

Discrete changes happen as distinct green blobs. We can take them, split them into smaller, more frequent changes, and spread them out. With enough splits, we could have smooth, continuous change.

So, discrete changes can be modeled by some equivalent, smooth curve. What does it look like?

The natural log finds the continuous rate behind a result. In our case, we grew from 1 to 2, which means our continuous growth rate was ln(2/1) = .693 = 69.3%. (The natural log works on the ratio between the new and old value (frac(text(new))(text(old))).)

Mathematically,

In other words: 100% discrete growth (doubling every period) has the same effect as 69.3% continuous growth. (Continuous growth requires a smaller rate because of compounding.)

Now here's the question: how should we talk about growth? It depends on the scenario:

- If growth happens in a man-made system, discrete growth works better (2^x, 3^x)
- If growth occurs a natural system, continuous growth is better (e
^{x})

Let's take a look.

Let's say we flip a coin. What are the possible outcomes?

- 1 flip: 2 outcomes (H or T)
- 2 flips: 4 outcomes (HH, HT, TH, TT)
- 3 flips: 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)

You see where this is going. I'd describe the number of possibilities as 2^{n} where *n* was the number of flips.

I'm using "n" (not x) by convention: x could mean any value on the x-axis (-3, 1.234, √(14)), while n represents an integer (1, 2, 3, 4).

*Could* we say the number of outcomes was e^{ln(2}x), where x was the number of coin flips? Yes. But it's confusing: in a man-made system, where we have change *events*, I'd use the discrete version to describe the possibilities.

Binary numbers follow the same pattern: if we have *n* bits, we get 2^{n} possibilities. For example, 8 bits have 256 possible values, and 16 bits have 65536.

(There may be some cases where intermediate values make sense, like representing the number of bits required, even though we need a whole number of bits in practice. This is similar to saying the average family has 2.3 kids.)

When radioactive material decays, we often talk about its half-life: how long until half the material is gone?

For example, the half-life of Carbon-14 is 5700 years. We could write it like this:

If we wait 5700 years, we expect (1/2)^{1}= .50 of the carbon remaining. If we double that and wait 11,400 years, we'd expect (1/2)^{2} = .25 of the carbon left.

However, this equation is written for our convenience. Carbon doesn't decay in jumps, politely waiting around 5700 years and suddenly decaying by half. We use (1/2) as the base because *we humans* want to count the number of halvings (decaying into half, decaying into a quarter, decaying into an eighth...).

The radioactive material is changing every instant. From a physics perspective, a continuous rate is more telling. We can find the *continuous decay rate* by converting the discrete growth into a continuous pattern:

This helps me understand why the natural log is *natural* -- it's describing what nature is doing on an instant-by-instant basis. None of this "wait until we decay by 50% so humans can count it easier" nonsense.

In practice, you don't discover the half-life by waiting for carbon to decay 50%. You'd wait a reasonable about of time (a year?), use the natural log to find the continuous rate over that period, and work out the half life.

**Example:** Material X decayed from 53kg to 37kg over 9 months. What's the continuous decay rate and half life (in years)?

The ratio between new and old was 37/53, so ln(37/53) = -.359 = -35.9% continuous growth over our time period. This happened over 9 months, so the monthly continuous rate is -35.9/9 = -3.98%. Scaling this up, the yearly continuous rate is -3.98% * 12 = -47.9%. (Notice how the rate must be scaled to match the time period. Earning "12% interest" isn't helpful without a time period. "12% interest per day" is different than "12% interest per year".)

Now that we know the continuous rate is -47.9% per year, we can work out how long until we're at 50%:

The half-life is 1.44 years.

This is a tricky one: the stock market changes every day, so it seems like it'd continuous, but there isn't an underlying predictable rate. We see a lot of jumpy changes, and sample them at yearly intervals to see how we're doing. The market is usually described with an annual average growth rate:

A continuous rate of the form e^{x} doesn't really make sense for the system. We aren't trying to model our portfolio's value on a per-instant basis: we want to know what to expect in 30 years.

Population is tricky: depending on the animal, discrete or continuous model can make sense.

