## Metadata - Author: [[Brent Davis]] - Full Title: Teaching Mathematics - Category: #books ## Highlights - To elaborate, in constructing the world on the foundation of the cogito, Descartes articulated more than the separations of mind from body, self from other, and representation from reality—all of which might be described as manifestations of a mind/body dualism. In addition to the essential distinction between mental and physical objects (with the consequent priority being assigned to the former), Descartes also contributed to the foundation of a host of other dichotomies, including knower versus known, organism versus environment, human versus nature. Further, the Cartesian orientation contributed to a view of the Self as a unified coherent subject: an autonomous entity that is isolated from others, independently constituted, essentially static, and able to maintain its integrity through diverse experience. ([Location 614](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=614)) - Seeking a middle ground between the mental and the physical (the inner and the outer), Merleau-Ponty suggests that the body is that which renders the mind and the world inseparable. Far from representing a discrete demarcation between subject and object, one's body is simultaneously of oneself and of the world. For Merleau-Ponty, then, the body is our means of belonging to our world—a world that shapes us and a world that we participate in shaping. ([Location 684](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=684)) - One's structure, in contrast, is thought to be fluid, temporal, and necessarily undergoing change. As Maturana and Varela put it, “Ongoing structural change of living beings … is occurring at every moment, continuously, in many ways at the same time. It is the throbbing of all life.” 18 Unlike modern conceptions of identity whereby one's self is regarded as a product, then, one's structure is product, producer, and process. Structure is thus a fluid notion that is analogous to our experience of the object/event of sound— an idea that might be reflected in the presence of sona in the word “person.” ([Location 698](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=698)) - With regard to the nature of the individual knower, Varela, Thompson, and Rosch suggest that the basis of cognition is not to be found in the Rationalist “I think” nor in the Empiricist “I observe”—both of which are founded on the premise of the detached knower (or disembodied I/eye)—but in the enactivist “I act.” Acting encompasses both thought and observation; acting presumes both actor (subject) and acted upon (object). In brief, acting demands reunions of mind and body and subject and object. It is this notion of embodied action that allows us to bypass the extreme positions of cognition as either recovering what is outer or projecting what is inner without seeking recourse in the supernatural or in metaphysics. ([Location 742](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=742)) - The upshot is that cognition cannot be a matter of internalizing or forming accurate representations of things of the world. Rather, cognition is inseparable from and fundamental to perception and action. Perceptions guide actions; actions enable perceptions. This inseparability is expressed in Maturana and Varela's aphorism, All doing is knowing, and all knowing is doing, 25 and the term “enaction” is intended to remind us of the primacy of action in shaping our experiences, our perceptions, and our world. ([Location 748](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=748)) - The individual's cognition, in this conception, is analogous to the evolution of a species, whereby an idea or an action comes about not because it is “correct” or optimal, but because it is possible in the given context. ([Location 753](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=753)) - An important upshot of this move from “the world as given” to “the world as unfolding through a choreography of action” is that the modern Cartesian desire to know the world as it is is thoroughly frustrated. The universe is constantly evolving, forever eluding any attempt to fix and to know it. It is thus that, for the enactivist, the world is not preformed, but performed. We are constantly enacting our sense of the world—in the process, because we are part of it, altering it. Varela, Thompson, and Rosch refer to this phenomenon as the “fundamental circularity,” whereby they suggest that the universe changes when something as minuscule as a thought changes—because that thought is not merely in the universe; it is part of the universe. ([Location 782](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=782)) - complexity theorists distinguish between systems that are complex (i.e., which are self-organizing, adaptive, and spontaneously evolving) and those that are merely complicated (e.g., mechanical devices), in the process dislodging the “clockwork universe” metaphor that was initiated by Descartes and his contemporaries and replacing it with a more holistic, organic model of the cosmos that hearkens back to premodern conceptions. ([Location 816](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=816)) - No longer believed to be the foundations to or the building blocks of reality, mathematical objects are now offered as possible models of or metaphors for varied phenomena. ([Location 824](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=824)) - hermeneutics is the art of interpretation. It is interested in meaning, in understanding, and in application. More particularly, hermeneutics is concerned with investigating the conditions that make certain understandings possible. It asks not only, What is it that we think? but also, How is it that we have come to think this way?