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An Invitation to Critical Mathematics Education deals with a range of crucial topics. Among these are students' foreground, landscapes of investigation, and mathematics in action. The book is intended for a broad audience: educators, students, teachers, policy makers, anybody interested in the further development of mathematics education. The book discusses concerns and preoccupation. This way it provides an invitation into critical mathematics education.

One can identify at least three different types of relationships between mathematics and crises. First, mathematics can picture a crisis. This is in accordance with the classic interpretation of mathematical modelling, which highlights that a mathematical model provides a representation of a piece of reality, a reality that could be a critical situation such as, for instance, a pandemic. Second, mathematics can constitute a crisis, meaning that mathematics can form an intrinsic part of the very dynamics of a crisis. This phenomenon can be illustrated by the economic crises that spread around the world in 2008. Third, mathematics can format a crisis. This final formulation refers to a situation where a mathematical reading of a crisis brings about ways of acting in the critical situation that might be adequate, but also counterproductive, if not catastrophic. This is illustrated with reference to the potential crises due to climate changes. As a conclusion, the paper addresses the politics of crises, which refers to the power that can be acted out through a crisis discourse in which mathematics may come to play a deplorable role.

By a politics of meaning I refer to the social, economic, cultural and religious conditions for experiencing meaning. I refer as well to the layers of visons, assumptions, presumptions and preconceptions that might construct something as being meaningful. By addressing different politics of meaning in mathematics education, I want to show how meaning becomes formatted. In order to do this, I provide a foreground-interpretation of meaning. The basic idea is to relate meanings and foregrounds, acknowledging that foregrounds are formed bya range of factors, as well as by the person's experiences of such factors. Politics of meaning can be analysed with reference to sexism, racism, instrumentalism, the school mathematics tradition, critical mathematics education, and the banality of expertise. ; By a politics of meaning I refer to the social, economic, cultural and religious conditions for experiencing meaning. I refer as well to the layers of visons, assumptions, presumptions and preconceptions that might construct something as being meaningful. By addressing different politics of meaning in mathematics education, I want to show how meaning becomes formatted. In order to do this, I provide a foreground-interpretation of meaning. The basic idea is to relate meanings and foregrounds, acknowledging that foregrounds are formed bya range of factors, as well as by the person's experiences of such factors. Politics of meaning can be analysed with reference to sexism, racism, instrumentalism, the school mathematics tradition, critical mathematics education, and the banality of expertise.

The referential theory of meaning as well as the use-oriented theory of meaning have huge impacts on the discussion of meaning in mathematics education. Here, I present a third theory in terms of an intentionality interpretation of meaning, which provides an alternative departure for the discussion of meaning in mathematics education. The importance of intentionality for understanding human phenomena was emphasised by both Brentano and Husserl. They associated intentionality with a pure stream of consciousness, which constitutes an a priori to any human experience. I agree that the notion of intentionality is important; however, I find it crucial to provide a paradigmatic uprooting of this notion and to consider it as being structured by economic, political, cultural, and discursive factors. Such real-life intentionalities constitute the basis for an intentionality interpretation of meaning. I explore this interpretation with respect to mathematics education by addressing imaginations, possibilities, obstructions, hopes, fears, stereotypes and preconceptions. I relate meaning in mathematics education to far away horizons of students' life worlds, to negotiations, to political issues, to diversity and to instrumentalism.

A radical form of postmodernity is presented with reference to Nietzsche's ideas with respect to truth, knowledge, sciences, progress, democracy, and ethical values in general. Thereafter is presented Foucault's archaeology of knowledge. This brings us forward to the notion of genealogy, which is a defining idea for the postmodern conception of critique. However, it is emphasised that a critique can address the generativity of mathematical rationality by considering mathematics-based fabrications. Finally, imagination is presented as yet another feature of a critical enterprise. It is illustrated how such a three dimensional critical enterprise is relevant for both mathematics and mathematics education. In this way the paper suggests moving beyond the postmodern outlook.