Technical writing is a big part of a software engineer’s job. This is more true the more senior you get. In the limit case, a principal or distinguished engineer might only write technical documents, but even brand-new junior engineers need to write: commit messages, code comments, PR descriptions and comments, Slack threads, internal announcements, documentation, runbooks, and so on. Whether you write well or badly matters a lot.

Keep it as short as possible

The primary rule about technical writing is that almost none of your readers will pay much attention. Your readers will typically read the first sentence, skim the next one, and then either skim the rest or stop reading entirely. You should thus write as little as possible. If you can communicate your idea in a single sentence, do that - there’s a high chance that people will actually read it.

What if you can’t communicate all the details in so few words? In fact, this is a feature not a bug. You should deliberately omit many subtle details.

This is the biggest difference between technical writing and code. Each line of code you write is as important to the computer as any other. The compiler or interpreter will methodically go through every single detail you put into your codebase and incorporate it into the final product. If there’s a subtle point you want to make - say, that prorated monthly billing should account for the fact that some months are longer than others - you can and should spend a paragraph of code articulating it. But each point you make when talking to humans consumes a limited attention budget¹. Because of that, it’s usually wise to leave out subtle points entirely.

One consequence of this is that you should frontload all important information. If you spend a paragraph providing context and throat-clearing, many of your readers will have stopped reading (or started skimming) by the time you make your actual point. Try to get to your point in the very first sentence. It’s even better to get your point into the title, like an academic paper does.

Lower your expectations

Many engineers refuse to deliberately leave out information, because they think they shouldn’t need to. They believe their technical colleagues are thoughtful, conscientious people who will pay careful attention to everything they read and attempt to come to a full understanding of it. In other words, many engineers have far too high expectations of what their technical writing can accomplish.

For instance, your technical writing is not going to transplant your understanding of a system into somebody else’s head. It simply does not work like that. Understanding of technical systems is won only through painstaking concrete effort: you have to interact with the system, read the code, make changes to it, and so on. At best, good technical writing will give someone enough rough context to get a fuzzy understanding of what you’re suggesting. At worst, good technical writing will convey to the reader that at least you know what you’re talking about, so they may as well trust you even if they’re still confused.

Your technical writing is also not going to get everybody on the same page. Disagreement and confusion in large organizations is not dysfunction but function: it is the normal baseline of operation, just like some constant degree of failure is part of the normal baseline of a complex distributed system. You should not expect your technical writing to fix this problem.

What can good technical writing do?

So what can good technical writing do? It can communicate a very simple technical point to a broad audience. For instance, you can explain “adding new settings requires a database migration, so it cannot be done dynamically” to your Product org, and thus prevent them from suggesting impossible things². However, be aware that the actual point being communicated is probably even simpler than that: something like “adding new settings is hard”, or even “settings are complicated”.

This is a fairly pessimistic view about the usefulness of technical writing. But I do think that even this limited goal is surprisingly important in a large organization. The baseline level of technical confusion is so high that communicating even obvious things is high-leverage. I wrote about this in Providing technical clarity to non-technical leaders: even the people leading engineering organizations are often operating on a really low level of technical context, because they simply have so many other things to worry about.

Good technical writing can also communicate a reasonably complex technical point to a tiny audience. For instance, you can write an ADR that goes into many subtle details about a planned piece of work. This is a great way to get excellent feedback on a plan, but the effective audience for writing like this can be in the single digits. Sometimes the effective audience for an ADR is two: you, the writer, and one other engineer who has the requisite context to understand it.

Thinking clearly

Of course, to write clearly you first must think clearly. I haven’t written about this here because it’s an entirely separate topic (and one I have less concrete advice for). For some engineers, this is the main obstacle to condensing their point into a key sentence or two: they simply do not have a clear enough understanding to do that, and must instead express a series of related ideas and hope the reader gets the overall message.

I did write about this almost a year ago in Thinking clearly about software. I stand by most of that, particularly the parts about sitting with your uncertainty and focusing on what you can say for sure, but I think there’s probably much more to be said on this topic.

