The Relationship Between Proof and Certainty in Mathematical Practice

Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced mathematics doctoral students were given a task-based interview in which they were presented with various sources of evidence in support of a specific mathematical claim and were asked how convinced they were that the claim was true after reviewing this evidence. In particular, we explore why our participants retained doubts about our claim after reading its proof and how they used empirical evidence to reduce those doubts.

Fixing the Crooked Heart: How Aesthetic Practices Support Sense Making in Mathematical Play

We build on mathematicians’ descriptions of their work and conceptualize mathematics as an aesthetic endeavor. Invoking the anthropological meaning of practice, we claim that mathematical aesthetic practices shape meanings of and appreciation (or distaste) for particular manifestations of mathematics. To see learners’ spontaneous mathematical aesthetic practices, we situate our study in an informal context featuring design-centered play with mathematical objects. Drawing from video data that support inferences about children’s perspectives, we use interaction analysis to examine one child’s mathematical aesthetic practices, highlighting the emergence of aesthetic problems whose resolution required engagement in mathematics sense making. As mathematics educators seek to broaden access, our empirical findings challenge commonsense understandings about what and where mathematics is, opening possibilities for designs for learning.

The case for a sub-element ‘measuring matter’ within the Australian national numeracy learning progression

Australia has a National Numeracy Learning Progression (NNLP) that is strongly aligned with the Australian Curriculum: Mathematics. This article examines how a sub-element within this progression could be impacting students’ learning of Science. This sub-element is firmly based on Mathematics education research as to how students build their understanding of geometric measurement (the structure of length, area and volume). Mathematics educators subsequently researched children’s measurement of mass and included it within the same sub-element of the NNLP. The contexts in which mass and volume are measured in Mathematics are different to those used in teaching Science. This article presents two studies that used variation theory and task-based interviews of children in Years 5 and 6, to explore their thinking about mass and volume in a Science context. The findings suggest that mathematical constructs in geometric measurement could be constraining the development of scientific ideas about matter. This research has implications for furthering the development of the NNLP to encompass scientific aspects of measuring matter.

Disputes in a Pre-Service Mathematics Teachers’ Programme: An (Im)Possible Narrative

Background: The curricula of the undergraduate programmes for pre-service mathematics teachers’ education have been debated (and disputed) in Brazilian academic communities over the past decades. Objectives: To investigate actions and disputes among mathematicians and mathematics educators which took place during the curricular changes and creation of the night undergraduate programme for pre-service mathematics teachers’ education at UFRJ. Design: Fictional dialogues were built to present and analyse data from individual interviews. Setting and Participants: Interviews were conducted with seven lecturers, five retired and two in office, who have played central roles in the institution or in designing curricula for the programme. Data collection and analysis: Data analysis and production were conducted through the re-storying methodology. Results: The dialogues indicate that the modification in the priorities of the group of Mathematics Education teachers at the IM-UFRJ moved the faculty away from the discussions that culminated in the curricular changes of 2001 and 2008, either from the understanding of what the laws and resolutions said, or in internal spaces for debate, such as the Fundão Project. Conclusions: Our analysis indicates that disputes take place in a landscape that transcends teachers’ education and reaches more complex political and epistemic terrains, partially related to tensions between mathematics and mathematics education, but that cannot be reduced to this binarism.

Unfettering discussions about social justice: the role of conversational prompts in discussions about mathematics education for Indigenous students

To increase possibilities for listening respectfully to Indigenous educators, there is a need to identify conversational prompts which are used to raise alternative views of social justice about mathematics education for Indigenous students. Using Nancy Fraser’s description of abnormal social justice, an analysis was made of transcripts from round table sessions, at an Indigenous mathematics education conference. This analysis identified a number of conversational prompts that enabled shifts from normal to abnormal discussions about social justice. Normal discussions exhibited assumptions in which mathematics was valued as a Western domain of knowledge; cultural examples could be used as vehicles to teach mathematics; and decisions about education for Indigenous students should be made by external authorities. In abnormal discussions, these assumptions were queried and alternative possibilities arose. The conversational prompts, which initiated this querying, occurred in a number of ways, including the telling of stories and the asking of questions that either directly or indirectly challenged normal justice discourses about Indigenous students’ learning of mathematics. Identifying conversational prompts can assist non-Indigenous mathematics educators, who wish to be allies, to challenge their own and others’ assumptions about normal social justice issues related to mathematics education for Indigenous students.

