Innovación Tecnológica
5.4 Procedimientos de recolección de información
The preceding presentation of themes naturally includes a measure of discussion, but in this section it seems advisable to interpret, at least briefly, the four themes more holistically. Because this is a study of community connections with mathematics curriculum and instruction, the interpretation will attempt to relate the themes in that context.
One anecdote, contained in fieldnotes and not in the interview transcripts, bears telling at this juncture. The grandfather of one of the lutherie-class assistants, near death at the time and in a nursing home, asked that his grandson use wood from a fallen buckeye tree in his front yard to make something special before he died. The raw buckeye lumber became the ukulele displayed in Figure 4. At the time the researcher talked with the assistant, the grandfather was still alive. The ukulele is suspended on a makeshift hanging rack after having been sprayed with sealer.
This story speaks to the core of the previous theme, and strongly echoes the “life of its own” observation so reverently offered by a parent of one of the students. This ukulele, produced in the class by a community member (not
a professional educator and not a student) is a tangible link connecting land, family, community, and school. Such manifestations may or may not be very rare, but they are, at least in the experience of the lead authors, seldom honored or celebrated. It is well at this juncture to remember that this connection was enabled by a teacher of high-school mathematics.
FIGURE4 //Buckeye Ukulele
True, little formal high-school mathematics is on view in the lutherie class. Students are not learning the content of Algebra I, Geometry, or Algebra II there. Possibly some of the students in the course are not even “taking” the full American triumvirate, or even any part of it. Nonetheless, the study heard convincingly from students that they learned a range of practical performances previously inaccessible to them. The most lofty of these practical performances was problem solving: not just the capacity to solve the problems materializing in the lutherie class, but, seemingly, an overall disposition toward problem solving as an approach to everyday life. It’s not clear that one could actually learn much that is more valuable, and certainly not in school; but the ghost of John Dewey surely nods approvingly at such learning.
What is one to make of the students’ confessions about mere arithmetic and incapacity with the use of measurement tools? Some mathematics teachers (a small minority, but evidently including Tony Perdue and his superintendent) do seem to regard arithmetical competence in everyday life as mathematical per se (e.g., measuring, computing proportions, reading blueprints, and perhaps a disposition toward problem solving). But many readers of this report would not be convinced. Indeed, mathematicians often regard school mathematics as easy math, and wonder what the big deal might be in teaching it (Robert Klein, personal communication). The reported mathematical connections would seem trivial to them. Mathematics
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57 educators (professors of math education), for their part, also want high
school mathematics instruction to move students well beyond arithmetic and computation to a range of “mathematical objects” and to a grasp of mathematical ideas, logic (i.e., proof), and problem-solving. These aspects of mathematics, however, are not logically the province of school math mostly or even in part, but must exist inherently (immanently) in everyday adult life,21 where they remain for the most part almost inexplicably hidden—as seemingly invisible there to math teachers as to everyone else. Flushing them into the open and exploiting them for instruction would seem to be excellent work for teachers in league with local adults, perhaps with some help from professors (Civil, 2005; Brenner & Moschkovich, 2002).
The story of the logarithmic scale underlying the layout of frets in lutherie work is of course an example: people mention it as really existing, but the making of instruments trumped engagement with the idea and the history of its application to music-making (see, e.g., Wardhaugh, 2008, and Isacoff, 2001, for the relevant mathematics and the musical context of temperament). The educators, however, seemed clear that the lutherie class was not a math class, but rather that math and science were implicated in the class, in much the same way in fact that significant mathematics remains implicated but hidden in everyday adult life. The class involved students, at least obliquely, in applications of math and science, and it was the applications per se and not instruction in math and science that had everyone’s attention. It’s also evident from transcript data that at least some, and probably many, students regarded the experience in the class as memorable and even influential, far beyond their other experiences in school, and the testimony deserves repeating:
Such testimony from students is itself significant (and much in line with what the study heard from students at other sites), and those concerned to see “significant mathematics” engaged in formal schooling should pay attention. What does such testimony mean? Most of what it means is well represented in the progressive pedagogical literature of the past 100 years: the familiar learning-by-doing argument, and the many reports of students’ and teachers’ memorable experiences of such engagements. Jack Shelton calls it “consequential learning” (title of his 2005 book). It might help to state the principle of this sort of learning: the ends dignify the means. This aphorism works at several levels, as follows.
First, at the classroom level, the ends in the lutherie class were musical instruments, objects that commanded great respect locally: dulcimers, guitars, fiddles, and mandolins. The means were techniques, tools, and dispositions—
the most notable being those considered in the second theme: “frustration, patience, and problem-solving.” Implicit in the means lay significant math and science content. This content was not formally engaged, but the point is that in such a context, under the influence of such ends, it could be—and not that it wasn’t. As a project-based pedagogical project evolves, it harbors the immanent possibility of dignifying formal mathematical content incrementally as part of the evolving pedagogy. And why not? Inventive teachers don’t invent all their routines overnight from whole cloth.
Second, the resident dignity inheres (again: is immanent) in the relationship between context and content. “Decontextualized learning” is an oxymoron because authentic competence requires a context, not only as it emerges, but for its exercise. As a performance becomes increasingly competent, moreover, the line between the emerging-instructional phase and the competent- performance phase vanishes. The insight here is that the learning and the competence are one and the same: one doesn’t learn and then experience a transformation—say, during the flawless completion of the items on a valid and reliable test—that results in competence. No. The application is essential to the learning and the competence because they are the same thing. Schooling that banishes context banishes a great deal of learning thereby. The invisibility of this loss is quite similar to the invisibility of mathematics in the everyday: we’re used to it, we don’t much care.
Third, the question of ends is momentous. The ends in the lutherie class may seem to have been (at one level) dulcimers and guitars. As with the parent who attributed “a life of their own” to these objects, however, these objects embody a meaningfulness widely acknowledged across the community. In this sense the lutherie class aimed at engaging a broad range of students in musical competence. But at a much higher level of abstraction, the ends of the lutherie class as a high school course may have been something like the meaningful participation of the school in sustaining local culture.
This observation brings the discussion to the topic of rural schools serving their communities. One might observe that the lutherie class, in its small way, could evolve as part of the strategy commended by Carr and Kefalas (2009), precisely because it helps middle-class youngsters get to know their equals from the “stereotypical redneck culture.” This is a strange observation to make of a contemporary comprehensive high school—the kind of high school that was intended to make such acquaintances normal (Conant, 1959), but didn’t. In any case, to continue to serve this larger purpose, of course, the lutherie class would have to be sustained, and similar courses would have to be added: as the superintendent worried, much remains to be done.
21That is, in order for any formal curriculum (even a dated one) to claim them.
I’ll take the knowledge with me, whereas in other classes I won’t; it’s just useless, another credit.
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