Philosophies of Teaching and Learning

What Does it Mean to “Know”?

On Saturday mornings, after my teacher’s 8 AM T’ai Chi practice, many of us head to a bagel shop and breakfast together. One October morning, a group of us stumbled onto the subject of knowing.
Chris said, “for example, I know 7x7 is 49” and I responded, “Well, one might know that 7x7 = 49 to write it down, but one might not have any real depth of knowledge about it. For example, somebody could know the correct answer, but they might not know that it’s a square number.” Chris, and two others who were sitting at the table, including Chris’ wife, Lisa, immediately piped up, a little bewildered “it is? What do you mean? How do you know that?” “Well, it’s a square number because it makes a square,” I replied. Someone said, “But that’s not what they mean when they call something a square number.” “That’s exactly what they mean,” I replied. Chris then said, “No, it couldn’t make a square, because it’s not an even number.” The others immediately agreed.
Taking out some pennies I explained to them: “Okay, lets say the corner of this table has one penny in the corner (figure 1). You can imagine this as a small square (figure 2): 1x1=1 and 12 = 1. So, one is a square number in that sense. Now, what’s the next square number?”
They all replied simultaneously “four.”
“That’s right,” I said, “so now I’ll add three more pennies to make a square of four,” and I did so (fig.3), adding, “2x2 =4 and 22 =4. And you’re right, it’s an even number. Now, what’s the next square number?”
“Sixteen,” came back the answer from one of the men. But Lisa, (who, incidentally, had said earlier that she had trouble with math in school), said “nine.” “No, it couldn’t be nine,” one of the men said, “that’s an odd number.” “Okay,” I said, “lets try it,” and I did (fig.4), adding five more coins. “You see?” I asked, “it’s a picture of 3x3=9, and the numbers are all odd, but it makes a square, and so it’s a square number.”
These were three college educated adults in their mid forties, at least one with an advanced degree. All of them had come through the traditional US educational system. They had learned about square numbers, yet none of them had a practical understanding of what a square number was.
Thinking later about when Chris said “No, it couldn’t make a square, because it’s not an even number.” I was fascinated. We inherently seem to see a square and know a square as an “even” thing as opposed to an “odd” thing, such as a triangle. The shape itself has four sides, an even (and a square) number. You can easily see how, without practical understanding of the matter, one might arrive at the question: how could you make a square, which is an even thing, out of an odd number? Incidentally, Chris is a twenty-year, devoted Go player, a game that takes place on a board with a 19x19 grid of squares, onto which round stones are arranged in rows and columns. Chris deals with stones arranged in squares all the time, yet he had never perceived these things in this particular way.

What is it to know, learn, understand or become aware of a thing? How do these conditions or actions come to pass, and how can they be nurtured or supported, especially in consideration of those instances in which a learner does not respond well to traditional teaching methods? In the eight years I’ve been working in elementary and middle schools, I’ve heard many terms and theories about teaching and leaning discussed. Sometimes terms become popular so quickly that they become removed from their original definition and source, and disassociated from meaning. In order to develop an understanding for myself of what kinesthetic teaching/learning is and what it might accomplish in music education, I first want to explore a wider field, that involving philosophies and theories of teaching and learning in general. In this section, I look at the dictionary definitions and histories of a few basic terms as a departure point for exploring theories that relate to kinesthetic teaching/learning.

