### Blogging 104. Exam 1.

The first exam was last week. I’ve posted grades into the appropriate intranet doo-hickey; students can look up their own grades, administrators can tell I’m actually doing the part of the job that matters most (to the machine), and I have some insurance against the almost-unthinkable fate of losing my gradebook. I haven’t been posting Quiz or Homework grades since these are subject to tweaks. The “daytime” classes have Lecture and Recitation meetings, with Q’s & HW’s in the recitations; in my “evening” class, I’m both Lecturer and Recitation Coach (both “sage on the stage” and “guide on the side”… holy moley, I’m beside myself). Anyhow, this situation, as you can imagine, improves the communication between Lecturer and Recitation Coach immeasurably and it’s, well, better for morale if I hold back on posting the “recitation” grades.

Alas, there’s been a marked tendency on the part of a sizable fraction of the class to treat Tuesdays (when I administer Q’s and pick up HW’s) as the “recitation” section and Thursdays as the “lecture”… and then skip the lectures. Attendance is a lot better on Tuesdays, in other words. For this and other reasons, I didn’t do much “post mortem” work (“going over the Exam”) when last I met the steady-attenders on Thursday. Here’s a rundown now.

Very pleasant for me overall. We use the same Exams as the Day Class versions of 104, so until the exact day, I didn’t know exactly what to expect. It’s a real good test: do-able in an hour by appropriately-prepared students. The temptation to use tricky questions has been suppressed to my relief. This is somewhat in defiance of a claim in the syllabus to the effect that any HW problem is as likely to appear on an Exam as any other. So be it. A custom more honored in the breach than the observance, sez I. I got a pretty typical distribution… too small (n = 15) to be a good fit to “the curve”, but with the much-to-be-expected “lots in the middle and a few at each end” property just the same. One perfect score; two failing scores. Mean $\mu = 75.4$; Median $Q_2 = 78.5$.

Solutions to “two linear equations in two variables”; check. The class-as-a-whole has, anyway, learned the basic moves for both the “addition method” and the “substitution method”, and proved it on the first page. One variable in one solution had a value of 0; this causes more confusion than one would like; the “no solution” and the “infinitely many solutions” cases are typically more confusing still… so I don’t say we’ve mastered the methods. But everybody’s at least prepared to have a conversation about how this stuff works (if there were world enough and time).

On the more abstract interpret-the-graphs version of the “systems of two linear equations” situation, there is, predictably, more confusion. The same page revealed more of the zero-versus-nothing bug… a common difficulty for learners of math (ever since the introduction of “zero”… and yet, confusing as this seems to be for the laity, “zero” is one of the best ideas of all time…). A classic instance of the classic general complaint of teachers about students, “they don’t want to think”. But we’ve got, anyway, kind of a grip… somewhat tenuous I have to admit… on translating from data presented graphically to equations and inequalities. (Going the other way one has the Graphing Calculator so this is the hard way.)

Speaking of inequalities: interval notation was the source of the worst difficulties here. Our examiner very tastefully avoided absolute value inequalities altogether. I don’t say that this topic is more trouble than its worth; I do say I’ve seldom seen it done right. Our text has rather a high-concept presentation that I liked… but I sure didn’t feel like we’d taken enough time to’ve improved the overall skill level in this area by much if at all. I note with pleasure that we skip the section on graphing-systems-of-inequalities. This topic was very badly handled at Crosstown Community College (in a mostly-very-different course, also called 104): in particular, there was a flat-out mistake, for years, on the key to the Final Exam (concerning the “shading” of a certain “boundary point”); nobody seems to have been scandalized by this but me: this is one of those areas (like “set-builder notation”) where the instructors quite often aren’t qualified to present the standard treatment of the material. The industry is aware of the situation and appears to like it that way.

The slope of y=3x+7 is 3. Not “3x” (dammit). If the variable were part of the slope, we couldn’t get the slope by “plugging (four) numbers into (all four variables of) the well-known formula for the slope”. (Now, could we? Think, doggone it! Think!) But… of course… the question of “how variables work” is one of the trickiest of all: this takes practice. Of course the classic area for confusion-as-to-the-nature-of-variables is “word problems” and we’re still seeing quite a bit of it here. It pleases me much more than it should that we had a “mixture” problem since I laid a lot of stress on those; my class would have done even worse on some of the other problem “types” (precisely, on my model, because of the nature-of-variables issues; problems-by-type is essentially a way of “routing around” students’ astonishingly-stubborn refusal to discuss what variables mean and how).

Intercepts are points, not numbers (when we’re being careful). It’s common enough when talking to say, for example, that “3” is “the intercept” for “5x + 3”, when we mean “(0,3) is the y-intercept for [the graph of] the equation y = 5x+3″. The fact that we require more precise language in certain contexts than in others should create no confusion.

But “should” has nothing to do with much of anything, and it turns out that this “be more formal in work to be handed in than when you’re banging away talking and calculating” thing has been a major problem for a sizable fraction of any class I’ve taught at this level of the game. They don’t want to do it and think I’m just being mean. “Well, I meant…” [such-and-such], they’ll tell me, refusing to believe that it’s my duty to grade what they wrote and not what they meant. This misses rather a big part of the whole point of bothering with “mathematical precision” at all: code can be perfect.. and “perfect” is a lot better than “almost perfect”. It should be helpful to think of computer interfaces here: one wrong mousetweak can botch the whole environment. But somehow it never is. Helpful, that is. “Should”. Feh.

Hey. Madeline just woke up. See you later.

1. vlorbik

http://blog.hiremebecauseimsmart.com/tagged/√2

the square-root-of-two file
at _mathematical_poetics_

2. vlorbik

http://math-blog.com/2011/01/13/developing-math-intuition/

plug for _math_better_explained_
(e-book version).

3. /* the instructors quite often aren’t qualified to present the standard treatment of the material. The industry is aware of the situation and appears to like it that way.*/

ah, me. i *used* to understand the situation
so much better than i do today…

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)