Why should I learn geometry

Silke Ruwisch

Clear justification in different ways in geometry lessons in elementary school

Children are curious and want to know how certain aspects are related. This curiosity can be put to good use, especially in geometry lessons, as the children can explore phenomena themselves and make assumptions. They have to be stopped to explain and justify, but it is precisely this language that is particularly fruitful for geometric learning.

Even if justification, argumentation or even proof are often considered to be the highest levels of mathematical penetration and textbook authors and teachers take the view that “this is at most something for the very strong”, all students should be encouraged, encouraged and challenged to justify . The search for good reasons, the question of general validity, questioning your own solutions - these are all profoundly mathematical ways of thinking and acting. These run through virtually the entire mathematical work and can therefore not only be offered in elementary school lessons from time to time, as the icing on the cake, so to speak. In addition to learning mathematical working methods, the aim is to develop a critical mindset, a questioning and discussion culture in which others are convinced of the correctness or falsity of a statement by means of rational arguments. The search for reasons and the questioning of statements contribute significantly to the mathematical maturity of children and adults.
Justification as a building block of argument
In mathematics lessons, justification is embedded in the process-related competence of reasoning. With Betzold (2010) four building blocks or components of argumentation in mathematics lessons in elementary school can be distinguished: Substantial tasks stimulate the discovery of mathematical phenomena. The discoveries must then be described. This is followed by the questioning: Is this always the case? Could it be that ...? Before finally justifications have to be developed. In the overview (Fig. 1), the competency areas of argumentation named in the educational standards are assigned to these four components. It is easy to see that in elementary school - above the four building blocks - work is carried out in a more exemplary manner, i.e. arguments should be understood, while in lower secondary level each of the four steps is justified, i.e., primarily consciously carried out by the pupils should. These argumentative competencies in the narrower sense of the word are framed and supported by other communicative skills, such as the use of technical language.
Justify in geometry lessons
Geometry is particularly suitable for introducing all children equally to a fundamental attitude. In elementary school lessons, geometric tasks always include different levels of representation, which offer such different approaches. When giving reasons, the pupils can also rely on sequences of actions with material, on drawings, on showing concrete objects, etc. (cf. Ruwisch “Basics”). In this respect, geometry gives all children access to justification. While geometry has long been considered a particularly suitable subject area for learning and practicing proving in secondary school (e.g. Weigand et al. 2018), we tend to think of arithmetic relationships or stochastic tasks when we focus on reasons in mathematics lessons in elementary school. From the beginning of school, children can be encouraged in geometry lessons to look for reasons for a situation and to give them (cf. Franke & Reinhold 2016).
How this can already be made possible in the beginning of the lesson should be ...