I can only see the excerpt at Amazon, which is just a few pages from the book.
The portion which is available shows repeated use of a numeric technique for approximating limits: calculating average rate of change over increasingly small intervals about the point in question.
This is very intuitive, but it's not the only way to evaluate limits. If this is the only technique a student knows, their knowledge will be a limiting factor, no pun intended, when they are confronted by problems that call for better mastery of the concept.
For example, later in the calculus series they will be confronted by limits that require L'Hospital's Rule or a similar approach. Someone who can look at a limit like (x^2+x)/ln(x+1) as x approaches infinity, and realize that the growth rate of the polynomial numerator is faster than the decay rate of the logarithmic denominator will probably have superior mastery relative to some one else who relies on a mechanical application of L'Hopitals Rule, or (even worse) a numerical treatment. Later still, they may encounter those limits inside of other problems, like evaluating an improper integral, and in that case a numerical treatment is probably not going to serve them very well.
I believe that if you're going to teach calculus at the high school level, you should teach it like university calculus. There is a reason that calculus is hard: it requires students to marshal unprecedented (to them) amounts of information. For example, when you learn about a function in calculus, you need to know the derivative and the anti-derivative. This is not at all like algebra class, and when calculus is slowed down, dumbed-down, and watered down, it's not helpful to high school students. They come to the university thinking that they already know calculus, when in reality, probably all they know is a few formulas for derivatives, or how to obtain a limit using their calculator. But they can't think. If you give them the graph of a function, for example, and ask them to sketch a graph of the derivative, they can't. Or if you ask them to apply the derivative in an application, they can't. If you ask them to take a placement test that checks what they know, they'll probably place well below the calculus level (because in all likelihood, they don't remember algebra very well, either).
I can only see the excerpt at Amazon, which is just a few pages from the book.
ReplyDeleteThe portion which is available shows repeated use of a numeric technique for approximating limits: calculating average rate of change over increasingly small intervals about the point in question.
This is very intuitive, but it's not the only way to evaluate limits. If this is the only technique a student knows, their knowledge will be a limiting factor, no pun intended, when they are confronted by problems that call for better mastery of the concept.
For example, later in the calculus series they will be confronted by limits that require L'Hospital's Rule or a similar approach. Someone who can look at a limit like (x^2+x)/ln(x+1) as x approaches infinity, and realize that the growth rate of the polynomial numerator is faster than the decay rate of the logarithmic denominator will probably have superior mastery relative to some one else who relies on a mechanical application of L'Hopitals Rule, or (even worse) a numerical treatment. Later still, they may encounter those limits inside of other problems, like evaluating an improper integral, and in that case a numerical treatment is probably not going to serve them very well.
I believe that if you're going to teach calculus at the high school level, you should teach it like university calculus. There is a reason that calculus is hard: it requires students to marshal unprecedented (to them) amounts of information. For example, when you learn about a function in calculus, you need to know the derivative and the anti-derivative. This is not at all like algebra class, and when calculus is slowed down, dumbed-down, and watered down, it's not helpful to high school students. They come to the university thinking that they already know calculus, when in reality, probably all they know is a few formulas for derivatives, or how to obtain a limit using their calculator. But they can't think. If you give them the graph of a function, for example, and ask them to sketch a graph of the derivative, they can't. Or if you ask them to apply the derivative in an application, they can't. If you ask them to take a placement test that checks what they know, they'll probably place well below the calculus level (because in all likelihood, they don't remember algebra very well, either).