Teacher learning through teaching and researching: the case of four teacherresearchers in a Masters programme

There is a large body of research and on-going discussions about mathematics teachers’poor content knowledge in South Africa, with many suggesting that teachers need moreopportunities to increase their content knowledge. In this article I consider one importantopportunity that does not seem to be exploited – that of teachers’ learning in the classroom.By considering the learning experiences of four teacher-researchers who were enrolled for amasters degree, I explore how these teachers’ mathematics teacher knowledge developed asa result of their research inquiry. The findings indicate that all four teachers have deepenedtheir mathematical knowledge for teaching in the various domains. However, learning in theclassroom is enabled by the presence of supportive and knowledgeable colleagues. Ifauthorities want to encourage such forms of learning, then attention needs to be directed toproviding intensive classroom support in order to maximise the opportunity for classroomlearning

Exploring student teachers’ perceptions of the influence of technology in learning and teaching mathematics

Rapid global technological developments have affected all facets of life, including the teaching and learning of mathematics. This qualitative study was designed to identify the ways in which technology was used and to explore the nature of this use by a group of 52 mathematics student teachers. The participants were pre-service Mathematics students who were enrolled for a Mathematics module at a South African university. The research instruments were an open question and a semistructured interview schedule. Saxe’s framework was used to analyse the data. Some benefits of mathematics software were found to be the provision of different representations, dynamic visualisation of concepts and variation in mathematical situations. It was also found that students used technology more often in their own learning than in their teaching, because the schools did not have many resources. It is recommended that the education department prioritise the provision of specialist mathematics software that can be used to improve learning outcomes in mathematics.Keywords: Dynamic geometry software; mathematics learning; mathematics teaching; Saxe’s framework; technolog

Understanding the contextual resources necessary for engaging in mathematical literacy assessment tasks

Mathematical Literacy (ML) was introduced in South Africa as a compulsory school subjectfor all learners who are not studying mathematics in Grades 10 to 12. In an ML classroom,mathematical skills are used to explore the meaning and implications of information incontext. In this article, the notion of context in ML is interrogated by identifying particularconstructs that can be used to illuminate the focal events in a contextual setting. It is arguedthat each context constitutes a particular domain with specific contextual resources whichset out the parameters of engagement with the focal event of the context. It is shown that thecontextual resources often differ from the corresponding constructs in the mathematicsdomain. It is then argued that in order to fulfil the life-preparedness mandate of ML, thedifferential purposes of the mathematics and contextual domains needs to be acknowledgedand the corresponding implications with respect to the nature of ML assessments needs tobe considered by ML practitioners

An Exploration of Regularities in Mathematics Education Pre-service Students' Responses to Variation in Number Sequences

Abstract The study of number patterns is an essential part of learning mathematics; however, the shift required from noticing patterns to expressing the patterns symbolically may require sophisticated algebraic techniques. The study reported on in this article focused on a group of 57 pre-service mathematics students in order to explore their interpretations of the mathematical symbolism embedded in pattern descriptions as well as their proficiency in using this symbolism to generate descriptions of the patterns. Four tasks were designed in line with Mason's theory of variation which asserts that carefully structured variation within learning activities can be used to enhance learning. The results show that although students were able to produce correct responses to the more direct questions, some students could not handle the added dimensions of variation. The study identified different strategies used by the students to reduce the elements of variation. Many students were unable to generate terms of the sequence which contained repeating cycles, and also struggled to generate a description of the general term of such sequences. It is recommended that such types of sequences may require additional scaffolding especially with respect to the use of the modulo n function

Insights into the Reversal Error from a Study with South African and Spanish Prospective Primary Teachers

[Abstract] 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 taughtThis work was supported by Spanish Government through the project PGC2018-096463-B-I00 as well as the National Research Foundation (NRF) grant number UID 118377South Africa. National Research Foundation (NRF); UID 11837

The impact of a community service learning project on the professional growth of pre-service teachers

This article reports on a qualitative study based on a group of third- and fourth-year science student teachers, who participated in a community service learning (CSL) project. This study followed five students at three community drop-in centres in order to explore how their participation contributed to their learning. Data was generated from the lecturer’s field notes and reports, as well as from the students’ reflections, summative reports and presentations. The findings indicate that students found the experience academically, personally and professionally enriching. Working with poor orphans and community workers also helped the students develop a greater awareness of social responsibility

