A GENERAL PROBLEM-SOLVING METHOD
[A.INTRODUCTION] [B.OVERVIEW] [C.EXAMPLES] [D.PEDAGOGY]
A serious difficulty experienced by weaker students in high school is a lack of skills in solving word problems. One pedagogical approach has been to have the student "learn" to recognize problem types and to memorize the corresponding solution steps (constructing a solution). It then appears that the student is a good problem solver because he or she can answer the questions asked by the teacher. However, when the student faces problems that are new and novel, he or she is at a loss to know what to do: there is no "construction" that exactly fits the problem under discussion. The method presented here attempts to overcome this shortcoming by providing students with a linear-thinking approach to problem solving.
The process of problem-solving can be divided into four tasks, in the following order: determining the GIVEN, defining the OBJECTIVE, working out the PATHWAY that leads to the OBJECTIVE and, finally, doing the actual calculations required for the SOLUTION.
The last step (SOLUTION) is purely mechanical, involving arithmetic and algebraic operations only: all the analysis and thinking have been done in the GIVEN, OBJECTIVE and PATHWAY steps. It is the PATHWAY which is the key to this linear-thinking approach.
The whole process will first be outlined in a cursory fashion in order to get an overview of the method, with two examples from different fields of Science. Another example will then be presented in more detail, with notes indicating the pedagogy of each step.
B. An Overview of the Method
The first task is to determine what is Given. To do this, the student must read the problem, looking for information about variables and quantities. The student will thus initially transform the information into numbers and units and record these under the heading "Given". In some circumstances, the Given will include a diagram or picture (for example, in electricity circuit diagrams may be necessary). The student will attempt to avoid using words as much as possible.
The second task is to define where we want to go with this information: to set out the "Objective". Again, the student will use symbols and/or units rather than words wherever possible.
Once the Objective is defined, the next task becomes to determine how we can get from the Given to the Objective. This is the point at which this method becomes different from others. Instead of "learning" a pathway ("when you have this kind of problem, this is what you do"), the student will build, step by step, a logical pathway from the relationships which he or she understands, beginning with the Objective. Each step will be based on a definition, a chemical, physical or mathematical principle or on a formula expressing an experimentally verified relationship. The reason for beginning with the Objective is that there may be many items of information in the Given, making it difficult to know which should be used as the starting point in the "construction" of a solution. There will, however, be only one Objective: it is thus a sure starting point. For this reason, the "Pathway" will be built backwards, going from the Objective step by step back to the Given.
Once the logical pathway has been developed, the student will implement it, now beginning with the element of the Given that is found at the end of the Pathway. The "Solution" may be carried out in a single step if factor-units are used (as may be appropriate when the relationships are direct or inverse proportionalities), or in several individual steps, depending on the particular Pathway being followed. Students may be taught the methods of dimensional analysis (factor-units), as a tool in the implementation and verification of the Solution
The following is a simple example of how the method works (note that the essential is not the use of factor-units, but the linear approach to the thinking):
Note that the Pathway step includes notations of what relationships will be used to carry out the operations indicated by the arrows. These notations can be left out if the relationships become so familiar that the student no longer needs to be reminded of these details. For the novice however, these relationships make it much easier to do the last step, the actual calculations indicated as "Solution". The student has thought it all through in the Pathway and does not have to repeat the thinking in the calculation step.
The following is a re-working in Pathway format of an example from a paper by Howard C. McAllister ("Problem Solving and Learning", 1993) 21st Century Problem Solving :
2. Show that the total energy of a satellite of mass m in a circular orbit of radius r about the earth of mass M equals half its potential energy.
Given: Total energy (E), U, K
Objective: E = 1/2 U
Pathway: determine E and compare with U:
Solution: Substitute and solve.
F/m = GmM/r2m = GM/r2
ar = GMr/r2 = GM/r
K = 1/2 m(GM/r) = 1/2 GmM/r
E = 1/2 GmM/r + -GmM/r = -1/2 GmM/r = 1/2 U
D. Pedagogy of the Pathway Method
The following is a problem which will be used to show a possible pedagogy of the method outlined above. The problem is presented in the final form first: this is what the student's notes will look like. It is followed on the next page by a step-by-step examination of each section: I have called them "pedagogical notes".
