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.

Why the Lecture Isn’t Obsolete

feature article in Harvard Magazine (March-April 2012) asserts that the traditional lecture method of instruction is giving way to something far better. The author, Craig Lambert, waxes euphoric over this development. He tells how Eric Mazur, professor of physics at Harvard, had an “epiphany” that led him to overturn his teaching methods. Finding that his students dutifully memorized formulas but lacked a grasp of fundamental physics concepts, he had them engage in “peer instruction” during class and saw dramatic results.

Mazur began experimenting more with peer instruction, gathered data on the results, published a book on the subject, and became a coveted speaker on the subject around the world. Other professors have followed suit, replacing lectures with “innovative” methods. According to Lambert, “active” or “interactive” learning is in, and “passive” learning out. “Interactive pedagogy,” he writes, “turns passive, note-taking students into active, de facto teachers who explain their ideas to each other and contend for their points of view.”

Lambert’s error (by no means his alone) lies in his assumption that students who listen to a lecture are less active than those who confer with each other. The reverse may be the case. In treating the lecture as an unwanted relic of the past, as the locus of passivity, we may set ourselves up for serious loss.

When a teacher or professor gives a lecture (including a physics lecture), he or she is not only delivering information, but also shaping, questioning, and recasting it. A lecture is a work of imagination and insight as well as an exposition. The student listening to the lecture may put it together in his mind, relate it to the reading or to specific problems, think of questions, enjoy the lecturer’s style, and more. A single phrase in the lecture may lead the student to an insight; one insight might lead to another. Lectures are not always this invigorating, but if they’re reasonably good, a student can find room for rumination in them. In addition, they offer respite from peer noise. There is plenty of time for talking with peers; during the lecture, this is not necessary.

Peer instruction, by contrast, can deaden the spirit and lower the level of instruction. Many subjects require quiet, extended thought; if there is no room for this in the classroom, if the room is usually “abuzz” with students talking to their neighbors, headaches may increase and insights decrease. Although students look active in such a situation (to outsiders, at least), they may be insidiously passive—relying on peers’ explanations instead of thinking about the problems on their own, or providing explanations to peers who haven’t done their homework. Of course, peer instruction need not always be stultifying—but it can be.

Oh, but the research shows… Let’s stop right there. Education research rarely “shows” what the researchers or the media claim it shows. (See, for instance, an egregiously flawed study that purportedly shows the superiority of “deliberate practice” to the lecture method.) Moreover, to determine what “works,” you need to establish what you want to accomplish in the first place. Otherwise the findings may not apply to your goals at all. (This point often gets lost in education discussion.)

If you wish to teach a subject richly, if you want students to grapple with its fundamentals and tackle difficult problems, then you need to present these fundamentals and problems, period. There are different ways to do this; the lecture is a particularly appealing method, since it brings everyone together in the same room. It brings responsibilities; students must learn to take the lectures in and work with them in their minds. If they don’t know how to do this, if a lecture strikes them as boring because it’s a lecture, then they need more practice listening to lectures, not less.

In addition, a lecture sets an example of scholarship. A fifty-minute lecture approximates certain scholarly articles in scope and length. When listening to the lecture, students learn what kinds of topics might fit into that time frame. They recognize interesting explanations and examples; they light up over an insight; they enjoy a good joke or allusion. The lecture carries a certain honor; just as the students listen respectfully to the professor, they imagine a time when they themselves might speak to an audience. The challenge (for professors and students alike) is to live up to that honor, not destroy it.

So, if we want students to grasp both the substance and shaping of a subject, we want something like a lecture. Of course, we also want them to do well in the subject and to understand it. How to accomplish all of this? Well, first of all, take them into the lecture format gradually, during the K-12 years of school, so that they know what to do with it. Second, pair it with a contrasting instructional format, such as a seminar or discussion group. (The lecture is usually insufficient on its own.) Third, provide books, problems, and other resources.

Don’t get rid of the lecture. Properly prepared, delivered, and received, it gives students something substantial and allows them to think about it. At its best, it offers insight and illumination; it may stay in the memory, for years, as both detail and gesture. Better not to spurn such gifts.

  • “To know that you can do better next time, unrecognizably better, and that there is no next time, and that it is a blessing there is not, there is a thought to be going on with.”

    —Samuel Beckett, Malone Dies

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  • Pilinszky Event (3/20/2022)



    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.


    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.


    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.

    When I revise a piece substantially after posting it, I note this at the end. Minor corrections (e.g., of punctuation and spelling) may go unannounced.

    Speaking of imperfection, my other blog, Megfogalmazások, abounds with imperfect Hungarian.

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