The Plena of Participation: How an Algebra Lesson with My Daughter Revealed the Fullness of Knowledge

Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.”

– Paul Lockhart

One recent morning, a few winter rays of sun beamed through the kitchen window and lit up the top of my black coffee. In that moment, I felt a little more of my being. I experienced, however faintly and briefly, a certain calm and joy. I reflected upon the coffee and the sunbeam; I could see them both in new and different ways. In this moment of contemplation, I gained a slightly deeper insight. Let’s face it, we were all in this together: the sun, the coffee, the kitchen, the window; my visual observation, my thought and reflection.

I mention this rather trite experience merely as a setup for a second, more profound, personal story. You see, the kitchen experience contains some key elements in the art of knowing, but it lacks others—so I offer it merely as a forerunner. I propose the following story as an epistemological model, a study in the fullness of knowledge and meaning.

As I was getting ready to write this article, I insisted that my 13-year-old daughter, Charlotte, sit down with me at the kitchen table to catch up on her prealgebra lessons. With my wife and other children out for the day, the house was unusually quiet. I sensed a good opportunity for us to spend some time together by working on our own materials next to each other. Because I work outside the home, my wife teaches the lessons to the kids; opportunities to study with my children are sparse. Shortly into my canon of inventio, Charlotte began to ask a multitude of questions about chapter 3 in College of the Redwoods Prealgebra Textbook, “The Fundamentals of Algebra.” Her questions revealed her frustrations and insecurity in this new terrain: Why do these equations vary? How do I work with these negative numbers. Should I multiply or divide? Which part of the equation should I attack first?
Now it had been several years since I had solved such equations—and I confess, my memory had grown a little fuzzy with algebra. And I really wanted to work on my article. So when she asked her initial questions about the function and operation of the coefficient, variable parts, like and unlike terms, the communicative, associative, and distributive properties, and ultimately, how to solve equations involving integers with variables on both sides, I declared that it would be best for her to simply read closely the four pages of detailed explanations and examples—and surely she would come to understand the fundamentals of basic algebra and be able to solve the 68 equations. In my body language and tone, I probably conveyed that I had work to do, and that algebra is a subject that you “just have to get through at your age.”

Yet, this clinical suggestion that she simply “read the objective facts of the chapter and acquire the factual knowledge for the equations” was not working very well. She was far from gaining true knowledge, and even further from the meaning of algebra in its essential nature. My selfish fortitude did not last long, for she and I both knew the advice was cheap. Besides, her questions had piqued my curiosity and wonder. I was soon sitting next to her, determined to answer her questions by reading the chapter along with her. Now we both wanted to know what to do at each step. For me, it was an exercise in memory; I had learned these long ago in school, and some years later for the GRE exam. For Charlotte, it was all new.

As we read the explanation pages together, each part of the various operations became intelligible. After each example, we would turn to the assigned equations and begin solving them. Step by step we proceeded. After a few rounds of this, I returned to my seat to resume my studies. After all, our own agendas die hard. But as the problems became more advanced, she asked more questions. And yet again, I found myself sitting next to her, learning with her, sharing in the act of discovery. As she gained confidence, we gained joy. I was teaching and learning. We were learning together. After an hour or two, Charlotte reached an epiphanal moment, exclaiming with pure joy, “I get it! I can solve this equation on my own without a mistake!”

So my article was off, but life was on.

After reflecting on this experience, I was struck by how rife it was with the fullness of knowledge—so much so that it seemed the perfect model to serve as a central example. Indeed, our lesson embodied knowledge and meaning, for we employed and adhered to no less than twenty-five different tools and principles in the process of attaining knowledge. (We shall define knowledge as justified true beliefs.) I will list them here, with each one followed by a brief explanation of its use in our lesson.

My purpose for offering these twenty-five tools and principles is to provide a brief exposure to all that is involved in the art of knowing, to reveal just how much is at stake. After these, we will look at three fundamental concepts in greater detail.

