Strategies for

Problem Solving

by Craig Rusbult, Ph.D.

In some parts of this page which was written in 1989 to be Section 20.2 of
Physics: Tools for Problem Solving but not in other parts, the focus is mainly on
"word problems" of the type often found in K-12 or undergraduate science textbooks
and exams, but there are also general principles that are more widely applicable.

 I still think it is useful to master these types of problems, but 
now my perspective on problems has widened to include
 Design Method and Scientific Method. 

20.2  Developing Problem-Solving Skill

Two definitions of "problem" are "a source of difficulty or trouble" and (as it is used in this section) "a question proposed for solution or discussion."  Most people enjoy the stimulating challenge of a good problem and the satisfaction of solving it.  You will feel this satisfaction more often when you master the tools of problem solving.

      Get a Good Start by Understanding the Problem
      Get "oriented" by using all available information (words, pictures, and free) to form a clear, complete mental picture of the problem situation: what is happening, and what are the questions?  Like a good detective, be aware of what is in plain view and aggressively search for clues.

      WORDS:  Read the problem statement carefully, to get accurate comprehension.  Study the word meaning and sentence structure, to gather all of the important facts.  Most problems are written clearly; the problem-writer isn't trying to mislead you, so just use standard reading techniques to accurately interpret what is written.  You will probably find it useful to "skim" a problem to get an overview, and then re-read it carefully for details, using the "successive refinements" method that is described in more detail later.
      Occasionally a problem contains useless information — a "decoy".  In Section 3.3, Problem 3-A shows how decoys "give you practice in deciding relevance, so you can learn to recognize what information should be used and what should be ignored."   { note: Section 20.2 is from the final chapter of Physics: Tools for Problem Solving, a supplementary textbook for introductory college physics that I wrote in the late 1980s.  Occasionally I'll refer to ideas in Chapters 1-19 of this book, such as "Section 3.3" above. }

      PICTURES:  If there is a diagram, study it.  Or make your own picture; when the problem information (lengths, angles, forces, velocity,...) is visually organized on paper, it is easier to understand.  This also decreases your memory load, leaving your mind free to do creative thinking.
    Section 3.4 discusses how tvvax tables and force diagrams help you translate information between thinking modes [words, pictures, equations] and recommends that you treat drawing as a thinking tool, not as art.   { note:  I first read about "modes of thinking" in Conceptual Blockbusting by James Adams.  Later, I adapted and applied this idea to problem solving in physics. }

      FREE INFORMATION:  The problem-writer may expect you to assume certain reasonable things about the problem situation, or to use data that isn't "given" in the problem but is available in textbook tables or in a special part of the exam.

      Planning and Action
      Here is a simple but useful model of problem-solving strategy.
      1) Orientation:  As described above, translate the problem's words, pictures and free information into a clear idea of NOW (the situation that is defined by the problem-statement) and GOAL (what the problem is asking you to do).
      2) Planning:  Figure out how to get from where you are NOW to the GOAL.
      3) Action:  Start doing your plan, and continue until you've reached the goal.
      4) Check:  Ask yourself, "Have I answered the questions that were asked?  Have I reached the GOAL?"

      After orientation you may be able to plan the entire solution from NOW to the GOAL.  It is often useful, however, to plan in stages by looking for a point where you can say "After I get to here, the rest of the way will be easy."  This halfway point is a subgoal.  Here is a summary of the overall strategy:

Step 1:  Orientation — form a clear idea of the NOW and GOAL situations.
Step 2:  Plan a subgoal — "If I can get to here, I'll be able to reach the goal."
Step 3:  Do it — first reach the subgoal, then continue on to the goal.






      Try to keep sub-problems and the overall problem in your mind at the same time, and make smooth transitions between these levels.  If you can say "after I do this step I'll know __ and then I can do __", it is easier to plan overall strategy, because you won't "miss the forest while you're busy looking at the trees."
      A closely related strategy is go-and-improvise.  After orientation, start doing something productive, anything that looks like it might be useful (ask yourself "What can I do with what I know?") until you reach a point where a solution plan becomes clear.  { If you discover how the problem fits into a familiar pattern, the solution is easy.  Here's an example travelers will recognize.  During a trip north on Interstate 5 from Los Angeles to Seattle, you stop to eat dinner in an Oregon city and get lost.  But while driving around the city you see an "I-5 North" sign, and you can suddenly say "I'm not lost anymore.  Now I know what to do!" }

