The effect of context and individual differences in human-generated randomness

Many psychological studies have shown that human-generated sequences deviate from the mathematical notion of randomness. Therefore, the inability to generate perfectly random data is currently considered a well-established fact. What remains an open problem is the degree to which this (in)ability varies between different people and can be affected by contextual factors. In this paper we investigate this problem. We focus on between-subjects variability concerning the level of randomness of generated sequences under different task descriptions. In two studies, we used a modern, robust measure of randomness based on algorithmic information theory to assess human-generated series. We tested hypotheses regarding human generated randomness vis-a-vis effects of context, mathematical experience, fatigue, and the tendency to engage in challenging tasks. Our results show that the activation of the ability to produce random-like series depends on the relevance of contextual cues which rather help to avoid production of trivially non-random sequences than increase the rate of production of highly complex ones. We also show that people tend to get tired very quickly and after first few attempts to generate highly random-like series their performance decrease significantly and they start to produce more markedly patterned, sequences. Based on our results we propose a model that considers two main areas (intellectual and cognitive) of possible psychological factors related to the variability of ability to produce random-like series.

Mathematical Mystery in a Cultural Game

This study described the mathematical depth in a mathematical activity carried out in a village in Turkey’s Eastern Anatolia Region. This activity presented in the context of the game reflects a cultural situation of doing mathematics over time. In this context, it can identify as a study of ethnomathematics. Therefore, the cultural game was introduced first, and then the mathematical depth behind this game was uncovered in all its aspects. Finally, the mathematical relationship behind the game was analysed in terms of mathematics education. The case study, as one of the qualitative research methods, was used in the study. The participants of the study consist of 1 person who knows, transmits, and teaches the cultural game. The game process and semi-structured interview that constituted the research data were recorded with a camera and a voice recording device. Descriptive analysis was used in the analysis of the interview. Findings of the study suggest that the cultural game is played without considering its mathematical depth, but that there is a rich mathematical depth behind it. The results also indicate that such games offer an effective way for adults learning mathematics. On the other hand, the study revealed that there could be different ways of thinking between school mathematics and ethnomathematics. It is thought that synthesizing mathematics with games that include ethnomathematics has the potential to provide students at diverse levels with an excellent mathematical experience.

Epistemology and rationality of intuition and imagination in the field of mathematics

This paper aims to contribute to the clarification of the role of mathematical intuition and imagination in the constitution of mathematical knowledge, evidencing its epistemological and procedural characteristics. For that, an "epistemology of intuition and imagination" in the field of mathematics is outlined (suggested) emphasizing the need to adopt a dynamic conception of mathematics. In this context, intuition and imagination present themselves as forms of mathematical experience that give access, through paths that are not purely logical, to mathematical knowledge. Its epistemological and rationality characteristics, a rational of a non-logical nature, are highlighted by several examples, resources for moving the ideas involved. The epistemological study of intuition and imagination also allows highlighting its ontology, constituted of more relations than objects. From the pedagogical point of view, we discuss the formative character of philosophical studies involving intuition and imagination, mainly related to creativity in mathematics. Keywords: Mathematical knowledge; Mathematical experience; Epistemology of intuition and imagination; Creativity in mathematics.

Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

LEARNING MATHEMATICAL MODELLING WITH AUGMENTED REALITY MOBILE MATH TRAILS PROGRAM: HOW CAN IT WORK?

The aim of this study is to investigate how an augmented reality mobile math trails program can provide opportunities for students to engage in meaningful mathematical modelling activities. An explorative research design was conducted involving two mathematics teachers and 30 eight grades in Semarang, Indonesia. An Augmented Reality Mobile Math Trails App was created, and several math trail tasks were designed, then students run the activity. Data were gathered by means of participatory observation, interviews, questionnaires, tests, and worksheets. Data analysis began with the organisation, annotation, description of the data and statistic tests. The findings indicate that an educational program was successfully designed, which offered students a meaningful mathematical experience. A mobile app was also developed to support this program. The mobile app with augmented reality features is helpful for students as a tool that bridges the gap between real-world situations and mathematical concepts in problem-solving following the mathematical modelling cycle. The program thus contributes to a higher ability in mathematical modelling. The study identified a link between instrumented techniques in programs and mathematical modelling, as built during the instrumentation process. Further studies are essential for project development and implementation in other cities with different situations and aspects of study.

Mathematics support for science: a reflection of a blended and online development project

The diverse range of backgrounds that students bring to university has many advantages, but also some challenges such as a wide range of mathematical experience and ability. A particular issue identified by staff teaching mathematics to the Science Fundamentals cohort was lack of engagement due to the material presented being too easy or too difficult, with the main concerns directed towards students who are weak at mathematics and others who lack some of the basic skills necessary for a successful undergraduate experience at Glasgow.Our experience from the Science Fundamentals course is that traditional lectures are a poor way to motivate the weaker students and the more collaborative models of teaching such as online and blended learning may be more appropriate. Following a two-year project, a suite of online resources was developed to supplement course material; students completed a ‘Mathematics Confidence Test’ to determine the level of support required and the number of mathematics lectures was cut down and replaced with tutorials aimed at the weaker students.We will discuss our experiences of running this project, reflect on feedback and discuss further plans for supporting the Science Fundamentals cohort in subsequent years.