When are we ever going to use this stuff?

When are we ever going to use this stuff?  Why do I have to study math? 


Common questions from students.  These questions usually come with at least a hint of disrespect, so I'm usually tempted by a snide quick reply, like: Probably never.  You don't have to study math...there's the door!   If my mood is more patient I might reply with some bizarre, artificial example the textbook, like: When will two trains meet? How much  is 5 hotdogs, 3 hamburgers, and a small coke, if 2 hamburgers, 1 hotdog and 3 small cokes cost $10? - or -  If you're in a rowboat 2 miles offshore, what heading should you take to get to the lighthouse quickest?  These are all good problems, but none of them really answer the student's question.


In general, instructors are quick to respond to these types of questions, and usually from a defensive posture. We've chosen mathematics as a career!  If the students are right, (that there's really not a "good" reason to take math), then we're sunk!  It's time for some real soul searching, and more importantly, a real meaningful answer to the question.  


There are as many answers to this question as there are people who teach or study mathematics.  My answer to this question is ever-changing and evolving, so check back with me in a few days if you want a different answer.  


Of course, if we do a little searching, we can find practical applications for almost any topic in mathematics.  The "why or when" questions will invariable pop up whenever we mention complex numbers.  One option here is to stand smugly in front of the class, strike the aforementioned defensive stance,  and state: "Complex numbers are used extensively  when working with partial differential equations, signal analysis, and quantum mechanics."  At this point it's best to quickly continue with the lecture before the students' eye rolling quits and they ask follow up questions.  My point here is that there really are real world applications for most math topics, but this won't satisfy many students.


This brings up another topic:  Do students really deserve a satisfying answer?  Is this one of those things "you'll understand when you're older"?  I think students do deserve, at least an attempt, at a satisfying answer.  It may be difficult to give a student, who just scored 12% on an Algebra exam, a satisfying answer, but I think they do deserve an attempt.


Here's my attempt:


I think it's a good idea to bring up some applications in different fields occasionally, but let's be honest....beyond arithmetic, a typical undergrad student just won't use mathematics that much.  According to the U.S. Department of Education, only 14% of post-secondary students were in science, technology, engineering, or mathematics.  100% of post-secondary students need to take some type of mathematics, and all 100% will need to use some arithmetic occasionally.  But what about complex numbers?  Algebra?  Calculus?  


People attend college for education, not for training.  The purpose of colleges and universities is not to train people how to do a job.  This is done at vocational schools or on - the - job.  There are parts of a university curriculum that might be considered as training, such as student teaching, practical nursing programs, physician residency, and others, but this is not the main purpose of the university or college.  The most important thing education can do for an individual, in my opinion, is to help that person become a better thinker.  If you can read, analyze, discuss, and think about an essay or novel, you'll be a better thinker.  If you can create a poem or painting, you'll be a better thinker.  If you can solve a problem using "non-real" complex numbers, you'll be a better thinker. 


Mathematics is a great tool to help teach critical and logical thought.  Even if mathematics had no practical application, a person who could understand and think about calculus or number theory has demonstrated an ability to think in a critical and logical way.  To be completely honest, other than arithmetic, which I try to avoid when possible, I rarely use mathematics outside the classroom, but I often use the critical and logical thinking I learned by studying mathematics.


We need to continue teaching and encouraging students to study mathematics, and we also need to continually remember why we're doing that.  I sometimes need to remember that my students and I are on the same "side".  I need to be completely honest with them and with myself.  "Good" math applications rarely show up in the undergrad classroom, so I make a big deal when they do, but more importantly, I often remind my students that we're here so we can learn to think, maybe learn to shift gears on our brains and think in new and better ways.


Too often I think we fail our students.  We teach them the steps involved in solving an equation, or which buttons to push to get an answer.  How can we help them to not worry so much about the steps, solutions, and answers, and help them become better thinkers?