Exploring grade 11 learners’ functional understanding of proof in relation to argumentation in selected high schools.

Doctoral Degrees. University of KwaZulu-Natal, Durban.Research has established that understanding the functions of proof in mathematics and argumentation ability provide learners with a firm foundation for constructing proofs. Yet, little is known about the extent to which learners appreciate the functions of proof and whether an association between functional understanding of proof and argumentation ability exists. Guided by van Hiele’s and Toulmin’s theories, this study utilised a sequential explanatory design to randomly select three schools from a cluster grouping of ten Dinaledi high schools in the Pinetown district. Three survey questionnaires, Learners’ Functional Understanding of Proof (LFUP), self-efficacy scale, and Argumentation Framework for Euclidean Geometry (AFEG), were administered to a sample of 135 Grade 11 learners to measure their understanding of the functions of proof and argumentation ability, and to explore the relationship between argumentation ability and functional understanding of proof. Then, Presh N (pseudonym)—a female learner who obtained the highest LFUP score despite attending a historically under-resourced township school—was purposively selected from the larger sample. In addition to her responses on the questionnaires, a semistructured interview, and a standard proof-related task served as data sources to explain the origins of her functional understanding of proof. Statistical analyses were conducted on data obtained from questionnaires while pattern matching method was used to analyse the interview data. The analyses revealed that learners held hybrid functional understanding of proof, the quality of their argumentation was poor, the relationship between functional understanding of proof and argumentation ability was weak and statistically significant, and the collectivist culture and the teacher were the two factors which largely accounted for Presh N’s informed beliefs about the functions of proof. In addition, although she constructed a deductive proof, she did not perform the inductive segment prior to formally proving the proposition. The recommendation that Euclidean geometry curriculum needs to be revamped for the purpose of making functional understanding of proof and argumentation explicit and assessable content has implications for two constituencies. Instructional practices in high schools and methods modules at higher education institutions need to include these exploratory activities (functional understanding of proof and argumentation) prior to engaging in the final step of formal proof construction. Future research initiatives need to blend close-ended items with open-ended questions to enhance insights into learners’ functional understanding of proof. This study not only provides high school teachers and researchers with a single, reliable tool to assess functional understanding of proof but also proposes a model for studying factors affecting functional understanding of proof. Overall, the results of this study are offered as a contribution to the field’s growing understanding of learners’ activities prior to constructing proofs

Buying Votes in Indonesia: Partisans, Personal Networks, and Winning Margins

How many voters sell their votes in Indonesia? My PhD research starts with this question that has haunted scholars for the last 15 years. Using data from a nationally representative survey, which included an experimental survey, my study demonstrates that vote buying has become central to electoral mobilisation in Indonesia. If we use the highest estimate, one out of three Indonesians was personally exposed to vote buying in Indonesia’s most recent national election, making the country the site of the third-largest reported sum of exchange of money for votes in the world, as indicated by voter surveys taken over the last decade. My nationwide survey and massive dataset of local election surveys also show that, among other things, partisanship is a significant predictor of vote buying. The closer the ties of a voter to a political party, the more likely that voter is to receive offers of vote buying (or to be accepting of the practice). Puzzlingly, however, the number of partisan voters in Indonesia is comparatively small. Only 15 percent of my national survey respondents admitted being close to any political party and this limited number of party loyalist are highly contested among candidates from the same party in the context of Indonesia’s open-list proportional systems. When we connect partisanship and distributive politics, we arrive at the centre of a lively scholarly debate that involves two competing camps: the so-called core- versus swing-voter models. The former says vote buying when parties or candidates try to mobilise their core supporters, viewing the practice as being above all about increasing turnout. The latter interprets vote buying as an electoral strategy to sway uncommitted voters. What types of voters do Indonesian politicians target? At first glance, the data I collected from low-level candidates and brokers provide more proof in support of the core-voter strategy than in support of the swing-voter strategy. My in-depth interviews with high-level politicians also reinforce the notion that they prefer to target partisan voters in their vote buying operations. Yet my voter surveys clearly showed that although in relative terms such voters are more likely to be targeted, in absolute numbers vote buying mostly happens among non-partisans. How do we explain this combination of features —actors’ insistence that they are targeting partisan voters with the reality that they are mostly providing cash and gifts to non-partisans? This study proposes an addition to the scholarly debate between the core- versus swing-voter models by combining an emphasis on the core-voter strategy and reliance on personal networks. It argues that in Indonesia, candidates and brokers actually intend to target partisan voters, but in reality they mostly distribute benefits to voters who are politically rather indifferent, but who are embedded in personal networks through which they are connected to the candidate and their brokers. This study offers the concept of ‘personal loyalist’ strategy, which targets people identified through personal networks. While the personal loyalist model still recognises the importance of partisan voters, it highlights that candidates seek voters who are not only loyal to the party, but who are also, or instead, loyal to the individual candidate within that party. However, given that partisan voters are not only limited in number but also highly contested among competing co-partisan candidates in the context of the open-list system, politicians seek to expand their electoral base by making use of personal connections mediated by non-party brokers. Given their reliance on personal networks, most candidates and brokers typically misidentify non-partisans as loyalists because they misinterpret personal connections as partisan leanings. In addition, many of the people who are identified through personal networks mediated by brokers are in fact not even loyal to the candidate. Indeed, some of the brokers are themselves not particularly loyal. These two factors– confusion of personal connections with loyalty, and agency loss– in combination contribute to another element of vote buying in Indonesia which I identify in this study: the provision of payments to large numbers of uncommitted voters who receive benefits yet do not reciprocate with their votes. If vote buying is tremendously inefficient, how can vote buying have an effect on electoral behaviour? Why do candidates still do it? Utilising multiple data sources and various methods, I provide strong empirical evidence that gifts of money ‘only’ influenced the vote choice of roughly 10 percent to 11 percent of the total electorate. In these seemingly low numbers, however, lie the key to understanding vote buying’s attractiveness. Across Indonesia, the average margin of victory for successful candidates in legislative elections when defeating their party peers (i.e. candidates who were on the same party list) was only 1.65 percent. In this context of such highly competitive elections, candidates therefore enthusiastically pursued vote buying because they see that it can be critical for determining electoral outcomes. By showing that vote buying helps generate narrow but sufficient victory margins, my study explains how and why vote buying is so prevalent in Indonesia