Solving problems is the only way to learn population genetics... Remember that you cannot expect to know the answer to a problem instantly, or merely by looking up the relevant page in the text; it may take time and effort. (J.M. Smith, p. vi)
However, in most research into introductory-level science education, it has been realized that for students to gain conceptual understanding, the instructor must teach conceptual understanding. Hence, the focus of much recent physics education research has concentrated on what concepts students have of the world around them, and on finding ways to bring these concepts in line with those held by physicists.
However, it is important that problem-solving skills not be neglected in the search for improved physics education.
Perhaps the most compelling reason is given by the results of an American Institute of Physics survey from 1994 (Blake). Members in all fields and with varying levels of degree completion were asked to rank how useful they found various skills in their lines of work. Along with interpersonal skills, computer skills, management skills and others, the importance of problem solving skills and knowledge of physics were ranked on a five point scale from "almost always used" to "almost never used."
The results were somewhat surprising. While almost all AIP members involved in teaching physics ranked knowledge of physics concepts very high on their list, those in non-teaching fields such as industry or basic research ranked such knowledge much less important. Less than half of industry professionals of all levels of degree completion found knowledge of physics concepts to be a frequently used skill. On the other hand, all groups across the board had nearly 100% of respondants ranking problem-solving skills as being frequently used. These results strongly suggest that it would be a good idea to expand research into problem-solving skills. Anything so broadly useful should be given some consideration. Secondarily, problem-solving skills are often a limiting factor on students. They may understand the concept...or think they understand it...but are blocked by inability to do the problem itself. Researchers in various fields of science education have pointed out how students often seem to have great difficulty with problems that are simply concatenations of several exercises the students can already work (Bodner, p 25, Waddling 1983, p 230, Gabel, p 849, Plants, p 26). By improving the problem-solving skills of the student population, it may become easier to spot conceptual difficulties the students have. As cognitive science researcher J.G. Greeno said, "The processes used to generate concepts and procedures in novel situations probably correspond to general problem-solving skills..." (Greeno, p 266). In other words, as Stewart so ably paraphrased it, "...all problem solving is based on two types of knowledge: knowledge of problem-solving strategies and conceptual knowledge." (Stewart, p 734) If we can be confident the students possess the former, it's easier to see where they need help in the latter. Thus, even if the main interest of one's research is to draw out conceptual difficulties, improvements in problem-solving education will make this easier.
The goal of this paper will be to examine a particularly useful problem-solving strategy, that of the Strategy Map. Such maps are used in various forms in many fields for aiding in the solving of problems, and some of these examples will be examined in this paper. Maps from outside the physics field will also be scrutinized for possible adaptation to use in physics education.
A prescriptive method for helping problem solvers organize their knowledge efficiently for the solution of a particular class of problems. Should be possible to organize in diagram form.
Of course, for that definition to make sense, some of the terms used in it must also be defined.
Instructions - A list of operations the solver is told to perform in order to solve a specific problem. If the solver follows the instructions properly, the problem will be solved, but the instructions may not help at all with a similar problem.
Things like assembly plans for a bicycle or cookbook recipies for Chicken Kiev are sets of instructions. While they may present difficulties to the inexperienced, they are meant to be learned and performed rote, much like exercises are. Instructions are completely prescriptive, and don't leave room for individual judgement. Skill in following instructions is useful as a tool in problem-solving, as is skill in working exercises, but it is not the sum and substance of problem-solving.
The simplest form of Strategy Map consists of a list of instruction sets or prescriptive sets and a separate method for choosing the best set for the problem, like the cookbook example given below. More complex maps will more closely resemble computer flowcharts as students navigate a maze of decision gates and checkpoints. But they all share in common the fact that they help users figure out which way to go and then how to get there, just as a road map does.
The front of the book, however, would contain the actual map. It would ask various questions and use the answers to narrow down the list of recipies to those the student can use. Here's a few possible questions:
Answers to these questions would help eliminate recipies that the student wouldn't be able to get any use from and narrow the search. Other question sets might discuss dietary preferences such as vegetarianism or keeping kosher. After answering all of the questions, the cook can then find the set or sets of instructions that can solve the problem of what to have for dinner. Without the map, the cook would have had to pore over the book recipe by recipe and mark off the ones that could be made with the available resources.
GIF of sample Cookbook. Note how the top half uses a simpler progressive map (i.e. if you can use Chapter 3, you can automatically use 1 and 2) while the second half uses more of a search map.
From the kitchen to the classroom, it's now time to look at some Strategy Maps used for solving various other problems.
Sample subgoal page of Stewart's model.
