A musician and a gentleman

A musical interlude

As you may know, I bestowed my extra ticket to the San Francisco production of Wagner's Ring on the son of long-time friends. “EF” is a recently-declared music-composition major and I got to play the part of a patron of the arts by introducing him to the sui generis landmark of operatic composition. My friend Gene O'Pedia weighed in with a vigorous endorsement: “Neat that it's all in the space of a week, a concentrated dose of Wagner. Could be transformative. Like if EF starts the next school year as an engineering major.”

Good point. The experience could confirm the young student in his career goals or scare him off into some different field entirely. As it happened, the former seems more likely than the latter. An important factor in EF's opera adventure was his opportunity to converse with one of the performers in the orchestra pit.

We arrived at the War Memorial Opera House early enough on the evening of the performance of Die Walküre to catch most of the talk that was scheduled one hour before curtain time. It turned out to be an unfortunately dull affair—a flat and uninflected reading of an analysis “by some great expert for the edification of other great experts”—and I was glad we had missed part of it. However, the talk was presented on the opera house's main floor, so EF and I got a different perspective than the one we normally had from our regular seats in the balcony level. After the talk ground to its eventual end, I led my guest over to the orchestra pit so that he could check out the disposition of musical forces and ogle the conductor's stand.

It was still half an hour before curtain, so there were very few musicians in the pit, but EF was in luck. Although he is studying several instruments, his principal instrument is the cello, and there was a cellist at his post in the orchestra pit. EF promptly leaned over the railing and asked the musician about the cello part for Die Walküre. I was concerned that EF was committing a faux pas by bothering one of the performers, but the cellist seemed not in the least perturbed.

He came up to the rail and informed EF that the cellos had 79 pages of music to get through for that night's performance. He added that Walküre worked the cellos especially hard, having the same number of pages for them to perform as the massive Götterdämmerung, despite being 45 minutes shorter in overall duration. Upon finding out that EF was studying music composition, the cellist gently suggested that his future compositions might give the cellos a break by not emulating Wagner too much.

Although we did not know it at the time, EF's friendly advisor was David Kadarauch, the opera's principal cellist. He took several minutes to chat with EF and was generous in sharing his informed perspective on composition and performance. When he found out that we were sitting in the lower balcony, Kadarauch congratulated us: “I always tell my friends to sit there. It's the best location in the house for appreciating the music.” My young companion soaked it up like a sponge and it started the evening on a high note for him.

I am confident that Mr. Kadarauch does not follow this blog, so he may never see this. However, I feel that I witnessed something significant and praiseworthy. He probably did not think it was a particularly big deal to take a few minutes to encourage a young music student who hung on his every word. He was simply exchanging his performer's hat for a moment for that of a teacher. But it was a big deal. EF may be at the beginning of a long musical career. The kindnesses of those who have gone before him will shape and inform that career. Thank you, David Kadarauch, for giving that career an encouraging and appreciated nudge at its very start.

And I'm pretty sure it won't be in engineering.

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Fixing California education

USC puts in the fix

Saturday morning's edition of the Sacramento Bee treated us to an opinion piece by Dr. William G. Tierney, director of the Center for Higher Education Policy Analysis at the University of Southern California. Naturally it caught my eye, especially when I noticed the title: Simple changes would make college degree easier and cheaper. My eyebrow quirked with skepticism and I steeled myself for disappointment.

I was not disappointed—about being disappointed, I mean. Tierney is not completely out to lunch, but he certainly overreaches and oversimplifies. Here are some pertinent excerpts from his article, along with my comments:

Viewpoints: Simple changes would make college degree easier and cheaper

By William G. Tierney

Special to The Bee
Published Saturday, Jun. 11, 2011

How can California produce the number of college graduates its future economy will need when its public higher education system is staggering because of the ongoing budget squeeze? Unfortunately, the state's public universities and colleges won't receive any of the unexpected surge in new tax revenue and will continue to scale back their enrollments. If the tax extensions sought by Gov. Jerry Brown are not approved, enrollments will likely shrink further.

California's private nonprofit and for-profit colleges and universities, by contrast, are in relatively good financial shape. Enrollments in most institutions are holding steady or are up. Endowments and philanthropic giving are on the upswing. Tuition is still higher than in the public schools but is rising at a slower pace.

We should be careful not to overstate the situation relative to public versus private college education in California. The Great Recession has caused a dramatic spike in tuition costs at the California State University and the University of California. The steep rates of increase cannot be sustained without the destruction of these institutions (so I predict it won't happen). It would be misleading to make too much of the “slower pace” of private-school tuition in the state.

If these two higher-education systems would put aside their long-running competition for students, faculty and resources, and cooperate to boost graduation rates, they could go a long way toward turning out the 1 million more credentialed individuals—according to one study—the economy will need in 2025. Heresy? Hardly.

Here I pause to climb onto one of my favorite hobbyhorses: I hate the expression “according to one study” and similar unhelpful non-references. What study? I realize that this is an opinion piece published in a newspaper and not a peer-reviewed research article in an education journal, but the Bee falls short in its mission to inform the public when it expects us to take unsourced statements at face value. I don't know whether to blame Tierney as well. Did he try to include a citation, only to have the Bee editors complain about the fusty academic prose?

In any case, I have done the leg-work for you, should you want to check whether the claims are well supported. Tierney is referring to the work of Hans Johnson, who published two papers in 2009 with the Public Policy Institute of California:

Johnson, H. (2009). Educating California: Choices for the future. San Francisco, CA: Public Policy Institute of California.
Johnson, H., & Sengupta, R. (2009). Closing the Gap: Meeting California's need for college graduates. San Francisco, CA: Public Policy Institute of California.

Johnson's more recent paper may also be of interest:

Johnson, H. (2011). California workforce: Planning for a better future. San Francisco, CA: Public Policy Institute of California.

While I'm at it, I'll point out that Tierney's article appears to be a public-consumption version of a more extensive report titled Making It Happen: Increasing college access and participation in California Higher Education: The role of private postsecondary providers. Tierney coauthored it with Guilbert Hentschke, a colleague at the USC school of education.

Now let's get back to Tierney's argument:

There are three important ways the public and private sectors can work together to produce more graduates.

• Shifting the remedial burden to the private sector: California's public schools and universities are lousy at remedial education. Sixty percent of entering Cal State students have to complete at least one remedial course when they arrive at college. It's a task that consumes professorial and student time, and is ill-suited to the mission of graduating students.

For certain private nonprofit and for-profit schools, however, remedial education is a forte. They have experience in dealing with learning deficiencies and are adept in tutoring and some forms of special education. Unencumbered by competing missions, they can focus on the remedial task at hand. And monitoring their success rates would be as easy as grading exams.

Did a warning flag pop up when you read that? Here we have a professor at a private university recommending that more of California's education program be shifted to private institutions. Of course, he's not suggesting that the remediation work be allocated to the University of Southern California, which is presumably above all that. Instead, Tierney is arguing that certain profit-based schools excel at making up educational deficiencies and should be encouraged to do what they do best. I have my doubts.

