Mesa and Instructor’s Mediation of the Math Textbook

Posted on September 22, 2012 by

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Mesa, Vilma, and Brett Griffiths. “Textbook Mediation of Teaching: An Example from Tertiary Mathematics Instructors.” Educational Studies in Mathematics 79, no. 1 (2012): 85–107.

Methods:

  • Drawing on data from interviews with mathematics faculty in three different types of undergraduate institutions and using Rabardel’s model of instrument use (Vérillon & Rabardel 1995), we describe three ways textbooks mediate college faculty work regarding instruction. The model anticipates epistemic and pragmatic mediations between the instructor and teaching, others, and self, with the textbook playing a significant role. We provide illustrations of each of these mediations as described by the participants. Pragmatic rather than epistemic mediations were more common. (85)
  • we identify three ways instructors described using textbooks to mediate instruction. Although grounded on empirical data (primarily interviews, field notes, and classroom observations), this work is mostly conceptual because it seeks to flesh out the distinct role of the textbook in mediating instruction. (86)

  • we listen to how faculty in mathematics departments describe their use of textbooks for teaching and then anticipate how the textbook mediates the relationship between instructors and tertiary mathematics teaching. We focus on instructors’ own descriptions of their use because we recognize instructors as potential agents of reform in teaching and, as such, we wanted to include their voices and rationales for textbook use in this study. (88)
  • This paper uses data from a year-long data collection process designed to generate a survey of textbook use among faculty and students in undergraduate settings. The first author interviewed 15 instructors during October and November of 2006; she observed seven instructors teaching 11 classes between January and April of 2007 and their students responded to a few questions regarding their use of textbooks. In this paper, we concentrate on the data from the interviews, in which we explicitly asked participants about the ways they use their textbook for teaching. The analysis we present here is neither intended as a comparison between what the instructors say and what they actually do, nor are we interested in imposing our own definitions of teaching onto the data. Rather, our intention is to understand from the users’ perspective the relationship they developed with their textbooks (Speer, 2005). (90)

  • The interviews followed a semi-structured protocol (see Appendix 1) that covered three main areas: background (academic credentials and ways in which they used their mathematics textbooks as students); current teaching (descriptions of courses, ways in which they were using the textbook in those classes, perceptions of how their students were using the textbook); and textbook features (useful and not useful features and changes they would incorporate in order to make the textbook useful to new faculty). Instructors answered questions freely. We followed an inquiry-driven interview process, and did not redirect instructors to answer questions, though we did repeat questions upon request and after long answers (Weiss, 1994). The open nature of these interviews often invited instructors to discuss any number of related insights they had into the teaching of mathematics. This became especially evident in the coding and analysis of their responses. The interviews lasted from 45 min to 2 h and were transcribed, noting only the pauses in speech. The first author observed 11 different lessons from 11 courses taught by seven participants; these lessons ranged in length from 21 to 100 min and covered different levels of content from one course in remedial mathematics to one graduate course in complex analysis. Most, however, were courses in the calculus sequence. In all, about 100 students responded to the short survey administered at the end of each class. (92)
  • Coding was guided by our interest in capturing how instructors described and understood their use of textbooks. In assigning a code, we selected a full speaker turn, in order to maintain the full context that would give meaning to the code. Excluding a category related to instructors’ personal background, the coding resulted in six general categories: descriptions of processes in which the textbook was used for instruction, elements of the environment in which instructors teach, teacher obligations, and the nature of textbooks, students, and mathematics (see Appendix 2). We then generated a report that listed the frequency with which the codes were assigned by the number of cases (instructors) that received the codes, and focused only on those codes that were assigned at least 14 times and to at least 11 instructors. This excluded from our analysis all codes devoted to the Environment and Mathematics categories, and also a number of other codes within each of the remaining categories, resulting in a specific analysis of text assigned to 13 codes (see Table 2). This process allowed us to attend to, and report on, those aspects of textbook mediation that were common to most instructors and that were most frequently mentioned, putting aside codes that were less frequent and that may have been more idiosyncratic. (92)
  • We then generated reports for each individual code and through reading the quotes determined whether they could describe the different mediations anticipated by the model. In some cases the quotes for some codes referred to all three mediations, and in others they

    referred to only one. We then engaged in a “vertical” analysis, reading the quotes under the columns Object, Other, and Self, to identify whether the type of mediation was epistemic or pragmatic, and to identify themes that run across all these quotes.

