Adonis Diaries

Archive for July 18th, 2022

How to Teach Math and Science

You already know who Feynman is and why he’s here. Credit: ProfFeynman on Twitter

Rebecca Lea Morris

 Daubechies published an article called Ingrid Daubechies and the Importance of Mathematical Communication.

She tells the story of how Ingrid Daubechies published a paper on wavelets that led to them becoming an industry-standard in several industries.

Daubechies cites her attempt to make it easy to read and put into practice as one of the main reasons it had such an impact. (I don’t want to quote too much because it’s a short article.)

Where I Come In

When I read the article, there was a line that stuck out to me.

Daubechies’s background in physics helps her communicate mathematics in a way that scientists and engineers can understand. But for mathematicians without that background, it may be hard to figure out what will make their research more understandable to non-specialists.

I started writing a short response to this line.

Like everything I write, it grew quickly (The Road to Quantum Mechanics, the last three articles of The Limit Series, and the Entropy Trilogy were going to be one article for each series.).

At the end of the response, I had written roughly a thousand words in an hour or two, which is fast even for me. At that rate, I realized that I could make an article out of the response if I continued to flesh out the ideas.

This is that article.

An Experiment

Here’s an excerpt from System of a Down’s song Toxicity:

You, what do you own the world? How do you own disorder, disorder?
Now, somewhere between the sacred silence, sacred silence and sleep
Somewhere between the sacred silence and sleep
Disorder, disorder, disorder

I want you to write down or type up what you think of when you read these lyrics. Make sure it’s something you can’t change so you don’t change your answer.

The Three Archetypes of Science Communication

I want to start by talking about the different levels of science communication.

As in many fields, coming up with rigid classification systems is bound to lead to some problems. For this reason, I’m going to talk about archetypes instead of trying to force examples into boxes.

Each archetype has a role, creating a whole greater than the sum of its parts.

Pop Science

Pop science refers to science for the masses as entertainment.

It can be a great way to get people interested in science and expose them to new ideas. For example, one of my first encounters with Physics came from this video.

Before you judge the video, remember that Game Theory made it in 2013.

In the video, Matthew Patrick (a.k.a. MatPat) talks about a game mechanic from the game series Assassin’s Creed, known as a leap of faith.

It involves jumping off of high buildings into hay to escape enemies and return to the ground without having to climb back down.

Feel free to scroll down if you start getting motion sickness.

In the video, MatPat talks about different topics, but one, in particular, stood out to me.

Based on the height of the fall and the time it took to fall, MatPat calculated the gravity using standard kinematics (Δs = at² / 2). He found that Assassin’s Creed 2 had a gravitational acceleration of around 12.4 m/s² without accounting for drag. I wanted to account for drag.

I spent a week or so doing research on the topic. I looked up how to calculate drag coefficients. I found the equations for drag.

After assuming linear drag, I got a gravitational acceleration of around 12.5 m/s².

This story is pop-science at its best. The video got me interested in an idea. I then used free resources online to learn about differential equations and concepts like a drag.

I was able to put my knowledge to the test by calculating some value. While the experience didn’t make me into a full-blown Physicist, it gave me a head start when I took classes in college and high school.

The Pitfalls

I’m hardly the first to talk about pop science and the video below is great for understanding the problems with pop science.

As I’ve said in multiple articles, pop science often simplifies topics beyond the point of usefulness.

In some cases, it can even introduce more confusion and even contribute to scientific disinformation campaigns so common that people have written papers about specific examples.

Another video on a specific example of bad pop-science.

To make matters worse, pop science is supposed to get people into science, but it provides no way to do stuff with what you’ve learned.

Introduce Technical Descriptions

As a start, pop science shouldn’t be afraid of introducing some technical descriptions.

A few relevant math equations here and there or links to more detailed resources won’t hurt anyone. As a benefit, seeing math in the safe context of pop science can help overcome the math anxiety that haunts so many people.

Moving Forward

When talking about Physics in one of my first articles, I said

People often talk about physics as if it were merely a set of facts and equations: F=ma, energy can neither be created nor destroyed, E = mc², stars are giant balls of hydrogen fusing into helium and heavier elements, etc.

These facts and equations can either form the base of our understanding or result from our understanding, but to do physics is to connect the two. In other words, learning physics requires learning techniques and guidelines alongside facts and equations.

