Acknowledgement to reviewers

No abstract available.

Reasoning under uncertainty within the context of probability education: A case study of preservice mathematics teachers

Interpreting phenomena under uncertainty stands as a substantial cognitive activity in our daily life. Furthermore, in probability education research, there is a need for developing a unified model that involves several probabilistic conceptions. From this aspect, a central inquiry has been raised through this study: how do preservice mathematics teachers (PSMTs) reason under uncertainty? A multiple case study design was operated in which a purposive sample of PSMTs was selected to justify their reasoning in two probabilistic contexts while their responses were coded by NVivo, and corresponding categories were developed. As a result, PSMTs’ probabilistic reasoning was classified into mathematical (M), subjective (S), and outcome-oriented (O). Besides, several biases emerged along with these modes of reasoning. While M thinkers shared equiprobability and insensitivity to prior probability, the prediction bias and the belief of Allah’s willingness were yielded among S thinkers. Also, the causal conception spread among O thinkers.

Novice and expert Grade 9 teachers’ responses to unexpected learner offers in the teaching of algebra

In South Africa, limited studies have been conducted investigating responsive teaching and little is known about how teachers respond to unexpected events ‘in the moment’ that did not form part of their planning. In this article, we report how a Grade 9 novice and expert teacher responded to unexpected learner offers during the teaching of algebra using a qualitative case study approach. Three consecutive lessons for each teacher were video recorded, transcribed and analysed. Our units of analysis for episodes were teachers’ responses to unexpected learner offers and we coded the responses as ‘appropriate’ or ‘inappropriate’. Indicators used to highlight the degree of quality of the response were ‘minimum’, ‘middle’ and ‘maximum’ if a response was coded as appropriate to a learner’s offer. Once lessons were analysed, the first author conducted video-stimulated recall interviews with each participant to gain insight into the two teachers’ thoughts and decision-making when responding to unexpected learner offers. The findings from this study illustrated that the novice teacher failed to press learners when their thinking was unclear, chose to ignore or provided an incorrect answer when faced with an unexpected learner offer. Conversely, the expert teacher continuously interrogated learner offers by pressing if a learner offer was unclear or if she wanted learners to explain their thinking. This suggests that the expert teacher’s responses were highly supportive of emergent mathematics learning in the collective classroom space.

Constructing mental diagrams during problem-solving in mathematics

In mathematics, problem-solving can be considered to be one of the most important skills students need to develop, because it allows them to deal with increasingly intricate mathematical and real-life issues. Often, teachers attempt to try to link a problem with a drawn diagram or picture. Despite these diagrams, whether given or constructed, the student still individually engages in a private discourse about the problem and its solution. These discourses are strongly influenced by their a priori knowledge and the given information in the problem itself. This article explores first-year pre-service teachers’ mental problem-solving skills. The emphasis was not on whether they solved the problems, but rather on their natural instincts during the problem-solving process. The research shows that some students were naturally drawn to construct mental images during the problem-solving process while others were content to simply leave the question blank. The data were collected from 35 first-year volunteer students attending a second semester geometry module. The data were collected using task sheets on Google Forms and interviews, which were based on responses to the questions. An interpretive qualitative analysis was conducted in order to produce deeper meaning (insight). The findings point to the fact that teachers could try to influence how students think during the problem-solving process by encouraging them to engage with mental images.

Insights into the reversal error from a study with South African and Spanish prospective primary teachers

This study, using a quantitative approach, examined Spanish and South African pre-service teachers’ responses to translating word problems based on direct proportionality into equations. The participants were 79 South African and 211 Spanish prospective primary school teachers who were in their second year of a Bachelor of Education degree. The study’s general objective was to compare the students’ proficiency in expressing direct proportionality word problems as equations, with a particular focus on the extent of the reversal error among the students’ responses. Furthermore, the study sought to test the explanatory power of word order matching and the static comparison as causes of the reversal error in the two contexts. The study found that South African students had a higher proportion of correct responses across all the items. While nearly all the errors made by Spanish students were reversals, the South African group barely committed reversal errors. However, a subgroup of the South African students made errors consisting of equations that do not make sense in the situation, suggesting that they had poor foundational knowledge of the multiplicative comparison relation and did not understand the functioning of the algebraic language. The study also found that the word order matching strategy has some explanatory power for the reversal error in both contexts. However, the static comparison strategy offers explanatory power only in the Spanish context, suggesting that there may be a difference in curriculum and instructional approaches in the middle and secondary years of schooling, which is when equations are taught.

