Using Concept Mapping as a Visual Problem-Solving Strategy ...



Appendix B: Problem-Solving Strategies

Polya (1957)

(1) Understand the Problem

a. restate the problem

b. select appropriate notation

c. make a sketch, a drawing, or table

(2) Devise a Plan

a. look for a pattern

b. make a simpler problem

c. make a guess and check it

d. use appropriate labels

(3) Carry out the Plan

a. check special cases

b. verify the details of the plan

(4) Look Back

a. generalize

b. find another method of solution

c. study the method of solution for future reference

Merrill and Hansen (1970's)

Solving problems in physics can become much easier for you when you have a regular pattern to

follow. 3D BE SNUB is the mnemonic used at BYU to help students remember the steps used by

experienced problem solvers everywhere.

D1 Diagram or draw the situation described in the problem. Label the picture with symbols (you may include numbers too) to represent the information given and needed.

D2 Define the symbols when the diagram doesn't explain them enough. List them in a short table, giving their numerical value. Identify the ones you are solving for.

D3 Describe what is happening, and what needs to be found in the problem. One line will do. List any assumptions that you are making about the situation.

BE Identify the Basic Equation that you will use for your solution. At this step we require that you write a very short essay. Describe why this is the correct equation by identifying the assumptions it makes, and the concept it uses; examine your choice carefully; why does this equation work and not others? . . . Select the equation that fits the physics involved and box the variable you are looking for, and check the variables whose values you know.

If you are not sure which equations to use, list all of the equations that are likely, boxing, checking and examining each equation as above.

S Using the basic equations(s) and math, rearrange the equation until you have Solved for the variable needed. It is indeed usually a bad mistake to plug numbers in at this stage.

N After S, then plug in the Numbers. Every number must include its units. Plug in the numbers as given, and keep three significant figures.

U Check the Units to make sure they match the expected answer. If you have not done the algebra on the units earlier, do it here.

B Check the numerical answer for Ballpark accuracy. If you prefer to call this step "Getting a Fermi Answer," then the mnemonic becomes 3D BE SNUF, for snuffing the problem. Use a "B " to indicate if you believe the answer is reasonable, and a "B?" to indicate that you have no experience with this kind of situation that would help you judge whether the answer is reasonable.

Expert problem solvers constantly apply B to each step; does what I am doing fit? Does it make sense? Does it really describe my situation?

Youmans (1971)

1. First ask what the problem requires as a solution. This requires a careful and thorough reading of the problem. Study the meaning and content of each word, phrase, and clause. Write things down when possible. Think with pencil and paper as tools.

2. Next ask what information the problem contains that will help in its solution.

3. What do you know that might help in the solution of the problem? Formulas, data, solutions to similar problems, factual relationships, etc.

4. What other resources do you have at your command? Handbooks, texts, references, friends, etc. Use these.

5. How might all your information be fitted together to produce a solution to your problem? Think of possible solutions. Try them. Keep looking for relationships that will produce a solution. Don't let the problem stun your brain into immobility. Keep turning things over and looking at them differently until you have a satisfactory solution.

6. Check your solution for reasonableness and correctness.

7. Consider what new or similar problems your solution presents or suggests.

Rubenstein (1975)

Stage 1: Preparation. You go over the elements of the problem and study their relationship.

Stage 2: Incubation. You "sleep over" the problem. You may be frustrated at this stage because the problem has not been solved yet.

Stage 3: Inspiration. You sense a spark of excitement as a solution (or a route to a solution) suddenly appears.

Stage 4: Verification. You check the inspired solution against the desired goal.

Chorneyko, et.al (1979)

1. Identify the system to be analyzed.

2. Identify the objective of the analysis—what meaningful relationship are we seeking and how do we know when we achieve this?

