You Don’t Need a Million Different Explanations

by Justin Skycak (@justinskycak) on

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One of the most common misconceptions about learning is that students need a million different explanations of the same topic until one “clicks” for them.

They don’t.

What they need is a single great explanation that’s been repeatedly battle-tested, analyzed, and refined across a large number of students, until it’s rock-solid.

And they need to have mastered all the prerequisite material that’s being leveraged in that explanation.

That’s it.

If you have to explain something a ton of different ways to a student before they can follow that explanation well enough to successfully engage in active problem-solving, then either

  1. your original explanations were not good in a pedagogical sense, or
  2. the student was lacking prerequisite knowledge and the explanation that "clicked" managed to circumvent that prerequisite knowledge (which often indicates that it's reducing the topic to a simpler case that doesn't involve the prerequisite -- which means the curriculum is watered down and the student will only be able to solve cherry-picked problems).

When you have

  • highly scaffolded, carefully curated content,
  • that has been battle-tested over a large number of students,
  • and continually analyzed to detect and further scaffold any areas where more than a sliver of students fall off the rails,
  • and has gotten to a point that 95% of students pass lessons on the first try (and 99% within two tries),

if you give a lesson to a new student who has mastered all the prerequisite material, then there’s really no excuse for them not to be able to learn it.

For lessons that have undergone this much data-driven refining, on the rare occasion that a student does struggle with it, it doesn’t mean that the lesson needs to explain things in a different way.

Usually, all it takes to rebound is a bit of rest and a fresh pair of eyes. And then the same exact content will “click” the next time around.

Follow-Up Questions

Q: But a what’s a “single great explanation” for some people isn’t necessarily one for others.

A: What’s a specific counterexample of a granular math topic and 2+ different explanations that it would require?

Constraints:

  • Topic must be an atomic cognitive "chunk" (e.g., if a theorem has multiple proofs then each of those proofs would be a separate topic)
  • The 2+ different explanations can't reasonably combined into a single better one
  • The student only needs to understand 1 of the explanations to have a comprehensive understanding of the topic
  • The other explanation(s) are inaccessible to the student for a reason other than missing prerequisite knowledge

(I am making a more general point, but precise debate tends to get lost in generality and tying it to the specific context of math, knowledge graph, mastery learning, etc., helps keep it grounded.)


Q: Here’s a counterexample: describing vectors as both arrows versus lists. Some students find it easier to think about vectors as arrows; other students as lists.

A: These are two different topics, two separate cognitive chunks, and the student needs to understand both. If the student fails to master one of the interpretations, prevents them from understanding postrequisite topics.

For example:

  • If the student has not mastered the list (algebraic) interpretation of vectors then they will struggle with (e.g.) matrix multiplication.
  • If the student has not mastered the arrow (geometric) interpretation of vectors then they will struggle with (e.g.) cosine distance.

These two topics (list interpretation and arrow interpretation) get united in many downstream topics such as PCA. If you don’t understand the list interpretation of vectors, you can’t understand the PCA computations. If you don’t understand the arrow interpretation of vectors, you can’t understand the geometric interpretation of the principal components.


Q: How about limits of a function? Epsilon-delta (formal) suits analytical minds, but visual learners need a graph-zooming analogy, and others numerical tables.

These are different topics, separate cognitive chunks, and the student needs to understand them all. If the student fails to master any of the interpretations, it will prevent them from understanding some postrequisite topics. The issue with this counterexample is similar to that with the “arrow vs list interpretation of vectors” one above.

Also, there is no such thing as a “visual learner.” While students may have learning style preferences (e.g., visual vs. verbal), they do not actually learn better when receiving information via their preferred learning style (this has been empirically tested over and over again). What you’re probably thinking of is matching instructional design to the content, not matching teaching style to the learner. Lots of math topics benefit from images/diagrams regardless of what the learner might claim is their learning style.


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