Learning of combinatorics

We describe the procedures used by 11- to 12-year-old students for solving basic counting problems in order to analyse the transition from manipulative strategies involving direct counting to the use of the multiplication principle as a general procedure in combinatorial problems. In this transition, the students sometimes spontaneously use tree diagrams and sometimes use numerical thinking strategies. We relate the findings of our research to recent research on the representational formats on the learning of combinatorics, and reflect on the didactic implications of these investigations.

The work on combinatorics at school is restricted in many cases to the use of formulas that limit the development of reasoning. English (1991) and Fischbein and Gazit (1988) emphasized the interest of students’ reasoning processes when solving combinatorial problems and their educational implications. At the time of such investigations, Piagetian theory affirmed its relevance to cognitive psychology, considering combinatorics as an essential component of reasoning. Currently, the interest in discrete mathematics and, particularly in combinatorics is increasing the research on this content (Jones, 2005).

Much previous research related to our interests has focused on early education, and the detected strategies emerge from a context where students solve counting problems whose solution is usually a number small enough to be obtained by enumerating all possibilities, and counting one by one afterwards. (Empson and Turner, 2006; English, 1991; English, 1993; Steel and Funnell, 2001). Within this context, English (2007) concludes that 7- to 12-year old students “with no prior instruction and receiving feedback only through their interaction with the physical materials, the children were able to apply their informal knowledge of the problem domain to their initial solution attempts.” (p. 152) She suggests that activities with tree diagrams and systematic lists lead 11- and 12- years old children to derive the basic formula for combinations (p. 154).

From a cognitive perspective, Holyoak and Morrison (2005) emphasize the relationship between problem solving and representations performance of subjects and conclude that the representation used to solve a particular case is a key factor in solving the general problem. Rico (2009) characterized the notion of representation as all those tools -signs or graphics- which are present mathematical concepts and procedures and with which the subjects dealt with and interact with the mathematical knowledge, i.e., record and communicate their knowledge about mathematics.

There is agreement in mathematics education to distinguish between internal and external representations. Although both types of representation should be seen as separate domains, from the genetic viewpoint, external representations are characterised by acting as a stimulus for the senses in the process of building new mental structures and allow the expression of concepts and ideas to individuals who use them. Ideas must be represented externally in order to communicate them (Hiebert and Carpenter 1992). We focus our attention on the external representations as those that have a trace or tangible support even when this support has a high level of  abstraction (Castro and Castro 1997).

In the specific case of combinatorics, Kolloffel et al (2008) focus their research on three representational formats: (a) arithmetic, (b) text and (c) diagrams. Diagrams are considered to help students to understand new situations. Its functionality and, particularly, tree diagrams, has been analysed in several studies. For example, Fischbein and Gazit (1988) give the maximum benefit to the tree diagrams in its instructional programs. They consider that tree diagrams are representative of a state of maturity in the counting strategies. The effectiveness of this type of graphical representation has been questioned by Kolloffel (op.cit), arguing that the benefit is restricted only to conceptual learning, and taking into account the possibility of combining two or more representations. Moreover, we will deal with a new category of representational format which also integrates two or more representations, but under the additional condition that none of them by themselves make sense of the problem. We'll call this new type of representation synthetic representation.

The research responds to theoretical and teaching interests. From a theoretical point of view, it provides specific information concerning the use of inductive reasoning in solving combinatorial problems. Concerning representational formats, we confirm that the use of tree diagrams allows students to abandon manipulative strategies and move towards generalization. Furthermore, we have analysed how such diagrams are used to construct the multiplication principle, in some cases extending recent research findings. In particular, we have seen that the diagrams have been used by students at different stages of cognitive processing spontaneously, using them to represent particular cases from which the multiplication principle is derived efficiently without

specific instruction.

The importance of using materials in problem combinatorial problems. As we observe in the analysed cases, students can use them till they feel comfortable with written representation. This has been observed in different stages of the inductive procedure, which lead students to generalise the multiplication principle and justify their conjectures. The study support that experience with two-dimensional combinatorial problems help students to adopt more efficient strategies for three-dimensional combinatorial problems, as English suggested in previous studies.

The findings have a direct impact on instruction in the area of combinatorics. Although we do not question the importance of instructional programs including the use of tree diagrams to generate algorithms for enumeration and counting, we suggest that they should not be the first approach to combinatorial problems. We base this recommendation not only on the fact that some students are able to produce for themselves this type of representation, but on the fact that some students face basic combinatorial problems using inductive reasoning and not tree diagrams. Although the process to generate the scheme of the multiplication principle is slower, students advance significantly towards generalization from particular cases.