Here are some of my thoughts on instructions.
I hope that you try to rise above the level of simply following instructions. Some students expect the professor to supply them with detailed instructions for every particular type of problem that they may encounter on an exam or in a homework assignment. Indeed, much of the power of calculus and other branches of mathematics lies in the fact that by following simple instructions, one can do many important and useful things. For example, grade school students learn to multiply three-digit numbers by following a relatively simple set of instructions. Algebra students use a very simple formula (the quadratic formula) to solve quadratic equations. Calculus students use simple rules to find the derivatives of functions.
At the university level, however, it becomes more important for students to make decisions about how to do things. They need to do this without receiving detailed instructions from a teacher. This is important for two reasons. The first reason is simply the complexity of higher-level mathematics: the teacher simply does not have sufficient time (or knowledge, for that matter) to provide students with detailed instructions for every conceivable mathematical problem. The other reason is the complexity of the world as a whole. In the course of your lives, most of you will encounter complex problems on the job and elsewhere. You may not be aware of any particular “recipe” or standard procedure for the solution to such a problem. Your only choice will be to devise a solution of your own.
In many jobs in the U.S. and around the world, of course, one can survive by simply following instructions. Many assembly line jobs or common labor jobs require little decision-making on the part of the employee. The employee simply does what he or she is told. And as a society we most certainly need people to perform these types of jobs. But university students should aspire to a higher level of thinking than this.
When I teach introductory mathematics courses, I devote considerable
time to the
discussion of step-by-step instructions for solving specific types of
problems. And if you have questions about these techniques, feel free
to ask them. But I also try to teach my students some of the logic
and theory of mathematics. If you understand some of this logic
and theory, then you will be better able to solve a wide variety of
problems. If you encounter a problem different from anything you have
ever seen, then you may be able to solve the problem by using some of
the logic and theory that you worked with during the course. The
ability to do this may help you in future employment as well
as in other activities. So I hope that you pay attention and
participate in class, even if our class discussions occasionally go a
little bit beyond step-by-step instructions for solving particular
types of problems.
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