—all with a view toward affecting how we act in the world. Hermeneutics is thus concerned with past, present, and projected understandings. ([Location 870](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=870)) - a motto for hermeneutics might be that “truth keeps happening.” 3 ([Location 882](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=882)) - Etymologically, hermeneutics is derived from the name and character of the Greek god Hermes, the messenger of the gods—and, not insignificantly, a trickster. The term thus echoes with senses of revelation, of coming to more profound understandings of the previously perplexing or paradoxical, and of a wariness of being overly certain. ([Location 886](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=886)) - An etymology of “text” is helpful here, in part because the term is conventionally understood to refer strictly to written works. Originally, however, “text,” like “web,” was used to describe things woven, and so the metaphor of “life as text” does have a particular richness. Considered alongside the more popular “literature as text” metaphor, for example, the image of intertwined linguistic threads forming a tightly woven narrative fabric foregrounds the roles of language, of storytelling, and of rereading in the construction of our respective understandings and identities. The textual metaphor also offers an image of the interweaving of our selves in the fabric of our culture. Like the microscope and the telescope, which demonstrate that complexity is not a function of scale, the metaphor of text applied at different conceptual planes reveals that the web of existence is as tangled at the individual level as it is on a planetary level. And, perhaps more importantly, the references to text, textuality, and texture remind us that we ourselves are woven into the fabric that we seek to understand. It is from inside our traditions that we interrogate them. ([Location 901](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=901)) - We did not design our world. We simply found ourselves in it; we awoke both to ourselves and to the world we inhabit. We come to reflect on that world as we grow and live. We reflect on a world that is not made, but found, and yet it is also our structure that enables us to reflect upon this world. Thus, in reflection we find ourselves in a circle: we are in a world that seems to be there before reflection begins, but that world is not separate from us ([Location 923](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=923)) - David Smith notes that, beginning with the contribution of Schleiermacher, the three common themes in hermeneutic inquiry have been “the inherent creativity of interpretation, the pivotal role of language in human understanding, and the interplay of part and whole in the process of interpretation.” 9 This third theme, the back-and-forth movement between the particular and general, is more popularly referred to as the “hermeneutic circle.” As one moves between the specific and the broad, one's understandings of both are deepened, and all other understandings are also affected. ([Location 932](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=932)) - the “object” of the hermeneutic inquiry is a moving target. As we study our conception of mathematics teaching, for example, our understanding of teaching—that is, the very “object” of our inquiry—changes … in part because of our efforts to understand it. 12 One of the central concerns of hermeneutics, and one of the reasons for a cyclical mode of inquiry, is the question of how one might go about inquiring into a phenomenon in which one is immersed, entangled, and complicit. ([Location 949](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=949)) - hermeneutic inquiry cannot be conceived of as a linear process. While we as yet lack a word to describe the sort of path that might be taken through the research, terms such as recursive, circular, and reflective provide some sense of the process. ([Location 953](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=953)) - Hermeneutics, then, seeks to undo our habit of “writing backward”—that is, of weaving narrative strands that serve to impose structure on and that enable us to extract meaning from an amorphous mass of, at the time of living through it, largely unformulated experience. ([Location 955](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=955)) - Hans-Georg Gadamer 16 has provided us with a provocative exploration of this issue, arguing that the relationship between the research question and the phenomenon under study is not uni-di-rectional, but reciprocal. Briefly, his suggestion is that the topic of investigation, at least in part, reveals the manner in which it should be investigated. Further, the phenomenon is shaped by the way we inquire into it—that is, by how we structure our question. Physicist Werner Heisenberg made essentially this same point in his famous statement, “What we observe is not nature itself but nature exposed to our method of questioning.” ([Location 982](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=982)) - The