Pub Date : 2018-10-31DOI: 10.4102/pythagoras.v39i1.363
Million Chauraya,Karin Brodie
The growing perception of professional learning communities as an effective professional development approach needs to be supported with knowledge of how such communities create learning opportunities for teachers. Activities in professional learning communities are underpinned by collegial conversations that foster learning, and in this article we analysed such conversations for learning opportunities in one professional learning community of mathematics teachers. Data consisted of audio-recorded community conversations. The focus of the conversations was to understand the thinking behind learners’ errors, and teachers engaged in a number of activities related to learner errors and learner reasoning. Our analyses show how opportunities for learning were created in identifying the origins of learners’ errors as well as learners’ thinking underlying their errors. Results also showed that the teachers had opportunities for learning how to identify learners’ learning needs and in turn the teachers’ own learning needs. The teachers also had opportunities for deepening their own understanding of the conceptual meaning of ratio. The learning opportunities were supported by the following: having a learning focus, patterns of engagement that were characterised by facilitator questioning, teacher responses and explanations, and sharing knowledge. Such mutual engagement practices in professional learning communities resulted in new and shared meanings about teachers’ classroom practices. Our findings also show the critical role of a facilitator for teacher learning in professional learning communities.
{"title":"Conversations in a professional learning community: An analysis of teacher learning opportunities in mathematics","authors":"Million Chauraya,Karin Brodie","doi":"10.4102/pythagoras.v39i1.363","DOIUrl":"https://doi.org/10.4102/pythagoras.v39i1.363","url":null,"abstract":"The growing perception of professional learning communities as an effective professional development approach needs to be supported with knowledge of how such communities create learning opportunities for teachers. Activities in professional learning communities are underpinned by collegial conversations that foster learning, and in this article we analysed such conversations for learning opportunities in one professional learning community of mathematics teachers. Data consisted of audio-recorded community conversations. The focus of the conversations was to understand the thinking behind learners’ errors, and teachers engaged in a number of activities related to learner errors and learner reasoning. Our analyses show how opportunities for learning were created in identifying the origins of learners’ errors as well as learners’ thinking underlying their errors. Results also showed that the teachers had opportunities for learning how to identify learners’ learning needs and in turn the teachers’ own learning needs. The teachers also had opportunities for deepening their own understanding of the conceptual meaning of ratio. The learning opportunities were supported by the following: having a learning focus, patterns of engagement that were characterised by facilitator questioning, teacher responses and explanations, and sharing knowledge. Such mutual engagement practices in professional learning communities resulted in new and shared meanings about teachers’ classroom practices. Our findings also show the critical role of a facilitator for teacher learning in professional learning communities.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":"19 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138535986","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 : 2018-10-29DOI: 10.4102/PYTHAGORAS.V39I1.347
M. Ledibane, Kotie Kaiser, M. van der Walt
Mathematics has been defined by researchers as a ‘second or third language’ and, as a result, it should be taught as a second language. Results of the literature reviewed from the theories on the teaching of mathematics and English as a second language, as well as on mathematics learning and English as a second language acquisition, have resulted in the emergence of four themes, which are similar to the ones on the teaching and learning of both mathematics and English as a second language; these are: comprehensible input, language processing and interaction, output, and feedback. In this article, the themes are illustrated in a theoretical model and discussed to show how English as a second language and mathematics can be acquired simultaneously. (English as a second language in the South African context is referred to as English as a first additional language.)
