What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?
I’d say it’s an eye for simplification, capturing the essence of an idea.
For example, let’s say we want to understand Ed O’Neill:
A literal-minded artist might portray him like this:
While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.
Wow! The cartoonist recognizes:
The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.
The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.
So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.
Numbers Began With Cartoons
Think about the first numbers, the tally system:
I, II, III, IIII …
Those are… drawings! Cartoons! Caricatures of an idea!
They capture the essence of “existing” or “having something” without the specifics of what it represents.
Og the Cavemen Accountant might have tried drawing individual stick figures, buffalos, trees, and so on. Eventually he might realize a shortcut: draw one buffalo symbol to show the type, then a line for each item. This captures the essence of “something is there” and our imaginations do the rest.
Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8″ instead of “three plus five is equal to eight”. Much better, right?
So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence.
Technically Correct: The Worst Kind Of Correct
We agree that multiplication makes things bigger, right?
Ok. Pick your favorite number. Now, multiply it by a random number. What happens?
- If that random number is negative, your number goes negative
- If that random number is between 0 and 1, your number is destroyed or gets smaller
- If that random number is greater than 1, your number will get larger
Hrm. It seems multiplication is more likely to reduce a number. Maybe we should teach kids “Multiplication generally reduces the original number.” It’ll save them from making mistakes later.
No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior of multiplication is misleading, can you imagine what happens when we study the formal definitions of more advanced math?
There’s a fear that without every detail up front, people get the wrong impression. I’d argue people get the wrong impression because you provide every detail up front.
As George Box wrote, “All models are wrong, but some are useful.”
A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)
Takeaways
I wrap my head around math concepts by reducing them to their simplified essence:
Imaginary numbers let us rotate numbers. Don’t start by defining i as the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.
The number e is a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.
The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape.
Euler’s Formula makes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.
Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yet accurate truth of the idea.
Happy math.
PS. Here’s my cheatsheet full of “cartoonified” descriptions of math ideas.
Other Posts In This Series
- How to Develop a Mindset for Math
- Developing Your Intuition For Math
- Learn Difficult Concepts with the ADEPT Method
- Brevity Is Beautiful
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Intuition, Details and the Bow/Arrow Metaphor
- Finding Unity in the Math Wars
- Why Do We Learn Math?
- Math As Language: Understanding the Equals Sign
- Learning math? Think like a cartoonist.
- Learning To Learn: Intuition Isn't Optional
Khalid, could you better explain to me the concept of “i” being a 90 degree rotation? I would be really interested. I just love connections and concepts
Thanks
Diane
Definitely. Try checking out the article here, there’s some diagrams and a video as well:
http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
In a nutshell, if we consider positive numbers to be forward, negative numbers to be backward, then imaginary numbers are sideways.
The main idea for imaginary numbers is that we don’t need to be stuck on the number line, we can take our numbers into 2d :).
Enjoy your articles——–> keep them coming! Thanks.
Thanks Jeffrey, I appreciate it.
What’s funny is that all of these concepts are available in high school, where Euler should be taught in freshman year.
Hi Tim! I totally agree – the practical essence of Euler’s formula can be taught early on. Later, we can dive into the details (if needed).
Your articles are excellent! So useful for me as a maths teacher in Italy. I want my students to create cartoons on any concept I teach them – I’m sure it will help them remember as well as making it fun!
Thanks Dora, that’s a great idea. Trying to visualize ideas on your own is a good way to come up with new explanations, and one of your students might have a neat cartoon that helps the whole class!
Always a treat to see your insights !
Kalid, you’re the best!
Has someone ever asked you for permission to translate your articles to Portuguese? Sometimes I think about it. Could I do it with all the references? Actually, all your site, your book and your course deserve a full translation.
Kalid, please keep doing what you do, it’s amazing.
Neat summary!
@Pierre: Thank you!
@Frederico: Thank you! Yes, feel free to translate any articles you like and send me the link (or the text, and I can post them here). I’m keeping a list here: http://betterexplained.com/translations/
@John: Appreciate the encouragement, hope to keep going for a long, long time.
@Steve: Thanks!
Awesome. Are the Drawings yours?
@Mark: No, but I wish! They are from the blog post linked: http://www.tomrichmond.com/blog/2008/02/14/how-to-draw-caricatures-1-the-5-shapes/
Thanks for sharing this! We love the content and hope to keep seeing more great posts!
@notAprodigyInc
http://www.facebook.com/notAprodigy
Wonderful as usual. I try to get students to think of written math as pictograms, like Chinese writing. Sadly, I don’t reach everyone with this analogy but those that get it seem to love it.
Peter
@Peter: Thanks! Yep, not every analogy can work for everyone (no more than a food that everyone loves), but it’s at least a starting point. There might be other variations that people like if the first version didn’t click.
