Research suggests that troubleshooting activities that require students to reflect on teacher-crafted erroneous examples; i.e., erroneous solutions to problems that correspond to widespread naïve ideas, are beneficial to learning. One possible explanation to these beneficial effects is that troubleshooting activities encourage students to test the…

The aim of this study is to investigate how students graphically represent qualitative and quantitative aspects of co-varying quantities in an embodied digital learning environment that provides feedback in the form of an animation showing what kind of motion their graph represents and how this feedback helped them to overcome typical graphing…

Graded Troubleshooting (GTS) is a powerful routine that teachers can use easily to engender students' metacognitive thinking and boost their understanding of mathematics concepts and procedures. This article describes a new GTS activity designed to prompt students to efficiently exploit worked examples when asked to diagnose erroneous examples…

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum…

Mathematics is a core academic subject, regardless of grade level or setting. Everyone uses mathematics in their everyday life, so being competent in basic mathematics is critical to independent living. One thing teachers can do to ensure learners are learning the mathematics concepts being taught is to diagnose and remediate the errors they are…

Barlow et al. (2018) discussed three types of mistakes worthy of inspection: procedural errors, inappropriate solution processes, and misconceptions. Here, the authors focus on procedural errors, as these often led the teachers in their professional development project to limit their inspection of mistakes to correcting. Despite this narrow focus,…

Teachers can make number talks more ambitious and increase learning opportunities for students by debugging errors, promoting multi-student thinking, and orienting to make connections. This article describes the basic number talks routine and examines some common tendencies, such as "Serial Sharing." It concludes with a vignette to…

Problem-based learning puts students in situations that suggest problems without providing instructions and available knowledge. Therefore, when using problem-based learning, students need to be flexible, self-disciplined, active and self-occupied with knowledge and turn the knowledge the teacher intends to impart into their knowledge. For locus…

This study illustrates how two secondary mathematics teachers used students' incorrect answers as they supported students' engagement in collective argumentation. Three ways of supporting argumentation when students contributed incorrect answers are exemplified, and the structures of these arguments are investigated. Then, by focusing on the…

Engaging with learner errors in mathematics classrooms is an important aspect of teacher pedagogy which can support learner identification with mathematics. This study examined the opportunities that teachers provided for the construction of learner identities based on how the teachers spoke about and interacted with learner errors in two…

This paper presents an initial analysis of 10 Preschool children's responses to a look-draw-trace-draw task. The findings suggest that figure-tracing helped half the children to produce a more accurate representation of the geometric figure presented to them, in their second drawing.

Error analysis is an instructional strategy that can assist teachers to identify learners' areas of weakness in mathematics and that can point to remediation of those errors. This article explores the errors learners exhibit when solving quadratic equations by completing the square using Newman's Error Analysis Model. A research study explored the…

The aim of this study was to investigate the effectiveness of GeoGebra software in preservice mathematics teachers' process of correcting and making sense of errors regarding the concept of the asymptote. Case study method, one of the qualitative approaches, was adopted in the study. The study group consisted of 11 preservice teachers studying in…

Practice-based pedagogies, such as representations of practice and approximations of practice, are increasingly common in mathematics teacher education. More needs to be known about how teacher candidates' (TCs') productions through such activities make visible the resources they bring to the work of teaching. In this paper, we highlight our use…

The development of statistical literacy has become a key outcome of mathematics curricula in various countries. The purpose of the study was to explore primary pre-service teachers' (PSTs') knowledge of aspects of measures of central tendency, which is a central concept in statistical literacy. The 183 PSTs' written responses to four tasks were…