1. Experiential and Intellectual Input – Input acquired through the five senses, along with our conceptual ideas, provided the necessary data for us to solve the equations. For example, through the use of our sight we were able to read the information on the page, and then process the ideas derived from that sensory and intellectual information.

2. Memory and Imitation – Because I learned algebra in the past, I used my memory to recall many of the details; as well, Charlotte used her memory to recall a variety of arithmetic facts. Additionally, the ancient Greeks suggested that all learning happens by imitation, the creative impulse to reflect what is already there. We imitated the steps portrayed in the examples.

3. Reason and Logic (dialectic and hypotheses construction; formal and informal) – The equations involved the use
of reason, the means by which we move from one idea
to another, by means of logical inference. We combined

like terms by dividing. We negated terms in the sum.
We divided both sides of equations. We multiplied and simplified. These were logical and reasonable moves that we knew would help us solve the problem. We also used dialectic, the “question and answer” dialogue in our joint discussion to solve the problems.
4. Verbal and Mathematical Language – We used a fairly complex verbal language to communicate with each other and to describe and explain the mathematical language.
5. Pattern Discernment and Recognition – Because our

minds are able to recognize visual patterns, cause-and-effect patterns, and other structural patterns, we noticed that a pattern exists in each equation, a pattern similar to other equations.

6. Adherence to Order – Each step in the equation needed to be solved in the proper order otherwise we would not have arrived at the truth (right solution).
7. Practice and Repetition – To arrive at the truth (right solution) consistently on our own, we needed to practice and repeat the steps several times.
8. Association – By associating one idea with another, and one experience with another, we were able to understand increasingly complex ideas by reasoning from one concept to the next.
9. Belief in Objective Truth – the mathematical numbers, laws, and principles in these algebraic equations are objective, eternal, and immutable. There is one right solution; anything other than the right solution is wrong.
10. Effort and Discipline – In order to arrive at the truth (right solution), we needed to put forth effort and to be disciplined. Though challenging, we believed that truth can be discovered, and that finding the truth is worth the effort. 11. Invention – Invention involves creativity; it is the activity of inventing ideas and arguments. This includes hypotheses, explanations, and interpretations. We interpreted the explanations of the algebraic equations.
12. Experimentation – On a few occasions we were inspired to think in different ways to solve the equations. If we generated hypotheses that produced different results from the method taught, we used dialectical reasoning to compare hypotheses and to determine which ones were correct.
13. Form, Structure, and Parts – It was important that we honored and adhered to the proper form, structure, and parts of the equations.
14. Evidence and Proof – Charlotte’s answers would have meant little or nothing if she did not show her work: how she arrived at the solution. Similarly, most assertions (theses) are meaningless without supporting proof.
15. Penmanship – Beautiful penmanship is a sign of elevated and ordered thoughts. I insisted that Charlotte use neat penmanship to reflect the quality and facility of her thinking and problem solving abilities.
16. Intuition – This can carry a variety of meanings,
but it usually stands for thoughts that are immediately, necessarily, or self-evidently true. Though we didn’t rely much on intuition, some of the mathematical concepts seemed “intuitively” right.
17. Relationship – Forming a relationship with Charlotte propelled her into true knowledge. If I would have insisted that she learn it on her own because I was busy, she would have struggled longer with the task, and she would not have known it as well. The relationship manifested in our activity has implications that are transcendent and eternal. 18. Participation – If I had insisted on looking at the algebra lesson from the outside, from a distant, objective vantage point, and made assertions from my outside perspective without participating as a subject in the activity, I would

