As an undergraduate, I took five semesters of calculus. The last semester on partial differential equations supposedly gave me a head start for grad school in my chosen field of chemistry.
But, unlike my friends in engineering programs, I had little need for calculus except in a few physical-chemistry labs and classes. I can't remember using calculus in grad school, and since then I can't remember using it more than a few times to solve electronics problems. Instead of having to take all those calculus classes, two semesters would have provided sufficient knowledge. Classes in scientific data analysis, instrumentation techniques, and statistics would have made more sense, but at the time either the courses didn't exist, or no one suggested them.
I looked at the math requirements for an engineering school, and found the following requirements for a BS degree:
Calculus I and II
Ordinary Differential Equations
Discrete Mathematics
Statistics
That seems like a better series of course requirements (one semester each) for engineers than the "exposure" I got in college. My biggest impediment to learning calculus centered on the lack of a relation between it and practical problems. After the math professors got past a few simplistic examples, it was all x-this, y-that, and everything between plus and minus infinity. Practical examples of why nascent scientists and engineers might need a triple integral might have helped us understand calculus better. Or perhaps I'm just more of a practical than theoretical mind.
We now have many computer programs that can help analyze data, and that provide sophisticated mathematical tools. And using such tools takes the drudgery out of tedious mathematical analyses and equation evaluations. So, does it still make sense to load engineering and science students with more than an introduction to calculus? If so, what level of calculus should they reach? (Of course, math majors and students interested in calculus could still take more courses.)
During college, I had a dull-as-dirt calculus text I was glad to throw out. Since then, I discovered The Calculus Tutoring Book by Carol and Robert Ash. The IEEE Press published this book in the mid 1980s, and although now out of print, AbeBooks.com lists many copies at reasonable prices. I recommend this book highly to students who struggle with calculus. One summer I went through about half the book on my own, and worked many of the problems. Finally, I found calculus interesting and clear. It only took an extra 20 years to "get it."
What level of calculus should engineering students reach? Discuss in the comments section below.
My first ever lab session in University was about fourier transform on image smoothing duirng 2nd year first semester. But I was supposed to take signal processing course on next semester. It was the same for my classmate and our instructor had one of the busiest days. I had no idea what is going on with this lab. But when I reach final year, doing some mini project about computer vision applications, then only my first lab session is crystal clear.
I like the idea of students taking their major first and then learn backward for necessary subjects. It is more precise and efficient. I think it is impractical in this day due to insufficient data or information about current technology, job perspective and economic.
You certainly convinced me – I just ordered Elements of the Differentialand Integral Calculus, by W. A. Granville on amazon.com. My teenage son is entering the math, engineering, technology science academy for high school next year so hopefully this will help him out...Thanks for the tip!
I once asked an engineering professor if any school had ever tried teaching on an "as-needed" basis, where you'd start with the things you wanted to learn and then move backwards into pure science and calculus classes as needed. For example, if I want to learn how to design a bridge, I would start with basic bridge design, then learn calculus and structural mechanics when the complex analysis finally called for it. To my surprise, the answer was , "Yes, we tried it." The school was Illinois Institute of Technology. They reportedly had several students go on for Ph.D.s after their undergrad coursework. But the program didn't work for many reasons, mainly because it was too hard to manage and too many students didn't really know what they wanted to major in, so they didn't know where to begin. My point is, that's the way we all learn in life. I think if I could have learned that way, the application of calculus would have made more sense to me.
We had the same professor for the entire sequence of P&S. He seemed to rush through the text & core of the class, assigning a lot of "self-study-solve" problems. In the 2nd half of the course, at one class meeting he mentioned a book titles, INFERENCE & DISPUTED AUTHORSHIP, THE FEDERALIST PAPERS. From that moment on, we knew we were doomed!!!! The rest of the semester was devoted to the mathematical analysis of the text of Alexander Hamilton's treatise, THE FEDERALIST PAPERS. Some of us in the class pondered whether the professor was somehow misplaced..... that he should have been assigned to the HISTORY DEPT. instead of the Mathematics Dept. Although I did well in the class (as a grade point statistic), I learned little of practical value. The only saving grace is that in my career, P&S has never been an issue of concern. And, unfortunately for me, I was to meet up with him again in senior year in a course titled, MODERN ANALYSIS, with an accompanying text about 600 pages, which seemed to be a compendium of mathematical oddities & questions that no one really wanted to investigate. In fact the ramblings in class were so far off topic that most of the 20 students in the class didn't attend on a regular basis, a heretical concept in the early 1960s, when attendance in class was part of the "college experience." The final exam was 10 questions taken from the end of various chapters from that book. We were given the question numbers about 2 weeks before the official end of the semester.
