Exploring a Bottom-Up Approach to Learning Math, Part II of IV: When to Practice?

In our last article we discussed the idea that a Bottom-Up approach to teaching math could do more for lasting skill mastery than the more traditional Top-Down one.  Our next few articles will look at What, How, and When to practice to take advantage of this approach.  Here, we will delve more deeply into what the When part would look like.

The key element to this strategy is that repeated practice of a skill—and a quite a bit of it—should precede longer explanations and discussions about that skill.  This gives the brain a chance to do what it does best: recognize patterns and assimilate them into its bank of experience.  The goal here is to make those patterns intuitive, as natural to the learner as his or her spoken language.  THEN, after the skill feels comfortable, we go back and do all the explaining and conceptual learning. This makes the underlying reasons actually make SENSE; there’s something in your brain for them to stick to.

The most obvious objection to this is: Repetition is boring!  It’s monotonous.  It’s drudgery.  True, true, and true. Repetition can actually be mind-numbing and hinder your efforts at getting a skill to Mastery, BUT not if it’s done the right way.  Instead of endless pages of the same kind of problem (the rote busy-work we all love to bash), repetition can and should mean repeated exposure over time.  Let’s examine what that would look like.

Say you were introduced to the skill of converting a mixed number to an improper fraction.  You do a handful of practice problems, it begins to feel easy, and so it appears that you’ve got that skill down. In that moment, when it began to feel easy, you DO have that skill down.  That’s called Relative Mastery.  You have that skill in that particular snapshot in time.

A week later, let’s say you’re presented with the skill again, and you need a reminder in order to be able to do the first problem.  After the reminder and the first problem however, you’re off and running and any extra practice at that point is really not doing all that much to advance the lasting knowledge of the skill.  The extra practice at that point would fall into the rote, boring, mindless repetition arena.  Sure there’s some muscle memory development happening, but THAT FIRST PROBLEM– the one where you were like “oh yea!!! I know this!”– is where most of the magic happened.  I call that the “first-problem-moment”  and that’s the moment we want to repeat, which you won’t be able to do again until some more time has passed for you to “forget” it a little. We make this first problem moment happen again and again, strategically over time, and before you know it, you have that skill at Lasting Mastery (not merely Relative, or Snapshot Mastery).

Research has prompted many learning systems take advantage of this strategically-timed repetition of skill practice, and with great success. (Check out spaced repetition, Anki, and Pimsleur to name a few). In language acquisition, using strategic repetition means a student is asked to recall vocabulary after enough time for him to have almost forgotten it.  When the term gets roped back onto the radar just before having gotten too far off it, the result is strongly cemented understanding.  This can be just as useful in math studies.  After all, math is a language, and we’re going for fluency.

What a good “spaced repetition” learning program does is build into it as many of those first-problem-moments as it takes to get a skill to LASTING mastery.  This means that the first problem, after a long interval of NOT practicing it, feels as easy as the 10th, with no re-teaching required and no hesitation.

Stay tuned for our next article where we’ll explore the What and the How aspects of practice that can elevate a student’s math skills to mastery.