Distributive law as object of learning through direct and inverse tasks
Authors: Olteanu Lucian 
Published: January 2017

PurposeThe purpose of this paper is to report how the use of a direct and an inverse task gave students an opportunity to discern the structure of a distributive law that they could apply to expand or to factorise algebraic expressions. The paper discusses a teaching sequence (of about 40 minutes) in a Grade 8 (14-15 year-olds) algebra class and it focusses on how the use of inverse tasks opens the dimension of variation. For instance, if the distributive law has always been used asa(b+c), factorising the expressionsab+acmeans opening up the dimension of distributive law. The analysis showed that two central processes, transformation and variation, improved communication in the classroom.Design/methodology/approachThe data used come from a longitudinal study conducted in Sweden. The methodology is grounded in educational design research. Two secondary school teachers conducted a lesson with variation theory as a guiding principle, supervised by a researcher. The relationship between teaching and learning was analysed in the enacted object of learning. The critical aspect for students’ learning was identified by asking questions to probe the students’ understanding.FindingsThe use of a direct and an inverse task gave the students an understanding of the structure of the distributive law that they could apply to expand or to factorise algebraic expressions. The teacher opened up a dimension of variation by similarity that gave the students the opportunity to discern the commonality in direct and inverse tasks as well as to relate the direct and inverse tasks to each other. Without an identification of similarity that might help students to compare underlying meanings, or to match one representation to another, students may not experience variations because there is not concordance among the relationships between the representations.Research limitations/implicationsTeachers can produce new knowledge as well as communicate successfully in the classroom when creating teaching activities that promote the discernment of similarity and difference that might help students to compare underlying meanings, or to match one representation to another.Practical implicationsThe study represents an example of research which has the aim of improving teachers’ practices by using research results from mathematics education whilst keeping in mind that learning must be improved.Social implicationsThe central educational problem is to have students make sense of sophisticated ways of reflecting on the general laws used in mathematics in relation to the algebraic ways of acting and reflecting. Variation theory sees learning as the ability to discern different features or aspects of what is being learned. It postulates that the conception one forms about the object of learning is related to the aspects of the object one notices and focusses upon.Originality/valueThe commutative law for algebraic generalisations is not characterised by the use of notations but, rather, by the way the general is dealt with. Algebraic generalisations entail: the grasping of a commonality related to the discernment of whole-parts relationships, the generalisation of this commonality to two types of transformations: treatments and conversions, and the formation of direct and inverse tasks that allows one to discern the relationship between the whole, the parts, the relations between the parts, the transformation between the parts and the relationship between the parts and the whole.