As a math grad student, I go to a lot of math colloquiums, which always have the property that everyone understands the first 10 minutes and noone understands anything after that. It makes for a funny in-joke around the department, but it doesn't make for very great pedagogy ("pedagogy" is a fancy-schmancy word for "teaching", of course)
It's a grave mistake to base a lecture around covering a certain number of pages. A lecture should be based around conveying as much understanding and knowledge as possible in the alloted time. If that means spending the entire time on one page from the book, so be it. Even if the curriculum calls for certain deadlines in the textbook, it's useless to just read the textbook while noone comprehends it. Better to be behind and have the class understand 20 pages out of 100, than be on schedule and have the class understand 0.
In the sense of deadlines, it's an example of a rule I live by: "never cover your ass" In many jobs, there are certain requirements, and people follow the requirements to the letter, just to "cover their ass". If you find yourself in a position where you're doing something just to "cover your ass", then you are not adding value to the world. In teaching, if your department says you have to cover ten pages today, but you know none of your students will understand them, then reciting those pages to a stunned audience is something you'd do to "cover your ass", just so you can tell the department, "well, I taught them, it's their fault if they didn't understand". That's not being a teacher, that's being a robot.
One of my greatest teachers was Dr. Oma Hamara at the University of Arizona, who taught me a grad level linear algebra class. Dr. Hamara would go through every proof in excruciating, almost painful detail, writing out *every line* and explaining what justified it. It made the lectures a little painful, but looking back now, I see how much I learned from him. Because I watched him go through Zorn's Lemma arguments so many times, I now have a titanic grip on the mysterious Zorn's Lemma. Because Dr. Hamara went through the details of so many tensor-product diagrams, I'm very good at understanding abstract algebraic tensor products, a subject that staggers many mathematicians with its arcaneness.
A fatal pattern you might see in many classes: the lecturer asks some question about the material just covered, and the class responds with stony silence or some hemming and hawing. And then, the lecturer goes on with the lecture, moving on toward even tougher stuff, when the class is obviously already lost. It's a waste of everyone's time. Don't waste your students' time, and don't waste your own time.
When people come to me for tutoring in calculus, the first thing I ask is how good their algebra is. If their algebra isn't good enough to do calculus, they'll get nothing from calculus tutoring. Ten or twenty minutes of algebra review, though it cuts into the calculus tutoring time, can end up giving the tutoree a lot more understanding of the calculus we do go over.
Everyone is capable of learning anything. People believe they are unable to learn some subject, because they've had teachers who taught out of a calendar instead of teaching out of a thirst for understanding and enlightenment. Far from adding value to the world, such teachers have actually destroyed value, making people fear academic subjects. Create value for the world, teach for understanding. Better that the students understand one single paragraph, than that the lecturer recite a whole textbook if noone understands any of it.
Here are some other articles I wrote. Please let me know right away if you understand everything alright.
How To Be A Better Teacher
Activism Goes Away After Graduation
"Problems" In Mathematics
The Higher Infinite
It's a grave mistake to base a lecture around covering a certain number of pages. A lecture should be based around conveying as much understanding and knowledge as possible in the alloted time. If that means spending the entire time on one page from the book, so be it. Even if the curriculum calls for certain deadlines in the textbook, it's useless to just read the textbook while noone comprehends it. Better to be behind and have the class understand 20 pages out of 100, than be on schedule and have the class understand 0.
In the sense of deadlines, it's an example of a rule I live by: "never cover your ass" In many jobs, there are certain requirements, and people follow the requirements to the letter, just to "cover their ass". If you find yourself in a position where you're doing something just to "cover your ass", then you are not adding value to the world. In teaching, if your department says you have to cover ten pages today, but you know none of your students will understand them, then reciting those pages to a stunned audience is something you'd do to "cover your ass", just so you can tell the department, "well, I taught them, it's their fault if they didn't understand". That's not being a teacher, that's being a robot.
One of my greatest teachers was Dr. Oma Hamara at the University of Arizona, who taught me a grad level linear algebra class. Dr. Hamara would go through every proof in excruciating, almost painful detail, writing out *every line* and explaining what justified it. It made the lectures a little painful, but looking back now, I see how much I learned from him. Because I watched him go through Zorn's Lemma arguments so many times, I now have a titanic grip on the mysterious Zorn's Lemma. Because Dr. Hamara went through the details of so many tensor-product diagrams, I'm very good at understanding abstract algebraic tensor products, a subject that staggers many mathematicians with its arcaneness.
A fatal pattern you might see in many classes: the lecturer asks some question about the material just covered, and the class responds with stony silence or some hemming and hawing. And then, the lecturer goes on with the lecture, moving on toward even tougher stuff, when the class is obviously already lost. It's a waste of everyone's time. Don't waste your students' time, and don't waste your own time.
When people come to me for tutoring in calculus, the first thing I ask is how good their algebra is. If their algebra isn't good enough to do calculus, they'll get nothing from calculus tutoring. Ten or twenty minutes of algebra review, though it cuts into the calculus tutoring time, can end up giving the tutoree a lot more understanding of the calculus we do go over.
Everyone is capable of learning anything. People believe they are unable to learn some subject, because they've had teachers who taught out of a calendar instead of teaching out of a thirst for understanding and enlightenment. Far from adding value to the world, such teachers have actually destroyed value, making people fear academic subjects. Create value for the world, teach for understanding. Better that the students understand one single paragraph, than that the lecturer recite a whole textbook if noone understands any of it.
Here are some other articles I wrote. Please let me know right away if you understand everything alright.
How To Be A Better Teacher
Activism Goes Away After Graduation
"Problems" In Mathematics
The Higher Infinite
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Like the website man
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