A bacteria colony is made of billions of organisms. Although *each bacteria cell* grows discretely (it has to wait until it splits before splitting again), the entire *colony* grows smoothly because so many bacteria are in different stages of growth.

Like the radioactive decay example, we can sample the colony at different time periods and work out how long it takes to double. We might have a continuous rate (e^{x}) that expresses the colony's instant rate, and a discrete rate (2^{x}) that helps us humans count the doublings.

One of my pet peeves were problems like "A bacteria colony doubles after 24 hours...". Argh! Are you telling me the bacteria colony just *happens* to have a continuous rate of precisely ln(2) over the course of a day?

I'd prefer you told me the colony doubled while a grad student stared at a petri dish for 24 hours straight. (*1.98kg... 1.99kg... 2.00kg. I found the doubling time, I can go home! What's that Professor? I...ok, I'll work out the continuous rate after an hour next time.*)

Rant aside, how about modeling a tiger population? Tigers have breeding seasons. They aren't having kids throughout the year, so the population changes in a discrete event.

(The model gets more complex as you account for how long it takes for cubs to have children of their own.)

I wrote this post because my video on e had questions about how 2^x represents "staircase growth". Isn't that a smooth curve too?

Sure, but most of the time we use 2 as a base to model discrete patterns. 2^{n} (where n is an integer) models discrete scenarios like coin flips or binary digits. If your system does change continuously, why not provide the continuous rate and write e^{ln(2} x)?

There's no right and wrong here, just the message we convey. A whole-number base (2^{x}, 3^{x}) implies you want people to think about whole-number values of x (and half-life is a good example). Using e as a base (e^{rate · time}) implies you want people to think about change that happens at every moment.

Either way, be fluent in both models and learn to hop between the two.

Happy math.

]]>And the formula we found was:

It seems that regular arithmetic, algebra, geometry, or even statistics could help work out the equation.

But how about Calculus? Is this bringing a nuclear missile to a gun fight?

Let's find out.

The sequence to add (1 2 3 4 5 6 7 8 9 10...) looks a lot like f(x) = x. At every position on the x-axis, we put in a number and get the same one out.

Intuitively, the integral is "repeatedly adding a bunch of stuff" -- it seems like we could put it to work. From the rules of Calculus (or using Wolfram Alpha) we get this:

Intuitively: Add up things following the f(x) = x pattern and you end up with frac(1)(2) x^{2}.

Well, let's see: the actual sum from 1 to 100 is 5050. But using the Calculus equation we get:

Uh oh: there's a difference. What's going on?

**Calculus works with continuous patterns, and we used a discrete one**.

Here's what's happening:

Calculus was built to measure *smoothly changing functions*, like a line, parabola, circle, etc. The pattern we have is a jumpy staircase (going from 1 to 2 without ever passing through 1.5, or 1.1, or 1.0001). In math class, books harp on analyzing whether a function is "continuous", aka changes smoothly enough for Calculus to work.

So when a pattern changes smoothly, Calculus works great. If a pattern changes suddenly, Calculus can only give an approximate answer. So what's the plan?

Use Calculus where possible, on the smooth part, and adjust for errors in the jumpy part.

The area under the line is the integral. We a bunch of triangles above the line we need to include.

- How many of them? 1 for each item (x)
- How big are they? They're half a of a 1x1 square, so they have area 1/2.
- What's the total area to add back in? x frac(1)(2) = frac(x)(2)

So our final formula should be

Aha! Learning Calculus doesn't mean we hunt around for Official Calculus Problems.

Nope. Take your scenario (adding 1 to 100) and realize what Calculus brings to the table: finding patterns in smoothly changing functions. Use Calculus on the smooth parts and adjust (or ignore) the other parts.

(Ironically, Calculus works by making jumpy approximations for smooth functions, and is in fact "jumpy" under the hood. If you are planning on working with jumpy patterns, use Discrete Calculus.)

Let's take this further: what's your guess for the sum of the first 100 square numbers?

Hrm. Getting the exact formula is tricky. But maybe we don't need the exact count, just an estimate.

With Calculus, we'd say: The pattern isn't continuous, but it looks like f(x) = x^{2}. Let's integrate x^2 from 0 to 100.

