People who design applications, like simulators and CAD programs make heavy use of math. The hands-on aspects of robotics (and Vex in particular) make it appealing for students, but moving from users of applications to designers of them requires "leveling up." Unfortunately, not many students have the patience to stay with it to the upper levels.

Over 20 years ago, I worked on an early version of cochlear implant at U. of Michigan. While animal studies were used, we tried to minimize animal use by creating mathematical models of the transducer properties of parts of the ear. Your ear converts mechanical motion (ear drum vibration) to electrical signals (nerve impulses), which your brain then converts to a perception of hearing of certain sounds. Modeling this process required the use of Fourier transforms and Wigner distributions, advanced math concepts beyond calculus.

In the reverse direction, we collected large amounts of data (with sample rates of up to 100Khz), requiring signal processing (more Fourier transforms) and statistical analysis. Doing the “right” analysis yields patterns which can represent the information in meaningful ways, while the “wrong” analysis makes it look like gibberish. Voice recognition and face recognition depend on filtering/recognizing the “right” patterns. I imagine the folks at Facebook who design the applications to tag photos are using algorithms that depend on some fairly sophisticated math.

You can always create simple models using “easy math”, but sometimes these simple models are crude approximations which need refining, comparable to a stick figure vs. a photograph as a representation of a real person. For example, a car’s value can be approximated with the linear model

v(t) = 20000 – 3000t, with original value at $20K, losing $3K/year

This crude model is a poor representation beyond ~ 4 years. A 7-year-old car doesn’t generally have a value of negative $1K, as calculated using this model.

A better model is an exponential model:

v(t) = 20000*e^(-0.15t), with 15% depreciation annually.

Using this model to calculate value, a 10-year-old car has a value of $4463, closer to real life. The constant (15%) can be tweaked accordingly – a Honda Civic depreciates more slowly than a Dodge Caravan.

But even the exponential model breaks down under certain conditions. For example, iPod sales may show exponential growth in the early years, but will level off as the number sold approaches the total world population. In this case, a more sophisticated model like the logistical model might be appropriate:

v(t) = c/(1 + a*e^(-bt)), where c is the “level off” value

And so it goes. Life is complicated, so the formulas that represent it have to be, too. The upside is that once a good representation is made, math can manipulate the data in a way of your own choosing without the physical limitations of real life.

A simple example of this using only Algebra II is the transformation of parabolas, as used in the movie Avatar. The movements of the Blue people were created by modeling the movement of cats. Rather than tossing cats from a 100-ft cliff, they were dropped from more reasonable heights (like 10 ft), and their motion was recorded, modeled, then stretched and shifted accordingly. There are lots of computer calculations in CGI, which require math as the foundation.

Finally, some students encounter difficult formulas and think that complicated means obscure, as in, "It's so complicated that I'll never use it." One example is the two-sample z statistic, an rather nasty-looking formula:

z = [(x1 – x2) – (m1 – m2)]/sqrt[(s1^2/n1) + s2^2/n2)], where many of those variables are not a just single value, but means or sum of squares means calculated from hundreds or thousands of data points.

It computes the difference of difference of means of 2 populations, a concept I encountered in Time magazine (a relatively non-mathy publication) in an article about a cholesterol medication which appears to be more effective for men than women. Two populations: men and women. Difference of means: average cholesterol level before and after medication (aka, improvement). Difference of difference of means: men’s improvement is greater than women’s improvement. I took a look at the original data to decide whether this medicine was worthwhile for me (it wasn’t, as I'm a woman).

There are many connections between math and the scientific/technical world, but it takes some knowledge and awareness to find them.