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Teacher questioning and invitations to participate in advanced mathematics lectures

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Abstract

We were interested in exploring the extent to which advanced mathematics lecturers provide students with opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers’ questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent to which lecturers provide students with opportunities to generate mathematical contributions and to engage in reasoning that researchers have identified as important in advanced mathematics. Our findings highlight that, although lecturers asked many questions, lecturers did not provide substantial opportunities for students to participate in generating mathematical content and reasoning. Additionally, we provide several examples of lecturers providing students with some opportunities to generate important contributions. We conclude by providing implications and areas for future research.

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Notes

  1. In the United States, mathematics students usually take these courses after completing a calculus sequence and an introduction to proof course. In other countries, university mathematics students take these courses in their first year.

  2. We point the reader to Gabel and Dreyfus (2017) and Lew et al. (2016) for such reviews.

  3. Researchers have examined teacher questioning in K-12 classrooms (e.g., Boaler & Brodie, 2004), community college classrooms (Mesa, 2010; Mesa, Celis, & Lande, 2014), and calculus classrooms (Viirman, 2015). As the pedagogical goals of these courses tend to be similar (i.e., helping students develop conceptual understanding and/or procedural competence) and this literature has produced findings that are largely consistent, we collapse our review to K-14 mathematics.

  4. Fukawa-Connelly et al. (2017) also discussed modeling mathematical behavior as occurring when a lecturer emphasizes the types of practices, questions, and appraisals that are common or natural in mathematics. However, this is not relevant for the purposes of this paper.

  5. Because we focus on the mathematical contributions, questions that were non-mathematical (e.g., Do you know when your exam is?) were excluded from our analysis.

  6. It is possible that, due to the socio-mathematical norms in the class, lecturer questions were soliciting contributions that went beyond what was literally stated in the question. However, we observed few instances of students providing answers that went beyond what was explicitly solicited in the question.

  7. For more on how each contribution was counted across the lectures, we refer the reader to Fukawa-Connelly et al. (2017).

  8. Because L1 did not provide a formal definition, cardinal addition, multiplication, and exponentiation are not part of the 21 total definitions presented by the lecturers.

  9. The number of Proof framework questions per lecturer is less than those counted for the entirety of the lecture,s as some Proof framework questions occurred outside of the context of formally writing a proof (e.g., considering how one would start a proof without ever actually writing the proof).

  10. S1 was the first student to respond in this excerpt but was the 18th student called on at this point in the lecture.

  11. We coded L2’s statement as an invitation to participate as S19 responded to this statement by providing the next step in the proof.

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Authors and Affiliations

  1. Mathematical Sciences Department, Montclair State University, 1 Normal Ave., Montclair, NJ, 07043, USA

    Teo Paoletti

  2. Graduate School of Education, Rutgers University, 10 Seminar Place, New Brunswick, NJ, 08901, USA

    Victoria Krupnik, Joseph Olsen & Keith Weber

  3. Department of Mathematics, Drexel University, 15. S. 33rd St, Philadelphia, PA, 19104, USA

    Dimitrios Papadopoulos

  4. College of Education, Temple University, 1301 Cecil B. Moore Ave, Philadelphia, PA, 19122, USA

    Tim Fukawa-Connelly

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  1. Teo Paoletti
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  2. Victoria Krupnik
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  3. Dimitrios Papadopoulos
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  4. Joseph Olsen
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  5. Tim Fukawa-Connelly
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  6. Keith Weber
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Correspondence to Teo Paoletti.

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Paoletti, T., Krupnik, V., Papadopoulos, D. et al. Teacher questioning and invitations to participate in advanced mathematics lectures. Educ Stud Math 98, 1–17 (2018). https://doi.org/10.1007/s10649-018-9807-6

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