    The initial coding was developed in several stages in which several codes were generated collectively, then assigned individually, then refined through comparison and discussion. We set aside one interview to check the consistency of the coding between us; we then refined the meanings of the codes to help ensure such consistency. In order to corroborate our classification and seek realistic validity (Maxwell, 1992), we shared our initial code assignment and the emerging interpretations with the participants; most of them responded, indicating agreement with our interpretations of the relations described herein. (92-93

Findings:

  • we found that the mediations seem largely dependent on instructors’ categorical perceptions of their students, either as “math students” or as “undergraduate students”. These alternate perceptions resulted in different descriptions of schemas for using the textbook with each type of student. The analysis generates further questions for research regarding either a developmental component or a curricular component that could explain this categorization of students. (85)
  • textbooks remain a ubiquitous course component with various implications for the teaching and learning of mathematics at the tertiary level. Specifically, new textbooks continue to be produced and be required for undergraduate classes. They are expensive for students and in general departments take the process of textbook adoption very seriously. Yet we know little about how instructors and students use textbooks for the purpose for which they were written: to assist in teaching and learning. (86)

  • According to Rabardel and colleagues, the designers’ role is to elaborate
    instrumental propositions in the form of artefacts and in terms of anticipated operations (design for use). Users may take advantage of these propositions (totally, partially, or not at all) in order to develop their own instruments—instruments that fulfill their own characteristic needs, depending on the organization and the situations, where they are to be used (design in use). (Rabardel & Waern, p. 643)…

    There exists a similar situation with textbooks. The authors, as designers of instruction, anticipate potential uses for the textbook that support instructors’ work. The instructors (and students) can use the textbook as intended by the authors totally, partially, or not at all in order to develop an instrument out of the textbook that fulfills their own needs, depending upon the context in which instructors work, their situation, and where the textbooks are to be used. Two consequences of this conceptualization are that “‘development-through- use’—or the transformation from textbook-artefact to textbook-instrument—must be considered as an intrinsic characteristic of human activity” (Rabardel & Waern 2003 p. 643) and that the genesis of instruments is “NOT an effect of deficient design, but rather an expression of the concept embodied by the artefact in all ways instantiated by the user” (capitals in original, p. 643). These two consequences are fundamental, because both recognize that users bring agency to the relationships with their artifacts (Remillard, 2005) and acknowledge that new uses, unanticipated by the designer, are expected. That is, many schemes of use may emerge from user relationships with the same artifact. The term, scheme, is used here in the sense given by Vergnaud (1996), as “invariant organization of behavior for a given class of situations” (as cited by Gueudet & Trouche, 2009, p. 789). (88)

  • Imagine a subject who is teaching a course for the first time; he or she will need to perform a series of tasks that all together will define a course for the students, including ideas for presenting content and strategies for ensuring that students will learn such content and for assessing whether students have indeed learned it (Brousseau, 1997). In mediating these processes, the textbook might allow for the subject to learn about designs for homework or designs for lectures, and it can help instructors anticipate whether those designs will indeed assist students in learning. The textbook may also assist instructors in making changes to their designs so they fit better with the demands of the content and the students’ needs. In assessing the potential of the design, instructors engage in pragmatic considerations that are mediated by the tool (see Artigue, 2002; Trouche, 2003), in this case the textbook.
  • The second mediation—interpersonal mediation—acknowledges that the subject’s activity is oriented toward others (students, colleagues, authors, etc.), and it can also be epistemic (because there is an interest in knowing others) or pragmatic (there is an interest in having them do something). In the case of mathematics teaching, instructors might use the textbook to learn the extent to which a particular presentation of a theorem in the textbook is too difficult or too easy for the students or whether a given assignment generated the progress and understanding expected (89)

  • Instructors did well in describing how they used textbooks for generating syllabi, homework, exams, or lecture notes, but did not reflect on these designs as objects that were the result of many transformations catalyzed by the textbook, except when describing their use of lecture notes. The textbook assisted in the creation of the lecture notes, which in turn generated similar mediations, as lecture notes were intended for interpersonal mediation between senior and junior instructors and the students and for reflexive mediation for instructors to remind themselves about important points or teaching strategies for the future (94)
  • In general, the textbook was described as providing a basic structure that instructors could employ in different ways; the textbook was not seen as an overly constraining resource for the design process (Colin, FN 85–91, Ian, 200–202). (95)

  • Instructors used their textbooks to design homework, quizzes, exams, and projects. Instructors considered the difficulty of the tasks, the availability of the answers, their perception of students’ ability to handle the work, and the need to practice with the material. Instructors took the problems as they appeared in the textbook or created new ones as needed (95)
  • According to the instructors, “math students” read the text in the textbook before attempting the problems, whereas “undergrad students” start with the problems, and when stuck, leaf back to find a similar example to complete the assignment.2 Instructors appeared to be more inclined to teach “math students” to read the textbook and to follow it closely. In contrast, with the “undergrad students,” instructors described preparing lectures containing many examples, either taken from the book or designed by instructors, to give the students as many examples as needed to be successful in solving the homework. (96)

  • Instructors described “undergrad students” (including students in remedial courses) as lacking the skills for reading the textbook. Either the textbook was “too chatty” or the instructors had no control over students’ reading compliance. (96)
  • When referring to classes for lower division students, instructors described the students as “needing” the instructor, “needing” to attend class, and being unable to work on their own, which translated into more reliance on the textbook for homework and examples. However, instructors described the importance of having enough difference between the lectures and the textbook content to ensure students’ class attendance (a pragmatic concern). (97)

