It came up again a few days ago, that question every teacher dreads and every student obsesses over, especially when having a hard time on a problem or assignment: “When will we ever use this, anyway?” It is an understandable question. Abstract concepts like quadratic curves and complex numbers are foisted on students seemingly out of nowhere, and the process of mastering those concepts seems to begin and end in the classroom. What’s more, even though this question may be triggered by frustration with a particular problem, I have noticed that my students do genuinely care about linking their classroom knowledge with their other pursuits.
Unfortunately, many of the answers offered are unsatisfactory, either in and of themselves or in terms of their effectiveness. In the latter category, for example, many well-meaning teachers will say lofty things about mathematics as a supreme achievement of the human mind, and the intrinsic value of being able to appreciate its beauty. This is not a bad sentiment, but at this stage in students’ intellectual development it is like expecting someone unfamiliar with the rules of chess to appreciate an hours-long game between world masters. Furthermore, merely saying something is beautiful does not kindle an appreciation for it. So telling students that mathematics is beautiful and worthwhile is quite ineffective: they just can’t see that it is, and the promise that if they work hard now, they will in the future, is too abstract to be compelling.
Also in the latter category, textbook writers try to connect abstract mathematical concepts with the everyday lives of students by peppering the textbooks with glossy, full-color photos of everyday situations and presenting problems using simplified real-world situations. Unfortunately, many of these problems use the real-world situation as a source of data for answering otherwise typical textbook puzzles and challenges. My students very quickly realize that, despite the ‘realistic’ gloss, they are rarely if ever going to face such puzzles in their daily lives. The only truly relevant math for everyday life is simple arithmetic and percentages for balancing check-books and calculating interest rates, etc. . ., as well as a little statistics.
As an example of an answer that’s unsatisfactory in and of itself, some teachers will just throw up their hands and tell the students that they understand how frustrated the students are, but they have to learn the material to pass a standardized test. It is of course true that students need to master certain content to pass standardized tests, but it is a rather poor motivator. How disheartening it must be for a student to have to deal not only with his or her own struggles with comprehension, but with the teacher’s own lack of enthusiasm and cynical pragmatism.
So if we can’t talk our students into appreciating the beauty of math and cannot promise them that it will even for the most part be useful in their daily lives, what can we tell them?
When asked by students when they are ever going to solve quadratic equations in their daily lives, we should be honest and say, “Probably never.” But then again, we engage in many activities that aren’t directly ‘useful’ in some strict sense, but which nevertheless offer significant secondary benefits. Take basketball, for example, which we would never play if the only reason to do so was mastering the specific skills of basketball, like dribbling, passing a ball, performing a lay-up, etc . . . When if ever would we use those skills off the court? But this is to miss the point. Playing basketball makes us fit, allows us to practice our tactile coordination, builds leadership and teamwork skills, and allows us to relax and unwind after a long day of work. And surely those are all of immediate practical use. Or consider weightlifting: what chore or profession (except professional weightlifting, of course) requires a person to lie flat on a bench and raise and lower a weighted barbell over and over? What we’re after is the strength and conditioning that weightlifting confers, not the specific skills of performing particular exercises.
Similarly, when it comes to mathematics, what educators should really be attempting in teaching math is to instill those valuable transferable skills that all students will need, regardless of their profession. The most important of those skills is the ability to represent a problem situation numerically or graphically and manipulate that representation until it yields up the solution. An important associated character trait is perseverance and the willingness to approach a problem from many different angles. While the specific mathematical content varies from year to year, this is the overarching idea that is (or should be) reinforced at each grade level. As a teacher put it in the Heath Brothers’ book “Made To Stick,” math is mental weightlifting. There’s no better subject to cultivate analytical precision and rigor.
Finally, as teachers we should never forget that intellectual passion is often slow to develop. In high school I hated every minute of Ancient Greek, but as a religion student in college I was very grateful for the head start it gave me in reading ancient texts, which I began to see as fascinating in their own right. Even though it doesn’t make for a good student motivator, the prospect of a later intellectual flowering should inspire teachers to persevere, even as they hear that perennial question over and over again.