Identifying Rhetorical Purpose on Paper about Mathematics

Introduction to a paper I’d like to write
We have all gone to grade school, and learned arithmetic, geometry, algebra, and perhaps even more mathematics than that. Deep down, though, what does it really mean to know mathematics? This question is complicated by the existence and significance of what Michael Polanyi has termed “tacit knowledge” – knowledge that we possess and can act on, but which cannot be expressed explicitly regardless of how eloquent we are.
This first paragraph seeks to entice the reader. It creates a connection between the reader and author, giving a common experience that both has experienced. It also introduces the focus of the essay, which is about possessing tacit knowledge.
When your goal is to use mathematics as a tool, a healthy dose of tacit knowledge can be a good thing, as it enables all of us to make our way through our mathematical days without fuss. But in some contexts, having tacit knowledge of mathematics simply won’t be sufficient. For instance, what is a teacher to do if their young student argues “I got the problem right, why do I have to explain it,” and then proceeds to do the next problem, which is entirely similar, incorrectly? How can the teacher effectively probe the student’s knowledge and understanding when that student lacks the expressive language to adequately convey the subtleties involved in their problem-solving activity?
From the introduction, the narrative of the essay expands to accommodate the understanding of the reader. It proposes questions to make the reader think and put their attentiveness towards the writing. By presenting something to wonder upon, the author is grabbing the interest of the reader.
Professor Paul Ernest of the Exeter University School of Education has written extensively on the interaction of rhetoric and mathematics education. He assures us that “adequate warrants can be provided for tacit knowledge” (71), and in particular argues that the rhetoric of mathematics “is central to both the context of learning and the context of instruction and assessment” (79). There is no great sophistication in the assertion that what a person says about what they are doing reveals information about what they are thinking. But what Professor Ernest is pointing out goes deeper than this. He is saying that developing a full understanding of how mathematics is expressed can improve mathematics pedagogy, because it can provide a mechanism for probing a student’s tacit knowledge.