  • Having inadequate time to cover the material as presented in the textbook becomes a source for reflection about how to organize the course every time she teaches it, a process she is still struggling to resolve in the three semesters she has taught the course. (99)
  • First, we suggested that a better understanding of how instructors of first year mathematics use textbooks can inform current calls for instructional reform. Second, the research describing the complexity of teacher and textbook relationships in primary and secondary education anticipates a similarly complex relationship between tertiary mathematics instructors and their textbooks. The distinct contexts for the conditions of instruction for teaching at the tertiary level require further, situated analysis. Finally, the textbook remains an unexamined piece of our educational systems in spite of the tremendous investment in time and cost around its use. (99)

  • Our analysis suggests that as these instructors gain experience teaching with a particular textbook, the mediation of the textbook with instruction changes, as anticipated by the literature on document genesis. Instructors described making fewer adjustments over time and appeared to rely on the textbook less as they became familiar with what the textbook did or did not offer. (100)
  • they perceive textbooks to be written for students. Instructors in our sample did not allude to ways in which the textbooks would help them learn more about the various activities of planning instruction, they did not comment on the need to have features in the textbook that could assist them in making decisions about which problems to choose or how to sequence the topics in constructing the syllabus, and they did not mention that the textbook would help them understand mathematics in a different way so they could teach it to their students differently. (100)

  • In contrast, it appears that much of the work on writing school textbooks, especially reformed ones, is predicated on the need to re-train teachers and to help them learn the mathematics they need to teach (Davis & Krajcik, 2005; Remillard & Bryans, 2004). (100)
  • making sure that the textbook contains features that the teachers will use is fundamental. Our impression is that our instructors of tertiary mathematics in general resisted this possibility. Textbooks that offered more explanation than needed were labeled “chatty”, hard to read, difficult to follow, verbose. When asked about features in the textbooks they felt could help them teach better, they mentioned problems and examples. (100)

  • The distinction between the “undergrad students” and the “math students” so clearly outlined by our instructors is fueled by how they use their textbooks in planning their courses for each type of student. With “undergrad students,” instructors start with the main skill goals, then seek exercises that would give students opportunities to practice those skill goals, and then seek examples that would illustrate ideas and processes students would need for doing the homework. (101)
  • Instructors may change the examples a little, so that students will have more illustration, but the focus is on learning to solve the problems assigned. In contrast, for “math students,” instructors read the textbook from beginning to end, maintain coherent notation, and expect students to read the textbook. Each type of student is, in turn, described as using the textbook very differently. There are accounts that suggest that students in lower division courses rely on examples for doing the homework (Weinberg, Wiesner, Benesh, & Boester, 2011). We contend that this perception is likely to be a direct consequence of the different schemes that instructors have for using the textbook. (101)

  • However, we found it noteworthy that instructors often described their desire that students use the textbooks to learn for themselves—that is, according to Rabardel’s model, epistemically to better understand mathematics. Yet instructors’ own practices in designing instruction appear to work against their hope that students will learn to use the textbooks in that manner. This is consistent with suggestions that students’ use of textbooks may develop from the way they are taught to use them (Love & Pimm, 1996; Sierpinska, 1997). This hypothesis would benefit from further investigation of textbook mediation by instructors and students in real time, which, in turn, would benefit from a more complex modeling of student and instructor perceptions and use of textbooks (Weinberg & Wiesner, 2011). Such a model would anticipate how a designer of instruction (the teacher) might use an artifact to get another subject (the student) to use the same artifact in particular ways. (102)
  • A second area of investigation concerns lecture notes and their creation and use for planning and mapping curriculum. The textbook is an instrument in the creation (genesis) of lecture notes (documents) as artifacts for teaching, but we know very little about these documents. Some of the instructors indicated that they have given lecture notes to their students or that they have used lecture notes to write textbooks. Mapping the different mediations of this new artifact would give us unprecedented information about how instructors conceive of teaching and would help us to better understand the relationship between instruction and textbooks. Such mapping would afford a greater understanding of some of the unique ways that intellectual freedom and perceptions of institutional contexts may inform instructors’ interpretations of texts through use. As we noted in our introduction, there are various contextual differences between college instruction and primary and secondary mathematics instruction. Perhaps the little evidence that we found regarding textbook mediations with self is due in part to instructors’ use of lecture notes for mediation rather than textbooks. (102)

  • A third area concerns understanding how textbook use changes over time. By studying faculty designing a new course, it could be possible to describe the multiple mediations in which the textbook participates in the design and mapping of the curriculum for the first time. This study could also help us understand how those processes change as faculty gain experience teaching the course, and how teaching the course informs and determines how the textbook is used the following time the course is taught. Such knowledge would be important for learning how faculty conceive of courses, which in turn would be useful knowledge for new faculty or for graduate students who are learning to teach tertiary mathematics. More importantly, such knowledge would shed light on how faculty gain an understanding for teaching tertiary mathematics courses and would have potential for our theorization of mathematics teaching at the tertiary level. (102-103)