It’s Time For An Intuition-First Calculus Course

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Summary: I’m building a calculus course from the ground-up focused on permanent intuition, not the cram-test-forget cycle we’ve come to expect.

Update: The course is now live at http://betterexplained.com/calculus


The Problem: We Never Internalized Calculus

First off: what’s wrong with how calculus is taught today? (Ha!)

Just look at the results. The vast majority of survivors, the STEM folks who used calculus in several classes, have no lasting intuition. We memorized procedures, applied them to pre-packaged problems (“Say, fellow, what is the derivative of x2?”), and internalized nothing.

Want proof? No problem. Take a string and wrap it tight around a quarter. Take another string and wrap it tight around the Earth.

Ok. Now, lengthen both strings, adding more to the ends, so there’s a 1-inch gap all the way around around the quarter, and a 1-inch gap all the way around the Earth (sort of like having a ring floating around Saturn). Got it?

Quiz time: Which scenario uses more extra string? Does it take more additional string to put a 1-inch gap around the quarter, or to put a 1-inch gap around the Earth?

Think about it. Ponder it over. Ready? It’s the… same. The same! Adding a 1-inch gap around the Earth, and a quarter, uses the same 6.28 inches.

And to be blunt: if you “learned” calculus but didn’t have the answer within 3 seconds, you don’t truly know it. At least not deep down.

Now don’t feel bad, I didn’t know it either. Only one engineer in the dozens I’ve asked came up with the answer instantly, without second-guesses (my karate teacher, Mr. Rose).

This question has a few levels of understanding:

  • Algebra Robot: Calculating change in circumference: 2*pi*(r + 1) - 2*pi*r = 2*pi. They are the same. Calculation complete.

  • Calculus Disciple: Oh! We know circumference = 2*pi*r. The derivative is 2*pi, a constant, which means the current radius has no impact on a changing circumference.

  • Calculus Zen master: I see the true nature of things. We’re changing a 1-dimensional radius and watching a 1-dimensional perimeter. A dimension in, a dimension out, it’s like making a fence 1-foot longer: the initial size doesn’t change the work needed. The gap could be made around a circle, square, rectangle, or Richard Nixon mask, and it’s the same effort for similar shapes. (And, silly me, I’d forgotten the equation for circumference anyway!)

We can be calculus warrior-monks, cutting through problems with our intuition. Notice how the most advanced approach didn’t need specific equations — it was just thinking about the problem! Equations are nice tools, but are they your only source of understanding?

See, according to standardized tests and final exams, I “knew” calculus — but clearly only to the beginning level. I didn’t immediately recognize how calculus could help with a question about making a string longer. If you asked someone for the amount of cash in a wallet with six $20 bills, and they didn’t think to use multiplication, would you say they’ve internalized arithmetic? (“Oh geez, you didn’t tell me this would be a multiplication question! Could you set up the problem for me?”)

I want you to have the intuition-first calculus class I never did. The goal is lasting intuition, shared by an excited friend, and built with the test of “If you haven’t internalized the idea, the material must change.”

How Can We Make Learning Intuitive And Interesting?

With Progressive Refinement. You may have seen these two methods to download and display an image:

  • Baseline Rendering: Download it start-to-finish in full detail
  • Progressive Rendering: Download a blurry version, and gradually refine it

baseline vs progressive

Teaching a subject is similar:

  • Baseline Teaching: Cover individual concepts in full-depth, one after another
  • Progressive Teaching: See the big picture, how the whole fits together, then sharpen the detail

The “start-to-finish” approach seems official. Orderly. Rigorous. And it doesn’t work.

What, exactly, do you know when you’ve seen the first 20% of a portrait in full resolution? A forehead? Do you even know the gender? The age? The teacher has forgotten that you’ve never seen the full picture and likely can’t appreciate that you’re even seeing a forehead!

Progressive rendering (blurry-to-sharp) gives a full overview, a rough approximation of what the expert sees, and gets you curious about more. After the overview, we start filling in the details. And because you have an idea of where you’re going, you’re excited to learn. What’s better: “Let’s download the next 10% of the forehead”, or “Let’s sharpen the picture”?

Let’s admit it: we forget the details of most classes. If we’ll have a hazy memory anyway, shouldn’t it be of the entire picture? That has the best shot of enticing us to sharpen the details later on.

How Do We Know If A Lesson Is Any Good?

With the Pizza Box Test. Imagine you pass a dumpster while walking home. You see a message scrawled on a discarded pizza box. Is the note so insightful and compelling that you’d take the pizza box home to finish reading it?

Ignore the sparkle of a lesson being digital, mobile-friendly, gamified, interactive, or a gesture-based hologram. Would you take this lesson home if it were written on a pizza box?

