On Tuesday nights this semester, my nights are spent teaching a group of 21 students for one of KU's four astronomy lab sections. Since my lab section and the Monday section get to meet 15 times within the semester, but the Wednesday/Thursday sections only meet 14 times due to long weekends, my class had the luxury of getting the first week to have a bit of extra time to do a math review before actually starting a lab.
This math review covers pretty basic things. The first thing we went over was how to do unit conversions such as converting feet to meters using a conversion factor. I think I do a pretty good job of this, showing exactly why you can multiply/divide by the factor and it's ok to do so (ie, because a conversion factor is essentially multiplying by one). I very carefully show how if you have the same units in the numerator as you do in the denominator, it cancels out, leaving you with the new units you need.
We discuss ratios as a means of comparing things and determining how many times larger one quantity is than another. The lab packet each student has even has a big bold statement saying "Notice that a ratio does not have any units!"
We also cover a bit of scientific notation and how it relates to the metric system. Should be simple enough. After all, it's base ten. If you need to go from meters to centimeters, just move the decimal point two places to the right.
We discuss triangles. Not even in the depth a high school student had in which they have to learn how to do sine and cosine, but just the realization that there's a base, height, and some angles.
So the first week of lab, I go through all of this, taking what is a ridiculous amount of time doing so given that it's so basic. Every step along the way, I ask if there are questions, if it looks familiar, if they think they could replicate what I'm doing, etc... I don't continue until I see, at the very least, some general head nodding from the majority of the class.
All in all, if people can't do this, there's no reason they should ever have graduated high school. Yet last night in my lab, a sizeable amount of students demonstrated that they lack these skills as well as any sort of basic algebra (junior high type stuff).
Our lab last night was over angles and parallax. Since the angles involved are generally pretty small, we can skip the nasty trig functions and just use the small angle approximation (angle = 57.3ºBase/Height). In this equation, there are three unknows: angle, base, and height. Thus, if you know two, you can find the third.
Since the angle is the measurable quantity in all cases we study, this means that they will have to solve for the base and the height. Out of 21 students, at least three were completely unable to do this (possibly more if they worked with a partner that explained it to them). To be fair, one is a non-traditional student and admitted she hadn't used algebra in any recognizeable form in at least a decade.
In one section, they had to measure the height of a classmate, and knowing this, derive the distance after measuring the angle. For some reason, my students have no concept of what a "meter" is and were getting heights of 30m before I peered over their shoulders and asked if their partner was really as tall as the building we were working in.
We also measured the height of boxes having known distances and measured angles. The answer would come out in meters, but I gave them the true value, so they could calculate their error, in centimeters. It should be obvious that you're going to need to convert them all to the same units, otherwise the dimentions don't come out right. To make sure they would recognize this, I told them to write their units on everything. Some still didn't do so, and wondered why their errors came out to be several hundred percent. After figuring out that units were the problem, a good two or three still didn't know how to convert within the metric system.
One of the questions at the end of the lab, asked them to compare their observed angle for a certain measurement, to the largest oberved stellar parallax angle in astronomy (.76 arcseconds). The point of this question is to drive home the fact that stuff is so far away, that even the largest shift is still about 50,000 times smaller than what they were measuring in class. To make this comparison, the students need to put things in the same units. They measured in degrees, the given angle was in arcseconds. The conversion was given (1º = 3,600 arcseconds). A full 3/4 of the class was unable to answer this question without assistance. So much for them understanding what I'd said at the beginning of the semester about convserion factors. And apparently they missed the bolded statement about ratios not having units too.
Admittedly, this is a 100 level course, taken by primarily freshman wanting to get their lab science requirement out of the way so they never have to think about that awful "science" thing again. But admittedly, this is a 100 level course, taken by college freshman. The skills we use here are ones that are taught to students(with the exception of those with developmental disabilities) in 10th grade or before.
And yet somehow, many seem to be making it through America's public school system, and accepted to a university without even a smidgeon of proficiency. I suppose this is the true meaning of "No Child Left Behind": We'll drag them kicking and screaming along, whether or not they understand anything.