Sine of Learning

Learning acceleration is the order of the day. 😉 When would it not have been? Today, the fastest connections and the speediest professors are available online at any time for this dream. I wonder, however, whether such a dreamed-of linear acceleration of learning is feasible.

I’m doing a test run and try a fresh boost for an old hobby: “Calculus: Single Variable”, a Massive Open Online Course from the learning accelerator Coursera. I had already started this course once and abandoned it due to lack of time and energy, so at least I know it’s well done and interesting. Anyway, I also already know that acceleration and time are closely related when it comes to learning.

The course promises to be tough, and it is suggested that the preliminary refreshing of some rusty knowledge would be promising. I am already caught in the tension between being allowed to learn and having to learn, which often leads to the dilemma of learning without stress vs. stress without learning, only to end up in non-learning at all. I first read a bit about the sine function and finally find a fascinating site that gives me the boost I’m looking for, in a slightly different way. The author, Kalid Azad, attempts an intuitive explanation of the sine. He starts with a detour, a quick analogy:

You: Geometry is about shapes, lines, and so on.
Alien: Oh? Can you show me a line?
You (looking around): Uh… see that brick, there? A line is one edge of that brick.
Alien: So lines are part of a shape?
You: Sort of. Yes, most shapes have lines in them. But a line is a basic concept on its own: a beam of light, a route on a map, or even–
Alien: Bricks have lines. Lines come from bricks. Bricks, bricks, bricks.

Most math classes are exactly this. “Circles have sine. Sine comes from circles. Circles, circles, circles.” Argh! No – circles are one example of sine. In a sentence: Sine is a natural sway, the epitome of smoothness: it makes circles “circular” in the same way lines make squares “square”.


So line would be independent of bodies and sine independent of circles? Can’t be, for many extraterrestrial and terrestrial brains! From this caricature of stubbornness, we begin and the subject grips me for several hours: “Sine is a natural rhythm, the epitome of the smooth.” Sine is speeding up and slowing down toward a maximum, and then back to the middle, to the other side and back to a middle and on and on. There is a simulator on the site to explore sine and a guided tour of definitions, forces and reaction forces. “Sine is acceleration relative to my current position.”

What makes this site so engaging? The explanations are learner-friendly, they rely on insights, and they stay at a natural pace without sidetracking the reader. There are reflections on this on the site itself, such as the successful formula “From a blurry to a sharp image”.

Suddenly, however, I have the further insight: The sine also determines the rhythm of learning. The natural momentum of curiosity and satisfaction does not unwind linearly, but in a smooth, faster or slower interplay of forces, which is, however, susceptible to stress and boredom, joy and enthusiasm, to information overload and disorientation, to structure and dialogue. For every learning action there is a reaction force, which demands the necessary patience in learning and rhythmizes progress and hard periods.

Alien (a brain): So I can’t accelerate learning linearly?
You: Exactly, acceleration is not possible, but different rhythms are possible.

Sine is the acceleration or deceleration with respect to my current position. This sway of learning always needs the confidence that it will last. If I am permitted, then in a good scenario, I identify the threat of overdrive, retain curiosity and continue reading at my pace.

First written and published in German.



PS: betterexplained by Kalid Azad is very helpful, it undoubtedly explains the sine better than this text can.

Original Picture by Alex Brogan

3 Replies to “Sine of Learning”

  1. Lars Harrysson says: Reply

    In Swedish we use the word “takt” to illustrate how we align to situations. English words are time and beat, but the idea is a cultural connection between doing and being. Sine has to be looked up. Very interesting.
    A co-Alien

  2. The image of a rollercoaster that you choose to illustrate this post is most apt. They are all about since curves, acceleration and rhythm. Does your description of the Calculus MOOC reflect the approach you take to teaching or learning or does it describe an ‘ideal’ version’?

  3. Thank you for another set of fascinating insights Andy. What seems the most important for me is the pulse, or the tempo of learning. I think that each person has different capability of absorbing information, considering it meaning and feeling its impact on our life. Sometimes it comes later, after some hours or days, when we recall some conversations, discussions or texts we’ve read. I wonder what will still touch us tomorrow, next week or decade?

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