hermeneutic critique of conventional scientized approaches to research is not that those methods are wrong or inappropriate, but that they are narrow and inevitably lead to a particular (i.e., abstracted and ostensibly objective) sort of truth; they do not allow the researcher to be aware of, let alone to move outside of, a particular interpretive frame. Such research, we might say, is guided by the reductive question, “What is … ” rather than the expansive, “What might be… ?” In contrast, successful hermeneutic research does not seek to close the doors of inquiry by arriving at some “answer” or uncovering some Truth. Rather, it seeks to open the doors wider, permitting both writer and reader to see their positions in a more open way. ([Location 989](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=989)) - “Hermeneutics” refers neither to a particular (instrumental) approach to research nor to a unified field of inquiry. Radier, it addresses a broad range of topics and issues. The unifying theme in hermeneutics is a persistent questioning of our taken-for-granted modes of speaking and acting. ([Location 1036](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1036)) - It is important to draw an initial distinction between a conversation and a discussion at this point. (The distinction is elaborated upon in the next section.) In the conversation, all of the participants are oriented toward deepening their understanding of the issue at hand. In a sense, then, the subject matter conducts the participants and there is a quality of self-forgetfulness as all concerned come to understand that they share in the truth of the interaction. The goal of the discussion, in contrast, is more toward the articulation of pre-formulated ideas, and so the subjects endeavor to exert some measure of control over the subject matter. The emphasis in the discussion is placed on the subjects' conceptual differences rather than on achieving a consensus. Rather than a forgetting of selves, there is a concretizing of subjective positions; horizons are not placed at hazard in the discussion. ([Location 1054](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1054)) - Quite unlike the discussion, then, the conversation is fluid, meandering its way toward a destination that is not specific, but that will be commonly known. That the destination is unspecified and unanticipated is the strength of the conversation for, by being unconcerned with reaching a particular point (i.e., relinquishing the modernist desire for control)—by allowing the path to be laid down in walking—the participants are able to listen to the particularities that shape that path. The goal of the participants in a discussion, much in contrast, is often to remain rigidly in place, to be unswayed. ([Location 1061](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1061)) - we can never be aware that a conversation is taking place. We can, however, be aware that one has taken phce. When understandings have changed, when a new commonsense has been established—when self and other have been altered—it has happened. ([Location 1079](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1079)) - The movement away from culturally privileged research methodologies toward a hermeneutic inquiry might be described as a shift from looking to listening. We tend to use visual metaphors in our descriptions of scientific and scientized approaches. (A glance in a thesaurus for synonyms of such terms as “understanding” and “investigation” is most revealing.) Many science texts use the image of a disembodied eye to represent the scientific attitude, reinforcing the notion that, through vision, the subjective observer is separated from objective reality. The necessity of such separation is hardly surprising since the word “science” is derived from a term that meant “to cut apart.” This scientific gaze, insofar as it is applied to issues in a social context, also tends to “freeze” phenomena. (This tendency is powerfully revealed in the pervasive use of statistics—derived from the Greek states, one that stops or steadies—as a basis of interpretation.) 23 ([Location 1104](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1104)) - In contrast, the hermeneutic text endeavors to avoid these distancing and fixing activities, and this is why explications of hermeneutics tend to employ auditory metaphors. The notion that conversation is foundational to the hermeneutic inquiry, for example, has a figurative as well as a literal relevance as we seek to locate ourselves in the wider conversations of history and context. Further, the goals of hermeneutic research are toward attunement and harmony amid the noise of existence. One is concerned with theme, with tone, with rhythm, with resonance. As such, one is constantly reminded that understanding, like sound, is fleeting and unfixable. Further, unlike the isolating tendencies of vision, sound incorporates. Sound pours into the listener, whereas the object of sight exists outside the observer. Walter Ong elaborates on this point: Vision comes to a human being from one direction at a time: to look at a room or a landscape, I must move my eyes around from one part to another. When I hear, however, I gather sound simultaneously from every direction at once…. You can immerse yourself in hearing, in sound. There is no way to immerse yourself similarly in sight. By contrast with vision, the dissecting sense, sound is