I like to think of Mathematics as food for the mind, precisely because it deals with abstractions. Constructing proofs and transforming one expression into another are useful building blocks for other things. I don't regret any of the Mathematics I had to learn.

I agree with your comments on math but I am left handed and right brained. I had to really struggle with the required extreme math, compared to the minimal tuition we got on electronic basics. It consisted of theory on resistors, capacitors and inductors and some very basic information on electronic emission. There was nothing about the application of electronics.
In those days, you had to be an advanced mathematician to become an engineer. At the end of the course, our math instructor proudly announced that there was no more to learn except some specialized branches of math.

 

I agree with your comments on math but I am left handed and right brained. I had to really struggle with the required extreme math, compared to the minimal tuition we got on electronic basics. It consisted of theory on resistors, capacitors and inductors and some very basic information on electronic emission. There was nothing about the application of electronics.
In those days, you had to be an advanced mathematician to become an engineer.

That is the advantage of pursuing a degree in Engineering Physics. You focus on the mathematics and then apply it to engineering problems. I once told our Dean of Engineering that a graduate in Engineering Physics is someone with the solution looking for a problem. He loved it!

 

I like to think of Mathematics as food for the mind, precisely because it deals with abstractions. Constructing proofs and transforming one expression into another are useful building blocks for other things. I don't regret any of the Mathematics I had to learn.

I tend to think that excessive abstraction is one of the root causes of math learning difficulties for those with "hands on engineering" brains. The need to link physical electrical/electronic properties/circuits and actions to the mathematical is glossed over in many required higher engineering math courses. Math is not engineering, math is a general method (with many ways to get the same solution) for the expression and communication of engineering problems and solutions in the abstract so we can mentally build things using physical rules before they are physically constructed. Math is absolutely needed to say when your physical intuitive is wrong but it also allows you to wallow in a sea of non-practical, beautiful fantasies much too easily when actual work needs to be done.

 

Sadly, not so much any more. Many geometry curriculums have gone out of their way to eliminate as much of the "proofs" content as possible. Often the proofs are simply presented, often in a rather handwavy form, to the students who are then asked to simply use the resulting theorems to solve problems. I've encountered only a few students over the past couple of decades that have had any exposure to formal proofs in which each line of the proof must follow from the preceding line using a single axiom or previously-proved theorem. The exceptions tend to be home-schooled kids.

Certainly, as a post-secondary instructor your experience is more recent than mine. My experience was from the 1962-63 academic year, and I would be the first to admit that it just might be out of date. OTOH, we were the beneficiaries of the post-Sputnik surge in emphasis on mathematics education which was a radical turn from tradition for the purpose of beating the Russians.

 

I tend to think that excessive abstraction is one of the root causes of math learning difficulties for those with "hands on engineering" brains. The need to link physical electrical/electronic properties/circuits and actions to the mathematical is glossed over in many required higher engineering math courses. Math is not engineering, math is a general method (with many ways to get the same solution) for the expression and communication of engineering problems and solutions in the abstract so we can mentally build things using physical rules before they are physically constructed. Math is absolutely needed to say when your physical intuitive is wrong but it also allows you to wallow in a sea of non-practical, beautiful fantasies much too easily when actual work needs to be done.

Despite my facility for Mathematics, I ended up practicing various kinds of Engineering for over half a century. I'm retired now since I decided to let my modest savings work harder than I ever did.

 

I agree with your comments on math but I am left handed and right brained. I had to really struggle with the required extreme math, compared to the minimal tuition we got on electronic basics. It consisted of theory on resistors, capacitors and inductors and some very basic information on electronic emission. There was nothing about the application of electronics.
In those days, you had to be an advanced mathematician to become an engineer. At the end of the course, our math instructor proudly announced that there was no more to learn except some specialized branches of math.

I'm not sure what you mean by "required extreme math", so I'm not in a position to comment on that. I'm also not sure which days are "those days", but I don't think there has ever been a requirement to be an "advanced mathematician" to become an engineer. As for your math instructor's pronouncement, I wish you had been fortunate enough to have a different instructor.

