Practice? Practice What?
OK, practice is an important aspect of learning, but in a problem-solving context, what needs to be practiced? To paraphrase one of the seven principles of learning1 in the context of a problem-solving course: to develop problem-solving mastery, students must acquire component skills and knowledge, practice integrating them and know when to apply what they have learned.
So, a prerequisite to developing problem-solving mastery is the acquisition of component skills and knowledge. The are several kinds of “component skills and knowledge” that are necessary. One is a knowledge of definitions of terms associated with the problem type. Another is familiarity with the equations used when solving the problem type. If the students are required to memorize them, that is part of the component knowledge; if not, the ability to identify them is necessary. The component knowledge also includes understanding the assumptions that are inherent in the equations used to solve the problem type. Most problem-solving courses have pre-requisite coursework. Equations, definitions, etc. from pre-requisite courses that may be used for solving the problem type are also included in the component knowledge. In most cases, knowing how to perform the mathematics and/or write the computer code for solving the problem type is a component skill. Simplification of the equations (elimination of neglible terms, etc.) is another component skill.
All of those component skills and knowledge are fairly obvious. There are two others that are not as obvious, and, unfortunately, they are often not taught in problem-solving courses. They relate to the last part of the learning principle: knowing when to apply what they have learned. Knowing how to identify each of the different problem types encountered in the course, and, having identified the type of a problem, knowing the general prodcedure that is followed when solving that type of problem are essential component knowledge items. As I said, they often are not taught, perhaps with the assumption that they will be learned when completing homework assignments. I believe that they should be presented at the same time the other forms of component skills and knowledge are presented, and they should be emphasized.
As an example, in one problem-solving course that I teach, there are five basic types of problems. The same general equations are used when solving four of the five problem types, but the equations are used in very different ways with very different objectives. When I present each of the five basic problem types, I talk about how that type of problem can be identified and I list a set of distinguishing characteristics that can be used to differentiate that type of problem from other types of problems. I follow that immediately with a description of the general steps that need to be taken when solving that type of problem.
Next, when I present examples of solving that type of problem, I try to always start by explaining how I identified the problem type and knew what to do to solve it. As an instructor, it is easy to leave this out because being an expert in the subject, the instructor does this almost sub-consciously. It is a form of “instructor blind spot” where something is so obvious to the instructor that they don’t mention it, forgetting that it is not necessarily obvious at all to a student. The same problem exists in the examples presented in many textbooks. After presenting the assignment, they jump immediately into the solution and never mention how one would know where or how to start.
The problem is exacerbated by the general structure of courses and textbooks. Both tend to proceed sequentially from one type of problem to the next. When a student is assigned a homework problem, they never need to identify the type of problem involved because the assignment is associated to specific classes in the course and specific chapters in the book where that type of problem is presented. They know the problem type based on the chapter in the book it comes from or the classes they attended during the past week.
In contrast, it is not uncommon for exams to include more than one type of problem derived from different classes or book chapters. If students have not been taught how to identify a problem’s type and had an opportunity to practice it, the exam may be the first time they need to do so.
A similar issue is associated with knowing the general procedure used to solve a given type of problem. A student may understand everything that the instructor presents in class while solving a given problem type. Most students probably won’t even realize that the instructor never explained how they (the instructor) knew what to do. Later, when they are assigned that type of problem to complete for homework, they may realize after reading the problem statement that they don’t know where to start or how to proceed. To find out, they likely will look back at class notes or textbook examples and see what steps the expert took when solving the problem, and they will take the same steps.
I refer to this as the mimicry approach to solving problems. One issue with this approach is that they are practicing solving problems using available solutions as a “recipe.” Eventually, after solving enough problems of a given type, the student will associate the recipe with the problem type, but not through a conscious effort to do so. A more efficient and effective approach is to explicitly tell the students that they need to know the “recipe” for each type of problem they will solve and to teach them that recipe. That is, to make sure they understand that the “recipe” is an important part of the component knowledge they must consciously acquire in order to become a proficient problem-solver.
So what does needs to be practiced in a problem-solving course? First students need to practice identifying a problem’s type, and to do that they first must have acquired the knowledge of how to identify a problem’s type. Second, students need to practice knowing which of the component skills and knowledge that they have acquired are needed for solving a particular problem type. Again, the general approach for solving a particular type of problem should be part of that previously-acquired component knowledge. Finally students need to practice actually using their knowledge and skills to solve problems.
Putting it differently, students won’t automatically know when to apply each bit of knowledge they have acquired. Instructors can help them by teaching them how to identify different problem types and a general approach for each problem type that inchldes which bits of knowledge are needed and how to use them. Instructors can further help students by giving them opportunities to practice all of these things.
Footnotes
S. A. Ambrose, M. W. Bridges, M. DiPietro, M. C. Lovett, and M. K. Norman. How Learning Works: Seven Research-Based Principles for Smart Teaching. The Jossey-Bass Higher and Adult Education Series. San Francisco: Jossey-Bass, 2010.↩︎