Undergrad Math Students: Extracting the “Useful”, Ignoring the Meaning-Making

Posted on May 31, 2012 by

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Weinberg, Aaron, Emilie Wiesner, Bret Benesh, and Timothy Boester. 2011. “Undergraduate Students’
Self-Reported Use of Mathematics Textbooks.” Primus 22 (2): 152–175. doi:10.1080/10511970.2010.509336.

Methods:

  • 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook (p.152)
  • We recruited nine undergraduate students to keep “textbook-use journals,” and an additional eighteen students to participate in structured interviews (the journal template can be found in Appendix A, and the interview protocol can be found in Appendix B). These students were enrolled in a wide range of mathematics courses (including precalculus, first- and second-semester calculus, introductory statistics, and several upper-level mathematics courses) and participated voluntarily (p.157).
  • The goal of this study is to examine how students in introductory mathematics classes report using their textbook. Additionally, the study investigates the interactions among students’ values, their perception of their instructor’s implementation of the curriculum, and their self-reported textbook use. (p.153)
  • The classes surveyed were primarily intended for students pursuing a four-year B.A. or B.S. degree, who were not mathematics majors; the students enrolled (including those in teacher-education classes) were predominantly first- and second-year college undergraduates. All the classes were taught in small sections of up to 30 students.
  • Research Questions:
  • 1. To what extent do the ways students use textbooks follow the carefully laid out paths that (closed) textbooks prescribe? Specifically, what text components
    do students use?When do students look at each component? What are their reasons for doing so?
  • 2. If a student values certain characteristics of a textbook, is he or she more likely to use the book in a specific way? 
  • 3. In what ways does the instructor’s implementation of the curriculum affect the student’s textbook use? (pp. 154-155)
  • To address these research questions, a framework was developed to describe students’ textbook use in terms of the textbooks’ structural components and the factors affecting their textbook use. The framework has three principal
    elements:
    1. a description of the structural components of textbooks;
  • 2. the purposes and situative conditions under which students use their
    textbooks; and
    3. other potential influences on students’ textbook use. (p.155)
  • Components of a textbook:  The chapter introduction is located at the beginning of each chapter or
    unit. It describes the content that will follow, possibly giving motivation for
    including the content and drawing connections with other topics in the book.
    ● The chapter text contains the exposition and content kernels—the definitions,
    theorems, procedures, formulas, and descriptions of how each of these is
    related to the others and the topic of the chapter or unit.
    ● The examples are frequently embedded in the chapter text or are placed
    immediately after the chapter text but before the homework problems.
    ● The homework problems are typically included after the chapter text and
    examples. These consist of problems that can be solved using the ideas and
    techniques described in the chapter text, and are frequently similar to the
    examples.
    ● The chapter summary is a recapitulation of the content kernels included at
    the end of the chapter or unit. It is generally composed of a list of terms,
    phrases, or questions that a student could use as a basis for reviewing the
    chapter, but that contains little or no new exposition in its own right.
  • The answers to exercises (or solutions manual) is frequently included at the
    end of the textbook, and consists of either short answers to a subset of the
    homework problems or a brief outline of how to complete these problems (p.155-156)
  • The following list of purposes represents students’ potential reasons for
    using each part of the text.
    1. Read for better understanding.
    2. Make sense of definitions or theorems.
    3. Look up definitions or theorems.
    4. Rephrase/summarize text (for notes, homework, etc.).
    5. Read the homework problems to see what ideas come up most frequently.
    6. Use the answers to exercises to check homework.
    7. Use extra problems and answers to exercises to check understanding of
    problems that weren’t assigned.
    8. Read or copy homework problems to complete homework assignments.
    9. Look up answers without solving the problems. (p.156)
  • Based on Schoenfeld’s description and Lloyd and Behm’s [22] list of values, this
    framework includes five primary beliefs students have about their textbook:
    1. A textbook should explain the “big ideas” of the course.
    2. A textbook should explain the “underlying concepts” of problems.
    3. A textbook should give examples to explain the material.
    4. A textbook should give examples that can be used to complete homework.
    5. A textbook should highlight important equations and definitions. (p.157)

Findings:

