Research in mathematics education as a DESIGN SCIENCE in line with Wittmann (1995) puts emphasis on a constructive element, i.e. designing substantial learning environments. As outlined in Sect. 3.2, these are based on fundamental mathematical content and represent fundamental objectives (Wittmann 2001a). One pending issue still to be resolved is identifying content, which is fundamental in this sense. This is where key ideas of mathematics come into play.
In this chapter the different meanings of these key ideas are specified. Afterwards, some brief insights into two examples of using key ideas in research projects are given. Furthermore, in the projects’ descriptions the interweaving relation between key ideas as normative settings and empirical research is addressed.
3.3.1 Theoretical Reflections on Key Ideas of Mathematics
In the early 1970s—around the same time as the founding of the German Society of Didactics of Mathematics (GDM)—different studies on fundamental ideas of mathematics and mathematics education started to appear in Germany. The papers at that time try to assure and reassure the newly established community of its purpose and aims. Some of the studies focus on mathematical behaviour and mathematics as an activity. Others address mathematics education as a recently encountered independent research field. And last not least, studies put an emphasis on the scope of mathematics contents and topics.
The concept of structuring the ‘body of knowledge’ by fundamental ideas dates back to Bruner (1966a, p. 41). In Bruner’s work various terms, such as ‘basic idea’, ‘general idea’, or ‘fundamental idea’, can be already identified (Bruner 1966b). In subsequent studies and literature the various terms are taken up by others or even the scope is broadened by big ideas, core ideas, etc.
The understanding of key ideas in the current studies seems to depend on miscellaneous underlying meanings. It is indeed futile to attempt to nail down each term in its specific meaning, although the respective focus on (key) ideas can be identified. Bruner (1966b) has already used such ideas with the aim of at least two objectives: He merges thoughts on basic subjects and the grasping of general principles or attitudes of mathematical thinking by the learner whilst working on these basic contents. In a nutshell, the aim of using key ideas is twofold: a focus on specific mathematical thinking processes on the one hand, and a focus on fundamental mathematical content on the other.
184.108.40.206 Focus on Key Ideas of Mathematical Thinking
In different papers, mathematical thinking is attributed to special objectives and behaviour (for more detailed information, see Vohns 2016). For instance, Bender and Schreiber (1985) record inter alia ideation and exhaustion to be fundamental for geometrical thinking. Vollrath (1978) characterizes no specific mathematical thinking types but defines ideas to be fundamental indirectly by the impact of such ideas within an individual thinking process:
When I speak of ideas in the following, I mean the crucial thought of a theme, the substantial core of a consideration, a fruitful inspiration while solving a problem, the leading question of a theory, the central statement of a proposition, the underlying relations of an algorithm, and the images linked to conceptualization. (Vollrath 1978, p. 29, translation by the author of this paper)
Other studies outline the understanding of mathematical ideas in the context of learning and mathematics education, e.g. Winter (1975) identifies general ideas of learning mathematics in using heuristic strategies, proving, mathematising, formalizing, and using mathematical skills. The attempt to describe certain cognitive activities to be typical of mathematical thinking is still ongoing. Current studies mostly reflect on the register that the Organisation for Economic Co-operation and Development (OECD 2013) identifies as so-called mathematical literacy and the therein described mathematical processes “formulating situations mathematically; employing mathematical concepts, facts, procedures, and reasoning; and interpreting, applying and evaluating mathematical outcomes” (p. 9) and fundamental mathematical capabilities. Moreover, in this view it is common to differentiate mathematical thinking gradually in competence levels which are denominated to be reproductive, connective, or reflective.
In summary, this perspective tries to outline the specific components of thinking which define thinking processes and attitude to be mathematical. This meaning of fundamental or key ideas is taken up in current standards and curricula as so-called process goals or principles. Hence, they define fruitful teaching and learning interactions, attitudes, and beliefs towards favourable terms of learning mathematics.
220.127.116.11 Focus on Key Ideas of Mathematical Content Cores
In a different perspective, it is not the thinking and interaction processes that are focused upon but the core content areas. The approach used here follows a constructive orientation in order to provide practical and concrete designs (Wittmann 1974).
