This study analyzes the representation of a conception of tangency in different geometry textbooks. Seven textbooks were sampled from one publishing house at approximately 15-year intervals, from 1903 to 2001. Within each textbook, exercises that involve reasoning about lines that are tangent to circles were identified and analyzed according to the conceptions-knowing-concept (ck¢) model of student conceptions (Balacheff & Gaudin, 2010).  According to the ck¢ model, a conception is a state of equilibrium between students and a learning milieu—the set of epistemic guideposts students use to make sense of (and decisions about) the mathematical acts they do. In this way a conception can be modeled as a quadruplet: a set of problems, a set of operators, a semiotic register, and a control structure. Textbooks are one aspect of a student’s milieu that give access to the mathematical knowledge that would be possible for a student to learn, were a student to read, complete, and understand the material presented in the text. In this way, textbooks are valuable records of intended curriculum (Mesa, 2004). The textbooks in the sample were taken as repositories of problems (the exercises in the books) and control structures (the axioms, definitions, and theorems that concern lines drawn tangent to circles) for conceptions of tangency that modal students could exhibit, were they to use that textbook to learn geometry. The study tracks how a specific conception of tangency—that of the “point of contact” or “point of tangency”—is represented in 20^th century geometry textbooks.

This study takes a cross-section of the point-of-contact conception of tangency as it is developed in different geometry textbooks. This conception was chosen as the organizing knowledge item for this study for three reasons. First, the tangent-circle relation is fundamental in plane geometry, so conceptions of point-of-contact were likely to be treated in any secondary geometry textbook. Second, previous studies of conceptions of tangency suggest that there is a divide between “empirical” and “mathematical” conceptions of tangency (Schoenfeld, 1997), with the specification of the point-of-contact being the point of departure for this distinction. This study builds on these observations by tracing the history of how the point-of-contact has been represented in geometry textbooks that span the 20^th century. Third, the specification of the point-of-contact conception could involve all three semiotic systems that written mathematics texts use to make meanings: words, symbols, and mathematical pictures (O’Halloran, 2005). This makes it possible to track differences across time in the point-of-contact conception by examining the differences in the semiotic systems (verbal, symbolic, visual) used to represent it in textbooks.  The possibility of tracking changes in the conception by examining its changing semiotics in textbooks raises the question: How is the point-of-contact conception represented in secondary geometry textbooks across time?

Data:

The textbooks in this sample of geometry books come after those textbooks that are in “the era of the text” (Herbst, 2002). University Libraries do not keep school textbooks—in general—in their holdings.  Thus it was not possible to randomly sample 20^th century geometry textbooks; rather, the sample was determined by tracing the history of the McGraw-Hill-Glencoe publishing house. Using a combination of electronic (e.g., Worldcat) and physical bibliographic records (the Publisher’s Trade-list Annual), geometry textbooks from the McGraw-Hill-Glencoe publishing house were identified from 1903, 1914, 1933, 1949, 1972, 1984, and 2001.

Method:

From each textbook, problems and controls that involved the point-of-contact or point-of-tangency were identified and extracted from each textbook in the sample. Problems were defined as the numbered (or otherwise marked) exercises that appeared at the end of a unit, section, or subsection of the textbook. Controls were defined as those statements in each textbook that were identified as axioms, postulates, definitions, theorems, corollaries, or other statements that established the geometry facts of the tangent-circle relation. These controls were identified and extracted from each of the seven textbooks in the sample.

The problems and controls were coded according to their type, using a categorical coding scheme. Four different categories of problem (prove, find, measure, construct) were identified, as were three different categories of control (definition, theorem, declaration). The controls were further coded according to the semiotic system that was used to communicate the control: verbal, symbolic, or visual, with the possibility that one control could communicate its meaning through multiple semiotic systems working in parallel.  Once the controls for the point-of-contact conception were identified in each book, their differences in meaning were analyzed using a systemic functional linguistics approach to the verbal, symbolic, and visual semiotic systems of mathematical discourse (O’Halloran, 2005).

Results:

The first phase of coding tracked the difference in the semiotic systems used to state the controls for the point -of-contact conception. Across the sample, the six textbooks from 1903 to 1984 showed some consistency in the semiotic systems they used to communicate these controls. For these books, the verbal system was used for every control, while the symbolic and visual systems were used infrequently (less then 20 percent of the time per book, on average). This situation changes in 2001: every control except one—which is a declarative statement about tangents found in the introductory text of the chapter—uses all three semiotic systems. The 2001 text also makes the differences in the semiotic systems explicit: each control is set aside in a textbox that states the control in three different ways, each identified with a different semiotic system.  The second phase of coding will analyze the shift in meaning in the control structure for the point-of-contact conception in each textbook.

Scholarly significance:

Examining the evolution of a foundational geometry conception across 20^th century textbooks gives a measure of both current and past norms for representing mathematical knowledge in written texts. Understanding how these norms have changed (or are changing) is essential to a big-picture understanding of the ongoing efforts at math curriculum reform.

References:

Balacheff, N., & Gaudin, N. (2010). Modeling Students’ Conceptions: The Case of Function. Research in Collegiate Mathematics Education, 207.

Herbst, P. G. (2002). Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century. Educational Studies in Mathematics, 49(3), 283-312.

Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach. Educational Studies in Mathematics, 56(2), 255–286.

O’Halloran, K. L. (2005). Mathematical discourse: Language, symbolism and visual images. Continuum Intl Pub Group.

Schoenfeld, A. H. (1997). Mathematical problem solving. Academic Press Orlando, FL.