Opening Day Thoughts on Goals and Effort

Before we launch into a semester of exciting mathematics, I’d like to discuss with you some matters that go above and beyond mathematics - matters that could determine how successful this course will be as a learning experience for you. They may also change the way you look at other courses.

The first matter concerns goals. Students attend college and take courses for many different reasons. At one end is a student whose sole goal is to do whatever it takes to earn high grades in order to land a good job after graduation; psychologists call this a performance goal, and it is certainly a laudable goal. At another end is a student, perhaps returning for another degree, whose sole aim is to engage in the learning process with little concern about grades and outcomes; this is called a learning goal, and it is also a worthy goal. Said differently, some students have extrinsic motivation and are more concerned with outcomes, while other students have intrinsic motivation are more concerned with the process. Most students probably study with a combination of both performance and learning goals, which is healthy. It’s a good idea to be aware of this spectrum of goals and occasionally stop and take a “goal measurement” to determine where you are on the spectrum.

Another matter - and it is particularly serious in mathematics courses - is your interpretation of effort. What does it mean when you find yourself working hard in a course? One interpretation, which I call the measure theory, is that effort is a measure of your intelligence: smart students don’t need to work as hard as less gifted students. Another interpretation, which I call the alternative theory, is that there are many alternative explanations for high effort. And now it is important to be clear: there is little research or experience that justifies the measure theory. Here are a few reasons to believe the alternative theory:

• History provides many instances of brilliant people who had to work very hard. Thomas Edison claimed, “There is no substitute for hard work. Genius is one percent inspiration and ninety-nine percent perspiration.” Albert Einstein is known to have struggled with mathematics courses as a student. And it's easy to attribute the success of athletes such as Tiger Woods (or artists) to rare natural ability. In fact, these people are usually the first to admit that they owe their success to intense motivation and perhaps obsessive effort.
• If we want to use effort as a measure, let’s be sure that everyone uses the same reference point. A student who excels in a course with little effort may have previous experience or familiarity with the course material. And if we want to use effort as a measure, let’s be sure that everyone uses the same tools. It is well documented that students learn in a variety of different ways. In mathematics, some students think and learn abstractly; others think and learn visually (geometrically); others excel at short bursts of insight, but cannot assemble multi-step solutions (sort of like a sprinter); and others have an ability to organize many pieces and formulate a complicated solution (sort of like a long-distance runner). Effort cannot distinguish among these many types of ability.
• The relationship between the amount of effort required and the quality of the learning experience is not well understood. There is reason to believe that what is quickly learned is also quickly forgotten, while a more deliberate learning process implies a greater assimilation and understanding of the subject.

The point is that effort is an extremely unreliable yardstick when it comes to measuring something as complex as ability and intelligence, and you should not try to use it! If you are working hard in a course, it might well mean that you are in the midst of a vigorous and successful learning experience. It does not necessarily mean that you are unfit to do mathematics. More telling, in terms of your future as a mathematician, is your enjoyment of the effort. It’s safe to say that most practicing mathematicians work hard and enjoy their effort.

So what does this all mean? I would summarize it as follows. First, research suggests that at times of challenge, students with learning goals and the alternative theory of effort tend to have better persistence and resolve. More generally, your success in this course depends very much on your outlook on goals and effort. Be aware of the goals that motivate you and try to find a healthy balance between performance and learning goals. And don’t defeat yourself by measuring yourself by your effort; expect effort to a be natural part of learning and let it be a part of your enjoyment of mathematics.

Don’t hesitate to contact me if you want to talk about any of these ideas. Have a great semester.

December 2002