{"title":"Acquiring mathematics as a second language: A theoretical model to illustrate similarities in the acquisition of English as a second language and mathematics","authors":"M. Ledibane, Kotie Kaiser, M. van der Walt","doi":"10.4102/PYTHAGORAS.V39I1.347","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.347","url":null,"abstract":"Mathematics has been defined by researchers as a ‘second or third language’ and, as a result, it should be taught as a second language. Results of the literature reviewed from the theories on the teaching of mathematics and English as a second language, as well as on mathematics learning and English as a second language acquisition, have resulted in the emergence of four themes, which are similar to the ones on the teaching and learning of both mathematics and English as a second language; these are: comprehensible input, language processing and interaction, output, and feedback. In this article, the themes are illustrated in a theoretical model and discussed to show how English as a second language and mathematics can be acquired simultaneously. (English as a second language in the South African context is referred to as English as a first additional language.)","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.347","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49608984","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 : 2018-10-24DOI: 10.4102/PYTHAGORAS.V39I1.424
Piera Biccard
Learners often view learning mathematics as non-sense-making (Dienes, 1971; Schoenfeld, 1991). Non-sense-making is distinct from nonsense (no meaning is possible) and is closer to the term senseless (having no meaning). Schoenfeld (1991, p. 316, 320) coined the phrase ‘suspension of sensemaking’ or ‘significant nonreason in students’ school mathematics’ to describe learners’ disengagement with mathematics. The senselessness experienced by learners when trying to engage with mathematics may stem from a disconnection between the learners’ procedural and conceptual understanding. Teachers also mistake procedural competency for conceptual understanding where they see the latter as a natural consequence of the former. Often the senselessness of mathematics comes from this assumption, especially when the problem changes from ‘basics’ (manipulation) to ‘application’ (word problems). Curricula are also often set up to mask procedural ability for conceptual understanding.
{"title":"Mathematical sense-making through learner choice","authors":"Piera Biccard","doi":"10.4102/PYTHAGORAS.V39I1.424","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.424","url":null,"abstract":"Learners often view learning mathematics as non-sense-making (Dienes, 1971; Schoenfeld, 1991). Non-sense-making is distinct from nonsense (no meaning is possible) and is closer to the term senseless (having no meaning). Schoenfeld (1991, p. 316, 320) coined the phrase ‘suspension of sensemaking’ or ‘significant nonreason in students’ school mathematics’ to describe learners’ disengagement with mathematics. The senselessness experienced by learners when trying to engage with mathematics may stem from a disconnection between the learners’ procedural and conceptual understanding. Teachers also mistake procedural competency for conceptual understanding where they see the latter as a natural consequence of the former. Often the senselessness of mathematics comes from this assumption, especially when the problem changes from ‘basics’ (manipulation) to ‘application’ (word problems). Curricula are also often set up to mask procedural ability for conceptual understanding.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.424","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45712763","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 : 2018-10-18DOI: 10.4102/PYTHAGORAS.V39I1.376
J. Alex, K. J. Mammen
After a long six-year lapse, the Curriculum and Assessment Policy Statement introduced in 2012 included geometry as part of the South African Grade 12 Mathematics Paper 2. The first cohort of matriculation students wrote Paper 2 in 2014. This article reports on the understanding of geometry terminology with which a group of 154 first-year mathematics education students entered a rural South African university in 2015; 126 volunteered to be part of the study. Responses to a 60-item multiple-choice questionnaire (30 verbally presented and 30 visually presented items) in geometry terminology provided the data for the study. A concept’s verbal description should be associated with its correct visual image. Van Hiele theory provided the lens for the study. An overall percentage mean score of 64% obtained in the test indicated that the majority of the students had a fairly good knowledge of basic geometry terminology. The students obtained a percentage mean score of 68% on visually presented items against that of 59% on verbally presented items implying a lower level thinking as per Van Hiele theory. The findings of this study imply a combination approach using visual and verbal representations to enhance conceptual understanding in geometry. This has to be complemented and supplemented through scaffolding to fill student teachers’ content gap.
{"title":"Students’ understanding of geometry terminology through the lens of Van Hiele theory","authors":"J. Alex, K. J. Mammen","doi":"10.4102/PYTHAGORAS.V39I1.376","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.376","url":null,"abstract":"After a long six-year lapse, the Curriculum and Assessment Policy Statement introduced in 2012 included geometry as part of the South African Grade 12 Mathematics Paper 2. The first cohort of matriculation students wrote Paper 2 in 2014. This article reports on the understanding of geometry terminology with which a group of 154 first-year mathematics education students entered a rural South African university in 2015; 126 volunteered to be part of the study. Responses to a 60-item multiple-choice questionnaire (30 verbally presented and 30 visually presented items) in geometry terminology provided the data for the study. A concept’s verbal description should be associated with its correct visual image. Van Hiele theory provided the lens for the study. An overall percentage mean score of 64% obtained in the test indicated that the majority of the students had a fairly good knowledge of basic geometry terminology. The students obtained a percentage mean score of 68% on visually presented items against that of 59% on verbally presented items implying a lower level thinking as per Van Hiele theory. The findings of this study imply a combination approach using visual and verbal representations to enhance conceptual understanding in geometry. This has to be complemented and supplemented through scaffolding to fill student teachers’ content gap.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.376","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46989603","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 : 2018-09-26DOI: 10.4102/PYTHAGORAS.V39I1.378
Kathleen M. Mellor, Robyn Clark, Anthony A Essien
Textbook content has the ability to influence mathematical learning. This study compares how linear functions are presented in two textbooks, one of South African and the other of German origin. These two textbooks are used in different language-based streams in a school in Gauteng, South Africa. A qualitative content analysis on how the topic of linear functions is presented in these two textbooks was done. The interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions, and the links created to other mathematical content areas and the real world were considered. It was found that the German textbook included a higher percentage of content that promoted the development of conceptual knowledge. This was especially due to the level of cognitive demand of tasks included in the analysed textbook chapters. Also, while the South African textbook presented a wider range of opportunities to interact with the different representations of functions, the German textbook, on the other hand, included more links to the real world. Both textbooks linked ‘functions’ to other mathematical content areas, although the German textbook included a wider range of linked topics. It was concluded that learners from the two streams are thus exposed to different affordances to learn mathematics by their textbooks.