For-profit schools tend to report high success rates, but these statistics can be misleading. Such schools have a vested interest in retaining their paying customers. Students and colleagues of mine who have taught at profit-driven schools are amazed at how difficult it can be to maintain standards or drop non-performing students. (“Hey, I paid for this class. Now give me my passing grade!”)

Are public schools any better? Tierney says we “are lousy at remedial education.” In my long experience as a college teacher who often teaches elementary algebra (a course that used to be standard high school freshman fare), I can report that my success rate hovers between sixty and seventy percent. In general, my colleagues and I find that one-half to two-thirds of our students pass algebra.

It's shocking, I know. I think our success rates would be higher if we had fewer students in each class and more time to give them individual attention. Perhaps that's what private schools could do (for a price). However, I also want to point out that open-admission institutions like community colleges have to take on all comers, ready or not. We strive mightily with the twin tools of assessment and placement to figure out what students already know and what courses they should take to maximize their probability of success. Still, even the best instructors lose a quarter of their students.

It's my opinion that we can't do much about it. That might be a defeatist attitude, but I'm not one to casually acquiesce in failure. The reality is that every semester brings us students who are placed as best we can manage but who lack any real interest in education. These are the students who are marking time till they find something better or more interesting to do. They may be living rent-free under a parent's roof as long as they're enrolled in school, so sitting in class is like the price they pay for shelter. It would be rude to also expect them to work at the subject material.

Other students have life problems or emergencies that predictably or unpredictably sabotage their academic progress. Many of these people will regroup and try again (and succeed) under better circumstances. Still, they go into the “failure” column when we tote up the statistics. In general, you can assume that ten to twenty percent of your students are doomed to fail because of attitudes or circumstances. As instructors—at least if you are serious about doing the greatest good for the greatest number—you have to guard against snap judgments. Try to foster success for every student, even if you know you are fated to fall short in a unknown number of instances.

Good schools in the private and for-profit sector might also be serious about helping students. I expect that most are. However, it's often apples and oranges when we make these comparisons and I can't quite get on board with Tierney's assertion that public schools are inherently worse at remediation than private schools. Community colleges, in particular, do a lot of remediation. Furthermore, to stand things on their heads, consider that we get over half of our algebra students to succeed. In algebra! The math class from hell!

• Making it easier to complete required courses: Currently, a student seeking to transfer credit to another school faces too many institutional and faculty hurdles. An “A” in English 101 at Los Angeles City College isn't automatically credited at UCLA.

The state took a baby step this year toward clearing up the uncertainty with the Student Transfer Reform Act, which guarantees junior status at Cal State schools to community college students who earn an associate degree. There is no reason why such a relationship should only exist between community colleges and Cal State.

To facilitate transfers, all accredited institutions would adopt a common course-numbering system that ensures that students learn similar things regardless of where they took the class. For example, credit for completing English 101 at a community college would automatically transfer to a UC or a private college or university. Not only would general education requirements be part of this system but preparation courses for students' majors as well. Arizona has set up such a credit-transfer system, and initial reports are that it is producing more graduates faster.

I agree unreservedly with Tierney's recommendation concerning the transferability of college courses. It should have happened yesterday.

I know, however, why it didn't. And Tierney plows right into the problem without apparently realizing it: “all accredited institutions would adopt a common course-numbering system that ensures that students learn similar things.” How much experience does Tierney have in higher education? Has he been paying any attention at all? California's colleges and faculty will fight tooth and nail against a uniform statewide curriculum. Hardly anything is so precious to a college as its own curriculum. Losing control of course definitions to some centralized authority is tantamount to becoming merely one small cog in a monolithic educational machine.

No thanks!

Community colleges, especially, tailor curriculum to their communities and colleges in general cherish the right to tweak their own courses and experiment with their own curriculum. Ceding that ability to a central authority is a non-starter. Of course, you can never tell what the California legislature will do—or try to do. Years ago the legislature passed and the governor signed a measure mandating that all California community colleges use a uniform course-numbering system. The requirement is still on the books:

66725. (a) It is the intent of the Legislature to facilitate articulation and seamless integration of California's postsecondary institutions by facilitating the adoption and integration of a common course numbering system among the public and private postsecondary institutions. The purpose of building and implementing a common course numbering system is to provide for the effective and efficient progression of students within and among the higher education segments and to minimize duplication of coursework.

It never happened, probably because the legislature failed to allocate funds to pay for it and to establish the mechanism by which it would occur. We do, however, at least have the California Articulation Numbering system, which provides an intermediary for the comparison of courses at different institutions in the community college system. Something along these lines might be a way to advance the positive aspects of Tierney's recommendation without falling into the trap of statewide course uniformity.

• Encouraging private colleges to admit more students, especially through online learning: To get private colleges to admit more students, the state might pick up a portion of the tuition difference between private and public schools. That, no doubt, would bring howls of protest—taxpayers giving money to well-heeled privates. But consider UC's newest campus in Merced, currently with 4,000 students. The state could surely find cheaper seats for those 4,000 students in California's 79 private institutions than pay $500 million and counting to complete the campus.

While shilling again for his own segment of California's postsecondary education system, Tierney takes an ill-considered slap at the University of California. The Central Valley is a major growth center for the state. The establishment of the tenth UC campus in that region was long overdue. Tierney is recommending a penny-wise and pound-foolish approach of shutting down a growing institution that will be sorely needed by the burgeoning San Joaquin population. Does he want higher education in that region ceded to private institutions? Perhaps so.

But more private admissions can't begin to close the graduate gap. A significant state-led effort to increase online education would have far more impact—and the private nonprofit and for-profit sectors are best qualified to lead it because they are doing it now and want to grow. Given their checkered history, participation by the for-profits would have to be tightly regulated.

I support the expansion of on-line education, although I have reservations about quality control and identity verification (who is taking those on-line exams?). It's interesting that Tierney felt obligated to cite in passing the “checkered history” of profit-driven schools in the on-line sector. Here's an area where we are best advised to hurry slowly.

California's persistent budget squeeze and anti-tax mood erect a high hurdle to increased graduation rates. Only a coordinated effort of its five higher-education systems—three public and the nonprofit and for-profit privates—can produce the number of graduates the economy will need. There's still plenty of room for spirited competition, but California's economy needs all five on the same team to remain competitive globally.

I can't argue with Tierney's team metaphor for addressing the problems in Caifornia's postsecondary education. I'm not sure, though, that I want him to be the captain.

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From the time vault

As it was in the beginning

There may very well be a hoarding gene in my family tree. I seem to have inherited it from my parents. Dad exhibits this behavior in his workshop, which is cluttered with the detritus of decades, including a large collection of old-fashioned vacuum tubes—perfect for fixing up that old Curtis-Mathes television of yours. Mom keeps boxes of stuff in the basement, including virtually every scrap of school work ever brought home by one of the children. In my case, this includes bundles of material from my initial foray into grad school back in the seventies.

