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On the Levels and Types of Students’ Inquiry: The Case of Calculus

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Practice-Oriented Research in Tertiary Mathematics Education

Part of the book series: Advances in Mathematics Education ((AME))

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Abstract

The learning of mathematics organized around carefully designed questions is vital for students’ quantitative literacy development at both elementary and advanced stages. Deliberating on how the inquiry-based instruction could be supported, we further theorize about the idea that inquiry comes in many levels and through different types of activities. We use the Herbartian schema and related constructs of the Anthropological Theory of the Didactic in order to embrace the continuum of levels of inquiry, labeled as confirmation, structured, guided, and open. We review calculus textbooks’ descriptions of these inquiries and their roles. We also suggest additional activities of the types that complement those found in the textbooks. The new types include tasks for comparison and recognition of objects, evaluation of the validity of statements, modification of questions, and evaluation of reasoning - the actions common in mathematical research. We conclude by commenting on possible relations between the activity’s inquiry level, type, and its learning potential (e.g. adidactic, linkage, deepening, and research).

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Notes

  1. 1.

    Following the ATD tradition, we mark by the rhombus (Q♢, A♢, Π♢, etc.) the pieces of information given to learners and by the heart (Q♡, A♡, Λ♡, etc.) - the elements produced by them. The subscript y symbolizes the teacher y ∈ Y  who supplies the information and the subscript x symbolizes the learner x ∈ X who produces the items while possibly working with other students and teachers.

  2. 2.

    For ATD-based analysis of project-based learning see Markulin et al. (2020).

  3. 3.

    Students may be given either only the praxis block \( {\Pi}_y^{\diamondsuit } \) requiring own explanations \( {\theta}_x^{\heartsuit } \) or both praxis and logos blocks \( \left({\Pi}_y^{\diamondsuit },{\Lambda}_y^{\diamondsuit}\right) \) requiring a validation.

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Acknowledgements

I am indebted to Carl Winsløw, Marianna Bosch and anonymous referees for many useful and constructive comments.

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Authors and Affiliations

  1. Memorial University of Newfoundland, St. John’s, NL, Canada

    Margo Kondratieva

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  1. Margo Kondratieva
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Correspondence to Margo Kondratieva .

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Editors and Affiliations

  1. Institute of Mathematics, Paderborn University, Paderborn, Nordrhein-Westfalen, Germany

    Rolf Biehler

  2. Institute of Mathematics, Paderborn University, Paderborn, Nordrhein-Westfalen, Germany

    Michael Liebendörfer

  3. University Paris Saclay, Orsay, France

    Ghislaine Gueudet

  4. Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA

    Chris Rasmussen

  5. Faculty of Science, University of Copenhagen, Copenhagen, Denmark

    Carl Winsløw

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Kondratieva, M. (2022). On the Levels and Types of Students’ Inquiry: The Case of Calculus. In: Biehler, R., Liebendörfer, M., Gueudet, G., Rasmussen, C., Winsløw, C. (eds) Practice-Oriented Research in Tertiary Mathematics Education. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-14175-1_23

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