{"title":"Affordances for learning linear functions: A comparative study of two textbooks from South Africa and Germany","authors":"Kathleen M. Mellor, Robyn Clark, Anthony A Essien","doi":"10.4102/PYTHAGORAS.V39I1.378","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.378","url":null,"abstract":"Textbook content has the ability to influence mathematical learning. This study compares how linear functions are presented in two textbooks, one of South African and the other of German origin. These two textbooks are used in different language-based streams in a school in Gauteng, South Africa. A qualitative content analysis on how the topic of linear functions is presented in these two textbooks was done. The interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions, and the links created to other mathematical content areas and the real world were considered. It was found that the German textbook included a higher percentage of content that promoted the development of conceptual knowledge. This was especially due to the level of cognitive demand of tasks included in the analysed textbook chapters. Also, while the South African textbook presented a wider range of opportunities to interact with the different representations of functions, the German textbook, on the other hand, included more links to the real world. Both textbooks linked ‘functions’ to other mathematical content areas, although the German textbook included a wider range of linked topics. It was concluded that learners from the two streams are thus exposed to different affordances to learn mathematics by their textbooks.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.378","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46858298","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 : 2018-08-13DOI: 10.4102/PYTHAGORAS.V39I1.396
D. Jagals, Martha Van der Walt
Awareness of one’s own strengths and weaknesses during visualisation is often initiated by the imagination – the faculty for intuitively visualising and modelling an object. Towards exploring the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, four secondary schools in the North West province of South Africa were selected for instrumental case studies. Understanding how mathematical objects are modelled in the mind may explain the transfer of the mathematical ideas between metacognitive awareness and the rigour of the imaginer’s mental images. From each school, a top achiever in mathematics was invited to an individual interview (n = 4) and was video-recorded while solving a mathematics word problem. Participants also had to identify metacognitive statements from a sample of statement cards (n = 15) which provided them the necessary vocabulary to express their thinking during the interview. During their attempts, participants were asked questions about what they were thinking, what they did and why they did what they had done. Analysis with a priori coding suggests the three types of imagination consistent with the metacognitive awareness and visualisation include initiating, conceiving and transformative imaginations. These results indicate the tenets by which metacognitive awareness and visualisation are conceptually related with the imagination as a faculty of self-directedness. Based on these findings, a renewed understanding of the role of metacognition and imagination in mathematics tasks is revealed and discussed in terms of the tenets of metacognitive awareness and imagination. These tenets advance the rational debate about mathematics to promote a more imaginative mathematics.
