The above question is asked by nearly every single freshman at Johns Hopkins. I'm going to attempt to explain to you why calculators are forbidden on exams and generally discouraged for use in most math classes here and at other institutions of higher education.
The feeling that calculators hinder students is in no way unique to JHU. Prof. Wilson has compiled an ad hoc list of math professors around the world who agree.
The calculator is a machine. It doesn't know anything about mathematics. A smart engineer has programmed it to give some pretty good answers to some math questions, but these are usually approximations, and, in the strictest sense, very often incorrect.
For example, once you've become acquainted with elementary manipulation of exponents in algebra, you, a human, can tell that 20x×30x=600x so 20x×30x – 600x=0.
Look at what calculators think the graph of 0 looks like when you write it as above:
|TI-92: $60||Maple: $995||Michelle - age 13: Priceless|
This problem is called “overflow error” and it's one of the many ways calculators lie to you if you don't carefully guess real answers based on their approximations. One instance of this error may have cost over $370,000,000.
Humans can understand that the functions
Calculators understand this:
|TI-92||Maple||Esther - age 15|
|TI-92||Maple||Catherine - age 14|
Depending on a machine to do your intellectual work is not just degrading, it also limits your understanding. If you want to do any kind of science from Engineering to Biology to being a practicing physician, you're going to have to understand mathematics. Understanding comes from doing.
Your graphing calculator is a precision instrument that will probably not have a real bug for years to come. But, if you only learn how to do math with a calculator, you're going to have to depend on whatever machine happens to be around when you're doing your job. That machine might not be as dependable:
Please ask me about any aspect of mathematics that's escaping you. You can come by my office anytime. Don't have shame about anything. Teaching you is my job. Understanding your coursework is your job. I mean it. If you can't add 1/3 to 1/2 ask me how!