I once assigned a problem asking students to calculate the speed of an electron. One of the students gave a solution of approximately 1.0 × 1069 m/s. Although I don’t remember the exact numerical value, I vividly recall my shock at seeing such an astronomical order of magnitude. Honestly, I would have preferred if the student had skipped the problem altogether. Unfortunately, while not always as striking as this example, it is not uncommon for students to report nonsensical results. This is a serious concern because it suggests a complete lack of understanding of the problem at hand.
When solving problems in science and engineering, it is crucial to remember that all numbers, along with their units, carry important physical meaning. Mathematically, 1.0 × 1069 is a perfectly valid number; however, what kind of speed is 1.0 × 1069 m/s? How is that even possible?
Another, more subtle issue is the use of an unrealistic number of significant figures. For example, consider a bond length reported as 2.12456891 Å (I’m making up the number here, but you get the idea). What an impressively precise number! The problem is, is there any instrument capable of measuring with such precision? Is it even necessary to be that accurate?
In both cases, I believe the student performed some calculations, copied the results from their calculator, and moved on. On the surface, this approach seems fine—the problem was solved, and a solution was found. However, a critically important step is missing: reflecting on the results and the solution process. From an instructor’s perspective, merely completing an assignment or reporting a result holds little value; what we are looking for is an understanding of the problem. Reporting a value that is orders of magnitude off or using an excessive number of significant figures certainly does not convey a sense of understanding.
This is where reflection comes into play. Instead of jumping to the next problem immediately, take a moment to examine your result and ask yourself: does it make sense? Is my result consistent with known values (which can be easily obtained through a simple online search) or expected trends? If not, what might be wrong? Sometimes, it is a simple calculation error that can be quickly fixed; other times, it is a problem with the solution procedure. The latter can be more challenging to identify and correct, but as you revisit the solution process, your understanding of the problem will gradually improve.
Even if your result makes perfect sense, do not move on to the next problem just yet. Ask yourself: what does the result mean within the context of the problem? Did I learn anything new from solving this problem?
Reflecting on your problem-solving process certainly takes time, and I understand the temptation to skip it altogether. However, reflection is the best investment you can make in your learning, arguably more important than solving the problem itself. It provides an opportunity to develop your critical thinking and problem-solving skills, which ultimately leads to improved learning outcomes. Consider this: if you solve a problem in ten minutes but do not understand it, or you spend twenty minutes and fully grasp the problem after solving it and reflecting on the result, which do you think is a better use of your time?
In short, reflection is your best friend in learning. It won’t let you down as long as you consistently practice it in your problem-solving.
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