Last week I made a quick trip to the old homestead to participate in a nephew's birthday. At one point I took a break down in the basement and ended up riffling through the family archives. I hit a rich vein from my early teaching days, including bundles of punch cards from student surveys. In addition to filling in bubbles on the front of the card, many students wrote comments on the back. One of the first to catch my eye made me sad:

Z very rarely came prepared for class. His lectures were all straight out of the book, word for word. He also wasn't able to answer various homework problems in class.

Damn. That didn't square with my recollection of my first teaching assignments. Did I really get stuck that much? Did I parrot the book? That's hardly my style today. How things have changed! Or have they? The next card said:

Z is usually a well organized lecturer who presents the subject in a clear and illuminating manner. He also takes time in class to discuss difficult problems, as well as to answer questions on the reading.

Ah! That's more like it! Here's another:

Z will sometimes introduce a proof of his own to supplement the book's proofs. Usually, his make more sense.

Ha! So much for “word for word”! Of course, it's not all 100% positive:

Works very hard to prepare for the lectures & exams. Z tends to joke around too much sometimes.

Well, that sounds accurate. What was that first kid thinking, anyway? It happens every school year, of course. You have students sitting side by side in the same class and their reports of their experience sound as if they were on different planets. (Helpful hint, kids: Try really hard to find an instructor on your wavelength!)

For sheer perversity, the following comment is one of my favorites. Could it be a joke?

Your timing on exams needs some help. I forget the material by the time the exam comes around, therefore, I actually need to study the material.

It might very well be serious. I have had students in the intervening thirty-five years who say very similar things. (Poor babies! Having to study!)

Then there's this:

Z is dressing very well this quarter.

That's what a TA gets for occasionally wearing a tie in the seventies. (It didn't take much to impress them.)

Fortunately, since I couldn't get anything from the multiple-choice side of the punch cards by inspection, I found a folded-up copy of the student questionnaire stuck between two of the card stacks. Most of the questions were perfectly straightforward, dealing with such matters as clarity of presentation, fairness of grading, punctuality, and so on. I racked up good scores on all of these until question #16, when my numbers took a steep nose-dive. Here's what it said:

EXAMS EMPHASIZE, LECT., READINGS, LAB DISC., HOMEWORK, LECT. & TEXT EQUALLY

Whoa! For one thing, lecture is cited twice (and we didn't have a lab discussion session either), but the real point is whether this is even a reasonable standard. Homework as important as exams? You're kidding! I didn't do it then and I still don't do it now. I guess I'm just a rebel.

Still crazy after all these years.

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The terrible, horrible, no good, very bad math problem

A boon for Lucky Larry

Teachers typically take pains to ensure that their exam problems are clearly stated and have unambiguous answers. This is especially true in math, where precision is at a premium. I try to make it clear exactly what I want and what form it should be in (such as “exact” or “rounded to the nearest hundredth”). The last thing any exam grader wants is a problem that is easily misinterpreted, because that just leads students astray. Even worse, however, is the problem where an incorrect calculation can produce the right answer. Then you're really stuck. You have to read the solution especially carefully to make certain the student didn't get the right answer by a lucky fluke.

I have put myself in that situation. It happened in a suite of problems designed to test my students' ability to compute the perimeter, area, or volume of certain standard geometric shapes. Having quizzed them on various rectilinear figures (break it up into rectangles, kids!) and circles and boxes, I began to challenge them with combinations of the basic forms.

You may be familiar with the classic Norman window, which is a semicircle mounted atop a rectangle, the semicircle's flat side coinciding with one side of the rectangle. In finding the perimeter, the student must make a judicious use of the circumference formula for a circle and the perimeter formula for the rectangle. The answer is the sum of half the circumference of the full circle and the three exterior sides of the rectangle (omitting the one adjoined to the semicircle). Not too hard. The computation of the area is even easier: half the area of the full circle plus the area of the rectangle (that length times width business beloved by all).

I wasn't surprised that the results were a mixed bag, but I felt we were making progress. Each time I quizzed them on a variant of the basic shapes, more students were picking up on the standard formulas and the tweaks required to apply them to the hybridized shapes. With an excess of confidence, I then reached a bit too far and outdid myself: I adjoined a quarter-circle and a triangle. To make matters much, much worse, I made some very bad choices for the dimensions. Can  you see my mistake(s)?

As before, I asked my students to find the perimeter and the area of the shape. You can check that the perimeter is (6π + 30) ft and the area is (36π + 30) ft². Thanks to my carelessness, I got lots of quasi-correct solutions. First of all, the perimeter of the 5-12-13 triangle is numerically equal to the area: the perimeter is 30 ft and the area is 30 ft². That meant students were able to get the “right” answer by employing the perimeter formula when the area formula was required. And, yes, some of my students did exactly that.

It gets better. For the area computation, they needed to compute πr² and take one-quarter of it. A full circle of radius 12 ft would have an area of 144π ft², so a quarter-circle's area would be 36π ft². Several students, however, apparently reasoned thus: The arrow labeled with the “12 ft” goes all the way across, so it must be a diameter. I need to use the radius, which is half of that, so I'll use 6 ft. It's a circle, so the area is πr². Therefore I get 36π square units.

¡Caramba!

In grading this problem, it really didn't much matter if the answer happened to be right. It was much too likely that a right answer could be obtained by accident. I corrected this exercise very carefully and very slowly.

I hope I learned my lesson.

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A bag of tricks

Stop making sense!

I remember when “Massha” enrolled in my algebra class. Recognize the name? She's a character created by Robert Lynn Asprin in his Myth Adventures fantasy series. He introduced her in the third volume, Myth Directions, well before the long string of pun-obsessed novels became rather labored.

My student lacked the bright orange hair or heroic girth that characterized Massha, but she reminded me of Asprin's creation because of the way she preferred to do algebra. The fictional Massha was described as a “mechanic”—or even “no-talent mechanic”—by other characters in the novels because she used amulets and other physical trinkets to cast her spells. She possessed no actual magical talents, but relied entirely on magical devices.

The Massha in my class drove the point home time and again whenever I tried to explain a procedure and she would counter with a memorized algorithm, as if that pre-empted any further discussion. It was an occasional irritant, but she was entirely sincere in her approach. She had found success in treating math as a collection of miscellaneous tricks and she resisted any attempt to explore below the surface. As far as our Massha was concerned, that was just a dangerous distraction.

A particularly clear case arose while we were discussing the solution of rational equations. These are nothing more than equations that contain rational expressions. For example,

is a rational equation. One of the glorious principles of algebra is that you can do just about anything to one side of an equation as long as you do the same thing to the other side. In the case of a rational equation, you can use this principle to eliminate all of the denominators. Just multiply both sides of the equation by the least common denominator of all of the rational expressions! For the given example, the least common denominator is x(x − 1). Multiplying both sides results in massive cancellation and simplification:

Since algebra students seem to regard operations involving division with more trepidation than anything else, I can usually engage their enthusiasm for a process that destroys all denominators. It has its moment of messiness, but the results are clean and rational (no math joke intended). I leave it as an exercise to the reader to complete the demonstration that x = −2.