Commercial hydrochloric acid has a density of 1.19 g/mL and a mass percent of 38% HCl. Determine the molarity of the solution.
|The Given is a symbolic representation of the information of the problem. But it is
more than a transcription of that information. The information is also transformed by
expanding the units given in the problem, making them explicit; for example, the density
is not expressed as simply "g/mL" but rather as the ratio (or factor)
|The Objective is also expressed in as "revealing" a way as possible. Instead of writing "Molarity" or even "M" as the Objective, we indicate composite units like molarity in terms of the units that go into making them up. The reason for doing this is that it clarifies what we are really looking for. A clear concise Objective is essential to good problem solving. At this point in the process of problem solving, all the essential information of the problem has been analyzed and is on paper in symbolic form. There is no further need to refer to the original problem.|
|[PATHWAY]||Here we will take it step by step, as the teacher would do in developing the problem with the students.|
|This first step begins with the Objective: the teacher writes the Objective on the board and then questions the class: "what do you know about moles of HCl, in terms of what is in the Given?" The class will already have learned about the Mole and its relationship to Grams through the Molar Mass. Now the students must use that information, by realizing that if they knew the number of grams of HCl, they could get the number of moles of HCl. The backwards arrow represents the mathematical operation involved in going from g HCl to mol HCl that will be carried out in the last step (Solution).|
|At this point, it may be wise to write down the relationship or formula which will be
used to get from g HCl to mol HCl. The factor or the formula can be written under the
arrow. Once the relationships become automatic, as they usually do for molar mass, they
may be omitted from the Pathway, which will then be simply
|Once a logical connection has been established for the original objective, the newly
connected element becomes the new objective. The students are invited to think about what
they know about gHCl, keeping their eye on the Given. In this case, they find the
which gives a way of getting between g HCl and g solution. So we write another backwards arrow to join g solution to g HCl.
|Again, at this point, it might be wise to write the relationship under the arrow, as a reminder for the calculations that will be done later.|
|Finally, we realize that we do not know the g solution and must assume some mass. We may choose any mass we wish: the only constraint is that we must be able to get back to the same assumption in the other part of the Pathway, finding the L solution. And now we are finished with this Pathway! Note that since the problem-solver is invited to focus on only one objective at a time, he or she becomes more efficient: the elements that do not impinge on the immediate objective need not be considered. If they never enter the logical pathway, they must be "dummy data".|
|The second Pathway is begun the same way as the first, with the Objective. The question is "What do we need to know in order to get L solution?" And the answer might come back: "We could get the L solution if we knew the mL solution." We would then write the relationship under the arrow, if the students are not already very familiar with this identity. If they know it very well, the arrow will suffice.|
|The next question is: "What do we need to know in order to get mL solution?" And hopefully the students will answer: "We could get the mL solution from the g solution, using the density. We could then write the density under the arrow to make it very clear how we know that the arrow represents a true relationship.|
|The last step is to make the same assumption that was made for the other Pathway. Having got back to the same assumption, we can be confident that when we do the calculations, the assumptions will cancel (since we are determining something that is a quotient). And that's it!!! All that is left to do is the actual calculations...|
|We now begin at the end of the Pathway and work our way forward step-by-step. The
first step involves using the relationship
The solution shown here uses factor-units. It is also possible to do this step using ratio and proportion (la règle de trois), but it would be well to encourage the use of factor- units. It is a much faster method, it is self-checking (by the units) and it is mathematically equivalent.
|The last step in this Pathway is to change the g HCl to mol HCl, which is accomplished
by multiplying by the factor
or by using another ratio and proportion. Another advantage of using factor-units is that all calculations can be done at once, instead of in installments. At this point, the student is encouraged to check the units to ensure that no "silly" mistakes have been made: if we do not get the unit that is in the Objective, then we have made a mistake somewhere.
|Again, this Solution begins at the end of the Pathway with the 100 g solution, and proceeds first to change g solution to mL solution, and then to change mL solution to L solution.|
|The very last step is to put the results of our two Pathways together and report the answer to the question, to the proper number of significant figures (of course!).|
And that is all there is to it !!!
The author would like to acknowledge the support provided by lively
discussions with John Miller and other colleagues at St. Lawrence and with Howard
McAllister at the University of Hawaii (via the Internet). May we continue to bat this
ball back and forth for many moons! -->Top of this page