My two cents on this... A lot of STEM education is very formalized and test-oriented, and while this is good for evaluating teaching performance, it may not be the best approach to train mathematics for practical use. Here's a number of basic skills I think are revealing about one's "numeracy": 1. Understanding units, both in terms of orders of magnitudes, and the quantities they represent. For example, if I say a device consumed 2000 W for 10000 s, that is 2000 X 10000 Watt-seconds, or 20000000 Joules. So first, you must understand that a watt is a Joule/second. Secondly, for example, if someone asks you how many kilowatt-hours it is, a kilowatt-hour is 1000 W X 3600 s = 3600000 Joules, so that 20000000 Joules X (1/3600000 kWh/Joule) = 5.555556 kWh Notice that a number of skills were required here, for example, understanding units, what they mean, their conversions, and what prefixes such as kilo- mean (as well as others). You would be surprised the number of people who are unable to do this kind of math. Secondly, more basically, to understand that a Joule is a unit of energy. For example, a BTU (or British Thermal Unit) is also a unit of energy, and so that represent the same quantity (1 BTU = 1055.06 J). I could go on with more examples, like torque being the same "units" as energy, for example, Joules = Newtons X Meters an applied moment also being represented in Newton X Meters, but that one represents energy and the other an applied moment. 2. Understanding the rough order of magnitude of common quantities. For example, how big is a micrometer? If I am microfabricating a feature that is 0.25 micrometers, how big is that compared to a wavelength of visible light (roughly half, because green is about 0.5 micrometers wavelength)? How big is that compared to an atom? (For example, the lattice spacing in silicon is 0.000543 micrometers). Similarly, lets say one is surveying. One measures the parallax of a target from two vantage points as being offset by 0.1 degrees, with the vantage points being 10 meters apart. How far away is the object? 0.1 degrees = 0.1 degrees * Pi radials/180 degrees = 0.00174 radians. The object is 10 meters / 0.00174 radians distant = 5.73 km. See how the two magnitudes (offset and distance) are related by the small angle. This type of problem is exemplified by "Fermi problems" ( https://en.wikipedia.org/wiki/Fermi_problem ) where the goal is not necessarily to get a very accurate answer to the problem, but only to get an idea of what the correct order of magnitude of the answer is. Understanding the correct order of magnitude is important to knowing whether to reject an answer outright for being incorrectly scaled or in error. 3. Understanding the amount of error in a measurement or calculation. For example, carrying through the appropriate number of decimal places to ensure that a numerical answer has at least as much precision and accuracy as the original measured quantities it is based on. For example, if I want to calculate a circumference of a wheel to 3 decimal places, and I use 3.14 as my value of pi, the approximation may limit the accuracy, while the better approximation 3.14159 would not. Secondly, to understand how to propagate error, for example, if one is trying to estimate the error in an estimate in the area of a rectangle that needs to be painted, for example, of measures one side as (3 +/- 0.1 m) and (4 +/- 0.05 m), one can approximately the error as (3 m) (0.05 m) + (4 m)(0.1 m) = 0.55 m^2. You could use calculus to derive this result, but the more intuitive result is to draw the rectangle, change the length of each side, and calculate the difference in area due to the change, and add the changes in the area together. For example, take a 3 m X 4 m = 12 m^2 rectangle, and perturb the 4 m side by 0.05 m, so that the perturbation to that is (3 m) X (0.05 m), and likewise perturb the 3 m side by 1 m, so that the difference is (4 m) X (0.1 m). This provide a basic understanding of error analysis. 4. Understanding the "framework" and "limits of applicability" to formulas. The assumption here is that technicians are not going to be creating new procedures, only following ones set down by engineers or other qualified personnel. But the basics here is to understand that when one has an equation, what do each of the symbols mean and what actual physical measurable quantities to these represent? For example, if I ask a technician to calculate the travel time of a wave through a coaxial cable, a formula that uses the free-space speed of light does not apply here, and the technician should know the formula that uses the velocity factor. Another example would be to compute the temperature of an electronic device. A device might have a junction-to-package thermal resistance of 2 K/W, and a package to ambient of 100 K/W. If the device dissipated 1 W, the total temperature rise would be (2 K/W + 100 K/W) X 1W = 102 K without a heatsink. If the technician did not know how to add thermal resistances, or understand that the heat flow is impeded by both resistances, the technician might for example calculate the temperature rise is (2 K/W X 1 W) = 2 K, which a technician who knows the rough order of magnitude of the temperature rise of a device without a heatsink should reject as incorrect. The 2 K/W only applies when the package is held at a constant temperature, and so the technician should know when to apply the package-to-ambient, for example. These are some of the basic skills I think I technician needs to be able to be competent. There are lots of practical math examples that could be used to teach. 73, Dan KW4TI
J'ai travaillé dans un centre de recherche pure, le grand problème rencontré fut le décalage entre le chercheur technique pure et le praticien qui doit essayer de réaliser des projets Ubuesques !