People say pop science is great for getting people into science, but it can only get people into science if it has some way to go forward with the topics covered.

After you watch a video or read an article, there should be something you can do with what you’ve learned.

If you’re talking about climate change, try giving people a climate model to play around with. If you’re talking about a simple equation, you could start with the complex equation and show the simplifications.

What to Aim For

There’s a wide variety of topics you could go into, but I think any of the videos below are some of the best examples of pop science.

Note that while equations are present, they’re not the focus. You can read the comments and find people who enjoyed the video without understanding the equations.

This video breaks from standard depictions of general relativity to better illustrate the concepts. Furthermore, it uses a lot of multimedia skills to illustrate difficult to understand concepts.

Yes, I am a hack promoting my own content. Anyway, half of this video consists of a simulated explicit example.

This video does a great job answering questions students might have in math class. Why aren’t we taught the cubic formula? Why should I care about imaginary numbers?

The best answer to the question “Why?”.

Try to analyze what makes these videos work, how they’re alike, and how they’re different.


You’ll find expert science communication in scientific journals and papers.

Experts in the field have already seen all the stuff beginners have seen. They also have the background to not need everything explained in the field.

For these reasons, expert science communication often focuses less on the flashy parts and more on the technical parts.

The Pitfalls

Expert science communication often makes tenuous connections to background information needed to understand the field.

In an ideal world, you could read a paper and understand it with no effort. In a less ideal world, you can read the cited papers until you do.

In our world, you may Not understand what’s going on even if you read the cited papers and the papers cited by the cited papers.


Not every paper needs an application or a textbook of background, but you should explain how your paper fits into the larger field.

If all you’re doing is citing a simple fact like “CO₂ emissions went down by x% in 2020,” then you can state the fact and move on. If you’re citing a more complex idea like an algorithm, give a short worked example of the algorithm.

(Papers of experiments in “social/psychological” fields should include a section that explains how the results could be applied, especially in matters of safety/health consequences for re-design of objects/services)


The gap between pop science and expert science communication isolates scientists from the public. Beginner science communication often lacks a way forward while expert science communication lacks a path to it.

To solve this problem, we need to connect the two. Connecting the two is what I call intermediate science communication. Most of my work is intermediate science communication and textbooks are a typical example.

The Pitfalls

Intermediate science communication has the same pitfalls as pop science and expert science communication but to a lesser extent.

Credit: xkcd #2501

To do intermediate science communication, you have to know a lot like an expert, but you have to think like your audience.

It can also be quite difficult to research because most people prefer to do expert or beginner science communication.


To reach as wide an audience as possible, I often link to a large number of resources. Some resources are remedial while others are for experts. As for researching, you’ll often end up having to do your own derivations.

This is what every derivation of Lagrangian Mechanics looks like.

What to Aim For

These videos are similar to the videos given in the pop-science section, but they require a bit more mathematical background and get technical.

This video takes something people are interested in and uses it as a starting point to talk about Information Theory.

This entire video consists of explicitly worked examples and includes code to help viewers do something with the knowledge.

This video definitely leans towards expert science communication, but it does a great job connecting different ideas together.

This video shows a complete derivation along with an explanation. It helped me to understand a concept I needed to understand in my Road to Quantum Mechanics series.

This video contains most of a derivation.

As I’ve said before, most of my articles would also count, though I tend to lean towards expert science communication.

General Tips

Now, I want to talk about tips that work for all levels. Some tips work better for some levels than other levels.

I wrote part of this section of the article as a response to the original article, but I’ve rewritten parts of it to align with the rest of the article. I’ve also added several new parts that don’t show up in the original.

Respect and Humility

Say you decide to learn a new language. You pick up an app, hire a tutor, or do whatever you need to do. One of your lessons says that the language puts objects before the verb instead of after as in your native language. You decide that the language is being stupid, so you continue to do your lessons in Subject Verb Object order. You later learn that the language doesn’t use articles (a, an, the). You decide that the language is being stupid again, so you decide to invent your own articles for the language. After a year of lessons in this fashion, you visit a country in which they speak the language you’re learning. You try to talk with them and they look at you funny before leaving. Clearly, they don’t understand your brilliant new fixes to the language.

Photo by Hannah Wright on Unsplash

I hope everyone can see that you would be a complete jerk. Not only would you have an incredible disrespect for the language you’re learning, but your arrogance would prevent you from learning.