The indefinite integral is frac(1)(3) x^{3} , the running total for how much we have. From 0 to 100 it would be

That's our guess, without a calculator. And the actual answer? 338350.

How close were we? 99.9%. Not bad for something we worked out by hand in a minute!

Truly internalizing Calculus means it helps other elements of your math understanding, even regular addition problems.

Happy math.

PS. To keep building your intuition, check out the Calculus Guide.

]]>**Developer Tea Interview (Part 1, 43 mins)**

**Developer Tea Interview (Part 2, 1h 2m)**

**Developer Tea Interview (Part 3, 1h 13m)**

- 10 year history of BetterExplained
- Lessons on creating content that's fundamentally valuable
- Lots of real-time intuition gathering

- Developer Tea podcast / creating content regardless of "motivation"
- Shower time / unfocused thinking - where does your mind wander, what interests you?
- Creator mindset
- "Output" is not solely determined by willpower. Hidden variable of emotional state, not controlled directly
- Being gentle with yourself
- Exploring the role of fear
- Find values by working backwards from what's previously brought joy. "Feel good list" or "Jar of Awesome"
- Culture is what you celebrate
- Not taking feedback/criticism personally
- Goal: avoiding burnout or permanent demotivation
- Looking at absolute contribution (not relative)
- Separating worthy/unworthy from skilled/not yet skilled, creation from identity
- The unwinnable situation of seeking external validation
- Getting feedback without applying extra emotional weight

- Sharing 3 epiphanies
- 1: Role of "evergreen"
- Evergreen content (10 year old articles still popular)
- Evergreen motivation - being motivated for a day not as important as being motivated for a year, or lifetime.

- 2: Value of empathy
- Acknowledging when a concept is difficult
- Writing for a younger version of myself, share what I struggled with

- Confusion like finding holes in your roof to patch up. Inspector vs. repairman.
- Megaman vs. Tetris learning mindset

- Learning languages like Spanish, role of memorization vs. thinking
- Incorporate vocabuluary like a game of taboo

- Getting to the concepts behind math: "circle has these properties" just like the color "red" has certain properties
- Intuition behind e and logarithms. Seeing the theme behind the concept.
- Logs as a count of digits (6 figures, 7 figures...)
- Asymptotic progression (losing weight, baseball batting average)

- 3: Highway numbering system - encoding information

- "Learning to code" -> really about learning to think in a structured way
- How BetterExplained started
- Acknowleding your limitations - look for sources of evergreen motivation, warmth to returning
- Philosphy that anything can be understood intuitively eventually
- Learning from someone in your shoes vs. the world expert
- Building small things
- Being gentle with yourself so you can return to the material
- What would I do differently
- Intuition dictionary idea

What should I call it: My name? Stuff explained simply? Better Explained? (Hey, not looking for the best... just... better!)

10 years, 168 posts and 220k words later, here we are.

Below is a collection of thoughts on life and business after pursuing an idea for a decade.

Advice gets skewed by our personal experience, cognitive biases, and the presence of someone else (smile for the camera!). My strategy is to write for a single person -- a younger me -- as simply as I can.

**Evergreen content:**Work on what's still useful/enjoyable years later.**Evergreen motivation:**Find values that can inspire you for years. (Use a feelgood.txt list.)**Be kind to yourself.**Think about inputs, not outcomes. Try not to take things personally, or base yourself on these outcomes.

It's ok. Be gentle with yourself. Not "gentle now to work harder later" but genuine compassion. Delay, confusion or disappointment may have a role, or be a warning sign, and you don't know ahead of time how things will work out. BetterExplained exists because you didn't go to grad school as you'd hoped -- would you trade the two now?

There's an inner tug that knows what matters to you: *Ideas can be understood intuitively and truly enjoyed.*

What lit a fire in your belly in the past? BetterExplained began after overcoming immense frustration with a poorly-taught traditional math class. Concepts finally clicked after weeks, months or years -- argh, why couldn't they explain it simply in the first place?

Do you remember the exhilaration of being up all night and finally having an idea make sense? The anger of seeing friends discouraged from careers by poorly-taught, hoop-jumping math requirements?

Tap into what drives you.