      Pattern Recognition
      When you ask a computer to find a certain phrase in a word processing file, it searches through the file until it finds a "match."  This strategy can also be used for solving problems: form a clear mental image of the problem situation, then search your memory for a match.  Most problems are variations of previous problems, not duplicates, so it is rare to find a "whole problem" match.  Instead, look for matches at the level of principles, equations and sub-problems.  Then use these individual tools, modified in whatever way is needed to solve the present problem.
      Your pattern-recognizing skill will improve as you do more problems and as you learn to focus your attention on the relevant aspects of a problem situation.  Consider three motions: a rock-on-a-string moves in a circle, tire friction lets a car drive around a circular track, and gravity makes the earth orbit around the sun.  If you focus on the obvious differences, you'll miss an important similarity — each motion is circular.  This means that you can, as described in Section 5.2, use the same basic equations for all three situations.
      You should know the range of usefulness and the limitations of each physics tool.  For example, every equation has " if... then..." conditions and "cues" that tell you when the equation can (and cannot) be used.  Good problem solvers generalize an equation to all appropriate situations: no more, no lessOver-generalization will make you use the equation when you shouldn't, and you'll get an incorrect solution.  But with under-generalization you won't recognize a situation where the equation is needed, and you'll get no solution.

      How to develop skill in solution-planning
      TOOLS:  You must be able to understand physics principles, translate between thinking modes (words, pictures, equations), perform mathematical operations (as summarized in Chapter 18), know each equation (cues for when it can be used, its links, the physical meaning of each letter), and use problem-solving strategies.  These tools are emphasized throughout the first 19 chapters of this textbook.
      Your skill in solving problems will depend on whether you can use these tools quickly and easily (to free your mind for creative planning), reliably (with minimum error) and flexibly (in a wide variety of problem situations).

      MEMORY AND EXPERIENCE:  To use tools intuitively you must "internalize" them by working to develop an active memory, and learning from experience like the welder and basketball player in Section 20.1.  With practice, you will get better at planning complex multi-step solutions, and you'll be more comfortable when you must "improvise as you go."

      ATTITUDE:  Your problem solving is more effective — and more fun — when you have a positive attitude.  Attack each problem with enthusiasm (and good tools), confident that you will be able to find an answer quickly.
      Try to balance action and checking.  Avoid the extremes of passive inactivity (because you're afraid to make a mistake?) and unconstrained action (when you never check for correctness).  One useful strategy is the creative process of brainstorm-and-editfirst get ideas and then decide whether they're "good" ideas.  This is also a good way to solve problems.  Often, the hardest part of a solution is just getting started; after brainstorming gets you moving, you can edit the solution-plan and steer the movement in a useful direction.

      PROCESS:  In effective problem solving there is overlap between the phases of orientation, planning, action and checking.  During orientation you search for a solution plan and check it for correctness in quick brainstorm-and-edit cycles.  And with go-and-improvise you begin action before the entire strategy is planned.
      Expert solution finders explore problems with non-mathematical thought.  During orientation they "play with the problem", trying to fully understand its physical characteristics.  This is usually done by getting the big picture first and filling in the details later.  In "The Physics Teacher" of May 1981, Fred Reif uses an art analogy to illustrate a "successive refinements" orientation toward problem-solving process, by comparing two ways to paint a woman's portrait:
      "... One might begin by first painting completely the woman's eye, then her eyebrow, then her nose, and so forth until the entire portrait has been completed.  The same problem might be approached by a "method of successive refinements" which deals first with the major features of the portrait and then successively with more minor features.  Thus one might begin by first drawing a rough sketch of the entire woman, then elaborating this sketch by additional lines, and then successively elaborating with more lines and colors until the entire portrait has been completed.  Almost any painter would use the second method.  ...  In scientific problem solving, such a method of successive refinements can be implemented effectively by using successively more detailed levels of description of the same situation."

      During a successive-refinements orientation, an expert will often discover the equation that leads to a solution.  This occurs because he (or she) has good physical intuition, built on a solid foundation of understanding and experience.  Part of this intuition is equation knowledge; if there is a clear mental picture of the connection between a physical situation and the equations that describe it, understanding the physical situation (by exploring the problem with non-mathematical thinking) will naturally lead to the proper equation.
      In contrast, beginners notice that equations are used in most solutions, so their immediate goal is to find-an-equation.  Too much attention is focused on equations, the visible "tip of the iceberg," and not enough on the "main body of the iceberg below the surface" — the physical understanding that is needed to use equations wisely.

When the going gets tough — Perseverance or Flexibility?
If you get stuck sometime during the solution, you'll have to decide
whether to continue in the direction you're going or to change course.

is needed.


is needed.



If you keep your solution-goals in mind they can act as a "compass"
to help you keep moving in a useful direction.
Look for signs of toward-the-goal progress (or lack of it)
in the direction you're moving.
You may have a feeling, based on logic and experience,
that your solution strategy isn't working and probably never will.
Or you may feel that the goal is almost in sight and you'll soon reach it.