If yes, great! Clean it up and add in the glitz. But if the core lesson is not compelling without the trimmings, it must be redone.

Everyone’s “pizza box” standard varies; just have one. Here’s a few things I wish were written on the boxes outside my high school:

  • Psst! Think of e as a universal component in all growth rates, just like pi is a universal factor in all circles…

  • Hey buddy! Degrees are from the observer’s perspective. Radians are from the mover’s. That’s why radians are more natural. Let me show you…

  • Yo! Imaginary numbers are another dimension, and multiplication by i is a 90-degree rotation into that dimension! Two rotations and you’re facing backwards, aka -1.

How Do We Know What’s Best For The Student?

By focusing on what future-you would teach current-you.

Teachers, like all of us, face external incentives which may interfere with their goals (publish or perish, mandated curriculum, need to impress others with jargon, etc.). The test of “What would future-me teach present-me?” helps me focus on the essentials:

  • Use the shortest lessons possible. There’s no word count to meet. The same insight in fewer words is preferred.

  • Use the simplest language possible. It’s future-me talking to current-me. There’s nobody to impress here.

  • Use any analogy that’s memorable. I’m not embarrassed by “childish” analogies. If a metaphor excites me, and helps, I’m going to use it. Nyah.

  • Be a friend, not lecturer. I want a buddy, a guide who happened to experience the material before I did, not a pompous schoolmarm I can’t question.

  • Point out the naked emperor. Most calculus classes cover “limits, derivatives, integrals” in that order because… why? Limits are the most nuanced concept, invented in the mid-1800s. Were mathematicians like Newton, Leibniz, Euler, Gauss, Taylor, Fourier and Bernoulli inadequate because they didn’t use them? (Conversely: are you better than them because you do?). Most courses are too timid (or oblivious) to question the strategy of covering the most elusive, low-level topic first.

  • Learn for the long haul. The elephant in the room is that most math courses are a stepping-stone to some credential. Future-me doesn’t play that game: he only benefits when current-me permanently understands something.

Sign Up To Learn More

Let’s learn calculus intuition-first. The goal is a lasting upgrade to your intuition and storehouse of analogies. If that doesn’t happen, the course isn’t working, and it will be enhanced until it does.

Sign up for the mailing list and I’ll let you know when the course preview is ready, in November.

Happy math.

Kalid Azad loves sharing Aha! moments. BetterExplained is dedicated to learning with intuition, not memorization, and is honored to serve 250k readers monthly.

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26 Comments

  1. Fantastic pedagogy principles, I hope someday curriculm designers and educationist will follow these. Will the course be available as a series of small articles on better explained or on u tube or through email.

  2. Some of this is similar to the point of view we’ve taken with “mooculus” which you can find at https://mooculus.osu.edu/ Video of a physical person writing on paper evokes more “friend” than “lecturer.” From your progressive point of view, we start off with the metaphor of functions as transforming input into output, which later gets refined with the “covarying” perspective when we introduce derivatives. We start off with limits, do some derivatives, which then lets us go deeper into limits, and so forth.

  3. @gulrez: Glad the approach clicked with you, I hope the intuition-first philosophy can spread as well :). I plan on releasing the articles as a series on this site, but want to do a few rounds of feedback from email readers first. Eventually it’ll be something similar to railstutorial.org (free written content, with premium videos/workshops, etc. if you’d like more depth).

    @Jim: Cool, I have to check that out! I just glanced through your video about the derivative of x^n (https://www.youtube.com/watch?v=XIXctIdQxwg) and love that you show the process of a cube growing. We don’t want to memorize a rule about “bringing down n”, we want to imagine a few surfaces growing. (A cube has 3 surfaces, each growing by x^2, so d/dx x^3 = x^2 + x^2 + x^2 = 3x^2).

    I agree about the cycle of overview, sharpen, overview, sharpen.

  4. I have an honours degree in Applied Mathematics that I earned a long time ago. I recall that Calculus might have been my least favorite course and judging from the fact that I had absolutely no intuition on how to approach your “string” problem, I can see why. I am looking forward to trying this new course that you are developing. Thank you for sharing your excellent work so generously.

  5. Kalid -

    I love this original post about “intuition learning”.

    What is especially “intuitively-resonant” to me, as a 30+ year teacher who matches your teaching/learning preferences and have used them all my career despite “pushback”, is seeing the “global” (connected-whole) fuzzy picture first, then filling in the details and the connections to the rest as interest dictates. This not only seems to work much better than the fractured/chopped up pieces we teach/test now at elementary grades especially, but is actually how the brain prefers to learn as shown by increasing new educational neuroscience — the “global” Right brain grasping the whole picture and context, the Left focusing in the details that sparkle like diamonds in the Right’s perception, based on what is of most intuitive interest to your own understanding.

