Papers by Aisling Twohill
Zenodo (CERN European Organization for Nuclear Research), Oct 9, 2023
The aim of this study was to investigate the probabilistic thinking of young children, focusing i... more The aim of this study was to investigate the probabilistic thinking of young children, focusing in particular on the judgements that influence their identifications of the most and least likely outcomes of experiments. Research studies present conflicting results pertaining to young children's potential to engage in probabilistic thinking and a wide variance exists across international mathematics curricula regarding the age at which children receive formal probabilistic instruction. At present, young children in Ireland are not formally introduced to probability until Third class when they are approximately 8 or 9 years old. In this study, the probabilistic thinking of 16 children aged 5-6 years was examined using task-based group interviews. The results suggest that young children are capable of engaging in sophisticated probabilistic thinking and highlights that the current practice of formally introducing children to probability in Third class warrants further investigation.
Conference Theme: Mathematical Literacy, throughout and beyond education It is with great pleasur... more Conference Theme: Mathematical Literacy, throughout and beyond education It is with great pleasure that we present the following proceedings from the Seventh Conference on Research in<br> Mathematics Education in Ireland (MEI 7), which took place in Dublin City University in October, 2019. In these proceedings of MEI 7, we present papers that reflect a broad variety of mathematical research that is taking place in Ireland and further afield. Collectively, the authors seek to solidify and progress the research field of mathematics education, throughout and beyond Ireland. Lorraine Harbison and Aisling Twohill
Le Centre pour la Communication Scientifique Directe - HAL - Archive ouverte HAL, Feb 1, 2017
Generalisation is a key element of algebraic thinking, and children's developing thinking is supp... more Generalisation is a key element of algebraic thinking, and children's developing thinking is supported by engagement with algebraic activities when they attend primary school (Kaput, 2008). Shape patterns provide a context within which children may identify structure, and construct general terms. Within my research I presented children with shape patterns and observed their interactions as they discussed the structure of the patterns and sought to construct general terms. In this paper, I discuss the elements of the patterns to which they attended, and how their focus supported some children in constructing general terms, while other children experienced more challenge in their engagement with the patterns. Specifically I focus on children who observed both numerical and figural aspects, and I examine the varying levels of success they experienced.
STATISTICS EDUCATION RESEARCH JOURNAL
This paper reports on a study that investigated young children’s responses to a range of probabil... more This paper reports on a study that investigated young children’s responses to a range of probabilistic tasks. A central aspect of the study was our examination of the children’s use of subjective thinking. Most research that has been conducted in relation to young children’s probabilistic thinking has focused on the extent to which young children can identify the most and least likely outcome of experiments. There is, however, limited research into the types of judgements children use when making these identifications. For example, while a small number of studies have reported on children’s use of subjective thinking, there is an absence of research focusing on the role of subjectivity and the range of beliefs on which these judgements are based. In this research, the subjective thinking of children aged 5–6 years in Ireland was examined to address this gap in current knowledge. The data were collected through task-based group interviews and analysed using thematic analysis. Results...
Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds, 2017
Constructing general terms for shape patterns supports children in reasoning algebraically about ... more Constructing general terms for shape patterns supports children in reasoning algebraically about relationships between quantities and relative rates of change. This chapter describes a research project wherein I investigated the strategies children attending Irish schools used when asked to solve shape-patterning tasks. The research instrument was a task-based group interview, and the children’s interactions shed light on a number of catalysts for the broadening of their observations of the pattern structure. Such catalysts included peer interactions, concrete materials, and teacher prompts. In this chapter I draw attention to children’s observations of structure, and seek to trace the thinking of children whose observations broadened from an initially narrow or limited perspective.
International Journal of Science and Mathematics Education, Mar 16, 2022
Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education, 2019
Teaching for Social Justice and Sustainable Development Across the Primary Curriculum, Feb 11, 2021
Critical mathematical literacy empowers all citizens to robustly interrogate their lived experien... more Critical mathematical literacy empowers all citizens to robustly interrogate their lived experience, and to probe the accuracy of information presented to them. Educators play a central role in empowering children to apply mathematics critically, and this chapter aims to support such valuable work by unpacking theory relating to key components of critical thinking. In addition, the chapter includes a model for critical numeracy that children and teachers can apply to explore scenarios they encounter within and beyond the classroom. In this chapter, we highlight the potential of mathematics education to raise awareness of societal structures, and to contribute to the development of children’s positive social and cultural identities. We draw attention to the role of contexts in exploring mathematical concepts, and how educators have opportunities to prioritise meaningful socio-political contexts over normalisation of capitalism and consumerism. We outline how narratives may be shared with children that broaden their understanding of the global origins of mathematics. The chapter concludes with two lesson-exemplars that adopt a social justice context in order to provide a meaningful opportunity for learners to engage with the tools of critical mathematics literacy.