Massha was not content with my demonstration. She wanted to rephrase things in her own way, which I'm normally inclined to encourage, as it indicates the student is assimilating the knowledge. What she said, however, disturbed me:

“Do we have to show our work and cancel things or can I just do it the way I learned it? I was taught that you compare each term to the LCD and give the term the part of the LCD that it's missing. Is that all right?”

I looked back at the problem on the board. Massha knew the LCD was x(x − 1). She would consider 3/x, observe that it lacked the x − 1 and “give” it that factor. At the same time, presumably, she would drop the factor x, which it did have in common with the LCD.

“And you drop whatever it already has in common with the LCD?” I prompted.

She nodded her head. “Just do that to every term,” she said. “It's faster!”

Massha had a magic amulet from her bag of tricks. She had memorized a procedure that saved her from writing the messy cancellation step because it algorithmically led her to the same result without actually justifying it. The juice had been squeezed out of the algebraic process and the dried husk preserved the result if not the rationale.

I pondered.

“Since this is our first encounter with rational equations in this class, I want everyone to show a step-by-step justification for our simplifications. Later on, when we return to rational equations in terms of applications, I'll let people reduce the amount of work they show. For now, though, always write down the LCD and show the step of multiplying through by it and reducing.”

Massha scowled at me.

“But I already know how to do this!” she complained.

“I am confirming that your short-cut is a valid algorithm,” I replied, “but I will be holding everyone to the same standard of completeness in presenting solutions.”

Massha was unhappy much of the semester. She had a keen memory, had had algebra before (so why was she in my class?), and retained quite an array of solution gimmicks. I was happy for her (sort of), but her view of mathematics had been reduced to the rote application of algorithms. It was enough for her to do well in class, but she chafed at every requirement to justify her solutions. She would have found a kindred spirit in the algebra student who had been in one of my previous classes. When I finished demonstrating the derivation of the quadratic formula, that student rolled her eyes and said, “Oh, Mr. Z, are you explaining things again? Why didn't you just give us the formula?”

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A failure to communicate

She said ≠ She heard

My colleague and I were making light conversation in the faculty room as we checked our mail-boxes.

“I see you have a clique of my former prealgebra students in your compressed algebra class,” Professor Turin observed. “I saw them hanging together before your class.”

“Oh, were they yours last semester? Most of them are doing pretty well,”

“I'm not surprised,” she said. Then she hesitated. “But how is Kara doing?”

I sighed.

“Poor Kara. Not well. The pace of the class has her quite stressed and she makes lots of mistakes. She really should have picked a more regular schedule.”

“That is exactly what I told her,” said Turin. “She was keen to take your class because of the compressed schedule and I warned her that it was a bad fit. She freaked out several times during my prealgebra and it was always about her fear of falling behind. I wish she had listened to me and enrolled in a regular section.”

“Yeah, well, what can you do?”

It was less than a week later that Kara read the handwriting on the wall and visited my office hour to inform me that she was cutting her losses and dropping my compressed algebra class.

“I could really use the time better on my other courses, Dr. Z. The class goes too fast and it's hard to understand.”

“That's a perfectly reasonable decision, Kara. It's important to make the best use of your time. You should do better next semester in a regular section of algebra.” I paused before asking her a question. “Did you talk to your prealgebra instructor before enrolling in my class?”

I deliberately did not mention my colleague's name or otherwise indicate that I had already discussed the matter with her. Kara brightened up immediately.

“Oh, yeah! I did! Turin said I could definitely do well in your class. She said I was all ready for it, but I guess things just didn't quite work out as we had expected.”

My eyebrows wanted to go up and my eyeballs wanted to bulge out, but I think I managed to control my facial features and maintain a mien of serenity.

“Well, yes, Kara. Things didn't work out this time. Better luck next time.”

After Kara left my office, I stalked the hallways looking for my colleague. Turin was in her office. I recounted my conversation with her former student. She was dumbfounded.

“That doesn't sound anything like the conversation we had. I tried really hard to warn her she was making a mistake!”

We considered the matter for a while. Clearly Kara had a ferociously effective data filter that allowed only good news to impinge on her consciousness. Since it is Professor Turin's nature to be encouraging and as positive as possible, I was certain she had sprinkled her cautions with snippets of praise that had been the only things Kara had heard. Eventually, Turin reconstructed her comments and we identified Kara's post-production editing.

What Turin said:

“You're a good prealgebra student, Kara, but Dr. Z's compressed algebra class would be a tough challenge. I'm certain a regular algebra class would be perfect for you.”

What Kara heard:

“You're a good prealgebra student, Kara, but Dr. Z's compressed algebra class would be a tough challenge. I'm certain a regular algebra class would be perfect for you.”

There's no simple cure for this. Certainly Turin isn't suddenly going to stop offering her students positive feedback, even if only as mitigating factors in a negative review. Equally certainly, Kara is not going to stop selectively hearing what she wants to hear. I fear the set of solutions may be the empty set.

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Lumpers and splitters

Considering a “stupiphany”

The secondary school math curriculum used to be extremely predictable in the middle decades of the twentieth century. High school freshmen took elementary algebra, sophomores enrolled in geometry, juniors refreshed and extended their first-year curriculum with intermediate algebra, and college-prep seniors took trigonometry. That's just the way it was in many places across the United States (and certainly when I was in high school).

Eventually, though, many changes occurred. While that old-fashioned core curriculum survives in many ways, it certainly drifted. Algebra trickled down into middle school and high school seniors started taking introductory calculus (or something called “analysis”). Perversely, however, the old high school courses also migrated into the college curriculum, where we originally called them “remedial” and then relabeled them with the less pejorative “developmental” tag. I think that such remedial courses used to be the province of what we call “continuation” high schools, but today most developmental math is taught in community colleges. My college, for example, teaches more developmental math than anything else. We even teach basic arithmetic to those who failed to learn it in elementary school.

And we teach developmental math over and over and over again to students who fail it the first, second, or even third time. Success rates hover between fifty and sixty percent for most of the classes, indicating the degree of recycling that goes on. It's maddening to both students and instructors.

Most of my colleagues in the math department know the secrets to success in a math class. In fact, they're hardly secrets because we share them constantly with our students: attend class regularly, pay attention, study the material, do the homework, and ask for help when you're stuck. While luck plays a role (catastrophic illness, financial distress, and family emergencies can derail anyone), most student failure is based on the neglect of those fundamental guidelines.

Of course, we don't just enunciate the principles of successful math learning and then sit back and wait for our students to succeed. We try to meet them halfway (or more than halfway). We offer tutoring centers, accommodations for learning disabilities, on-line support, and different course formats. These days the traditional classroom-based lecture class is often supplemented with on-line instruction or hybrid classes that combine in-class and on-line elements. Students may be able to enroll in self-paced computer-based math labs, too.