Summary

The biggest mistake engineers make in their technical writing is setting their expectations too high. They try to communicate in too much detail and end up failing to communicate anything at all: either because readers are checked out by the time they arrive at the key point, or because they’ve simply assumed too much background technical knowledge and the reader is hopelessly confused.

When you are doing technical writing, you are almost always communicating to people with less technical context than you. You may have a crystal-clear understanding of what you’re talking about, but your reader will not. They likely won’t even be as interested as you in the topic: if you’re writing a paragraph to try and ask some other team to do something, that something is your project that you (presumably) care about, but the other team has their own projects and will probably only skim what you wrote.

If you can say what you want to say in one sentence - even if it means leaving out some nuance - you should almost always do that. If you have to write a paragraph, make it as short a paragraph as you can. If you have to write a page, make sure the first paragraph contains as much key information as possible, because many readers won’t make it any further than that.

The good news is that even if you’re overestimating how much you can successfully convey, you’re likely underestimating how useful it is to convey even a tiny amount of technical content. In large organizations, many technical decisions are made by people with effectively zero technical context³. Going from that to even a very rough “lay of the land” is a massive improvement.

Perhaps the most famous equation in pop mathematics is Euler’s identity:

The equation is deemed profound because it combines not one, not two, but five “special” mathematical constants: e, π, 0, 1, and the imaginary unit i.

The identity is a special case of the following equation (Euler’s formula):

The “identity” form is what you get if you choose an angle of α = π in radians (180°). This makes the cosine expression equal to -1 and the sine part equal to zero; the end result of the substitution is:

Now, four “special” values is still weak sauce, so we move -1 to the left to increase the profoundness factor by another 25%.

There are multiple proofs of Euler’s formula, but none of them are easy for a layperson to understand. I don’t have a proof that will fit on a napkin, but I think there’s a reasonably intuitive way to reason about what the equation does.

A geometric view of complex numbers

Imagine a point in a Cartesian coordinate system that lies on the horizontal axis at a distance l from the center. If you wish to rotate this point by an angle α in radians, you can calculate the new (x, y) coordinates using simple trigonometry:

Less obviously, there is also a way to rotate points without trigonometric functions. If we take point (x, y) and flip the signs of the individual coordinates — (-x, -y) — we always achieve what looks like a rotation by 180°:

The sign-flipping operation is equivalent to multiplication by -1. Changing the magnitude of the negative multiplier doesn’t result in a different rotation angle; it’s the “negative unit" itself (-1) that appears to be doing the hard work. If we want to achieve a rotation by 360°, we need to multiply the coordinates by -1 twice; the pattern extends in a pretty obvious way:

This leads to a peculiar realization: it would appear that the way to achieve rotations of less than 180°, we could raise -1 to a fractional power. In particular, for 90°, we could use an exponent halfway between (-1)⁰ and (-1)¹, multiplying the coordinates by (-1)^1/2. This multiplier, known as i, can be also written as √(-1).

The number i is not a real. That’s not to say that it doesn’t exist; it just isn’t a member of the set of real numbers, ℝ. This construct lets us represent the coordinates of any point (x, y) as a single compound (“complex”) number: z = x + iy. The first part corresponds to the distance along the horizontal axis. The second (i-coupled) value is a number that’s evidently “rotated” by 90° — i.e., it represents the distance along the vertical axis. This might sound unhinged, but our algebra system extends to this realm in a remarkably straightforward way. For the most part, you can just work on the two halves separately, keeping in mind that i · i = -1.

If we go back to the first method of rotation — using sin and cos — we can rewrite the formula as:

Equivalently, we can express rotation by m · 90° by multiplying a given number by a power of i:

In the first equation, the angle of rotation (α) is expressed in radians, so a full 360° turn is achieved when α = 2π. In the second version, it’s expressed in 90° increments, so a full rotation is m = 4. To reconcile these equations, we need to toss in the appropriate scaling factor. If we do it on the i^m side to settle on radians, we get:

This can be further simplified by choosing l = 1, essentially building a model of a point moving along a unit circle with a radius of one:

The equation is already in the same ballpark as Euler’s formula, but we’re not quite done yet.