The trend I have seen -- both directly and indirectly talking to many who came before me -- and which somewhat supports what you are saying, is that engineering education moved from being focused on the practice of engineering to being focused on the science of engineering. What is needed, of course, is a good blending of the two. There are a lot of factors that drove this shift, some technical, some political, and some cultural (both the broader national culture and the cultures within the engineering disciplines). Practical engineering can only take technology so far; beyond that, it becomes increasingly critical to have a science-based foundation. Those limits became pretty evident with the technological explosion in and around the second world war. Unfortunately, it is too easy for the proponents of a science-based engineering education to completely forget that real-world engineering is about practical implementations of solutions -- as a result, that is something that is learned by the folks that get an engineering degree largely only after they leave school and enter the real world. In and of itself, that's probably neither good nor bad (though I certainly consider it unfortunate). But since so many of the people that drive engineering education are people that are not only the product of a practice-deficient curriculum, but are also dominated by people that have never left the halls of academia and have never worked, in any substantial way, in the real world, it becomes an echo-chamber resulting in engineering education being pushed progressively further away from any practice-based content to the degree that attempts to incorporate such practical content is frowned up and discouraged on the basis that, "we are educating engineers, not training technicians." A phrase that I have heard multiple times over the years at multiple universities.

 

Certainly, as a post-secondary instructor your experience is more recent than mine. My experience was from the 1962-63 academic year, and I would be the first to admit that it just might be out of date. OTOH, we were the beneficiaries of the post-Sputnik surge in emphasis on mathematics education which was a radical turn from tradition for the purpose of beating the Russians.

My high school geometry was in the 1980-81 academic year and it probably didn't differ too much from yours. It was all about formal geometric proofs. We had been exposed to formal proofs previously in algebra, but geometry took it to a whole new level. It was sufficiently in-depth and, more importantly, of sufficient duration (pretty much the entire year) that it affected the way that you think about solving problems in general. This was reinforced by the occasional proofs required in trigonometry, and calculus and especially in the analytical geometry that our school required as the first semester of the AP Calculus course.

I witnessed a huge drop-off in mathematical literacy and problem solving skills starting with folks that graduated high school in the 1985 to 1987 time frame (and I was far from the only one to notice the steep decline that occurred in engineering classrooms in the 1990ish time frame) and long after I noticed it, was able to correlated it pretty well with the wholesale national shift toward introducing calculators in first grade. The trend has continued pretty much non-stop since then, but at a slower overall pace and with a lot more variability.

Since then, I have not only spoken to many students, but have run across the actual textbooks, that don't explain what things are at all, but rather only show pictures of which buttons to press on a certain model of calculator to evaluate expressions.

 

Despite my facility for Mathematics, I ended up practicing various kinds of Engineering for over half a century. I'm retired now since I decided to let my modest savings work harder than I ever did.

I've tried to retire (I'm also on full SSA and medicare) once because of a medical issue that I'm still working through. It's not the money that got me back as I've invested well since the 80's, own my current house (a boomer that paid a bag of turnips for a house per GEN-X) and have properties overseas and in the USA. It was the offer to setup my own EE shop, with my own budget, be my own boss and to help train the next generation of working manufacturing practical engineers and technicians that talked me back until the medical issue (knee replacements run in the family) is done.

 

I'm not sure what you mean by "required extreme math", so I'm not in a position to comment on that. I'm also not sure which days are "those days", but I don't think there has ever been a requirement to be an "advanced mathematician" to become an engineer. As for your math instructor's pronouncement, I wish you had been fortunate enough to have a different instructor.

Those days were the early 1960s in England.
My adverse comments on "advanced mathematics" refer to such required branches as hyperbolic functions. This consumed a full semester of math, deriving 13 different basic formulas, each one taking up several lines of text. They were so time consuming to derive that we were required to memorize all thirteen resultant formulas for our final exams.
Needless to say that I have never had an occasion to use them in my whole electronics career.