While this model was not intended to be taught to students directly, it would make a good prescriptive method for solving genetics problems. By following the model, a student would perform all the steps necessary to solve a basic genetics problem. No single step is beyond the ability of the students if you presume they've had adequate practice with the steps in previous homework, and most steps are very simple (such as choosing labels for the alleles).
However, at eight pages of flowcharts, this prescriptive method is rather bulky, and might be unfeasibly hard for students to learn. Stripping it down to the core, it uses the standard prescriptive technique of, "Figure out everything the expert does, then lay it out for the novice." Since this is the starting point of other methods, this one doesn't have anything new to offer.
It is common to find students who are competent at the various steps in a titration calculation; however, it is not unusual to find a large number of students unable to link up these steps. It is the "bridging" between existing knowledge and skills that is needed to lead to successful problem solving. (Waddling 1983, p 230)
Waddling's initial method involved a flowchart approach in which students were guided through a variety of steps and decision gates which sent them down the path needed to solve the particular problem they were working on. Rather than a linear set of items, this one branched out to cover various types of stoichiometry problems (see figure).
Assertions are made both in the paper where it was presented and in a later paper that this Strategy Map was useful in helping students work the problems. However, no data is presented to support this in either paper, and attempts of find other works by the author or contact the author have borne no fruit.
Despite the lack of hard data, the flowchart format used looks promising, especially since it fits onto a single page. This presentation style might be worthwhile to adapt for calculus-based physics courses, with perhaps one flowchart for each major class of problems.
Waddling's Map. Adapted from the version printed in Journal of Chemical Education.
In his second paper (Waddling 1988), the flowcharts were expanded on by the addition of diagrams to more clearly demonstrate which experimental steps the students were to perform. These charts were tailored to the particular experiment being performed, such as the reduction of copper oxide or the oxidation of magnesium, and as such are more like instructions. However, they do suggest that in building a Strategy Map for physics labs, one might do well to include illustrations where words don't convey meaning as well.
The analysis is broken into five major blocks (six in an earlier version): Dynamic Modeling, Dynamic Behavior Analysis, Controller Design, Response Visualization and Stability Analysis. The blocks are linked via several intermediate steps involving things the problem solver can do in attempting to solve the problem.
From each of the five blocks there are at least two paths the analyzer can take, such as "Physics Based" and "Empirical" as options for Dynamic Modeling. After the student has followed his path from Modeling, he can either attempt various methods of Response Visualization (analytical via equations or numerical via Matlab or Maple programs), go straight to Dynamic Behavior Analysis to see if it fits the model generated, or perform a Stability Analysis directly from the predictions of the model.
The map has one entrance and two exits: the student always starts with a model, and ends with either a visualization of how the model works or a design for controlling the system. If the results are unsatisfactory, the student begins again at the modeling step, using knowledge from the last run through the map to help guide choices on this run. If a Stability Analysis didn't yield a correct control system last time, perhaps a Dynamic Behavior Analysis should be tried this time.
Each of the subsidiary steps is something the student has already learned to do, such as the Routh Test for instability, or a numerical visualization using Matlab. The map helps them organize these blocks of knowledge into a solution for the larger problem, without leading them down a single path to The Answer.
Preliminary data suggest that use of this Strategy Map has allowed the class to shift focus away from theoretical/classroom aspects of the course without hurting test scores over that material. Early scores suffered, but as students actually got more of a chance to use their knowledge, scores improved and even slightly exceeded old levels.
In short, figure out what kind of physics you need to solve the problem, and solve it. Deceptively simple, it's what experts do. The real work comes in finding a way to train students in this.
Hestenes lays out a model-generation map for this process (Hestenes, p 447), but the terminology used is somewhat difficult to navigate. Below is presented a simplified version of this map.
Simplified Hestenes Map. Initial research with this method showed that even in a lecture situation there was some improvement in problem-solving skills (Halloun and Hestenes, p 460), and since then the modeling strategy has been taught in a more interactive setting on a wide scale (Model-Based Learning Homepage). While the article presenting the results of the modeling method in the classroom does not specify which archtypical problems the students were taught to work, it's not unreasonable to assume this list is related to the "ramified" models presented in the first paper (Hestenes, p 444). Uniform velocity, uniform acceleration (both one-dimensional and ballistic two-dimensional), uniform circular motion and simple harmonic motion.
Unfortunately, this method does have a weakness in terms of teaching problem-solving skills in general. Since it teaches students to try to reduce problems to exercises already in memory or at least available from outside sources, it doesn't address the possibility of encountering problems unlike the paradigm problems. While a great majority of daily experience involves problems that can be solved by referring to what is known already, not all problems can be solved this way. Thus, successes in the classroom won't necessarily translate into better problem-solvers outside the classroom.
