Learners’ Views on Asymptotes of a Hyperbola and Exponential Function: A Commognitive Approach

Abstract

Learners in South African schools often respond poorly in questions related to the asymptote. Despite the fact that there are only a few functions in the South African curriculum that actually explore the asymptote, learners still show some deficiency in their understanding of the concept. This research examined Grade 11 learners’ mathematical discourses about the asymptotes of the hyperbola and exponential functions. Data were analysed using the Realisation Tree of a Function, an adaptation of the Realisation Tree Assessment tool from Weingarden, Heyd-Metzuyanim and Nachlieli. While the Realisation Tree Assessment tool focused on teacher talk, the Realisation Tree of a Function focused on learner expression and responses. A qualitative research design was essentially adopted, with exploratory, descriptive and interpretive elements complementing both its data collection and analysis. A purposive sampling strategy was implemented. Data were collected by means of a test administered to a total of 112 Grade 11 participants from four selected secondary schools. Focus group interviews were conducted with 24 of the best-performing participants by using their responses from the written mathematical tests. The results revealed that the learners’ mathematical discourse is not coherent. While learners’ work on each representation was often mathematical there seemed to be a struggle when the task had an unusual orientation. Different expressions of the same mathematical object elicited different responses. The challenge is that learners exhibited a fragmented relationship between the mathematical objects of the function. Keywords: commognition, realization tree, ritualised learning, visual mediators.

Routines’ errors when solving mathematics problems cause cognitive conflict

“…The absence of these indications indicates that in learning mathematics, the initial routine carried out is an implementation of a ritual and does not change even in a state of cognitive conflict. This condition is different from the results of the study, which stated that the initial routine of bringing up student discourse appeared, which was initially implemented as a ritual [39].…”

“…Likewise, the research results describe a gradual change from visual abilities to more formal discourse, which are characterized by developments in routine abilities [40]. This fact is in line with those that said mathematics learning is enhanced through exploration, not just expecting students to follow routines procedures [39]. So that students can only rely on memorizing procedures and concepts and then apply them in solving the problems given.…”

Commognitive Siswa dalam Menyelesaikan Masalah Persamaan Linier Satu Variabel

“…Sedangkan subjek dengan strategi menemukan pola menggunakan komponen commognitive berupa word use, visual mediator, dan narrative. Menurut Mudaly & Mpofu (2019) bahwa perbedaan objek matematika yang digunakan dalam menyelesaikan masalah dipengaruhi oleh pemahaman konseptual yang dimiliki oleh siswa. Subjek dengan strategi menemukan pola terlihat menggunakan pemahaman konseptual pada tahap menyusun rencana karena subjek dengan strategi menemukan pola menggunakan word use berupa pemisalan deret aritmatika yang dijelaskan dengan narrative.…”

Rural Teachers’ Teaching of Algebraic Functions Through a Commognitive Lens

“…However, the sparsity of mathematics education research located within rural schools has not been able to offer insights into the nature of teaching and learning of mathematics in those contexts as well as teachers' and learners' lived experiences. There is an anthology (Berger, 2013;Brodie, 2014;Gcasamba, 2014;Mudaly & Mpofu, 2019), focusing on many aspects of mathematics teaching at different levels of schooling generally. However, there is sparse research that has explored the actual practice of mathematics teaching within rural and farm areas (Nkambule, 2017;Mbhiza, 2019).…”

“…This discussion resonates with Eisenberg's (1992, p. 153) iteration that developing learners' sense of functions "should be one of the school's main goals and collegiate curriculum" because developing learners' conceptual understanding of functions helps learners to work with other mathematics topics with ease. In view of the above discussion, functions can be considered a meta-discourse of algebra (Caspi & Sfard, 2012;Mudaly & Mpofu, 2019). My concern is that while such iterations are made about the importance of functions in the school curriculum, sparse research has been conducted within the South African context focusing on the teaching of the topic to explore the nature of teachers' classroom practices in order to understand how such practices enable and or constrain learners effective learning.…”