And then there are the “splits,” which try to slow down the pace of the curriculum by slicing the courses in half. Students having trouble with our one-semester elementary algebra might be permitted to take half of the course during fall semester and the second half during spring. You can spot these courses in college catalogs where they bear labels like “Algebra 1A” and “Algebra 1B.” Many schools have even done this with arithmetic. You can struggle during the fall to learn your times table and save fractions till spring.

I wish I were kidding, but I exaggerate only slightly.

Guess what? The students who enroll in the splits aren't particularly more successful than those enrolled in regular lecture classes, on-line classes, or math labs. Are we rescuing a few additional students with each new approach, or would they do as well (or as badly) if we just ushered them all into a classroom and made them sit in rows?

I believe we do achieve some marginal additional success with the multiple formats because students do learn in different ways and one-size-fits-all is almost never true. Still, I wish the benefits were more than marginal.

This reflection on student success and failure in developmental math was stimulated by a recent post by a pseudonymous community college dean. “Dean Dad” traveled to California last month to attend the San Diego meeting of the League for Innovation in the Community College. He was particularly struck by the remarks of a Bay Area faculty member:

Prof. Myra Snell, from Los Medanos, coined a wonderful word: “stupiphany.” She defined it as that sudden realization that you were an idiot for not knowing something before. The major “stupiphany” she offered was the realization that the primary driver of student attrition in math sequences isn’t any one class; it’s the length of the sequence. Each additional class provides a new exit point; if you want to reduce the number who leave, you need to reduce the number of exit points. If you assume three levels of remediation (fairly standard) and one college-level math class, and you assume a seventy percent pass rate at each level (which would be superhuman for the first level of developmental, but never mind that), then about 24 percent will eventually make it through the first college-level class. Reduce the sequence by one course, and 34 percent will. Accordingly, she’s working on “just in time” remediation in the context of a college-level course. There is definitely something to this.

Um. Under the given assumptions, I can't fault the math (0.70⁴ = 0.2401 and 0.70³ = 0.343), but it is just a tiny bit simplistic. If we squeeze all the remediation into one course, then we'll be rewarded with a 49% overall success rate at the end of the college-level course. Yay!

Except that it certainly wouldn't work.

This is a classic optimization problem—the kind that you see in calculus. Two countervailing factors have to be balanced in order to achieve the best possible outcome. For example, if you want to enclose the maximum possible rectangular area with a given length of fence, you have to balance the contributions of length and width, because one can be increased only at the expense of the other—yet both contribute equally to area. (Thus the ideal figure turns out to be a square. Big surprise!)

In the case of developmental math classes, the splits offer more failure opportunities. On the other hand, they reduce the curriculum to bite-size chunks that more students might be able to master. The more you cram into a course, the more likely the students are to be overwhelmed. The trade-offs are rather obvious.

(Frankly, I prefer that split classes be taught at the same pace as regular classes, because stretching them out to semester length attenuates the reinforcement that most students need. At the halfway point the successful student moves on to the second-half split while the unsuccessful student repeats the first-half class without having to wait till the next term.)

I don't think that Prof. Snell's “stupiphany” is quite as significant as suggested by Dean Dad, although I presume her presentation would be more nuanced at greater length than it is in a one-paragraph summary. (She did, apparently, couch her presentation in terms of timely intervention.) The tension between length of sequence and course content will continue. The experiments will certainly continue. In fact, I can even tell you the direction in which they will go. The splitters having had their day, the lumpers anticipated Snell's observation and are putting accelerated curriculum into place. Courses are being designed and curriculum is being implemented. Hang on to your hat as developmental math tries to speed up.

That's probably a future post.

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Identically different

You're not alone

Every semester begins in the same way. Students show up, a bit disoriented, and try to figure our their instructors and their classes. We instructors usually have the advantage of more experience (although some of our students have been here a long time), but we're a little disoriented, too. What will the new crop of students be like?

One thing is all but certain. Some of them will fail.

I know. That sounds like a defeatist attitude. Is the fact that it's true a defense? There are solid reasons why a college instructor cannot realistically expect to shepherd an entire class of forty (or more!) to passing grades at term's end. (Please take it as read that the instructor has actual standards and does not simply hand out C's for “trying.”) I'll enumerate some of them:

Circumstances

In a class of any size, you're going to have students who run afoul of emergencies, whether anticipated or unforeseen. I've had students distracted by health issues (all the way up to and including life-threatening physical conditions or debilitating emotional difficulties), family problems (divorce, custody disputes, offspring with behavioral issues), and legal matters (such as probation violations, lawsuits, restraining orders, evictions, and incarcerations). While some of these circumstances could be mitigated by high-functioning and responsible individuals, many would overwhelm any mere mortal. Severe illness, in particular, is not something easily managed. No one blames a student for not doing well in a class if he or she is simultaneously struggling with a debilitating illness.

Laziness

The lazy student exists. I seem to have a few every semester. They're apparently not quite sure why they're in school, but perhaps it was the path of least resistance. They like to sit in the back row and drowse—or play surreptitiously with their electronic toys. Each semester I fight the temptation to label students as indolent too quickly—their characteristics are often so overt—and instead give a good college try to getting them involved and learning. If they don't snap out of it, they're doomed. But they mostly don't care. At least, not enough.

Uniqueness

Perhaps this one is new to you, but it's a commonplace to me. My struggling students frequently suffer from singular situations—or so they think. No one has ever suffered as they do! It finally occurred to them that I should try to disabuse them of this notion.

Sure, absolutely everyone is unique. Even identical twins (DNA isn't everything). But people are unique in their assemblage of traits and experiences, not in their components. The various traits and experiences, when viewed individually, are part of the common legacy of humanity. In other words, you have more company than you realize.

The poet Hugo von Hofmannsthal said it in a way that impressed me back when I was in graduate school. The original German is not at my fingertips (as if it ever was), but the English sense of it is this:

No matter how embarrassing or isolated your seeming situation, you nonetheless have thousands of companions of whom you are unaware.

Quite right. And thus I try to get my students to understand that they are neither the first nor the last to have a problem with mathematics. Literally millions of other students have had problems with algebra, for example. No professor during office hours or tutor during drop-in assistance periods in the help center is going to recoil at a student's question and say, “Oh, my God! I've never heard that question before! No one has ever had this problem before!”

Been there. Done that. Students and teachers and tutors alike. (Okay, a few of the newer teachers or tutors might have that reaction, but they'll get over it pretty quick.)

You are unique yet the same. No one else has quite your special combination of characteristics, but every part of you is shared with others. Don't fall into the trap of thinking, “No one has ever been this confused before. No one has ever made such mistakes before. No one has ever been this bad at math.”

Plenty have, and they have done so in ways that are both different and the same as your missteps and failings. Many of them have found assistance and solutions that are also as different and as identical as the ones available to you. Go find them and swell the ranks of the successful.

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Reasonably accommodating

I said “reasonably”!