not have been able to arrive at a complete and accurate understanding. I would have given answers based solely on my memory, which is fallible and prone to error. I needed to step inside the activity of learning to read the information myself and attempt to solve the problems.
19. Commitment to Universals – We affirmed not only the universal axioms of mathematics, but eternal realities such as truth, goodness, love, and the soul.
20. Deference for Tradition – Mathematics is an old study; we honored its function and role in the universe and in the history of man. We endeavored to participate in the Great Conversation (about mathematics) with the past.
21. Humility – Humility was absolutely essential before we could learn anything. We had to acknowledge how much we did not know. I needed to admit that I had forgotten some of the strategies in solving the equations. Charlotte needed to admit that these new concepts were a challenge and that she needed help.
22. Imagination – Here we emphasize the importance of
the imagination for a fuller, more complete knowledge of ourselves and the world. We affirm the vital relationship between reason and imagination in the activity of knowing. 23. Wisdom – Though my work on the article was set behind, it was wiser for me to invest in the lesson with my daughter because it was the right thing to do. All knowledge has an ethical and spiritual dimension (all Truth is God’s truth). So all knowledge, in some way, relates to wisdom. Time spent with her was the wiser choice for many reasons, but to name two—we are a little closer now, and she is growing in her knowledge of math.
24. Faith – We needed faith in God, and in His eternal mathematical laws. By studying them, we believed that we might come to know reality a little more fully, and through that reality, know something more of Him and ourselves. 25. Love – Because I love Charlotte, and care enough for her to learn algebra, she now understands it. If I had insisted on her reading the pages on her own, as mere facts separated from reality, existence, and relationship, she would not have come to a full knowledge of it.

We shall now consider three salient concepts from above that are vitally important in the activity of knowing: Universals and Truth, Participation, and Language and Imagination. We will begin with universals and truth because they influence and inform the other concepts.

The first slip into modernism might well be located in the figure of William of Occam in the early 14th century. Occam established the doctrine of nominalism, which denies that universals and/or abstract objects have any existence or reality. The doctrine suggests that only particular, concrete things are real, and that universal terms and concepts have no existence (other than as mere names for classes of particular things). As Richard Weaver suggests, the issue at stake is whether a source of truth exists that is higher than, and independent of, man. The consequence of nominalism is that it banished reality perceived by the intellect and the spirit, and reduced reality to only what is perceived by the senses. And with this change in the assumption of what
is real, the entire orientation of culture took a turn toward modern empiricism.1

The effect of nominalism is the diminishment, if not the devastation, of our ability to know reality in a more comprehensive way. The denial of universals carries with it the denial of everything transcending sensory experience, and with this, the denial of truth. Astutely, Weaver recalls the story of the witches from Shakepeare’s Macbeth, who tempt Macbeth with the idea that man can realize himself more fully if he will only abandon belief in the existence of transcendentals.2 By denying transcendent reality
and objective truth, the witches spoke delusively and presciently—instead of man realizing himself more fully, he is actually sundered from knowledge and reality. For it is the transcendent entities that complete the fullness of reality and knowledge, giving life and being to all things.

James S. Taylor aptly states that the fullness of knowledge is a kind of natural, everyman’s metaphysics of common experience. It is a way of restoring the definition of reality to mean knowledge of the seen and unseen. Its restoration is essential for reawakening the intuitive nature of human beings who are able to know reality in a profound and intimate way that is prior to, and in a certain sense, superior to reductionistic, empirical knowledge.3

Let us now turn to the vital role of participation in knowledge. In “Meditation in a Toolshed,” C. S. Lewis relays an enlightening experience of standing in a dark toolshed. He says that the sun was shining outside and through the crack at the top of the door, a sunbeam pierced through. Everything else in the shed was pitch black. Particles of dust were floating in the beam. The beam appeared striking and beautiful. Importantly, he was looking at the beam, not seeing things because of the beam.

Then, Lewis moved into the beam so that the beam fell on his eyes. Instantly, he says, “the whole previous picture vanished. I saw no toolshed, and (above all) no beam. Instead I saw, framed in the irregular cranny at the top of the door, green leaves moving on the branches of a tree outside and beyond that, the sun. Looking along
 the beam, and looking at the beam are very different experiences.” The modern method of acquiring knowledge is akin to looking at the beam; but to partake in the fullness of knowledge implies standing in the beam and looking along the beam. Here are two different ways of knowing. Both are valid, yet the second way implies participation inside; it facilitates passage into the glorious realm of universals, the transcendent realities that comprise the fullness of our knowledge, being, and purpose. From mere matter to intellect, spirit, and truth.

