Abstract
The purpose of this paper is to propose a new ontology of reasons for inferentialism. The existing inferentialist approach to mathematics education has a methodological challenge in retrospective analysis and a noncollaborative issue stems from a narrow view of learning. The proposed ontology, built on a radical interpretation of the inferentialist idea of conceptual pragmatism, enables us to maintain philosophical consistency on the basis of the distinction between students’ reasons and the observer’s perspectives, on the one hand, and dealing with the collaborative nature of classroom mathematical activities without assuming the knowledge to be learned as given. In order to build the ontology, the authors will redefine the notion of conceptualization as a judgment and claim that only articulated reasons exist and that unarticulated reasons are vague and variable. The shift in the ontological perspective requires us to reject a kind of retrospective analysis. In this regard, we should consider the impact of the observer effect on the assessment of conceptual development: This effect implies that it is not until a person is asked to provide her reason that her reason comes to exist. Finally, the authors will derive implications from the ontology: the teachers’ dual roles of enhancing students’ conceptualization and of assessing their current conceptual development and the implicit connection between inferentialist approaches and design research.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the famous discussion between John McDowell and Herbert Dreyfus (cf. McDowell 2007a, b, 2013; Dreyfus 2007a, b, 2013), McDowell (2007a) claims that even absorbed coping is conceptual. Our philosophical position is similar to McDowell’s in that one’s recognition of a situation always depends on one’s concept use.
Although we argue that reasons come to exist when they are made explicit, there are two possible interpretations regarding whether or not reasons exist before they are made explicit. First, one can think that reasons exist before they are made explicit. In this case, it implies that one identifies a situation with someone’s reason for doing it, a situation where someone does something. The reason exists as the situation, though it is undetermined how the reason becomes verbalized. Second, one can think that reasons do not exist before they are made explicit (i.e., the position we take in this paper). In this case, it implies that one distinguishes someone’s reason for doing something from the situation where someone does it. Figuratively speaking, the difference between those two interpretations is which exists—we think, water or ice. If we take the first interpretation, then the water exists, but the ice does not exist until it freezes. The form of the ice varies, depending on how it freezes. Similarly, the situation exists, but the reason does not exist until it is made explicit. The verbalized form of the reason varies, depending on how to make it explicit. If we take the second interpretation, we strongly focus on ice, not on water: that is, on the articulated reasons, not on unarticulated ones. The reader may choose either of the two interpretations if the reader correctly recognizes that this paper talks about articulated reasons.
According to Bakhurst (2011), Dreyfus attacks McDowell in the famous discussion cited in footnote 1 “because he thinks of reasons as articulable, general, rule-like considerations, with reference to which the agent determines what to think and do” (p. 126). For this reason, some readers may think that the authors’ position is similar to Dreyfus’s one. Indeed, we also think of reasons as articulable, general, rule-like considerations. However, we never deny McDowell. We believe, on the contrary, that inferentialists also have an interest in how students describe their current situations as their reasons in their games of giving and asking for reasons. As we mentioned in footnote 2, we can simultaneously consider both “reasons” in Dreyfus’s and McDowell’s senses without inconsistency. The point the authors want to make is that we discuss the ontology of articulated reasons from students’ perspectives in this paper.
In this paper, we interpret a person’s knowing how in Ryle’s (1949) sense as information she implicitly holds about when and how to do something, while we specifically refer to a description of a person’s knowing how at a given moment as knowledge-how. For example, when we observed that a student calculates 7 × 4 × 25 = 7 × 100, her method of calculation is considered knowledge-how rather than knowing how. We cannot precisely generalize when she would perform such a calculation. She may first calculate 4 × 25 when she looks at this expression. She may also first calculate a numerical expression when she immediately understands that its result is a multiple of 100. Similarly, we cannot precisely discuss a piece of knowing how in principle based solely on our observation. A holistic view for concept use in inferentialism also supports this stance. Thus, we treat such an overgeneralized description of how to do something as knowledge-how. In fact, knowledge-how has the potential to be used in a future unknown context due to overgeneralization.
References
Bakker, A. (2018a). Discovery learning: zombie, phoenix, or elephant? Instructional Science, 46(1), 169–183. https://doi.org/10.1007/s11251-018-9450-8.
