A friend of mine brought this up about two months ago and so I want to approach it from a more thorough math perspective.

Occasionally you’ll hear that something has an extremely steep learning curve. Naturally we think steep and imagine scaling a sheer wall (or something along those lines) and the sense is that this must be an arduous undertaking. In reality, that’s actually pretty far from the case.

I’ve set up the “Learning Curve” as a graph, with skill level (y-axis), as a function of time spent learning/practicing (x axis). For non-math people this means we are observing what happens to your skill level as you spend more time practicing.

First of all, I will make the assumption that the relationship is positive – so the more time you practice, the better you get. It doesn’t seem very reasonable to think otherwise – at worst your true skill level should remain constant (even if it feels like a particular performance is uniquely horrible). At the very least, this is likely true for the average person, and so we’ll ignore the fact that you’re a freak with no real skills while you go cry in the corner.

I’m going to start with two people learning to yodel, with each one’s learning curve represented by one of two straight lines labeled 1 and 2. (See figure below.) Observe that line 1 is steeper than line 2. Mathologists will note this means 1 has a greater slope than 2, implying that A1/B1 > A2/B2. This implies that person 1’s skill level increases more for each unit of time spent practicing than 2. In nerd-light terms 1 is getting better at yodeling faster than 2 is; more accurately if 1 and 2 spend the same amount of time practicing, 1 will reach a higher skill level than 2.

So now we know that a steeper learning curve is a good thing, but how many skills have proportional returns? I certainly can’t think of any, so I thought about what some common learning curve would look like.

Below you’ll see learning curves that should hopefully be a bit more realistic. These two curves should represent the transition from complete beginner to advanced level rider for snowboarding and skiing.

Curve 3 is meant to represent snowboarding. The first few times you try, it is hard, painful and often progress trickles as freely as chilled molasses (speaking of molasses, I’ve seen a few people give up the attempt in favor of gingerbread cookies and apple cider… losers). As the curve describes, this slow start lasts for a little while, until suddenly you find that your board is underneath you, your balance is solid and you are comfortable on the hill. With each trip to a hill you get better and better, faster and faster until you’ve earned the right to stop terrorizing skiers and risk serious bodily harm in the terrain park. Increased risk of injury and paralysis = nirvana.

On the other hand, strap on some skis and you’ll be snow-plowing down the hill in no-time. You’re riding chair-lifts and going anywhere you want in a matter of days, and then you hit that rut known as parallel skiing. It takes seconds to learn the dynamics, and years to master its application – not to mention that it requires constant practice. Unlike the proverbial bicycle, if you don’t use this skill on a fairly regular basis you’ll wind up frisky with the snow and intimate with more than a few trees. Curve 4 tells this story quite well, you start improving quickly, but gradually it gets slower and slower until it may take you a full season to fix a handful of small problems.