Sometimes my students need a little extra assistance. I understand. In the past I have printed out tests in a large font for a student with poor vision. I have set up my classroom to make space for the sign-language interpreter for a deaf student. I have used the testing center to allow time-and-a-half on exams for students with diagnosed learning disabilities. That's what our Student Assistance Program is for. It helps students succeed where they might otherwise fail. We call it “reasonable accommodation.”

Unfortunately, some of our accommodated students appear not to understand how it's supposed to work. Occasionally we get someone who decides that “accommodation” means “whatever I want”:

“Dr. Z, I can take the exam next week.”

“But the exam is this week.”

“Yeah, but I need more time.”

“Yes, you get more time to take the exam, but you still have to take the exam when your classmates do. There'll be a copy of the exam in the testing center for you.”

“But I'm not ready!”

Math isn't the only thing the student has difficulty understanding.

But students like that are rare. Most students with learning disabilities leap at the opportunity to succeed and dutifully jump through the hoops my college requires in order to take advantage of its special student support services. Naturally enough, the learning disabilities related to math tend to manifest themselves most dramatically among the students taking the low-level developmental courses like arithmetic and prealgebra. Students taking higher-level courses either do not suffer from dyslexia or dyscalculia or have learned to control the problem, thereby leading to success in math.

Not too long ago, though, I ran into a striking exception to this general rule. It was in a multivariate calculus class. We were in the final weeks of the semester, with only one chapter left to cover in the textbook. (Line integrals, anyone?) One of my students came up to me after class and handed me a note. It was a memo from his private counselor, who was offering me some advice about why my student was struggling to maintain a C in the class. The counselor was a clinical psychologist who had seen my student twice. He had some specific observations and recommendations:

Based on my interviews and initial assessments, it is my opinion that Mr. X has above average intellectual capacity, but suffers from being overwhelmed with too much information after about 20 minutes. Therefore I suggest that Mr. X be granted at least two preliminary accommodations. First, he should be allowed to take frequent breaks. Second, he should be allowed additional time to complete timed assignments in class, especially exams. I would suggest he be given twice as long to complete such tasks.

A double-time accommodation on exams is quite unusual but not unheard of. The notion of frequent time-outs, however, is rather more daunting. Exactly how, pray tell, is this supposed to work? While we may shift gears multiple times during a class period as we solve problems, work quizzes, review homework, and present new material, but class time is at a premium and we can't take a break every twenty minutes. It doesn't work.

And double-time on in-class quizzes isn't particularly feasible either. I use them as highly focused evaluation and teaching tools. The students' results tell me, of course, how well they're keeping up. And my immediate presentation of the solution on the board takes advantage of their momentarily intense receptivity. The students who got it right preen a bit as they see my solution matching theirs. The students who got it wrong watch wide-eyed and often have “Aha!” moments (“Oh, is that all I had to do?”). Teachable moments.

But not if Mr. X had to be sent from the room to accommodate his extra ration of time on the quiz. By the time he came back he would have missed my presentation of the solution and missed the learning opportunity. (And even if I had sent an advance copy of the quiz to the testing center so that he could have his double-time before class began, the logistics were impossible. My class was an early morning class and the testing center wasn't even open until after my class began.)

On top of everything else, Mr. X was trying to make an end-run around the counselors and staff of our testing center, the people who evaluate students and make recommendations for accommodation. For fairly obvious reasons, faculty members don't welcome external evaluations by private counselors. We don't know the people who make them. We do know, however, that any desired opinion is available on the outside. (Court trials, after all, never seem to lack for experts on both sides of any given issue.) I told Mr. X to take his evaluation documents to the testing center for review by college personnel. He was not happy about that and said, correctly, that it was probably too late for the testing center to evaluate him before the semester ran out.

In most respects, though, Mr. X was lucky. He was pulling a solid C in my class and I was able to show him that he was in little danger of failing the class. He squeaked through with a modest margin to spare. What he wanted, of course, was a B, but I wasn't quite that accommodating.

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Words versus numbers

My students tell me my job

“You can't do that!”

My student's emotions were an admixture of horror and disbelief.

“No, really! You can't do that! This is a math class!”

Oh, really? I guess I had lost track of that. I inquired as to the basis for the student's convictions.

“Why do you think I can't give you an essay question to answer?“

The student goggled in disbelief at my question.

“Because that's what we do in English class, not in math!”

Another student chimed in.

“Yeah. Math is about numbers and calculations—not about words!”

I've had this conversation a few times now, mostly in intermediate algebra and precalculus. It tends to occur when I hand out a quiz or exam with the following kind of problem:

Rewrite the equation x² + y² + 4x − 6y + 9 = 0 in standard form and graph your results. Describe your graph in words.

It's a perfectly ordinary problem that occurs after the students have learned about the basic conic sections and the technique of completing the square. Upon rewriting the equation as

(x + 2)² + (y − 3)² = 4,

most of my students (if they've gotten this far) easily recognize that they have the equation of a circle with center (−2, 3) and radius 2. They quickly sketch the circle and then stare in hopeless confusion at the prompt, “Describe your graph in words.”

I've tried amplifying the prompt in an attempt to make it less intimidating:

“Think about how you would describe your graph over your phone to a friend so that your friend could graph it without having seen it.”

(These days I have to add the warning that it's no fair to just send the friend a quickly snapped image of the graph.)

Lots of students leave that part of the problem blank and move on. Others tentatively write “circle” (miffed that I didn't just ask for the name of the conic section and anxious that I used the plural “words”) and nervously move on.

And then there's the handful of students who write dissertations like this:

Subtract the 9 from both sides to isolate the variables. Look at the coefficient of the x term, which is 4, take half of it and square it. Add that to both sides. Change x² + 4x + 4 into (x + 2)². Now look at the y term...

Wow. A complete procedural guide to deriving the answer (though seldom as coherent as the mocked-up example above). Where did I ever ask for that? (I'm sure they get a prickly feeling that something must be wrong when they overflow the tiny space I allowed for their answer and they continue their discourse on the back of the page.)

Why do so few of them offer the brief and straightforward response that “The graph is a circle of radius 2 with its center at (−2, 3)”? Wouldn't that suffice to fully inform their imaginary friend at the other end of the phone conversation?

Hardly anyone is pleased when I unveil the answer. The typical reaction is exasperation:

“Is that all you wanted? Why didn't you say so?”

I thought I did.

“You're just confusing us. This isn't English comp!”

My students are like fussy eaters who get upset if their corn touches their mashed potatoes. Food should reside in carefully demarcated regions and college curriculum should reside in strictly disjoint sets. (They're not like my kid brother, who regarded his dinner plate as an artist regards the palette whereon he mixes his colors.)

Eventually, however, I break down my students' reservations and most of them start scooping up the relatively easy points I assign for complete one-sentence answers to simple prompts. By the end of the semester they are rather less startled by questions that require a written response. They still don't, for the most part, like them, but they can do them.

