However, this concern and its supported anecdotal evidence are somewhat misleading. Read on to see what I mean.
Even mathematicians use calculators of some sort. At some point or another in the last year, I have used a basic (a.k.a. four-function) calculator, scientific calculator, graphing calculator, computer, and even an antiquated adding machine to do what some may deem as simple arithmetic. I have also used some of these to perform more advanced mathematical investigations. Does this mean that I cannot do mathematics "by hand"? I'll let you decide.
"Are you saying that you support your students using calculators all the time?"
Absolutely not! However, there are moments where an instructor would be silly to take the calculator away. More often than not, however, there are situations and critical points within a student's mathematical career where forcing an absence of a calculator is required.
In which courses will you instructor likely allow calculators? Well, if you look at the list of courses below, try to imagine a bell curve (a curve that starts low, climbs to a maximum, and then tapers back down). You calculator usage per course will slightly mimic that curve.
Prior to Algebra (no calculators)
Since arithmetic is focused on getting used to the operations of addition, subtraction, multiplication, and division (as well as smaller exponents), you likely will not (and should not) be allowed a calculator. The same argument somewhat holds for prealgebra; however, in prealgebra you may have some application problems that require a calculator.
Algebra (some calculator usage)
The use of a calculator becomes increasingly useful as you go through both elementary and intermediate algebra. The vast majority of your computations will still be done without a calculator because the focus is less on terrible-looking arithmetic problems and more on algebraic expressions and equations. If you had an instructor who allowed you to use a calculator in courses prior to algebra, you will struggle in algebra. This is because of some of the subtleties in algebra that require knowledge of operations like fractional arithmetic (e.g., when working with rational expressions or functions).
I cannot stress enough that reliance upon calculators in courses prior to algebra really spells disaster for the student in future math courses.
Applied Algebra (lots of calculator usage)
These courses are becoming more common in the last decade. They are referred by many different names (intermediate algebra with applications, math for the consumer, and liberal math to name a few), but you can pretty much guarantee that calculator usage is expected and required in these courses. You will often have to purchase a graphing calculator (a completely useless device with modern day electronics).
Statistics (insane calculator usage)
A statistics course without a calculator of some type (even Excel) is completely devoid of reality. Suffice it to say that you must have a computing device in a good statistics course.
A Cautionary Tale About the Courses From This Point Forward
Although you are likely going to be allowed a calculator in each of the following courses, using a calculator for every exercise (or even most) will likely convey to your instructor that you can't do math without a calculator. It is very common for instructors of these courses (myself included) to build exams that don't allow calculators, or where calculators are of no help. More often than not, all of your lack of practice with doing work by hand will show up on exam day. Hence, try to do as much work by hand to stay in practice.
Trigonometry (some calculator usage)
In trigonometry, you spend a good chunk of the course not touching a calculator so you can get used to trigonometric functions. The rest of the course should involve applications of trigonometric functions... these require calculators.
Precalculus (heavy calculator usage)
At our college, we follow a full semester of trigonometry with a full semester of precalculus. I think of our precalculus course as the "sum up everything you have learned and put it on steroids" course. Deep theoretical problems and intense applications are the focus. As such, technology is used (or can be used) frequently to really round out a student's education on the non-calculus, lower division math courses.
Calculus (some calculator usage)
The entire calculus series is filled more with theory than application (with the exception of calculus for the life and social sciences, which is more applied). Thus, calculus naturally tends to be less calculator-focused. This does not mean that you will never use a calculator in calculus! In fact, to a certainty you will be required to used a calculator at some point - just don't rely on it too much.
Differential Equations (heavy calculator usage)
Differential equations (affectionately known as DEs) can be done without a calculator, but that would make it somewhat bland and lifeless. The whole point of DEs is to do some heavy, applied problems. Well, that's not the whole point, but it is to the vast majority of the students taking DEs.
Linear Algebra (light calculator usage)
Linear algebra is my favorite course! There is so much beautiful theory that goes on in this subject and it cannot be captured on a calculator; however, there are some really cool uses for linear algebra in applications. So if you have an instructor who bends a little towards the applied side, you may need a calculator at some point.