This paper presents results of a qualitative study that explored teachers’ instructional practices, including the discourses they use to teach mensuration, which could be associated with learners’ erratic discourses or appropriate mathematical discourses. It also explored the errors and erratic discourses committed by learners as they worked on mensuration problems. In the context of this study, erratic mathematical discourse means the communication between learners or between a teacher and learners that is inconsistent and deviates from known norms of mathematical reasoning and expression. This communication can be noticed from unclear explanations around mathematics concepts and problems, incorrect logic, and a lack of shared understanding among the learners. Teachers play a crucial role in shaping the discourse and conceptual understanding of mathematics in a classroom. Teachers verbal and non-verbal communication strategies can facilitate or hinder learners' mathematical understanding and expression. These discourses may, however, contain errors in the use of mathematical terms and symbols, procedural mastery, and deep conceptual understanding.

The two research questions are.
- How do teachers’ instructional communication and activities influence learners’ mathematical discourse and problem-solving processes?
- What types of conceptual errors and erratic discourses are committed by learners as they solve mensuration-related problems?

Four teachers from four different high schools in Maseru central area were purposely selected and were observed teaching the concepts of Area and Perimeter, where four groups of learners were randomly selected and were observed while they worked on the mathematics problems provided by each teacher as part of the lesson activities. All the observations were video recorded and were later transcribed verbatim. Guided by the Communicative Competence theoretical framework, the study analyzed both sets of data thematically to understand the connections and disconnections between the teachers’ instructional discourse and learners’ erratic and non-erratic discourses around the concepts of Area and Perimeter.

The analysis revealed a number of instructional practices and communication that could be related to learner errors and communication that reveal these errors. These include an authoritative interactive teaching approach that emphasises memorization of the formula Perimeter = 2L + 2W as a universal procedure for perimeter regardless of the type of polygon. Learners, on the other hand, were found to commit errors and exhibit erratic discourses, including conceptual, computational, and procedural errors, which indicate their level of understanding of the concepts, computational procedures, and formulae, as well as real-life problems related to these concepts. For instance, as they worked on a problem requiring the calculation of the perimeter of a 5-sided irregular polygon, a group of learners struggled to identify the correct sides to substitute into the formula 2L + 2W, hence obtained an incorrect solution, while their discussion indicated their adherence to teacher’s authoritative emphasis on the memorization and rote use the formula, and their lack of understanding of the concept of perimeter and the related computational procedures and formulae. The study concludes that there exist instructional practices that influence learners’ errors and erratic discourse on concepts of Area and Perimeter.

Keywords: Mathematics, area, perimeter, errors, erratic discourses, teachers, learners.