      Checking Your Answer.  To find out whether you've reached the goal, ask:
      Have I answered all of the questions that were asked?  Are my answers reasonable?  Are the units correct?  If the correct answer is available (on a teacher's solution sheet or in the back of the textbook), compare it with your answer.
      You can also ask: Is there more than one possible answer?  { In science class there is often, but not always, only one solution to a problem.  But for many problems in life, and some in science, there can be alternative solutions. }
      Or you can ask:  Is there more than one method that will lead to the same answer?

      Find The Trouble and Fix It
      A troubleshooter is "an expert in discovering and eliminating the cause of trouble in mechanical equipment." (Random House Dictionary)
      You can do troubleshooting with ideas, too.  If you can't solve a problem, find what is causing the trouble and fix it.
      Check the orientation:  Have you correctly interpreted all available information (written, visual and free) to get the proper problem-situation?  Do you know what questions are being asked?  What progress have you already made toward the goal?  Where are you now, and what still needs to be done?    { Orientation is important.  When a student can't do a problem, I'll often just say "Read the problem again; have you missed anything?"  or "look at the picture", and he can easily solve the problem on his own.}
      Have you ignored any "cues" that tell you to use a particular equation?
      How certain are you of your overall solution plan?  Of the individual parts?
      If you think the method is correct but you didn't get a correct answer, check the details.  Is the substitution, algebra and arithmetic correct?  Is everything in the correct units?  Have you done all of the necessary steps?
      If all else fails, try dividing the correct answer (if it is available) by your answer, or vice versa.  If you can recognize anything special about the division result [if it equals 10 raised to some power, or 2 or 1/2 or 4 or..., or pi or 2 pi, or some constant of nature, or...] this may give you a clue about what is wrong with your solution.
      If you still can't solve the problem you may be missing an essential tool.  Re-read the appropriate part of your textbook to search for this tool.  Or ask someone for help: ask a fellow student, or a tutor, T.A. or teacher.
      But you may just need a change of pace.  Sometimes the best thing you can do is get away from the problem for awhile.  This option is discussed next.

      INCUBATION — using "vacations" to solve problems
      If you stop work on an unsolved problem and return to it later, you may quickly find the answer.  Why?    a new approach: If you get stuck in an unproductive rut, it may be easy to break loose (and move onto a better path) when time has weakened the rut's hold on your thinking habits.  an alert mind: If fatigue has decreased your productivity, a break can restore the fresh, clear thinking and confident attitude you need for success.  new knowledge: Between attempts, you may discover a key insight that lets you easily solve the problem the next time you see it.  subconscious thought: Even when you don't actively work on a problem, thinking can still occur "under the surface of your mind" and this may help you find a solution later.
      Using an analogy between hatching eggs and ideas, the waiting period between solution attempts is called incubation.  Here are two useful tips:
      1) To hatch ideas, you must give your mind something to work with.  The closer you come to a solution on your first attempt, the more likely it is that you'll succeed on the second attempt or third attempt or...  The incubation process is ON-OFF-ON: active thought (with solution as the goal), a break, then more active thought.
      2) Expect an answer.  When you take a break, tell yourself "The next time I see this problem, I'll be able to solve it."
      Time:  An incubation break can be short or long, lasting minutes, hours, days, ...
      Activities:  You can skip to the next problem, study another subject, lay down and rest, go for a run, do the dishes, listen to music, talk with a friend, relax in isolated silence, sleep (think about the problem when you wake up, before you get involved in other activities),...
      Don't be lazy (or cowardly) by using incubation as an excuse to procrastinate, to avoid a challenge you should face.  But don't waste time, either.  If you're making progress, keep going!  If not, decide whether doing something else for awhile is a better use of your time, or whether you should persevere.

      how to LEARN MORE
from your problem-solving experience:
      When you finish a problem, review what you've done and, like the expert welder, ask: What can I learn from this problem that will help me do better in the future?
      If you solved the problem, ask: What did I do, and why did it work?
      If you initially had trouble but eventually solved the problem, ask:  What was wrong with my original approach?  Was my orientation incomplete or inaccurate?  Did I lack skill in using the necessary tools?  How can I change my approach so the next time I see a similar problem it will "look different" — more like it should for inspiring a solution?
      If you couldn't solve it, use the strategies in "Troubleshooting" and "Incubation" above.
      If possible, do problems with answers so you have feedback.  A correct answer gives you confidence, and a wrong answer says "change what you're doing."

      When you make a rough draft summary (as described in Section 20.3) and use it for problem-solving practice, two good things happen:  1) the summary improves your problem-solving skill, and  2) you'll discover ways to make the notes better so they can help improve your skill even more.
      Notice how each problem fits into the logical organization of your summary notes and look for patterns:  Do you recognize any reoccurring sub-problem combinations?  Can the summary's organization be changed to show strategy relationships more clearly?  Should any tools (or suggestions for using them) be added to your notes?  What will make the notes more useful for your problem-solving needs?

