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Perhaps counterintuitively, it’s not by actually increasing their mental “jumping” ability. Instead, it’s by helping them build knowledge “bridges” that reduce the distance to jump.
There’s a mountain of empirical evidence that you can increase the number of examples & problem-solving experiences in a student’s knowledge base, but a lack of evidence that you can increase the student’s ability to generalize from those examples (by doing things other than equipping them with progressively more advanced examples & problem-solving experiences).
As described by Sweller, Clark, & Kirschner (2010):
- “In short, the research suggests that we can teach aspiring mathematicians to be effective problem solvers only by providing them with a large store of domain-specific schemas. Mathematical problem-solving skill is acquired through a large number of specific mathematical problem-solving strategies relevant to particular problems. There are no separate, general problem-solving strategies that can be learned.”
- — Sweller, Clark, & Kirschner (2010) in Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics
Now, to be clear, this does NOT imply that students learn/generalize at the same rate.
Different students have different levels of generalization ability; you can give a group of students with similar background knowledge the same problem-solving experiences and some students will walk away with a more general understanding than other students.
It’s just that you can’t really train the implicit generalization ability so much as explicitly equip students with a larger underlying knowledge base.
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