Are College Professors Responsible for Student Learning?

aliceI learn a heck of a lot from Andrew Gelman’s blog–not only his own posts, but the many interesting and substantial comments. It’s one of my favorite places on the internet right now (granted, I have low tolerance for “surfing” and tend to focus on a few sites). That said, I find myself questioning some of his arguments and views, particularly about measurement in education. Now, I am not about to say “learning can’t be measured” or “tests are unfair” or anything like that. My points are a bit different.

In an article for Chance, vol, 25 (2012), Gelman and Eric Loken observe that, as statisticians, they give out advice that they themselves do not apply to their classrooms; this contradiction, in their view, has ethical consequences:

Medicine is important, but so is education. To the extent that we believe the general advice we give to researchers, the unsystematic nature of our educational efforts indicates a serious ethical lapse on our part, and we can hardly claim ignorance as a defense. Conversely, if we don’t really believe all that stuff about sampling, experimentation, and measurement—it’s just wisdom we offer to others—then we’re nothing but cheeseburger-snarfing diet gurus who are unethical in charging for advice we wouldn’t ourselves follow.

They acknowledge the messiness and complexity of education but maintain, all the same, that they could improve their practice by measuring student learning more systematically and adjusting their instruction accordingly. “Even if variation is high enough and sample sizes low enough that not much could be concluded,” they write, “we suspect that the very acts of measurement, sampling, and experimentation would ultimately be a time-efficient way of improving our classes.”

I agree with the spirit of their argument; yes, it makes sense to practice what you proclaim, especially when this can improve your teaching. Of course assessment and instruction should inform and strengthen each other.  Still, any measurement must come with adequate doubt and qualification. I think they would agree with this; I don’t know, though, whether we would raise the same doubts. I see reason to consider the following (at the college level, which differs substantially from K-12):

While still moving toward independence, students are more in charge of their own learning than before. Ideally they should start figuring out the material for themselves. What is the class for, then? To introduce topics, organize the subject matter, illuminate certain points, and work through problems … but perhaps not to “produce” learning gains, at least not primarily. On the other hand, the course should have adequate challenge for those at the top and support for those at the bottom (within reason). Introductory courses may include additional supports.

Also, a student might deliberately choose a course that’s too difficult at the outset (but still feasible). Some people thrive on difficulty and are willing to let their grade drop a little for the sake of it. The learning gains may not show right away, but this does not mean that the teacher should necessarily adjust instruction. If the student puts in the necessary work and thought, he or she will show improvement in good time. Students should not be discouraged from the kind of challenge that temporarily slows their external progress.

In addition, there are inevitable mismatches, at the college level, between instruction and assessment. (This may be especially true of the humanities.) If you are teaching a literature, history, or philosophy class, your students will probably write essays for a grade, but your teaching will address only certain components of the writing. Students have to learn the rest through practice. Thus you will grade things that you haven’t explicitly taught. (Your course may not deal explicitly with grammar, but if a paper is full of ungrammatical and incoherent sentences, you still can’t give it an A.) This may seem unfair–but over time, through extensive practice and reading, students will come to write strong essays.

Since September 2015 I have been taking classes part-time, as a non-matriculated student, at the H. L. Miller Cantorial School at JTS. In my first class, I was far below the levels of my classmates. That was what I wanted. I studied on the train, in my spare moments, and at night. (I was teaching as well.) I flubbed the final presentation, relatively speaking, not because I was underprepared, but because I prepared in the wrong way. I ended up with a B+ in the course. The next semester, my Hebrew had risen to a new level; the course (on the Psalms) enthralled me, and I did well. This year, I have been holding my own in the course I longed to take all along: a year-long course in advanced cantillation. If the professors had worried too closely about my learning gains, I wouldn’t have learned as much.

On the other hand, in the best classes I have taken over the years, the professors did great things for my learning. I wouldn’t have learned nearly as much, or gained the same insights, without the courses.  The paradox is this: to help me understand, the professors also let me not understand. To help me progress, they sometimes took me to the steepest steps–and then pointed out all the interesting engravings in them. It wasn’t just fascination that took me from step to step–I had to work hard–but they trusted that I could do it and left it largely in my hands.

Granted, not all students are alike, nor are all courses. In an introductory course, students may be testing out the field. If they are completely lost, or if the course takes extraordinary effort and time, they may conclude that it’s not for them. A professor may need to respond diligently to their needs. There are many ways of looking at a course; one should work to become alert to its different angles.

In short, college should be where students learn how to teach themselves and how to gain insights from a professor. While helping students learn, one can also hope, over time, to simulate Virgil’s last words to Dante in Purgatorio, “I crown and miter you over yourself” (or to accompany them to the point where, like Alice, they find a crown atop their heads.)

