Features of written proofs from New Zealand IMO students

Mathematics;proof;writing;New Zealand;International Mathematics Olympiad

Norrish, Jordana Susan Lea ; Holton, Derek ; Ingram, Naomi

University of Otago

关键词: Mathematics;    proof;    writing;    New Zealand;    International Mathematics Olympiad;   

Others  :  https://ourarchive.otago.ac.nz/bitstream/10523/5953/1/NorrishJordanaSL2015PhD.pdf

美国|英语

来源: Otago University Research Archive

【 摘 要 】

This research investigated the linguistic features of written mathematical proofs, and partial proofs, from a small group of secondary school mathematics students in New Zealand.The linguistic features included are outlined below.The students were part of a training camp for the purpose of selecting six students to represent New Zealand at the International Mathematical Olympiad.Micro-level and macro-level linguistic features of the students’ writing were analysed through a sociocultural lens.Using this lens, language was viewed as being influenced by cultural, social, and situational factors (Moschkovich, 2007) and the students’ language was observed in a naturally occurring context.Furthermore, during the training and selection camp, the tutors and lecturers were viewed as experienced members of the mathematical community of practice (Wenger, 1998). The different linguistic features investigated were: personal pronouns, tense, causal connectives, abbreviations, mathematical equations and expressions, and argumentation.Personal pronouns and tense can indicate people’s views about the nature of mathematics (Burton & Morgan, 2000) as well as their perceptions about how people should talk about mathematics.They can also indicate the degree of generality (Rowland, 1999) involved in the author’s reasoning.Causal connectives serve to connect the parts of the reasoning to form a coherent argument.Where different connectives have a similar meaning, the choice of connectives by the author can indicate the language patterns of the community of practice.Abbreviations are also an interesting linguistic feature which can reflect the taken-as-shared sociomathematical norms (Yackel & Cobb, 1996) of a particular community of practice.Abbreviating words or phrases indicates that the author believes the reader will be able to understand and decode the abbreviations through mutually accepted knowledge and practices.The language patterns of the community of practice are further reflected by the density of mathematical equations and argument structures present in a piece of writing.Students’ written examination answers from the conclusion of the training camp and the six students’ answers at the International Mathematical Olympiad were the main source of data collected.Furthermore, lecture sessions and solution sharing sessions were video-taped and transcribed, and field notes taken in order to understand the situation, teaching methods, and taken-as-shared socio-mathematical norms (Yackel & Cobb, 1996) during the training camp.Quantitative methods were employed to analyse the different linguistic features and link these to the training camp community of practice as well as the conventions of the wider mathematical community.These methods included descriptive statistics, chi square testing and the use of Fisher’s Exact Test.Significant chi square results were followed up with a post hoc Cramér’s V calculation in order to determine the strength of the association between variables.The Benjamini and Hochberg (1995) procedure was used to control the false discovery rate, and any results remaining significant after this procedure were followed up further with odds ratio and confidence interval calculations.For these students, the training camp community includes the other students attending the camp (who were not selected for the IMO), and the former IMO competitors and university lecturers who mentored them at the camp.The wider mathematical community includes textbooks, journal papers, university lectures and so on.Results indicate associations between some linguistic features and both Topic and Score.Results also indicate that these students have accommodated some of the conventions from the training camp and the wider mathematical community.For example, often these students expressed themselves using the personal pronoun we or no pronoun at all, combined with the present tense, which is typical of textbooks and journal articles (Burton & Morgan, 2000).The examination responses provided rich data with numerous aspects to explore.There are several features yet to be investigated with this data set, and also several ways to extend and enhance this research to other settings.This research has developed a profile of written proofs in the New Zealand IMO setting as well as investigating the features of written proofs associated with success.It has also investigated the influence of the training camp community of practice as well as the wider mathematical community of practice.This study has addressed the need for more research on the mathematical communication within the secondary school age group, and has also addressed the call from previous mathematics education researchers to investigate the wider context of mathematical communication, rather than just one aspect in isolation.