This paper discusses authenticity from the perspective of mathematics education. Often, school mathematics offers students inauthentic word problems, which don’t show the authentic usefulness of mathematics in real life. In some tasks, authentic aspects are combined with inauthentic ones (e.g., an authentic context, but the question is artificial and different from what people within that context would ask). Several studies show that students are more motivated by authentic questions than by authentic contexts. Embedding these findings, I discuss issues associated with defining authenticity in education. A first issue is that philosophers use the term to characterize a person’s existential expressions (e.g., being true to oneself), whereas in education, we use the term for learning environments, artefacts, etc. Second, some researchers define authentic learning environments according to criteria (being open to different approaches, simulate a real-life activity, etc.), but I will illustrate that inauthentic activities can comply with such criteria as well. Alternatively, I suggest using the term for separate aspects in a learning environment (contexts, questions, etc.), and define authenticity as a social construct rather than as a subjective perception. In this way, a community (teachers, students, out-of-school experts) can reach agreement on the nature of this characteristic. For an aspect to be authentic, it needs to have: (1) an out-of-school origin and (2) a certification of originality (e.g., by bringing artifacts physically into a classroom or by testimony of an expert). This approach is illustrated by a study on students’ project work during an excursion to a mathematics research workplace.

In this paper we explore how students can experience the relevance of mathematical modelling activities. In the literature we found that relevance is a connection among several issues (relevance of what? to whom? according to whom? and to what end?). We framed this concept in terms of Cultural-Historical Activity Theory (CHAT), a theory for analysing how individuals engage in activities within social environments. We designed modelling activities within a mathematics course for engineering students: there were ample mathematical modelling tasks, a guest lecture by an employee from an engine company who used mathematical modelling in his job, and a group work modelling assessment with a presentation to the whole group. After the course, we interviewed ten students with a wide range of final grades in the course. We analysed the interview data in light of the theoretical framing of the concept of relevance. Our analysis showed that, generally, students experienced the modelling activities as relevant, and that they imagined themselves working in professional practices for which mathematics is relevant. However, doing mathematics was also judged as being relevant only to obtain grades, leave school and enter professions for which mathematics might not be needed. We offer recommendations for making mathematics education more relevant to more students.

This article reports on a longitudinal observation study about students’ development in their use of procedures to calculate instantaneous rate of change. Different procedures for solving tasks on rate of change are taught in mathematics and physics classes, and together they form a repertoire. Our study took an actor-oriented perspective, which we operationalized as a search for students’ personal constructions of relationships between (1) learning from mathematics and physics classes and (2) interview tasks. We followed 10 students for 2 years (from grade 10 to 12), during which we administered 4 task-based interviews. We analyzed the breadth and connectedness of students’ repertoire of procedures and report on the long-term development thereof. We conclude that often procedures are not part of students’ repertoire shortly after the first introduction of this procedure in class. Students need time to acquire single procedures, and much more time to develop a broad and connected repertoire. In the development of their repertoire, there are major differences between students. From an actor-oriented perspective, many personal constructions are visible between learning and interview tasks. Students often use procedures that differ from procedures that are most appropriate from an expert’s perspective. We also observed from an actor-oriented perspective that words such as velocity, steepness, or slope act as bridge for creating relationships between situations and procedures.

This volume draws on research to discuss these topics and highlights some of the differences in terms of challenges, issues, constraints, and affordances that accompany large-scale and classroom assessment in mathematics education as well as some of the commonalities.