Reflecting on dilemmas in digital resource design as a response to COVID-19 for learners in under-resourced contexts

The coronavirus disease 2019 (COVID-19) pandemic and the resulting school closures in South Africa necessitated a major shift in how to support learners’ ongoing mathematics learning. For 10 weeks learners were strictly confined to their homes with restrictions that prohibited seeing any person outside of their household. The only means to access learners and parents in their homes was to reimagine our South African Numeracy Chair Project work and transform it from predominantly face-to-face interventions to digital modalities. As a result, we initiated a project of digital resource development and distribution, particularly focused on our local community in the Eastern Cape. Twenty-two existing resources and 36 purpose-designed resources were shared via Facebook. Through in-depth post hoc reflection of the rapid digitalisation of our materials and ways of working we address these questions: (1) In relation to learners’ new ‘ecology of learning’ during lockdown what digital access modality and platforms were most fit-for-purpose in sharing mathematics learning resources? (2) What principles informed resource design and adaptation for digital distribution and use? (3) What dilemmas were confronted in making decisions about resource design and distribution?. These questions are answered through a document review and post hoc reflections on the noted dilemmas. We share some feedback received and discuss implications of our work and the dilemmas confronted for the provision of quality digital resources for supporting mathematics learning in historically disadvantaged and under-resourced communities in a post pandemic world.

Grade 9 learners’ understanding of fraction concepts: Equality of fractions, numerator and denominator

Our research with Grade 9 learners at a school in Soweto was conducted to explore learners’ understanding of fundamental fraction concepts used in applications required at that level of schooling. The study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions. A qualitative case study approach was followed. A test was administered to 40 learners. Based on their written responses, eight learners were purposefully selected for an interview. The findings revealed that learners’ definitions of fraction were neither complete nor precise. Particularly pertinent were challenges related to the concept of equivalent fractions that include fraction elements, namely the numerator and denominator in the phase of rational number. These gaps in understanding may have originated in the early stages of schooling when learners first conceptualised fractions during the late concrete learning phase. For this reason, we suggest a developmental intervention using physical manipulatives to promote understanding of fractions before inductively guiding learners to construct algorithms and transition to the more abstract applications of fractions required in Grade 9.

Design principles to consider when student teachers are expected to learn mathematical modelling

This article sets out design principles to consider when student mathematics teachers are expected to learn mathematical modelling during their formal education. Blum and Leiß’s modelling cycle provided the theoretical framework to explain the modelling process. Learning to teach mathematical modelling, and learning to solve modelling tasks, while simultaneously fostering positive attitudes, is not easy to achieve. The inclusion of real-life examples and applications is regarded as an essential component in mathematics curricula worldwide, but it largely depends on mathematics teachers who are well prepared to teach modelling. The cyclic process of design-based research was implemented to identify key elements that ought to be considered when mathematical modelling is incorporated in formal education. Fifty-five third-year student teachers from a public university in South Africa participated in the study. Three phases were implemented, focusing firstly on relevance (guided by a needs analysis), secondly on consistency and practicality via the design and implementation of two iterations, and lastly on effectiveness by means of reflective analysis and evaluation. Mixed data were collected via a selection of qualitative instruments, and the Attitudes Towards Mathematical Modelling Inventory. Through content analyses students’ progress was monitored. Results analysed through SPSS showed significant positive changes in their enjoyment and motivation towards mathematical modelling. Student teachers require sufficient resources and opportunities through their formal education to participate regularly in mathematical modelling activities, to develop competence in solving modelling tasks, and to augment positive attitudes. This study adds value to the global discussion related to teachers’ professional development regarding mathematical modelling.

Professional development for teachers’ mathematical problem-solving pedagogy – what counts?

Problem-solving is of importance in the teaching and learning of mathematics. Nevertheless, a baseline investigation conducted in 2016 revealed that mathematical problem-solving is virtually missing in South African classrooms. In this regard, a two-cycle design-based research project was conducted to develop a professional development (PD) intervention that can be used to bolster Grade 9 South African teachers’ mathematical problem-solving pedagogy (MPSP). This article discusses the factors that emerged as fundamental to such a PD intervention. Four teachers at public secondary schools in Gauteng, South Africa, who were purposively selected, participated in this qualitative research study of a naturalistic inquiry. Teachers attended PD workshops for six months where PD activities that were relevant to their context were implemented. Between the PD workshops, teachers were encouraged to put into practice the new ideas on MPSP. Qualitative data were gathered through reflective interviews and classroom observations which were audio-recorded with teachers’ consent. Data were analysed through grounded theory techniques using constant comparison. The findings from the study suggested that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their professional growth in MPSP. The major recommendation from the study is that facilitators of PD must acknowledge these factors to promote teachers’ professional growth in MPSP. If PD processes and activities are relevant to teachers’ personal meaning, reflective inquiry, and collaborative learning, teachers find the PD programme fulfilling and meaningful. This study contributes to the PD in MPSP body of knowledge by having worked with teachers in an under-researched context of historical disadvantage.