Then the school term ends and I have to start all over again with a new batch. And I know what words will be coming out of their mouths.

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A new kind of student?

Constant negative slope

“I didn't understand that.”

“What part of it didn't you understand?”

“All of it.”

We were in an algebra class. We were solving a simple linear equation. A simple linear equation. Integer coefficients. Integer solution. Stuff at the prealgebra level of difficulty. Piece of cake. But not for everyone.

“Okay. How about the instructions at the beginning? Do you understand what we're trying to do?”

“No.”

“We're trying to solve for x. What does that mean?”

“I don't know.”

She was matter-of-fact about it. I didn't get any sense that she was being deliberately or provocatively obtuse. She was a serene icon of incomprehension, exhibiting none of the stress or anguish that usually accompanies such stark confessions of ignorance.

“It means we want to find out what x is. It means we want x all by itself on one side of the equation and a number on the other side of the equation. We want to end up with x equal to a number.”

“Oh. Well, I didn't get that.”

The equation was so simple that it could have been solved with the techniques taught at the end of prealgebra (the prerequisite the student had supposedly satisfied in order to enroll in algebra). This particular student, however, acted as if she had never seen any of the techniques or had had an extremely successful brain purge since her last class.

Students do forget, of course, but we hope that they recognize and relearn things as we review them and progress to new topics. My algebra student instead remained at a complete loss. What's more, unlike students of the past, she was not willing to suffer in silence. In a way, I guess, this is good. When you need help you should ask for it.

My joy in her recognition of her need for help was, however, not unalloyed. My joy was incomplete because we were more than four weeks into the semester and she had not once bothered to darken my office door during office hours. She had never visited the class's assigned tutor. And I had had trouble learning her name because she was often missing from class.

Yet she was expecting me to abandon my planned progression through the day's topic in order to back-fill the profound abyss where her prerequisites were supposed to be.

“It's a sense of entitlement,” said one of my colleagues. She shrugged as she told me this. “We are now viewed as part of the service-sector economy. If they don't know something, we must spoon-feed it to them on demand.”

I was afraid that my colleague was right. My student had been completely unabashed while announcing her total ignorance to the entire class and then waiting calmly for me to do something about it. I'm a teacher and she's a student. That makes her my client and I must service her needs. In the meantime, she demonstrated no interest in lifting a finger.

“But students used to keep quiet when they were that lost, didn't they? In fact, they'd try to hide it and then either slip away to our tutorial services or drop the class. I'm not used to such overt pronouncements of ignorance or prerequisite amnesia—whichever it is. This is new and depressing behavior to me.”

“Don't worry, Zee,” said my colleague, junior to me in years but advanced in her blithe wisdom and patience. (Her years teaching high school probably helped.) “These things resolve themselves. You'll see.”

She was right. After skipping the following week of class sessions and meetings with the tutor, my student skipped the next exam. She was still on the class roster, so I dropped her. Naturally she couldn't be bothered to drop herself.

Problem solved. At least till next semester, when she wanders into another class having had even more time to forget what little she ever knew. I couldn't connect with her at all. Will her next instructor manage to rescue her? I don't see how.

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Small steps, big goals

Big talk, small deeds

Despite the variations, every semester strikes the same themes: how do you learn? how do you study? how do you succeed? how do you pass?

And there's always a few students who want the secret—the magic short-cut that will put them on the royal road to learning math easily. Apparently they think we are hiding some sort of trade secret that enabled us to become math teachers. We must guard it jealously lest we lose our strangle-hold on algebra!

While I disdain one-size-fits-all solutions to students' problems, there are some fairly uniform basic principles. For example, success is usually proportional to effort. Different students must put in different amounts of effort to achieve equivalent success, but the basic principle still holds. Thus I tell my students they have to gauge for themselves how much effort (which translates significantly into allocation of time) they must put in.

In an attempt to encapsulate the idea in a simple bite-size aphorism, I have told my classes, “Do a little every day.”

They blink at this, of course. It sounds deceptive.

“Only a little?” they ask plaintively.

“Gosh, no,” I tell them. “At least a little.”

It's a good lesson, and one that I learned later than I should have. During my last encounter with grad school, trying to teach a full-time load at my college while carrying a full-time academic load as a university student, there wasn't a lot of slack in my schedule. Determined not to let things slide (at least, not too far), I solemnly resolved to do a little work on my dissertation every day. Every day. No exceptions. Without fail.

I didn't slice my hand with a subtle knife and swear a blood oath, but I did the next best thing.

I told my grad school classmates.

It became an enjoyable game that anyone could play.

“So, Zee, what did you do last night?”

Occasionally I was reduced to paraphrasing Oscar Wilde:

“I was working on the methodology section of my dissertation all day, and took out a comma. In the evening I put it back again.”

And, sadly enough, that was often too close to the truth. Other times, however, an interesting thing happened. I'd be exhausted and ready for bed and then I'd realize I still had to discharge my solemn obligation to tinker on the dissertation. At least a little. Resolving to postpone sleep for just a few minutes, I'd sit at the computer and fire up the word processor. A paragraph. Let's just get one more paragraph in there.

And then, magically, I'd awaken from a trance to discover that half a dozen pages had been written in a two-hour altered state. (Perhaps elves had slipped in and moved my fingers over the keyboard.) Not that this occurred every time, of course, but still often enough to provide the quantum jumps that really advanced my progress toward graduation.

As with any project, completion won't occur without participation. You have to do something.

And that's what I tell my students.

“So choose whatever works best for you. It might be at eight o'clock each evening. It might be when you get home from your morning classes. Whatever. Just plan to sit down with your book and read the latest section or solve the first three, four, or five homework exercises. Pick something and do it. Maybe you'll promise yourself to do at least fifteen minutes of work at that same time every day. Yeah, that's not enough. Fifteen minutes. Maybe if you're a math genius that would be enough. But that's not the point. The point is that you commit yourself to working on it every day. And some days you'll find yourself digging in and getting a lot more done than you might have thought you would—maybe even a couple of hours. Now you're talking! Build a pattern of working every day and keep it going.”

When it's early in the semester, students tend to be in a hopeful mood and try to nod their heads agreeably when their instructor says something. However, the last time I gave an algebra class a harangue similar to the above, one of my students seized on the “math genius” comment.

“But what if you are a math genius?”

The speaker, a lanky kid sitting near the back of the room, was instantly the center of attention.

“If you're a math genius,” I replied, “then you should definitely talk to your math teacher. I know very few and I welcome the opportunity to meet another. Are you indeed a math genius?”

I neglected to point out that the density of math geniuses in college-level introductory algebra classes is extremely close to zero, but no doubt a math genius would have understood what I meant by that.

He nodded his head.

“Yeah, I am,” he modestly admitted. “This stuff is very easy. Very easy. No one should complain about having to take algebra because it is, like, totally easy to do.”

His classmates regarded him with keen curiosity. So did his instructor, for that matter.

