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Fluency with mathematical language is important for students’ engagement in many disciplinary practices such as defining, conjecturing, and proving; yet, there is growing evidence that mathematical language is challenging for undergraduate students. This dissertation study draws on two design experiments with pairs of students who were supported to encode their mathematical meanings with more formal language. I aimed to investigate the teaching and learning of mathematical language, and particularly the language in statements with multiple quantifiers, by engaging students in this type of activity. In the first paper, I investigated the complex ways in which the students in my study made sense of the grammar of statements with multiple quantifiers. This study contributes to the research literature in two ways. First, I showcased grammatical features that influenced the ways that students constructed and interpreted these statements. Second, I complemented the current static descriptions of statements with multiple quantifiers found in the literature with a dynamic understanding of these statements, which grew out of exploring the students’ thinking. The second paper is a case study of students reinventing the dynamical processes that are encoded with statements with multiple quantifiers. This paper offers a proof of concept of an instructional approach that engages students in defining multiple concepts and then reflecting on the resulting set of definitions to learn about the grammar of statements with multiple quantifiers. This paper also takes initial steps toward articulating a local instructional theory that frames the students’ reinvention process in terms of the emergent models construct from the theory of Realistic Mathematics Education. In the third paper, I explored the teachers’ role in ii supporting students in encoding their mathematical meanings with more formal language as the students engaged in defining and conjecturing. I used the conceptualization of the teacher’s role as a broker between the local community with students and the broader mathematics community. I offer episodes of brokering that supported students in refining their language. This paper’s contribution is a conceptual framework of brokering by leveraging pedagogical content tools and intellectual needs to support students in progressive mathematizing.

Guided Reinvention as a Context for Investigating Students' Thinking about Mathematical Language and for Supporting Students in Gaining Fluency