Extending exponentiation

In the realm of real numbers, exponentiation that uses one positive base other than 1 can be rewritten in another base simply by tossing in an appropriate scaling factor in the exponent. For example, we can write the following:

The scaling factor is just the logarithm of the old base in the new base: log₂(8) = 3.

Complex exponentiation is defined so that roughly the same principle extends to the imaginary unit. That is to say, we can go from base i to base 10 by including some scaling factor in the exponent:

The scaling factor is necessarily an imaginary number; the value of log₁₀(i) happens to be roughly 0.682·i, but we don’t need a way to calculate it. The point is just that the two forms — i^x and 10^{<something>·i·x} — are equivalent and produce the same result. They both represent rotation by a chosen angle in two-dimensional space.

Back to rotations

Before the segue into the properties of exponentiation, we established the following formula that equated two methods of rotating a point:

On the left side of the equation, we needed to toss in a 2/π scaling factor to convert from 90° increments to radians. It would be nice to switch to a different, real exponent base n that naturally has the same rotation speed as the cos + sin expression. This would simplify the formula to:

The cos + sin expression is drawing a circle with a radius of 1, completing one rotation every 2π radians. The circumference of the unit circle is 2π, so there is a 1:1 correspondence between the parameter of the expression (α) and the distance traveled by the moving point.

As luck would have it, there is a well-known real base for which there is a 1:1 correspondence between x and n^x in the vicinity x = 0. It is, by definition, the mathematical constant e. If you’re unfamiliar with this property, it’s easy to show it numerically; if we choose a small Δx = 0.0001 and calculate the rate of change near x = 0, we get:

In the domain of real numbers, the rate of change of n^x increases in tandem with the value of x; that’s the nature of exponential growth. But in the realm of complex numbers, we have already asserted that a real value raised to an imaginary power results in constant-speed rotation, not runaway growth. We must conclude that the rate of change is dialed in solely by the real part, which is zero in the formula we’re trying to build. This means that the scaling-factor-free solution we’re looking for is just:

This also gives us the value of log_e(i) = ln(i). Per the earlier discussion, switching from i to base e entails multiplying the exponent by log_e(i). In this instance, the multiplication evidently cancels out the 2/π factor, so ln(i) must be π/2 · i, or about 1.571 · i.

That’s it. Again, it’s not a proof: the argument contains a couple of appeals to intuition. That said, it may be better for getting a hang of the formula than an appeal to the properties of infinite series and function derivatives.

As to why all these constants come together, the basic answer is that complex numbers are a two-dimensional geometry — and that radians, e, and π describe similar things.

👉 For a more in-depth treatment of complex numbers, click here. If you like beavers, check out this article. And you enjoy the content, please subscribe. I don’t sell anything; it’s just a good way to stay in touch with the authors you like.

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Previously on the Internet

I have a theory: Most people from mine and slightly older generations (early 80s kids) still remember the first time we went online unsupervised.

It was late 2001, I was 18 years old, which was an admittedly belated entry into cyberspace compared to my peers, but the fact that I remember when and where it happened, and what websites I visited, should underscore my point, especially to younger readers: the internet felt like a revelation.

Why would I bestow such gravitas and import to that one hour over two decades ago, in a tiny internet cafe, on Internet Explorer of all things?

This was when I had finally decided what I was going to do with my life: I wanted to be a filmmaker. But I was in Sri Lanka, and had little access to the resources I would need; what films and filmmakers to study, how films were made in the first place, such things were mysterious and secretive in my pre-internet life.

On that day in 2001, in that one hour, I realized how wrong I was. Everything I wanted to learn about film was just a Yahoo! search away. The internet had lived up to its hype: it was the promised land for the insatiably curious. Today, the kids would call it a nerdgasm.