Learning mathematics is not an event but a lifelong process. Mathematical key ideas are one possible answer to identify strands—aligned to age and level of development—but also to keep in mind the big picture and connectivity. Bruner recommends to present knowledge as a connected set of facts in a sufficient structure to be re-examined throughout both primary and secondary school (Bruner 1966b).
The scope of mathematical contents, which is considered to be crucial and substantial, constitutes key ideas. Of course it is necessary to condense mathematical contents in such ideas which are continuative and expandable. This concentration in key ideas allows one possible orientation for researchers and teachers.
In order to provide teachers with an orientation beyond substantial learning environments, it is useful to summarize basic knowledge about mathematics, learning, and teaching mathematics in didactical principles. One principle, for example, is “orientation on fundamental mathematical ideas”. (Wittmann 2016, p. 26)
In accordance with Whitehead’s (1929) view on mathematical education, Wittmann follows the idea to restrict the teaching contents (Table 3.1) and not to choose any subject but the mathematical important ones in line with Freudenthal (1983). Ideally, key ideas are never out of fashion because mathematics and its structural important subjects do not change.
Key ideas make it possible to get an overview of important topics from kindergarten up to grade 12. They allow for the understanding of content areas at a glance. This possibility should not be underestimated especially by both teachers and researchers. Focusing on the relations and connections of topics in mathematics education stops the whole picture from being put on the line and from creating isolated or disconnected (Whitehead 1929) teaching-learning-environments, which would be useless for mathematical literacy or the development of fundamental mathematical thinking. At best, key ideas, metaphorically spoken, function as the backbone of the living body of the lifelong mathematical education process.
For research, key ideas function as a framework for designing substantial learning environments. Lifelong learning in terms of a spiral curriculum allows individuals to deepen their understanding while working on these designed tasks in continuous strands of key ideas.
3.3.2 Two Examples of Using Key Ideas
Key ideas can be seen as normative settings determined by mathematics itself, even though empirical research is consistent with these ideas. On the one hand, mathematics education as a DESIGN SCIENCE (Wittmann 1995) needs researchers and expert practitioners to translate the ideas into suitable learning environments and tasks. In doing so, the design naturally takes into account empirical findings according to learning conditions which are psychologically and educational sound. On the other hand, mathematics education research is requested to evaluate the effects and impacts of the implementation of the environments on students’ abilities and mathematical development.
The key ideas serve as designing principles for substantial learning environments. The research responsibility is to identify crucial key ideas and learning trajectories and to implement these ideas into tasks and SLEs. Consequently, these activities provide access to the key ideas and allow sensibility for the main subjects:
The language in which substantial learning environments are communicated is understandable to teachers, so reflective practitioners have good starting points to transform what is offered to them into their context and to adapt, extend, cut, and improve it accordingly. (Wittmann 2016, p. 25)
The projects briefly presented in this chapter are assigned to the essential phases of transition concerning primary school, i.e. the transition from kindergarten to school and the transition from primary to secondary school. Both approaches aim to support the smooth transition phases by pinpointing mathematical key ideas.
18.104.22.168 Mathematical Key Ideas in Kindergarten
Early mathematics in kindergarten is commonly regarded as being an important lifelong learning process. Nevertheless, the scope of core areas has not yet been entirely agreed upon. The role of key ideas here is to suggest one possible approach of an overview of important subjects.
In our design and research project MaiKe (Mathematics in Kindergarten), we take into account the wide range of competencies which are considered to promote a successful school beginning, different content areas such as number and operations, geometry and spatial sense, measurement, pattern, etc. as described in the learning paths or the big ideas (e.g. NAEYC & NCTM 2010; Wittmann 2009).
In this project, the design of a little application for tablet use is the specific vehicle for allowing children, parents, and kindergarten educators to gain access to the normative set of important key ideas. The design idea is to provide awareness of the wide range of suitable mathematical contents (for both children and adults). The purpose of the digital feature is to tempt children and adults to explore real life mathematical objects and analogous situations.
