Publication: Proof and proving in the Singapore secondary school mathematics curriculum
Abstract
<p>A mathematical proof is an unbroken sequence of steps that establish a necessary conclusion based on truth preserving rules of logic. However, in practice it may be a series of ideas and insights rather a sequence of formal steps. It has been a common belief that this abstract concept is out of reach to students and even some teachers and hence not the focus of the general mathematics curriculum.</p><p>This study treats proof in a broader sense, recognising that a narrow view of proof neither reflects mathematical practice nor offers the greatest opportunities for promoting mathematical understanding. It explores proof and its aspects, proof and its place in the curriculum, students’ and teacher’s conceptions and beliefs about proof in the context of the Singapore mathematics curriculum in secondary schools.</p><p>Two teachers and four students from their respective classes in a secondary school in Singapore participated in the study that was carried out in three phases. In the first phase all the subjects (both teachers and students) did a questionnaire (conceptions) comprising modified secondary four national examination questions on mathematics (elementary and additional mathematics). In the second phase the teachers did a survey (beliefs) on their beliefs about teaching proofs while the students did the same about their beliefs of learning proofs. In the third phase, based on the data from the first two phases (conceptions and beliefs), subjects were interviewed for clarifications and further elaborations.</p><p>The findings related to curricular materials show that both Elementary Mathematics (EM) and Additionally Mathematics (AM) textbooks are structurally similar in terms of the proportion and type of proof tasks available. There is a higher concentration of proof tasks in the AM textbook compared to the EM textbook, suggesting a possible bias towards AM in containing proof tasks. Most proofs were found in the Geometry strand for both textbooks, suggesting a possible bias towards Geometry in containing proof tasks. There is a lack of variety of proof tasks in both the textbooks. Regarding teachers' conceptions related to teaching proof, it was found that there is diversity and depth of proof strategies used by teachers. Also, teachers' conceptions of proof were heavily influenced by their own mathematical knowledge and understanding of concepts and theorems. Students mainly perceived proofs as difficult problems to work on. Challenges related to the learning of proofs, stemmed from textbooks and school notes lacking in clarity and articulation using complex language.</p>