Unit Analysis Game!

One of the concepts that the freshmen need to pick up very early on is referred to sometimes as “unit analysis” (AKA “factor analysis” or “dimensional analysis”). It’s basically the idea that (1) a lot of chemistry “word problems” are really conversions from one unit to another, or involve conversions; and (2) the units themselves tell you how to work the problem.

For instance, they had trouble with this problem: Neon atoms are arranged in a line 2.54 miles long. There are 5.76 x 10^13 atoms in the line. What is the diameter of the neon atom, in Angstrom units (10^-10 m)?

(Yes, that wording is completely unrealistic, but that’s not the point. This avoids the density, solid structure, and mole concepts that would show up in a problem like “a cube of iron weighs 1.24 g, calculate the diameter of an iron atom”.)

It’s an easy problem, but about half of them had some trouble with it, and about 10% of the class is still pretty shaky on it. Basically, convert miles -> meters -> angstrom, and then divide by the number of atoms in the line. See? Unit conversions; there’s no “chemistry” needed to solve the problem. (Some of them picked it up when I pointed out that it’s exactly the same as “how big is a golf ball if 10 golf balls form a line one foot long?”)

The “unit analysis” part of it is based on multiplying the original ratio (2.54 miles / 5.76 x 10^13 atoms) by different constant ratios of units: 1.6 km = 1 mile, so multiply the original ratio by (1.6 km / 1 mile) gives you the same ratio expressed in km / atom. Then change that to meters / atom (using 1000 m / 1 km), finally to angstrom / atom. Without including the units, students can get confused: do I multiply or divide by 1.6? With the units, they can’t do this: If you multiply the wrong way: 2.54 miles * 1 mile / 1.6 km = mile^2 / km, which isn’t a useful unit.

It’s pretty trivial at this stage, and there’s other ways to get it right – as some of them point out. But later on, they can use the same approach with stoichiometric ratios to work out all kinds of problems that do involve chemistry.

Here’s the game part: Showing it on paper or on the blackboard doesn’t enforce the idea of fixed ratios that we can string together at will to reach the right answer. I want a set of plastic tiles that have the different unit ratios on them (e.g., a tile showing “1.609 km / 1 mile”, on the reverse it shows “0.6213 mile / 1 km”), that they can actually move around. (Tiles that just say “mile” and “km” separately wouldn’t work.) Also “prefix conversions”, like “1 / 1000 milli-”.

Oh, and I need them cheaply enough to be able to get about a hundred copies.

Found a procedure for “DIY shrinky dinks” – polystyrene sheet can be heated to shrink down to about a 1/3 of its original width/length, and 5-6 times its original thickness. Smallparts.com sells high impact PS sheet that should shrink up to about the right size… Too bad the hobby shop in town is closed on Sundays!

Guilt-Free Grade Assignments

One of the hardest things to do, I’ve found, is figure out someone’s final grade.

Putting “D-” on one lab report is relatively painless… double check to make sure that yeah, it really is that bad, but it’s only one grade out of many.

Entering “C” into the official records takes a little more work. It’s particularly hard for the ones that I know have tried consistently all semester. “C” could mean that the student will have to repeat the class. It could prompt them to change majors or careers. It could affect their scholarship or financial aid. Then again, turning all “D”s into “C”s and all “C”s into “B”s isn’t doing them any favors, either, because it overestimates how prepared/apt they are for the next class. So I have to check my sympathetic wish to nudge them up, just a hair, they’ve worked so hard for that “C” and is there really THAT much difference between their work and that of the lower-end “C” students?

Ignoring the names, though, it’s an interesting problem in classification. I’ve got 119 samples, who’ve been tested in various ways with the intention of sorting them into 5 categories, which just happen to have alphabetic labels. Now all I have to do is work out the best way to classify the samples, starting with some pre-determined borders for the categories. There are two types of measurements available, which can be broadly described as measuring independent understanding (“exam” type measurements) and as measuring consistent work to gain understanding (which includes homework, lab work, and attendance grades). The easiest of these to work with are the averaged exam grades and the work-over-time-except-lab. Plotting the exam average against the other average doesn’t take into account the lab grade, however. Should that go into the exam average (because it includes some lab quizzes) or into the other average (because it’s mostly work over time)? Let’s average the lab into both grades. (Losing some orthogonality, but I don’t feel like shoving the data into Matlab for PCA.)

