Cognitive load when learning mathematics: Difference between revisions

Revision as of 23:15, 4 August 2018

This page lists some math-specific stuff relating to cognitive load.

Some things that increase strain on working memory:

If a proof has steps $P_{1},P_{2},P_{3},\ldots ,P_{n}$ the proof is much easier to follow when the justifications for a step $P_{j}$ are close to $P_{j}$ . When much earlier stages of the proof are required to understand the current step, the reader has to go back to remember what was done early on. If $J(j)$ is the set of justification numbers used on step $P_{j}$ (e.g. maybe $J(5)=\{1,3\}$ ), then the quantity of interest is something like $\max _{j\in \{1,\ldots ,n\}}(j-\min J(j))$ , which is the most number of steps the proof stretches backwards (or maybe it's the average of this quantity, or the average length of "stretching back").
Level of nesting: nested sums, nested products, nested quantifiers, nested loops, combinations of nesting.
Number of variables in use. As a special case, the number of indices in use.
Numerical references, like "Lemma 5.8.9" and "Equation (63)". This is similar to the justifications one for proofs, but it also involves going outside the proof to look up statements.

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 Some things that increase strain on working memory:
-* If a proof has steps <math>P_1, P_2, P_3, \ldots, P_n</math> the proof is much easier to follow when the justifications for a step <math>P_j</math> are close to <math>P_j</math>. When much earlier stages of the proof are required to understand the current step, the reader has to go back to remember what was done early on. If <math>J(j)</math> is the set of justification numbers used on step <math>P_j</math> (e.g. maybe <math>J(5) = \{1,3\}</math>), then the quantity of interest is something like <math>\max_{j\in \{1,\ldots,n\}}(j - \min J(j))</math>, which is the most number of steps the proof stretches backwards.
+* If a proof has steps <math>P_1, P_2, P_3, \ldots, P_n</math> the proof is much easier to follow when the justifications for a step <math>P_j</math> are close to <math>P_j</math>. When much earlier stages of the proof are required to understand the current step, the reader has to go back to remember what was done early on. If <math>J(j)</math> is the set of justification numbers used on step <math>P_j</math> (e.g. maybe <math>J(5) = \{1,3\}</math>), then the quantity of interest is something like <math>\max_{j\in \{1,\ldots,n\}}(j - \min J(j))</math>, which is the most number of steps the proof stretches backwards (or maybe it's the average of this quantity, or the average length of "stretching back").
 * Level of nesting: nested sums, nested products, nested quantifiers, nested loops, combinations of nesting.
 * Number of variables in use. As a special case, the number of indices in use.
 * Numerical references, like "Lemma 5.8.9" and "Equation (63)". This is similar to the justifications one for proofs, but it also involves going outside the proof to look up statements.