Alex Friedlander

The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

Understanding Equivalence of Symbolic Expressions in a Spreadsheet-Based Environment

Use of spreadsheets in a beginning algebra course was investigated mainly with regard to their potential to promote generalization of patterns. Less is known about their use in promoting understanding and learning of transformational activities. The overall purpose of this paper is to consider the conceptual aspects of learning a transformational skill (use of the distributive law to produce equivalent algebraic expressions) in a learning sequence composed of both spreadsheets and paper-and-pencil activities. We conducted a sequence of classroom activities in several classes, and analyzed the students’ work on a spreadsheet activity and on an assessment activity by both qualitative and quantitative methods. The findings indicate both encouraging benefits and some potential sources of difficulties caused by the use of spreadsheets at initial stages of learning symbolic transformations.

Improving Mathematics Instruction and Learning in the Junior High School: An Israeli Example

Abstract A 4‐year project is described which was designed to improve mathematics instruction and achievements in two Israeli junior high schools. Project activities focused on the individual teachers, the mathematics staff, and the organization of instruction at the school level, under the assumption that more efficient teaching would lead to greater pupil learning. Evaluation data gathered from classroom observation, questionnaires and tests indicated a general positive impact: many teachers improved their teaching strategies, more curricular material was being covered in class, a school‐wide program for mathematics instruction was established, teamwork among teachers was formalized, and pupil achievements improved.

Instances of Promoting Creativity with Procedural Tasks

The learning of algebraic procedures in middle-school algebra is usually perceived as an algorithmic activity, achieved by performing sequences of short drill-and-practice tasks, which have little to do with conceptual learning or with creative mathematical thinking. The goal of this chapter is to explore possible ways by which all middle-grade students can be encouraged to apply higher-order thinking in the context of tasks that integrate procedural work, conceptual understanding and creative thinking. Each of the five instances presented in this chapter was intended to promote creative thinking in the context of procedural tasks. An a-priori task analysis and data collected in some of our previous studies indicate the presence of many learning competencies and high levels of mathematical creativity in the participating students’ work. Thus, we conclude that certain procedural tasks have a strong potential to promote higher-order, and creative thinking.