      Searching for insights and principles plus practice are discussed in Searching for Insight, using a skier and welder as examples.  The diagram below shows the "levels of learning" interactions between specific strategies (like those in Chapters 1-19) and general strategies (as in Chapter 20):  PRACTICE with "searching for insight" ANALYSIS helps you learn SPECIFIC STRATEGIES and GENERAL STRATEGIES, and also helps you combine both of these levels (as shown by the <--> ) into a flexible, powerful problem-solving system.

The most effective problem-solving strategy is to "know your stuff"
by mastering and using STUDY SKILLS and POWER TOOLS.



20.92  More Problem-Solving Strategies

      Flexibility and Perseverance
      Robert McKim, in his book "Experiences in Visual Thinking," discusses the relative merits of perseverance and flexibility: "When should you abandon one strategy to try another?  When is perseverance a virtue, and when is flexibility?  Sometimes dogged persistence in the use of a single strategy yields a solution: despite frustration and fatigue, the thinker rattles the same key in the door until it opens.  On the other hand, it may simply be the wrong key.  When staying with one strategy pays off, we call it "perseverance";  when it does not, we call it "stubborn inflexibility."  Genius is often associated with the ability to persevere or, in Edison's terms, to perspire.  Creativity is also linked to the ability to be flexible.  Clearly, we are facing a paradox.  Perseverance and flexibility are opposites that together form an important unity."

      Playing with Problems
      Modes of Thinking:  To be a good problem-solver, you must translate ideas fluently between three modes of thinking: verbal, visual and mathematical.  I like to think of these modes more concretely, in terms of their most common use in physics problems: words, pictures (on-paper or in-the-mind) and equations:
    Beginning students often try to translate directly from words to equations.  They have noticed that equations are used in most solutions, so their immediate goal is to "find an equation."  By contrast, expert solution-finders explore a problem with non-mathematical thought, "playing with the problem" until words have been translated into a clear, complete mental picture of the physical situation, often by using a successive refinements orientation that aims for getting the big picture first and filling in the details later.
      In the context of problem solving, "pictures" is anything that helps you  (1) form a clear idea of the problem situation, and  (2) bridge the gap between words and equations.  This could be a force diagram [drawn from a word-description, then used to substitute into the F=ma equation] or a tvvax table [whose structure makes it easy to store information so you can see what is "known" and "unknown"] or an actual drawing, a mental image, or...

      A picture-idea often leads to a useful equation.  This is why it is important to know the "physical meaning" of equations:  If you have a strong, clear idea of the connection between each equation and the physical reality it represents, understanding a physical situation (by exploring the problem with non-mathematical thinking) will naturally lead you to a true-and-useful equation.
      As emphasized in Section 2.3, PHYSICS MATH is easy.  It is the mastery of PHYSICS CONCEPTS that will make you a problem-solving expert.  { The main math we have so far is right-triangle trig (easy), tvvax equations (easy), and F=ma (very easy), but problems can still be challenging if they require that you interpret complex physical situations, or use ideas that are not yet internalized-and-intuitive [for example, if you're still "fighting" the independent analysis of x and y motion, it may be difficult to remember this tool when you need it to solve a problem]. }
      Even though math isn't the problem-solving "key", it is important and shouldn't be ignored.  In fact , one goal of this book is to help you be totally comfortable with math so your mind is free to think about words and pictures.  If you are confident about coping with math, you can relax and focus your attention on physics!

      DRAW TO PRODUCE LOGIC, NOT ART:  When you solve a physics problem, use drawing as a thinking tool, to help you translate words into clear picture-ideas.  Relax and let the ideas flow;  aim for logical creative thought, not "art".


The main ideas from Section 20.8, about productive thinking, are summarized in another page.

This website for Whole-Person Education has TWO KINDS OF LINKS:
an ITALICIZED LINK keeps you inside a page, moving you to another part of it, and
 a NON-ITALICIZED LINK opens another page.  Both keep everything inside this window, 
so your browser's BACK-button will always take you back to where you were.
Here are some related pages:

provides an overview of (and links to) my web-pages
about education for thinking skills, including:

Motivations (and strategies) for Learning
goal-directed personal motives for learning;  teamwork;
how a friend learned to weld, and how I didn't learn to ski

Aesop's Activities for Goal-Directed Education
a creative coordinating of goals and activities will
help students gain experience and learn from it

An Introduction to Problem-Solving Design
how to design a product, strategy, or theory
(this includes almost everything we do in life!)

This area of THINKING SKILLS has sub-areas of

This page, written by Craig Rusbult, has a URL of

Copyright © 1989 by Craig Rusbult, all rights reserved