Each of these steps is further broken down into tasks to complete on the way to the solution. This is not a map in the same way the stoichiometry and system analysis strategies above are, since it's fairly linear in appearance. However, it works for all sorts of physics problems and helps the student establish his own path to the solution, so it falls under this paper's definition of a Strategy Map.
Slightly expanded version of Heller's method. Still VERY condensed.
This method, like most prescriptive methods, generates a great deal of paperwork for even a relatively simple problem, because no step is left unstated. As a result, students being taught this method are given problems specifically designed to require a strategy, to reduce student resistance to the method (Heller and Hollabaugh, pp 639-640). Expert problem solvers often combine steps or perform them without consciously thinking about it (for example, Stewart p 735), and it is hoped that eventually a student will internalize most of the elements of this map. In an effort to help reduce the imposing load of the method on any one student, small groups are formed to work on problems in this method. Students take turns in different roles: Manager, Recorder, Skeptic and Summarizer. Each role emphasizes different parts of the method, allowing students to focus on part of the outline at a time.
Experimental data (Heller, Kieth and Anderson, pp 632-635) indicates that when this method was taught in a group problem-solving environment, individual problem-solving ability was improved in students of all skill levels. The study did not test to see if teaching the strategy independent of groupwork improved problem-solving ability, nor did it test for improvement based solely on group work without a specific strategy. Work on "Problem-Based Learning," or PBL, suggests that simply coordinating small student groups in the right way can improve problem-solving and student attitudes (Duch, p 3), but no data is available on whether individual problem-solving skills are improved by working in a group. As a result, it is difficult to tell if the strategy itself was the cause of the improvements, or if simply working in a properly managed group improved the skills of the students. Interviews with students (Heller and Hollabaugh, pp 639-640) suggest that the strategy itself is indeed useful, and the fact that the method was included in the 1994 textbook further suggests that more recent data does support the utility of the five-step method.
The results were that, in this case, more is better. The control group showed no real difference before and after, while the third (less prescriptive) group showed improvement in some areas but not all. However, those given the most detailed set scored significantly better in all areas, getting perfect scores in the preliminary parts of the problem and nearly perfect on the later steps.
Given that prescriptive methods seem to work quite well, Reif and his collaborators refined the specific methods used over the next decade, developing a multi-level flowchart that is more compact than the list of questions used in the initial work. Chapter 6 of Understanding Basic Mechanics is devoted to teaching the students this method, as each part of it is examined in more detail in the accompanying text. The chart is only presented in complete form in a very schematic way in the textbook, but a more detailed version compiled from the chapter can be found here.
Reif's Map in one page format.
Why chapter 6, and not chapter 1? The first five chapters are all very basic material, such as vectors, velocity and one-dimensional motion, and don't present particularly elaborate problems. As noted in Heller and Harbaugh (pp 639-640), students chafe at using more prescriptive methods on simpler tasks, since the strategies often involve far more steps than a simple problem requires. By the end of chapter 5, however, students are equipped with the tools needed to solve the subproblems that more complicated problems can be broken into. Chapter 6 shows them how to organize their new tools into a toolbox of skills that should serve them well as the text moves into more complicated matters.
The method used in this case has three basic steps: Analyze Problem, Construct Solution, Check (and Revise if need be). The first and third steps are broken down into a list of questions the student needs to ask about the problem and factors that should be taken into account. The second step, the meat of the method, concerns itself with finding appropriate subproblems that resemble the exercises the students are already capable of working, or can easily figure out how to work.
In constructing the solution, the student first determines what needs to be done: is there missing information? Are there unknowns that might be removed by proper combination of relations? Once that has been determined, the student is helped along the path to accomplishing the subgoal. The text goes into greater detail on each of these steps, the flowchart is meant to be an aid to students who are learning the full strategy as opposed to the complete strategy itself. This Strategy Map is meant to work at several levels of detail, depending on the user's familiarity with the territory: the version below is the mid-sized map, while the entirety of chapter 6 is the full-sized map.
The next paragraph involves some research conducted on the Physics 131 students here in Winter and Spring quarters, and how it ties in to the Analyze Problem section of Reif's Strategy Map.