  • The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic (p.152)
  • students do report using it more productively when they believe they are asked to do so (p.152)
  • instructors should carefully choose text materials to promote mathematical reasoning, and actively encourage their students to read the text in a way that supports the development of this reasoning.
  • Cowen [11] contends that learning to read mathematics should be a fundamental goal of undergraduate mathematics courses, as it provides a path for understanding mathematical theory as opposed to only mastering procedural skills (p.152)
  • Much of the existing research describes the linguistic or structural features of textbooks themselves (e.g., [6], [16], [35]) and how some of these features may affect student learning (e.g., [30]). (p.153)
  • Lithner [21] analyzed the strategies used by college students, as they worked through a set of textbook calculus exercises selected by the researcher. Lithner described the prevalence of an “Identification of Similarities” strategy in which the student focused on identifying superficial similarities between the exercise and earlier portions of the textbook. (p.153)
  • Eco defined a closed text as one that seeks to elicit a “precise response” from a reader at each step along a preconceived path [14, p. 8]. Love and Pimm have posited that all mathematics textbooks are
    essentially closed, and that typical mathematics textbook components, such as explanations, examples, and exercises, act as “devices used to organize the reader’s work within the text” [24, p. 386], a position that is supported by Rezat [32]…A closed text is left open to inappropriate interpretations by readers who do not follow the precise path laid out for them. (p.154)
  • a large percent of students also reported not reading the chapter
    text. (p.160)
  • Students reported using the worked examples more than any other part of the textbook (see Table 2). They reported using examples primarily while completing homework and studying for exams (see Table 5). (p.161)
  • Some instructors may believe that students simply look up the answers to assigned homework problems, and turn them in. Roughly a third (37%) of all students who looked at the solutions manual (which corresponds to 28% of all students) reported at least “sometimes” copying homework solutions before attempting to solve the problem on their own. Few students reported using answers in this way “often” or “always” (see Table 10), although this varied among the course-school groups (e.g., 2% of calculus students at school C reported using the answers “often” or “always,” but 18% of pre-service teachers at school A reported doing this). (p.163)
  • Students neglect to read the chapter introduction and—to a lesser extent—the chapter text. These are the portions of the text in which the author attempts to develop a deeper understanding of the mathematical concepts. In addition, students primarily report using the text when doing homework problems or studying for exams, and not as an ongoing resource for understanding material from class sessions. (p.166)
  • This is consistent with students’ stated preferences for textbooks that contain useful examples, and separate the content kernels (such as definitions and theorems) from the exposition. These results suggest that students are looking for algorithms and shortcuts, which mirrors Lithner’s [21] description of his subjects
    relying on an “Identification of Similarities” strategy to solve mathematics problems. (p.167)
  • As Love and Pimm note, when the text so clearly signals the important results of the textbook,
    by extracting the kernels from the exposition and using examples as a model for homework problems, it is natural that students become “impatient with the exposition” and “skip to the ‘essential’ results” [24, p. 387]. This is precisely what many students report doing in this study: they are less likely to
    read the introduction and chapter text than the other components, and these are generally the components that the author intends to help the students generate meaning. In doing so, students are less likely to use the lessons “as intended by the authors,” [32, p. 486] and, thus, to miss the “precise response” [14,
    p. 8] planned by the author. (p.167)
  • Suggestion #1:  it would be helpful to give space for open response on the survey to investigate other response categories students would create. Related to this, the results of this survey may be clouded by students’ interpretations of the terminology used on the survey, such as “read for understanding” or “rephrase.”
  • Suggestion #2:  The survey also does not describe how the textbook is incorporated into the class, from the instructor’s perspective.  This could be addressed by supplementing the students’ assessment with a form for instructors to describe how they incorporate the textbook into the course. (p.168)
  • Suggestion #3:  In addition, it would be informative to gather data that describe the tools instructors use to assess their students; these assessment instruments may affect students’ goals in their mathematics
    class and, consequently, the ways they use their textbook. (p.168)
  • Suggestion #4:  it would be useful to consider students’ perceptions of the effectiveness of class lectures and discussions. If students feel that their instructor creates opportunities to make sense of the material in class, they may view the examples as the only useful component of the textbook (p.168)
  • Conversely, students who do not feel that the class discussion and lecture are sufficient to help them understand the material may be more likely to turn to their textbook. To investigate this further, items that ask students to describe the ways their instructor creates learning opportunities in class would be a useful supplement to this survey. (pp.168-169)
  • students seem to value textbooks that provide them with clear examples
    similar to problems on their homework and exams (p.169)

Personal Thoughts:

Based on the suggestions, I’m wondering how a lot of instructors in the US, from K-16, have adopted the practices they have.  Where is the pressure to teach like “they have” coming from?  “Teach[ing] like they have” meaning, teaching math in a way that many people want to avoid and/or somehow disengage.  Is it just based on their experience, professional or communal norms?

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