Igor’ Kontorovich

The answer depends on your lecturer

ABSTRACT This article is concerned with the approaches to the root concept that lecturers in calculus, linear algebra and complex analysis employ in their instruction. Three highly experienced university lecturers participated in the study. In the individual interviews the participants referred to roots of real numbers, roots of complex numbers, roots as real functions and roots as complex functions. However, participants’ approaches to the root concept were mathematically not equivalent and the lecturers resisted to the approaches of their peers. The choices of approach were shaped by different types of pedagogical considerations, such as the mathematical ideas that appear in the courses that lecturers teach, textbook approaches and lecturers’ interpretations of students’ further academic needs. Several cases were indicated where the participants consciously compromised mathematical rigour and limited the scope of their instruction when teaching general mathematics courses.

Response to Mahmood and Mahmood (2015)

ABSTRACT In the article published in the International Journal of Mathematics Education in Science and Technology in 2015, Mahmood and Mahmood suggested an explanation for defining 0! as 1. In this response, I argue that their reasoning is flawed.

We All Know That a0 = 1, But Can You Explain Why?

This article is concerned with exploring the mathematical knowledge and reasoning that can be engaged in providing explanations for the convention a0 = 1. Posts from online open forums are used to illustrate the variety of mathematical ideas that can be utilized in the explanations. The explanations are examined in terms of quality criteria of definitions and proofs.RésuméCet article vise à explorer les savoirs mathématiques et les raisonnements mis en jeu lorsqu’on cherche à expliquer la formule a0 = 1. Nous nous servons de messages publiés dans des forums en ligne pour illustrer la variété des idées mathématiques qui peuvent être utilisées dans ces explications. Les explications sont ensuite analysées sur la base de la qualité des définitions et des preuves.

Teachers Unpack Mathematical Conventions via Script-Writing

This chapter is focused on mathematical conventions and their unpacking. Conventions account for the choices of the mathematics community regarding the ways concepts are defined, named, and symbolized. By unpacking, I refer to the act of offering plausible explanations and arguments for the choice of conventions. A normative practice of unpacking conventions has not been established, which creates a special opportunity for teachers to engage in a special type of rhetorical persuasion: the one that is less biased towards the perspective of a more authoritative rhetorician. Script-writing can be used as a format for designing convention-unpacking tasks. Using responses of prospective teachers to such a task, I illustrate how scripted dialogues and reflections can be used to promote teachers’ knowledge.

Math Lessons: From Flipped to Amalgamated, from Teacher- to Learner-Centered

We introduce a collaboration framework for teachers and teacher educators, who are interested in designing learner-centered mathematics lessons that amalgamate instructional videos. The framework is spiral, when each of its rounds consists of four phases: understanding the teaching context, developing a plan of an amalgamated lesson, carrying out the lesson and looking back. The implementation of the framework is expected to foster teachers’ technological pedagogical content knowledge. To exemplify the framework in action we present a case of an experienced high-school teacher. The case highlights the complexity of designing learner-centered lessons even for a knowledgeable teacher with predispositions towards integration of technology in the classroom.

Evoking the Feeling of Uncertainty for Enhancing Conceptual Knowledge

This paper is focused on mathematical conventions, which account for the decisions of the mathematics community regarding definitions, names and symbols of concepts. We argue that tasks that request learners to create and discuss not necessarily historically valid, but convincing explanations of mathematical conventions, provide them with opportunities to enhance conceptual knowledge. Specifically, the tasks are designed to evoke the feeling of uncertainty that can be resolved through active engagement with mathematical concepts. We analyze the tasks using different theoretical lens and exemplify two responses of teachers who engaged with one of the tasks. We conclude by suggesting avenues for using the tasks in research and practice.