abstract: Industry, academia, and government have spent tremendous amounts of money over several decades trying to improve the mathematical abilities of students. They have hoped that improvements in students' abilities will have an impact on adults' mathematical abilities in an increasingly technology-based workplace. This study was conducted to begin checking for these impacts. It examined how nine adults in their workplace solved problems that purportedly entailed proportional reasoning and supporting rational number concepts (cognates). The research focused on four questions: a) in what ways do workers encounter and utilize the cognates while on the job; b) do workers engage cognate problems they encounter at work differently from similar cognate problems found in a textbook; c) what mathematical difficulties involving the cognates do workers experience while on the job, and; d) what tools, techniques, and social supports do workers use to augment or supplant their own abilities when confronted with difficulties involving the cognates. Noteworthy findings included: a) individual workers encountered cognate problems at a rate of nearly four times per hour; b) all of the workers engaged the cognates primarily via discourse with others and not by written or electronic means; c) generally, workers had difficulty with units and solving problems involving intensive ratios; d) many workers regularly used a novel form of guess & check to produce a loose estimate as an answer; and e) workers relied on the social structure of the store to mitigate the impact and defuse the responsibility for any errors they made. Based on the totality of the evidence, three hypotheses were discussed: a) the binomial aspect of a conjecture that stated employees were hired either with sufficient mathematical skills or with deficient skills was rejected; b) heuristics, tables, and stand-ins were maximally effective only if workers individually developed them after a need was recognized; and c) distributed cognition was rejected as an explanatory framework by arguing that the studied workers and their environment formed a system that was itself a heuristic on a grand scale. / Dissertation/Thesis / Doctoral Dissertation Curriculum and Instruction 2015

To make mathematics more significant for the beneficiaries, the problem studied in this thesis is to investigate how to connect mathematical daily practices with educational contexts. The overarching aim is to investigate how to contextualise school mathematics within Rwandan cultural mathematics practices. The content of the thesis reports on the characteristics of mathematical organisations in three workplace settings (taxi driving, house construction and restaurant management) which in turn serve as source for the design of contextualised mathematical activities for student teachers in a teacher education programme. Three levels of mathematical practices are described: (1) mathematical practices that are performed by workers within their respective workplaces, (2) mathematical practices that are performed by student teachers while solving and posing contextualised mathematical tasks for secondary school students, (3) mathematical practices that are carried out by secondary school students. Data gathered from individual and group interviews, transcripts of group discussions and students’ written reports of mathematical work were analysed from the perspective of both activity theory and anthropological theory of didactics. Findings from workplace settings revealed that mathematical organisations performed by workers are characterised by techniques which are functional to the problem at hand, the cultural constraints and the educational background of the workers. As long as they are pragmatic towards the goals of the activity no further justification of the techniques used is needed, resulting in a mathematical organisation with undeveloped know-why (logos). On the contrary, at university and secondary school settings, students justified the used techniques throughout the related taught content of the subject mathematics. Also from each category of mathematical practice, it is shown that while connecting workplaces and educational settings the didactic transposition process was much influenced by the institutional conditions and constraints. / För att göra matematiken betydelsefull för avnämarna är problemområdet som studeras i denna avhandling hur den matematik som finns i samhället kan överbryggas till en undervisningskontext. Syftet med avhandlingen är att undersöka hur man kan kontextualisera skolmatematik i kulturella praktiker i Rwanda. I avhandlingen belyses först matematisk organisation på tre arbetsplatser – i en taxiverksamhet, hos en byggmästare och hos en restaurangägare. Matematik i dessa verksamheter utgör underlag för att konstruera uppgifter för lärarstudenter inom ämnet matematik som först löser uppgifterna och sedan i sin tur konstruerar uppgifter för elever motsvarande årskurs nio i grundskolan. Uppgifterna konstrueras med utgångspunkt i den information studenterna fått om de tre verksamheterna. Datainsamlingen skedde med hjälp av individuella intervjuer, gruppintervjuer och bandinspelade gruppdiskussioner samt studenters och elevers nedtecknade lösningar på respektive uppgifter. Data analyserades med hjälp av aktivitetsteori och antropologisk didaktisk teori. Resultaten från arbetsplatserna visade att matematisk organisation kännetecknades av tekniker som är funktionella för de problem som behövde lösas, de kulturella villkor som förelåg och deltagarnas utbildningsbakgrund. Så länge som teknikerna ledde till önskade mål för verksamheten fanns inga behov att utveckla tekniken som kännetecknades av en matematisk organisation med outvecklad logos. I kontrast till denna strategi sågs studenter och elever i respektive miljöer redovisa de tekniker som användes och motivera dem i enlighet med vad som krävs inom matematikämnet. Den matematiska transpositionsprocessen som utfördes av deltagarna i de olika miljöerna influerades i hög grad av rådande institutionella villkor och begränsningar.