thus a unifying sense. A typical visual ideal is clarity and distinctness, a taking apart…. The auditory ideal, by contrast, is harmony, a putting together. … Knowledge is ultimately not a fractioning but a unifying phenomenon, a striving for harmony. ([Location 1114](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1114)) - We tend to take for granted, for instance, that we are insulated and autonomous individuals, that a mysterious substance called “information” can flow between us as we interact, that we are somehow in control of what is said and what is heard. In some current educational discourses, these ideas have coalesced into such notions as voice and empowerment— notions that acknowledge the ineffectiveness of communication in today's settings, but which reify instead of dismantle the modernist separations that underlie this ineffectiveness. Rather than seeking to promote conversation, we are compelled to perpetuate the model of competing monologues. The goal of such emancipatory discourse, it appears, is to promote listening—but by force. The reasoning seems to be that by developing more powerful voices, we will be able to reach across separations and we will be able to compel others to see things from our perspective. In brief, we want people to think the way we think. ([Location 1196](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1196)) - I contend that we do not need to amplify our voices in an effort to overcome chasms and walls. Rather, we need to realize that those barriers are not really there, and a deeper understanding of listening will enable us to dispel those pervasive illusions. ([Location 1204](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1204)) - within our culture, there is a pervasive use of visual metaphors to describe the facets of education. We see learning as gaining insight, intelligence as brightness, investigation as looking, understanding as seeing, opinions as perspectives or views, hopes as visions, and (very often) teaching as supervision. More broadly, tendencies to associate truth with light, believing with seeing, and objectivity with the distance afforded only to the observer, point to the overwhelming domination of vision over the other senses. ([Location 1237](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1237)) - Given the different manners in which hearing and sight situate us in relationships with others, it is perhaps not surprising to note that relationships among those who are hearing-impaired are often more strained that among those who are visually impaired. Berendt, 7 for example, comments on the very different sorts of interactions that occur among residents of homes for the deaf relative to those among residents of homes for the blind. The former tend more toward aggression and discord, the latter toward caution and compromise. And Helen Keller, in her letters and journals, frequently commented that the problems of deafness are “more complex” than the problems of blindness, for the deaf person is cut off from others in a much more profound way. ([Location 1255](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1255)) - There is a particular bodily aspect to listening, a visible orienting to the subject of the discussion. When two persons converse, for example, it can be seen that they are listening to one another as the actions of their bodies become bodily interactions. They lean into and reach out for one another, momentarily unaware that they are violating the Western taboos on proximity, touch, and extended eye contact. They seem to focus in a way that suggests they are oblivious to the noise around them; they attend to each word and to each action as though nothing of importance occurred prior to the discussion and nothing of importance awaits them at its end. They are unconcerned that their voices are perhaps too loud, that their bodies are too animated. Listening, then, need be neither motionless nor silent (although more often than not, it seems, it is precisely this sort of inactive attention that is demanded of students by teachers). Of course, the listener may assume this posture, but it is something other than an audience's lack of motion or their silence that makes us aware that they are listening. In the classroom, for example, as the novel is read or the mathematical principle emerges, the teacher knows the students are listening not because they have ceased to move but because a certain rhythm or harmony is established—there is an awareness that each is immersed in and conducted by the same subject matter. The gazes are fixed not on the teacher nor on one another, but on that which is among them. That the listener should not be thought of as the quiet partner in an interaction is suggested by our use of the term “sounding board.” Just as the sounding board in a musical instrument is intended to resonate with, echo, and amplify the sounds generated by other parts of the instrument—that is, the sounding board participates in the music generated—so the listener who acts as a sounding board participates in the emerging conversation. ([Location 1269](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1269)) - To discuss originally meant to “shake apart,” and its emphasis on separation continues to echo in our current use of the term. To converse, much in contrast, had a meaning more toward “to live with” or “to keep company with.” This is why we use “conversation” when referring to interactions