This is an interesting collection of essays from maths education, using ideas from semiotics (which is a little badly-spoken-of in my field of pragmatics, but interesting nonetheless). There are essays covering Saussure, Peirce and Vygotsky.
Owen McNamara writes on Saussure, and quotes Kaput drawing an analogy between “the way that the architecture of a building organises our physical experience, and the way the architecture of our mathematical notation system organises our mathematical experience,” in Notations and Representations as Mediators of Constructive Processes.
Adam Vile criticises Rotman somewhat, and draws links between his work, Peirce and his focus on practices, and Wittgenstein.
Paul Ernest says a great many interesting things in “The epistemic subject in mathematical activity”.
“In this process [of mathematical learning] the learner is internalising some of the central functions and structures of the number system. However, this is not just a case of internalisation, because the learner is all the while engaging in public performances and deriving feedback, incorporating confirmations, and corrections in his or her conceptions or functioning, which is shaping the child’s powers. The learner is also learning to read, understand, and respond to the social contexts of arithmetic.” p. 85
He also pays homage to and criticises Rotman, noting that “It is not clear that the mathematician does in fact perform actual operations as opposed to imaginary ones.” p. 91
Candia Morgan in “Social identities in mathematics communities” takes a similar focus on persons in mathematics, noting that “…semiotic activity in mathematical practices, as in other practices, is not just about communicating ‘content’ but is also about the maintenance and contestation of power.” p. 112. She discusses mathematical authority at length.
There’s a lot of Kress, social semiotics, and multimodality in here. Interestingly i found a review of Kress’ Multimodality by Charles Forceville advocating for the use of relevance theory in place of social semiotics.
This looks interesting: https://www.researchgate.net/publication/243635357_The_Visual_Syntax_of_Algebra
Leslie P. Steffe argues for taking childrens’ mathematics to be legitimate, not wrong:
Adalira Saenz-Ludlow includes a paper that covers similar concerns to this one:
Jay Lemke notes: p. 232