As much as I love discovery learning, there is a dirty little secret I really hate to have to admit: discovery is, in some ways, a cheap thrill.
It's easy to get excited about something new. It's harder to get enthusiastic for the middle. And this creates problems when it comes to developing fluency and deep conceptual understanding. I think that's why my students (who are all superb students, probably in the top top tier of all public high school students) run into troubles in math.
Sustaining enthusiasm through the MIDDLE of mastering something that is conceptually very difficult is what's really hard. But it's also where there is the greatest payoff. Getting students over their resistance and into flow is where situational motivation and projects can work wonders.
Sometimes this means helping students learn how to practice — how to develop what I tell them is called "mental motor skills." With my Geometry students this week, that amounted to giving everybody an index card first thing every day and playing "Let's Pretend" (i.e., let's pretend this is a quiz) and write down the five ways we know of proving lines parallel. I set the timer for 90 seconds and called out, "GO!." We all wrote down the five techniques we now know — corresponding angles, alternate interior angles, same-side interior supplementary angles, both lines parallel to a third line, and both lines perpendicular to a third line.
We traded and "graded" each other's index cards, remembering to find something positive that others did and circling their errors.
Then I yelled, "Flip over your card and do it again."
The first day we actually did this four times. I handed out new index cards and offered to destroy the evidence of any previous mistakes. And then we did it again — two more times.
With my Precalculus students, I had them flesh out Quadrant 1 of the unit circle on their index card.
And again 3 more times.
From years and years of piano training, I'm used to having to practice a new figure multiple times. I know how to repeat it multiple time across multiple different days so I can burn it into my mind and body. But many students seem to have missed that learning episode.
So I am giving it to them now.
With my Precalculus students, we then practiced mental event rehearsal through guided imagery afterwards. "Close your eyes and visualize Quadrant I. Find pi over 4 and visualize a point there on the circle. Picture its cosine and sine. Now go back to (1, 0) and travel to pi over 4 in the NEGATIVE direction. Where is the traveling point now? What are its cosine and sine? What is sine of pi over 4?"
"New point: now visualize 2 pi over 3. What is the cosine of 2 pi over 3?" I pause. "Now go back to (1, 0) and travel to NEGATIVE 2 pi over 3."
Over and over, to help them learn how to engage their will in the service of their own best interests.