10 years ago I started keeping a "feel good" list: if you have a great day, when you feel truly alive, write it down. A few scattered entries:

12/3/16 - WOW! I am cranking like insanity on instacalc. I have that fire in my belly where I just want to make things. Create what I can and see where it goes.

10/26/16 - Feeling great about myself for finishing the Coursera class! (machine learning). Itâ€™s motivating me to do other classes (Complex analysis.)

5/20/16 - from Chris XXXX. Loves getting the BetterExplained emails.

Review the list for patterns. Even better, put your notes into a tag cloud tool and see what themes pop out:

My core values, things that make me feel alive, are:

- Curiosity and learning
- Helping others
- Getting in the zone, flow state, cranking away on a project
- Having fun / playful irreverence

Find an outlet that engages your values.

Based on your values, find things that make you come alive. What grinds your gears? What do you desperately wish someone told you when starting?

General writing advice:

**Pick evergreen topics**: Imagine 5 years have passed since publishing your article. Is it still useful or interesting? If not, add strategy/insights that go beyond current events. (E.g., a post on simplicity vs. complexity might use an old video as an example, that's ok.). Fortunately, math is about as evergreen as it gets, since we study 2000-year old theorems. Even there, make connections and present general strategies.**Use more diagrams**: Most technical content doesn't have enough analogies, diagrams, or examples (interactive is even better). For evergreen content, this effort pays back since the post is read for years. A 10-year old post on version control, ancient in the tech world, shows up in many presentations because of its images.

**Embrace improvements over time**: Evergreen text with evolutionary updates is a sweet spot. It can be updated years later with a fresh diagram, example, or embedded video. With video as a*primary*source, you have to redo the whole thing. (Where's the video version of Wikipedia?)**Empathize with your audience**: Matter-of-fact textbooks sweep decades of math debate under the rug (imaginary numbers, Fourier Transforms). Share your personal struggle and how you overcame it. Highlighting your confusions makes the content both approachable and helpful. What would you share with a younger version of yourself?

Aim to genuinely help your readers in the long run. (You become your own reader when you revisit your post years later.)

My goodness, what I would have paid for *one* satisfying intuition on imaginary numbers in college. Read the article once and you're done. Boom, you have an insight that sticks in your mind and makes every math class so much easier.

Today we see parades of clickbait "15 easy ways to learn imaginary numbers" articles. Maybe this week's lesson will finally work!

I call it the Men's Health problem. In theory, you should read one magazine and become superhuman. Yet somehow there's hundreds of issues, over years, detailing the same goal. Either the tips don't work, or the problem is with *applying* the tips, not getting more of them. (Issue 1: *Here's 10 essential tips.* Issue 2: *Did you read and apply issue 1*?)

Help your readers graduate beyond you. Google search is useful because it sends you *away* as fast as possible. And that's why you come back with new questions.

With evergreen content, you're building nets to throw into the ocean. More nets = more fish.

With read-once content (like a newspaper), you're spear-fishing. More effort = more fish, but yesterday's effort doesn't help you today. It's exhausting.

Just keep writing content you'd read yourself, and over time you'll see a buildup:

- Monthly traffic: ~450k visitors, 600k pageviews
- Lifetime traffic: 21M visitors, 32M pageviews
- Average annual growth: 23%

**Raw Traffic Data**

Analysis comes from this Google sheet.

**Monthly Views vs. Year of Publication**

Old posts are just as good as new ones for generating today's traffic. Let that back catalog work for you - 45% of my traffic comes from 2007-era posts. (At that time, I had a lot of back content to import. I also had more time to spend after leaving my first job.)

**Sleeper Hits**

Each evergreen post has a chance to become a hit years later. A 2015 post on cross products was published quietly, and later became a top 5 article:

Maybe one of the 2016 posts will pick up steam next year. (In 2016 I tended to write more "strategy" posts, which are fun but don't get as much traffic as tactical "How to learn X" articles.)

**Post Wordcount**

Evergreen content doesn't have to be long. The average wordcount was 1310 (median 1273, max 3604).

Let's see how wordcount impacts traffic:

There is a modest positive correlation (.26) between wordcount and traffic. After an article reaches 1000 words, shorter articles (Combinations vs. Permutations, 1000 words) and longer articles (Fourier Transform, 3600 words) have similar popularity.

