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A qualitative descriptive case study in the interpretive paradigm uses document analysis and students’ written tasks from four assignments given to 30 pre-service teachers. Findings show that engagement and reasoning improve when tasks target key concepts before introducing total surface area formulae.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/pre-service-teachers-mathematical-engagement-in-learning-about-the-total-surface-areas-of-geometrical-solids/297742/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/pre-service-teachers-mathematical-engagement-in-learning-about-the-total-surface-areas-of-geometrical-solids/297742.png","ImageObject",300,407,{"name":92,"@type":93},"Aldword","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the study mainly investigating?","Question",{"text":112,"@type":113},"The study explores how pre-service teachers communicate conjectures, justifications, and generalisations to develop formulae for geometric solids’ total surface areas.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How was the data collected and analyzed?",{"text":117,"@type":113},"A qualitative descriptive case study collected data through document analysis and students’ written tasks. 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Despite several attempts at improvement, the poor performance of South African learners in mathematics persists. This is attributed to instructional approaches. In the study reported on here we explored how pre-service teachers communicate conjectures, justifications, and generalisations to develop formulae for geometric solids. We employed a qualitative descriptive case study within the interpretive paradigm. Data were collected through document analysis and students’ written tasks. Four tasks were administered to 30 pre-service teachers to enable the researchers to reflect on their performance. Students’ written tasks were analysed with the aid of the model of mathematical knowledge for teaching, which served as the theoretical underpinning of the study. The findings of the study reveal that students can develop mathematical engagement and reasoning when appropriate tasks are designed to facilitate understanding of key concepts that are the cornerstone of learning about geometric solids. Certain concepts, notably, circles, radii, pi, rectangles, cones, Pythagoras’ theorem, slanting height, congruence, and prism, were crucial elements that should be explored prior to the introduction of the topic of the total surface areas of geometric solids. The study was an eye-opener to South African policymakers, mathematics teachers and lecturers in terms of identifying students’ weaknesses at pre-service level on how to develop logical methods to make sense in the learning of geometrical solids.  \nKeywords: areas of three-dimensional shapes; areas of two- dimensional shapes; conjectures; deductions; formulae; generalisations; justifications; mathematical concepts; pre-service teachers; understanding  \nIntroduction and Background  \nIn the South African context, the poor performance of learners in mathematics gives stakeholders cause for concern. In this context, the term “stakeholders” refers to people who are interested in education systems worldwide.  \nIn this study we focused on the development of mathematical engagement among first-year pre-service teachers to enable them to make sense of learning about geometrical solids, which includes cylinders, cuboids, triangular prisms, and cones.  \nThe Department of Basic Education, Republic of South Africa (2011:12) recommends that “teachers should facilitate conjecture about the properties of special triangles, quadrilaterals, and other polygons, and test conjectures using any logical method.” In this context “logical method” refers to students’ ability to identify relationships among shapes and use appropriate reasoning to justify their solutions. Siyepu (2018) suggests that teachers should facilitate the processes of deriving formulae of finding total surface areas of complex solid shapes.  \nThe poor performance of learners in mathematics has been attributed to poor instructional approaches that do not make sense (Feza & Webb, 2005; Siyepu, 2013; Skemp, 1976) . This problem does not only apply in South Africa; it is a global issue. Mathematics researchers in Colombia such as Pino-Fan, Assis and Castro (2015:1429) refer to “the importance of teachers knowing school mathematics in depth and breadth, with the general consensus being that this knowledge in turn impacts upo","cbCaihO11RyFT3GS","https://ap.wps.com/l/cbCaihO11RyFT3GS","pdf",679759,13,"English","# Introduction and Background\n## Mathematical engagement and reasoning focus\n## Instructional approaches and learner performance\n## Key theoretical and contextual references","[{\"question\":\"What is the study mainly investigating?\",\"answer\":\"The study explores how pre-service teachers communicate conjectures, justifications, and generalisations to develop formulae for geometric solids’ total surface areas.\"},{\"question\":\"How was the data collected and analyzed?\",\"answer\":\"A qualitative descriptive case study collected data through document analysis and students’ written tasks. The tasks were analyzed using a model of mathematical knowledge for teaching as the theoretical underpinning.\"},{\"question\":\"Which concepts were reported as crucial before introducing total surface areas?\",\"answer\":\"Key concepts include circles, radii, pi, rectangles, cones, Pythagoras’ theorem, slanting height, congruence, and prism.\"}]","Pre-service teachers’ mathematical engagement in learning about the total surface areas of geometrical solids | PDF",33]