This survey addresses the following questions: Which kinds of transitions have been considered by mathematics education research? Which research questions were studied in connection with transitions? Which theoretical approaches and associated methods were used? Did the studies lead to the identification of continuous processes; of successive steps; or of discontinuities? Are difficulties attached to the discontinuities identified, and does the research proposed in such a case means to reduce the gap to create a transition (seen here as a transitional state)?The volume begins with a literature review, describing the main perspectives on transition in available research on mathematics education, the questions they study and the results they produce: namely the epistemological, cognitive, and sociocultural perspectives. It then deepens the respective perspectives in the following four chapters. Chapter 2 concerns learning processes in the case of difficult concepts. These processes can be viewed as discontinuous, with a gap between naive and scientific knowledge. Here the volume introduces a view of conceptual change as pieces and processes, using in particular the concept of Knowledge in Pieces. Chapter 3 focuses on double discontinuity between secondary school mathematics and university mathematics, as introduced by Klein. Students entering university to learn mathematics experience a first transition; if they become secondary school teachers at the end of their studies, they encounter and undergo a second transition. These transitions are both cognitive and institutional, and they are linked with the development of Mathematical Knowledge for Teaching (Ball "et al.

" 2005). Chapter 4 introduces an institutional perspective, viewing mathematical practices as shaped by the institution where they are employed. This perspective illuminates the process of change taking place when students enter a new institution: the primary-secondary transition, the secondary school-university transition, etc. The study of the discontinuities found in the transition between educational institutions can be organised according to different levels of specificity; accordingly, the concept of levels of co-determination is introduced. Lastly, Chapter 5 mainly considers two kinds of transitions and corresponding research works: the transition between prior-to-school and school mathematics, and the transition between school and out-of-school mathematics (for students, and in both directions). It shows that, for young children and for students alike, many mathematical experiences take place out-of-school. For young children, their mathematical performance at school can be linked with these early experiences. In the case of students, the research focus is more on differences and difficulties in the transitions between informal and formal school mathematics, and on attempts to overcome those difficulties.

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