Abstract:

Several studies have shown that many students experience difficulties in learning combinatorics, a topic that is typical of the secondary school mathematics curriculum. More often than not, students remain at an instrumental level of understanding combinatorial operations, which may contribute to explain those learning difficulties.

This study was developed within the context of the student teaching internship of the second author and it aimed at knowing how students understand combinatorial operations and mobilize that knowledge in problem solving and problem posing activities. Aiming at developing a relational understanding of combinatorial operations, we designed a teaching intervention with a class of 12th graders (16 to 17 years-old), in a public, urban, secondary school in northern Portugal. The intervention was developed completely at distance given to the COVID-19 restrictions that were in place at the time of the study. Google Meet was the communication platform chosen by the school, taking advantage of some of its affordances, such as breakout rooms, pear deck, and jamboard. We privileged an inquiry-based teaching approach, emphasizing students’ collaborative work on challenging tasks and collective discussions of solutions and approaches.

This study was guided by the following research questions:
1) how do students intuitively solve simple counting problems?
2) how do they understand and apply combinatorial operations in problem solving activities?
3) how do they understand and apply combinatorial operations in problem posing activities? In this text, we focus on the latter research question.

Data were collected through field notes and the audio-recording of a lesson in which the students worked in small groups (six groups in total) in a problem posing activity. Each group was given a combinatorial expression and was asked to formulate a problem that could be translated into that expression; after working collaboratively in breakout rooms, students engaged in a collective discussion about their proposals. Students’ productions were also gathered.
The students were not very familiar with problem posing activities, but they engaged significantly with the task posed. Three groups posed problems that corresponded to the combinatorial expression that had been given, and a forth group only had to make a slight change in the problem formulation so that the desired correspondence was attained. One group was not able to complete the task; yet, during the collective discussion, the group was able to overcome the difficulties and pose a problem that met the conditions. Another group posed a very confusing and unclear problem statement, and even the teachers had serious difficulties in understanding the students’ ideas. Unfortunately, there was no time to discuss the problem suggested by this particular group.

Data analysis suggests that most students evidenced a relational understanding of the combinatorial operations, despite some persisting difficulties in some of them. Students were quite creative in formulating problems, resorting to diversified contexts and situations, and they also revealed some critical thinking through the questions they raised during the collective discussions. This particular problem posing activity allowed them to mobilize their knowledge of the combinatorial operations and allowed the teacher(s) to better access their understanding of the whole topic.

Keywords:

Combinatorics, problem posing, inquiry-based teaching, relational understanding.