Learning from previous experiments: readability in national mathematics assessments

This article aims to investigate the challenges associated with understanding the instructions in the Common Tasks for Assessment (CTA) by a Grade 9 mathematics class of second-language English speakers. The design of the CTA was such that a series of tasks was set using an extended context. The Flesch-Kincaid readability test judged the instructions of some tasks to be beyond the readability levels of an average Grade 9 learner. Some instructions had high lexical density, showing that these were difficult to understand. It is recommended that education authorities note the challenges associated with contextualised assessment such as the CTA, and ensure that the challenges associated with understanding instructions in national assessment should not outweigh the benefits of accessing the mathematics in different settings

An exploration of learners’ theorems-in-action used in problems on ratio and proportion

The purpose of this study is to explore Grade 9 learners’ understanding of ratio and proportion. The sample consists of a group of 30 mathematics learners from a rural school in the province of KwaZulu-Natal, South Africa. Data were generated from their responses to two missing value items, adapted from the Concepts in Secondary Mathematics and Science test set in the United Kingdom over 30 years ago. The study utilised Vergnaud’s notion of theorems-in-action to describe the learners’ strategies. It was found that the most common strategy was the cross multiplication strategy. The data reveal that the strategy was reduced to identifying and placing (often arbitrarily) three given quantities and one unknown in four positions, allowing the learners to then carry out an operation of multiplication followed by the operation of division to produce an answer. The study recommends that the role of the function underlying the proportional relationship should be foregrounded during the teaching of ratio and proportion

An investigation of mathematical literacy assessment supported by an application of Rasch measurement

Mathematical Literacy (ML) is a relatively new school subject that learners study in the final 3 years of high school and is examined as a matric subject. An investigation of a 2009 provincial examination written by matric pupils was conducted on both the curriculum elements of the test and learner performance. In this study we supplement the prior qualitative investigation with an application of Rasch measurement theory to review and revise the scoring procedures so as to better reflect scoring intentions. In an application of the Rasch model, checks are made on the test as a whole, the items and the learner responses, to ensure coherence of the instrument for the particular reference group, in this case Mathematical Literacy learners in one high school. In this article, we focus on the scoring of polytomous items, that is, items that are scored 0, 1, 2 … m. We found in some instances indiscriminate mark allocations, which contravened assessment and measurement principles. Through the investigation of each item, the associated scoring logic and the output of the Rasch analysis, rescoring was explored. We report here on the analysis of the test prior to rescoring, the analysis and rescoring of individual items and the post rescore analysis. The purpose of the article is to address the question: How may detailed attention to the scoring of the items in a Mathematical Literacy test, through theoretical investigation and the application of the Rasch model, contribute to a more informative and coherent outcome?http://www.pythagoras.org.zaam201

Lesotho Grade R teachers’ mathematical knowledge for teaching numeracy

Background: It is important that mathematics teachers have a robust knowledge of the mathematics they teach. They need more than just knowledge of the content because they are expected to facilitate understanding of the content with their learners. This study focused on the knowledge of practising Grade R teachers from Lesotho. Aim: The purpose of this exploratory study was to explore practising Grade R teachers’ mathematical knowledge for the teaching of numeracy. Setting: The study was conducted with 48 practising Grade R teachers while they were enrolled in an in-service programme at a College of Education in Lesotho. Methods: Data was generated through the use of questionnaires. The questionnaire included items focusing on the four domains of knowledge, namely common content knowledge, special content knowledge, knowledge of content and students, and knowledge of content and teaching. Results: The findings revealed that some teachers were unable to carry out division problems accurately and many struggled to explain the possible modification of a teaching plan. Only 27% of the group were able to explain that there were levels of understanding numeracy. With respect to ordering a sequence of three topics, only 15% of the teachers produced a reasonable sequence. Conclusion: These teachers’ difficulties raise concerns about the effectiveness of their teaching of numeracy. Despite the Lesotho government’s commitment to improving the learning of mathematics at the Grade R level, much more work is required to be conducted with teachers so that their mathematical knowledge for teaching (MKT) numeracy can be improved