Daoism, differential equations, and death: Offerings from the seasons

February is ending, 56 degrees latitude, east side island, eastside Atlantic. The world is still shivery but from sky to sky it is blossoming, and my inwardliness is also unfolding outward. The crocuses are sending orange and violet flames from the grass, but this time of growth does not belong to the plants. This blossom is of the Sun and the swift elemental energies it circulates into expression. An aggressive vigour develops the air, hurtles windy momentum, and light breaks new angles between buildings, new geometries cast almost daily now onto the city’s granite floor. My eyes can feel it even if my skin is attuned to a different spring, one that will only appear once lakes and oceans embrace these energies and contribute to them in turn.

Days have been getting longer with the northern tilt’s accelerating plummet toward the fire orb. The stretching of daylight is at first undetectable to my eyes and nose. Measured in seconds added to the day, the solstice holds winter in what seems a catatonic lull, ambiguous in its beginning and end -until one day it isn’t. The shift is felt in ways I live but do not always acknowledge. The alleyway I pass on my morning walk now glitters in light rather than shadow. The difference is felt but not cognised, the stroll feels brighter even if I do not pinpoint why.

The first climax of light will come this year on March 20th at 9:58 pm, but this climax like all celestial transitions, comes with its paradoxes.

Between March 10th and March 26th, days will be getting longer by 4 minutes and 40 seconds every day. Between winter solstice and this time, days were getting longer more and more quickly, after march, days will be getting longer more slowly. We are now rushing towards this peak in vim, after which will be its ageing and senescence. The official first day of birth is the first day of death. Ostara is an inflection point.

Growing up, our summer break always seem to coincide with my father grumbling that autumn had arrived: the days were now getting shorter (his unsolicited astronomical realism never seemed to raise its head during winter solstice). But as I contemplate the hurtling unfolding before me, it seems summer solstice is already the second autumn of the year (my father might be proud of this pessimistic observation). The rate at which days get longer begins to slow in March, decelerating to a standstill in June.

In other words, we are observing changes, and we are also observing changes to the rate of changes. The latter is what mathematicians call “first derivatives” of differential equations. The first derivative of velocity is acceleration. Let’s follow the seasons with an eye on such rates of change. From June until September, death will gain increasingly in vitality, days not only getting shorter but getting shorter increasingly quickly (and like the initial hiddenness of January’s Spring, this death is hidden for awhile by the slow transition and the lagging heat). When the heat begins to catch up a bit, the loss of light is in free fall and Mabon lurks. His first Spring a hidden bloom: beneath the autumnal gloom, with everything about us is withering back into the ground, the accelerating darknesses pivots and pulls away.

But does it end there?

The cos x curve depicts the rates of change of the sin x curve

If changes in daylight hours etch a sine curve into the galaxy, high school mathematics tells us that changes to these changes is a cosine curve. This is represented as f’(x)sin x = cos x. But the rate of change of the cosine graph is an inverse sine curve (-sin x). There is a technical term for this. The rate of change of acceleration is called “jerk” -a term which is almost certain to confuse any intuitive sense we have of what is going on in the passage of the seasons. Because all derivatives of the sin curve are oscillate with the same amplitude and frequency as the original curve, the only difference we should expect is phasic. But can we experience this?

I think we can. In fact, I have already alluded to it above. Between March 10th and March 26th, days get longer by 4 minutes and 40 seconds every day. This approximation hid the fact that the rate of change at which days were getting longer was itself changing extremely slowly. Days are getting longer more quickly, but longer more quickly more slowly. Compare this to the difference between December 23rd and 24th last year. On December 23rd there was 9 seconds more light than the 22nd. On December 24th, there was 17 seconds more light than on the 23rd. The rate of change in March is almost constant even though it is dramatic. The rate of change is in rapid flux in December even though it is imperceptible. There is a lull at the equinoxes too, but unlike the lull during the solstices, it is a lull where we stop feeling the days accelerating. For a week or so, they just seem to be in an almost steady acceleration.

I believe this is the end of what can be perceived (and this only with some difficulty). Who knows what if further derivatives are picked up on and responded to by other creatures? But the mathematics suggest that the extent to which life and death are intertwined in the seasons goes well beyond what has so far been suggested. The derivative of -sin x is -cos x, and the derivative of -cos x is sin x. After four derivatives, we return full circle. Although this cannot be felt or witnessed, it suggests something very powerful and subtle. Superficially, it evokes the four seasons that are themselves also indications of cyclicity. But the more luminous point is that with sin curves there are an infinite number of rates of rates of change, each an oscillating wave. There is no end in sight. In March, the rate in which the rate of change in daylight is changing, is itself changing too, this rate too is changing, and so on ad infinitum. Wrapped into the dynamics of the sine curve is an infinitely intricate conjoinment of slowing downs and speeding ups. Every inflection point is a turning point governed by a new inflection point.

The fourth derivative of sin x is sin x

The solstices and the equinoxes are all moments where solar expansion and withdrawal switch hands, one birthing process now dying, but within that dying another birth. This reveals in the most acute way a concrete and infinite dialectical contradiction.

