“It’s Understandable Enough, Right?” The Natural Accountability of a Mathematics Lesson

ABSTRACT

A fundamental problem for all educational reform efforts is one of specifying in practical terms just what the reform requires of teachers and students. In addressing it, educational designers have introduced the term “accountability” to characterize something missing in classroom discourse. This term, however, held special significance in the writings of the American sociologist Harold Garfinkel. Livingston applied Garfinkel’s treatment of accountability to understanding the lived-work of doing mathematics, particularly proving work. We examine an 8th-grade student’s presentation during a Japanese geometry lesson as a proof-account. Within it we see elements of both classical and natural accountability placed on display.

ACKNOWLEDGMENTS

We thank Jim Stigler and Brian MacWhinney for facilitating access to the materials upon which this analysis was made. We also thank Christian Greiffenhagen, Patricio Herbst, Gerry Stahl, Alan Zemel, and two anonymous reviewers for helping us work out what this article was actually about.

Notes

¹ The term Lebenswelt comes from Husserl (1970) and can be literally translated as ‘life-world.’ It is “the mundane world of lived experience already existing as a product of the unreflecting cognitions of ordinary actors” (Heritage, 1984, p. 44). Note the discovery that proofs have this paired structure is an ethnomethodological discovery. It was one of the principal findings of Livingston’s thesis (1983).

² Examples would include: “read*” (p. 146), “revealed details*” (p. 187), “worksite details*” (p. 187), “naturally accountable details*” (p. 188), “precise description*” (p. 188), and most importantly “order*” (p. 146, FN1).

³ This reflexive definition closely resembles Garfinkel’s (2002) definition of an “instructed action” (pp. 105–106) and this is not an accident. Both proof*s and “instructed actions,” as Garfinkel conceptualizes them, are Lebenswelt pairs. Indeed, a proof-account might be considered to be a special kind of instruction.

⁴ Garfinkel (2002) referred to this as “praxologizing” (p. 149) the description. For a critical take on Garfinkel’s notion of tutorial problems and of “mis-reading” accounts, see Wilson (2003).

⁵ Another example of a proof* presented as a tutorial problem can be found in Livingston (1987, Chapters 14–16).

⁶ Classes in Hong Kong were mostly conducted in English; classes in Switzerland were conducted in German, French, and Italian. Other languages included Czech, Dutch, and Japanese.

⁷ http://talkbank.org/media/ClassBank/TIMSS-Math/Japan-unlinked/744/

⁸ The transcripts were prepared by one of the authors, a native speaker of Japanese.

⁹ The full set of conventions is described in Jefferson (2004). In summary, numbers enclosed in parentheses represent periods of silence measured to a tenth of a second. Brackets are used to mark talk or other forms of action delivered in overlap. Use of standard punctuation marks such as periods and question marks denotes delivery with falling or rising intonation resembling that ordinarily heard at the end of a sentence (or question). Colons are used to display sound stretching. Text enclosed between degree signs represents talk delivered at diminished volume. Annotations are enclosed in double parentheses and italicized.

¹⁰ The following symbols are employed for Japanese function words: Cop (copula), FP (final particle), LK (linking marker), Neg (negative), Nom (nominalizer), Top (topic marker), O (object), Q (question particle), QT (quotative particle), and S (subject marker).

¹¹ These pseudonyms come from the transcript in the TIMSS database.

¹² Or, maybe not. Were this to occur, it is likely that the teacher would challenge the class to develop an alternative solution, there being another available.

¹³ As an alternative way of appreciating Mr. Manabe’s solution, try superimposing his various demonstrations at the board captured in Figure 8 upon the theorem illustration shown in Figure 3. Recall that in Figure 6 the teacher had the students change their frame of reference in order to more easily visualize how the theorem illustration related to the solution figures. You can accomplish the same by rotating Figure 8 counter-clockwise 90°. Mr. Manabe’s first referenced triangle (Figure 8a) can then be seen as corresponding to triangle ACB in Figure 3 and the conforming triangle (Figure 8c) to ADB. Continuing on this basis, the line segment DB would represent the new property line demonstrated by Mr. Manabe at [0:20:50;14]. Because, by the stated theorem, triangles ADB and ACB have the same area, Chiba’s property holdings must remain the same when using the newly straightened property line. And, if Chiba’s holdings remain constant, it would stand to reason that the same is true of Bando’s.