Right attitude towards working on hard problems
Often, when one is not able to solve a problem, one feels
frustrated. Natural tendency is to be disappointed as 'ego' feels
hurt. At an early stage of problem solving process, one may be stuck while solving a problem.
As you are stuck, you may not know of any action you can take to make
progress on the problem. However, you may feel that teacher is expecting
you to do some work. Therefore, you feel unhappy about the situation.
Furthermore, when you are stuck and not able to think of ways to
progress, you anticipate that you are likely to fail in solving the
problem. This adds to the frustration of the situation. This explains why it is common to see students with a negative attitude towards hard problems.
Attitude that helps students enjoy work and persist in effort includes some of the following elements:
(1) Acceptance of the process: Acceptance of the process of
solving hard problems in which you work for a long time and you are
not always sure if you will be able to solve problem and that 'being
stuck' is a normal state and that such a process includes mixed
(2) Thrill of taking on challenges: When one works on an easy task, not solving is viewed as something of concern whereas solving it is not a big accomplishment. In contrast, when one works on a challenging problem, not solving is not a concern as the problem is inherently hard for anyone. When one does solve a challenging problem, there is tremendous satisfaction and a sense of accomplishment. Despite this, it is natural to feel frustrated when you are stuck. When this happens, you can start by trying to identify what is hard about the problem and writing down about the stuck state. Learn a few approaches (e. g. try simpler problem) that can always be used when you are stuck and when you don't know what approach you can try next. Initially, keep the goal 'to try to make progress
on solving the problem' instead of setting the goal of completely
solving what seems like a very hard problem. Thus, one would set many
short term objectives in the process of solving a hard problem and one
would succeed in many of these even if one does not succeed in the
overall goal. In particular, when you use the strategies of working on a
simpler version of the problem or working specialized cases of the
problem, realize that you are actually solving some problems in the
process and making progress. Making progress involves gathering
information, noticing patterns and gaining insights about the problem.
This way, you would have a sense of accomplishment if you work on the
problem and progress without completely solving the problem. Sometimes, after initially feeling frustrated, one is able to make progress on the problem and solve the problem. That can give
(3) Attitude towards failures: Do not be discouraged by failures. Read this quote from the famous scientist, Edison. An assistant asked, "Why are you wasting your time and money? We have had failure after failure, almost a thousand of them. Why do you continue to pursue this impossible task?" Edison said, "We haven't had a thousand failures, we've just discovered a thousand ways to not invent the electric light." Failure often offers a bigger opportunity for learning than successes.
(4) Thirst for learning: Furthermore, have a clear objective of
trying to learn from successes and failures in problem solving process.
To learn the most, you need to reflect both on successes and failures.
And, you are going to learn the most if you are working on the kind of
problem that you are not always capable of solving.
(5) Appreciation of beauty in mathematics: Appreciate particularly neat insights and your 'AHA' moments as you
progress on problem solving. These may be interesting patterns and
surprises you encountered in problem solving. Ingredients of beauty in mathematics include surprise at the unexpected, the perception of unsuspected relationships and alternation of perplexity and illumination. Mathematical beauty is found in patterns. Famous mathematician Hardy wrote "a mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. .. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the idea, like the colors or the words, must fit together in harmonious way."
(6) Interest in mathematical communication: It helps to
write the insights you learn as you work on the problem and those you
learn when you reflect on your successes and failures. Communicating
about these to others helps as well. If you learn a mathematical trick or a puzzle in class, you may want to share it with your friends or siblings.