In Trinidad and Tobago, there have been substantive efforts to reform mathematics education. Through the implementation of new policies, the reformers have promoted changes in mathematics curriculum and instruction. A focus of the reform has been that of increasing opportunities for students to engage in reasoning and proving. However, little is known about how these policies affect the opportunities for reasoning and proof in the written curriculum, the teaching of proof, and students' learning. Furthermore, we are yet to know how the high-stake assessment measures interact with these new policies to impact the teaching and learning of proof.
In this dissertation, my overarching question asks: What are the implications of reform on the teaching and learning of secondary school mathematics in Trinidad and Tobago? To answer this question, I conducted three studies, which examined the content, teaching, and students’ conceptions. All the studies are situated in the teaching reasoning and proof when introducing geometry concepts. In the first study, I conduct a curriculum analysis focused on examining the opportunities for reasoning and proof in the three recommended secondary school textbooks. In the second study, I conduct classroom observations of teachers’ geometry instruction focusing on opportunities for engaging students in reasoning and proof. In the third study, I examine geometry students’ conceptions of proof. The three studies are intended to provide an overview of the impact of reform on instructional issues in relation to the dynamics between teachers, student, and content (Cohen, Raudenbush, & Ball, 2003).
For the first study, I adapt a framework developed by Otten, Gilbertson, Males, and Clark (2014) to investigate the quality and quantity of the opportunities for students to engage in or reflect on reasoning and proof. The findings highlight some unique characteristics of the recommended textbooks such as, (a) the promotion of the explanatory role of proof through the affordances of what I define as the Geometric Calculation with Number and Explanation (GCNE) exercises, (b) the necessary scaffolding of proof construction through activities and exercises promoting pattern identification, conjecturing, and developing of informal non-proof arguments, and (c) the varying advocacy for Geometry as an area in the curriculum where students can experience the work of real mathematicians and see the intellectual of proof in their mathematical experiences. All these characteristics align with the reformers’ vision for the teaching and learning of reasoning and proof in secondary school mathematics.
In the second study, I examine the nature of the teaching of reasoning and proof in secondary school. I use classroom observations along with pre- and post-observations interview data of three teachers to determine (a) the classroom microculture (i.e., classroom mathematical practices and sociomathematical norms), (b) teachers’ pedagogical decisions, and (c) teachers’ use of the Caribbean Secondary Examination Certificate (CSEC) examination materials and textbooks. I also determine whether the teachers’ instruction demonstrate the four characteristics of reform-based mathematics teaching (Hufferd-Ackles, Fuson, & Sherin, 2004). My analysis of classroom observations of the three teachers suggests that their instructional practices exhibit elements of reform-based instruction. These include teachers’ use of open-ended and direct questions to solicit students’ mathematical ideas and teachers’ consideration of students as the source of mathematical ideas. Each teacher established sociomathematical norms that governed how and when a student can ask questions. In this case, questioning helped students articulate their ideas when responding to questions and clarifying their understanding of other's ideas when they posed a question. Teachers also established sociomathematical norms that outlined what counts as a valid proof and what counts as an acceptable answer during instruction. The aforementioned norms supported the expectation that students must always provide explanations for their mathematical thinking, which is another characteristic of reform-based teaching. Teachers used group work and whole class discussions to offer opportunities for collaborative learning, which facilitated their creation of a social constructivist environment for learning reasoning and proof. Teachers used the reform-oriented curriculum materials to provide opportunities for construction of proofs. However, the textbooks and curriculum were limited in their support for proving some geometrical results. Overall, the teachers emphasized the making and testing of conjectures, which afforded students with authentic mathematical experiences that promoted the development of mathematical knowledge.
In the third study, I use the six principles of proof understanding (McCrone & Martin, 2009) to examine 21 students’ conceptions of proof. I use semi-structured interviews to gather students’ perspectives of (a) the roles of proof, (b) structure and generality of proof, and (c) the opportunities for proof in the curriculum materials. The findings indicate that the students considered proof as serving the roles of explanation, verification, systemization, and appreciation in mathematics. The latter role helps students see the value and purpose of the mathematical results they learn (a) for applications during problem solving and (b) within the axiomatic system of Geometry results. The aforementioned roles also help students see the intellectual need for reasoning and proof in their mathematical experiences. Students’ talk suggests that, their teachers’ and the external examiners’ expectations of the structure, generality, and validity of proof influence their notions of what constitutes a proof. Students also consider the examination opportunities that require the development of reasoned explanations as possible opportunities to construct proof arguments.
The combined findings of these three studies could help researchers understand the implications of the recent reform recommendations on the teaching and learning of proof in Trinidad and Tobago. Firstly, these findings can be useful to policy makers and education stakeholders in their future efforts for developing the national curriculum, revision or development of instructional policies, and recommendations of textbooks and instructional support materials. Secondly, these findings can help curriculum designers, examiners, and teachers in creating future opportunities in the national curriculum and CSEC mathematics syllabus to support students’ learning of proof in Trinidad and Tobago. Thirdly, these findings can help educational stakeholders understand the type of support that is needed for teachers’ future professional development and students’ competency with reasoning and proof on CSEC examinations. This international study is a case of the larger issues surrounding reform implications in a centralized governed educational system, which offers uniform prescriptive guidance for teaching and uniform curriculum support for learning. Furthermore this work potentially adds to the ongoing discussions in mathematics education about the interplay between policy, practice, and student learning
