Categories and Their Relationships Among Socially Open-Ended Problems

Values play a significant role in selecting careers and solving problems in daily life. However, when considering problem-solving in mathematics education, values are not typically addressed but are regarded as noise (Iida et al., Res J Kyushu Math Educ 1:32–43, 1994). In this context, Baba (J Jpn Soc Math Educ 89:20–27, 2007; J JASME Res Math Educ 15:51–57, 2009) proposed appreciating values in mathematical problem-solving and developed the concept of socially open-ended problems by extending the renowned open-ended approach (Shimada, in Open-ended approach in mathematics. Toyokan Shuppan, Tokyo, original version 1977, new version 1995). Subsequently, Shimada (Arithmetic and mathematics education and diverse values: approaches through socially open-ended problems. Toyokan Shuppan, Tokyo, 2017) developed examples of socially open-ended problems and experimental lessons utilising them. In addition, he discovered that these problems can be inductively and experientially grouped into four categories: Sharing, Selecting, Rule-making, and Planning/Designing. For example, in Hitting the target (Shimada, in Arithmetic and mathematics education and diverse values: approaches through socially open-ended problems. Toyokan Shuppan, Tokyo, 2017; Shimada and Baba, in Proceedings of the 40th conference of the international group for the psychology of mathematics education. PME, 2016.), students develop a rule for when a ball hits the border between two areas; hence, this is classified as a Rule-making category. This study considered these categories systematically, including the manner of their interrelation, to further develop the theory and practice of socially open-ended problems. A document analysis was employed in two stages. The first reflected the characteristics and relationships among previously formed categories, whereas the second collected problems from major studies of mathematical modelling and analysed them in terms of these categories. Consequently, this research (1) identified the characteristics of inductively and experientially developed categories (Shimada, in Arithmetic and mathematics education and diverse values: approaches through socially open-ended problems. Toyokan Shuppan, Tokyo, 2017; Shimada and Baba, in Open-ended approach in mathematics lesson to enrich diversified values and mathematical viewing and thinking. Meijitosho Shuppan, 2022) in relation to six universal activities (Bishop, Mathematical enculturation: a cultural perspective on mathematics education. Kluwer Academic Publishers, 1988) and discovered that (2) some categories belong to the meta-level, whereas others belong to the object level; (3) tasks in normative and socio-critical mathematical modelling have the potential to be socially open-ended problems; and (4) these tasks can be grouped into four categories of socially open-ended problems.