Hur påverkas elevers förståelse för multiplikation av areamodeller som representation? En designstudie i en grekisk grundskola med elever i Åk 2, där undervisningsprocessen fokuserar multiplikativt tänkande samt kommutativa lagen för multiplikation

Abstract

This design study explores the use of the area model as a representation of multiplication, providing opportunities for second school grade pupils to develop an understanding of multiplication and build multiplicative thinking. Furthermore, the study aims to give pupils the opportunity to discover the commutative law of multiplication with the help of the area model. The Realistic Mathematics Education (RME) is used as the theoretical framework. Also, through with the help of the area that are close to the pupils’ experience and meaningful to them, pupils are sup ported to develop and strengthen their understanding of multiplication. This study combines qualitative and quantitative analyses. The study includes 20 Greek pupils in second grade of primary school in Greece, more precisely in Chania - Crete. For the qualitative part of the study, mathematics lessons were recorded and analyzed. For the quantitative part of the study, specific tests were carried out before and after the lessons and the results were analyzed using the Excel program. The results of the design research show that the area model representation of multiplication contrib utes to the development of understanding of multiplication and to the development of multiplicative thinking of pupils from the beginning of elementary school. The area model representation for mul tiplication expressions also helped children to be able to focus on the three quantities involved in the multiplicative situation at the same time: the number of equal groups (the multiplicand), the number in each group (the multiplier) and and the total amount (the product) and be able to coordinate the grouping structure. Also, through the results of the teaching experiments, the five developmental phases of multiplicative thinking ware identified: In the first developmental phase, students count one by one. The second phase of development is the additive strategy where students understand that counting is a fixed in dicator for measuring quantities. The third development phase is identified as the pupils used the multiplicative strategy without success and the pupils understand that the additive strategy is not enough. At this stage they understand that sets can be counted and that they can keep track of two things at once, the number of sets and the sum of parts of each set. In the fourth stage of development, pupils successfully use multiplicative thinking and can describe the relationship between numbers. The pupils understand that multiplication situations involve three aspects (the multiplication factors and the product). The fifth development phase is the proportional strategy where the student can introduce a new quan tity as a uniting factor and with the help of multiplication provides a solution to the problem. Through the results of the multiplicative problem-solving strategies investigated in students, solution strategies are identified: direct counting with modelling, double counting and repeated addition and multiplica tive operation, which students use to solve the identified multiplicative problems.