I don't have a wordcount goal, just a "helpfulness" goal: include an analogy, diagram, example, plain-english description, and technical description (ADEPT method).

(For what it's worth, this article is now the longest at about 4000 words.)

Be willing to write for years. But if your content aligns with your values, and is interesting to you, it's a win-win. Traffic grows, you'll feel fulfilled, and you'll improve your writing and subject matter expertise.

It's not the fast path to Mega Blogging Success. But whether it takes 2 years or 10 doesn't matter, you're building something that fulfills you, and the time will pass anyway.

I wrote 220k words, averaging 60 words a day over the past 10 years. At 95 words per minute, that should only take 38 seconds a day (right?).

A desire for consistency, while helpful, isn't necessary and certainly shouldn't invoke a feeling of dread ("I shouldn't start if I can't be consistent").

In 2015, I wrote 4 posts (3 months per post) but hopped back on the wagon in 2016 (16 posts, 3 weeks per post). Being gentle with myself meant I looked forward to writing more without guilt.

There's a saying that long after you forget what someone did, you remember how they made you feel. Long after you forget what you wrote about, you remember how coming back to your project made you feel.

Have an account on a social service, use it normally, and learn the norms. If your content shows up, reply to questions and take feedback. It's ok if everyone doesn't love your cooking. (*But seriously, why don't you?*)

In 2006, I posted articles to smaller communities (like dzone.com), while today that'd be a small subreddit or forum. Participate, make things you'd want to read yourself, and stick with your values.

More pratical tips:

- Monitor live traffic with a browser plugin (I use Clicky). When a spike happens I can see it and respond quickly. When I hit reddit, I saw 3000 simultaneous readers on the site -- time to see what's happening!
- Make your site static. For WordPress, get a caching plugin that doesn't require the database to be active. (To verify: turn off your database and make sure your site still works.) This has burned me a few times.
- Have an "elevator pitch" for your site ready to go. What's it about? What's your best content?
- Examples of high-traffic social media events (12/22/2015)
- Reddit front page and discussion (I'm "pbzeppelin")
- 591k users, 968k pageviews (1.6 pages/visitor). 2:51 average session, 41% bounce rate.
- 2786 newsletter signups. (0.47%)
- Hacker news front page and discussion (I'm "kalid")
- 15k visitors, 39k pageviews (2.5 pages/visitor). 6:02 average session, 14% bounce rate.
- 277 newsletter signups (1.88%)
- For both discussions, I had rapid-fire replies to comments. Threads can become a mini-interviews and stay engaging.

- Useful content gets reposted. Ideally, someone you don't know is sharing your article with someone else you don't know. For example, the ELI5 reddit frequently asks about imaginary numbers, and my evergreen article usually gets referenced. The article has a lasting, visual intuition that
*imaginary numbers are like rotations in 2d*which won't go out of style.

You don't need to merge the two. If you want to make money, it's *much* easier to just get a day job, live simply, and invest your savings. However, financial independence is alluring and a side business (or other project) can speed that up.

For BetterExplained, my goals are:

Goal #1: Do not lose my passion for learning (evergreen motivation)

Goals 2-1000: Help improve education, make a living, make friends/connections, have a portfolio, improve my writing, explore new technologies, get better at design, marketing, copywriting, and business.

This project is a lifelong mission and I'm ok if it accelerates slowly while enthusiasm is preserved.

I fear burnout because math enthusiasm is what I peddle.

There's plenty of standard details out there, I don't need to be the 105th site that introduces Calculus using the slope of the tangent line. (*If your Calculus class completely ignores Archimedes, I'm going to... suggest you read this guide*.)

I'm slowly coming to grips with the risks of diving deeper. If I lose interest, for whatever reason, it'll be ok. I'll forgive myself for a mistake made in the pursuit of something I care about. One door closing may open another.

But I really like the room opposite the door that's open now.

Patio11's bingo card creator reports inspired me about what was possible with my own products, and I share stats in that spirit. Given the traffic so far, the site more than covers my expenses, coming from:

- Amazon books on kindle (about 350 copies/month) and print (about 100 copies/month): $3000-4000/m
- Direct course/book sales from website: $1000-1500/m
- Math, Better Explained (full course)
- Calculus, Better Explained (full course)

- Amazon affiliate links: $300-500/m
- Newsletter sponsorships and other promotions: $700-1500/m

I think there's a lot of growth potential. But I don't want a growth *requirement*, or the mission warped by the grow-or-die monster.