I start this essay with this flashback because I want to carry out a thought experiment: All other things about me being equal, what would an 18 year old me dreaming of films and film-making, encounter on the internet in 2025? I encourage my younger readers (those born in the 2000s) to do the opposite: imagine if you were old enough to encounter the pre-social media, pre-SEO spam, pre-AI sludge filled internet.

The Dead Internet

In their paper The Dead Internet Theory: A Survey on Artificial Interactions and the Future of Social Media (Asian Journal of Research in Computer Science, 18(1) 67-73), Muzumdar, et al., trace the genesis of the theory to online communities in the late 2010s:

"The origins of the Dead Internet Theory (DIT) can be traced back to the speculative discussions in online forums and communities in the late 2010s and early 2020s. It emerged as a response to the growing unease about the changing nature of the internet, particularly in how social media and content platforms operate. Early proponents of the theory observed that much of the internet no longer felt as vibrant or genuine as it had in its earlier days, where user-generated blogs, niche forums, and personal websites created spaces for online interaction."

In Is Google Getting Worse? A Longitudinal Investigation of SEO Spam in Search Engines Bevendorff, J., et al,. showed there was empirical evidence to back these observations.

What does that look like at a macro level? On the surface, it means more than half of all internet traffic is bots.

This seems almost inevitable.

Around 2005, I was working as a copywriter for a web development firm that specialized in the hospitality sector. Our clients were some of the largest brands in the industry, and every week our job was to ensure their websites would rank above the competition. My employer was a well-known service provider to the entire sector, which meant we worked on brands that were competing against one another.

One half of the day would be spent ensuring Hotel X in New York City ranked higher than Hotel Y, the former's competitor in, say, the luxury hotel space for New York. The second half would be focused on—and I wish I was joking—ensuring Hotel Y would rank over Hotel X. This mercenary approach to winning Google search rankings for clients, drove me to quit. When my boss at the time asked why I was quitting, I could not adequately express my misgivings. It only took me twenty years to crystallize my thoughts on the matter.

The Costs of A Dead Internet

The research carried out by Bevendorff, et all., restricted itself mostly to websites that focused on product reviews. We don't require advanced comprehension of statistics to extrapolate these findings into more critical areas such as political and social discourse; as AI generated news combines with SEO Spam and bots, the stakes are enormous.

The evidence shows that AI misinformation is leading to an erosion on a common, shared truth. Is it any wonder that the last decade has seen increasing polarization in our societies?

Reviving the Revelatory Internet

The study by Campante et al., 2025 offers a way forward:

"While exposure to AI-generated misinformation does make people more worried about the quality of information available online, it can also increase the value they attach to outlets with reputations for credibility, as the need for help in distinguishing between real and synthetic content becomes more pressing."

Reviving the internet has to be a collective fight. Everyone of us can play their part in ensuring a more vibrant internet. Then we don't have go into survival mode and opt for devices like Prepper Disk for a post-apocalyptic, offline internet knowledge. Excellent idea, by the way.

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Here are some ways we can still resist for a more human internet:

Spam Protection and Authenticity

• mosparo: AI-powered open-source spam filtering for website forms, avoiding intrusive CAPTCHAs and preserving genuine user interactions.
• ASSP (Anti-Spam SMTP Proxy): Open-source email firewall using Bayesian filtering, greylisting, and AI spam detection.
• Anubis: Blocks AI scrapers with proof-of-work challenges, protecting self-hosted sites from bot scraping.
• CAI SDK (Content Authenticity Initiative): Open-source tools for verifying content provenance and checking if media/news is authentic and unaltered.

Disinformation Detection and Curated Search

• iVerify: Fact-checking and false narrative alerting tool with transparent code, useful for journalists and regular users.
• Disinfo Open Toolbox: Suite of open-source tools to verify news credibility and monitor fake news/disinformation sources.
• Codesinfo: Set of open-source civic journalism tools for fact-checking, evidence gathering, and author attribution.
• phpBB, Discourse: FOSS forum platforms for authentic, moderated human communities.
• OSINT tools (Maltego & others): Free open-source tools to investigate online identities, emails, and website authenticity.

