What does it mean to be good at mathematics? Certainly one would say that a student who earns a good grade is likely more capable than a student who earns a poor one. But what characteristics does a good mathematics student have that a poorer one does not? And, are there differences within the students who earn good grades? While this list is in no way comprehensive and may be debated even among teachers, here are some of the characteristics I have observed in students who understand mathematics well. They tend to

• have an adventurous spirit.
  We all know the phrase, “Nothing ventured, nothing gained.” It's OK to venture out and make a mistake. Often we learn best through our missteps, and the students who know mathematics best can often say from first-hand experience why a particular approach to a particular problem is more fruitful than other approaches.
• focus in on detail.
  At first look, two types of problems may look very much alike, although they are solved in different fashion. Students who can (or train themselves to) see the sometimes miniscule differences have a leg up on others who cannot (or do not). This attention to detail is easiest to cultivate when one does not blindly memorize a method for solving certain problems, but rather comes to understand how each step is helping to make progress toward a solution, and is or is not necessary based on specific features of the given problem.
• have good verbal skills.
  We hear a lot about learning styles nowadays, and it is good to inform learning with pictures/diagrams for the visual learners, provide hands-on activities for the active learners, etc., whenever these things are possible. Still, mathematics is a language, and there is no substitute for learning how to speak and write it in such a way as to express ones ideas clearly. The better mathematics students either have a natural tendency towards acquiring language and communicating well verbally/in writing, or they recognize their weakness in this area and pay special attention to developing these skills in themselves.
• be curious.
  Not every mathematical topic appeals to one's intellect with the same strength. The truth is, one never knows just how rich or superficial an idea is until it is fully explored. The better mathematics students readily investigate ideas that, before class that day, they had not even considered. They play “what if” games, and see where their musings lead, even if it is not apparent how such an endeavor will help them finish homework or get a better grade. They seek a coherent understanding that only comes from fitting the new ideas neatly into place with prior ones.
• be humble.
  The best learners are people who are filled with awe and love for their Creator as they gaze on the vast intellectual landscape which He has provided them to explore. They know their teachers are only guides along the way, having knowledge which required a lot of work to aquire, but who are still exploring that landscape as well. They know that past experiences may not necessarily point the direction for current ones, and they readily shed old perspectives when new ones are required. And, they are too happy about the delights of the journey to demand quick transport from one destination to another.

• Am I able to speak about the individual concepts and how they fit together in terms similar to those used in the textbook and/or by the professor? Am I using the language of mathematics properly, expressing my problem solutions in a clear flow from start to finish?
• Is my general trend away from a “give me the map” frame of mind (i.e., “Just tell me how to do the problem”) and towards an “explorer's” mindset (i.e., “I want to be able navigate my way through unfamiliar situations”)?
• For those concepts that were already familiar to me, am I shedding ideas I thought I knew to be true but could not say why, and replacing them with ones that I can support?
• Am I learning to see the little differences that matter — the ones that make one problem far different from another — as well as the (seemingly) big differences that do not?
• Am I learning to recognize those spots during an in-class example where detailed notes are most important?
• Am I getting better at reading the textbook? Am I beginning to anticipate questions that seem important to the author and/or professor? Can I work through book ”Examples” without actually reading their “Solutions”?

This page maintained by:
Thomas L. Scofield
Department of Mathematics and Statistics
Calvin College