Let us conclude with language and imagination. Remember the opening anecdote where I was sitting in the kitchen with the morning sun and my coffee? By the active use of language and imagination, I imbued the experience with meaning. With modern reductionism, it is usually assumed that there is little connection between the physical causes of things and their meaning. But, as Owen Barfield illuminates, the meaning of a process is the inner being which the process expresses.5 And it is language and imagination, through symbol and metaphor, that connect the inner beings of things to their processes and to man.

So then, a thing functions as a symbol when it not only announces, but represents something other than itself.6 We owe the existence of language to this process: memory and imagination convert the forms of the physical world into mental images, images which function not only as signs and reminders of themselves, but as symbols for concepts.
If this were not so, they could never have given rise to words, which make abstract thought possible. If we really think about this, it implies that this symbolic significance is inherent in the forms of the outer world themselves.7

Thus, Barfield reveals, if language is meaningful, then nature is also meaningful. He quotes Emerson, “It is not only words that are emblematic; it is things which are emblematic… Man is placed in the center of beings and a ray of relation passes from every other being to him. And neither can man be understood without these objects, nor these objects without man. It is precisely in this ‘ray of relation’… that the secret of meaning resides.”8

Perhaps it is just this ray of relation dispersing through each other and the world, our experience and our soul—the interaction of coffee, sunlight, algebra, and spirit—the joy of participation and the fullness of knowledge—which grants meaning to all that we hold dear: that which we write, that which we hope to know, and those whom we love.

“What’s Going On?” as an Essential Question: Jesus, Socrates, and the Imagination

The use of essential questions to guide both curriculum and lesson planning is characteristic of classical Christian education. As John Milton Gregory proposes in The Seven Laws of Teaching, questioning is an artful science that invigorates the learning process by drawing students into an active posture of inquiry, rather than relegating students to passive receptacles of information.1 The use of essential questions is taken to be synonymous with the Socratic method and the teaching style of Jesus, placing the practice squarely at the center of classical and Christian education. The desire of the Socratic and Christological method, however, is concerned with a reality deeper than the (truly gratifying) moment in which a student, rather than the teacher, answers the question. In short, to take a cue from the pedagogies of Socrates and Jesus is to be occupied with a form of comprehension more primary than rationality—namely, the imagination—and to seek after this precise form of comprehension demands an appropriately precise form of questioning. I propose, in the words of a teacher who has had a great influence on me, that Socrates and Jesus teach us to interrogate the imagination by taking the time to ask Marvin’s question, “What’s going on?” before Lenin’s question, “What is to be done?”2 In order to develop what is meant by using “What’s going on?” as
an essential question and to demonstrate its use, I will first examine the pedagogies of Socrates and Jesus, reflect upon the philosophical underpinnings of an imagination-centered pedagogy, and, finally, provide two case studies drawn from the seventh and twelfth grade classes I teach.

The Socratic method of interrogation deals with one ethical question by asking a series of other, seemingly unrelated, ethical questions. In order to answer, “What’s to be done?” Socrates demands that we address what’s going on. If we were to ask, “Is it good to know oneself?” Socrates would barrage us with a whole host of other questions: is this knowledge the whole or a part of virtue, is this knowledge teachable, is it enough to make us happy? The purpose is not simply to elicit right answers to a list of discrete questions, but to destabilize our sense of knowledge by demonstrating the unity of these answers—that, in fact, their usefulness is not as distinct definitions or separate units of knowledge, but as they cohere together by their reference to wisdom in its unity.3 To answer one is to find oneself approaching another; to fail to answer one of the many is to fail to know anything at all. It is in this sense that Socrates challenges others by his claim to know only that he knows nothing.4 Socratic questioning resists the fragmentation of knowledge by building connections among different forms of analysis and disciplines of study. To the extent classical education models itself upon the Socratic method, classical educators are committed to rigorous, interdisciplinary interrogation.