3. Identify and compare the elements in the system to identify the basis for dividing the system into parts.

4. Divide the system into parts.

5. Look back.

Ashmore, Frazer and Casey (1979)

|Stage |Comments |

|Stage 1 |i. Many people fail at problem solving because they are not clear |

|DEFINE the problem. |about the objective. |

| |ii. Most frequently a problem is put in the form of a statement or |

| |question. It often helps to rephrase the problem (into one or more|

| |questions). |

| |iii. Often a problem needs structuring (i.e. subdividing into a |

| |number of smaller problems). |

|Stage 2 |This stage has to be alternated with stage 3. As progress is made,|

|SELECT appropriate information. |pieces of relevant chemical information may need to be incorporated|

| |and/or become meaningful to the solver. |

|Stage 3 |i. The simplest problem requires combining two pieces of |

|Combine the separate pieces of information. |information (A and B) to reach solution (S). |

| |[pic] |

| |ii. Most problems require piecing together several pieces of this |

| |type to form a network or “tree”…. The way the problem is |

| |structured (see Stage 1, comment (iii)) will often give information|

| |on the way the tree is formed. |

| |iii. Stages 2 and 3 are alternated until a solution is reached. |

|Stage 4 |It is essential always to check that the solution is: (i) a |

|Evaluate |solution to the problem defined in Stage 1; and (ii) consistent |

| |with the information stated in the problem. |

Program of Actions and Methods (Mettes, et.al., 1980)

Phase 1: Analysis of the Problem

1.1 Reading the problem carefully, e.g. by putting a slant line after every datum.

1.2 Transformation of the text of the problem into a scheme, using paper and pencil to develop an image of the problem situation and to get a schematic survey of the data and unknown. All data should be mentioned in the scheme, in correct symbols and units (e.g. boundaries and characteristics of the system, variables of state, angles, characteristics of the processes). In some cases, plotting or sketching a graph may help.

1.3 Writing down the unknown in symbols, if possible.

1.4 Estimation of the answer, probable sign, magnitude, dimension, special cases.

Phase 2: Transformation of the Problem

2a. Establishing whether the problem is a standard problem. If so, the problem solver can go on to phase 3.

2b. Writing down possibly useful relationships

2b.1.Splitting up the problem into sub problems.

2b.2.Writing down possibly useful relationships from the following sources:

a. Charts with key relationships for this subject.

b. Charts with relationships for other fields.

c. Relationships which follow from data, directly and indirectly.

d. Relationships which the problem solver at this stage only can indicate in general terms.

2b.3.Checking the relationships found for their validity in this problem situation.

2c. Conversion of the problem to a standard problem

2c.1. Trying to interrelate unknown data by applying the relationships to the problem situation and by linking them.

2c.2. If it is not possible to arrive at a standard problem by the actions in 2c.1., the following actions might be tried:

a. Trying to simplify the problem.

b. Trying to restate the problem or to consider it from a different point of view.

c. Trying to solve an analogous problem in a different field.

d. Letting the problem rest for some time.

Phase 3: The Execution of Routine Operations

3.1 Writing down the routing operations and the answer in a well-organized way.

3.2 Checking very frequently whether all signs, powers, and units are taken along, and whether the results still make sense.

Phase 4: Checking the Answer and Interpretation of the Results

4.1 Checking the answer by comparing it with the estimation that has been made in the analysis.

4.2 Checking if the answer is the correct answer for the question asked.

4.3 Checking if all sub-problems have been solved.

4.4 Looking back at the way the problem has been solved to improve problem-solving skills, writing down conclusions.

a. Is the way the problem has been transformed to a standard problem useful in other cases?

b. What mistakes have been made? How could they be prevented next time?

c. Which (key) relations have been used? Should a relationship that has been used probably be incorporated into the list of key relationships because of its importance in solving this type of problems?

Genyea (1983)

Step One: Creation of a clear picture of the physical situation involved.

Step Two: Determination of a method for solving the problem.

Step Three: Algebraic manipulation and arithmetical operations.

Step Four: Verification that the answer is reasonable.

