## Abstract

How should we expect growing understandings of the nature of mathematical practice to inform classroom mathematical practice? We address this question from a perspective that takes seriously the notion that mathematics education, as a societal enterprise, is accountable to multiple sets of stakeholders, with the discipline of mathematics being only one of them. As they lead instruction, teachers can benefit from the influence of understandings of mathematical practice but they also need to recognize obligations to other stakeholders. We locate how mathematics instruction may actively respond to the influence of the discipline of mathematics and we exemplify how obligations to other stakeholders may participate in the practical rationality of mathematics teaching as those influences are incorporated into instruction.

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## Notes

- 1.
To define the didactical contract in terms of differential roles in coming to know the content highlights the notion that instruction is only one of the activities that teachers and students participate in. Other activities, such as mentoring youth, are not only legitimate and expected ones for a teacher to engage in, but also activities that compete with instruction for resources such as time and rapport.

- 2.
- 3.
Chevallard (1991) calls

*mathematical*notions those that are explicit objects of study in mathematics. For example, any mathematical idea that receives a definition and whose properties are studied is a mathematical notion.*Paramathematical*notions, in contrast, are those which are explicitly used as tools in the study of mathematical notions, but are not themselves constructed as objects of study (e.g., the notions of*definition*or*parameter*). - 4.
Chevallard (1991) also speaks of

*protomathematical*notions, which are implicit, subject-specific notions that support the functioning of mathematical practice. For example, the notion of strength of a proposition or the notion of pattern. - 5.
Foucault (2005) uses these similitudes to examine the evolution of scientific discourses.

- 6.
For example there are characteristics of individual students that are more salient in mathematics than in other areas, such as intelligence, given the societal assumption of a correlation between calculation speed and intelligence. There are social expectations farmed out to school mathematics such as helping decide on merit for college admissions (through college aptitude tests such as the US’s SAT). And there are program evaluation functions that schooling institutions farm out to school mathematics, such as growth in mathematics test scores as an indication of school effectiveness. These three may be questionable, but they are part of the reality that mathematics teachers, especially, have to contend with.

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## Acknowledgements

The writing of this paper has been supported in part by Grants DUE 1725837, ESI-0353285, and DRL-1316241 from the US National Science Foundation to the first author. All opinions are those of the authors and do not necessarily represent the views of the Foundation.

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Herbst, P., Chazan, D. Mathematics teaching has its own imperatives: mathematical practice and the work of mathematics instruction.
*ZDM Mathematics Education* **52, **1149–1162 (2020). https://doi.org/10.1007/s11858-020-01157-7

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### Keywords

- Mathematical practice
- Instruction
- Curriculum
- Instructional exchanges
- Professional obligations
- Practical rationality
- Didactical contract
- Didactical transposition
- Professional community
- Mathematicians