A few scenarios I'd like to avoid:

- HowStuffWorks - One of my favorite sites when young. But it got bought, and is now filled with ads, "relevant links" (celebrity gossip), and articles split over 6 pages to maximize pageviews. There's still some good content but you have to hold your nose and dive in.
- Google Knol - an attempt to monetize crowdsourced information (Wikipedia with ads). The incentives for quality weren't there and it wasn't built to last. A few years later all its articles were gone.
- About.com, eHow, etc. - designed to pump out uninspired copy/pasted material for pageviews. Again, incentives drive behavior.

I'd prefer a slower, harder-to-kill path to sustainability while always being proud to send my Mom to the site.

The most important business advice: start an email list. Other services (RSS readers, twitter, facebook, etc.) are ephemeral and not really in your control. In 10 years, who knows what we'll use. But email will probably remain as a direct conversation to your audience. At a minimum, you can use email to publicize whatever platform comes next.

Currently there's 38k email subscribers (growing about 1k/month) and I wish I started in 2006 not 2013. The best time to plant a tree is 20 years ago, the second best time is now.

The next business-y thing: practice making products. An ebook, guide, book, video tutorial, in-person lesson, anything. Overcome the trepidation of creating something, selling it, getting feedback, and iterating -- there's only so much you can read about swimming.

After reading and replying to thousands of comments, you realize that any external judgment (whether positive or negative) is not necessarily true, and you can't tie your worth to it.

It's wonderful when an article helps someone. Maybe it was the right time, the right place, they were in the right mindset and they saw the right analogy. I'm happy for them. Maybe I can learn to do more articles like it.

Sometimes someone hates an article. They found it boring, or confusing, or they didn't have the right background, or were expecting something else. That's too bad, and maybe a different approach will help them.

But "Another person loved it" and "Another person hated it" aren't in your control. Put in an honest effort and separate your sense worth from the outcome. (Like playing poker: the right play can still lose, the wrong play can still win.)

You'd think the "no blame, no praise" mindset removes joy, but it doesn't -- my feelgood list is ever growing! I'm vicariously happy when someone enjoys an article for the simple reason they enjoyed it. Their Aha! makes me smile. It's "I feel good" not "I am finally good".

Not taking things personally doesn't mean you need to *like* it. When something bothers you, it means you're basing some element of self-worth on that opinion. It's a "teaching moment", however frustrating. Writing online gives you many teaching moments.

I read somewhere the Dalai Lama will cut short interviews with people who aren't making a genuine effort. You can be polite, compassionate, and respect your own time.

Meditation, CBT, and other techniques helped me immensely with very negative thoughts (I like Alan Watts and Noah Elkrief). Subjectively, 90% of issues that used to bother me don't anymore. They were typically a variation of *I'm not good enough because of XYZ*. (Protip: If XYZ makes you good enough, then losing XYZ means you're no longer good enough. Uh oh. You've got the tiger by the tail and can never let go.)

- Example Hacker News discussion with criticism. (This resolved well, it's often just a misunderstanding.)
- Example Reddit discussion with criticism

I'm still working on this one. My internal chatter is usually something like:

*Inner critic: You're not sharing real knowledge, just generalities. Without rigorous oversight, your readers will build a nuclear reactor based on a flawed understanding.*

First: Someone read an article about math and liked it enough to apply it? Wow! That's objective #1.

More seriously, I want to mitigate a false sense of understanding by including real examples and sanity checks. It's not intuition *or* technical, it's intuition **for** the technical. I'll make mistakes, sure, and thankfully text is easy to update.

*Inner critic: You started in 2006? It should be a giant-mega-business by now.*

Perhaps I could have pushed things faster while staying motivated. But is there an expiration date on sustainably aligning with your passion? My goal isn't growth for its own sake, it's to uncover insights that wouldn't have been shared otherwise. Hitting a number in 2017 (not 2007) is ok with me if I enjoyed the journey.