Children’s engagement with algebraic thinking has traditionally commenced in senior primary or se... more Children’s engagement with algebraic thinking has traditionally commenced in senior primary or secondary school, preceded by primary school curricula that prioritised computation and understandings of number (Kieran, et al., 2016). Increasingly during the latter decades of the 20th century, educators and researchers identified that such an approach may contribute to insurmountable challenges for some children when they first encounter formal algebra, typically after six to eight years of school. In this paper I present the research underpinning the Early Algebra movement that arose from a motivation to address such challenges, and the implications of Early Algebra for the Irish Primary School Mathematics Curriculum (IPSMC). While the 1999 IPSMC included algebra as a content strand, key concepts of Early Algebra, such as generalisation, and exploration of structure, are absent (Twohill, 2013).
The intervention described in this paper facilitated mathematical discussions between parents and... more The intervention described in this paper facilitated mathematical discussions between parents and children within the context of picture books. Parental involvement has been shown to have the potential to impact significantly on a child's attainment in school (Epstein, 1995; Anthony & Walshaw, 2007). The intervention took place in a rural school in County Kildare and lasted for three weeks. The research focus encompassing the intervention was the parents' involvement with their children's mathematical learning. Data collection included parent interviews and a reflective journal maintained by the teacher-researcher. Findings indicated that the majority of parents felt they were more involved in their child's learning of mathematics through the intervention. Furthermore, the participants noticed a number of benefits when using the picture books, including a greater understanding in children's mathematics, the children having a greater motivation to do mathematics, ...
International Journal of Science and Mathematics Education, 2022
This paper reports on research which explored the mathematics teaching efficacy beliefs of preser... more This paper reports on research which explored the mathematics teaching efficacy beliefs of preservice primary teachers, where efficacy beliefs describe individuals’ beliefs in their potential to enact teaching to promote learning. Efficacy was conceptualised as a bi-faceted construct consisting of personal efficacy and outcome expectancy. This research sought to establish the extent to which differences in efficacy are explained by students’ mathematics attainment level prior to entry into teacher education; the educational level of the students (whether postgraduate or undergraduate); students’ sense of preparedness to teach mathematics on school placement; and students’ gender. A total of 186 students responded to a questionnaire designed to measure their efficacy beliefs after completing one taught mathematics education module in university and one teaching practice placement in primary schools. Bivariate and regression analysis pointed to complex relationships between the explan...
International Journal of Science and Mathematics Education, 2022
This paper reports on research which explored the mathematics teaching efficacy beliefs of preser... more This paper reports on research which explored the mathematics teaching efficacy beliefs of preservice primary teachers, where efficacy beliefs describe individuals' beliefs in their potential to enact teaching to promote learning. Efficacy was conceptualised as a bi-faceted construct consisting of personal efficacy and outcome expectancy. This research sought to establish the extent to which differences in efficacy are explained by students' mathematics attainment level prior to entry into teacher education; the educational level of the students (whether postgraduate or undergraduate); students' sense of preparedness to teach mathematics on school placement; and students' gender. A total of 186 students responded to a questionnaire designed to measure their efficacy beliefs after completing one taught mathematics education module in university and one teaching practice placement in primary schools. Bivariate and regression analysis pointed to complex relationships between the explanatory and outcome variables. On bivariate analysis, findings included statistically significant associations between gender, mathematics attainment, preparedness to teach, and one or both of personal efficacy and outcome expectancy. In the regression analysis, gender was statistically significantly associated with outcome expectancy, while preparedness to teach and mathematics attainment were statistically significantly correlated with personal efficacy. Personal efficacy and outcome expectancy were significantly correlated on bivariate analysis, but significance was not retained after controlling for other factors in the regression models. This research has implications for teacher educators in understanding factors explaining mathematics teaching efficacy and therefore helping to better prepare preservice teachers to teach mathematics in the primary school classroom.
