I will now provide some examples since I am obligated to do so. Computer programming, whether it is using Microsoft Excel or graphing calculators, is a great way to help students thoroughly understand procedures and algorithms. Say for instance that you are studying regular polygons. You could give students problems such as, "find the area of a regular octagon with sides that are 12 inches long," and they could solve it by hand and that's great. Next, they could work on how to type that program into a graphing calculator. The nice thing about computers for this purpose is that the students have to "teach" them what to do, and the computer can't guess what the student means if their explanation is incomplete. If their program comes up with the wrong answer, they have to go back and fix any errors until it is perfect; they can't fudge anything or take shortcuts. After that, you could work on generalizing the formula for any regular n-gon with sides of length x. This gives them practice manipulating variables, and taking a concrete shape like an octagon and abstracting it to something with an indeterminate number of sides. The final result in this example would be a program in which the user inputs the number of sides and their length, and the computer gives the area of that regular polygon. They could also develop programs for finding roots, generating prime numbers, or approximating logarithms. Pretty much anything you want to study, they can write a program about it.
Computers can also be used to do multiple calculations quickly and accurately. For example, using a recursive formula in Excel to generate a Fibonacci-type sequence. This takes a lot of the boring, menial work out of the way, and allows students to focus on the bigger picture and get more in-depth into the topic. However, this has many drawbacks if you try to use it in place of student understanding. Using the polygon example above, students would have an extremely difficult time writing that program if they did not have experience themselves executing that procedure (it's not easy to write even if they do). You see this often in some classrooms, where students can do something like use a graphing calculator to find the roots of a polynomial but don't understand what it really means. The computer becomes too much of a crutch, and their understanding is truly just in their short-term memory. The technology should be used to expedite procedures and algorithms students are already comfortable with, not relied upon from the outset.
I agree that it is important not to use technology to replace any sort of understanding that students have of the material. Technology should give students the opportunity to explore math in a deeper way, rather than give them another tool (other than their brain) that does the work for them. There is already a line that divides teachers who tell the rules, or teachers who let students discover the rules and technology has the potential to further distance those two perspectives. As math teachers, I think we will need to be very thoughtful with our use of technology to make sure that we use it to allow students to delve deeper into material rather than skim even less of the surface.
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