Building and Joining Authentic Communities

• Fediverse platforms (e.g., Mastodon, Lemmy): Decentralized open-source social networks emphasizing moderation and organic growth.

Protect Your Browser

• Browser privacy extensions and alternative search engines (Searx, DuckDuckGo): Reduce SEO spam and filter content farms.
• RSS aggregators and curated open-source communities: Bypass algorithmic feeds for direct access to trusted sources.
• FOSS moderation, spam filtering, fact-checking, and media verification: Ensuring content authenticity and reliable engagement.

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Next On the Internet

The easy thing for someone like me—a writer of speculative fiction—is to veer this column towards the dystopian. I could, for instance, liken a future internet to a zombie apocalypse where AI powered spam and content bots bury thriving virtual communities run by actual people.

This isn't a feat of imagination even: just take a gander at blogging sites like Medium (which began with a promise to make writing and writers on the internet feel seen); almost all the site's tech writing is clearly AI generated, while some of its writers in the paid partnership write repetitive pieces on how AI has allowed them to supposedly make six-figure incomes.

In such a case, I should end this with a eulogy to an internet that I no longer recognize.

Or I could write this note to the imaginary 18-year-old me using the internet in 2025. In which case, I would tell him: there is a better way, and that better way is within your grasp.

By request, here’s a summary of my lessons from the last week. Hopefully this gives a bit more detail to my post earlier this week about teaching with multiple strands.

A quick note about the structure of this post: Below is a basic summary of where I’m at in the yearlong sequence for each of the three strands. Next there’s a quick outline of my class each day this week. Then I give a few notes reflecting on how I structure my class and sequence curriculum. I don’t want to clog up the post with too many details about the nitty gritty of my routines so there are a ton of footnotes. If you want to learn more about all those details, enjoy!

The Strands

I have three main strands of my curriculum this year. This image shows the rough idea:

In the top strand we are almost finished with proportional relationships; this week we’re focused on analyzing graphs of proportional relationships and then we’ll move on to percent increase/decrease in the top strand.1 In the rational numbers strand we’ve finished a big review of adding/subtracting/multiplying/dividing fractions, a lot of practice with the basics of number lines, some work on different contexts where negative numbers come up, and adding positive and negative integers, fractions, and decimals.2 This week we started working on multiplying with negatives. In the expressions and equations strand we are working on one-step multiplication equations. We’ve finished a lot of practice with addition and subtraction equations, including addition and subtraction word problems.

A Few Everyday Routines

Before I get into my week, I have a few everyday routines so students know what to expect at the beginning and end of class. At the start of class I welcome students in, they drop their backpacks in a designated area in the back corner of the room, grab a half sheet for their Do Now, and sit down. Below is one example of a Do Now I used on Tuesday.3

While students answer the questions and try an optional puzzle, I take attendance, make sure everyone has a pencil, and deal with any issues that come up. I aim for three minutes flat for work time. I do a quick circulation around the room, often focusing on one question I anticipate will be the toughest. Then we quickly go over the answers. If there’s one question I see a bunch of students getting wrong, I make a note and return to it when we pull out mini whiteboards in a few minutes.

Next, on the back of the Do Now half sheet, we do an estimation routine.4 Here’s an example question:

The final everyday routine is DeltaMath. DeltaMath is a practice website that offers a lot of customization for teachers. Every day students have a DeltaMath assignment, and we spend the last 5-10 minutes working on these.5 Students finish anything they don’t get to for homework or in a separate study hall period.

Also, note that my classes are 50 minutes long each day. There are a lot of bullets under each day. The timing varies a bit from class to class but in general each bullet represents 5-10 minutes of teaching. We move fast! There’s lots of math to learn, and not much time to learn it.

Monday

• On Monday we started with the Do Now and estimation.

• Then, I had students grab a whiteboard, marker, and rag to erase with.6 I asked a few quick questions to check understanding of prerequisite skills for the day’s lesson. I write each question on the board, have students answer on their own mini whiteboard, and hold up their boards on my signal so I can see their answers. I checked whether students could identify if a relationship was proportional from a table, whether they could solve a one-step multiplication equation, and if they knew that the name of the point at (0,0) was called the origin. A few students mistook the multiplication equation for addition, so we did a bit of clarification and extra practice with that. This was maybe 5 minutes total.

