One of the things that mathematics educators often talk about is the idea of teaching the norms of the discipline of mathematics to students, starting at a fairly young age. In Jo Boaler and Cathlee Humphreys’ book Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning, the authors have compiled several examples of teaching the basics of mathematical argumentation to middle school students. Students are told that when making a mathematical claim, they need to first convince themselves of the claim, then convince a friend of the claim, and then convince an enemy of the claim. The idea is that students can engage in mathematical practices in the classroom which approximate the kind of practices that mathematicians use in their work. There have been many attempts to systematize these practices; most recently, with the publication of the Common Core State Standards for Mathematics and its eight mathematical practices.
The idea that we can inculcate students into the practices of a discipline like mathematics relies heavily on theories that were developed in the early 1990s. In their 1991 book Situated Learning: Legitimate Peripheral Participation, Jean Lave and Eitenne Wenger propose an idea called legitimate peripheral participation. They suggest that learning is the process by which one comes to be a part of a community of practice; learning is a process of coming to negotiate the social meanings, moving one not towards central participation (since there is no definitive center) but rather to full participation. Legitimate peripheral participation, Lave and Wenger propose, is a theory of how people learn rather than a theory of how to teach. Schools teach many things other than just subject matter, and students participate in many different communities of practice, not just that of the mathematics classroom.
There is something qualitatively different about the mathematics in the mathematics classroom and the mathematics that professional mathematicians do. A professional mathematician works primarily to generate new knowledge rather than to merely learn past knowledge. Although students taught with discovery learning methods have occasionally discovered new theorems or invented new mathematical ideas, that is not the primary purpose of mathematics education, particularly at the K-16 level. A professional mathematician does not work in groups of three to four for fixed periods of time like the way a student might work in a Complex Instruction- or groupwork-based classroom; they have to learn how to structure their own time and to attend professional colloquia and conferences in order to present their ideas. Professional mathematicians also use very different tools than students; they would be more likely to be seen reading a professional journal than a textbook, and use more advanced specialized software such as GNU Octave or Sage instead of (say) a handheld graphing calculator.
The idea of legitimate peripheral participation, therefore, is tricky when you try to apply it to the K-16 classroom. We are not necessarily enculturating students into becoming mathematicians; our students come to us with many different goals. The classroom as a community of practice is a very different space in which full participation means to take part in an active role in classroom discourse, to make and defend ideas, and to begin to develop one’s own ideas while also mastering the fundamental concepts of mathematics. The norms we establish and the experiences that we give students in the classroom help to make them full members of the mathematics student community of practice. By instituting the Standards for Mathematical Practice in our classroom, we provide the foundation for students to begin as legitimate peripheral participants in their future studies and careers.