This research was initiated as a proactive response to a major reform of the structures of initia... more This research was initiated as a proactive response to a major reform of the structures of initial teacher education (ITE) in Ireland to support best practice (Teaching Council, 2017). We are undergoing an iterative process of design research with the intention of investigating and refining the mathematics education modules undertaken by students on our undergraduate Bachelor of Education programme (Borko, Liston & Whitcomb, 2007).The first phase of the research involved the design of a module on Measures for 460 first year ITE students. Students’ perceptions of the module and their mathematics teacher efficacy beliefs were investigated using an online questionnaire based on Enochs, Smith and Huinker’s (2000) Mathematics Teacher Efficacy Beliefs Instrument (MTEBI). The second phase of data collection will commence soon. This involves inviting the same cohort to participate in focus group interviews.
Recent changes in the landscape for primary initial teacher education (ITE) in Ireland prompted t... more Recent changes in the landscape for primary initial teacher education (ITE) in Ireland prompted the first stage of this longitudinal design-research study on Mathematics Education which sought to establish a model of teaching and learning based upon iterations of research upon practice (Borko, Liston, & Whitcomb, 2007). The challenge was to design a research led undergraduate Bachelor of Education (B.Ed.) mathematics programme of study that would focus on the development of students’ efficacy, and more particularly in Year 1, on students’ knowledge of and approaches to the teaching of Measures and perceived implementation of same on their first teaching placement in schools. This first phase of the study gathered students' views retrospectively on the B.Ed. 1 programme, and explored their perceptions of the skills and abilities they possessed whilst teaching mathematics on School Placement. The entire cohort of 430 first year ITE students were invited to participate in an online...
Generalisation is considered by many to be a highly significant component of algebraic thinking (... more Generalisation is considered by many to be a highly significant component of algebraic thinking (Carpenter & Levi, 2000; Kaput et al., 2008). In particular, constructing general terms for shape patterns supports children in reasoning algebraically about covariance and rates of change (Rivera & Becker, 2011). Generalisation is not mentioned within the current Irish Primary School Mathematics Curriculum, and shape patterns are not presented as learning activities, beyond simple repeating patterns explored with very young children (Government of Ireland, 1999). This thesis reports on a research study that explored the strategies Irish children adopted in seeking to construct general terms for shape patterns. Within the context of task-based group interviews, patterning tasks were presented to groups of three or four children (aged nine or ten years old), where the children were asked to describe and extend the patterns, and to consider near and far generalisations. Following the interv...
The purpose of my research is to explore to what extent children in Irish primary schools are dev... more The purpose of my research is to explore to what extent children in Irish primary schools are developing skills in algebraic reasoning. In addressing the challenge faced by students encountering algebra in secondary school, one recommendation is to commence algebra instruction early in a child’s schooling (Kaput, 2008; Carpenter and Levi, 1999; Mason, 2008; among others). Underpinning my research will be a framework of growth points, which outlines a possible developmental pathway in algebraic reasoning for primary school. In this paper, I present five growth points in algebraic reasoning with interim trajectories which may be supportive in exploring children’s emerging understanding.
In the research project presented, I categorise the strategies employed by four Fourth class girl... more In the research project presented, I categorise the strategies employed by four Fourth class girls and four Sixth class girls when asked to construct a general term for a visual spatial pattern. The approach adopted by each girl is categorised as either explicit or recursive, and the mode of generalisation is categorised as either linear or numeric.
This research study seeks to explore the Mathematics Teaching Efficacy Beliefs of student teacher... more This research study seeks to explore the Mathematics Teaching Efficacy Beliefs of student teachers as part of a longitudinal design-research study, whereby research findings will guide planning for future modules. The Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) of Enochs, Smith and Huinker (2000) was employed to measure the overall efficacy beliefs of teachers, and also the beliefs within two subscales, the Personal Teacher Efficacy, and Teaching Outcome Expectancy. A convenience sample of 40 undergraduate students participated in a questionnaire based upon the MTEBI with additional questions relating specifically to the content of a Mathematics Education module recently completed by the students. The sample was drawn from a year group of 440 students by invitation.
This paper will discuss the mathematics teacher efficacy beliefs (MTEB) of primary initial teache... more This paper will discuss the mathematics teacher efficacy beliefs (MTEB) of primary initial teacher education (ITE) students. We are interested in studying how ITE students’ MTEBs are influenced (or not) by mathematics education modules undertaken as part of an undergraduate Bachelor of Education (BEd) programme. We will detail how approximations of practice (Grossman, Compton, Igra, Ronfeldt, Shahan, & Williamson, 2009) have been incorporated into mathematics education modules to create opportunities for the development of MTEBs and will report on focus group interviews which explored MTEBs of ITE students
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Papers by Aisling Twohill