I worked in a fundamental research center, the big problem encountered was the gap between the pure technical researcher and the practitioner who must try to carry out Ubuesque projects!
Dan, great thanks, and I enjoyed reading this. I see that you're an EE. Might you consider taking the survey that we've oriented toward technicians and giving us your suggestions about anything that's unclear, missing, or irrelevant? Again, here's the link: https://tinyurl.com/technicaleducators Tnx. Michael, K2CA
I worked in the auto repair sector. As an older man in A 20 something work place, I have run in to "out of the way"situation , just to have them ask me for advice how to work the problem. World class tech for G.M.
I took the survey. I understand what the survey is getting at, but I am not sure a pure emphasis on mathematics is going to be sufficient. Especially for technicians, the connection with practice should be frequent, and it is not even clear to me that the current separation of academic subjects serves students, and especially technicians, well. Unfortunately I think reform is constrained by both existing ideas about how subjects should be separated and administrative requirements to separate these subjects. But it's certainly a worthy goal and I hope significant improvement is possible. 73, Dan KW4TI
Dan, I heartily agree with you. My belief is that when students are challenged to solve interesting design problems they have to synthesize knowledge from different domains to find solutions….and they learn about the need to make tradeoffs in order to optimize their design. In fact, age-appropriate engineering design problems can necessitate the use of math and help kids see it’s value. Thanks a lot for your help with the pilot test of the survey. 73s, Michael
One of my techs took it and I watched over his shoulder. This person was a manufacturing tech for a bout 20 years and then advanced into R&D. Part of the problem with technicians and math skills is that the better math students usually move on to 4 year or higher degrees. They become engineers, scientists, etc. By nature, most techs have lesser math skills because they weren't taught in ways that worked for them or they simply don't have the aptitude. Once they get on the job, there usually isn't an obvious path to get more education on the related math. Some are self-motivated and find a way to self study. There was a time a few decades ago that many companies did invest in their employees. But much of that went the way of the budget cuts. I have seen co-op programs work very well as students get a feel for job requirements and the hands on helps them to relate when they go back and finish school.
Thanks very much indeed. I will be keenly interested in reading over any comments. The key question is whether schooling prepares technicians for the math they need once they’re on the job. Appreciate your help! Michael
Finally another who knows about 'pataphysics, "the science of imaginary solutions". https://en.wikipedia.org/wiki/'Pataphysics 73/Karl-Arne
The term "technician" covers a lot of ground. I have worked with engineering technicians, who are basically engineers without degrees or without advanced degrees. Certain technicians are trained to perform a single moderately detailed task on a repetitive basis, lots of medical technicians fall into this category Somewhere in the middle are people like automotive technicians, who have a good grasp of practice over a considerable field but no true need for the math behind the design. It comes down to Bloom's Taxonomy. At what level are the technicians in question required to function? An engineering technician may well be at the "Create" level, while a medical technician may be at the "Understand" level. That makes all the difference where training is concerned.