This actually works out pretty nicely:

Classification of 119 samples using two parameters

There’s a fairly clear line between most of the categories. Which means…. that it’s quite easy to draw straight lines between those categories as grade boundaries. This also agrees well with the grades based on the overall average (collapsing the two dimensions here into one dimension, so no surprise)… but it makes it a little easier to show a student that “yeah, you really do fall right into the middle of the ___ pack, so better luck next semester”.

Now to finish up grading the senior class… this is much harder to do with 7 samples than with 119!

Spring ’09: What worked

The class I taught was an upper-level (graduate students only) class. Mostly a lot of fun.

Parts that I (and they) seemed to like:

Active Learning for the win

Started talking about separations methods today.  Had a nice little lecture all planned out – “Abstract Separations”, I called it, since we were talking about separations in general rather than any particular technique.  I didn’t intend it to last the whole class period, and it didn’t.  I noticed the usual percentage of blank looks and nodding-off.

Not one of the most successful class periods.  And then…

MATLAB: not so hard?

Multivariate analysis techniques like principal components analysis and partial least squares are an important part of modern analytical chemistry.  Partly that’s because they’re trendy; partly it’s because the problems that can easily be solved with existing detection methods and univariate methods (‘easy’ statistics, like just the average and standard deviation) are pretty well exhausted.

So, it’s also been an important part of the “advanced” analytical class I’m teaching this semester.  Tough going for the class and for myself!  This is the first time I’m actually explaining some of this material to anyone else, and it’s definitely stretched my own understanding.

The in-class lectures have been a mix of abstract theory (on multidimensional stuff in general, and PCA, clustering and PLS specifically) and practical examples in the form of some recent papers.  We’re done covering these specifically for now – future looks will only be as encountered in interesting papers.  Which means they’ll come up again and again…

Anyway, yesterday for lab we did a MATLAB tutorial.  MATLAB is a large and complex program (or environment, really), but it’s the way I learned to work with this stuff and so it’s the only way I know to show them… Of course, only one of them had seen it before, so there was an hour or so of introductory “this is a variable, this is a vector, this is how we do this particular thing…” etc.  We got through importing the data from a CSV file, plotting it up several different ways, PCA (using princomp only, no rotation) and clustering, and linear discriminants (classify).

The format – 13 students and me – was alright, although I wouldn’t have minded having someone else around who could’ve circulated to help people get the syntax right.  (MATLAB is, of course, tremendously particular about little things like spelling and where commas go, and I think most of the errors students encountered were caused by one or the other.)

It wasn’t terribly well organized. We started out with basic things like creating variables in the workspace – after defining ‘variables’ and ‘workspace’, but most of the other elements were introduced “along the way” to specific goals like “plot this set of data as a surface plot”.  I think they understood the row-column notation for addressing – or they did at some points, but then came up blank when I asked “and how do we plot column 2 against column 3?”

We could’ve used better organization particularly about the commands to plot data.  PCA got a little sketchy – somehow I skipped over scree plots as a way of identifying the proper number of eigenvectors/components, for instance, although we did talk about them (with an example) during class.  The linear discriminants bit was well received; we used the builtin “fisheriris” dataset, and I showed at the end that yes, it really can be used to classify unknowns into more than 2 groups.

I’ve been writing an extended “howto” guide, covering mostly the same content as the live tutorial but with more detailed explanations.  It’s up to 19 pages, and still not done.  I think instead I’ll write a series of shorter ones – maybe 2-3 pages each, focused on a specific task or question.  With actual chemical data, as much as possible.  Examples that match the sort of data that they’re actually collecting will be that much more useful…