In Reif's method, students are asked to identify "significant times" in the problem. Any point that stands out as being when something changes or happens or otherwise looks important. Points such as the top of a ballistic arc or the beginning of a frictionless section of a surface being moved on. By finding these points, the student can assign subproblems that consider only the behavior either at a point or between points, rather than trying to solve the problem in one chunk. Early work with a mechanics lab last year suggested that student groups have trouble choosing these "Zones" of interest when not specifically guided through the process of identifying them. This raised the question of whether students were simply not thinking to look, or if they were looking and not finding. A series of multipart problems was designed and given to students with specific instructions on identifying the significant points and the Zones of interest. With the exception of some apparent misconceptions on the working of springs and the importance of collisions, students were able to break up the problems into manageable pieces when instructed to do so. This suggests that students can be successfully taught a method for identifying and solving subproblems, such as Reif's.
It is worth noting that while the textbook's most elaborate version of the map focuses on mechanics problems, the basic form of the map presented above is applicable to a very wide range of problems that involve a mathematical description of some physical situation.
Richardson lays out the core schema as a concept map, with the main concepts of charge, field and potential linked by various laws and equations. Then the student is to go through a list of questions much like Reif's early prescriptive method while using the core schema as a guide. This is similar to Model-Based Learning, if less well-developed, in that it attempts to guide students to a problem-solving schema right away. It is worth noting that Richardson's use of the term "schema" isn't quite the same as Gick's, but Richardson's use is more general. The method has been condensed into a single page map by this author, similar to the Reif map above in that there is a level of detail beyond the one-page map that students would need to have access to in order to learn the strategy.
Richardson's Electrostatics Schema Map built from references in his article.
In each of the main areas of the core schema, there are three lists of questions for the students to answer and steps to follow as they work through the method. For example, under Charge, the detailed map has:
1a) Charge Geometry: indicate on a figure of the situation at hand the sign(s) and, if appropriate, the magnitude and distribution of any and all charge(s), including any induced charges. 1b) Qualitative Description: Qualitatively, what approximate or actual charge configuration(s) can be substituted (if any) for the configuration depicted as a function of position either near to or far from (or within or on) the depicted situation? (This is closely related to 2b.) 1c) Quantitative Relations: What quantitative relationship(s) can you specify which determine the magnitude and/or density of the charge(s) in the situation before you? Give actual values, if possible, for charge(s) or charge densities in the situation. (Richardson, p 7)
There are equivalent lists for Electric Field and Electric Potential, as well as for the checking and implementation stages of the map (Richardson, pp 7-8, 11-14).
Unfortunately, due to errors in training of teachers, the strategy failed to create a significant improvement in problem-solving ability in the experimental group (Richardson, pp 25-26). The method is still believed by the researcher to be valid, but the difficulty in transporting it from the researcher to the student prevented the data from supporting this opinion. No further results could be found, as Richardson moved into the field of Artificial Intelligence shortly afterwards and apparently did not pursue this line of research. Email has been sent to Dr. Richardson, and if a reply is received before the oral presentation, the results of this will be included then.
One page solution of problem using Reif's method.
"You're out sledding with friends, and it's late in the day. The side of the hill has become icy from all the sledding and the afternoon sun, so it's almost frictionless. At the bottom of the hill, about 10 meters of flat ground are also icy, followed by softer snow. Unfortunately, 20 meters after the snow stops being icy, there's a set of baseball bleachers which you dont really want to sled into. You decide that rather than experimentally determining if you'll crash, it's a good idea to do some rough calculations first. It's about 50 meters down the hill at a slope of what you guess is 30 degrees. You and your sled together mass 60kg. You don't know the coefficient of friction between fresh snow and toes dug into it, but you figure it can't be more than .7, since that's what tires on dry concrete get. Is it safe to try sledding down the icy slope, or would you have to bail out at the bottom to avoid becoming part of the bleachers?"
Solution is three pages, and even that is somewhat trimmed.
Before establishing other criteria, it is worth noting that all of the strategies presented have generated at least anecdotal successes in their areas of use (the biology strategy being a special case...while not taught as a strategy, it was found that successful problem-solvers used it or something like it). The question becomes, would the method be worth teaching in a large calculus-based physics course? This leads to a number of criteria to evaluate each method by.
The importance of criteria 1 and 1a are fairly self-evident, since a non-functioning method isn't one to recommend adopting. Criteria 2 and 2a are important to examine because a method is more likely to be adopted if it doesn't require major modification of the class structure to implement, especially in larger schools where changing the nature of a classÕs scheduled times can take years. Criterion 3 is important in light of the previously-mentioned importance of problem-solving skills in the workplace...a method that only helped solve textbook problems would fail this test.
One page review of all the methods, rating them out of four stars in the various criteria. For a more complete evaluation, read on.
