How much math(s) do we really /need/ to learn? 23

[GregLocock]

This is a continuation of a previous thread that was going OT.

At university something like 15% of my course was maths. It was the hardest part of the degree for me (and I was in the top 2-5% in maths at high school).

Since this is an international forum I guess it complicates things, but I'd like to get an idea of how much maths people (a) think should be taught in, say, a decent mechanical engineering degree, and (b) of that what they actually use, and (c) what they wish they had learned but didn't (d) what they had to learn but wish they hadn't.

Here's my (non complete) guess:

(a)

Calculus to say the level of double integrals and surface integrals, Taylor series and so on.

Fourier Analysis

ODEs, preferably a bit more than I did

PDEs, to a very simple level

Stats, sufficient to design experiments and test hypothesese

Complex numbers of course

Trig - I wouldn't get too hung up on trig, just the basics seem enough to me

Matrices with hand worked examples up to say 3x3, or 4x4, inverting, transposing, adding and so on, but not Gaussian elimination or any of the other tricks we needed before PCs

Vector maths (I didn't do enough of this)

Laplace, to a very basic level.


(b)I've used all the above, to some extent, since leaving uni.

(c) Green's function, Bessel functions, more Laplace, more statistics

(d) Lots of matrix stuff and numerical methods. Some of the calculus.

Cheers

Greg Locock

 

[chris9]

I think Gauss2k has hit the nail on the head here. It's about your approach to learning and solving problems. As an engineer if you don't know something you can teach yourself.

Engineers need to appreciate all areas of maths in the beginning and then apply the bits that are relevant to solving real world problems. Very few people can remember much of what they were taught at school a decade ago.

 

[moltenmetal]

Pure math is a poor tool to teach people how to solve engineering problems- there are far better ways, and more relevant problems to set before engineering students to teach them problem-solving skills than teaching them analytical integration.

Yes, considerable knowledge of mathematics is essential to an engineer's education- nobody's debating that! But the sheer amount and complexity of the math in our courses of study also seems to be some kind of macho rite of passage for graduates of engineering programs, and even more so for professors of engineering. We keep hearing the same tired argument, "It was basically useless, but it taught me how to think." Well, bully for you!- but there are far better ways to teach engineering students to reason logically and apply their rational analytical skills than by having them perform the mathematical equivalent of running around inside a squirrel cage!

As to the argument that a Bachelor's degree in engineering is just pre-training for an academic Master's or PhD, let's get real- most graduates of engineering do not immediately go on to grad school (thank God!). They go out into industry to serve society- and we have a responsibility to ensure that they know enough about what they'll actually be doing for a living so that they provide a service rather than disservice. That's every bit as important as preparing them for grad-level classes.

Let's be reasonable here: why do we teach analytical integration with the rigour that we do? Much of it has been rendered essentially obsolete by the advent of modern computing technology. It's as if we were still teaching students how to use slide rules- a great way to get them to learn logarithms, but hardly necessary any more despite being an essential tool forty years ago. And yes, I know that analytical solutions are more "elegant" than bashing away at something with Euler's method, and that there are certain types of problems that frustrate the numerical methods- but they're relatively few and far between. It hardly seems logical to continue to beat students' collective heads against that particular wall while leaving important, essential curriculum untaught because there's so little time!

IMHO there needs to be more focus on model-building and the corect use of modern computational methods and tools, and far less on analytical solution of the resulting differential equations. The vacuum created by dropping out one or two courses worth of useless analytical solution-oriented mathematics could easily be filled by other essential course material which the students will actually find useful in their careers. This approach will be far more effective in the quest to build a better engineer.

 

[Lorentz]

Hi moltentmetal!

I couldn't agree with you more. In an ideal world I would have people like you and Greg Locock redesign engineering programs to create a better balance between math and other essential subjects.

In the famous Feynman lectures on physics, the Nobel prize winner cautions students that while math is essential to physics, math IS NOT physics. He does this in the volume introducing electromagnetics. Feynman noted that many mathematically adept people took the approach that since electromagnetics is summed up by Maxwell's equations, they would simply analyze the equations from every angle to learn EM theory. In his observations, people who took this approach rarely contributed much to either the field of physics, or the field of mathematics for that matter.

What Feynman observed in physics students seems to me to be even more pronounced among engineering professors. And IMHO, the first responsibility of engineering professors is to produce competent engineers to serve the needs of society - not to prepare the next generation of graduate students!

Having said all this, I will state that there are times when analytic solutions are valuable, as in confirming or benchmarking numerical codes. No one is advocating removing ALL math from engineering programs, but many have observed (in this thread and others) that the current balance is so math heavy that other essential topics are being neglected.

 

[2dye4]

How about turning this around a little bit.
What math courses have you studied that are not providing
any real world benefit and could have safely been left out
of your degree.
Or what math taught in Eng is of little practical use ??