    I just loved this explanation — and will use it in my work. Thank you deeply for this!

  6. @Gerry: Thanks for the note! I’m looking forward to collecting a bunch of feedback, then getting the course out there :).

    @Harish: Thanks!

    @June: Really appreciate the comment, so glad it helped! I love hearing from teachers in the field who have experience with what truly works. (For me, I try to explore what seems to *actually* work for me, and hope it is effective for other people too).

    You’re right on the brain learning as well. We try to teach foreign languages via the ‘chop-up’ method (here’s the vocab, here’s the grammar, plug the vocab into the grammar), and we wonder why people still can’t order a hamburger after 4 years :). We need to appreciate the details and big picture together (and my preference is big picture, then details!).

  7. Thank you Kalid for all the material you have posted on you site. I teach in a school that would love to be progressive. Large open learning spaces, introduction of ipads, etc. The trend in Aus at the moment in education is to concentrate on style and delivery, in fact, anything that does not relate directly to the content. Unfortunately, I also teach in a school where the clientele are inclined to be algebra robots. many of their eyes glaze over if I launch into an intuitive approach. The overwhelming sense I get from many of them is: “stop trying to make us understand it. Just teach us how to pass the exam!” It is my ambition to try to break down this resistance, with things like: “what do you need u and v for when you do calculus. Just think about the rules in words (as well as why the rules work!)” I once introduced integration before differentiation using a strips approach. Later on, we made the connection between the 2. It didn’t work because of the built in resistance of the students. “Why is he doing it the wrong way around? Should we trust him?”
    However, I will keep trying.
    Sincerely,
    Adrian.

    PS: My analogy for fog and gof (composite functions) f is a kitchen blender, g is a cement mixer. You could put any of the output of f into g without any dire consequences (so gof exists) However, some of the output of a cement mixer would not be appropriate as input for the blender. It would break it, just as negative numbers break the square root function and zero breaks the hyperbola (so fog does not exist). Hope you like it

  8. Hi adrian, thanks for the note! It’s really unfortunate, we’ve been conditioned to just cram and pass the test because, quite often, there’s not much intuition being taught. I think it’s going to be a gradual process of showing people “Hey, this stuff can actually click” — almost like approaching a scared animal! It’s awesome that you’re going to keep pushing forward, the world needs more teachers like you.

    That’s a great analogy with the f o g and g o f. I might adjust it further by making the two operations quite different, say, blending and baking. If you blend a cake mix then bake it, you’ll get different results than if you bake the mix then blend it :).

    As you write, certain inputs might break one item or another — the cement in the blender is a good visualization of that!

  9. I happen to be one of those High School mathematics instructors who enjoy, as one of the recent posts noted, to digress from topic to explore the nature of zero, discuss infinity, measuring time,etc… and then under the onus of district demands to”complete the designated curricular map, administer the designed assessments…”, only to get back on topic and complete the proofs. Over the past few years, however, I have been spending more time with intuitive based instruction, attempting to allow students to understand the “why’s” of mathematical concepts rather than simply progressing through the content in the traditional a priori manner. I have noticed two results. I have increased my own understanding and appreciation of mathematics and my students are more engaged. Secondly, once students are so engaged, their actual comprehension and applications of formulas increases. I am looking forward to joining your Calculus course. A good discussion ensued when I shared your string example with an engineer friend ( he didn’t get it right away either!). The only alteration I used: a marble, a basketball, and the earth; which made the example easier to visualize for students as opposed to a ‘flat’ quarter. Great stuff!

  10. I think this is brilliant. Seeing the goal of learning
    as the creation of intuition is empowering.
    I remember teachers using the baseline method answering my questions about the bigger picture of a topic with
    “You don’t need to know that” and I wish they had had the freedom to give us the big picture first and then flesh it out.

  11. @james: Thanks for the thoughtful comment! You nailed it, getting an intuitive understanding can deepen the technical one. They go hand in hand. And yep, once students are interested, future learning becomes that much easier.

    The Calculus course should open up in September (for the text version) and I may run another “guided tour” then also. Stay tuned :). By the way, I love the marble example — it’s better to compare a sphere to a sphere! Having 3 is also nice, since you can actually perform the experiment with a marble and basketball.

    @Sam: Glad you liked it. Exactly, if we aren’t building lasting intuition, what’s the point of learning? A backup copy of dry facts stored in human brains, instead of on paper? :)

    Spending just a few minutes on the big picture can make the entirety of the resulting class so much more pleasant.

  12. I enjoyed reading this so much. I can’t wait to go through the material you write for this course.

    I think that your mind thinks about things in a very clear way, and furthermore you have the ability to express them that way also. It’s beautiful!

    Thanks a lot.

