Words of wisdom
On rigour and intuition
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient)
From Terence Tao's weblog. The whole post is worth reading.
Will I ever use this?
If the treadmill is not seen during the actual game, was it just a waste to use it? Were all those trainers wasting their time? Of course not. It produced (if it was done right!) something of value, namely stamina and aerobic capacity. Those capacities are of enormous value even if they cannot be seen in any immediate sense. So too does mathematics education produce something of value, true mental capacity and the ability to think.and
Education is built up with facts, as a house is with stones. But a collection of facts is no more an education than a heap of stones is a house.
From an online essay by Robert H. Lewis, which is worth reading, albeit with a sceptical alertness to rhetoric and anecdote.
Learning from lectures?
We do not learn to play the violin by playing the violin or rock climbing by climbing rocks. We learn by watching experts doing these things and then imitating them. Practice is an essential part of learning but unguided practice is generally useless and often worse than useless. People who teach themselves to program acquire a mass of bad programming habits which (unless they wish to remain hackers all their lives) they then have to painfully unlearn. Mathematics textbooks show us how mathematicians write mathematics (admittedly an important skill to acquire) but lectures show us how mathematicians do mathematics.
From T. W. Körner's 2004 essay In Praise of Lectures, available on his webpage under the blunter title "How to listen to a maths lecture".