When I was in middle school, China was still relatively undeveloped with very limited investment in education. Most of us didn’t have access to a calculator, and I never felt the need for one: many calculations were symbolic, and when numerical calculations were involved, they were done by hand. I finally could afford a calculator in college, but rarely did I use it because it was often not needed.
This is why I was surprised when I first came to the United States. Calculators had become a necessity. Every time I assigned a problem in class, the first response from students was to bring out their calculators. It was like a reflex, even though many times a calculator was not needed at all. Trust me, I literally saw people using a calculator to find 2×5. In another instance, a student was supposed to find the average of 6.4 and 6.6. The result reported was 13. So, I told him that 13 could not be the average of 6.4 and 6.6, and he was like: why? I used my calculator. Honestly, I was a little speechless at that point.
These days, calculators are also becoming a lot fancier. I was surprised to see many students have calculators that can generate graphs of various functions. Surprised again when I realized that most of these calculators can also perform numerical integrals. It is nice that people have access to educational resources such as calculators. Sometimes they are very helpful, and I get that. However, do we really understand the subject if we rely on the calculator to do every single calculation? It troubles me that one cannot tell that the average of 6.4 and 6.6 must lie between these numbers. It troubles me that students treat derivatives as if they are divisions. It troubles me that many students can evaluate a definite integral using a calculator but are totally lost if the same integral is given in symbolic form.
From what I see in my teaching, unfortunately, my observation is that the use of calculators has impeded, not facilitated, learning. It seems that many students do not even bother to try to understand a mathematical concept because they believe they can simply resort to their calculators. Why bother with functions and graphs if I can plot all functions using my calculator? Why do I need to understand integrals if I can get the result by pressing a button? Yes, your calculators can do all those calculations, but where is your understanding? What about operations that your calculators cannot handle? (For the same reason, I’m extremely concerned about the use of AI and its impact on learning, i.e., using AI to solve a problem without any understanding, but that’s a different topic.)
Mathematics is the most important tool in solving science and engineering problems. Before we can apply mathematical techniques to address real-world problems, we must understand the tool that we are using. You don’t need a calculator to develop such understanding. In fact, resist the temptation of relying on a calculator. Below are my recommendations:
- When solving a problem, try to start with symbolic calculations. Do not plug in numbers unless a numerical value is needed. Do not use a calculator unless you need to plug in numbers.
- Try to understand the variables, expressions, and equations that you are working with.
- Simplify your expressions and/or equations as much as you can using mathematical and/or physical reasoning. Make sure to understand the mathematical techniques you use for your simplification.
My book contains plenty of examples and exercise problems. Try to follow a similar solution procedure as described in my book and keep practicing. After all, you will be surprised to see that you can survive without a calculator in many, if not all, cases.
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