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Often in a Calculus or Linear Algebra class, my students may ask - when am I ever going to use this? The answer I try to give them is "You're not. And that's totally OK. But you do need to learn how to think logically and solve problems. Mathematics provides a safe place to learn these thinking skills."

But increasingly, the focus of the students is still "Is this going to be on the test?"

They seem unhappy when questions they haven't seen before end up on the test. My thought is that if they learn the process for solving problems, then they should know how to approach a new (but similar) problem.

I'm not interested in seeing them regurgitate facts, but I want them to develop their thinking skills and use them to solve new problems. This often causes backlash. I get it - learning how to think is harder than memorizing facts.

How do I effectively communicate that it's not the individual questions or ideas that are important, but it's the learning and thinking processes that are key?

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    In the assignments and on the tests, are you just asking them for the answers, or do you ask them to show the steps they took? Do you let them use any solution to find the answers, or limit their solutions to a specific approach which you explicitly taught them? Lastly, are you showing them the math just as numbers and equations on the board, or are they exposed to the real-world situations where these would be used? – Village Sep 7 '14 at 0:00
  • Check out the short "The Five Elements of Effective Thinking", by Burger and Starbird, two math professors. They include a lot of stories they use to inspire better thinking habits in their students. – bevanb Sep 7 '14 at 8:23
  • What country are you teaching in, and for those of us not familiar with your education system: how do these students end up having to study something they don't really want to study? In my country e.g. you choose and apply for an undergraduate programme/specialization, which mostly determines what courses you must take during the duration of the 3 or 4 year programme. I'd argue that the specific specialization may be relevant for the answer. E.g. economics students tend to revolt against linear algebra, but it's not hard to argue why it's very relevant for economics. – Szabolcs Sep 7 '14 at 17:22
  • What I'm getting at is that, as J.R. pointed out, sometimes students can grossly misjudge what they really need and what they really might need to use in the future, especially considering that the required skills for a given field change over time. With the ubiquitousness of computers and data analysis math skills are becoming needed in more and more fields. – Szabolcs Sep 7 '14 at 17:27
  • I said that to my calculus teacher repeatedly...and have regretted it ever since as I work as a computer programmer now. It was the abstraction that I found annoying and so I think you can't go wrong with "real world" examples. One other thing is no one ever told me what algebra, trigonometry, calculus actually were. Never once was I given a clear definition of these areas of mathematics or the kinds of problems they could be used to solve. So I heartily suggest you start there. – Dave Kanter Jul 7 '15 at 18:03
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I disagree with your fundamental premise: that the concepts won't be used, and therefore the main benefit of learning calculus is to hone a student's problem-solving skills.

Many engineering problems do require an understanding of calculus. Perhaps not all your students are destined to be engineers or physicists, but, if you explain that there are people who do need and use this knowledge to advance in their career field and solve real-world problems, you might get them to sit up and realize they aren't trudging through their homework as a mere academic exercise. Fact is, some of these courses are taught to freshmen and sophomores because they will need those problem-solving skills as juniors and seniors – not just to know how to solve a problem, but to also understand the underlying theory and mathematics. If all the students in a particular demographic will never use calculus in the future, then they probably don't need to take calculus now.

I'd try to find some real-world problems that require the knowledge taught in your course to solve. Perhaps that might spark an interest and help students realize that the are not just solving integrals for integrals' sake.

If I was at your university, and I taught an upper-level engineering course in heat transfer or thermodynamics, I'd be disappointed to learn that my students' calculus teacher thought the main reason to teach my engineers calculus was to provide a "safe place" for them to "learn how to think logically and solve problems."

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    I consider it an implicit premise of the question that the students asking this are the ones who likely aren't going to use this knowledge - not the hard science and engineering majors, but people in fields like English, history, and politics, and others where calculus is not involved. Or at least, my point is that there are students to whom the reasoning in this answer doesn't apply. – David Z Sep 7 '14 at 6:30
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    @David - My answer could apply if one answered the class in the aggregate. In other words, maybe Bobby, Susan, and Joey aren't going to ever use calculus, but their classmates Terry and Leslie might. Moreover, it's people like Terry and Leslie who give us cell phones that work, bridges that don't collapse, and roads that don't flood. Maybe I don't personally use much of the trigonometry I once learned, but knowing there are people who do use it can make the lessons seem less futile. And if that doesn't motivate Joey and Susan, they probably shouldn't have signed up for calculus. – J.R. Sep 7 '14 at 8:24
  • Thanks for your answer! I guess I should say that they're never going to solve a problem exactly like the ones I give - but, as you point out, they will likely solve similar problems. I will think about how to get some actual engineering (or other application) inputs from today's engineers and scientists. – Luke Oeding Sep 21 '14 at 20:45
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While it's indeed likely that many of your students will not extensively use the material they learn in your class, it may not be encouraging to them to hear that they won't use it. And indeed, they do learn important critical thinking skills, but they may be wondering what is special about the sort of critical thinking in calculus and linear algebra that they couldn't get out of another course.

In this vein, one solution may be to present real-world examples of how calculus and linear algebra are used today, get the students involved as you present, and relate the topic to the students' lives.

For instance, after you present a basic optimization example (maximize fence perimeter given the area it contains) you might present a slightly more complicated problem (building a window of a particular shape that maximizes the amount of light coming through with a fixed amount of building material, for instance), and get the class involved in trying to solve the problem. Ask them what's different about it, maybe have them briefly discuss a strategy with the person next to them. (This is not usually done in a large class, but at least one very popular professor at my school does this in his large classes.)

Continuing with the example of optimization, there's a great topic on matheducators.stackexchange about real-life optimization problems here. In particular, I've linked to a problem that video-game players might find to be really cool and related to their everyday lives. Of course, it may also help to branch out and briefly discuss some current, advanced applications of calculus and linear algebra. An excellent example where ideas from calculus and linear algebra are essential is the story of how UPS route optimization involves the minimization of left turns. This last example might inspire a lively discussion in your classes, because it shows how calculus and linear algebra can trump everyday intuition... and how pushing beyond a basic understanding can lead to new, interesting ideas! (I'm sure if you asked students in class, they could come up with reasons for why UPS's strategy might make sense.)

On the other hand, some students might not be moved by industry applications and might benefit from seeing mathematics as the kind of "safe place" you suggest where perfect formulations lead to perfect results. Strogatz has some interesting words in a Math Horizons article about the importance of enabling others to experience the beauty of math, and he provides a great example of why math is beautiful. I also just found a TED talk discussing the use of calculus in some parts of modern architecture. Though this last one is admittedly again an industry application, it at least sheds some light about how aesthetics and mathematics can interact in interesting ways.

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