*Inner critic: You should be writing more. You have dozens of half-finished drafts lying around.*

Got me on that one. I have to constantly give myself permission to share quick insights.

What went well:

**I wrote more.**Thanks to Marquis Butler (intern / Kalid Wrangler), I've written more in 2016 than I have in several years.**Traffic grew.**The site appeared organically on reddit, hacker news, and total readership is at an all-time high.**Passive income still worked**. Math books and courses are pretty passive, and they continued to hum along.**Mentally and physically I'm in a good place.**I've finally figured out what worked emotionally (not taking things personally, see above), physically (low sugar / low carb) and I'm the healthiest I've been in years. In 2016 we moved into our first house, and a dedicated office for work (and rooftop for parties) made me feel energized.

Things I'd like to work on:

- I wanted to release new editions of my Calculus and Math courses but didn't.
- I wanted to add more community features to the site. This is a long-standing vision, with several attempts, but I find myself stymied by the (imagined) overhead of maintaining a community.
- I wanted to add to my YouTube channel / start a podcast, but didn't.
- I'd like to do more "topic" posts (on specific math concepts) vs. "strategy" posts (how to learn). I'm wary of getting too meta and writing about learning without learning things myself.
- I'm usually behind in my personal communication.
- I still hesitate quite a bit before publishing. I'd like to reframe writing as a skill to constantly practice, vs. a performance.
- I have a Patreon which I haven't promoted (I fear being externally obligated, which needs to be reframed)
- I feel held back by concerns about losing my passion, a contributor to many of the above.

These are things to observe, not beat myself up about. I've tried the beat-up approach and it works for days, not years. When I'm in the zone I do weeks of work in days while having a blast. I'd like to foster that.

Questions suggested by newsletter subscribers:

I primarily wanted a place for creative expression, and secondarily wanted it to grow large enough to make a living. BetterExplained has made more than projects I started with the purpose of making money.

Write about something that makes you come alive, for which you can share truly helpful information. What does everyone overlook? What do you wish someone told you? What don't you see anywhere else?

There are tools to find underserved niches, but for a lifelong project it's hard to get motivated by a report telling you to write about X. It's ok to separate the "make money" and "learn a skill" projects.

My strongest drive is that intuitive learning works: I *know* an idea can be understood if we have the right approach. It's a fact of life for me -- you know gravity works because you've seen things fall over and over. I know an Aha! moment will eventually help because I've experienced hundreds of concepts snapping into place with the right analogy or example.

Now, finding this Aha! moment isn't easy or even pleasant. But there's an optimism that however long it takes (even years), I'll be thrilled when I finally figure it out. (It's the Mega Man philosophy, where difficult ideas become allies after they are understood.)

I don't have this motivation in other parts of life, but experiencing it in one domain makes me think it's possible. Maybe there's a mindset shift that can work similarly; pulling towards my values seems to be on the right track.

I use the ADEPT method.

I tend to write once I'm both confident about my intuition, and have verified its usefulness in practice problems and simulations. Of course I've had plenty of corrections, suggested improvements, and miss subtleties that true experts would see. But the key insights seem to be generally OK (always happy to fix them, yay for text).

I have certain philosophical differences with a typical math curriculum. For example, I casually introduce Linear Algebra as a type of "math spreadsheet". This is a casual description compared to the official "vector space" abstraction, but spreadsheets are familiar and what I wish someone had showed me first. I won't pretend an approach is helpful to me when it's not.

Typically 10-20 hours per post, but sometimes 2 hours and sometimes years. Certain ideas can sit in my head (or idea notebook, or idea text file) for months before a satisfying Aha! hits me. I try to capture it as soon as possible, with the fresh enthusiasm of "Whoa! Here's how I finally figured out XYZ".

It's hard to express how much a role BetterExplained has had in my life. This project is more than I could have asked for as a kid looking to publish his college insights online.

I feel deeply grateful to every BetterExplained reader for the encouraging and thoughtful blog comments, emails, and contact form submissions. From marked-up books with countless improvements, to class lessons based on the analogies, to mind-blowingly detailed extensions to existing articles.

Thank you for helping the site (and me) develop these past 10 years. Hope you'll join me for a decade more :).

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