https://medium.com/khan-academy-early-product-development/mastery-learning-and-creative-tasks-97c4a11f8364
Our most interesting and pressing problems require people to apply their understandings creatively in unfamiliar contexts.
Maybe we have some sense of what it means to “master” a specific skill — like “classifying triangles by type.” But what does it mean to “master” the skill of creatively manipulating understandings in novel situations?
At Khan Academy, we hope to help students build real understanding around what they’re learning. We’ll have failed if learners only feel confident applying concepts in routine situations; instead, they should be able to use what they’ve learned flexibly and fluently, adapting and connecting their ideas in novel settings.
As we develop activities for students on their path to thorough understanding, we want to create a smooth ramp for students: knowing simple facts, applying routine procedures, multi-step reasoning, extended inquiry. Instructional designers often find it useful to label tasks to make sure they’re appearing at a reasonable spot in that ramp. We often use two related axes for those labels: depth of knowledge, and transfer demand.
When students have to strategize or use creative, extended thinking to tackle a task, we say that the task requires high depth of knowledge. When students have to combine familiar facts, procedures, and concepts in unfamiliar ways or situations, we say that the task has far transfer demand.
The magic in these tasks comes from their unfamiliarity. If there were a lesson about “how to solve problems where the mass of an oscillator suddenly changes,” then this task would no longer demand deep knowledge or any transfer: it would become routine application of a memorized procedure.
But we can imagine shades of gray, too. What if we gave the student a set of three related far-transfer problems like the one above, each with the parameters slightly permuted? The first problem would definitely demand far transfer — but imagine the student couldn’t figure it out. So we show them a worked solution or give them some hints to help them see how the concepts combine in this new context. Now, once we’ve done that, if we show the student a second problem (identical except for some permuted parameters), they can just find-and-replace values appropriately in the worked solution we showed them. That’s much more like applying a routine procedure. It no longer requires far transfer.
Or consider a problem like this:
If students had never seen a problem like this before, the task would certainly require extended reasoning and demand far transfer of several prior ideas from geometry. But in fact, this exercise immediately follows a lesson on Khan Academy which explicitly teaches the procedure students can use to solve for the unknown side of a triangle for which two sides and an angle are known: the “Law of Cosines.” After watching that video, students only have to execute a routine procedure to solve this problem, so we’d say it doesn’t demand any transfer at all.
All this is to say: the same problem could function successfully at a variety of depths of knowledge. Is the problem a challenging puzzle or a rote plug-and-chug question? It all depends on how the student has “chunked” their prior understanding.
The notion of “chunks” in learning originates with The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information, in which George Miller introduces the term to describe the level of granularity which is perceived as a single unit. Two people might “chunk” information at different levels! For instance, a toddler sees words in terms of individual letters, while adults typically read words or groups of words at a glance. A toddler grasps the number “5” by counting up, one-by-one, on their fingers; eventually, with experience, “5” becomes its own “chunk” with automatically retrievable properties.
Let’s return to our triangle problem for a moment.
Students who have never seen this type of problem before must creatively combine a few chunks they might already have internalized, e.g.: