Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Abstract

Students' apprenticeship into proof-oriented mathematical practice requires that they become aware of a range of conventions and assumptions about mathematical language, reference, and inference that rarely appear as explicit components of the content of undergraduate courses. Because these matters of interpretation are often preconscious for students and taken for granted by instructors, students may fail to apprehend the problems at hand and thus misunderstand the mathematical community's solutions. So, we argue that proof-oriented instruction must simultaneously help students understand problems of ambiguity and referencewhich requires them to engage in what we call metalinguistic and metamathematical reasoningas well as provide some informal language for discussing possible solutions to these problems. We provide three illustrative episodes from real analysis classrooms to highlight our common way of framing this class of learning challenge and the pedagogical difficulties they entail. In each case, we highlight the use of analogies as a method of helping students apprehend a problem of mathematical language, reference, or inference and discuss possible solutions. This paper's contributions are thus both practical and theoretical: the instructional method is intended to serve as one possible approach to training students in mathematical conventions and assumptions, and it helps demonstrate our theoretical perspective on this class of teaching and learning phenomena.

The use of creative problem solving model to develop students’ adaptive reasoning ability: Inductive, deductive, and intuitive

“…They were not familiar with the concept that algebra can also be used to extend and support logical reasoning, and its purpose in problem-solving is not to describe a solution procedure that has already been constructed mentally. Consequently, the majority of students had a lack of adaptive reasoning abilities despite the fact that the process of adaptive reasoning is one of the learning objectives since junior high school years (Dawkins & Roh, 2016). In addition, some findings revealed students' difficulties in solving adaptive reasoning problems, namely at the stage of ideas generation, solution finding and acceptance of finding.…”

“…The current educational process seems to favor rote learning (Rofiki et al, 2017), looking for one correct answer without discovering other solutions nor promoting higher-order thinking (Blakemore & Frith, 2005). Consequently, the majority of students are lack of adaptive reasoning abilities despite the fact that the process of adaptive reasoning is one of the learning objectives since junior high school years (Dawkins & Roh, 2016). A preliminary study illustrates the problem related to the proof given to the students to examine the initial adaptive reasoning ability of students in this study.…”

Everyday Examples About Basis From Students: An Anti-Deficit Approach in the Classroom

“…Studies have shown that connecting mathematics to students' daily experiences provides more meaningful engagement, particularly for underrepresented minority students [11], and fosters continued interests in mathematics [6]. Studies in undergraduate mathematics have shown the utility in leveraging intuitive knowledge in learning [1,8,14,18].…”

“…The word "basis" and its nonmathematical uses in the language also provide an additional connection to students' intuition [19]. This need for a grounding imagery of an abstract content could also be seen in topics like the formal definition of a limit [1,8]. Thus, other topics with tightly related core concepts that can be used to build on each other, but are initially difficult to distinguish for learners are good topics with which to adapt the task in this paper.…”

Adaptive Reasoning Based on Microsoft Mathematics

“…struggle to use their reasoning power and consequently, they cannot formulate, present and solve problems efficiently, precisely, and logically. Even though the adaptive reasoning process is one of the goals of learning mathematics since students enter junior high school (Ansari et al, 2020;Dawkins & Roh, 2016). The lack of student's adaptive reasoning aspect is also caused by the mathematics learning model that is not facilitating student to learn actively and not stimulates the student to use reasoning (Yanti et al, 2020) With the complexity of these problems, a learning model is needed that can generate students' adaptive reasoning skills.…”