Thanks for this post!
I visit the site every once in a while during the year and find something interesting always; you seem to have a gift for teaching.
By the way, this is a little out-of-topic, but there’s a interesting course on Coursera called: “Learning How to Learn: Powerful mental tools to help you master tough subjects”, that started recently and seems to promote similar ideas to learn/understand more effectively complex topics, amongst other things.
hey brother will u please explane us about sequence and series, as welll as its relation with complex no., i mean both in real and complex part
Hey Khalid, I really like the way you teach and explain things. Can you please help me visualize dot product and cross product of vectors?
@Roberto, thanks for the pointer! I’ve just gotten in touch with Prof. Oakley, who is running the course ;).
@abhisek: I’d like to cover this topic too, thanks for the suggestion!
@Yash, I have an article on the dot product here: http://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/
I don’t have one on the cross product yet, but would like to cover it down the line.
I teach statistics grad students to communicate and collaborate with non-statisticians. One module in my class is explaining statistics terms to non-statisticians. I make a big effort to distinguish an “explanation” from a “definition” and I give examples, but many of my students still “explain” terms via a definition. Ironically, I don’t think I explain “explanations” well enough, i.e., I can’t think of any analogies or diagrams to differentiate the two, just examples and technical differences. Do you have any tips for explaining the difference between an explanation and a definition?
I am so glad that I came across this site. I think this is the place that would be for my 4 year old. So glad that he and me will learn together mathematics
Hello Kalid,
Keep up the good work.
@Eric: this Ted talk by Tyler DeWitt partially addresses your concern. http://www.ted.com/talks/tyler_dewitt_hey_science_teachers_make_it_fun/transcript?language=en
Hi;
I like a lot of your articles, but i have a notice, i am Egyptian, and i think that your writing style is nearer to English speakers than others, please take this point into your consideration.
Thanks
Thanks – love your writing, and the way that you express math ideas. Thanks for sharing.
With this particular one I feel compelled to share a corollary. I try to have kids transform all problems into visual problems (from word problems to picture problems). I want them to visually see the problems.
We do this with the operations as well. From basic arithmetic upward I have them draw their math ideas out. It leads to some amazing discussions and helps me know what math concepts students need to be retaught.
Students are often shy at first about drawing, so I model for them what they say, and over time they become better at it. I teach HS math, and I find that they often want to revert to the symbols since that’s all that math has typically been reduced to for them.
Thanks again for sharing!
@Ehab: Ah, thanks for the feedback! I often write conversationally, which means a lot of slang gets in — I’ll try to keep that in mind :).
@Dan: Really appreciate the note, thank you. I like the idea of trying to find some (any) visualization for the ideas being presented. Not all can be visualized, but the exercise gets you thinking about how to put a concept into your own words. I love helping out teachers, so it feels great if this article sparked something useful!
Kalid,
You are…..AWESOME.
I loved these words of yours”So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence.” Your ‘idea of
mathematicians really being cartoonists’ is really great.
LOVE YOUR WEBSITE!!!!!
@Varun: Thank you!
Thanks for the posts. Keep them coming. These are opening my eyes and I have been teaching Algebra for 6 years. Thank you.
Thanks Ryan, really glad to hear :).
Nice article as usual Kalid. Learning a lot everytime I get your newletters…thank you!
Hi Kalid,
Thanks for the article. Interesting and thought-provoking, as usual.
In the example of multiplication, wouldn’t it be more accurate so say “multiplication changes the size of things”? This would then apply better to negative numbers and numbers between 0 and 1. Less need for future “concept refactoring”
My point is that some caricatures are better that others…
@sb: Thank you!
@Massimo: Good feedback — I’d say multiplication changes the properties of a number. Normally we think numbers only have one property, their size, but they can have another (a direction: positive or negative), which can be changed by multiplying by -1. Later, we learn that numbers can have other directions (up/down, with imaginary numbers) and multiplication changes that too.
That said, totally agree that some caricatures are more expressive than others! (A stick figure caricature is too reductive to be useful.)
Khalid,
Your mail has now replaced my early morning coffee , much more delicious !!
Love it and request to please keep it going.
Thanks!
Thanks Malini, glad you’re enjoying it!
Good point.
I think we should never forget that we are human beings, and human beings are different from computers. Our brain can integrate things in a way we still cannot figure out till now. We should follow the way the brain works. Clearly, simplification and abstraction is one of the special properties of our brain. It will make us absorb things easily and efficiently.
I find ur teaching methodology quite fine and impressing by the way you argue out even in tough ideas you make them look simplified than ever. I do enjoy ur coaching. I would like to learn mre abt series an finding sum to infinity. Thnks.