Learning , Knowing, Understanding and Awareness
The Oxford English Dictionary (OED) defines learning as “the action of receiving instruction or acquiring knowledge; ... a process which leads to the modification of behaviour or the acquisition of new abilities or responses, and which is additional to natural development by growth or maturation” (2002). The word learn is defined as meaning to “acquire knowledge of (a subject) or skill in (an art, etc.) as a result of study, experience, or teaching”(2002). Also, it can mean to commit to memory, such as of passages of prose or verse or in phrases such as “to learn by heart.” I find it interesting that the learner is presented exclusively as a passive receiver, and never as a creator or even co-creator of what is being learned. It is also interesting that the OED’s etymology of learning relates to the word lore, which is something learned or that can be gained through knowledge or experience.
If learning is the acquisition of knowledge, then what is knowledge? What is knowing? It’s not surprising that the word know earns a rather involved discussion in the OED. What is a surprise is that, “ etymologically related to Gr. gignwskein, L. (g)noscere and (g)novisse,F. connaître (:-L. cognoscere) to `know by the senses'” (2002). This is distinguished from those definitions “...which belonged to the archaic vb. to wit, Ger. wissen, and are expressed by L. scire and F. savoir, to `know by the mind' (2002).
Again from the OED, when we understand something, we “grasp the meaning” of it, we “have thorough or technical acquaintance with or expertness in the practice of” the thing, or we have become “thoroughly familiar with the character and propensities of” it (2002). But do we grasp that meaning through our senses or with our minds? Indeed, is there even a tangible difference between these two things, or just a cultural one? Our language seems to make a distinction between these two kinds of knowing, yet simultaneously weaves in and out between them. Merriam Webster has awareness as the quality or state of being aware, consciousness, being informed, cognizant, conscious, sensible, to have cognizance, to know, to be mindful. Note that with sensible the OED has knowing by the senses coming in again whereas with consciousness we’re brought once again to con- together + sci- knowing, as in scire, to know, by the mind (2002).

Examining this distinction further while looking back again at knowing, the etymological treatment of the word, and the uses to which it has been put, differ essentially from a logical or philosophical analysis of the notion of knowing. The etymology puts knowing in the body, or at least involves the senses of the body. Might this indicate a historic consciousness of knowing as involving the body fundamentally, one which does not extract it to be strictly a thing of a separated brain or mind?
What is remarkable to me here is the fact that knowing through the senses is the first definition. It is then pointed out that this is different from its most common current usage. Perhaps the etymological development of the word itself tells us something about how we as a culture have moved from sense-based, to brain/mind-based notions of what knowing is and should be, to the perspective of dualism.
But a new awareness of knowing and learning is emerging. Elliott (1995) explores issues of consciousness and knowledge in laying out his argument for praxial music education, and presents dualism as fading, while a theory of materialism or naturalism emerges in our culture. This theory regards consciousness as a part of our very nervous system, the outcome of biological process. In presenting his theory Elliott quotes both Dennett, who says “the mind is the brain,” and Johnson, who claims “the body is in the mind” (p. 51) I was reminded of psychoneuroimmunologist Pert’s (1997) conclusion that “the body is the unconscious mind.” Pertinent to my investigation into kinesthetic teaching/learning, Elliott concludes that thinking and knowing may be manifested as action (p.53).
Small (1998) writes from a perspective that seeks to transcend dualism as well, and looks to the work of neurologists and neurobiologists when he claims that “mind is not substance at all but process...and is thus inseparable from the living matter of whose operation it is the outcome” (52). But Small’s real hero is Bateson, whom Small says regards mind simply as the ability to give and to receive information, the most important of which concerns relationships (p. 57). This information is physical in nature (p. 56). Elliott’s and Small’s perspectives put knowing in the substance, processes and actions of the body. Perhaps all learning is kinesthetic learning.
Are traditional definitions of knowing an indication of how we’ve gotten away from the kinesthetic aspects of learning? If so, is the recent trend away from dualism and the recent interest of the educational community in concepts related to kinesthetic teaching/learning a sign that finally some ancient pendulum is beginning to swing the other way?