If you had approached the language with respect and humility, you would have learned to communicate more effectively.

How to Learn Respect and Humility in Communication

If you want to learn to communicate effectively, you have to start with respect for the subject. You also need the humility to understand what you don’t know. In this section, I want to put you on a path to becoming a better communicator.

Read What the Masters Have Written About Writing…

I recommend picking up a style guide like Strunk & White’s The Elements of Style.

It’s a little old, so you might have to throw out some outdated advice. I only know English (and Latin), so I can only recommend resources for English (and Latin).

Every language should have its own specific guide on writing well and some advice can differ from language to language. For example, you should avoid using the passive voice in English but it doesn’t matter in Latin. I’ve also heard that Medium might have an article or two on writing advice, so you might want to check there. I’ve heard good things about Stephen King’s On Writing, though I have never read it. Lastly, read Politics and the English Language.

…Until You Learn When to Break the Rules

You should follow their advice to the letter long enough to break your bad habits. After a while, you’ll start to realize why they said what they said and you won’t need to follow strict rules.

For example, authors recommend using as few adverbs as possible for beginner writers. (You can use the Hemingway Editor to help.) Doing so forces beginners to use strong nouns and verbs.

Sounds Tedious

Like all skills, writing requires work and discipline. You’re going to spend a lot of time when you start agonizing over every detail. As time goes by, you’ll learn which details to focus on.

Look at What Other People Have Done

There’s nothing new under the sun. To quote T. S. Elliot:

One of the surest of tests is the way in which a poet borrows. Immature poets imitate; mature poets steal; bad poets deface what they take, and good poets make it into something better, or at least something different.

The good poet welds his theft into a whole of feeling which is unique, utterly different from that from which it was torn; the bad poet throws it into something which has no cohesion.

A good poet will usually borrow from authors remote in time, or alien in language, or diverse in interest.

These ideas apply to science communication (and science in general). When you want to explain something, look for every explanation you can find. Take the parts of each explanation you think are best and combine them into a cohesive narrative.

Understand the Meaning of Words

In my AP Literature class, we’d often have to write essays and we’d often have to edit the essays others had written. I was awful at timed writing, but man could I edit.

You gave me four pages of timid or weak sentences and I’d give you three pages so strong and streamlined that they would go on to win a state swimming championship.

Photo by Gentrit Sylejmani on Unsplash

I was editing a paper once and I found a sentence that had the word “adhere” in it even though it didn’t fit the sentence.

To make my point, I looked up the definition of “adhere” and wrote every possible meaning on the back of the paper. When I returned the paper to the author, I gave them some commentary and explained why I made the edits I made.

When I got to the word “adhere,” I showed them the definition and respectfully asked them which definition fit the sentence. They looked over the definitions and couldn’t find one, so I told them to use a different word.

Credit: xkcd #1860

Science communicators use words that don’t fit for many reasons. They may not know a better word. They may not understand the material well enough to find the right word, so they have to rely on what others say. In either case, they need to learn more to make sure what they’re saying fits what they mean.

Don’t Force an Explanation

Someone once described love like a fart: If you have to force it, it’s probably crap. Writing is similar.

Bad writers often fixate on bad explanations. I want to give you an extreme example so you can see the problem.

Say someone asks you about radio waves. You have to compare them to something the person asking knows, so you say radio waves are “like red light, but redder.” This explanation can make sense if you define “redder” to mean “redshifted” in the sense that radio waves have less energy and a longer wavelength.

Credit: Inductiveload on Wikipedia. I added a white background.

If you explain it, it’s a bad comparison. People understand “redder” to mean “closer to a pure, intense red” like how the image below gets redder as you travel from left to right.

For this reason, the concept of being redder than red doesn’t make sense.

Credit: xkcd #1984

Metaphorical language should be part of the explanation and not need to be explained. To make matters worse, you could have used the explanation of the simile instead of the simile and it would have been easier to understand.

Don’t Force a Word

You should have written down what you thought of when you read the lyrics to Toxicity at the start of the article. In case you haven’t, here are those lyrics again.

You, what do you own the world? How do you own disorder, disorder?
Now, somewhere between the sacred silence, sacred silence and sleep
Somewhere between the sacred silence and sleep
Disorder, disorder, disorder

Maybe you thought of a futile attempt to control an uncontrollable world.

