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> After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In part

> After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In particular, the teacher’s summary makes sense and is meaningful, because students have had the experience of playing around with these ideas on their own before coming together to formalize them as a class. It's important for students to get their hands on examples and play with the ideas we're trying to present on their own. One issue I have is that this takes so much time. If you're introducing concepts that the students don't have good intuition about, you have to go so slowly. Even then it feels like some students can't follow. I think it's beneficial when it's possible to get students to engage with the material on a daily basis as reading or homework. Hard to do when they're expected to take 4 classes (college) or 6-8 (high school) and dedicate study time to each of them.

gizmo686 9y ago
When you use this approach, you generally cover less material, but cover it better. For most classes, this seems like a good tradeoff. 4 classes isn't really that much. You can essentially replace the time students would normally spend "studying" with them spending that time working on the problems. Further, in my experience, this type of work is far more engaging than traditional studying, so it is easier for students to spend time on it; and the time they do spend tends to be more thoughtfull.
- jacobolus 9y ago
  Moreover, as people practice, they can get generically better at meta-skills such as (in no particular order) skimming, close reading, constructing mental outlines of arguments, drawing diagrams and pictures, inventing examples to match given specifications, testing concrete examples against abstract statements, generalizing specific relationships discovered in the examples to abstract laws and proving them generically, discarding or modifying abstractions to better match a wider collection of examples, working backwards, discarding dead-end problem-solving strategies while mining them for partial useful results, breaking problems down into manageable pieces, deductive logic, generating hypotheses likely to be interesting, abandoning whole problems which turn out to be too hard and switching to something else, cross-applying solution strategies from one field to another, simplifying complex arguments by discarding or separating extraneous or duplicated steps, writing up explanations in clear and coherent way for various audiences, researching past work in an efficient way, diving into new fields with completely alien notation and terminology and quickly taking the lay of the land, and so on. These skills are more generically useful than recipes or collections of facts about any particular subject, because they serve as multipliers for quite generic learning and problem-solving efficiency. One big problem with trying to teach a single specific course in a very problem-centered / student-driven / socratic style is that often students have not been sufficiently trained in any of these meta skills, and as a result move at much slower pace than their potential because they waste a lot of time on inefficient problem-solving methods, get side-tracked, throw away useful work, explain things poorly, don’t get around to crystallizing their useful results, and so on. The more courses get taught in this style, the faster each subsequent course can move as students figure out how to learn effectively.
  - katdev 9y ago
    That's a nice list of meta-skills. Do you have any further reading to recommend on identifying/teaching meta-skills?
    - jacobolus 9y ago
      You could try Polya’s How to Solve It, and follow up with Schoenfeld’s Mathematical Problem Solving (which cites a lot of other material). These are rather focused on problem solving per se, and don’t really discuss larger-scale strategies for research or learning. You can try just diving into child development and education literature. Not sure what out there really tries to be comprehensive in summarizing the best techniques for teaching and learning arbitrary meta-skills.
gjulianm 9y ago
I think this approach is better suited to lower level schooling, where kids already spend several years learning and practicing a small number of concepts. Instead of making them do countless calculations and mechanical problems, invest some of that time in abilities that will actually stick with them and be useful in the future.

gizmo686 9y ago

When you use this approach, you generally cover less material, but cover it better. For most classes, this seems like a good tradeoff. 4 classes isn't really that much. You can essentially replace the time students would normally spend "studying" with them spending that time working on the problems. Further, in my experience, this type of work is far more engaging than traditional studying, so it is easier for students to spend time on it; and the time they do spend tends to be more thoughtfull.

jacobolus 9y ago
Moreover, as people practice, they can get generically better at meta-skills such as (in no particular order) skimming, close reading, constructing mental outlines of arguments, drawing diagrams and pictures, inventing examples to match given specifications, testing concrete examples against abstract statements, generalizing specific relationships discovered in the examples to abstract laws and proving them generically, discarding or modifying abstractions to better match a wider collection of examples, working backwards, discarding dead-end problem-solving strategies while mining them for partial useful results, breaking problems down into manageable pieces, deductive logic, generating hypotheses likely to be interesting, abandoning whole problems which turn out to be too hard and switching to something else, cross-applying solution strategies from one field to another, simplifying complex arguments by discarding or separating extraneous or duplicated steps, writing up explanations in clear and coherent way for various audiences, researching past work in an efficient way, diving into new fields with completely alien notation and terminology and quickly taking the lay of the land, and so on. These skills are more generically useful than recipes or collections of facts about any particular subject, because they serve as multipliers for quite generic learning and problem-solving efficiency. One big problem with trying to teach a single specific course in a very problem-centered / student-driven / socratic style is that often students have not been sufficiently trained in any of these meta skills, and as a result move at much slower pace than their potential because they waste a lot of time on inefficient problem-solving methods, get side-tracked, throw away useful work, explain things poorly, don’t get around to crystallizing their useful results, and so on. The more courses get taught in this style, the faster each subsequent course can move as students figure out how to learn effectively.
- katdev 9y ago
  That's a nice list of meta-skills. Do you have any further reading to recommend on identifying/teaching meta-skills?
  - jacobolus 9y ago
    You could try Polya’s How to Solve It, and follow up with Schoenfeld’s Mathematical Problem Solving (which cites a lot of other material). These are rather focused on problem solving per se, and don’t really discuss larger-scale strategies for research or learning. You can try just diving into child development and education literature. Not sure what out there really tries to be comprehensive in summarizing the best techniques for teaching and learning arbitrary meta-skills.

gjulianm 9y ago

I think this approach is better suited to lower level schooling, where kids already spend several years learning and practicing a small number of concepts. Instead of making them do countless calculations and mechanical problems, invest some of that time in abilities that will actually stick with them and be useful in the future.