Understanding Before Fluency
Despite a century of critiquing "mere memorizing", schools still routinely ask students to commit ideas to memory without understanding them.
It's not enough to remember — you have to understand.
But the inverse of that is true, too: it's not enough to understand — you have to remember.
Schools often have a hard time getting students to do both of those, and in the right order.
Our basic plan:
When it comes to concepts that students have to struggle to understand (e.g. math formulas), imagine three stages: understanding, fluency, and automaticity.
First, help students aim for complete understanding. Help them ask a flurry of small questions about the concept, help them see different ways of representing it. Build ladders for them.
Only after students demonstrate complete understanding should they aim for fluency — the ability to explain the concept quickly.
And only after students can routinely reach fluency should they attempt automaticity — the ability to see the entire concept in their head at once.
Adults who understand things fully — and know when they don't.
Adults who understand the practice it takes to get concepts from understandable to useable — tools for solving larger problems.
If you walk into a classroom, you might see:
If three kids are working on a math problem like — let's say 6 × 5 — one might be working on understanding, and drawing out five rows of six dots, then counting them: "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15..."
Another might be working on fluency, and counting by sixes: "6, 12, 18, 24, 30."
Another might be working on automaticity, and holding all the above in her head when she reports: "6 times 5 is 30."
Taking concepts in stages is crucial for math. Where else might it be helpful?
(This idea is currently in beta! If you've thoughts on how to make it better, please shoot an e-mail to Brandon at email@example.com.)
Deep Practice Book
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Actually, I'm not sure — typically math educators advocate for a focus on memory or a focus on understanding. I'm certainly not the only person to say both is necessary, but I'm not sure who else talks about the need to systematically progress through stages.