Pub Date : 2021-11-09DOI: 10.4102/pythagoras.v42i1.623
Benjamin Shongwe
{"title":"Early career teacher’s approach to fraction equivalence in Grade 4: A dialogic teaching perspective","authors":"Benjamin Shongwe","doi":"10.4102/pythagoras.v42i1.623","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.623","url":null,"abstract":"","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48784655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-21DOI: 10.4102/pythagoras.v42i1.602
Methuseli Moyo, F. Machaba
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.
{"title":"Grade 9 learners’ understanding of fraction concepts: Equality of fractions, numerator and denominator","authors":"Methuseli Moyo, F. Machaba","doi":"10.4102/pythagoras.v42i1.602","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.602","url":null,"abstract":"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.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47011133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-29DOI: 10.4102/pythagoras.v42i1.618
R. Durandt
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.
{"title":"Design principles to consider when student teachers are expected to learn mathematical modelling","authors":"R. Durandt","doi":"10.4102/pythagoras.v42i1.618","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.618","url":null,"abstract":"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.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48404415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-25DOI: 10.4102/pythagoras.v42i1.532
Brantina Chirinda
{"title":"Professional development for teachers’ mathematical problem-solving pedagogy – what counts?","authors":"Brantina Chirinda","doi":"10.4102/pythagoras.v42i1.532","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.532","url":null,"abstract":"","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43235687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-13DOI: 10.4102/pythagoras.v42i1.593
K. Brodie, Deepa Gopal, Julian Moodliar, Takalani Siala
The coronavirus disease 2019 (COVID-19) pandemic supported an investigation of ongoing challenges as to whether and how to make mathematics relevant to learners’ lifeworlds. Given that COVID-19 created major disruptions in all learners’ lives, we developed and taught tasks that attempted to make links between their experiences of the pandemic and disciplinary mathematical knowledge. We located our investigation in current debates about the extent to which disciplinary knowledge can be linked to learners’ out-of-school experiences. We developed and analysed two tasks about COVID-19 that could support link-making and productive disciplinary engagement, and analysed one Grade 10 teacher teaching these tasks. We found that linking mathematics to learners’ lifeworlds is both possible and extremely difficult in relation to task design and how the teacher mediates the tasks. In relation to task design, we argue that teachers cannot do it alone;they need to be supported by the curriculum and textbooks. In relation to mediation, we saw that teacher practices are difficult to shift, even in the best of circumstances. We articulate the complexities and nuances involved in bridging powerful knowledge and lived experience and thus contribute to debates on how to teach powerful knowledge in relation to learners’ lifeworlds.
{"title":"Bridging powerful knowledge and lived experience: Challenges in teaching mathematics through COVID-19","authors":"K. Brodie, Deepa Gopal, Julian Moodliar, Takalani Siala","doi":"10.4102/pythagoras.v42i1.593","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.593","url":null,"abstract":"The coronavirus disease 2019 (COVID-19) pandemic supported an investigation of ongoing challenges as to whether and how to make mathematics relevant to learners’ lifeworlds. Given that COVID-19 created major disruptions in all learners’ lives, we developed and taught tasks that attempted to make links between their experiences of the pandemic and disciplinary mathematical knowledge. We located our investigation in current debates about the extent to which disciplinary knowledge can be linked to learners’ out-of-school experiences. We developed and analysed two tasks about COVID-19 that could support link-making and productive disciplinary engagement, and analysed one Grade 10 teacher teaching these tasks. We found that linking mathematics to learners’ lifeworlds is both possible and extremely difficult in relation to task design and how the teacher mediates the tasks. In relation to task design, we argue that teachers cannot do it alone;they need to be supported by the curriculum and textbooks. In relation to mediation, we saw that teacher practices are difficult to shift, even in the best of circumstances. We articulate the complexities and nuances involved in bridging powerful knowledge and lived experience and thus contribute to debates on how to teach powerful knowledge in relation to learners’ lifeworlds.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41472779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-10DOI: 10.4102/pythagoras.v42i1.575
M. M. Moleko, M. Mosimege
Flexible teaching of mathematics word problems is essential to improve learning. Flexible teaching is vital in terms of providing meaningful learning, creating inclusive learning spaces and making content accessible. As such, teachers need to strive to provide flexible teaching of mathematics word problems in order to optimise and maximise learning. In line with this notion, therefore, the qualitative case study reported in this article aimed to explore the implementation of one aspect of universal design for learning (UDL), namely multiple means of representation (MMR), to guide flexible teaching of mathematics word problems. Data were collected using focus group discussions, reflection and observation sessions in which five high school mathematics teachers and a Head of Department were involved. The teachers participated in a mini-workshop on the application of the UDL principles which was organised to introduce and induct them to the approach. The study showed that MMR can be used to help guide flexible teaching of mathematics word problems by providing varied options for comprehension: options for language, mathematical expressions and symbols, as well as options for perception. The findings of the study recommend the need for teachers to adapt their teaching by considering the application of the MMR principle to guide and promote flexible teaching of mathematics word problems.