with friends and “discussion” when speaking of meeting with strangers, professional colleagues, and business contacts. The conversation-discussion distinction also points to a, perhaps unfortunate, trend in teacher-student interactions: as recent developments have alerted us to the importance of active interaction while learning, we have tended to opt for “classroom discussions” rather than conversations. Not surprisingly, there doesn't seem to be much listening happening within such settings, ([Location 1300](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1300)) - [Conversations] move beyond mere coordination and have a common rhythm. The interlocutor not only listens but participates with head nodding and “unh-hunh” and the like, and at a certain point the “semantic turn” passes over to the other by a common movement. The appropriate moment is felt by both partners together in virtue of the common rhythm. ([Location 1308](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1308)) - Unlike the discussion, which lacks the qualities of rhythm and of “living with the other,” the conversation's path is neither predictable nor controllable. Nor would we want to prescribe its route or its outcome because, again in contrast to the discussion, the “purpose” of the conversation is as much the act of conversing (i.e., “living with others”) as it is the development of a deeper understanding. There is no winner, no gaining of the upper hand, no final word, no compulsion to stick with the topic. Rather, the conversation allows us to move freely and interactively toward those questions that animate us while enabling us to explore not just the topics that emerge, but why such topics capture our interest in the first place. ([Location 1316](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1316)) - “hearing and touch meet where the lower frequencies of audible sound pass over to tactile vibrations (at about 20 hertz). Hearing is a way of touching at a distance.” ([Location 1398](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1398)) - As for the belief that listening is passive, Straus 27 reminds us that this pervasive notion is founded on a troubling distinction that tends to be drawn between perception and action. The inappropriateness of that idea is evidenced by our bodily actions in listening: we raise, tilt, and turn our heads, we move closer, we hold still other parts of our bodies. Listening—likes smelling, tasting, touching, and looking—is a reaching activity, a motor activity, an en-activity. ([Location 1406](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1406)) - Hearing and listening, then, are different phenomena. Consider, for example, the difference in intended meaning between the two statements: “I can't hear” and “I can't listen.” In uttering the former, my concern is that the sound isn't loud enough. It is something I say when I want to hear but, for whatever reason, cannot. The concern is strictly sensory. In uttering the latter, however, I am suggesting that I am able to hear the sound without difficulty. When I say, “I can't listen,” I'm not saying that I can't hear but that I won't hear. To make this point clearer, the statement “I can't hear” is often followed with “Turn up the volume.” But the statement “I can't listen” tends to be accompanied by requests more along the line of “Turn down the volume.” Listening is thus a capacity which is founded upon hearing but which goes beyond hearing. It is orienting (we listen to something) and oriented (we listen for something). Hearing, in contrast, lacks such intentionality. ([Location 1436](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1436)) - Hearing presumes understanding and, when we cannot comprehend someone, “we can't hear a word he's saying.” In contrast, the statement “I'm listening” implies a recognition of the preeminent role of interpretation in our interactions. It is when we perceive gaps in our understandings that we are compelled to listen. ([Location 1461](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1461)) - Quite apart from the capacity to “pull in sound,” hearing is more a process of imposing order on noise—of bringing forth a sound—and there is physiological evidence to support this contention. There are approximately 10 percent more neural connections running from the brain to the ear than from the ear to the brain, indicating that sensory organs do not merely “take in” incoming information; they go fishing for it. To listen, then, is to subject our hearing perceptions to scrutiny, endeavoring to disrupt the “taken for granted” which precedes, constrains, and (in effect) determines those perceptions. “Listening” does not suggest an ability to transcend such constraints, but a willingness to “unfix” our selves or to position our selves differently. ([Location 1520](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1520)) - Tags: [[favorite]] - Ong argues that knowledge in oral traditions exists only in action. 