Image: Sir John Tenniel, illustration for the eighth chapter of Lewis Carroll’s Through the Looking Glass (1865).

Note: I revised the fourth paragraph for clarity and made a minor edit to the last sentence.

Daydreams, Lectures, and Helices

What do daydreams, lectures, and helices have to do with each other? Quite a bit.

One of my favorite parts of Dante’s Purgatorio is at the end of Canto XVIII, when Dante starts dozing off. Here is Allen Mandelbaum’s translation of those lines:

aaaThen, when those shades were so far off from us
that seeing them became impossible,
a new thought rose inside of me and, from
aaathat thought, still others–many and diverse–
were born: I was so drawn from random thought
to thought, that, wandering in mind, I shut
aaamy eyes, transforming thought on thought to dream.

I read this as a tribute to daydreaming (though Dante is on the verge of sleep and a nightmare). To be “so drawn from random thought / to thought” (in the original: “e tanto d’uno in altro vaneggiai”) is to have an expanse and few restrictions; I love this kind of expanse, though of course I can’t have it all the time.

As I have said elsewhere, that is one thing I enjoy about lectures: they not only take my mind to unexpected places, but they send it wandering off to the side and back, or backwards and forwards. While listening to a lecture, I may do with my mind what I please; if the lecture is good, then my mind dances with it, sometimes spinning away, sometimes drawing close. If the lecture is bad (or dreadfully dull, as lectures sometimes can be), then my mind can go off on its own. This, too, has its benefits.

Lecture or no lecture, I need time to let my mind go where it wishes. A few days ago I took out a textbook of three-dimensional calculus and started reading the chapter on vectors. The vector equation for a helix immediately made sense:

helixr(t) = cos t i + sin t j + t k

where i = , j = , and k = . (These are unit vectors along the x-, y-, and z-axes, respectively.)

If you omit the z-axis, you can see that you have the vector equation for circular counterclockwise motion:

r(t) = cos t i + sin t j

Adding the component t k turns the circle into an upward spiral.

I toyed with this in my mind for a while. The next day, I encountered a helix again, when reading Taking the Back off the Watch: A Personal Memoir by the astrophysicist Thomas Gold (1920–2004). Before the helix passage, there was a wonderful comment on the possibilities for thought during a dull lecture:

A dull lecture is like an experiment in sensory deprivation. You are sitting in your seat, you can’t leave the room because that would be too rude, you are carefully shutting out the incoming information because you have decided you don’t want to hear it, and your mind is now completely free from external disturbances. It was during this lecture that I suddenly saw how all the facts of the case would fall together.

Yes, during this dull lecture he figured out why a sound entering the cochlea produces a “microphonic potential”–an electric potential that both amplifies the sound and mimics its waveform. He took his theory to Richard Pumphrey, with whom he had been investigating this matter; they published their papers in 1948. But that’s an aside here (though interesting in itself). I bring this up because his words about the lecture rang true, so to speak, in my mind. Then, a few pages later, I came upon his description of an experiment with a helix and an eel.

The eel can move forward along a sinusoidal curve, both horizontally and vertically. Thomas Gold and the zoologist Sir James Gray found that it could move swiftly and easily through a sinusoidal tube. Sir James Gray posited that the eel could therefore move through a helical tube; a helix, after all, is the addition of the vertical sinusoid to the horizontal sinusoid in three-dimensional space. Thomas Gold disagreed; he was convinced that the eel could not move through the helical tube. He was right.

Very well. But I was momentarily intrigued with the problem that would be elementary to mathematicians: is the vector equation

r(t) = cos t i + sin t j + t k

equivalent to the addition of two traveling sinusoidal waves, one horizontal, one vertical, in three-dimensional space? I grasped that it was but spent a little time explaining it to myself. Yes, and the two sinusoids must be a quarter-cycle out of phase with each other.

The first traveling sinusoidal wave has the equation r(t) = cos t i + t/2 k.

The second traveling sinusoidal wave has the equation r(t) = sin t j + t/2 k.

So, unless I’m missing something, these sinusoids are twice as scrunched as the resultant helix, their sum.

These have been my daydreams, or a fraction of them, over the past week or so. There were no lectures involved, but there were memories of lectures and the liberty I found in them.

Note: I corrected one term and made a minor revision after the initial posting.

Squaring the Circle

Since the summer, I have been obsessed with the problem of squaring the circle—that is, finding the square whose area is equal to that of a given circle, with no tools other than a straight edge and compass.

I took interest in the problem when reading the ending of Dante’s Paradiso (in Allen Mandelbaum’s translation):

 

 

As the geometer intently seeks
to square the circle, but he cannot reach,
through thought on thought, the principle he needs,

so I searched that strange sight; I wished to see
the way in which our human effigy
suited the circle and found place in it—

and my own wings were far too weak for that.
But then my mind was struck by light that flashed
and, with this light, received what it had asked.