Mathematics Educators’ Perspective on Pre-service Mathematics Teachers’ Professional Competencies

This research was designed to study pre-service mathematics teachers' professional competencies to assist student learning by using Lesson Study and Open Approach innovations from mathematics educators' perspectives. A total of 35 mathematics educators have more than three years of experience not only in terms of utilizing the Lesson Study and Open Approach innovations but also in providing training to the pre-service mathematics teachers were selected. The researchers employed three data collection methods, namely document analysis, a survey using a questionnaire, and interviews. The obtained data from three sources was designed with the principle of triangulation. The findings of this research were presented under the three steps of the Thailand Lesson Study Model. In the first step, “Collaboratively Design Research Lesson Plan”, pre-service teachers can create problem situations that associated with the students' real world, can analyze the context of the problem situations, can analyze keywords that initiate students' ideas, can anticipate students' ideas, and can prepare teaching materials to support students' ideas. This is followed by the second step as “Collaboratively Observe Research Lesson”. The findings revealed that pre-service teachers can observe students’ ideas when their students were solving mathematical problems, can notice students’ difficulties in their learning, can give feedback using words that match with students’ proficiency level, give students opportunities to show how to think and present their ideas, listen to and accept students’ opinions, and taking notes on students’ ideas or pieces of learning evidence. The findings of the final step namely “Collaboratively Reflect on Teaching Practice” showed that pre-service teachers could reflect the learning outcomes by correlating students’ ideas with the instructions.

Not Another Computer Algebra System: Highlighting wxMaxima in Calculus

This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students’ understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers.

Investigating insight and rigour as separate constructs in mathematical proof

In this paper we investigate undergraduate mathematics students' conceptions of rigour and insight. We conducted comparative judgement experiments in which students were asked to judge different proofs of the same theorem with five separate criteria: rigour, insight, understanding, simplicity and assessment marks. We predicted, and our experiment found, that rigour is a reliable construct. We predicted that insight is also a reliable construct but asking students to judge on the basis of ``which proof gives you more insight into why a theorem is true'' did not result in a reliable judging scale. Our analysis suggests two distinct dimensions: rigour and marks contribute to one factor whereas simplicity and personal understanding relate to a second factor. We suggest three reasons why insight was related almost equally to both factors. First, while comparative judgement was suitable for assessing some aesthetic criteria it may not be suited to investigating students conceptions of insight. Second, students may not have developed an aesthetic sense in which they appreciate insight in ways which are regularly discussed by mathematics educators. Lastly, insight may not be a coherent and well-defined construct after all.

MATHEMATICS EDUCATORS’ PERSPECTIVES ON CULTURAL RELEVANCE OF BASIC LEVEL MATHEMATICS IN NEPAL

The main purpose of this paper was to explore mathematics educators’ perception of the cultural relevance of basic level mathematics in Nepal. The design of this study involved an interpretive qualitative approach by administering in-depth interviews with five purposively selected mathematics educators teaching at five higher education institutions in the Kathmandu valley. Each interview was audio-recorded and transcribed for coding and constructing themes. The major themes that emerged were teaching in a mother language, contextualized Ethnomathematics, and the local knowledge in the curriculum as a teaching approach. The findings of the study can be helpful to curriculum designers and teachers at the basic level of mathematics. The study also adds to the literature of cultural aspects of mathematics teaching and learning and curriculum design.