What Does Understanding Mathematics Mean for Teachers? by Handa Yuichi

Author:Handa, Yuichi
Language: eng
Format: epub
Tags: Education
Publisher: Taylor & Francis
Published: 2011-02-27T16:00:00+00:00

Understanding as Verstehen : “Standing before” so as to Unlearn Mathematics

In this chapter, I have explored the meaning and nuances of a will as would be joined with grace in the image of “staying with a problem,” in negative capability and Dewey’s notion of perception (vs. recognition), in the seeking and welcoming of difficulties, in intentional learning, and finally, in a manner of mindful repetition (vs. a rote one). As I have suggested at the beginning of this chapter, each affords an experience of undergoing—the topic of the previous chapter—whether as in an affective or aesthetic response, or as an experience of understanding or “aha.” The point being that, as Buber suggests, it is when “will” and “grace” are joined that one arrives at true relation—not one without the other.

In closing the last chapter, I suggested an image of understanding as connaitre, or as an undergoing event animated, informed, and structured by personal involvement. Yet, just as “grace” without “will” offers only partial description of the phenomenon of entering into relationship, understanding as an undergoing, I would suggest, presents also only a “half-picture.” Thus, in the spirit of pondering the other “half”—that is, the complement to understanding as undergoing—I consider the notion of understanding as the modern German verstehen (deriving from Middle High German, verstan, meaning, “to stand before, or in front of”).9

Note the closeness of its etymological imagery to Gattegno’s “staying with a problem,” as well to singer-songwriter Jack Gladstone’s “(Bison) facing the storm.” Further, verstehen’s “standing before” describes well what underlies mindful repetitive action. In such a repetitive act, a person brings his or her own being, again and again, so as to “stand before” some idea in order to experience the unveiling of meaning and understanding. All suggest an exertion of will and effort coupled with an affordance of agency toward the other that one is facing, standing before, staying with, so as to make allowance for emergence. It is will as would join with grace.

At the same time, there is another shade of meaning implied by understanding as “standing before.” It is that to “stand before” is also to put oneself on the line. Standing before does not just imply a persevering through trials, and overcoming them. It also intimates a putting of oneself “out there”—not slinking away in a comfortable hideout by sitting near—but standing oneself in a place that is vulnerable to the influence of the other. It is the risk-taking and courageousness required of learning and of passion, which I spoke of in Chapter 2. In mathematical terms, it could take the form of sharing one’s preliminary thinking with a colleague, or of unearthing one’s own assumptions to oneself.

Part of what can happen when one learns something—not just in an additive sense of tacking more knowledge onto what one already knows (the savoir knowing), but in a transformative sense that would involve a qualitative shift in one’s knowledge state (understanding as connaitre)—is that one loosens one’s hold of old ideas and concepts for sounder ideas and concepts.

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