Laying the groundwork: The grounding metaphors that build quotient groups

Quotient groups are a foundational topic within abstract algebra courses, and they provide a context in which rich, powerful, and vivid language can be used to convey an understanding of the mathematics at play. We examined the language mathematicians used to describe their instruction on quotient groups and found that mathematicians draw on several source domains and use various metaphorical expressions when discussing their teaching practices. Of prominence in these findings was the use of a construction source domain which spanned all major aspects of quotient groups. The use of this source domain afforded the use of many metaphorical expressions that are tied to various formal content goals. We detail these findings and discuss how the metaphorical expressions used relate to the formal mathematics content that mathematicians are trying to convey.