{"title":"Flexible teaching of mathematics word problems through multiple means of representation","authors":"M. M. Moleko, M. Mosimege","doi":"10.4102/pythagoras.v42i1.575","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.575","url":null,"abstract":"Flexible teaching of mathematics word problems is essential to improve learning. Flexible teaching is vital in terms of providing meaningful learning, creating inclusive learning spaces and making content accessible. As such, teachers need to strive to provide flexible teaching of mathematics word problems in order to optimise and maximise learning. In line with this notion, therefore, the qualitative case study reported in this article aimed to explore the implementation of one aspect of universal design for learning (UDL), namely multiple means of representation (MMR), to guide flexible teaching of mathematics word problems. Data were collected using focus group discussions, reflection and observation sessions in which five high school mathematics teachers and a Head of Department were involved. The teachers participated in a mini-workshop on the application of the UDL principles which was organised to introduce and induct them to the approach. The study showed that MMR can be used to help guide flexible teaching of mathematics word problems by providing varied options for comprehension: options for language, mathematical expressions and symbols, as well as options for perception. The findings of the study recommend the need for teachers to adapt their teaching by considering the application of the MMR principle to guide and promote flexible teaching of mathematics word problems.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49663618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-29DOI: 10.4102/pythagoras.v42i1.586
T. Marange, S. Adendorff
{"title":"The contribution of online mathematics games to algebra understanding in Grade 8","authors":"T. Marange, S. Adendorff","doi":"10.4102/pythagoras.v42i1.586","DOIUrl":"https://doi.org/10.4102/pythagoras.v42i1.586","url":null,"abstract":"","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45159883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-21DOI: 10.4102/pythagoras.v41i1.538
I. Mostert
Word problems are a central, yet hard-to-teach, aspect of early grade mathematics. For example, in South Africa word problems have been identified as a recurring weakness in the South African Annual National Assessments (ANAs) (Department of Basic Education, 2012, 2014, 2015). Research has shown that the relative difficulty of word problems differs: learners are more likely to solve certain types of word problems than others. For additive relation word problems, in other words any word problems involving addition and subtraction, compare type problems have been shown to be the most difficult for learners to solve. Compare type problems are of the form ‘Sbu has eight bananas and Sive has five bananas. How many more bananas does Sbu have than Sive?’ While there has been some research into early grade word problems in South Africa (e.g. Petersen, McAuliffe, & Vermeulen, 2017), and some research into word problems and African languages in higher grades (e.g. Sepeng, 2013), there has been little research into early grade word problems in African languages. This is problematic as more than 75% of learners are taught mathematics in an indigenous African language in the first four years of formal schooling (Spaull, 2016).
{"title":"Relative difficulty of early grade compare type word problems: Learning from the case of isiXhosa","authors":"I. Mostert","doi":"10.4102/pythagoras.v41i1.538","DOIUrl":"https://doi.org/10.4102/pythagoras.v41i1.538","url":null,"abstract":"Word problems are a central, yet hard-to-teach, aspect of early grade mathematics. For example, in South Africa word problems have been identified as a recurring weakness in the South African Annual National Assessments (ANAs) (Department of Basic Education, 2012, 2014, 2015). Research has shown that the relative difficulty of word problems differs: learners are more likely to solve certain types of word problems than others. For additive relation word problems, in other words any word problems involving addition and subtraction, compare type problems have been shown to be the most difficult for learners to solve. Compare type problems are of the form ‘Sbu has eight bananas and Sive has five bananas. How many more bananas does Sbu have than Sive?’ While there has been some research into early grade word problems in South Africa (e.g. Petersen, McAuliffe, & Vermeulen, 2017), and some research into word problems and African languages in higher grades (e.g. Sepeng, 2013), there has been little research into early grade word problems in African languages. This is problematic as more than 75% of learners are taught mathematics in an indigenous African language in the first four years of formal schooling (Spaull, 2016).","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45360463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-18DOI: 10.4102/pythagoras.v41i1.572
Erna Lampen, K. Brodie
becoming mathematical about mathematics, (3) being mathematical and making mathematics, and (4) mathematics. We argue that through such a curriculum, we can develop mathematical reasoning on the basis of learners’ everyday reasoning in ways that support their mathematical proficiency, identities and agency.
{"title":"Becoming mathematical: Designing a curriculum for a mathematics club","authors":"Erna Lampen, K. Brodie","doi":"10.4102/pythagoras.v41i1.572","DOIUrl":"https://doi.org/10.4102/pythagoras.v41i1.572","url":null,"abstract":"becoming mathematical about mathematics, (3) being mathematical and making mathematics, and (4) mathematics. We argue that through such a curriculum, we can develop mathematical reasoning on the basis of learners’ everyday reasoning in ways that support their mathematical proficiency, identities and agency.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44650932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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