The increasing number of technological devices available in schools, aligned with curriculum guidance, set an expectation for mathematics teachers to incorporate these devices into their teaching. This qualitative study investigated prospective teachers' use of TPACK and mathematical action technologies as they created screencast video lessons using iPads. Results showed prospective teachers' effective use of pedagogical techniques and the screencast app as an amplifier tool, according to the amplifier-reorganizer metaphor. Half of the participants used mathematics technology to confirm and expand the results they had found without technology. The other half had mathematics technology integrated into their solution exercising the balance among TPACK components. For some, their use of the mathematical tool had the potential of expanding the mathematical repertoire of virtual students. We conclude by making recommendations for teacher educators to implement cycles of learning for pre-service teachers to design, enact, and reflect upon the creation of screencast video lessons.

Understanding how one representation connects to another and how the essential ideas in that relationship are generalized can result in a mathematical theorem or a formula. In this article, we demonstrate this process by connecting a vector cross product in algebraic form to a geometric representation and applying a key mathematical idea from the relationship to prove the Shoelace theorem.

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The purpose of this study is to investigate secondary teacher candidates’ experience of mathematical modeling task design. In the study, 54 teacher candidates in a university-based teacher education program created modeling tasks and scoring rubrics. Next, the participants pilot-tested the tasks with students and had the opportunity to revise the original tasks and rubrics based on student responses. The data included participants’ statements, in which they described and reflected on the design and revision process of modeling tasks. The study describes six didactic revision strategies in revising modeling tasks and identifies five emerging pedagogical ideas from revising tasks and rubrics. The study also discusses the way modeling task design activities have the potential to support teacher candidates’ learning through a bottom-up modeling curriculum in teacher education.

Translation by a vector in the coordinate plane is first introduced in precalculus and connects to the basic theory of vector spaces in linear algebra. In this article, we explore the topic of collision detection in which the idea of a translation vector plays a significant role. Because collision detection has various applications in video games, virtual simulations, and robotics (Garcia-Alonso, Serrano, and Flaquer 1994; Rodrigue 2012), using it as a motivator in the study of translation vectors can be helpful. For example, students might be interested in the question, “How does the computer recognize when a player's character gets hit by a fireball?” Computer science provides a rich context for real-life applications of mathematics-programmers use mathematics for coding an algorithm in which the computer recognizes two objects nearing each other or colliding. The Minkowski difference, named after the nineteenth century German mathematician Hermann Minkowski, is used to solve collision detection problems (Ericson 2004). Applying the Minkowski difference to collision detection is based on translation vectors, and programmers use the algorithm as a method for detecting collision in video games.