Neeltje Cornelia Verhoef

Professional development through lesson study: teaching the derivative using GeoGebra

This study focuses on mathematics teachers’ professional development through elements of Japanese lesson study. The teachers designed a research lesson with regard to sense-making of the derivative using the integration of GeoGebra. In the second year of the four-year lesson study project, seven secondary school teachers – from different Dutch schools – worked cooperatively, building on the first-year experiences with the introduction of the derivative. The teachers designed a lesson that focused on encapsulation of the conceptual understanding of the derivative before solving problems with operational symbolism later in the course. They tried to make sense of the calculus using GeoGebra as a tool to realize surprise in which conceptual embodiment and operational symbolism blend together. The teachers integrated GeoGebra to consolidate the derivative using the visual idea of zooming in on the graph to see its local behavior (where a differentiable function looks ‘locally straight’). The teachers reported that they have learned to use visualizations and experienced the importance of student interaction. The teachers realized that this approach of the derivative – integrating GeoGebra – encouraged them to reflect on how the students made sense of learning activities in general.

Lesson Study: professional development (PD) for beginning and experienced teachers

Abstract Central in this study is the professional development of beginning and experienced teachers collaborating in Lesson Study teams. Two high school teacher teams participated, a chemistry and a multidisciplinary team. Each team consisted of a beginning and an experienced teacher. Both teams went through the Lesson Study cycle twice. What and from what the beginning and experienced teachers learned, differences in teacher leaning and what Lesson Study elements contributed to this learning were studied in a qualitative multiple case study using interviews, reflective journals, and recordings . The Extended Interconnected Model for Professional Growth was used to interpret teacher learning. Our results show that two Lesson Study teams materialized in which participants shared experiences, thoughts, and ideas related to teaching and learning. Lesson Study contributed to both beginning and experienced teachers’ PCK development. The combination of two phases in this professional development program proved instrumental for this PCK development: a development phase in which participants meet new pedagogies, discuss these in the perspective of student learning, design a lesson plan and prepare for class use. Followed by a class enactment phase where the designed lesson is enacted, students are observed, subsequently salient results are discussed and the lesson plan revised.

Combinatorial reasoning to solve problems

This study reports combinatorial reasoning to solve problems. We observed the mathematical thinking of students aged 14-16. We study the variation of the students’ solution strategies in the context of emergent modelling. The results show that the students are tempted to begin the problem solving process on the highest level and otherwise have difficulties transitioning from a lower to a higher level of activities. Qualitative analysis revealed some students’ preference of the use of formulas, while at the same time other students showed more insight by their systematic approach of the problems, leading to better results. We advocate matching emergent modelling with teaching combinatorial reasoning, stimulating students to create a relational network of knowledge.