On teaching oneself

It is for each teacher to figure out what kind of teacher they are.

At one point, I was very attracted to student-centred teaching, especially those models that broke down the teacher-student relationship in favour of a community of co-learners. I developed a school where the attempt to do this was itself a large part of the curriculum. I was attracted to Freire’s critique of banking education, and of the project-oriented Deweyan student. These approaches made sense. They seemed right, and not just in terms of my learning style. I also felt that such pedagogies were attempting to enact an attractive democratic vision that I’d like to see come alive writ large.

But I found very few, if any, situations where they seemed to ‘work’. Most often, students would complain about lack of structure, about excessive focus on process, and about not feeling like their errors were being corrected (students often have a healthy suspicion of the platitude ‘there are no right or wrong answers’). I confess to feeling that the aimless atmosphere evoked in these attempts often led me to lose confidence in my learning and in my educators’ capacity to facilitate such learning. The student centred teacher might be delighted to hear that their authority was being questioned. But I don’t think it was being questioned in the way they might hope. (They might be delighted in that too, but again for the wrong reasons).

I felt there was something missing but my conviction that such pedagogies must be right led me to fast and easy excuses: growing up in a system that so favours authoritarian approaches, I’d ask myself,  is it any wonder that there is some bumbling about on the part of teachers and students in recreating democratic pedagogies? I was reminded of how I would analogously defend those rudderless moments in my community activism work. In that work, I often witnessed an abundant need for people to debate microdetails. I saw that being suddenly able to have strong agency in a collaborative process meant we now needed to learn that responsible agency is partly knowing when not to exercise agency. In both the classroom and the community, certain types of engagement with democracy can quickly lead people to yearn for the vision of the ‘benevolent dictator’ that provides a goal and context that people can work within. I am still unsure to what extent time and experience might render such process-oriented approaches more effective and satisfying.

As time passes, I become less convinced that my students’ need for a teacher to explain things, correct errors, reveal interests and passions, and guide the classroom is merely a matter of students being unable to “understand their own rights because they are so ideologized into rejecting their own freedom, their own critical development, thanks to the traditional curriculum” (Shor and Freire, 1987, p. 21). In this same chapter, these authors discuss the aesthetics and artistry of being an educator. A key idea, but they seem to miss an essential point. Students want to experience an aesthetic phenomenon. This means an experience where there is a discernable integrity between the whole and the parts. They want to experience something crafted with mastery. Fraying ad hoc tangents, inexplicable eddies, a convoluted narrative arc that leads nowhere -there are many ways that an overly passive teacher can disrupt the relationship between the parts of the class. This forfeiture of responsibility is felt – it does not make sense rhythmically or melodically- and it reduce the student’s trust in the learning experience not because students need an authority but because they yearn to experience an author.

This is truer than ever in the dialogic student-centred classroom. I suspect that students will themselves be keen to collaborate in co-authoring the art of a developing classroom if they see their educator take up the challenge. In asking students to engage authentically in democratic and dialogical classrooms, we are asking them to make themselves vulnerable. If the educator hides their own vulnerability behind the opaque role of ‘facilitator,’ students will hardly feel free to open themselves up. By way of invitation, provocation, solidarity, and good faith, a teacher can show how and why their heart, and everyone else’s, matters. Their own fumbles can then be seen as parts of a whole they are trying to work on, the whole being of course their own integrity as people and as educators. The aesthetic nature of the classroom will come out in the relationship between the ways teachers present themselves as a whole and what they do to move towards this vision of themselves in their particular actions. Devoid of this part/whole relationship, the student is left without the content/context interplay that is so key to developing meaningful understanding of what is happening (this leads to hermeneutic analogies).

We can invite students into their hearts by showing our own in many different ways. When looking back at the various teachers that have made an impact on me, few ‘approaches’ are common to them. Some were strict, some playful. Some experimented pedagogically while others took traditional didactic styles. One especially exciting teacher was fiercely didactic. When Zev Friedman would stand at the podium, clutching both sides of it, his crooked reading glasses poised on his nose, my heart would thrum and shiver at the beautifully serious struggle, the momentous spiritual stakes at hand in details usually depicted as boring or souless concerns of canonical 17th and 18th Century philosophers. This was no academic brain game. A quiet profundity pervaded Dr. Friedman’s world and allowed him to reach into and draw out those same depths in the thinkers we studied. He dared to share it and I fell back in love with philosophy at a time when other course made it seem like egoistic head trips. What seemed to make a difference had nothing to do with some pedagogical principle he was applying. It was instead all about the extent to which I sensed his emotional engagement with his subject. It was an empathic response.

I remember being equally drawn to moments where a teacher would reveal excitement, fear, even despair.

Teacher educators often talk of ‘modeling reflective practice’. How might we model emotional response? How can this be anything but a contradiction? Might emotions be the very things that slip past our attempts to model, -glints and pulses of life breathing through the machine?

