The interplay between the guidance from the digital learning environment and the teacher in supporting folding back

Previous studies have proposed that students’ mathematical understanding develops dynamically through the process known as folding back, in which learners revisit earlier forms of understanding and use them to build even deeper levels of mathematical understanding. Digital learning environments, where students can manipulate representations, are often used to enable students to notice properties, patterns, or rules. When working in such an environment, students usually receive support from the environment and the teacher. The interplay between these diferent sources of support is important according to previous studies. In this study, we examine this interplay in the case of folding back. The study aims to understand how the teacher, together with the learning environment, can support the process of folding back. We collected data from second, fourth, and sixth grade students as they worked in groups to develop a rule for balancing a balance beam in a digital learning environment designed to support folding back. One pre-service teacher guided each threestudent group. Data were analyzed by identifying occasions for folding back and characterizing diferent ways in which the interplay between the teacher and the environment supported students’ folding back. We found diferent kinds of synergy between the two sources of support. The teachers followed up on and augmented the support from the environment, initiated supplementary folding back, and reinforced the support from the environment. We also found non-synergy between the two sources of support, when the teachers’ support was not aligned with support from the environment.