Of course the design of tasks and learning situations has to be accompanied by empirical research on the use, accessibility, and impact of the substantial learning environments: “The big ideas or vital understandings in early childhood mathematics are those that are mathematically central, accessible to children at their present level of understanding, and generative of future learning” (NAEYC & NCTM 2010, p. 6). For instance, first case studies indicate substantial differences between the abilities and competencies shown in an interview versus the digital play environment (Birklein and Steinweg 2018).
Key ideas in this research end up in designing digital learning environments. They thereby offer adults the chance to become aware of the mathematical contents and activities suitable for kindergarten children, and are helpful to overcome the widespread uncertainty of kindergarten educators, which subjects should be provided in early maths education. Furthermore, they may hopefully serve as an implicit in-service education to kindergarten teachers (and parents).
22.214.171.124 Key Ideas of Algebraic Thinking
In the field of algebraic thinking, the particular situation in Germany asks for key ideas for some other reason. Algebraic thinking is not mentioned in primary curricula and therefore the fundamental rule in the interplay of contents and topics is neglected (Steinweg et al. 2018). Hence, key ideas in this branch of mathematics pave the way to become aware of algebraic ideas as a possible subject in primary mathematics education.
International research indicates major ideas and core areas of algebraic thinking (e.g. Kaput 2008), even though these registers are not suitable for German teachers and thus have no influence on teaching-learning-situations in schools. Key ideas have to be made accessible in the specific cultural context. Furthermore, they have to take into account the existing ideas of mathematics and work out the interplay between the common and the supposedly new ideas. Only the connectivity of key ideas ensures dissemination and implementation in classrooms. Moreover, the sensible emphasis on the interweaving of contents protects classroom interaction against disconnected and isolated teaching.
The major branch of patterns and structures, which is given in the national standards (KMK 2004), is taken up as a possible link to algebraic thinking. This content area is controversially discussed and difficult to grasp for teachers. The offer of algebraic key ideas thereby gives one possible answer to the open question of which topics might be condensed in this twosome concept. Consequently, the key ideas of algebraic thinking are formulated in the spirit and wording of patterns and structures: patterns (and structures), property structures, equivalence structures, and functional structures (Steinweg 2017).
Exemplarily, one SLE in the idea of property structures is sketched here. Numbers have certain properties, which can be discovered and described. For instance, the divisibility relation between natural numbers is essential in mathematics. The abstract relation can be made accessible if the product is regarded as a rectangle area with a given length and width, k and a (see Fig. 3.7).
The rectangle has the area b which is—for given a and k—equal to the number of squares in the field on a piece of grid paper. Many mathematically sound activities arise from this idea of rectangles as a representation of factors as edge length. Special numbers that only have two dividers can be identified as numbers with only one possible representation (prime numbers). Numbers that can be divided by 3, 4, 5, etc. can be found and compared.
If the divisibility by two is investigated, odd and even numbers can be displayed. Moreover, not only the properties of numbers but the properties of additive operations on these numbers can be investigated by children. The introduced representation of rectangle areas allows the discovery of the remarkable behaviour of the sums of odd addends to be even (see Fig. 3.8).
This example illustrates the impact of key ideas put into concrete terms of learning environments and tasks. Key ideas allow teachers to become aware of main topics and fruitful tasks in order to initiate and enhance the chances of children’s learning processes.
3.3.3 Closing Remarks
The two illustrations given above indicate the possibilities of key ideas. First of all they serve as the designing principles for researchers in a constructive understanding of mathematics education. They function as a framework for designing substantial learning environments and adequate material. In addition to the area of research therein (identifying key ideas and design), research thereafter is essential. This research evaluates and eventually adjusts the designed environments on the one hand, and monitors and supports children’s learning processes and developments on the other.
For teachers, key ideas serve as guiding principles for classroom interaction. They allow for the awareness of core contents and to differentiate between important tasks and questions and less fruitful ones.
Teachers need to work with learners on the fundamental ideas behind topics. The Chinese teachers seem to me to be paying explicit attention and taking time over what I would call core awareness, or threshold concepts. Everybody can work at those, everybody can take that in, anybody who can get to school can comprehend them. (Mason 2016, p. 45)
Key ideas enhance the chances of children’s learning processes. At the same time they put emphasis on the core objectives of children’s developments. The important steps and milestones can be seen as being structured along the key ideas for both teachers and children.