Bakker, A. (2018b). What is design research in education. In A. Bakker (Ed.), Design research in education: a practical guide for early career researchers (pp. 3–22). London, UK: Routledge. https://doi.org/10.4324/9780203701010
Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. Mathematical Thinking and Learning, 13(1–2), 5–26. https://doi.org/10.1080/10986065.2011.538293.
Bakker, A., & Hußmann, S. (2017). Inferentialism in mathematics education: introduction to a special issue. Mathematics Education Research Journal, 29(4), 395–401. https://doi.org/10.1007/s13394-017-0224-4.
Brandom, R. (2000). Articulating reasons: An introduction to inferentialism. Cambridge, MA: Harvard University Press.
Bakhurst, D. (2011). The formation of reason. Oxford: Wiley. https://doi.org/10.1002/9781444395600.
Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20. https://doi.org/10.3102/0013189X023007013.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. Journal of the Learning Sciences, 10(1–2), 113–163. https://doi.org/10.1207/S15327809JLS10-1-2_6.
Confrey, J. (1991). Learning to listen: a student’s understanding of powers of ten. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111–138). Dordrecht, The Netherlands: Springer. https://doi.org/10.1007/0-306-47201-5_6.
Confrey, J., & Kazak, S. (2006). A thirty-year reflection on constructivism in mathematics education in PME. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present and future (pp. 305–345). Rotterdam, The Netherlands: Sense Publishers. https://doi.org/10.1163/9789087901127_012.
Derry, J. (2008). Abstract rationality in education: from Vygotsky to Brandom. Studies in Philosophy and Education, 27(1), 49–62. https://doi.org/10.1007/s11217-007-9047-1.
Dreyfus, H. L. (2007a). The return of the myth of the mental. Inquiry, 50(4), 352–365. https://doi.org/10.1080/00201740701489245.
Dreyfus, H. L. (2007b). Response to McDowell. Inquiry, 50(4), 371–377. https://doi.org/10.1080/00201740701489401.
Dreyfus, H. L. (2013). The myth of the pervasiveness of the mental. In J. K. Schear (Ed.), Mind, reason, and being-in-the-world: the McDowell-Dreyfus Debate (pp. 15–40). London, UK: Routledge. https://doi.org/10.4324/9780203076316.
Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117–146.
Ernest, P. (1998). Mathematical knowledge and context. In A. Watson (Ed.), Situated cognition and the learning of mathematics (pp. 13–31). Oxford: Centre for Mathematics Education Research.
Evans, J. (1999). Building bridges: reflections on the problem of transfer of learning in mathematics. Educational Studies in Mathematics, 39(1–3), 23–44. https://doi.org/10.1023/A:1003755611058.
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: a “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140. https://doi.org/10.2307/749505.
Hußmann, S., Schacht, F., & Schindler, M. (2018). Tracing conceptual development in mathematics: epistemology of webs of reasons. Mathematics Education Research Journal, 31, 1–17. https://doi.org/10.1007/s13394-018-0245-7.
McDowell, J. (1979). Virtue and reason. The Monist, 62(3), 331–350.
McDowell, J. (2007a). What myth? Inquiry, 50(4), 338–351. https://doi.org/10.1080/00201740701489211.
McDowell, J. (2007b). Response to Dreyfus. Inquiry, 50(4), 366–370. https://doi.org/10.1080/00201740701489351.
McDowell, J. (2013). The myth of the mind as detached. In J. K. Schear (Ed.), Mind, reason, and being-in-the-world: the McDowell-Dreyfus Debate (pp. 41–58). London, UK: Routledge. https://doi.org/10.4324/9780203076316.
Meyer, M. (2018). Using rules for elaborating mathematical concepts. In P. Ernest (Ed.), The philosophy of mathematics education today (pp. 297–308). Cham: Springer. https://doi.org/10.1007/978-3-319-77760-3_18.
Nesher, P. (1987). Towards an instructional theory: the role of student’s misconceptions. For the Learning of Mathematics, 7(3), 33–40.
Nilsson, P., & Schindler, M. (2018). The nature and use of theories in statistics education: looking back, looking forward. In M. A. Sorto, A. White, & L. Guyot (Eds.), Proceedings of the tenth international conference on teaching statistics, Kyoto, Japan. Voorburg, The Netherlands: International Statistical Institute.
