mathwonk said:
You may be right about your students' needs, but my own students are not like you and me, they did not do so many computations in lower school, and have almost no intuition for them. They have used calculators so long they never did acquire any skill or knowledge of basic operations.
E.g. many if not most of my students do not even realize that the reciprocal of a large positive number is a small number. In that situation, calculators have become monsters. I invite you to try this experiment on your own scholars. See if your assumptions about their knowledge are perhaps over optimistic.
I never had that experience in school (I skipped all the time). If you understand what you're doing, you should understand what solutions look like and stuff.
In the Calculus course that I TA in, I get students asking for help on assignments or to look at there work. I never look at the assignments, so when they show me their work or problem it's the first time I seen it. After I see the problem and then their solution without even working it out, because of understanding, I can already decide whether the solution makes any sense at all. If it doesn't make sense, I tell them and see how they did and throw a few hints on what they should work on and what not.
For the reasons above, this is why I choose courses like Calculus and Linear Algebra to TA in. It's because I understand them. I don't like getting nervous when somebody asks me a question and I have to pull out the textbook, like other TA's might have to do. I hate that. The only time I use the textbook is to check where they are so I don't show them knew stuff. Even for examples, if you understand it, you can create your own without too much difficulty. So, I don't understand why professors always use the examples given in the textbook it completely defeats the purpose or having the examples in the first place.
Anyways, I also have my weak spots, even in Calculus (I just don't tell anyone until I got it figured out.
).
In the end, understanding is key. Calculators won't get you there. Because you can plug in any information into your calculator but you'll never intuitively know whether the answer makes sense. You just have to hope that you did the steps correctly and inputted it in your calculator correctly.
Note: It's like that one time in Differential Equations where I pointed out two different ways to prove/solve a problem on the board. The professor dismissed both of mine, and went out with "easier" methods the other students came up with. As it turns out, he tried mine after the others, and it was done in 2 steps. He didn't bother trying the other one because he said it would be over their heads although it was the easiest method of all and could be done in one step. What were the concepts? The first was just finding the derivative of the equation and then using basic Linear Algebra, and the other method was using Taylor Series. Two things that you should know when in Differential Equations. Sad.
People wonder why they do bad in courses like Linear Algebra and Abstract Algebra. Maybe it's because a calculator won't tell you if a set with so-and-so operations is a vector space or a group!