Photo by Birmingham Museums Trust on Unsplash

Maybe you thought of some kind of mental disorder.

Photo by Aarón Blanco Tejedor on Unsplash

Maybe you thought of a zombie state between sleeping and death.

Photo by Sharon McCutcheon on Unsplash

You know what you didn’t think of? The logarithm of the number of microstates corresponding to a given macro-state. You know why? Because disorder has nothing to do with microstates or macro-states unless you define disorder to mean entropy.

At that point, saying entropy is a disorder is the same as saying entropy is entropy. Not helpful.

What Does X Get Me?

At every point where you have a choice to make in writing, you should ask yourself “What does doing X get me?” What does defining entropy as disorder get you? Is anyone going to hear you say entropy is a disorder and go “Ah, it must be proportional to the logarithm of the number of microstates corresponding to a given macrostate.”?

As with the forced simile, you can cut out the middle man. For entropy, you could use whatever definition you made up for disorder and lose nothing. Seriously, go find any video or article that talks about entropy and cut out any line that says “entropy is/measures disorder” and ask yourself what is lost.

On the other hand, if you say something like entropy measures the likelihood of a state or the number of ways a state can exist, then the audience at least knows that you’ll need counting or probability. Furthermore, it gives you a simple way to explain the origin of the Second Law of Thermodynamics.

Don’t Use the Word “Utilize”

It’s an exact synonym for the word “use” and it even comes from the same root word. You’re already talking about technical stuff. You don’t need to make yourself sound artificially technical unless you’re doing a bit like in the video below.

As a rule, prefer the simplest word that best fits what you want to say.

Learn Other Topics

Learning other topics can allow you to bring in ideas that aren’t normally associated with the original topic. These new ideas give you a unique approach to the topic that helps your work leave lasting impressions on readers.

Learn Other Skills

Learning other skills can allow you to present ideas in a new way. I can use my programming experience to make images like

See this article for the context.

and videos like

See here for 4k version and code.

I can also implement Numerical Methods to back up my claims about Physics as shown in the gif above. And the fact that I can make gifs out of videos is because I have experience with command-line tools like ffmpeg.

Use Worked Examples

If possible, worked non-trivial examples are one of the best ways to make a topic understandable to as many people as possible. I don’t just mean a plain English description of the topic, I mean getting into the weeds and doing the work.

For example, say a paper describes an algorithm for a new kind of matrix decomposition. Showing all the steps for the matrix decomposition for two or three non-trivial matrices will make the paper much easier to understand.

Structure of Examples

In general, the structure for the worked examples would be something like

  • Here’s a trivial example with all the steps worked out.
  • Here’s something less trivial with the steps that differ from the trivial example worked out.
  • Here’s a non-trivial example with all the steps that differ from the two previous examples worked out.

You can add examples if you need them, but you shouldn’t go below two examples.

Non-Existence Proofs

You can even give examples for non-existence proofs by giving examples that might seem to work, but don’t. These examples might be the most important. If you ever teach limits in Real Analysis, I’d recommend giving an example where you use the wrong value for a limit in an (ϵ, δ) proof as I did in a previous article.

An Example is Worth a Thousand Words

Worked examples can even include information not included in the body of the paper. For example, I found a paper that claimed that you could write every matrix in a finite field as a sum of a periodic and nilpotent matrix. I could use this decomposition in my series on finite fields if the nilpotent and periodic matrices commute.

The author did not mention whether the matrices commuted in the paper, but if the paper had a few worked examples, I could check if the example matrices commute. If any of them didn’t, I would know that I couldn’t use the decomposition.

Explicit, Searchable Names

Using explicit names that other people can look up is another great way to help people find more information. I spent several days finding something akin to a derivation of an equation in a paper because the author called the important parts of the equation “coupling coefficients” instead of using full names like “electro-striction tensor.”

I spent weeks trying to find proofs for certain properties of Sturm-Liouville operators. If any of the resources said “Sturm-Liouville Comparison and Separation Theorems,” I would have spent an hour at most. For this reason, I started writing searchable terms in bold in my two main series.

Conciseness is about density and clarity, not length.


To be concise is to give a lot of information clearly and in a few words. The purpose of conciseness is twofold.

First, using fewer words means the reader wastes less time.

Second, the more work a reader has to do, the more likely they are to misunderstand the text.