Characteristic of Jesus’ teaching is the way he revolutionizes, not simply modifies, understanding. This is because Jesus is not preoccupied with discrete objects of knowledge, but total ways of being in and seeing the world. When the Sadducees approach Jesus with a question about the resurrection, they seek to trap him with a “What’s to
be done?” question, but he responds with a revelation of what’s going on. To the question of whose wife a woman married seven times would be in the resurrection, according to the Sadducees’ plans, Jesus would have to either deny the resurrection or betray the law of Moses, which instituted Levirate marriage (see Luke 20:27–40). He does not answer their question when he says, “Those who are considered worthy of a place in that age and in the resurrection from the dead neither marry nor are given in marriage” for “they cannot die anymore.”5 He shows, instead, that their understanding of the resurrection continues to include a place for death—a profound misunderstanding of what the resurrection is. The law of Levirate marriage only makes sense in a world of death, wherein the death of a husband means a grief-stricken widow must be passed along and “given in marriage” to another. The condition for the possibility of their understanding of the resurrection includes death; death has so infected their vision that they are unable to imagine a world without it. It is precisely the good news of the gospel that a new imagination, a new way of seeing, comes with the new creation. To the extent classical Christian education models itself upon the teaching of Jesus, classical Christian educators must be committed to interrogating the imagination.

The imagination is like the structure and contents of a room that we enter. The furniture, the walls, and the carpeting are set: they are the setting in which we eat, read, or converse. A space’s form can have direct bearing on the sorts of encounters or behaviors that take place inside. The open concept living space of a modern home encourages fluid movement and interaction among those who would otherwise be occupying different rooms. A cathedral’s immensity, permanence, and verticality impart upon worshipers the awe that is appropriate for encounter with the eternal and wholly Other.6 Buried with books in the university library’s basement, the graduate student feels life and joy incrementally sapped away with each flicker of the fluorescent lights. To ask what’s going on of a particular situation is to start moving the furniture, tearing up the carpet, and examining the architecture of the rooms we inhabit. Forces beyond our immediate attention operate upon us and shape us; our very perception of particular situations is, in a sense, given prior to our rational engagement with that situation. Like a room, that givenness circumscribes, directs, and limits our engagement; it can make certain choices seem inevitable, and others unthinkable. The question of what’s going on engages this givenness.

James K. A. Smith has identified this imaginative givenness as a faculty that rests somewhere between instinct and intellect.7 It is a discipline deeper than a rationalist worldview that has been constructed over time and passed down by the incorporating and institutionalizing practices of our communities. The imagination is constructed and entered, given and inherited. Moreover, the imagination, not the intellect, is the motivating center of action; if action arose from intellect, academics would surpass all others in moral excellence. Instead, the seat of action is the imagination, or what French philosopher and social theorist Pierre Bourdieu calls habitus: a structure that structures our vision of the world and our moral place in it. Contrary to a common scholastic fallacy, there is no theoretical space above or behind the practices that shape our imagination; rather, we are fully embodied beings and our imagination reflects the social location of our bodies.8 The rooms we enter represent forces of desire and relations of power that shape identity. As Bourdieu carefully points out, this discipline is far from innocent—especially in the academy—for the way we see the world reflects our social location in it, and it is characteristic of this vision to be self-effacing. That is to say, it is all too easy to forget the conditions for the possibility of seeing the world from a position of scholastic privilege—“a site and moment of social weightlessness” wherein philosophical investigation is freed from the constraints of necessity.9 According to Plato, leisure (skholè) is the distinctive and requisite privilege of philosophers, the success of whose heavenly searching depends upon not being preoccupied with the hurried conditions of the world “at their feet.”10 Forgetting the privilege of that detachment, students and teachers re- inscribe the inequalities that support their studious position of sight.