14 It is thus local, current, practical, and fluid—as necessitated by the fact that ideas exist only in an oral milieu. ([Location 1711](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1711)) - By consequence, oral cultures are uninterested in definitions; meanings are implicit by usage and through enactment. Where there are gaps in understandings between persons, these are immediately negotiated. In contrast to the literate's location of meaning in language, the meanings of oral traditions are located in contexts and actions. Spoken words come into being in situations (whereas written words tend to be isolated from the setting in which they were recorded). It is thus that persons from oral societies resist—or may simply be unable to—provide definitions for familiar words; for them, words are not items but patterns of acting. ([Location 1716](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1716)) - The Inuit people in Carpenter's study had no concept of equal portioning (i.e., knowledge of fractions or processes that we would describe in terms of fractions); a universe structured auditorially need not be divided into “equal shares.” That is to say that the concept of ratio—the term from which “rational” and “reason” are derived—is not part of their oral culture. ([Location 1739](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1739)) - how the adjectives (and, later, verbs) became nouns— is an interesting question. Ong suggests that the ability to draw such abstractions is linked to the invention of the written word, a technology that, he argues, brought about a transformation of human consciousness. According to Ong, writing pushes the known into the visual field, detaching it from its author and its audience by assigning it permanence and reducing its mutability. ([Location 1772](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1772)) - In fact, in a very real way, words (like numbers) come into existence through the technology of symbolization. They become things with objective status, and thus objects that can be operated upon. Writing also demands that these fragments be presented linearly, and the resulting chains of reasoning mark a profound break with the all-at-once thinking modes of the oral traditions. In a further break with orality, definitions become important as one moves to the visual milieu of writing; lacking the space to interrogate the author, usage must be clear, unambiguous, and uniform. ([Location 1779](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1779)) - Tags: [[favorite]] - [Descartes] had been the first to embark upon a programme to establish a firm foundation for human knowledge of the world and had singled out mathematics as the only reliable route to unimpeachable knowledge. 34 In other words, Descartes called for nothing less than “the primacy of world mathematization” 35 whereby all truth—and not just that of mathematics—would be determined through a formal system, complete with a predefined and carefully articulated alphabet, grammar, set of axioms, and rules of inference. This transitional period is thus marked by three critical shifts in thinking: mathematical ideas came to be seen as something apart from human experience; the mathematical and the scientific were given identities distinct from the religious, the magical, and the spiritual; and mathematics acquired the status of the model of reasoning for a modern era. The overarching goal of this movement—or, perhaps more appropriately, its net effect—was prophesied in 1637 by Descartes in his Discourse on Method: [My discoveries] have satisfied me that it is possible to reach knowledge that will be of much utility in this life; and that instead of the speculative philosophy now taught in the schools we can find a practical one, by which, knowing the nature and behavior of fire, water, air, stars, the heavens, and all the bodies which surround us, as well as we now understand the different skills of our workers, we can employ these entities for all the purposes for which they are suited, and so make ourselves masters and possessors of nature. 36 All sense of our ecological embeddedness was cast aside in this formulation. ([Location 1868](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=1868)) - The key to the enactivist alternative is found in its challenges to two premises of the subjective-objective debate—namely, the belief that mental operations and physical actions (i.e., mind and body) are in some way independent and separable, and that individual knowing agents are isolated from one an-other and from the known world. ([Location 2053](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2053)) - Far from merely representing the universe (thus setting us apart from it), then, our mathematics presents 69 the rhythms of the planet and the patterns that are repeated in all forms and at all levels of life. It does not reduce the universe, but places us in conversation with it, hinting at the complex orders and the tangled relationships that inevitably exceed our attempts to understand and surpass our efforts to control. Conversely, our mathematics also presents us (i.e., “makes us present”) in these harmonies, enacting not the modernist separation, alienation, and exile from the natural world, but an attunement to the pulse of the planet. ([Location 2111](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2111)) - We should never forget that a stroll in the woods or a deep conversation with a new or old friend are beyond mathematics. And then, when we go back to our jobs as administrators, teachers, or whatever, let us still remember that numbers are only the shadow, that life is the reality. ([Location 2149](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2149)) - Unfortunately, like the static appearing glacier, the transitional rate of a culture's collective knowledge has traditionally been so much slower than that of an individual