How to work on problems?

NCTM recommended that problem solving be the focus of mathematics teaching because  it encompasses skills and functions which are an important part of everyday life.  Problem solving is an essential and central part of NCTM standards.  Mathematical thinking abilities are important in learning about yourself and about the world.
The focus on this site is on hard mathematical problems where the approach is solving the problem is not obvious to the problem solver at the outset of problem solving process.  Different aspects of mathematical problem solving process  include motivation, problem solving approach, attitude, beliefs, learning from reflection and relevant background knowledge of math concepts.


Problem Solving Approach


There are some problems for which students know the strategy to solve as soon as they examine the problems.  However, for particularly hard problems, they do not know right-away how they can solve the problem.  The progress on such problems often comes from heuristics or 'rules of thumb' that are likely to be useful, but are not guaranteed to solve problems. As a result, the progress on a problem takes the form of multiple explorations or search of different ideas. Progress on a typical problem would involve a student trying out a lot of different leads using such heuristics.  Work on the problem solving may go through different phases such as trying to understand the problem, working on a specific approach, getting stuck and trying to get unstuck, critically examining solutions or communicating. The work may involve going back and forth between these different phases of work.  On this site, we would now be providing a variety of different suggestions for attacking the problem.  Many of these are rules of thumb or heuristics.  These heuristics can be described in the form of form where conditions describe problem situations in which these should be applied and actions describe what should be done.

Are you about to start working on a problem?
If  you are starting the work on a problem or if you are stuck and you do not know how to progress on a problem, try to understand the problem. Ask the following:

  What is given and what is to be found?
  Is it possible to draw a picture or a diagram of the context described in the problem?
  Can you paraphrase the problem? 
  Can you come up with specific examples corresponding to the problem?


Have you thought out an approach to attack the problem?

If the general approach to solving the problem is obvious to you, create a plan to solve the problem based on this approach and carry out this plan.  
If you know a related or similar problem, you can use the knowledge of solution of the related problem  to come with a plan.
Otherwise, you may be feeling stuck.  


Are you feeling stuck?

Many different approaches can be tried to get unstuck. One approach is to try working a simpler version of the problem,  and use the solution to the problem to get insights that are useful in solving the original problem.
When you come a surprise or an 'Aha' moment, try studying the observations that triggered it in more detail and try observing how these could be used in progressing on the problem.  
Alternatively, you may just try to understand the problem better and use relevant suggestions.
If you are discouraged with a few failed attempts, 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." 

Are you busy working out details?

Monitor how you are progressing and backtrack if needed.  
Do not forget to look for patterns, the unusual and surprises. ( AHA! Insights)
Look for any  surprise,  understand it and its implication for the problem
 

Are you done solving a problem or a sub-problem or inferring a key conclusion?

Critically examine your hypotheses and solutions.  Done solving the problem? If it works, check each step.  Can you see clearly that the step is correct? Can you prove that it is correct? 

Learn from reflection
   •Specialize/ generalize heuristics (including meta-cognitive heuristics), Learn new heuristics
·       If the plan does not produce solution in a short time, then check from time to time: why are you doing what you are doing? are you progressing?  This is self-monitoring.
·       If your plan fails, examine why it did not work. Writing with a rubric or a template can help in recalling and studying what you have done so far. Organize the information.  Ask: Can you conclude about the approaches that won't work?  What else did you learn?  Do you see any patterns?

Are you about to communicate your conclusions to a teacher or to partners?    

Final part of your work on a problem is to communicate your conclusions.  What is communicated    may differ depending on the situation.  Sometimes, you are expected to report only the answer to the problem.  Sometimes, you are expected to show your work.  Sometimes, you may be doing collaborative problem solving.   In collaborative problem solving, it is important to be a good communicator.  Helping others on problems that you have solved can help you develop skills needed to become a good math communicator.  The aspects of such communication include  explaining your solution to someone else clearly,  understanding someone else's solution and  providing feedback on it at various levels of detail. After you create an explanation for your solution, examine carefully if you have justified each step in the work.
    Learning From Reflection
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The more you practice the better you will be.  Practice alone is not enough. Reflection over problem solving experience can help a student learn both about the problem situation and about problem solving process.  
Recollect how you progressed on the solution. Remember the important aspects of the progress.  Remember the stages when you were stuck and how you recovered. Also remember the 'Aha' moments if you encountered any.

What can you learn from your experience?
What made the problem hard? What worked? what did not work?   
What was the lesson learnt? 
Does it tell you about effectiveness of different approaches to problems of this type?
If you articulated particular rules of thumb or strategies, what is      
the reason these worked? In what circumstances, would these work? 
Are these specific cases of more general strategies?
 
An important part of reflecting on your problem solving experience is to get better understanding of strategies and rules of thumb that would be useful in future problem situations and, if possible, to come up with new rules of thumb.  This includes getting a better understanding of circumstances under which a heuristic would be applicable as well as specializing or generalizing heuristics

Motivation
Creative problem solving is needed to work on problems for which an approach to solve the problem is not clear initially. On such problems, chances of not being able to solve the problem are high. Our national instincts are to avoid such problems.  If problems are selected where at least partial progress can be done, then that is likely to be encouraging to students. There can be multiple objectives in working on problems:  to gain insights about the situation described in a problem, to learn about problem solving, to come up with a solution to the problem and to communicate problem progress to others on the team.  It is not necessary to have 'coming up with a complete solution' as the only purpose for working on the problems.   Given the right motivation, it is possible to enjoy the process of solving problems even if one does not succeed in it. 
Mathematical thinking can be motivated by surprise, contradiction or a gap in knowledge.  Kids can also be motivated by using appropriate problem solving contexts that interest them. 
Following are examples of problems that can be used to motivate students
Math puzzles
Math games
Math contests
Math in space and sports
Math magic
Math related everyday life activities of students: This can includes problems with situations with school, family, vacation, animals, and food. Math Forum  is one such site with rich source of everyday life problems.
Mathematical thinking can sustained using challenging, questioning and reflecting atmosphere. 
 
With the right attitude and practice, one can enjoy the process of mathematical thinking that involves thinking about mathematical problems, observing beautiful mathematical patterns, coming up with elegant insights,  facing hard problems that you may or may not be able to solve, thrill of progressing on such problems and solving these, reflecting mathematical thinking  and learning from your successes and failures.  Then, you have an activity you can enjoy wherever you areand joy of creative thinking is all you need to motivate yourselves to get going on any challenging math problem.

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 emotions.

(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.

Beliefs