One essential question I have been using with my seventh graders is, “What do stories do?” They have learned a simple answer: “Stories teach us how to see the world.” This is an inquiry along the register of the imagination, but what is the connection between stories, the imagination, and bodily discipline? As the Israelites entered Babylon in
the early sixth century B.C. and passed under the Ishtar Gate, they were submitted to an imaginative discipline. The imposing structure boasts extravagant wealth and power, not only by its sizable, artistic construction but also, and more seductively, by its brilliant, expensive blue hue. Images upon its walls tell the story of Marduk who, according to the Babylonian creation myth the Enuma Elish, created the world by destroying the gods who opposed him and then established the city of Babylon and the Ishtar Gate itself as testimony to his victory. It would be insufficient merely to note this point without attending to its formative power upon those exiled bodies passing under it. The Ishtar Gate serves as an entrance into the city, to Marduk’s temple,
and into the Babylonian imagination. It operates as a habitus, an imaginative discipline of domination over those shackled exiles subjected to it. Walls ask without rational argumentation, “Where is your wealth and power? Where is your city? Where is your god?” Bodies comprehend, “We are captive.”

Shackles tell a similar story in U.S. History. With my seniors, I ask the question more directly, regarding any given event we study, “What’s going on?” Thomas Jefferson is a particularly contested and paradoxical figure who well serves this analytic exercise. An agrarian antifederalist, Jefferson opposed the strong central authority created by the U.S. Constitution: he was one of the many who immediately recognized how subjecting local, state interests to national control would likewise consolidate power with the traditional, urban, moneyed, and property-holding elite. We cannot read the Constitution naively, but must ask with Jefferson what social arrangement the Constitution stabilized. After a century of class conflict—Shays’ Rebellion of 1786 only the latest flashpoint of tenant unrest—the elites sought finally to secure their position.11 James Madison proposed a national representative republican government as a mechanism capable of filtering out the vicious lower passions; according to the maxim of the federalist John Jay, “Those who own the country ought to govern it.”12 Once in office, Jefferson advocated for the poor landless whites. He would maintain education as a prerequisite for civic participation, yet he insisted that the poor were capable of being educated, growing in virtue, and joining the American experiment. There was one condition: the institution of slavery would have to be maintained. Who else would support the agricultural industry (America’s backbone, according to Jefferson) if poor whites were to seek advancement and full citizenship? The condition for the possibility of white prosperity was black enslavement—a channel in American imagination and society that flows throughout U.S. History to the present. Jefferson’s imagination was held captive by a racialized vision of the world. Even as Jefferson exposed the hierarchical social arrangement stabilized by the Constitution, we must inquire after what social arrangement had constituted his vision of American democracy.

The ultimate purpose of drawing attention to the ironically non-egalitarian author of the Declaration of Independence is not to cast judgment on an individual, but to demonstrate the way disastrous choices and skewed perception can seem completely reasonable within the horizon of a distorted imagination. “What’s going on?” sets us upon a trajectory capable of exposing that imagination. “What’s going on?” demands that we go deeper than asking what happened, who was involved, when it occurred, and why and how it came about. To follow the classical trivium, these are essential questions of grammar and logic, and they must be answered. We must identify the players, facts, and events in their entire social, political, and economic complexity; we must discern the logic of how those players, facts, and events are organized into relationships of causation. Yet, there remain questions of rhetoric: how are grammar and logic being deployed and whose interests are served? Following the Augustinian principle that societies are constituted by their loves, how does desire shape identity? Furthermore, with Augustine, how do relationships and systems reveal idolatries that play out in lust for domination?13 What is being desired, what relations of power are being established or stabilized, and how are identities being formed in the process? What identity is desirable and who is excluded from this identification? What are the social conditions for the possibility of satisfying what is socially desirable? What is the quality of the relationships being established—e.g., mutual, reciprocal, oppressive, violent? How do particular social arrangements validate or invalidate certain identities and desires? What’s going on? No question is irrelevant to study, as Socrates teaches us. Every question is relevant to exposing whole ways of seeing the world, as Jesus reminds us. Apart from this interrogation, we know nothing—the Ishtar Gate is just architecture, the Constitution is just a text, and slavery is just an institution.