 

[davefitz]

2dye4:
Some methods are being supplanted by numerical methods. For example, Perturbation theory was at one time required of grad students, but is generally supplanted by numerical methods. Likewise the manual mathematical techniques used by aerodynamicists is supplanted by CFD, and of course the use of finite elements for conductive heat transfer and lineat stress analysis has supplanted manual methods ( infinite series). Galerkin methods are used to solve nonlinear ( or non- self -adjoint) problems in heat transfer and stress analysis. Even the old Codes ( ASME, etc) are beginning to recognize the shift.

 

[BML]

It seems to me that a lot of the discussion seems to revolve around the math that doesn't work for X discipline. Would it be fair to say that there is some amount and type of mathematics that is common to all engineering disciplines? (Maybe calculus through multiple/surface integrals and linear algebra for example?) If so, then that should be the standard minimum courseload for college students. Beyond that, the types of problems encountered in the real world vary so differently that it may make sense to have additional math classes be discipline specific. For example, even within mechanical engineering there is a difference between the math needed for HVAC (thermo, heat/mass) and structural (advanced statics and dynamics). This way if you learn too much math, it is at least too much math that relates to your future job.

 

[QCE]

The math is more to show you can use your brain than for useful work. It is hoop to jump through to show you are smart enough to solve some advanced problems that are required in engineering. So deal with it don't cut it back. If it was easy to get an engineering degree everyone would have one.

 

[2dye4]

Is it possible that many of us work in industrial
enviroments in support positions and do not do much
pure design of new products?

Davefitz:
Those items sound a little far from what is commonly
taught in engineering undergraduate programs.
If so my question is what math that appears regularly in
undergrad programs can be done without.

Many engineering programs try to expose you to the things
that need the most explanation. In other words difficult.

They assume the student will pick up a great deal more
knowledge from self study and work experience.

 

[Lorentz]

As moltenmetal observed:

We keep hearing the same tired argument, "It was basically useless, but it taught me how to think."

A generation ago, when people with BA degrees began having trouble finding work, they voiced the same rationalization for the courses they studied.

It seems that the math is not meant to be a working tool, merely a hoop engineering students jump through to prove their intelligence.

It is interesting to note that doctors, dentists and lawyers study little math, yet they are generally considered to be smart, and they are usually employed and well paid (unlike too many engineering graduates of today). I think that one of their key advantages is that their training is highly focused and designed to produce graduates who are useful to society (well, at least doctors and dentists).

I do not advocate simply cutting math courses, rather I would like to see the mathematical examples taught in engineering courses to be based on actual real-world examples of problems working engineers face on a daily basis. Unfortunately, few engineering professors have enough experience to produce such examples, so hypothetical problems are used instead. I believe this is the real reason why their is so much math in most engineering programs.

Next week I'm off to the dentist for my semiannual cleaning and checkup. She is a very good dentist, even though I doubt she still remembers how to solve a quadratic equation.

 

[2dye4]

lorentz:
It is entirely up to you to see the applications of
the math that is taught in engineering classes.
If you doubt that applications exist I suggest visiting
your engineering school library and reviewing some of the
periodicals the ASME or IEEE or many other professional
publications to see that math is used at least to
publish articles.
Engineering professors do not have the time to carry you
through the application of the science

 

[Lorentz]

Engineering professors do not have the time to carry you
through the application of the science

I am so cynical that I do not believe most of them even know the applications. If you were to give a pop quiz on standard electrical code practices to a group of tenured Ph.D. electrical engineering professors and to a group of apprentice electricians, I am sure the professors would fare quite badly in comparison.

Sorry, but I know too much about college professors to revere them. I would suggest that the readers of this forum would look at their professors in a new light if they would:

1. Read "I. Asimov" the autobiography of the science fiction writer who was also a professor at the Boston University School of Medicine.

2. Rent the DVD of "The Wizard of Oz" and watch closely when Toto pulls aside the curtain.

 

[davefitz]

Of course Lorentz is right in most instances- the process of earning a PhD does not in any way reflect the practice of engineering. But then again, a PhD is not really an engineer- they are teachers and researchers.

By the way, it is my understandign that the Wizard of Oz was originanly written as a farce to describe the early 20th centtur debate about the gold standard or currency- and OZ was an ounce of gold, and the yellow brick road was of course the pathway to heaven if we stayed on the gold standard.

 

[moltenmetal]

Take it from someone who is DOING something about it: too many profs have little or no industrial experience. The truly good ones at least seek out practicing engineers like myself to provide some real-world grounding and insight to their students to supplement the NECESSARY and GOOD theoretical grounding they give the students. If I wasn't so busy working for a living as a professional engineer, I'd volunteer yet more of my time in this effort, because clearly the process of selecting and promoting professors doesn't place enough value on practical hands-on engineering experience as part of a professor's training.

I've worked with lots of PhDs in industry who DO have that grounding in reality, and they agree with me about the absurd focus on mathematics. And yes, I've met more than a few industrial PhDs who I WISH were in an academic ivory tower somewhere instead, but that's another story. There's even a joke on the subject:

How many profs does it take to screw in a lightbulb?

NONE- they get their grad students to do it for them, and then write a paper about it listing the prof as co-author!