• Next we did a bit of mixed practice on paper with whole-number one-step equations, including addition, subtraction, and multiplication equations.7

• Then we launched into a larger chunk of class focused on graphing proportional relationships. I modified an activity from the Illustrative Math curriculum about seagulls eating trash. It’s a bit of a weird example, but was good for building off of students’ knowledge of proportional relationships in tables to give examples and non-examples of graphs of proportional relationships.

• Then I had students open up their Chromebooks and try a short chunk of the DinoPops activity on Desmos.8 This is a chance to apply some of what we had just talked about with graphs of proportional relationships.

• Next we did some very quick practice distinguishing graphs of proportional relationships vs non-proportional relationships with mini whiteboards.

• Finally students finished with DeltaMath. DeltaMath included some mixed practice of addition/subtraction fluency, evaluating percentages, and solving one-step equations.9

Tuesday

• On Tuesday we again started with a Do Now and estimation.

• Then I had students answer a few questions on mini whiteboards. Some students had answered 0 for -5 + (-5) on the Do Now this day so we spent some time practicing adding negatives and distinguishing between problems that add to 0 and problems that don’t. I also asked students whether a graph of a relationship was proportional and why, I asked again for where the origin was, and another question about one-step multiplication equations.10

• Then we did a quick round of multiplication fact practice. This focused on the 11s fact family, which means half 11s and half the other facts we’ve focused on so far which are 2s, 3s, 4s, 5s, and 10s.11

• Next we did a handout extending multiplication equations to fractions. The goal was to extend the generalization that you can use division to solve a one-step multiplication equation to division of fractions. This honestly didn’t go that well, so I cut the discussion short and we returned to it the next day.

• Next we did a bunch of examples of proportional and non-proportional relationships. Some were about pizza — if you want to make twice as many pizzas, you need twice as much dough and cheese, but you don’t need to cook at twice the temperature. Others were some silly examples I made up about the relationship between the number of people and the number of arms, or students and teachers. The goal was to look at a bunch of graphs of relationships and think about why that graph is proportional or not.

• Then we did a five questions check for understanding of adding negatives, focused on questions like -3 + (-3) and -4 + (-4) + (-4), which is relevant to the lesson tomorrow. I followed up individually with a few students who made mistakes during DeltaMath.

• Finally students did some DeltaMath focused on questions about graphs of proportional relationships.

Wednesday

• On Wednesday we again started with a Do Now and estimation.

• Then I did a quick thing sharing the story of Srinivasa Ramanujan. We watched the trailer from the movie about his life, The Man Who Knew Infinity, and I added in a bit more of his story that isn’t in the trailer. We do one of these most weeks.12

• Next I did an introduction to multiplying negative numbers. I focused on the idea that 2x3 really means 3 + 3, or 3x4 means 4 + 4 + 4, so the same is true for negatives. 2x(-3) just means -3 + (-3), which we were just practicing yesterday.

• Then I had students grab mini whiteboards and we did some practice, with a mix of two positive numbers and one positive/one negative where the negative comes second. I also mixed in a check for understanding on fraction division, and did a quick model of fraction division and some extra practice.

• Then we did another handout building up to solving multiplication equations with fractions. This one moved more gradually from simple equations like 10x = 30 to equations with larger numbers, decimals, and then fractions. I did a bit of modeling again, and this went a bit better than Tuesday though students will need some more practice.

• Next we did a quick activity adapted from Illustrative Math connecting equations for proportional relationships (which we had already spent time on in the context of tables and word problems) to the graphs of proportional relationships.

• Next we did a 10-question quiz, which covered topics from all three strands as well as some review questions from weeks or months previously.13

• Finally students did some DeltaMath mixed practice, with questions on adding integers, solving equations, finding percentages, finding unit rates, and some fact fluency.

