Good at Math?
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
Of course, very few of us consistenly do/are all of these things all
of the time. There are plenty of students who earn good grades,
especially in 100 and 200-level mathematics courses, despite not fitting
the description above in a number of ways. But if you are willing to
press on toward the goal, then you should consider how
you might become more like these ideal mathematics students. To make
things more concrete, here are some specific questions that you should
grapple with throughout the semester.
- 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.
While the specific day-to-day skills of the course are to be mastered,
your current/future enjoyment and success in mathematics courses is
equally dependent upon growth in the areas listed above. Do not expect
this growth to occur just by doing what is required of you in the course.
It is likely possible to earn good marks in the course and still
answer No to the above questions at semester's end.
If you really wish to become mathematically adept,
however, you should begin immediately to think creatively about how
you can structure your study habits so that your answers are
increasingly Yes. Here are
ideas the professor has which may be helpful.
- 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
Friday, 03-Sep-2004 15:03:32 EDT
Department of Mathematics and Statistics,