Not every strategy works for every student but I found a few tips over the years in Engineering that I feel helped me make it through and I wanted to pass these suggestions on to other students who may find them useful.
Numbers are essential in engineering, as they are in almost every field, and we work with them every day especially when we are working to understand engineering concepts. When I was in high school I was one of the students that would always ask the math teacher if I could use a calculator and then look moderately distraught when they said one wasn’t necessary (you know the type of person I mean ). I really started looking at numbers when one of my teachers brought in a slide rule and was using it to illustrate a lesson. I thought that it looked pretty interesting and so I found one and learnt how it worked. After a while I was using the slide rule to help with some of my physics labs. Then I made a strange conclusion, the slide rule sped me up…
My slide rule has a precision of 3 digits, doesn’t sound like much, my calculator that has 15 digits. However, I found that most of my answers still fell incredibly close to those of my classmates. I started to think about how much accuracy was really needed in the problems we were tackling (in high school), in my case normally no more than three, almost never more than 4. Using the slide rule improved my addition and subtraction since I started doing it in my head, for simple problems I started to realize that I could omit some digits of lower significance. Please don’t get me wrong the idea isn’t to say the world is fine without precision, it is an art (which I certainly have not mastered), what I am proposing is that it is beneficial to get to know what numbers mean. By having to process equations in parts using the slide rule versus just typing a long expression into a calculator, I became more focused on the intermediate answers as opposed to looking at only the final result. The intermediate results give you insight (particularly in physics) on the components of the problem, often if a mistake is made you will spot the error before you get to the final result, with a calculator it can be difficult to see where the mistake entered into your calculations.
After using the slide rule, I wondered how people had historically accessed trigonometric functions. I quickly tracked down a couple of good books of mathematical tables, each of which includes valuable sections on calculus as well. Using tables helps you to naturally get a “feel” for the standard functions we use every day. I believe that this “feel” for numbers is essential in engineering, it gives you a sense of when something seems odd about a result and can help you track down the cause of errors.
In University many of my classes did not allow calculators for the final exams. Since most calculators can do far more than arithmetic professors want a level field. Whenever calculators were not allowed professors would confine themselves to arithmetic problems that you could solve in your head. Now I would look on in embarrassment as a small number of colleagues lamented the lack of calculators.
I love my calculator, I don’t think there is any shame in using the tools available to us. Especially if you understand why you are doing the calculations a calculator is a great tool. After hearing a professor extol the virtues of the HP-35 and reverse polish notation (RPN), I purchased a remake of the classic calculator, the HP-35S. RPN is in my opinion a fantastic way of interacting with a calculator. Instead of typing 2+3 you type 2 3 +, the display has two lines so it looks just as if you were doing it on paper. Once your operation is complete, your answer is entered as the first operand for your next operation. In this way, you see all of the intermediary results of your expression. When I am working with an RPN calculator I often spot errors early and can correct them by reversing the last few operations. I don’t have to deal with typing in brackets and so using an RPN calculator normally has fewer key presses than an algebraic device.
Numbers are very important in engineering, I think everyone can agree on that, and I feel it is very important to develop a “feel” for the numbers you are using. I suggest that on top of a complete understanding of arithmetic (yes, it is instructive to understand long division) it is useful to have experience looking up or calculating trigonometric functions, understanding number systems and always checking your intermediate values in a calculation.
I would love to hear what strategies others have used to improve and hone their basic mathematical skills.