Processing each word requires some work, so using fewer words means fewer chances for a misunderstanding.

What’s the Problem?

While pop-science often satisfies both of these goals by leaving out technical mathematical derivations, many papers satisfy neither of these goals. Having to read entire textbooks to find the few paragraphs of background information needed to understand how to implement a single algorithm is a huge waste of time, so the compressed nature of papers fails this goal.


Standard text in the English language can be compressed to around an eighth of the size. For example, I could compress 1,000 characters of text to around 125 bytes. You’ll find similar statistics for almost any language.

From the work of Claude Shannon, we know that this redundancy functions as a method of error correction or noise reduction.

Entropy is not disorder and you can compress text more than what’s said in the video, but eh.

I bring this fact up because scientists and mathematicians write papers in a compressed format. This format is so compressed that they’re borderline illegible unless you’ve trained to read the papers in the relevant field.

It’s like trying to read a zip file byte by byte. I could do it if I had the decompression algorithm and some paper, but it would be much faster to read if it were written in plaintext.

In this sense, using fewer words has increased the amount of work the reader needs to do, leading to a net gain of possible misunderstandings.


I’ve already talked about the gap between pop science and expert science, but expert science has gaps within itself. In this section, I want to suggest some ideas to help promote cohesion within expert science.

Background Information

Including more background information within a paper would help all audiences understand your paper. Each author has their own specialties, idiosyncrasies, and points they want to emphasize.

Having many authors cover the same material would allow the good parts of one author’s version to compensate for the flaws of another.

For this reason, I say that the best teacher is two teachers. (I also say it to give English teachers an aneurysm.)

Read the Citations

I know someone will say that you’re supposed to read the cited papers to find the relevant background information. Then, you read the papers those papers cite, and repeat the process until you understand the original.

Doing so would be like trying to learn a language by reading a dictionary and a grammar rule book. While dictionaries and grammar rule books have their uses, they’re no substitute for reading the language in its natural environment.

In their current state, papers are often written as islands connected to each other by single sentences. If you’re a programmer, you could draw an analogy between the way scientists write papers and spaghetti code.

To give you a sense of what it feels like, imagine how confusing this article would be if I mixed up all the sentences. Even if I put a link to the previous sentence in next to the sentence, you still couldn’t read it. While the information is still there and it is all related, there’s no cohesion.


Textbooks are part of the answer to the point about cohesion, but they’re only for papers written decades ago. In practice, going from a textbook to a modern paper can be quite difficult.

Modern Textbooks

For this reason, some people should collect all the ideas from related papers into a cohesive structure every few years.

It doesn’t need to have ideas from every paper ever written in the field, but it should try to put ideas into some kind of tree structure and establish some background information.

How Would You Even Write this Textbook?

Pick a few modern papers important to the field and put them at the end of the textbook. Then, write all the background information you would need to understand these papers. Put this background information in the first half.

Add related papers to the end of the textbook. Write the background information you would need for the new papers, then put it in the first half. Repeat this process until you have a decently-sized textbook.

Institutional Problems with the Science Communication Pipeline

I’ve talked about personal things you can do, but we often have institutional problems. Institutions pressure scientists to publish as many papers as possible. The science journalism machine pressures scientists to publish spicy, positive results.

Journals don’t look at retraction studies or negative results. Recommendation algorithms and our monkey brains pressure science journalists to write clickbait. This clickbait consists of exaggerating or adding claims not found in the source.

The audience eats up the clickbait, which leads to them being liable to read more clickbait. Their desire for clickbait makes writing clickbait more profitable. You can find a full discussion here.


Science communication is one part science and one part communication. You need both to do the job well. I’ve listed some of the major things you can do to become better in science communications, but I’ve left many things out.

I recommend you talk to science communicators you like and observe what made their communication effective. If you have any suggestions or explanations that made a topic clear to someone, let me know in response. Since I started with Feynman, I’m going to leave you with Feynman’s words on teaching.


If you liked this article, you’d probably like Ingrid Daubechies and the Importance of Mathematical Communication, the article that inspired this one.

As a bonus, one of my responses to that article has a lot of links about Information Theory. If you want to support me, you can read and share all my articles, which you can find on my website. You can also sign up for Medium and make me your referred member.

Lastly, I’m going to be streaming this week. If you have any questions about my articles or other topics or want to see some programming or article writing, feel free to drop into the stream.




July 2022

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