{"title":"Metacognitive awareness and visualisation in the imagination: The case of the invisible circles","authors":"D. Jagals, Martha Van der Walt","doi":"10.4102/PYTHAGORAS.V39I1.396","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.396","url":null,"abstract":"Awareness of one’s own strengths and weaknesses during visualisation is often initiated by the imagination – the faculty for intuitively visualising and modelling an object. Towards exploring the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, four secondary schools in the North West province of South Africa were selected for instrumental case studies. Understanding how mathematical objects are modelled in the mind may explain the transfer of the mathematical ideas between metacognitive awareness and the rigour of the imaginer’s mental images. From each school, a top achiever in mathematics was invited to an individual interview (n = 4) and was video-recorded while solving a mathematics word problem. Participants also had to identify metacognitive statements from a sample of statement cards (n = 15) which provided them the necessary vocabulary to express their thinking during the interview. During their attempts, participants were asked questions about what they were thinking, what they did and why they did what they had done. Analysis with a priori coding suggests the three types of imagination consistent with the metacognitive awareness and visualisation include initiating, conceiving and transformative imaginations. These results indicate the tenets by which metacognitive awareness and visualisation are conceptually related with the imagination as a faculty of self-directedness. Based on these findings, a renewed understanding of the role of metacognition and imagination in mathematics tasks is revealed and discussed in terms of the tenets of metacognitive awareness and imagination. These tenets advance the rational debate about mathematics to promote a more imaginative mathematics.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.396","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47198720","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 : 2018-07-25DOI: 10.4102/PYTHAGORAS.V39I1.393
M. Moloi, A. Kanjee
In this article we propose a framework for reporting mathematics results from national assessment surveys (NAS) such that effective use of the resulting reports can enhance teaching and learning. We explored literature on factors that may contribute to non-utilisation of assessment data as a basis for decision-making. In the context of South Africa, we identified the form and formats in which results of NAS are reported as a possible limiting factor to the effective use of summative assessment results for formative purposes. As an alternative, we propose a standards-based reporting framework that will ensure accurate measurement of, and meaningful feedback on, what learners know and can do. We illustrate how, within a properly designed reporting framework, the results of a NAS in mathematics can be used for formative purposes to enhance teaching and learning and, possibly, improve learner performance.
{"title":"Beyond test scores: A framework for reporting mathematics assessment results to enhance teaching and learning","authors":"M. Moloi, A. Kanjee","doi":"10.4102/PYTHAGORAS.V39I1.393","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.393","url":null,"abstract":"In this article we propose a framework for reporting mathematics results from national assessment surveys (NAS) such that effective use of the resulting reports can enhance teaching and learning. We explored literature on factors that may contribute to non-utilisation of assessment data as a basis for decision-making. In the context of South Africa, we identified the form and formats in which results of NAS are reported as a possible limiting factor to the effective use of summative assessment results for formative purposes. As an alternative, we propose a standards-based reporting framework that will ensure accurate measurement of, and meaningful feedback on, what learners know and can do. We illustrate how, within a properly designed reporting framework, the results of a NAS in mathematics can be used for formative purposes to enhance teaching and learning and, possibly, improve learner performance.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.393","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49431566","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 : 2018-06-28DOI: 10.4102/PYTHAGORAS.V39I1.377
Onyumbe Okitowamba, C. Julie, Monde Mbekwa
Various efforts are underway to improve achievement in high-stakes examinations in school mathematics. This article reports on one such initiative which focuses on the development of quality teaching of school mathematics by embedding it within an examination-driven emphasis. A quantitative approach was used to analyse the performance of Grade 10 learners in three consecutive end-of-year school-based examinations set by the initiative. Results indicate a trend in a positive direction over the three-year period. Nevertheless, there was a discernible decrease between the first and second administration of the examinations. It is concluded that examination-driven teaching holds a promise for enhancing achievement in high-stakes school mathematics examinations if sensibly and sensitively implemented.
{"title":"The effects of examination-driven teaching on mathematics achievement in Grade 10 school-based high-stakes examinations","authors":"Onyumbe Okitowamba, C. Julie, Monde Mbekwa","doi":"10.4102/PYTHAGORAS.V39I1.377","DOIUrl":"https://doi.org/10.4102/PYTHAGORAS.V39I1.377","url":null,"abstract":"Various efforts are underway to improve achievement in high-stakes examinations in school mathematics. This article reports on one such initiative which focuses on the development of quality teaching of school mathematics by embedding it within an examination-driven emphasis. A quantitative approach was used to analyse the performance of Grade 10 learners in three consecutive end-of-year school-based examinations set by the initiative. Results indicate a trend in a positive direction over the three-year period. Nevertheless, there was a discernible decrease between the first and second administration of the examinations. It is concluded that examination-driven teaching holds a promise for enhancing achievement in high-stakes school mathematics examinations if sensibly and sensitively implemented.","PeriodicalId":43521,"journal":{"name":"Pythagoras","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4102/PYTHAGORAS.V39I1.377","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46299633","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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