Here force failed my high fantasy; but my
desire and will were moved already—like
a wheel revolving uniformly—by

the Love that moves the sun and the other stars.

I, too, had difficulty reaching the principle—and it was precisely the principle that I needed to find. The difficulty lies in determining what the problem exactly is.

No one has found a way to square the circle with compass and straight edge alone; in fact, I believe it has been proved that it can’t be done. Yet I kept on trying, thinking that I would learn something from the attempt.

Yesterday morning, something close to a solution came to me—not a solution, mind you, but something that points in the direction of one. It’s probably old hat—or ancient hat—but it’s interesting, all the same.

We will consider the circle with radius r=1, since expanding or shrinking it (and the square) is then a trivial matter.

We know that if the circle has radius 1, its area is π. Thus, for a square to have that area, it must have sides of length √π. But how do you find length √π? There’s the puzzle.

When the square and circle have the same area, the ratio of the circle’s circumference to the square’s perimeter is a constant, 2π divided by 4√π ; that is, π/2 divided by √π; that is, √π/2.

In other words, the ratio of one-quarter of the circle’s circumference t0 one side of the square will always be √π/2 when the square and circle have the same area.

You could consider the ratio in terms of this figure:

|AO| is equal in length to the side of the square  (√π when the circle has radius 1); |OP| is one-quarter of the circle’s circumference (1/4 *  2π =  π/2), which you can measure with a string. (I know you’re only supposed to use a straight edge and compass, but this exception will prove helpful.)

Now, the ratio of |OP| to |AO| is equal to the ratio of|BC| to |AB|, where |BC| = √π = |AO|, and |AB| = 2. We already established this ratio a few paragraphs up.

(Again, the ratio of one-quarter of the circle’s circumference to the square’s side must be √π/2.)

So now the challenge is to tweak the figure until |AO|=|BC| (while keeping points A, P, and C on a single line and A, O, and B on a single line). When we get the two segments to equal length (without changing |OP| or |AB|), we have brought both |AO| and |BC| to √π. Now, you must do this by trial and error, by shifting the OP segment. If |BC| is greater than |AO|, then move OP closer to BC; if it is smaller than |AO|, then move OP closer to A. Your adjustments will be smaller and smaller until you either make |AO| and |BC| equal or come as close as you wish. The next step would be to find a mathematical way of doing this.

Once you have made |AO| and |BC| as close to equal as you wish, you then make AO the side of your square, which you superimpose on a circle of radius 1 (so that both have a common center). You will see that the square is mostly, but not entirely, contained within the circle. It will look like the first of the two figures above.

Although this is not a satisfactory solution, it results in a close approximation and seems to be on the right track.

Update (on a tangent, so to speak): The angle in the figure is the inverse tangent of √π/2, or 41.54822621257918513… Its common logarithm (1.6185524875…) is tantalizingly close to the golden ratio, but not close enough for the trumpets to sound.

Note: This is a reposting of a piece that originally appeared on October 19.

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    Diana Senechal is the author of Republic of Noise: The Loss of Solitude in Schools and Culture and the 2011 winner of the Hiett Prize in the Humanities, awarded by the Dallas Institute of Humanities and Culture. Her second book, Mind over Memes: Passive Listening, Toxic Talk, and Other Modern Language Follies, was published by Rowman & Littlefield in October 2018. In February 2022, Deep Vellum will publish her translation of Gyula Jenei's 2018 poetry collection Mindig Más.

    Since November 2017, she has been teaching English, American civilization, and British civilization at the Varga Katalin Gimnázium in Szolnok, Hungary. From 2011 to 2016, she helped shape and teach the philosophy program at Columbia Secondary School for Math, Science & Engineering in New York City. In 2014, she and her students founded the philosophy journal CONTRARIWISE, which now has international participation and readership. In 2020, at the Varga Katalin Gimnázium, she and her students released the first issue of the online literary journal Folyosó.

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    On April 26, 2016, Diana Senechal delivered her talk "Take Away the Takeaway (Including This One)" at TEDx Upper West Side.
     

    Here is a video from the Dallas Institute's 2015 Education Forum.  Also see the video "Hiett Prize Winners Discuss the Future of the Humanities." 

    On April 19–21, 2014, Diana Senechal took part in a discussion of solitude on BBC World Service's programme The Forum.  

    On February 22, 2013, Diana Senechal was interviewed by Leah Wescott, editor-in-chief of The Cronk of Higher Education. Here is the podcast.

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    All blog contents are copyright © Diana Senechal. Anything on this blog may be quoted with proper attribution. Comments are welcome.

    On this blog, Take Away the Takeaway, I discuss literature, music, education, and other things. Some of the pieces are satirical and assigned (for clarity) to the satire category.

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