Thursday

Day four: formative followup, CFU multiplying negatives and formative followup, 11s, multiplication practice, curriculum resources for graphs, stop and jot multiplication, DM focused on equations

• On Thursday we again started with a Do Now and estimation.

• The most common mistake on the quiz from Wednesday was a question about a snail traveling at 1/2 a foot per minute (students struggling with fractions is a theme this week). We talked about a simpler question with whole numbers, then the snail question.

• Next I had students grab mini whiteboards and try two more questions like the snail question, and did a check for understanding of multiplication with negatives (a surprising number of students added on the first one).

• Then we did a handout from the JUMP curriculum that focuses on patterns in multiplying negatives and the commutative property, and a handout I made with some practice multiplying negatives. No negative times a negative yet, though. This went really well!

• Next was another quick round of practice with multiplication equations and fractions.

• Then we did some problems from the Reveal curriculum about graphs of proportional relationships. These went alright, though students had a tough time reading unit rates off of the graphs so we’ll return to that next week.

• Next was a five-question check for understanding on multiplication equations. This is still a bit of a mess, need to figure out how to support the students who are having trouble with this next week.

• Finally students wrapped up with some DeltaMath focused on proportional graphs, adding integers, and solving equations with whole numbers.14

A Few Thoughts

I’m genuinely curious for people’s reactions when they read this. I want to emphasize that I’m not some huge expert on this type of teaching. I’ve experimented with these ideas for a few years but this is the first year I’ve gone all-in on teaching multiple strands at once. I was doing some of that experimenting last spring during my formal observation, which is also the only time I’ve received substantive feedback on my teaching in the last year. The one piece of constructive feedback was that the lesson felt scattered because there wasn’t one objective.

And sure, if you are a teacher who has only ever taught lessons with a single objective, this feels weird at first. It felt weird to me for a while. But my experience is that it doesn’t feel weird to students. They come to math class, we do math. Most activities are either practice, or taking something we’ve done and taking it one small step further. Is my sequencing and pacing perfect? No, absolutely not, I’m learning all the time.

I find that this style of teaching benefits students in multiple ways. Students who have trouble with focus and sustained attention benefit from the transitions and short chunks. Students who feel frustrated because math is hard for them benefit from spending a few minutes on a topic and then moving on to something different, rather than banging their heads against something for the whole 50 minutes. Students who have stronger skills benefit from the variety — it’s much more interesting to do a short chunk of review of something you already know pretty well mixed with new content than to spend the whole period on it. Students who struggle with long-term retention benefit from the spacing and repeated short chunks of practice.

I’m piecing together a few different curricular resources rather than following one resource. I do that because I’m not happy with the resources my district has adopted. But I think you could do this with any commercial curriculum. The biggest thing you would need to do for most programs is to add in a lot more review of content from prior years, and break the practice down into smaller, more focused chunks.

My motto is “go slow to go fast.” If you’re a 7th grade math teacher, you might read this and say, “wow, you have a lot left to get through this year.” True! While we’ve made some good progress, we’ve spent a lot of time on review and foundational skills early in the year. The goal — and this has mostly been true so far — is that if I take my time with the foundation, then learning the tougher grade-level content will go more quickly and smoothly. I have all the topics mapped out for the whole year. I adjust as I go, but so far I’m within a week of my goal for each of the three strands.

I wrote last year about the motivation fractal.

On the x-axis is time, and on the y-axis is difficulty. The blue ellipse is one lesson. So each lesson has a few activities, each going from simpler to more complex, each starting easier than the last one ended. Then the next day, we get a bit tougher and repeat.

The image above is the ideal. Practical challenges mean that my classes don’t always look like that. But it’s the goal. The reason I made this change to my curriculum is that I just haven’t been successful in past years. Too many students struggle to keep up, feel that the curriculum moves too fast, and don’t have time to build a strong foundation before going on to the next thing. This is the best solution I’ve found. It’s not perfect, but it’s way better than anything else I’ve tried.

If you have questions, please ask away!