“I'm glad to hear that you think so,” I said. “I'm inclined to agree with you, but most of your classmates probably have differing opinions. May I ask how you arrived at your conclusion that algebra is super easy?”

He nodded his head again.

“Yeah, sure. I took it during the summer at the extension center. Like I said, totally easy!”

Everyone paused for a long moment, considering what he had said.

“That raises another question,” I pointed out. “Why are you enrolled in this class if you've already taken algebra during summer school and found it totally easy?”

He regarded me with a trace of exasperation.

“I said it was easy, Dr. Z. I didn't say I passed!”

Apparently it was so easy that it failed to capture his attention.

“Oh, good point,” I said agreeably.

I turned my attention back to the rest of the class.

“Okay. Are we clear, everybody? Even math geniuses can occasionally have trouble with algebra, so commit yourself to working on it—at least a little!—every day of the semester. Let's see how successful you can be.”

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The questions answer themselves

Why can't I quit you? (You can!)

The student's first message at the beginning of the term was fraught with portents of doom. He had sent me a response to my initial assignment, which was to send me an self-introductory e-mail:

hello Mr Z this is Angus from your calc1 class.. I was the last one to leave your class this morning. Iam a social science/economics major at state u, and the reason i want to take calculus is, i really have an interest in math, even though iam kind of weak at it.

Calculus is not a course for the faint of heart or the weak of math. The message filled me with trepidation for the student's sake. Furthermore, he was enrolling in our heavy-duty calculus sequence for scientists, mathematicians, and engineers. Most econ majors are tracked into our social science calculus class. Perhaps Angus wanted to keep his options open, but that assumed he could surmount the challenge of grown-up calculus.

He faded gradually throughout the term. Occasionally he would seem to catch fire for a couple of days, but then he would fizzle out again. Just before the drop deadline, he came up after class and told me he had to bail. I commiserated, but agreed that he was probably making the right decision, both for himself and for the other courses he was trying to pass. Then he asked me one of those questions:

“Uh, do you mind if I kept coming to class?”

California community colleges have a strict rule against auditors. It has something to do with the fact that we are funded (when we are funded, that is) on average daily attendance—and ADA is accumulated only for enrolled students. Angus was clearly asking to do something that was not permitted.

“Sure,” I said. “It won't be as if I don't have room for you.”

He thanked me earnestly and went away—never to return.

I'm quite certain that he was sincere in his plans to sit in on the remainder of the class in hopes of giving himself an edge when repeating it during the next term. In reality, however, he quickly (instantly!) discovered that he couldn't force himself to roll out of bed in time to attend a morning class in which he no longer had a vested interest. Despite his teacher's willingness to allow him to flout the school's sacred rules, he never took advantage of it.

And to think I could have painfully explained the rules to him and turned down his request. He could have ended up nursing hurt feelings. This way, no harm done.

Perhaps you're thinking, “Oh, there goes bleeding-heart Dr. Z, running roughshod over the school regs with reckless abandon because of his tender feelings for the downtrodden.” Well, I do have tender feelings for those of my students who are downtrodden, but I answered Angus with the voice of experience. No student attends class more than once or twice after dropping. It just doesn't happen.

No need to bar the door when no one is trying to come in.

Jumping the gun

Another student approached me from a different direction. In his case, the semester had yet to begin. To give my potential students fair warning, I had e-mailed everyone the syllabus two weeks before the first day of class. (That usually creates some quick shuffling as a few decide that my approach is too rich for their blood.) A couple of students wrote me to express their thanks for the advance copy of the syllabus. A few other students send messages seeking clarification about the edition of the textbook we were using. (One asked if he could use a different author's text that he happened to already have. Sorry, buddy, that hardly ever works. You should have passed the course you purchased it for.)

The most interesting message, however, was the following:

Dr. Ferox;

Thank you for the Syllabus. I have already been working problems on Sec. 1. I have encountered a few questions. May I e-mail you my questions.

Sincerely
Rory

The answer is obvious, right?

“Dear Rory: I suggest you wait until after I try to teach you the material, okay? The semester hasn't even started yet. Your instructor is not in a position to provide individual tutoring to all forty of the students in the class. Sorry!”

And then I could embed a winking smiley face.

But that's not what I said. Nope. I send Rory this message instead:

You are welcome to contact me at any time, Rory, although my availability may be limited until after the semester actually begins.

-Z-

Once again, my reasoning is simple. No, there is no way I could find the time to provide individual hand-holding service for all of the students in my classes. Realistically, however, how many are going to be forging ahead on their own? In my experience, the number can reliably be expected to be less than two. In fact, it's usually less than one.

Rory did actually follow through with one homework question before the semester began. I answered promptly, taking only a few minutes. If Rory goes on to be a math whiz in his transfer university, I trust he will remember me kindly. One should always avoid discouraging the eager beavers. If they stretch a little too far, they'll regroup soon enough and fall into step with their classmates. If, however, they can maintain a racer's pace, then I want to give them free rein.

If you tell your students “yes,” the “noes” are likely to take care of themselves. You don't have to say them.

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What a deal!

An old dog learns an old trick

When I was merely a teaching assistant in graduate school, the university provided me with a paid homework grader for my calculus students. I collected homework every class day and had it to return within a day or two. Nice. Today, however, as a full-time college professor, I can only dream of such luxury.

Still, I think homework is important and that students need the practice that homework provides. I therefore encourage my students to do their homework by making it count toward their grades, even though I collect it only on exam days. I don't actually correct it. I just scan it for approximate completeness and dole out some points. Most of the students who hand it in do a decent job and get full credit:

10/10

These students are happy when they hand in their exams and pick up their high-scoring homework. (At least one part of the day has gone well.) Other students come up to me with tales of woe:

“I did my homework, but I forgot it at home.”

“I left my homework in my car.”

“My friend borrowed it and didn't give it back.”

I tell them all the same thing: Bring the homework to the next exam day and receive late credit for it. Late credit means half credit. Students did not generally seem to appreciate my generosity when I scored their homework:

Late credit: 5/10

“But I did the whole thing, Dr. Z!”

“Yes, but it was late. You should have handed it in on time.”

“That's not fair!”

“The rule applies to everyone. That is the epitome of fairness.”

“The what?!”

After more than three decades of teaching, I finally remembered a simple lesson from the retail sector. Most of us have undoubtedly heard a story about the impulse-purchase items near the grocery store checkstand. The grocer is trying to sell something—ball-point pens, or candy bars, or whatever—for 25¢, but they seem to be nailed to the countertop. No one is going for them. Then the grocer has a brainstorm. He marks them with a sign: 3 for $1. Now they fly off the shelf.

It's not the price. It's the perceived bargain.

I pondered.

The next time I scored late homework, I changed my tack:

Late credit: +5

That's right. The same number of points, but no denominator to remind anyone that it's half credit. I also made the plus sign nice and big. Students were pleased.

“Cool. Thanks, Dr. Z!”

“All right!”

“Whoa! Five more points! Thanks, Z-man!”

The Z-man says you're welcome.

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