An educator might slip unconsciously from honestly exposing themselves to their students one year to expressing the same idea in a formulaic way the next. I have found myself often tempted by this and have occasionally succumbed. I treat my moments of authentic engagement as having given rise to tools that can now be applied to generate authenticity. I have occasionally done this in full awareness that it is impossible and contradictory, knowing full well that authentic engagement is not about applying the past to the present but about showing up to what the current situation has to offer. I have even done this (ah, the absurdities of the human soul!) in describing this very problem itself.

Perhaps I shouldn’t be too hard on myself. Occasionally such stories can play a role because stories do have a life of their own, because experience is cyclical, because the present is often not so unique as to constantly require a radically new response. Sometimes I can respond to the current situation through how I tell the story, allowing it to be in reciprocal relation with what is around me (like replaying a piece on the piano in fresh awareness of it in all its immediacy). But the more times I tell it, the better I get at telling it, the more I sense it lose colour. Raw emotional expression is vulnerable to the same technocratic thinking that pervades all ‘what works’ approaches, mechanising itself to the purpose of mechanised relations.

The mask is fake… but the mask is also real. Looking back on times I have exposed some aspect of my inner horror or guilt about the ecological crisis, I find there to be something tentative in its first expression. In some way, it is was ridiculously honest and unfiltered. It had not been turned into a story or style or approach (yet). But it was still a performance. I knew I was being listened to, in some cases by two dozen or more people. I felt acutely the radial concentration of consciousness in the room, with me as the hub of the wheel. Such an experience sometimes draws out and magnifies the emotion I was am to convey, like a concave glass concentrating the rays of the sun. In a sense, I became even ‘truer’ than I am normally capable of. It became a piece of music that I was inside of. Per-form. “Per” means through, thoroughly, very, utterly, all around – how I can be so utterly formed through those all around me!

Such is the fate of ‘masks.’ Sometimes our performance brings out something authentic, indeed it develops our range of experiences and self-understanding. Other times it robs us of ourselves. Each performance has its fate, sizzling out of truth or groping towards it, sometimes hiding from it entirely.

How often a scholar’s ‘expertise’ lies precisely in the topic they seem to lack in skill and grace! An expert in care pedagogy fails to make eye contact with me when we meet. An advocate of interspecies interactions tramps loud and thoughtless in the forest. A theorist developing a concept of authentic student-teacher relations lectures behind a thick authoritative facade. Why is that? I believe it evokes a purity of heart long corrupted. Initially, we read and write because we yearn to grow. Oftentimes we are passionate about things we recognise are important but are not very good at. We write because we want to improve, knowing that in putting our strengths towards developing our sacred weaknesses, our heart’s ‘deep gladness meets the world’s deep need’ (as Parker Palmer would put it). But somewhere along the line, we confuse our expertise in writing and theorising about the topic with expertise in the topic itself! Piles and piles of “know that” fill the space still empty of “know how.” In this circuitous way, passion suffocates itself. A culture of academic posturing makes this slip into hidden inauthenticity all the easier.

I think we need to recognise and resist this transformation. As people, as teachers, as researchers, we need to recognise that vulnerability is our asset. We must even guard against becoming ‘experts’ in vulnerability. It is not a turf or territory to be defended, but an attitude to cultivate. It is an attitude that dissolves itself at high altitudes.

Immersed in such questions, I raised them in an exploration with a student, rich peaceful pauses. At one point in the conversation, she mentioned that she has some traits that she “prefers” in herself (one was kindness), and that she consciously reminds herself throughout the day to potentiate these dimensions of herself. It struck me how her list was different from mine (I find myself constantly wanting to work towards humility, astonishment, gratitude). Why were our lists different? Why was I attracted to certain types of experiences and enactments? At this moment, the glow of my desk light illuminating the room more prominently with the sun now set, I realised how pervasive the dialectics of the soul are at work. I know the power and beauty of humility because I have experienced it, but I have experienced its awesome power because I am not always a humble person. Someone who is often or always humble is immersed in the state, so it is nearly invisible in the accustomed background of their being. We contain within us the shadow side (this was her term, perhaps invoking Jung) of each of our most inspiring characteristics, a shadow which tortures but also teaches us the meaning of what we are capable of and the importance of striving for it. Chiaruscuro. I asked her about her kindness and she said she felt that she was sometimes kinder and sometimes much less kind than most people around her.

It often seems I have access to the feeling that life is an astonishing and wondrous event, as utterly beautiful as it is contingent and fleeting, and indeed this sense guides me in my pedagogy and my research. I am lucky that for me there is no choice but to let this feeling, both delicate and powerful, guide me into my best teaching and writing moments. But the excitement of these moments is contrasted by my passages through mazes of ungrateful doubt and self-deprecation. This darkness is exacerbated because I can still know the powerful lightness that suffuses everything, I remember it and can conceptualise it as a statement of fact all through my interminable journey into the night. But such knowing is not enough to pull me out.

We lean towards our vocation through acknowledging and being open to the generative tension between our poles, embracing the drawn string that we can pluck and strum to bring our tunes to the world, the gifts we’ve been given to give, modulating as music often must, between the minor and major key. I can teach this truth by struggling towards unification in front of my students, inviting them to assist, and supporting them (if and when they are ready) to identify and integrate the poles of their own soul. De-ossifying dualisms back into dialectics.