Elaine Simmt

Understanding Learning Systems: Mathematics Education and Complexity Science.

Complexity science may be described as the science of learning systems, where learning is understood in terms of the adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex learning systems, we explore some of the possible contributions of complexity science to discussions of the teaching of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of mathematical communities and the application of principles of complexity to the teaching of mathematics. Through the course of this writing, we attempt to highlight compatible and complementary discussions that are already represented in the mathematics education literature.

Teacher Expertise Explored as Mathematics for Teaching

The nature of the teacher’s encounter with mathematics has taken prominence in the last decade as teacher educators research the mathematics of teaching. In this chapter teacher’s expertise is articulated as mathematics for teaching (MFT). A model, theorized from complex learning systems, is discussed in this chapter. It posits MFT as multi-layered and nested knowledge involving subjective understanding at the core, enveloped by an understanding of the collective knowing that emerges from the interaction among individuals, which in turn is nested in knowledge of evolving and emergent curriculum structures, and further nested in a knowledge of the broader culture of mathematics. In this chapter the MFT model is read against the actions and interactions of a group of mathematics teachers in a professional development session to reciprocally explore the teachers’ encounter with mathematics while explicating the model.

The teacher's responsibility in whole‐class debriefing of mathematical activity

Prompted by recent moves in the United Kingdom to guide teachers’ practices in whole-class, direct interactive teaching, in this article, we offer an opportunity for North American mathematics educators to reflect on possibilities for whole-class teaching of mathematics. We focus particularly on the plenary aspect of lessons—what might be considered the debriefing of mathematical activity—and specifically on the teacher’s responsibility during those sessions, both to his or her students and to the authenticity of the discipline of mathematics. Drawing on data from a Grade-3 classroom and invoking complexity science as a theoretical lens to explore the classroom as a complex learning system, we present implications for teaching in whole-class debriefings of mathematical activity.RésuméDans cet article, en réaction à de récents développements en Grande Bretagne visant à guider la pratique de l’enseignement direct et interactif dans des classes nombreuses, nous souhaitons offrir aux enseignants des mathématiques en Amérique du Nord ‘l’occasion de réfléchir sur les possibilités que représente l’enseignement des mathématiques à des classes pleines. Nous centrons surtout notre attention sur les aspects ‘pléniers’ des leçons — ce que nous pourrions qualifier de débriefing des activités mathématiques — et surtout sur la responsabilité des enseignants au cours de ces sessions, en particulier envers les étudiants et envers l’authenticité qui caractérise la discipline des mathématiques. Notre argumentation se fonde sur des données provenant d’une classe de troisième année, et nous nous servons de la complexité des sciences comme lunette théorique permettant d’explorer la salle de classe comme système complexe d’apprentissage. Nous présentons ensuite certaines implications pour l’enseignement dans le cadre de débriefings de classe pour ce qui est des activités mathématiques.

Invited Lectures from the 13th International Congress on Mathematical Education

Practice-based initial teacher education reforms are typically organised around a set of core teaching practices, a set of normative principles to guide teachers’ judgement, and the knowledge needed to teach mathematics. Developing more than understandings, practices, and visions, practice-based pedagogies also need to support prospective teachers’ emergent dispositions for teaching. Based on the premise that an inquiry stance is a key attribute of adaptive expertise and teacher professionalism this paper examines the function and value of inquiry within practice-based learning. Findings from the Learning the Work of Ambitious Mathematics Teaching project are used to illustrate how opportunities to engage in critical and collaborative reflective practices can contribute to prospective teachers’ development of an inquiry-oriented stance. Exemplars of prospective teachers’ inquiry processes in action—both within rehearsal activities and a classroom inquiry—highlight the potential value of practice-based opportunities to learn the work of teaching.

Observing Observers: Using Video to Prompt and Record Reflections on Teachers' Pedagogies in Four Regions of Canada.

Regional differences in performance in mathematics across Canada prompted us to conduct a comparative study of middle-school mathematics pedagogy in four regions. We built on the work of Tobin, using a theoretical framework derived from the work of Maturana. In this paper, we describe the use of video as part of the methodology used. We used videos of teaching activities as prompts for discussions among teachers and the video recordings of such discussions became the data sources for our comparative research. Our use of video revealed a number of advantages and disadvantages which influenced the research.

Curriculum Development to Promote Visualization and Mathematical Reasoning

What first comes to mind? For you is 2 a number? Is it a radical? Is it a measure? Is it the length of the diagonal in a unit square? Is it the length of the hypotenuse of a unit-length right triangle? Maybe it is the secant of 45o? How you imagine 2 is dependent on the experiences you have had with it. The more limited our experiences with a mathematical object the more limited our understanding of it. The broader our experiences with the object the greater our understanding of it. 2 is an interesting object. What is it? What does it look like? How does it behave? What might a mathematics teacher do to provide opportunities for learner activity that broadens the learner’s relationship with 2, making possible greater understanding of it and greater understanding in general of radicals?