alt Hacker NewsAs someone teaching in higher education, I’d say that you can certainly incentivize the students to learn "understanding", although I agree that a lot is up to the student.
Some basic examples:
- Don’t give test and exam questions that are too similar to examples and problems in the text book and homework. Then they’ll know that learning to generalize is a better pay-off than memorizing the textbook problems, and may choose to change their strategy when studying for exams.
- Reduce the amount of curriculum. By studying in depth instead of in breadth, you have time to focus on how things really work instead of just rushing through material on a surface level, and in my experience that improves understanding more. (But I know many disagree with me on this one.)
- Focus on problem solving as part of the lectures (student-active learning). I’m not an extremist, like some advocating that we shouldn’t lecture at all, but the pedagogical literature is pretty clear that small doses of lectures interspersed with problem solving enhances understanding.
- Try to teach intuition and conceptual models, not just facts. For example, as a student, I really struggled understanding eigenvalues and eigenvectors because our linear algebra textbook defined it by Αv = λv but made no attempt at explaining what it means intuitively and geometrically. Similarly, integration by parts has a simple and beautiful geometric interpretation that makes it obvious why this is correct, but we were only taught the opaque symbolic version in my calculus classes. When I teach myself, I try to lean on such visualizations and intuitive pictures as much as possible, as I think that really enhances «understanding»; not necessarily being able to cough up a solution to a problem you’ve seen before as fast as possible, but being able to generalize that knowledge to problems you haven’t seen before.
But who knows, maybe I’m just biased by how I myself perceive the world. I know there are some people who for example eschew geometric pictures entirely and still do very well. My experience is that most students seem to appreciate the things listed above though.
Replies
I think you’re right and especially in regards to abstract concepts like linear algebra I don’t know anyone serious about learning who didn’t struggle with what turned out to be relatively simple things when viewed intuitively.
The problem as I see it
1. Professors themselves don’t understand it and are regurgitating pedagogy from books.
2. The material load is so high for your average bachelors degree you’d spend 8 years in school otherwise. I would hazard to say this is necessary and sufficient but schools wouldn’t get funding and our job oriented society would have it so then only the wealthiest could get education (like it was for centuries).
3. Tests are a benchmark and very expensive. You can consider a class’ total value to be loaded on the final exam. I recently wanted to go back to school casually. One of the cheapest universities available wanted 2600 dollars for a partial differential equations course. If I fail the course I lose 2600 dollars since I would need to retake it to proceed to higher mathematics. This alone does not allow a person time to explore - and that’s just one class!
4. Schools are simply a money laundering vehicle that takes money from students and moves it into administrator pockets. Education costs have skyrocketed yet education and pedagogy remains the same. This is money laundering by any other name.
- understanding leads intuition. There’s very little of either, anywhere.
You are absolutely right on all points!
Students need to take responsibility for themselves and Teachers need to point them in the right direction and help/steer as needed.
A Chinese Martial Arts saying which i keep in mind goes;
To show one the right direction and the right path, oral instructions from a Master are necessary, but mastery of the subject only comes from one's own incessant self-cultivation.
A good authoritative book can be the stand-in for a Master in which case there is more discipline and effort required of the Student.
These days different types of books/videos focusing on different aspects of the same subject are so easily available/affordable that the Teacher/Student can both work together and focus on understanding. A handful of real-world problems modeled and worked through beats pages of mere symbol manipulation. We need to start stressing quality over quantity i.e. deliberate effort via deliberate practice in the right way.