## Monday, June 3, 2013

### Semester-End Teachable Moments

At the end of a semester, math students become extremely eager (or anxious) to know about the details of how their grade will be calculated, what their chances are of a particular grade, and what they need on the final exam in order to achieve a particular grade. In the past I would just fire back the answer to such questions in order to have time for other matters, but now I realize that this may be the most fruitful "teachable moment" of them all. Such questions are, after all, algebra questions, and so any of our remedial-or-above students really should be able to answer them for themselves -- and nowadays I require them to do just that.

It's particularly advantageous, because the customary reaction to the incessant "what good is this math for?" question is usually to introduce application/word problems into a course, but frequently to students this looks even more tangential, abstract, and unearthly than the original math it was meant to demonstrate (particularly for any students lacking background familiarity with the given application area, which is almost guaranteed to be most of them). But here we have a concrete example of intense interest that is being brought up by the students themselves -- and therefore it represents a matchless opportunity to hammer home exactly what the utility of basic arithmetic and algebra is, in a way that is hopefully intense and thus memorable.

Likewise, I used to presume this was trivial for higher-level students and so I'd scribble out the math quickly to not be mutually bored, but then I'd find that even my college algebra and statistics students were stunned by what I was doing. So that's the kind of basic thing that warrants time to make totally ironclad. Two cases that come up in my remedial classes:

(1) The university elementary algebra final exam has 25 questions, and at least a 60% score is required to pass the class (among other requirements; link). Common inquiry: "How many questions do we need right to pass the final?" So my answer is now to write on the board "60% of 25" and ask the students to translate that to math as a word problem, and then compute the decimal multiplication by hand. Some are very rusty, but parts of that process are obviously on the final itself; so, good review.

(2) In my classes, I usually compute the weighted total for the overall grade by taking 15% of a quiz average, 50% in-class test average, and 35% of the final exam. Common question near the end of the course: "What do I need on the final exam to get a B grade?" (or whatever). So my response now is to say, "Well you're asking me an algebra question, and you should be able to solve that yourself", writing on the board "W = 15%Q + 50%T + 65%F", and then assisting them in substituting the decimals, desired W, and known Q and T values. Then I tell them to apply the basic solving process (likely with a calculator), and once they know F, then they have the answer to their question.

In fact, I feel that this latter case is such a golden opportunity that I've modified my class procedures in at least two small ways to highlight it. (a) I used to have a policy where I might possibly replace one test score with the final exam if it was significantly higher; but so as to make the test average definitely known prior to the final, now I just drop one test score outright for everyone (which is nicely handled by our Blackboard grade center). (b) I actually spend a half-hour block on this very topic in the early part of the course, as a prime application of the basic algebraic solving process; I get some resistance due to the longer-seeming equation (compared to simpler drill exercises), but -- you'll get resistance anyway for any application problems, and when it truly comes up at the end of the semester suddenly it seems a lot more relevant.

So in summary -- I used to think that these questions were trivial and uninteresting, but it turns out they're very much not. Most of my students, either in an algebra class or thereafter, can't recognize such inquiries as a basic application of algebra that they should be able to solve. Instead of firing off the answers as an aside so as to cover more literal coursework, I now take the opportunity to leverage that intense interest into making it abundantly clear what kinds of important questions can be translated to math and solved by algebra.

Can you think of other inquiries that you commonly answer in grading or course procedures, that are really opportunities for basic math reviews in disguise?