This Q&A was adapted with permission from the book Chalk Talk: E-advice from Jonas Chalk, Legendary College Teacher, edited by Donna M. Qualters and Miriam Rosalyn Diamond –
Question
Dear Jonas,
I’m frustrated with my students’ apparent inability to tackle basic problems in my course. Their approach to these problems appears to be haphazard and usually consists of writing a bunch of equations they think will help, and then choosing the equation that appears to have the right variable names, hoping that one of the equations will be solvable. Isn’t there any way I can get them to solve problems in a systematic way?
Signed: Soul-searching Solver
Answer
Dear Soul-searching Solver,
Problem solving is about the integration of two complementary skills: understanding a concept or principle and applying that concept to particular sets of circumstances. Generally learning takes place in increments with understanding enhanced by application and application enhanced by deeper understanding. Learning often does require some trial and error that leads to discovery.
In math, science and engineering, we generally present or derive a concept linearly and logically. We then illustrate that principle by applying it to a particular situation. It is relatively easy for a student to observe this process step by step and agree with the process without understanding fully why the steps were taken. We’re all familiar with a student saying, “I understand everything you did in class, but I don’t know how to even start the homework”.
We must constantly remind ourselves that we (instructors) understand the concepts fully and we intuitively know what solution strategy is best for the particular application we are presenting – but our students generally don’t. It’s a new experience for them. I could take a map of the Eastern US and highlight a route from Boston to New York that would work well and could be followed by someone with only the most rudimentary understanding of maps (e.g., roads are lines and have route numbers). If, however, I then asked him to highlight an efficient route from Hartford to Albany, he would probably suggest a continuous route but might include travel on secondary roads (that seem to be more direct) rather than highways. Had I, in the first instance, not just showed the route, but explained why I made certain decisions (e.g., highways are bolded while secondary roads are not) along the way, he would have been prepared to design routes, not just follow one. I could have improved his capabilities more quickly if I had illustrated and explained my selection strategy as I sketched the route from Boston to New York or engaged him as he traced the route from Hartford asking why he was making certain decisions (“Why include that link?” “Oh yeah, I forgot that the dotted line represents a dirt road.”). Intelligent trial and error is fundamental to learning and understanding. Our own efforts in research remind us of that.
Clearly, the first step in problem solving is to present the basic vocabulary and concepts. Then we move on to applications. Here we must remember to explain the “why” as well as the “how.” The primary method for “teaching” problem solving is modeling your thought process as you solve problems (i.e., showing students that there is a strategy for problem solving that is not random or based on luck).
Here are some suggested general strategies for teaching a logical problem-solving method:
- Present the problem, and be sure that it is understood (e.g., the task is to identify the fastest driving route from Boston to New York)
- Actively engage students in identifying graphical representations, tools, concepts and strategy (at least the first step or two) that would be potentially helpful in solving the problem. Often a suggestion will be put forth that you know will not work. It is sometimes a worthwhile learning experience to pursue the suggestion until it’s clear to all that it won’t work and why. This should improve understanding of the concept and lead to the selection of a more informed solution strategy. It will further illustrate the interconnected nature or problem-solving and understanding – not just the “hows” but the “whys”.
- Identify common pitfalls in solving certain types of problems (e.g., going back to our driving directions analogy, you could point out that exit numbers often change when one crosses state lines, and that this can be confusing; or, point out that people often miss the split in the road when I-95 and I-93 fork)
- After arriving at the correct solution, you might ask your students how the problem could have been changed to make the outcome different or require the use of a different approach to solve it. You might pursue a permutation of the problem to illustrate the altered solution strategy
- Homework involving multiple permutations will give students practice in applying the general concept to differing situations.
You have to use your judgment on how explicit to be in this step-wise approach, but be careful about assuming that your students recognize how you got from one step to the next. Showing students that there is a logical strategy for effective problem solving will reinforce the procedural aspect of the process, and make it less like an exercise in voodoo or something one needs years of experience to master.
Students don’t typically realize that we are trying to teach them general problem solving skills. They naturally expect to learn something about the subject of the course (“calculus”, “physics”, “chemistry”, “engineering”), but they might not see the bigger picture. A clear statement in the course syllabus that one of the goals of the course is to learn problem-solving skills would be helpful. You can also provide homework or quizzes and tell them in advance that the grading will be based entirely on how systematically they approach the problems. Whether we’re teaching engineers, scientists or mathematicians, technical problem-solving skills are perhaps the most valuable tool we can impart to our students.
Good luck,
Jonas
Quick Tip
In our research efforts, understanding is a result of our discoveries; it’s the same for the students in our classes.
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This content was adapted with permission from the book Chalk Talk: E-advice from Jonas Chalk, Legendary College Teacher, edited by Donna M. Qualters and Miriam Rosalyn Diamond.
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