within that culture that this knowledge tends to be “taken for granted”; it is the given that precedes the preparation of curriculum documents. Losing sight of its movement and the mass that lies behind it, we have tended to excavate bits from the front end of this glacier and to offer them as reasonable representations of the remainder. ([Location 2247](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2247)) - A complex and poorly understood phenomenon, human learning has proven itself to be tremendously adept (but wildly unpredictable) at adapting to the contingencies of existence: one never knows exactly what one will learn—just as, on a broader level, one can never predict the directions in which mathematics and other facets of collective knowledge might evolve. ([Location 2295](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2295)) - Temporally, in the movement from prescription to proscription, the concern shifts from a privileging of the future to a greater attentiveness to the present and to the immediacy of interpretation and action. One's focus is thus set not on the path (because the course has not been predetermined) but on negotiating a path: on currere, running; on the instant of interpretation; on doing. The contents of a mandated program of studies might thus be interpreted as outlining areas for exploration, rather than as specifying where each step will land. (“Curriculum” thus comes to be something that can only be discussed in retrospect, as the path that was taken, in all its experiential richness.) ([Location 2328](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2328)) - Alan Bishop, 28 in describing commonalties among mathematical systems across cultures, suggests the following: counting, locating, measuring, designing, playing, and explaining. While he does not take it to the level of individual activities, I would propose that an important feature of a mathematical task would be that it include most or all of these elements—all-at-once. ([Location 2428](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2428)) - In addition to being all-at-once, other features of a mathematical task are that it be variable-entry 29 (i.e., constructed so that learners, regardless of their backgrounds in mathematics, are able to locate themselves and negotiate the difficulty of the tasks they set for themselves), rich (i.e., presenting the possibility for diverse, unex-pected, and extensive mathematical interpretation), and open-ended (i.e., raising more questions than answers). As well, I believe that a mathematical task should involve more than a single mode of reasoning (i.e., rather than focusing exclusively on logico-deductive thought, one should be able, for example, to draw analogies between mathematical constructs and other categories of experienced phenomena and to locate mathematical activity in the context of stories), include opportunities for both independent and collective action, and present an opportunity for embracing the emotional/affective dimensions of mathematical inquiry (and, in particular, the possibility of profound personal engagement—and perhaps even ob-session—of the sort that one rarely witnesses in the context of a fragmented and decontextualized mathematics). Finally, and perhaps most importantly, a mathematical task should have the capacity to serve as a focal point for joint, collective action. That is, it should allow for both solitary and shared exploration: as a sort of common text around and through which learners can come together. In sum, a mathematical task should impose “liberating constraints” 30 which are intended to strike a balance between “complete freedom” (which would seem to negate the need for schools in the first place) and no freedom at all. ([Location 2432](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2432)) - of the many pages of records and field notes that were assembled over our lengthy collaboration, it was the interactions on the topic of the nature of mathematics (and on the companion project of planning for teaching) that were the most difficult to transcribe. They lacked the I-speak-then-you-speak structure of most of our discussions. Instead, they were filled with interruptions, pauses, incomplete thoughts, exclamations, laughter. In short, they were conversations, and they thus resisted the flattening process of transcription. ([Location 2850](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2850)) - Geometry, then, is not merely earth-measurement; it might be thought of as a pressing of one's body (metron) against the living world (Gaia). And so, in its remotest origins, geometry was, perhaps, an exemplar of ecological thought. Rising out of the efforts of our ancestors to, quite literally, measure the earth, one can envision those earliest mathematicians on their hands and knees—ears pressed to the ground, as it were—coaxing secrets from the living Earth. ([Location 2904](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=2904)) - Just as the movement from objectivist to subjectivist accounts of learning and understanding amounted to little more than a transference of monologic authority, the shift from the conventional teaching as telling to the constructivist teaching as orchestrating amounts to little more than a renewed attempt to prescribe or control the learning that is to occur. ([Location 5118](https://readwise.io/to_kindle?action=open&asin=B00B9KBOV4&location=5118)) - Tags: [[blue]]