What is differentiation in mathematics
Nina Berlinger & Timo Dexel
In a school context, natural differentiation describes a concept for tasks in the subject of mathematics. It is constitutive that all children receive the same learning offer, that it is sufficiently complex, that there is a technical framework, that the child is free to choose the level of difficulty, paths, aids and modes of representation and that learning together is possible (cf.Krauthausen & Scherer 2010, p. 5f).
Definition of terms
The concept to be explained implies, on the one hand, that learning opportunities for students differentiated i.e. that different students deal with a topic or completely different topics in different ways. On the other hand, this is what happens Naturally - the difference does not seem to be prescribed, but to happen by itself. These two aspects will be described in more detail below.
Usually, external (school structure, support classes, etc.) must be distinguished from internal (within a learning group) differentiation. Inner differentiation of learning content and paths is considered the standard repertoire of didactic activity, especially in primary schools (Krauthausen & Scherer 2007), and there is no lack of suggestions for implementation. At the same time, Trautmann & Wischer state that forms of differentiation are hardly widespread in the practice of secondary education (cf. 2011). Independently of this, various forms of internal differentiation can be identified, e.g. the number of tasks, the level of difficulty or the organizational framework (station learning, cooperative forms of learning, etc.) can vary. Käpnick (2014) points out that most of the forms are almost entirely organized by teachers, who assign them to ability groups. On the other hand, there is differentiation, which gives the children responsibility for their own learning, so that the children differentiate themselves, like natural differentiation. From a mathematics didactic perspective, this concept of differentiation can be included in the debate about open teaching in primary schools. Wittmann criticizes the fact that within this debate the opening up of teaching was discussed exclusively from a pedagogical point of view and demands that this opening must also be thought of from the “FACH” (capitalization ok, NB&TD) (cf. Wittmann 1996, p.11). Based on the thesis that mathematics is a comprehensive social phenomenon that has many references, e.g. to natural science, technology, art, etc., he explains opening as follows:
"Within technical frameworks that can" grow "from the lowest learning levels, problems and tasks of various degrees of difficulty can be formulated. These can be worked on from different conditions, with different means, at different levels and to different extents Wise scope for initiative and creativity. One can modify the problems posed, pose problems for oneself or locate them in the everyday world. The possible solutions are free. How certain tools are used and the results are presented is largely up to the problem solver. The mathematical language like any other language, it can be used flexibly within general conventions and rules. "(ibid.)
This explanation corresponds to the meaning of Naturally. It is crucial that the subject of mathematics offers opportunities for opening up, not just the organizational form (e.g. station learning, etc.). Didactically this is implemented differently, but mostly in the form of extended, substantial tasks. In the literature, this is called the learning environment (cf. Hirt & Wäslti 2008) or open, substantial problem or task areas (cf. Benölken 2016; Benölken, Berlinger & Veber 2017) and has been extensively worked out in practice, at least for primary schools (including Käpnick & Fuchs 2009). The basic intention of these open, substantial problem areas is to realize natural differentiations from the subject (see also Wittmann 1996; Häsel-Weide & Nührenbörger 2015; Scherer 2008). Such a problem area should contain a rich mathematical substance, offer every child the chance to deal successfully with his exploration, arouse curiosity and interest and offer openness with regard to the solutions, the aids and the presentation of results (including Benölken, Berlinger & Käpnick 2016).
Natural differentiation in the context of inclusive mathematics didactics
Fields of activity of this kind are considered to be a possible building block for the didactic implementation of inclusive mathematics lessons (Benölken, Berlinger & Veber 2017). They are particularly suitable for inclusive mathematics lessons, as all children have the opportunity to make mathematical discoveries; It is their own decision or their skills and preferences, how deep they penetrate the mathematical substance, in which way they solve the mathematical problem, which sub-problems they deal with or are interested in, which materials they use what level of representation (enactive, iconic, symbolic) they work and how they represent and present their results. This means that all of these decisions are not made by the teacher, but consciously or unconsciously by the child.