Noorloos, R., Taylor, S. D., Bakker, A., & Derry, J. (2017). Inferentialism as an alternative to socioconstructivism in mathematics education. Mathematics Education Research Journal, 29(4), 437–453. https://doi.org/10.1007/s13394-017-0189-3.
Radford, L. (2016). The theory of objectification and its place among sociocultural research in mathematics education. The RIPEM - International Journal for Research in Mathematics Education, 6(2), 187–206.
Radford, L. (2017). On inferentialism. Mathematics Education Research Journal, 29(4), 493–508. https://doi.org/10.1007/s13394-017-0225-3.
Ryle, G. (1949). The concept of mind. London: Hutchinson.
Schindler, M., & Joklitschke, J. (2016). Designing tasks for mathematically talented students. In K. Krainer & N. Vondrová (Eds.), Proceedings of the ninth congress of the European Society for Research in mathematics education (pp. 1066–1072). Prague.
Schindler, M., & Seidouvy, A. (2019). Informal inferential reasoning and the social: understanding students’ informal inferences through an inferentialist epistemology. In G. Burrill & D. Ben-Zvi (Eds.), Topics and trends in current statistics education research: international perspectives (pp. 153–171). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-030-03472-6_7.
Seidouvy, A., & Eckert, A. (2017). Designing for responsibility and authority in experiment based instruction in mathematics: the case of reasoning with uncertainty. Proceedings of the tenth congress of the European Society for Research in mathematics education, 3740–3747. Dublin, Ireland: DCU Institute of Education and ERME.
Seidouvy, A., Helenius, O., & Schindler, M. (2018). Data generation in statistics: both procedural and conceptual. An inferentialist analysis. In J. Häggström, Y. Liljekvist, J. Bergman Ärlebäck, M. Fahlgren, & O. Olande (Eds.), Perspectives on professional development of mathematics teachers: proceedings of MADIF11 (pp. 191–200). Göthenburg, Sweden: SMDF.
Seidouvy, A., Helenius, O., & Schindler, M. (2019). Authority in students’ peer collaboration in statistics: an empirical study based on inferentialism. Nordic Studies in Mathematics Education, 24(2), 25–48.
Seidouvy, A., & Schindler, M. (2019). An inferentialist account of students’ collaboration in mathematics education. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-019-00267-0.
Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715.
Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13. https://doi.org/10.3102/0013189X027002004.
Sfard, A. (2013). Not just so stories: practising discursive research for the benefit of educational practice. In V. Farnsworth & Y. Solomon (Eds.), Reframing educational research: resisting the “what works” agenda. Milton Park, Abingdon, Oxon ; New York: Routledge.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Hillsdale: Erlbaum.
Steffe, L. P., & Ulrich, C. (2014). Constructivist teaching experiment. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 102–109). Dordrecht, The Netherlands: Springer. https://doi.org/10.1007/978-94-007-4978-8_32.
Taylor, S. D., Noorloos, R., & Bakker, A. (2017). Mastering as an inferentialist alternative to the acquisition and participation metaphors for learning. Journal of Philosophy of Education, 51(4), 769–784. https://doi.org/10.1111/1467-9752.12264.
Ulrich, C., Tillema, E. S., Hackenberg, A. J., & Norton, A. (2014). Constructivist model building: empirical examples from mathematics education. Constructivist Foundations, 9(3), 328–339.
van Fraassen, B. C. (1980/2011). The scientific image. Oxford: Oxford University Press.
von Glasersfeld, E. (1995). Radical constructivism: a way of knowing and learning. London: The Flamer Press.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.
Acknowledgments
We would like to thank the anonymous reviewers for their critical comments that contributed to improving our manuscripts. We would also like to thank Editage (www.editage.jp) for English language editing.
Funding
This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 18K13162 and 18H05751.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Disclaimer
Any opinions, findings, and conclusions expressed in this article are those of the author and do not necessarily reflect the views of JSPS and Editage.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Uegatani, Y., Otani, H. A new ontology of reasons for inferentialism: redefining the notion of conceptualization and proposing an observer effect on assessment. Math Ed Res J 33, 183–199 (2021). https://doi.org/10.1007/s13394-019-00289-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13394-019-00289-8