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.