 

Those days were the early 1960s in England.
My adverse comments on "advanced mathematics" refer to such required branches as hyperbolic functions. This consumed a full semester of math, deriving 13 different basic formulas, each one taking up several lines of text. They were so time consuming to derive that we were required to memorize all thirteen resultant formulas for our final exams.
Needless to say that I have never had an occasion to use them in my whole electronics career.

There is a slightly amazing connection between hyperbolic functions and Chebyshev Filters. It sure surprised me when I first saw it.

 

Those days were the early 1960s in England.
My adverse comments on "advanced mathematics" refer to such required branches as hyperbolic functions. This consumed a full semester of math, deriving 13 different basic formulas, each one taking up several lines of text. They were so time consuming to derive that we were required to memorize all thirteen resultant formulas for our final exams.
Needless to say that I have never had an occasion to use them in my whole electronics career.

Yeah, have no idea what the situation was (or is) in England.

I've encountered hyperbolic functions on a few occasions and found myself wishing that I had had a stronger foundation in them instead of having to get by with a very superficial introduction just before getting to material that needed them. The first time that I encountered them (I think) was in statics when we were studying forces in suspended cables. The next time was in thermodynamics, where they tend to come up in solutions to heat conduction problems. But where I really encountered them was in the Advanced Electromagnetic Fields course, which is all about vector calculus and everything that goes with it. The statics course was required for all undergrads at my school, but the other two were required as part of the Engineering Physics program.

 

Yeah, have no idea what the situation was (or is) in England.

I've encountered hyperbolic functions on a few occasions and found myself wishing that I had had a stronger foundation in them instead of having to get by with a very superficial introduction just before getting to material that needed them. The first time that I encountered them (I think) was in statics when we were studying forces in suspended cables. The next time was in thermodynamics, where they tend to come up in solutions to heat conduction problems. But where I really encountered them was in the Advanced Electromagnetic Fields course, which is all about vector calculus and everything that goes with it. The statics course was required for all undergrads at my school, but the other two were required as part of the Engineering Physics program.

Did you ever need to use hyperbolic functions in real live engineering, or was it just in academic studies? Catenaries are cosine hyperbolic function which are symmetrical, and are the least difficult type to derive.

 

Did you ever need to use hyperbolic functions in real live engineering, or was it just in academic studies? Catenaries are cosine hyperbolic function which are symmetrical, and are the least difficult type to derive.

Not often, but a few times. Most of them have been when working with customers that wanted an ASIC designed to do some specific task and their end of things involved stuff that used hyperbolic functions. These usually involved various electromagnetic detection arrays of various kinds. When working with one of those customers, I was reviewing my part of the design with them and they had a hard time following it because I hadn't worked with hyperbolic functions, and so had a lot of fairly complex expressions involving complex exponentials. One of their folks spent about an hour recrafting my results in terms of hyperbolic functions and, after they collapsed into amazingly simple expressions, was able to show them to his other colleagues who spent less than five minutes reviewing them and said that they looks good, passing all of their sanity tests. At that point, it was really apparent to me how powerful and expressive hyperbolic functions were, but I was so unfamiliar and uncomfortable with them that I couldn't benefit from it. I still can't, because I still lack that familiarity and comfort.

Another area that I've read where hyperbolic functions are gaining ground is in post-quantum cryptography.

 

To begin, here is your basic course of Mathematics for Electronics Engineers.

Part 1
You need to become familiar and conversant with these basic math symbols, terms and expressions, and trigonometry functions.

<Snip>

End of Part 1

I was already familiar and comfortable with everything above except decibels which, though simple, I reasoned I'd get to in time with my electronics books. No need to get ahead of the curriculum.

Thanks for the sinewave-circle animated image. It's really a nice visual representation of the underlying concept.

Feel free to post part 2 at your leisure.