  13. @RB: Thanks for the interest, I’ll be announcing updates for the September release on the newsletter list. I haven’t determined the prebook approach yet but it should be open to a wider audience.

    @Peter: Really glad you liked it! I’m looking forward to getting this finalized as well :).

  14. I did not really understand how it is really about dimensions in the circle quiz until I read the “Calculus Zen Master” interpretation and subsequently tried the same thing out on a sphere, copying its surface and expanding it in such a way that we can have it a unit off the original sphere. Here, the amount added actually depends on the size of the unit!

  15. Hi Kalid,

    Thank you for this eye-opening article, especially the baseline vs. progressive methaphor. I’m a computer programmer and my current project requires me to learn about logistic regression. Alas, that topic is located at the end of every introductory statistic books I’ve checked so far. I need to make a presentation next week, but reading these “introduction to statistics” books really feel like watching a baseline download unfolds.

    After reading this article, I wonder if I can find a learning resource on statistics that can teach me basic things I need to know so that I can understand logistic regression. Not too deep, but not superficial either.

    But at this moment, I’m planning to fast read the book chapters all the way to logistic regression, and then go back to the beginning and fast read again while picking up the bits in more details. This is the closest I can think to achieve a “progressive download” using a learning resource intended for a “baseline download”.

    BTW, are you planning to use a MOOC platform? If so, I recommend edx-platform. It’s open source and used by Berkeley, MIT and Harvard. Also, check http://www.edx.org (hosted in Amazon cloud). Of course, you can host the edx-platform on your own server.

  16. @clarue: Ah! Yes, when we are dealing with higher dimensions like area, volume, etc. the size of the existing shape matters. In my head, I see the dimensions “interacting” with the other measurements (area is length * width, for example). But, a single dimension interacts with nothing else, it just grows on its own, so in my head it doesn’t need anyone’s “permission” to change (making a rope longer means we just add some to the end).

    @Ndaru: Thanks for the feedback! I’m on the lookout for an intuition-first stats class as well (after I finish up this Calc series… :)). A fast reading can be a great approach, giving you an idea of applications / more complex examples, so you can see what you’re building up to (vs. going step-by-step, staring at our feet).

    I’ve gone back and forth with the course format; in the short term, I might just host it myself (on this site) as simple webpages + videos. That way I can get the exact format I need, and branch out down the road. I’ll be curious to check out the EdX platform though. Thanks!

  17. Kalid,

    I am just about finished with your book, “Math, Better Explained.” I love it! The way you present the topics is incredibly easy to follow. I’m very fortunate to have stumbled upon this website (not literally) to see that there is a wealth more of your stuff to learn from.

    Certainly helps that you’ve got some wit and humor. Helps me maintain motivation to really grasp a concept. Look forward to perusing your website and am excited for your intuitive calculus course.

    Thanks,
    Derek

  18. Thanks Derek, I really appreciate the kind words. I try to write things as I wish a teacher told them to me — and if someone isn’t having a good time when they’re explaining, you’re probably not having a good time when learning! =). The course should be out in the very near future. Thanks!

  19. I need help in INTUITING DARK ENERGY. I know what it is. My phone is 301 774 9256 or send me an E-mail with a contact number. My wife looks at E-mails, I do not.

  20. I really don’t understand how the effort would be same when there is a change in shape there will be a change in effort too…you mean to say putting a fence around a 200 meter circle and 400 meter circle will require the same effort is same..

    Calculus Zen master: I see the true nature of things. We’re changing a 1-dimensional radius and watching a 1-dimensional perimeter. A dimension in, a dimension out, it’s like making a fence 1-foot longer: the initial size doesn’t change the work needed. The gap could be made around a circle, square, rectangle, or Richard Nixon mask, and it’s the same effort for similar shapes. (And, silly me, I’d forgotten the equation for circumference anyway!)

  21. Today was an important day in my understanding of calculus. It took me a long, hard year to understand the fundamental theorem, and then it took me three more months to understand Taylor’s theorem. But today I could roughly see how the nth derivative of c , always somewhere between a and b, decreases the error term in direct relation to the number of terms, and as the denominator becomes proportionately large.

    In my own experience, the fundamental theorem should be introduced in the beginning, maybe just a glimpse or two, so the student knows where the derivatives and integrals are going. Likewise, with the Taylor theorem, it wouldn’t hurt to see, however abstract, the remainder theorem first.

  22. Thanks Tim, I agree. I like introducing derivatives and integrals together — they’re a pair — and then the FTOC seems like a natural conclusion (they really are a pair!). I don’t like teaching limits, derivatives, and integrals in isolation and then “glue them together”. I think we can have a natural intuition that the parts fit together, without waiting for the FTOC to tell us.

    Taylor’s Theorem is a good follow-up too, and I like the philosophical implications of it (successively refining imperfect models to recreate the original).

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