It's Time For An Intuition-First Calculus Course

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.

This is a companion discussion topic for the original entry at

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.

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!

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!

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!

@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 :).

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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!

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.

Once again, You’re on it! and so am I :slight_smile:

Some of this is similar to the point of view we’ve taken with “mooculus” which you can find at 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.

@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 (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 ( 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.

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.

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.

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 (hosted in Amazon cloud). Of course, you can host the edx-platform on your own server.

@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!).

Can I prebook for the next session ?

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.

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

@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? :slight_smile:

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

@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!