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 <condition, action> 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.

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