I actually beg to differ with the article. By giving the students specific concrete examples instead of general equations; though of course they will grasp the concept less often and less strongly, it forces the students to go through a process of abstraction and generalization to induce the general formula. To be able to see several instances of something and then infer a pattern, common thread, or generally applicable method of solution is an invaluable skill, and of certainly more value than the specific material itself.
This is my favorite thing to do - abstraction - I remember there were once a lot of clock problems we had to do in geometry, determining what the angles between the hands were - it seemed to me kind of boring that we had so many - and i worked really hard to find a formula that based simply on the time would find the angle, and it was awesome. Did the same thing for all the different basic kinematics problems involving angles initial or final values of velocity, etc. This abstraction is (for me) the most terrific thing about learning, and necessary to infer things about life as well.
By giving the students specific examples and then testing them on their ability to apply inferred methods of solving them to new specific situations rewards the students with great abilities, and also teaches and reinforces those induction abilities. Unfortunately, it would seem obvious, fewer students do as well when they are expected to infer on their own the process.
The best method of teaching is probably a hybrid, where one starts by showing several examples and allow the students to try to solve slightly altered examples based on their knowledge and possible inferences. Then, eventually, show them the general formulae.
The hybrid teaching should be concept first the apply it to different examples. It think its too difficult for many of the students to infer patterns and deduce the formulas from the examples. That is a skill that is valuable but very difficult to teach.
2 comments:
I actually beg to differ with the article. By giving the students specific concrete examples instead of general equations; though of course they will grasp the concept less often and less strongly, it forces the students to go through a process of abstraction and generalization to induce the general formula. To be able to see several instances of something and then infer a pattern, common thread, or generally applicable method of solution is an invaluable skill, and of certainly more value than the specific material itself.
This is my favorite thing to do - abstraction - I remember there were once a lot of clock problems we had to do in geometry, determining what the angles between the hands were - it seemed to me kind of boring that we had so many - and i worked really hard to find a formula that based simply on the time would find the angle, and it was awesome. Did the same thing for all the different basic kinematics problems involving angles initial or final values of velocity, etc. This abstraction is (for me) the most terrific thing about learning, and necessary to infer things about life as well.
By giving the students specific examples and then testing them on their ability to apply inferred methods of solving them to new specific situations rewards the students with great abilities, and also teaches and reinforces those induction abilities. Unfortunately, it would seem obvious, fewer students do as well when they are expected to infer on their own the process.
The best method of teaching is probably a hybrid, where one starts by showing several examples and allow the students to try to solve slightly altered examples based on their knowledge and possible inferences. Then, eventually, show them the general formulae.
The hybrid teaching should be concept first the apply it to different examples. It think its too difficult for many of the students to infer patterns and deduce the formulas from the examples. That is a skill that is valuable but very difficult to teach.
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