kitchen table math, the sequel: the learning curve

Sunday, April 17, 2011

the learning curve

re: C's and my apparent learning "spike," here are Frank E. Ritter and Lael J. Schooler on the learning curve:
Most tasks get faster with practice. This is not surprising because we have all seen this and perhaps know it in some intuitive sense. What is surprising is that the rate and shape of improvement is fairly common across tasks. Figure 1 shows this for a simple task plotted both on linear and log-log coordinates. The pattern is a rapid improvement followed by ever lesser improvements with further practice. Such negatively accelerated learning curves are typically described well by power functions, thus, learning is often said to follow the "power law of practice". Not shown on the graph, but occurring concurrently, is a decrease in variance in performance as the behavior reaches an apparent plateau on a linear plot. This plateau masks continuous small improvements with extensive practice that may only be visible on a log-log plot where months or years of practice can be seen. The longest measurements suggests that for some tasks improvement continues for over 100,000 trials.


The power law of practice is ubiquitous. From short perceptual tasks to team-based longer term tasks of building ships, the breadth and length of human behavior, the rate that people improve with practice appears to follow a similar pattern. It has been seen in pressing buttons, reading inverted text, rolling cigars, generating geometry proofs and manufacturing machine tools (cited in Newell and Rosenbloom, 1981), performing mental arithmetic on both large and small tasks (Delaney, Reder, Staszewski, & Ritter, 1998), performing a scheduling task (Nerb, Ritter, & Krems, 1999), and writing books (Ohlsson, 1992).


Averaging can mask important aspects of learning. If the tasks vary in difficulty, the resulting line will not appear as a smooth curve, but bounce around. Careful analysis can show that different amounts of transfer and learning are occurring on each task. For example, solving the problem 22x43 will be helped more by previously solving 22x44 than by solving 17x38 because there are more multiplications shared between them. Where sub-tasks are related but different, such as sending and receiving Morse code, the curves can be related but visibly different (Bryan & Harter, 1897).


The learning curve has implications for learning in education and everyday life. It suggests that practice always helps improve performance, but that the most dramatic improvements happen first. Another implication is that with sufficient practice people can achieve comparable levels of performance. For example, extensive practice on mental arithmetic (Staszewski reported in Delaney et al., 1998) and on digit
memorization have turned average individuals into world class performers.

Draft version of:
Ritter, F. E., & Schooler, L. J. (2002). The learning curve. In International encyclopedia of the social and behavioral sciences. 8602-8605. Amsterdam: Pergamon.

Don't know how this relates to the experience of having a sudden jump in learning....


Glen said...

I assume you are experiencing a "phase transition". The "learning curve" describes reduction of difference phenomena, where you asymptotically approach perfection at a rate proportional to your distance from perfection. It's like a glass of cold water that warms up quickly at first, then more and more slowly as it approaches room temperature. How fast it warms depends on how far from room temperature it is.

Many phenomena where you zero in on perfection in steps that are bigger when you're badder and smaller when you're better produce a "Power Law of Practice." (Actually, I think the real curve is more exponential than power, and that the apparent "power" shape is a composite of exponentials, but that's technical detail. I'll call it a "power curve".)

Cognitive researchers base a lot of the power law literature on experiments involving simple, close the gap tasks.

But "phase transition" phenomena are different. They produce a sigmoid curve--slow rise, sudden growth spurt, then resumption of slow rise, usually as the result of changing "modes", such as a switchover from mainly reasoning to mainly recognizing. You could also be switching over from one form of memory to another.

Phase transitions also occur when systems have a lot of mutually dependent parts. Each part slowly improves toward the threshold where it becomes good enough to help related parts. As slowly improving parts start to cross the threshold of usefulness, the interconnected system as a whole suddenly springs to life.

If I had to guess, I'd guess that you transitioned from a mode of primarily relying on general math knowledge to figure out SAT problems to a mode where recognition of the problem type did most of the work, and you only had to finish off the details with general math knowledge.

This would be a phase transition and likely to resemble a sigmoid curve, not a power curve.

ChemProf said...

I've definitely seen this kind of "slow-fast-slow" curve in students learning new concepts. I usually describe it as "not being ready for the information," which is just another way of saying you need the schema first (slow) before you can fill that schema with information (fast). I'm not sure how much cognitive science research there is on these kinds of more complex tasks, though.

Allison said...

Really? The Cog Sci research isn't there?

The machine learning/agent-based AI community has been there for a decade or two. The cog sci people related to that seem to think it's old hat. The power law graphs are discontinuous and can't explain learning more over need something with smooth tops and bottoms.

Strange. Strange when something so obvious turns out to be not known by the people who should have noticed already.

Catherine Johnson said...


I doubt I'm switching from one form of memory to another since it takes 10 year to 'consolidate' memory....(have now forgotten what happens when memory consolidates, but as I recall you're probably switching parts of the brain)

I checked Gluck (Learning and Memory); the curve described isn't an S-curve.

Cognitive science has studied complex tasks like writing geometry proofs, not just simple tasks.

I haven't worked it out yet, but I'm **guessing** something related to chemprof's suggestion is involved with C. and me, which is that I think we were probably practicing problems we weren't ready to do ----

After each test, we analyzed and re-did every problem we missed, and we were probably nailing down the "pinpoint skills" or component tasks involved in the problems during that time.

I assume that if we'd been taking tests that measured increased learning of the component tasks we'd see a power curve (or exponential curve)....

I need to look at the study of people learning to do geometry proofs.

How much did they know about writing proofs when the study began?

That's the question.

Catherine Johnson said...

If I had to guess, I'd guess that you transitioned from a mode of primarily relying on general math knowledge to figure out SAT problems to a mode where recognition of the problem type did most of the work, and you only had to finish off the details with general math knowledge.

Not sure whether I'm contradicting myself...but this does seem right to me.

I've definitely been relying on general math knowledge 'til recently (& so has C., I presume) --- AND I had gaps in my general math knowledge. Haven't finished algebra 2, so quadratic functions were a mystery when I started taking practice tests.

So was any problem involving counting and probability.

C. and I also had brain freeze for any problem we lacked confidence on. At first, both of us, whenever we saw a function problem, would just collapse in a heap. Sometimes the problems would turn out to be so fantastically simple that we'd both be mortified.

C. is still having brain freeze every time he sees a circle problem.

I have now told him to cut that out.

Catherine Johnson said...

The thing is, when a test is timed and you see a problem and 'clutch,' it's difficult to un-clutch.

I've discovered that the best way to un-clutch is just to move on right away, then come back.

Every once in a while I'll clutch on two or even three problems in a row - and that's a super-clutch.

I still force myself to keep going, and I find when I come back to the problems my brain has 'relaxed.'

Catherine Johnson said...

I've definitely seen this kind of "slow-fast-slow" curve in students learning new concepts.

I'm going to ask the precision teaching people whether this is what they mean by "contingency adduction."