1a) The core of this method is to determine every step the expert takes in problem-solving, then teach those steps to the student. As shown in Reif and Heller's prescriptive work, this does work.
2) Not administered as a strategy to any student, used as an evaluation tool.
2a) As with Reif and Heller. Very prescriptive methods can be used on an individual basis, taught as part of a textbook.
3) Unknown, although the high specificity of this method suggests it wouldn't carry over to problems outside the class.
4) Probably not. Far too complicated for most purposes. Overall, a prescriptive method of this level of detail, practically a set of instructions, is not recommended for use in introductory physics classes.
1a) The principle of a one-page map to guide students through a class of problems is certainly attractive, although the other one-page maps seen have a lower level of detail than this one, suggesting all of mechanics at once cannot be treated exactly like this.
2) Anecdotally successful in teaching chemsitry students by simple presentation of the map and lessons in using it.
2a) If a similar map could be generated for mechanics, it might also be usable as a supplement to lecture.
3) While somewhat more general than the Genetics method, this is still fairly narrow, employing specific procedures rather than general tactics. Probably not very useful outside of chemistry.
4) Yes, Waddling indicated students did in fact use the chart. Overall, while a map of this sort would certainly be useful in helping solve a particular class of problems, it probably would only slightly increase overall problem-solving skills as it doesnÕt teach general strategies.
2) Shown to be mildly successful in a traditional lecture and recitation setting (Halloun and Hestenes, p460), but acknowledged to be more successful in a group-work setting that is more interactive (MBL Homepage).
3) Uncertain. However, it does teach students to try to simplify any problem into a "model" that can be more easily solved, and that should be useful in any context. However, it is less useful in situations where the problem cannot be reduced to a previously-learned exercise, and these situations are more common outside the classroom.
4) Given the widespread teaching of this method at the current time, it's fairly safe to say students do use it. Overall, while it may not be an optimal problem-solving strategy, it is at least a successful one in the classroom. This makes it better suited for the high school level education it is currently being applied to (MBL homepage).
2) Rigorously shown to work in group settings where the total class size was small enough that the teacher and TAs could effectively manage the groups.
2a) If applied to recitation sections, it should work. 100+ student lecture sections may be too large for this method, however, so there should be enough recitation time each week to implement this.
3) Since it teaches a general strategy with emphasis on the specific methods needed for physics problem-solving, this method should help overall problem-solving skills of students. Especially in the areas of focusing the problem and checking the results.
4) It is noted that students do not like using this method on simpler problems, so it is necessary to design problems complex and content-rich enough that students have to use the strategy to succeed (Heller and Hollabaugh, pp 639-640). This is worth noting as a general point, as students will have little patience with a complicated strategy when they can solve the problem by simpler methods they already know, even if their current methods are flawed (Heller and Hollabaugh, p 638). Overall, a method worth teaching in introductory physics classes, but only if it is feasible to spend a fair amount of time in small-group work.
2) The method is presented to students in the textbook, which allows for reinforcement by numerous methods that can fit a given classroom setting. The general prescriptive method was shown to be successful in individual work guided by written aids, without need for special arrangements of the class as a whole. The textbook and teaching guide simply emphasize use of the method itself, without groupwork or other factors. As a result, this method is very likely to work in the standard lecture-and-recitation class.
3) This method is a heuristic method, in that it teaches the student ways of thinking and learning. It should be quite usable outside of the domain of mechanics problems.
4) Repeated reinforcement of the strategy is applied throughout the textbook and the teachers' guide gives additional advice in encouraging use of the method. Problems are not tailored to need the method in the way Heller's method does, but this might be a useful addition. Overall, this is probably the best choice for teaching problem-solving to a large class of introductory physics students. Elements taken from other methods could certainly improve it, of course.
1) Research failed to show successful implementation, but the process of presenting this method may have been flawed (Richardson, pp 25-26).
1a) Given the similarity of this method to Model-Based Learning (teach core schema and then how to place a problem within the schema), it might work if taught correctly.
2) Attempted for use in a large course, failed in part because teachers and TAs were not properly trained.
2a) Again, given proper delivery to teachers, it could probably be taught in lecture with success similar to Model-Based Learning.
3) Marginal. Systematic search methods would be transferrable, though. Schema themselves apply only to the electrostatics material.
4) Students were graded in part on how well they applied the method, providing incentive for its use. However, this after-the-fact incentive is probably not as useful as the front-end incentive of having problems that require the method. Overall, this is essentially an underdeveloped electrostatics version of the same principles as Model-Based Learning, with the same benefits as that method.
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