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The study analyzes his visual representations and the function of “folding back,” arguing that apprentice images for basic concepts may be inflexible and offering guidance and caution about designing effective image modification in workplace contexts.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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\nSchool of Education and Lifelong Learning, University of East Anglia & University of British Columbia  \n\u003C[lyndon.martin@uea.ac.uk](lyndon.martin@uea.ac.uk)>  \nLionel LaCroix  \nDepartment of Curriculum Studies, University of British Columbia  \nLynda Fownes  \nBC Construction Industry Skills Improvement Council – SkillPlan  \nAbstract  \nThis paper presents some initial findings from a multi-year project that is exploring the growth of mathematical understanding in a variety of construction trades training programs. 1 In this paper, we focus on John, an entry-level plumbing trainee. We explore his understandings for multiplication, fractions and units of imperial measure as he attempts to solve a pipefitting problem. We consider the apparently limited nature of his images for these concepts and the role of ‘folding back’ in enabling his growth of understanding. We contend that it cannot be assumed that the images held by adult apprentices for basic mathematical concepts are flexible, deep, or useful in specific workplace contexts. We suggest that folding back to modify or make new images as needed in particular contexts is an essential element in facilitating the growth of mathematical understanding in workplace training, but also offer a note of caution about ensuring that this is genuinely effective in its purpose.  \nMathematical Understanding and Workplace Training  \nAlthough recent years have seen an increase in the attention paid by researchers to mathematics in the workplace there is still only a limited body of work that considers cognition and understanding in a vocational setting. Research that does exist in the area of adult mathematical thinking within a vocational setting is primarily concerned with the use of informal mathematics (e.g. , Noss, Hoyles, & Pozzi, 2000), or the place of mathematics within work (see Bessot, 2000a, 2000b; Eberhard, 2000; Harris, 1991; Millroy, 1992; Scribner, 1984a, 1984b, 1984c; Smith, 1999a, 1999b; Wedege 2000a, 2000b), rather than with the process of coming to understand mathematical concepts for the individual learner in the workplace.  \nStraesser (2000a) in his discussion of workplace vocational mathematics instruction noted two distinct kinds of pedagogies, the first being“modelling”and the second being “legitimate peripheral participation” (see also Lave & Wenger, 1991) . He notes that the first is usually employed in a classroom setting, where:  \nVolume 1(1)–June 2005 19  \nthe situation is to come from the workplace, the mathematical model rests upon mathematical structures and algorithms known before or taught on the spot and the solution of the model hopefully can be interpreted in a way to cope with the given professional situation (p. 70) .  \nIn contrast to this, legitimate peripheral participation relates to training on the job, where“learning takes place at the workplace whenever it is needed by the workplace practice and its problems” (p.70) . In this study we are interested in the first of these two learning situations, that of the workplace training classroom, where apprentices engage with contexts and problems from the workplace, but which require mathematical models in their solution. As Straesser (2000a) states “in most cases, modelling vocational problems by applying mathematics is a major difficulty for the future worker – especially the extraction of the mathematical model from a professional situation at hand” (p.70) . He does not, however, offer any detailed analyses of the kinds of difficulties faced by apprentices in extracting the mathematics and working with it, nor consider the ways in which they go about developing and using appropriate mathematical models.  \nPozzi, Noss, and Hoyles (1998) in discussing the use of artefacts and tools in the workplace note that the need for workers to understand “the models which underlie their artefacts, to sort out what is happe","cbCaiie4EnoTv2dO","https://ap.wps.com/l/cbCaiie4EnoTv2dO","pdf",301571,17,"English","# Abstract\n# Mathematical Understanding and Workplace Training\n## Pedagogical perspectives in workplace settings\n# The nature of mathematical understanding","[{\"question\":\"What is the main aim of the multi-year project described in the paper?\",\"answer\":\"To explore how mathematical understanding grows across a range of construction trades training programs.\"},{\"question\":\"Which learner and tasks are examined in detail?\",\"answer\":\"An entry-level plumbing trainee, John, working on a pipefitting problem that involves multiplication, fractions, and imperial units of measure.\"},{\"question\":\"How does the paper explain the role of “folding back” in mathematical learning?\",\"answer\":\"It proposes that folding back to modify or create new images in specific contexts helps facilitate the growth of mathematical understanding, while also warning to ensure the approach is genuinely effective.\"}]","Folding Back and the Growth of Mathematical Understanding - in Workplace Training | PDF",1792065396,43]