Krulik and Rudnick (1984)

|Read the Problem |Explore |Select a Strategy |Select a Strategy |Check |

|Subskills |Subskills |Strategies |Subskills |Subskills |

|1. Identify the facts. |1. Sufficiency of Data. |1. Pattern Recognition |1. Computation |1. Estimation. |

|2. Identify the |(a) What's Missing? |2. Working Backwards |Competencies |2. Is it Reasonable? |

|question. |(b) What's extra? |3. Guess and Test |2. Algebraic Skills. | |

|3. Understand |2. Organizing and |4. Simulation or |3. Geometric Skills. | |

|vocabulary. |Representing Data. |Experimentation | | |

|4. Visualize. |(a) charts |5. Reduction | | |

| |(b) tables |6. Organized Listing. | | |

| |(c) graphs |7. Logical Deduction. | | |

| |(d) diagrams | | | |

| |(e) algebraic statements | | | |

| |3. Operational concepts. | | | |

| |4. Estimation. | | | |

Bunce and Heikkinen (1984)

Statement of the problem in words.

Sketch of the situation described in the problem (if applicable).

Recall of rule(s), definition(s), equation(s), or principle(s) that might be needed to solve the problem

Solution diagram which identified what is given and asked for in the problem; lists the rule(s), definition(s), equation(s), or principle(s) needed to solve the problem; and outlines the steps used to solve the problem.

Mathematical solution which includes substitution of quantities into the solution diagram; dimensional analysis; performance of the mathematical operations indicated; and check for reasonableness of the answer.

Review which includes rereading the statement of the problem and the solution diagram.

Zoller (1987)

1. Understand and/or STATE and/or RESTRUCTURE the problem

2. Select appropriate INFORMATION and CONCEPTS

3. Combine the separate pieces of INFORMATION

4. SELECT the relevant STRATEGIES and PROCESSES

5. DECIDE or SELECT the ALTERNATIVES for constructing SOLUTION(S)

6. EVALUATE

Gendella (1987)

Stage 1: Use the information given in the problem to create a clear picture of the physical situation to which the problem refers and describe for yourself that situation in qualitative terms.

Stage 2: Consider the physical principles or mathematical equations that relate the quantities involved in the problem.

Stage 3: Devise a series of calculational steps that will enable you to determine what you want to know from the information that is given and the relationships among the quantities involved. (Algorithms are often of great value in obtaining answers to various part of the problem.)

Stage 4: Carry out the appropriate calculations.

Stage 5: Verify that the answer or answers obtained in Stage 4 are reasonable.

Stiff (1988)

1. Select Appropriate Notation

2. Make a Drawing, Figure, or Graph

3. Identify Wanted, Given, and Needed Information

4. Restate the Problem

5. Write a Mathematical Statement

6. Draw from Known Information

7. Construct a Table

8. Make a Guess and Check It

9. Exhaust all Possibilities

10. Make a simpler Problem

11. Construct a Physical Model or Experiment

12. Work Backwards

13. Look for Patterns

14. Generalize

15. Check the Solution

16. Find Another Way to Solve it

17. Find Another Result

18. Study the Solution Process

The Explicit Method of Problem Solving (Bunce, 1986)

1. Given: Information given in the problem

2. Asked For: Information asked for in the problem

3. Recall: Rule, equation or principle that is involved in the problem's solution.

4. Overall Plan: Simplified schematic diagram of the steps needed to solve the problem.

5. Mathematics: Mathematical ratios including the use of dimensional analysis where needed.

6. Review: Rereading the original problem and the first four steps of EMPS (Given, Asked for, Recall, and Overall Plan)

Heller, et al. (1992)

1. Visualize the problem. [“This step is a translation of the problem statement into a visual and verbal understanding of the problem situation. . ..”]

2. Physics description. [“This step requires students to use their qualitative understanding of physics concepts and principles to analyze and represent the problem in physics terms (e.g., vector diagrams).”]

3. Plan a solution. [“This step involves translating the physics description into an appropriate mathematical representation of the problem (equations of the principles and constraints), determining if there is enough information represented to solve the problem, then specifying the algebraic procedure to extract the unknown variable(s).”]

4. Execute the plan. [“Students use mathematical rules to obtain an expression with the desired unknown variable on one side of the equation and all the known variables on the other side. Specific values are then substituted into the expression to obtain a numerical solution.”]

5. Check and evaluate. [“Finally, the students evaluate the reasonableness of their answer—are the sign and units correct and does the answer match their experience of the world and/or their expectations of how large the numerical answer should be.]

Smith (2001)

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