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I wrote in more detail about my Do Nows here. One of this week’s Do Nows is below. The puzzle on the right is called a Shikaku puzzle from Naoki Inaba. For the first one, the goal is to divide the shape into three rectangles with sizes 3, 12, and 16. We rotate through different puzzles like these throughout the year, working on each puzzle type for a few weeks. It’s an optional challenge for students but a good way to challenge them and keep students who finish quickly busy.

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Pictured below is my mini-whiteboard setup at each table. Each table has calculators (only allowed for some parts of class), whiteboards, markers, and rags to erase with. It is unfortunately beyond my powers as a teacher to prevent students from drawing on the supply caddies. Mini whiteboards are a pain to manage sometimes but they are 100% worth it. There is no better way to quickly check student understanding.

Other than lawsuits, losing track of a child is every school district’s worst nightmare. We haven’t lost anyone yet, but an EdTech company has painted a compelling picture of this as a possible future problem that can be pre-solved with EdTech. Therefore, we’re pleased to introduce some cutting-edge yet profoundly user-unfriendly technology.

You’ve already been given an app for reluctantly volunteering to run the book fair, an app for viewing your kid’s hastily prepared report cards, and an app for lodging complaints about our apps. Effective immediately, we’ll also be tracking your child’s whereabouts on three additional apps.

First, if your child is going to be absent, log into your PITA, which does not stand for “Pain in the Ass” but for “Parents Informing Technology of Absences.” If you forget your login, call our wonderful school secretary, Marjorie. No, you can’t just call the office to tell Marjorie your kid is sick; you’ll need to call the office to retrieve your PITA login information, then log in to PITA to tell PITA your kid is sick.

Second, if your kid needs to leave school early for a doctor’s appointment or orthodontic torture session, log into your Dismal. (Dismal is an abbreviation of “Dismissal.”) You will also inform us of your child’s early dismissal when you arrive at the school to pick them up, at which point Marjorie will confirm that the dismissal has been logged in Dismal. If you forgot to log the dismissal in Dismal, everything will be the same, except that Marjorie will log it for you. Unless she forgets.

Third, if your child will be staying after school for an activity, log it in your Ass (“After-School Stuff”). Of course, you already completed the permission form in Piss (a shortening of “Permission”) and paid the activity fee in the other Piss app (“Paying Incessant Sums”—and yes, we know from the app complaint app that you’re confused by the two separate Piss apps). So when you go to log the activity in your Ass, we already know which activities your child is doing. You know, your child knows, the principal knows, the school secretary knows. Still, we need you to log in and tell a piece of Ass software what we all already know.

Since many after-school activities recur daily or weekly, we initially worried that parents would resent the repetitive task of re-inserting everything into their Ass every week, but the Ass app company assured us that once they get used to it, people love putting things in their Ass.

Yes, it’s true that one warm and competent woman used to do the work of all these apps with nothing more than a pen and a little notebook. But don’t worry, Marjorie is still our school secretary. Plus, more great news: We hired a few new part-time assistant secretaries to help Marjorie manage all the apps.

Is this time-effective? Of course not. We’re dispersing the labor previously done by one person across five paid employees and hundreds of parents and guardians. But is it cost-effective? Also, no, we’ll be paying for the apps, the part-timers, and Marjorie, who used to perform many interesting and varied tasks but now mostly just clicks things in apps, and she is no longer warm and feels incompetent.

So, will these PITA, Dismal, and Ass apps prevent your child from going missing? Absolutely they will, as long as all parents and guardians in the entire district, along with Marjorie, and Gill and Jen and Martin and Amy (the new admins), use all three apps consistently and no parent ever forgets to log their child’s whereabouts and the apps themselves don’t ever glitch or scramble information or lose service and no child ever does anything unpredictable.

And if, somehow, one of those ironclad links in the chain breaks and a child goes missing, don’t worry, the outcome will be the same as ever: We will find them in the bathroom texting on their phones. But this way, if we are ever sued, we can now mount the flawless defense “But we have all these apps!”

Or to explain it more succinctly: If a new technology exists, how could we not implement the most cumbersome version of it?