I was very successful in my early mathematical education. I’d get As with ease, take exams early, enter mathematics competitions, etc. I had a deep understanding despite doing very little work because all the concepts seemed obvious.
I continued in the exact same way and my performance declined to the point where I was struggling to get Cs. I was now meeting concepts that were not intuitively obvious (eg. limits, proofs, complex numbers), and because of my previous success I had not developed any techniques to gain deep understanding of them. I lost all sense of enjoyment of mathematics and convinced myself that it didn’t matter because a good CAS could do it all for me.
I have now started learning again, and there’s one realization which has made a big difference. As a student I was always told to solve “problems”. This is a terrible name and they should really be called “exercises”. The questions are obviously not problems because the teacher has the answer right there in his book. If they are problems then the correct way to solve them is to copy somebody else.
Thinking about the questions as “exercises” makes it clear why you’re supposed to solve them, and makes clear how much effort you should put into them. It’s analogous to physical exercises—you don’t lift a weight just once and declare it solved, and you don’t keep lifting the exact same weight when it becomes easy. I now take the exercises seriously and my understanding improves. I am starting to enjoy mathematics again. I wish somebody had explained this when I was a student.
I was very successful in my early mathematical education. I’d get As with ease, take exams early, enter mathematics competitions, etc. I had a deep understanding despite doing very little work because all the concepts seemed obvious.
I continued in the exact same way and my performance declined to the point where I was struggling to get Cs. I was now meeting concepts that were not intuitively obvious (eg. limits, proofs, complex numbers), and because of my previous success I had not developed any techniques to gain deep understanding of them. I lost all sense of enjoyment of mathematics and convinced myself that it didn’t matter because a good CAS could do it all for me.
I have now started learning again, and there’s one realization which has made a big difference. As a student I was always told to solve “problems”. This is a terrible name and they should really be called “exercises”. The questions are obviously not problems because the teacher has the answer right there in his book. If they are problems then the correct way to solve them is to copy somebody else.
Thinking about the questions as “exercises” makes it clear why you’re supposed to solve them, and makes clear how much effort you should put into them. It’s analogous to physical exercises—you don’t lift a weight just once and declare it solved, and you don’t keep lifting the exact same weight when it becomes easy. I now take the exercises seriously and my understanding improves. I am starting to enjoy mathematics again. I wish somebody had explained this when I was a student.