Math spoken here
Leland Teschler blogged about the wide use of calculators in engineering colleges and the lack of math knowledge in engineering grads. Readers see the connection and bemoan the disconnect between solving engineering problems and what is taught in colleges. Meanwhile, one reader asks for more math, not more calculations, to explain a problem.
Math? We don’t need no stinking math
Your blog on the prevalence of calculators in math classes (“We Are Turning Out Engineers Who Don’t Understand Math,” Nov. 14) and how they were dumbing down engineering students struck a chord with me. It reminded me of a calculus professor who was asked if calculators would be permitted for exams. His response was, “Yes, but I will accept no approximations of pi or e.” This forced some people to look at the problems and discover they could frequently be reduced to a fairly simple complex number and a function — the correct answer — without using a calculator.
The other classes which forced you to really understand the principles were materials courses where atomic distances were frequently less than 10-99 m, the limit of calculators at that time. There were a number of students who wrote on the exam that problems could not be done. The professor simply replied: “Incorrect, your calculator cannot do the problem.”
Knowing math and having a “feel” for it are two different things. Rote learning can give you a knowledge of math, enough to solve most common problems using a calculator. But in engineering you need more than just rote knowledge. You need a feel for the math involved, especially in design. And the only way to get that feel is by solving problems, lots of problems, applying the math knowledge you have learned in different ways to suit the information you have to work with. Hand calculations and aids such as a slide rule, trig tables, and log tables help develop the feel for the math in engineering problems.
This has always been an issue in engineering education and always will be.
No chalkboards, protractors, compasses, or even graphite pencils, only felt pens and a few overhead projectors — that seemed to be the norm for all levels of education in the U. S. A. just a few decades ago. Now our personal computers have taken over and left us with instant gratification but little reward for thinking the problem out first. We are all guilty of using the best and greatest — MathCad, CAD/CAM, and CNC — and need to use these tools, but it sure makes a difference if we had learned what it felt like to cut a piece of metal or wood with a file or saw somewhere along the line in our education.
Anecdotally, this is nothing new. Feynman wrote about this from his time at MIT in the 1940s. Some students were astounded when he showed them that certain classes of curves always had a horizontal tangent at their lowest point. Is it getting worse? I am skeptical, but I’d like to see some data comparing students now with students of yesteryear.
My best professor in college, who I was lucky enough to have for both statics and dynamics, structured most of his exam problems in a way that didn’t require students use a calculator to solve them. Instead, the problems contained similar triangles, 3-4-5 right triangles, and simple ratios and sums. He wanted us to focus on the concepts and not get bogged down in the math.
I remember him saying, when showing us how to solve the problems after the exams, things like, “Four equations, four unknowns, you’re home free.” If you had gotten that far on the exam, he would give you almost full credit.
Still, there would be many students frantically punching away at their calculators as they scrambled for an answer. Prior to going to college, I had only used a slide rule and trig tables. Five years later it was all different, but I think that background stood me in good stead both in college and ever since. I still do estimating and calculations in my head and usually arrive at the answer before my coworkers can find their calculators.
This is worrisome. I was in college during the slide-rule age and learned to know a long list of mathematical relationships very well. Calculators are only as accurate as the information keyed in, and the answers have to be understood, not just accepted as correct. We need people who can estimate, reason, and discern, not just mindlessly enter garbage. The old saying garbage in/garbage out was never more apparent.
This is nothing new. My dad told me a story about how, back in the 1950s, his employer did not want to hire any more unqualified contract help. It was decided to make up a simple math test to screen applicants. Nobody could pass the test. Then they started giving it to the people that already worked there. The less-than-stellar results explained a lot of problems they were having getting things to go together.
Maybe we need some math
I am working on a tank problem that relates to a recent article (“How Bolt Patterns React to External Loads,” Sept. 8). In doing so I have seen the formula: F = 4M/ ND numerous times and wondered how it was derived. Your article has shown me this and I appreciate that.
However, I do have a question: how F = M/Z comes out in pounds? Typically this is stress (psi). For some reason I am not seeing this.
Thank you for your interest in the article. Engineers use the term moment of inertia loosely for convenience. Generally, it means the second moment of something. Or the momentarm- square multiplied times something.
It could be a moment of inertia of masses, areas, lines or points. In treating bending stress in a cross section, moment of inertia actually means the “moment of inertia of the cross-sectional area,” which consists of Ai elements.
I = Σri2Ai (in.4 ), A = areas (in.2 ),ri= moment arms (in.)
And the section module of the crosssectional area is:
For moment load M in (in-lbf ), the maximum reaction stress in the cross section is:
|σ b =||M||[||in.-lbf||= psi]|
In the article, we were interested in the reaction of bolts as a group of points. The points have no physical unit. Therefore, the second moment or “the moment-of-inertia of the group of points” is:
I= Σri2(1)i = Σri2 (in.2), (1)i represent points (no unit), ri = moment arms (in.)
Therefore the reaction module of the group of points, which is equivalent to the section module in the bending stress problem, is:
|Z =||I||=|| |
|max(ri )|| |
For moment load M in (in-lbf ), the maximum reaction force in the group of points is:
|F =||M||[||in.-lbf||= lbf]|
— Moo-Zung Lee