However, except for these examples & one or two others, I'd go back to the same colleges, and the same curriculum again, given that I had a "time machine" to do so. I thoroughly enjoyed the learning experience from the first day I bunked in the dorm until the graduation ceremonies.
@OLD_CURMUDGEON: My probability and statistics course had the same issue, forty years later. The course I took was called "probability and statistics for scientists and engineers," so it was ostensibly focused on practical applications. In reality, it was just a condensed version of a two-semester sequence in the math department. It was taught by a math professor who was probably an excellent statistician, but who resented having to teach what he regarded as a dumbed-down course for non-math majors. We talked a lot about the central limit theorem (which admittedly is important), but very little about how to actually apply statistics to solve real-world problems. I think an applied statistics course taught by someone from the industrial engineering department would have been much more valuable.
Because the program that I was enrolled in at the time was a cooperative program (early 1960) w/ the University of Dearborn (a Jesuit school), my formal major was listed as Mathematics w/ a Physics concentration. The parenthetical major was "Pre-Engineering". So, due to the Mathematics concentration, the curriculum included a sequence of PROBABILITY & STATISTICS. Two courses of each, but they were presented from a pure Mathematics point of view, so they left little desire in most to pursue a more practical understanding in daily design & analysis in the "real world" atmosphere.
I had 3 semesters of calc and one of differential equations. The biggest hole was the fact that I didn't have any statistics (at least until I went for my MBA). I think the class list in this article does seem to be more practical.
I only have two..... my trusty K&E full-featured rule, and a no-name cheap one that only has the A,B,C,D scales printed on it. It was accurate enough for tests, and since it was so inexpensive, I didn't worry if it was stolen at school. I haven't used it for years, but I don't think I can get batteries for it any more..... used two PX-625 MERCURY batteries. (Ha! Ha!)
My first "electronic" slide rule was the TI SR-50 calculator. Was betwixt & between whether to buy brand new H-P 35 "scientific" calculator w/ its RPN entry method, OR stick w/ the more common TI model. Still have the TI, but ir doesn't work because the non-replaceable battery pack is kaput.
When I die, the world is gonna have tons of fun transporting all these relics into the Museum of Ancient History.......
Ah, the sliderule, my constant companion until about 1973 when I bought a Commodore scientific calculator. I love sliderules and have several boxes of them awaiting sorting, as well as three in my desk. My senior year in high school, Mom and Dad gave me a Post chemical-engineering slide rule (Model 1491) with temperature-conversion scales, atomic weights of elements, gas-pressure scales, and other scales I never learned to use. What a wonderful tool.
1) Jon- I, too, took two semesters of SCIENTIFIC GERMAN in undergraduate school. It was a course designed to aid students in reading technical literature written in German. Since I was the product of German immigrant parents, those courses were an easy A+. My first vehicle was a 1960 (GERMAN) FORD (TAUNUS) which my aunt & uncle brought back from an extensive trip to Europe in 1959. They unknowingly left the new car brochure under the driver's seat. When I got the vehicle and was cleaning it one day, I saw this pamphlet, and kept it for nostalgia purposes. When my German class professor gave us a course project, I showed him the brochure, and told him I would translate it into English. On the day the projets were due, I stood in the class, and read the entire brochure without missing a beat. After class, he said to me, "don't bother coming back here. You don't need my help." Not only did I get an A for that semester, but for the sequel course as well. Too bad that didn't also happen for E&M & a few other "toughies"!!!!
2) In the 1960s and beyond, just about every desk in an I-B-M office wherever you went there was a simple fixture on it. It was a shaped like a name plate, but all it said was "THINK". Surely a timeless admonition!!
3) Dave Palmer - I had the SAME problem in many of my Mathematics courses. It never seemed to fail, no matter how cautious I was that the batteries in my K&E slide rule ALWAYS failed the night before a final exam, and there were no stores open @ 2 in the morning to get replacement batteries. What a drag!!!!!
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