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.

Primary school pre-service teachers’ solutions to pattern problem-solving tasks based on three components of creativity

Education stakeholders and researchers in South Africa have emphasised the need to enhance teachers’ creativity through problem-solving tasks. Teachers’ creativity entails using new ideas of creative devices to solve problems, implement solutions, and make learning more effective. In the research reported on here, Guilford’s theory was used to explore primary school pre-service teachers’ solutions to pattern problem-solving tasks based on 3 components of creativity. The data for this research were produced from primary school pre-service teachers’ written responses to the pattern problem-solving tasks, and an extract from participants’ semi-structured interviews. The research involved a qualitative design using convenient purposive sampling to sample 62 pre-service teachers enrolled for a primary mathematics module at a selected higher education institution. Participants’ responses to the written tasks were analysed using content analysis, while the semi-structured interviews were analysed thematically. The result shows that 35 participants were able to draw patterns and express patterns in nth form, while 27 failed to do so. The most common method used to draw a new pattern was counting in 2s and 4s. Furthermore, the result shows that half of the pre-service teachers who participated in the study were not capable of producing varied solutions to pattern tasks. An indication that they did not have the creative potential to prepare learners even after they had been exposed to advanced mathematics content as part of their training process. We recommend that pre service teacher education programmes should include academic activities that could help pre-service teachers enhance creativity through tasks with divergent thinking.

Building an e-Learning Application Using Multi-agents and Fuzzy Rules

One of the biggest challenges in education is teaching mathematics, especially to children. It has been proven that difficulties students face when learning basic mathematics are often the result of previously acquired misconceptions. These misconceptions prevent the student from understanding new concepts and will eventually create a psychological barrier that prevents the student from learning more advanced mathematics. The conventional classroom environment does not provide the teacher with the most efficient means to detect and correct such misconceptions. The goal of our research is to develop an e-learning system for basic mathematics that is capable of providing each student with personalized content to overcome these misconceptions. The system uses a multi-agent architecture to monitor the activity of the student while simultaneously observing and modeling the student’s knowledge and misconceptions. Lessons and exam questions are chosen dynamically by the multi-agent system to cover the prerequisites of new lessons depending on the profile of the user.

Effective Pedagogy of Guiding Undergraduate Engineering Students Solving First-Order Ordinary Differential Equations

An alternative pedagogical design is discussed that aims to guide engineering students to solve first-order ordinary differential equations (ODEs), and is based on students’ learning weaknesses identified from previous teaching and learning activities. This approach supported student’s self-enrichment through exploration of relevant resources in ODEs, and guided students towards the choice of their own effective ways for solving ODEs for different problems. This paper presents the practices on designing and delivering solution techniques for first-order linear ODEs using this approach for more than 400 undergraduate engineering students at a regional university in Australia during 2014–2017. The timeline involved initial experimentation in 2014 and 2015, followed by refinements to the pedagogy based on student’s feedback. The refined pedagogy was then used for the advanced mathematics course in 2016 and 2017. Significant improvements were made in student’s learning outcomes in effectively and accurately solving the first-order linear ODEs over this period.