// 01
What the notebook is for
Some lessons use the label “copy to notebook” to pick out a definition, a piece of notation, or a rule worth keeping. It invites you to write it by hand in your own notebook. You can also leave it on the screen and continue; the lesson does not check your notebook or award progress for copying.
Copying has a straightforward use: it leaves you a reference. But suppose you want to find out what you took from the explanation. Then it is worth changing the task. Read the rule, look away, and try to reconstruct it before comparing your version with the original. The interesting part is what happens when they differ.
// 02
What a reconstruction can show
Suppose the card states a fact small enough to check with a pencil. The sum of the first n odd positive integers is n squared. For example, 1 + 3 + 5 + 7 = 16, which is 4 squared. You look away, and what comes out is “the sum of the first n numbers is n squared.” The word odd is gone.
That omission is a place to look closer. Perhaps you understood the condition and simply left out the word. Perhaps you did not notice that it mattered. Or perhaps you can remember the rule but have no idea why it is true. A mismatch alone cannot tell these apart, so the next move is to try a case.
Add the first four positive integers: 1 + 2 + 3 + 4 = 10. Compare that with the odd numbers above. Dropping “odd” changes which numbers you add, and the result is no longer 16. You now have a reason to keep the condition, beyond the fact that it appeared on the card.
There is a further question: why do the odd numbers build a square? Draw a square of three cells by three. To grow it into four by four, add three cells down one side, three along the bottom, and one in the corner. That border contains seven cells, the next odd number. For a square of any whole-number side length, the same construction adds two equal strips and a corner. Each new border is odd. The formula now has a picture and a reason behind it.
// 03
A passage recalled, a week later
There is evidence for practising recall, though it answers a narrower question than whether you understand a mathematical rule. In an experiment by Henry Roediger and Jeffrey Karpicke, students studied prose passages, then either restudied them or wrote down what they could remember without looking. The researchers later tested how much of the passages they could recall.
In a study of 120 university undergraduates, recall one week later averaged 56% of passage idea units after retrieval practice, versus 42% after restudy. This study did not test our notebook cards. [1]
The comparison concerns recalling prose after a delay. It does not establish an advantage for a pen over a keyboard, or show that remembering a formula means you can use it. That is why the notebook routine above continues past remembering the sentence, into trying a case and explaining what happens.
// 04
Leave yourself something useful
Keep the corrected rule beside the example that made its condition matter. Add the square if it helped you see why. When you return, cover the explanation and see whether you can reconstruct the next border yourself. If you get stuck, the notes give you a place to resume the work.
You started with a sentence someone else gave you. Now you have a condition you can defend and a construction you can continue. Close the notebook and draw the next square. Find the next border. See how much of the idea you can make happen with a pencil.
// notes
Notes
- Henry L. Roediger III and Jeffrey D. Karpicke, “Test-enhanced learning: taking memory tests improves long-term retention”, Psychological Science, 17(3), 249-255, 2006. Experiment 1 compared restudy with free recall of prose passages, without feedback. The later test measured recall of passage idea units. It did not compare handwriting with typing or test mathematical understanding. ↩