A Gentle Introduction To Learning Calculus

[…] Como no soy una persona de lo más iluminada ni inteligente, reiteradas veces necesito que me expliquen las cosas como si tuviese 2 años, entonces me puse a pensar ¿ qué no todo se puede explicar de esta manera ? ¿ no debería ser acaso la manera normal de explicar las cosas ? entonces me puse del otro lado y entendí que para poder explicar algo de esta manera hay que contar con un completo dominio de los conocimientos cosa que la mayoría de los docentes no tienen. Sobre todo para temas “difíciles” como matemática, álgebra, cálculo para las que el sistema de educación no puede ofrecer formas de entusiasmar a los alumnos, sino una bola de fórmulas las cuales nosotros tratamos de combinar para llegar a un resultado, perdiendo la dimensión del problema y la capacidad de resolverlo usando nuestra mejor herramienta, la imaginación (Einstein se revuelca en su tumba). Esta nota me levantó el ánimo viendo que no soy solo yo el único que piensa así, realmente creo que vale la pena leerlo (está en inglés) […]

@Doug: Thanks for the note, I’m glad you found it helpful! Good point on the note – I changed the wording a bit. It makes me chuckle when I see complex subjects (calculus) explained in terms of other complex subjects (limits, integrals, etc.), without at least some plain-english explanation. How is a beginner looking up what calculus means supposed to have an idea of what it does?

@Mark: I’m not sure I understand the connection to creation – the goal was to use evolution as an example of a simple, unifying theory that can explain a lot of natural behavior.

Animals that hated sugar, fat and other high-calorie foods probably starved when times were tough. But their siblings with a sweet tooth probably survived, which selected for that trait. Evolutionary pressure gives an explanation of why sugar would seem sweet to us today (I’m not a biologist, there may be other reasons too).

Anyway, the point is that calculus finds similar connections/underlying themes between math – there are nice (simple) reasons why the formulas are linked.

Without calculus, the similarity in the equations just looks like a happy coincidence, much like “sugar is sweet and spoiled food tastes bad” might seem like a lucky coincidence without the theory of evolution. Hope this helps clarify what I meant.

@Kat: That’s awesome! I love getting those “aha” moments and I’m happy you were able to get excited about calculus ideas (it’s a rare thing in this day and age).

You definitely can get a handle on math – I really believe it’s a skill like writing. Once upon a time, everyone thought reading & writing were “hard” and only for scribes; today everyone does it.

The hardest part about math can be staying interested and keeping your motivation, so hang in there! Seeing it as just another way to talk about an idea can help get the big picture. And you’re right, when you get it, even solving gigantic equations can be fun :).

[…] Anyway, this ramble came about as a result of reading A Gentle Introduction to Learning Calculus. […]

@Kalid:

Your implication appears to be that evolution is THE theory that provides the “aha” level of understanding the natural world. Yet the example you provided is just as easily explained by creation. It came off a bit preachy to me and detracted from an otherwise well-written article.

I always wanted to learn this calculus stuff. Tho I seemed to have survived the last 40 years of electronics and computer theory without it, I’ve always had a curiosity about just what all those squiggly lines were on the old chalk boards. I think you have succeeded in clearing up some of the fog. (so far so good anyway) Please keep up the good work you have been doing on this web site. I really have enjoyed all of your articles.

@Mark: Point taken, and happy for the discussion. I think the key point behind it all is that the sweetness of sugar serves a purpose (to help us survive) – but if we don’t notice this underlying theme then we miss many of connections that exist in the real world.

@Paul: Thanks for dropping in, and for your comment! Glad to make things clearer as I can – the funny thing is that despite using the squiggly lines many times, they tended to stay in the realm of abstract symbols without much inherent meaning. So I’m trying to go back and relearn the stuff with the viewpoint of “it has to mean something!”. I’ll keep writing as best I can :).

Absolutely magnificent. One of the best things I’ve ever stumbled upon. The analogy with finger painting only after learning chemistry/physics/anatomy is so very accurate.

Keep it up!!

Thanks Grey, I’m thrilled you enjoyed it so much! Yes, not letting people fingerpaint (with the absence of tests & grades) can destroy a child’s interest in a subject. “Drill & kill”, I’ve heard it been called.

Many Thanks for Sharing, such a valuable information.

Best Regards
Team
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Hi Kalid,

I’ve been waiting for this article/series about calculus for few months since started reading your blog. I tried to learn calculus myself few times. I’ve learned something I knew how to compute some simple examples but I’ve been missing that “Aha” moments so much. So I’ve decided that I’ll wait until you start to write about the calculus, since you explained so well every area you wrote about so far (exponential functions, natural log, complex numbers, …) and in the meantime I’ll spend my math time in other areas. I’m very lucky that I did such choice. Your article is, as always, so enlightening and clear. I’m very happy to have such a great math teacher! :wink: Thanks you so much for such material.

Also, I would like to mention the book by Keith Devlin, “The Language of Mathematics: Making the Invisible Visible”, which actually brings me to the interest in math, one or two years ago. Without that book I would probably not read this blog and would not believe in my bright math days :wink: So for others asking “Why Math?” or searching for a lot of “Aha!” moments, the Keith’s book is great reading during waiting for next Kalid’s article :slight_smile:

@Martin: Thank you for the wonderful comment – I’m glad you’re finding the articles helpful! I’ll try to keep them that way :).

Thanks for the book recommendation, I’ll need to check that out. I’m always interested in resources that can help people understand & appreciate math more.

WOW.

amazing stuff, when you first told me you were going to write an article how real world calculus I thought it’d be a stretch. this was very impressive, and made it easy to understand.

I like your approach, I used to learn very complex subjects by picking up the kids editions of things, it gave me the 80% i needed to know to be able to converse in very little time.

nice job.

It’s Paul from comment #1 again. Thanks for the reply Kalid. Again, the article is brilliant.

I wonder, does the triangle analogy also work with squares instead of discs? If the side length is x, the perimeter is 4x. I apply your awesome triangle procedure and get (1/2).x.4x which is 2x^2; but I was hoping for x^2.

Best wishes,
Paul

@Pham: Thanks man, glad you enjoyed it :). Yeah, it’s funny how explaining stuff “for kids” can force you to distill all the mumbo-jumbo into its most basic elements (and therefore making it more clear for everyone).

@Paul: Thanks for dropping by. That’s a great question – I think using a square should work. The tricky part is that even with “square rings”, we only want to take the radius (x/2).

Looking at the jagged triangle, you can see how you could bend the sides all the way around to make a circle. Thus, we’re only measuring the “outward” distance from the center, since the perimeter wraps around. Similarly for the square, you can imagine that we’re bending the jagged triangle into 4 corners – we move from the center to the right side, but the height of each line can wrap around the entire square. So we only go from 0 to x/2.

The equation turns into (1/2)(base)(height) = (1/2)(x/2)(4x) = x^2.

Hope this makes sense, I had to think about it for a bit. I think it’s weird because we aren’t used to talking about the “radius” of a square.

Keep up the fantastic maths analysis. Your diagramatic, pictoral explanations should be taught around the world.

Many thanks, glad you enjoyed it!

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Another great article from a great writer. :slight_smile:

Hi Ferenc, thanks for the support!

Amazing! Four semesters of mind-numbing calculus in engineering and I was blown away by the circle triangle example. Never really looked at such a basic relation in this light! Can’t wait for more!