Going back to the OED, the definition and history of know encompasses ways of knowing as well, and the complexity of this sheds light on how varied theories of learning, and knowing, can be. “Mr. James Ward, in Encycl. Brit. XX. 49 s.v. Psychology, assigns to the word two main meanings: `To know may mean either to perceive or apprehend, or it may mean to understand or comprehend... Thus a blind man, who cannot know about light in the first sense, may know about light in the second, if he studies a treatise on optics.' Others hold that the primary and only proper object of knowing is a fact or facts...and that all so-called knowing of things or persons resolves itself, upon analysis, into the knowing of certain facts about these...”(2002).
What comes to mind here for me is that there is knowing through the senses, which is direct, physical and personal, and there is a knowing which involves “the knowing of certain facts.” This divide bears direct relationship to how knowing is perceived traditionally in our schools, as the story above illustrates. In such a scenario, a student learns and knows a fact, and we know this is learned because when we ask for the fact to be repeated, it is repeated, as on a test. The mistake we make is when we then assume that this knowledge has any depth, understanding or immediacy for the student. Thus, a person can know that 3x3=9 and that 9 is a square number, but have no sense of what this really means beyond the written symbols that they have memorized how to put in the proper place. They may have no sense that nine pennies can be arranged in three rows of three to make a perfect square (a square number) and may even be surprised to find that this is the case. In this case, did this person really know this material?
Thus the examination of learning and knowing brings us around to awareness and issues of education. What is the best way to bring someone to an awareness, a knowing, a sense, of the thing we’ve decided is important to impart? How can we tell that someone knows something? Is a recitation of fact sufficient? Does a person have to experience something or imagine an experience of something sensorially in order to truly know it? Must the body necessarily and by definition be involved when it comes to knowing? This is where terms such as working knowledge and embodied knowledge and their overarching theory of experiential education come into play.
Although I have explored basic dictionary definitions of the above terms, ideas about knowing, understanding and awareness and how to teach and learn have been subjects of debate among philosophers and educators for centuries. Complete examination of them is clearly beyond the scope of this work. However, with dictionary definitions serving as a foundation, I turn to theorists in the field of teaching and learning to gain deeper insight.

Greenland and a Grapefruit
When she was in elementary school, my daughter brought home a map of the world. Her homework assignment was to color in all of the continents, and a list of these was provided. In the process, she asked why Greenland, which was obviously much bigger than Australia, was not on the list to be colored in. Possible answers flooded my mind (Is it because Greenland is not considered important politically? No, because Antarctica is a continent, and no one lives there...). Stuck, and thinking this was a very good point, I said to her, “Well, let’s find out.”
We looked up “continent” in the Encyclopedia Britannica and found “Continent: One of seven large continuous masses of land: Asia, Africa, North America, South America, Antarctica, Europe, and Australia (listed in order of size). Europe and Asia are sometimes considered a single continent, Eurasia.” Good enough, but this did not explain the Greenland problem.
We looked up the size of Greenland (2,130,800 SQ. KM) and Australia (7,682,300 SQ KM). I tried to explain that in reality, Australia is nearly four times the size of Greenland, and now my daughter was getting upset. Why, if Australia was so much bigger, would the map her teacher had given her be so wrong? How could a school do that?
My daughter was becoming increasingly teary: if Greenland isn’t that big, why did they make it so big? I had to explain to her that she was right. The map was wrong, and that I was truly sorry that she, (and everyone else) was given this wrong piece of information. Somehow the issue of map distortion came into my mind. I tried to explain the problem of trying to represent a sphere on a flat piece of paper. “They want to show you a picture of the world, but it’s hard to do on a flat piece of paper, so when they try, some things end up looking bigger than they really are.” This wasn’t really helping-it was as if her brain was short-circuiting.
Finally I got a different idea, and I sat her down with me at the kitchen table with a pen, a nickel and a grapefruit. The grapefruit was our model of the Earth, I explained, and I drew an equator around the middle. Next I traced the nickel with the pen, drawing perfect circles near the top, the “North Pole” of the grapefruit, to approximate where Greenland is. Then I drew one near the “equator” and finally in the “Southern Hemisphere” where Australia is.
I explained to my daughter that we were now going to try to take the skin off of the grapefruit and make it flat. I cut down the meridian lines from pole to pole, allowing the skin to stay connected around the equator, except on one meridian line. Finally I removed the skin and flattened it out on a piece of paper. There were significant spaces between the three segments of skin where the northern hemisphere nickel had been traced. I showed my daughter how, in order to make the Mercator projection map she had, they just filled such spaces in, creating a shape far bigger than the original nickel. That’s why Greenland looked much larger that it actually is.
I now know that my daughter, at around age ten, had discovered what cartographers call “the Greenland problem,” the distortion of size presented on a Mercator projection map. The Mercator projection map was developed in 1569 and is notorious for making Greenland and Africa, which is eleven times larger, look about the same size. Today, more than four hundred years after the development of this highly distorted map, it is still being used in schools. What’s worse, in this particular case, it was being used for an assignment which specifically dealt with the size of Earth’s land masses.

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