I came across this from an interesting blog on teaching Calculus:
There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this.
This reminds me of the time when I was doing Calculus in secondary school. For some reason, I still remember my first Calculus lesson where the teacher explained how differentiation works, and I distinctly remeber not understanding what he meant when he talked about “the limit to zero” this and “the limit to zero” that. Everyone else was equally lost, but no one was about to ask for a clearer explanation — the teacher’s body language made it clear that he was just going through the motions of covering the introduction, and it that wasn’t anything particularly important.
And we were right. Although the class never really understood how Calculus works, it made no apparent difference. We knew how to apply differentiation and integration to standard problems, we knew how to mechanically manipulate the numbers to arrive at the final answer, and that was good enough. In fact, most of the class managed to get top grades for that subject the following year in our GCE ‘O’ level exams. All this while without a feel for Calculus.
A year or so after that, I unearthed from a bookshelf dad’s ancient dust-coated Feynman Lectures on Physics. I liked Physics, so I flipped open one of the books in the series (if the title was Lectures on Calculus, things would have been quite different). The page I looked at was an explanation on how Differentiation works, using a distance-time graph of a car to find out its average and eventually instantaneous velocity at certain times, and how the method of Differentiation could be derived from there (are you still with me?).
I was amazed. It was less than two pages of explanation, but things finally clicked. Now, “limit to zero” made perfect sense, and I could see the elegance of alculus. I feel Calculus.
Unfortunately, countless students have gone through Calculus (and life) never having a feel for it. It’s not just Calculus, but most of Mathematics.
Math, for too many, is not much more than a mechanical manipulation of figures — learning and doing what the computer can do much faster and infinitely more accurately. Is there a problem with math education if our students go through their school careers with such an idea, and having no feel for all but the simplest mathematical operations? Most certainly.
I wonder what could or should be done to improve the situation.
It’s now late afternoon, and I’m feeling a little drowsy. Maybe it’s the heavy lunch?
I was just reading Dr William Dement’s article What All Undergraduates Should Know About How Their Sleeping Lives Affect Their Waking Lives:
Each of us has a specific daily sleep requirement. The average sleep requirement for college students is well over eight hours, and the majority of students would fall within the range of this value plus or minus one hour. If this amount is not obtained, a sleep debt is created. All lost sleep accumulates progressively as a larger and larger sleep indebtedness. Furthermore, your sleep debt does not go away or spontaneously decrease. The only way to reduce your individual sleep debt is by obtaining extra sleep over and above your daily requirement.[…]
[T]he size of your sleep debt determines the strength of the tendency or ability to fall asleep. If your sleep debt is zero, sleep is impossible. If your sleep debt is very low, only a small amount of stimulation is required to keep you awake. If your sleep debt is very large, no amount of stimulation can keep you awake.[…]
[T]he things we usually assume cause us to become drowsy or to fall asleep actually do not cause us to become drowsy or to fall asleep. Their true role is to unmask any tendency to fall asleep that is present already. If you believe that boredom, a warm room, or a heavy meal causes sleep, you are completely wrong! If boredom, a warm room, or anything else seems to cause you to feel drowsy, you have a sleep debt and you need to be stimulated in order to stay awake. If you frequently feel sleepy or drowsy in any dull or sedentary situation, you almost certainly have a very large sleep debt.
Unfortunately, many students, particularly those in higher education, tend to go to class with heavy sleep debts. Whether it’s caused by the sheer amount of school work they have to finish, or simply their late-night lifestyle, it’s just something teachers have to live with.
I remember bumping into a classmate on the bus to school years ago when we were still students. He was excited to see me, proceeding to describe to me how he spent the whole night solving a programming problem. As he was speaking, his speech started slurring, and his eyelids started drooping, before he fell completely asleep, mid-sentence! Talk about a heavy sleep debt.
But just because there are students around with very heavy sleep debt does not relieve you the teacher from all responsibility. If a third of your students have their heads resting on the desk and drenching their handouts with drool during your classes, it’s because your lessons are so unstimulating and boring that a minimal amount of sleep debt is enough to send them a-drooling.
Yes, all your students will be carrying some amount of sleep debt, which can get quite heavy by late afternoon. But if you provide enough stimulation, they will be able to stay awake. Sure, there are exceptions–some students may have been up all night working on an assignment, but you’ll recognize them. Just do your part, and don’t take things too personally.
As the (current) tagline of this blog suggests, this blog is for me to express and record my thoughts and views on matters relating to education. And being someone with wide interests, I added the “everything else” to the tagline as I’m probably unable to stick to the topic of education and nothing else. This should cover all bases, ensuring the longevity of that tagline – I’d rather be spending the time writing blog posts than thinking of a better blog tagline.