Although I'm only a chemical engineer, I believe I know enough about "hoop stress" to design adequate hoops for students to jump through in replacement of this ridiculous mathematical "rite of passage". The only difference is that my hoops would teach USEFUL skills on the way through.

 

[TheTick]

I don't think it's a question of how much math. It's a matter of schools doing a better job of tying math into "reality". Then again, most students do not care to connect the two on their ownsome.

Case in point: It happens every few months, where I have to explain "y= mx + b" to a fellow degreed engineer. This is usually in explaining that the spring behavior they are witnessing is exactly as predicted. I try to be gentle, as these people are our customers, but it is difficult to keep a straight face while explaining.

Due to illness, the part of The Tick will be played by... The Tick.

http://www.EsoxRepublic.com

 

[zdas04]

Where do you stop? Where do you start?

If an instructor of undergraduate fluid mechanics (for example) can't assume a certain level of competence in math, then how can he explain the stress-strain tensor (or a hundred other concepts)? If you can't rely on the math to describe the concept, then the whole exercise becomes arm waving and requiring that students take your assertions on faith--now there are a couple of stinko options.

Math ain't about hoops to jump through, and it isn't simply a tool to teach people to learn concepts (or time management). It is the language we use to describe the world engineers are trying to modify.

That being said, a huge percentage of engineers never use any math beyond algebra after graduation. In Oil & Gas Production (not necessarily refineries), engineering is all about economics (run inside a packaged program) and project management. I was once in a discussion that required an analysis of the assumptions behind Bernoulli's equation, so I derived it on my board and we analyzed each of the assumptions that went from one step to the next. At the end I didn't erase my white board. I had so many other engineers ask me "what the heck is that?" that I put a sign on it "can you name these equations?"--none could although each one of them used programs based on that equation every day.

David Simpson, PE
MuleShoe Engineering

http://www.muleshoe-eng.com

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.

 

[CanEngJohn]

Moltenmetal wrote -
Pure math is a poor tool to teach people how to solve engineering problems- there are far better ways, and more relevant problems...

It is only a poor tool when we are not taught how to bridge the pure math to the practical application.

I develop and use mathematical models about 1 to 2 times a month for a variety of problems I have on the manufacturing floor. Some involve predictive maintenance, some involve dynamic force loading on wear components that are wearing too fast (and no I do not have FEA at my disposal to help me with those). While I don't differentiate and integrate too often (second moment of area being the odd exception) I still use a lot of the anylytical tools and basic linear and nonlinear equations which I pulled from my education.

Also I had training on how to move from theory to practical. We called it modelling. It started with third year Heat Transfer which presented this in smashing fashion. Every example in class and every homework assignment was a perfect cylinder or a perfectly insulated something with a theoretical heat source.

Then the tests and exam were written .. A neighbor's well freezes up in January. To supply him with water you hook up a 50' hose between the two houses ...(data on temp and hose specs etc ..)... calculate the minimum flow rate required to ensure the water does not freeze while travelling between the two houses. Or Calculate the temperature setting for a thermostat in a toaster given that the toast is considered perfect when the surface temerature of the toast has reached ...

We very quickly learned how to model. I came to appreciate this when I got to industry and wished it was done for more subjects and courses in my degree than the three which did focus on models. No one can get enough practise or be too good at modelling.

 

[chris9]

When I first started FEA I always did hand calculations for a reality check and to give me more confidence in the results. I still do this when doing something new. Engineering is about choosing the right tool for the job and it really doesn’t matter how you get to where you want to be so long as meet your objectives. So if calculus is the best tool for the job then use that. In my experience using software is the quickest and most economical route. I may not know how to write the software that I use but I know how to use it to solve real problems.

 

[moltenmetal]

Again, nobody is saying that you can teach engineering without teaching adequate mathematics. Math is a key language of scientific understanding. And yes, I was a good math student, and I do use mathematics beyond arithmetic in the course of my job. But if I EVER have to use the method of Frobenius or to find the inverse transform of a convolution of hyperbolic functions again, I'll be happy to seek (and pay for) the skills of a mathematician.

Some math skills are like those of the slide-rule users or buggy-whip makers of days gone by- essential at the time, but rendered pointless by modern technology- and in my engineering education, those made up AT LEAST a full course, probably two, of the mathematics I was taught. Let's stop teaching these less-than-useful skills when they're, say, 20 years obsolete instead of waiting 50+ years.

Instead, let's teach students how to build mathematical models from the differentials up, then solve them using the best available tool- whether that be the analytical approach for the simplest problems, or a spreadsheet and Euler's method for slightly more complex ones, or a more complex mathematical software package for the tougher ones. Then let's teach kids how to parameterize and error-check their models, so they have some sense whether or not the computer is spewing bovine solid waste instead of useful information.

 

[davefitz]

I guess there is a lot of hysteresis in the educational system- the folks teaching engineering had been educated 20-30 yrs ago, and just can't let go of the times when they were spritely.

 

[Lorentz]

It seems to me that so far this thread has been dominated by people who work for OEMs. It would be interesting to have some input from engineers in the construction industry as well.