For the common mathematics lessons of all children, there is therefore no need for a completely "new" subject didactics, but the mathematics didactics offers various existing concepts that can be adapted and further developed for the realization of an inclusive mathematics lesson (ibid.). Even if the development of such approaches is only just beginning, the first publications can be recorded (e.g. Käpnick, 2016; Peter-Koop, Rottmann & Lüken 2015). Subject-specific didactics with regard to the design of inclusive lessons according to Lütje-Klose and Miller (2015, p. 20) have the great advantage that “from their respective subject-specific perspective they are dedicated to the analysis of the subject matter, the systematic build-up of knowledge, the development of the (to) employ the child's level of learning and the corresponding technical support in the learning process. ”From a mathematics didactic perspective, it is essential that children are allowed to experience mathematics; that they are allowed to make their own discoveries and that they are mathematically active themselves. Winter (1996) sets out the basic experiences that are elementary for mathematics didactics:
1. "Appearances of the world around us, which concern or should concern us all, from nature, society and culture, in a specific way to perceive and understand,
2. to get to know and understand mathematical objects and facts, represented in language, symbols, images and formulas, as spiritual creations, as a deductively ordered world of its own,
3. To acquire problem-solving skills in dealing with tasks that go beyond mathematics (heuristic skills). "(Winter 1996, p. 35)
If, on the other hand, students receive particularly simple or particularly complex tasks from the teacher that do not allow this, important basic experiences cannot be made. This is not necessarily the case with natural differentiation alone, but it is hardly possible without it. However, in order to do justice to principles for the design of inclusive lessons at the same time, the individualization of learning, the creation of a group structure based on solidarity, the cooperation of different professions, the orientation towards the child's living environment and an orientation towards potential should also be implemented (cf.Lütje-Klose 2011, P. 15).
Benölken, Ralf (2016): Open, substantial tasks. A possible key also and especially for the design of inclusive math lessons. In: Benölken, Ralf & Käpnick, Friedhelm (2016). Individual support in the context of inclusion. Münster: WTM, pp. 203-213.
Benölken, Ralf, Berlinger, Nina & Käpnick, Friedhelm (2016): Open, substantial tasks and areas of responsibility. In: Käpnick, Friedhelm (ed.): Different children. Inclusive support in mathematics lessons in elementary school. Seelze: Klett-Kallmeyer, pp. 157-172.
Benölken, Ralf; Berlinger, Nina & Veber, Marcel (2017): The project “Inclusive Mathematics Lessons” - conceptual approaches for teaching and teacher training. MNU Journal. [Accepted]
Häsel-Weide, Uta & Nührenbörger, Marcus (2015). Task formats for an inclusive arithmetic lesson. In: Peter-Koop, Andrea; Rottmann, Thomas & Lüken, Miriam M. (Ed.): Inclusive mathematics lessons in elementary school. Offenburg: Mildenberger, pp. 58-74.
Hirt, Ueli & Wälti, Beat (2008): Learning Environments for Mathematics Lessons. Natural differentiation for the arithmetically weak to the gifted. Velber: Kallmeyer.
Käpnick, Friedhelm (2014): Learning mathematics in elementary school. Heidelberg: Springer spectrum.
Käpnick, Friedhelm (2016): Different different children. Inclusive support in mathematics lessons in elementary school. Seelze: Klett Kallmeyer.
Käpnick, Friedhelm & Fuchs, Mandy: Math for little aces. Recommendations for promoting mathematically interested and gifted children in the 3rd and 4th school year (Volume 2). Berlin: Cornelsen.
Krauthausen, Günter & Scherer, Petra (2007): Introduction to Mathematics Didactics (2nd revised edition). Heidelberg, Berlin: Spectrum Academic Publishing House.
Krauthausen, Günter & Scherer, Petra (2010): Dealing with heterogeneity. Natural differentiation in elementary school. Handout of the SINUS program to primary schools. Kiel: IPN.
Lütje-Klose, Birgit (2011): Do teachers have to change their didactic craft? Including didactics as a challenge for the classroom. Learning school, 14, pp. 13–15.
Lütje-Klose, Birgit & Miller, Susanne (2015): Inclusive teaching - current research and desiderata. In: Peter-Koop, Andrea; Rottmann, Thomas & Lüken, Miriam M. (Ed.): Inclusive mathematics lessons in elementary school. Offenburg: Mildenberger, pp. 10-32.
Peter-Koop, Andrea; Rottmann, Thorsten & Lüken, Miriam M. (Ed.) (2015): Inclusive mathematics lessons in elementary school. Offenburg: Mildenberger.
Scherer, Petra (2008): Learning mathematics in heterogeneous groups - possibilities of natural differentiation. In: Kiper, Hanna; Miller, Susanne; Palentien, Christian & Rohlfs, Carsten (Ed.): Learning arrangements for heterogeneous groups. Bad Heilbrunn: Klinkhardt, pp. 199–214.
Trautmann, Matthias & Wischer, Beate (2011): Meeting diversity with diversity. Practice School 5-10, No. 1, pp. 4-8.
Winter, Heinrich (1996): Mathematics lessons and general education. Communications from the Society for Didactics of Mathematics, 61, pp. 37–46.
Wittmann, Erich Ch. (1996): Open mathematics lessons in elementary school - from the FACH. Elementary Education 43, pp. 3-7.
Nina Berlinger and Timo Dexel
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