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Abstract: Abstract For many decades, problem solving has been a focus of elementary mathematics education reforms. Despite this, in many education systems, the prevalent approach to mathematics problem solving treats it as an isolated activity instead of an integral part of teaching and learning. In this study, two mathematics teacher educators introduced 19 Irish elementary teachers to an alternative problem solving approach, namely Teaching Through Problem Solving (TTP), using Lesson Study (LS) as the professional development model. The findings suggest that the opportunity to experience TTP first-hand within their schools supported teachers in appreciating the affordances of various TTP practices. In particular, teachers reported changes in their beliefs regarding problem solving practice alongside developing problem posing knowledge. Of particular note was teachers’ contention that engaging with TTP practices through LS facilitated them to appreciate their students’ problem solving potential to the fullest extent. However, the planning implications of the TTP approach presented as a persistent barrier. PubDate: 2022-05-13
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Abstract: Abstract It is well established that spatial reasoning skills (i) support mathematics achievement, (ii) are malleable, and (iii) can be improved through training. More recently, there has been interest in using spatial training to causally support corresponding improvements in mathematics achievement; however, findings so far appear to be mixed. The current study explores the effect of a spatial reasoning intervention on Year 11 students’ spatial reasoning skills and mathematics achievement and considers the role of a pedagogical framework and the multidimensional nature of mathematics and spatial reasoning in the design of the intervention. The Experience-Language-Pictorial-Symbolic-Application (ELPSA) pedagogical framework was used to modify an existing spatial intervention program for delivery by high-school educators to Year 11 students (an important but understudied population). The spatial intervention involved training a range of spatial skills over an extended timeframe. Students were randomly assigned to the intervention condition or to a business-as-usual control (n = 73). Using a pre-/post-test design, we found the intervention was successful in improving participants’ spatial reasoning skills and performance on measurement and geometry items compared to the control condition but not on number and algebra items. These findings demonstrate that spatial training can support mathematics achievement in certain contexts, highlighting the importance of identifying how individual spatial skills support specific mathematics tasks. Consideration was given for how to use strong pedagogical techniques to scaffold transfer, finding utility in the ELPSA framework. Implications for how to embed spatial training within real mathematics classrooms, as done in the current study, are discussed. PubDate: 2022-04-25
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Abstract: Abstract This paper attempts to deepen our understanding of teachers’ selection and modification of mathematical tasks for their classroom instruction. In this paper, we focus on teachers’ orientations and resources that subtly influence teachers’ choices during their routine undertaking of task selection and modification. Interviews using stimulus material were used to induce teachers to expound their thought process as they deliberated the suitability of tasks and made choices related to task selection and modification. These thought processes are critical as it provides potential explanation for teachers’ task inclinations. The findings of this study show that whilst beliefs of the teaching and learning of mathematics and beliefs of students’ capabilities and capacities are preponderant as the teachers deliberate on which tasks they would use, the orientations within these beliefs differ amongst them. These orientations, productive or unproductive, influence their decisions during task selection and modification. Teacher resource specific to mathematical knowledge for teaching (MKT) is also critical in impacting teachers’ choice of tasks. Teacher orientations and expertise impact the range and extent the teachers draw on their MKT, resulting in dissimilar inclinations towards tasks for mathematics instruction. In addition, resources such as preparation time, curriculum time and materials teachers have at hand influence their choices when selecting and modifying mathematical tasks for instruction. PubDate: 2022-04-25
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Abstract: Children’s multiplicative thinking as the recognition of equal group structures and the enumeration of the composite units was the subject of this research. In this paper, we provide an overview of the Multiplication and Division Investigations project. The results were obtained from a small sample of Australian children (n = 21) in their first year of school (mean age 5 years 6 months) who participated in a teaching experiment of five lessons taught by their classroom teacher. The tasks introduced children to the “equal groups” aspect of multiplication. A theoretical framework of constructivist learning, together with research literature underpinning early multiplicative thinking, tasks, and children’s thinking, was used to design the research. Our findings indicate that young children could imagine equal group structures and, in doing so, recognise and enumerate composite units. As the children came to these tasks without any prior formal instruction, it seemed that they had intuitive understandings of equal group structures based on their life experiences. We argue that the implications for teaching include creating learning provocations that elicit children’s early ideas of multiplication, visualisation, and abstraction. The research has also shown the importance of observing children, listening to their explanations of their thinking, and using insights provided by their drawings. PubDate: 2022-03-31
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Abstract: Abstract The shift from additive to multiplicative thinking is challenging for students. A professional learning program was developed that focused on an identified area of need by teachers, namely multiplicative thinking. Program content focused on concepts underpinning multiplicative thinking, pedagogical approaches, challenging tasks, and application to classroom practice. It was delivered via six 90-min modules in 13 participating schools across terms 2–4, as part of regular after school professional learning. Whilst all staff participated, the research focus was year 3–4 teachers. Students’ historical data were collected across four years (2016–2019) to determine mean growth over time, in participating and non-participating schools. National Assessment Program Literacy and Numeracy (NAPLAN) and Mathematics Assessment Interview (MAI) data were used to determine the impact of the learning, as both assessments are administered annually. Analysis of year 4 students’ longitudinal data showed greater mean growth in student learning over a 2-year period in schools involved in the learning and additional in-class coaching support, than students in non-participating schools. Our findings showed that targeted school-based professional learning, with in-class support from a knowledgeable other, leads to teachers’ improved understanding of multiplicative thinking and subsequent pedagogical content knowledge to support student learning. PubDate: 2022-03-22
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Abstract: Abstract This study aimed at investigating how preservice teachers’ understandings of division and reasoning about ratios support and constrain their formation of proportional relationships in terms of quantities. Six preservice teachers from a middle-grade preparation program in the USA were selected purposefully based on their mathematics performance in a previous course. An explanatory case study with multiple cases was used to make comparisons within and across cases. Two semi-structured interviews were conducted with each pair. The results revealed that preservice teachers who did not explicitly identify different meanings for division struggled to differentiate between the two perspectives on ratios. The results also showed that those teachers had difficulty forming proportional relationships while solving the proportion tasks. These results suggest that explicit identification of the meanings for both types of division is critical to keeping the two perspectives on ratios separate, which is a key aspect for a robust understanding of proportional relationships. PubDate: 2022-03-22
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Abstract: Abstract Due to rapid immigration, many children worldwide are learning mathematics in a second or additional language. This language diversity can be challenging for both teachers and students and carries profound implications for mathematics educators. Research shows that teachers use various ways to support English Language Learners. Research on multilingualism in mathematics classrooms has often focused on qualitative research. This meta-analysis aims to explore the statistically effective successful teaching practices from the studies using quantitative or mixed-method research approaches and aims to inform the research field in a cumulative manner. The specific research question that guided this meta-analysis is: What is the evidence regarding successful teaching of mathematics for Year 1–10 English Language Learners from 2009–2019 in countries where curricula are delivered predominantly in English' Four successful intervention categories were identified: Dual Language Programmes, Curriculum integration, Teacher Professional Development, and Cognitively Focused Interventions. The paper concludes with recommendations for practice and further research in this area. PubDate: 2022-03-18
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Abstract: Abstract Teacher responses to student mathematical thinking (SMT) matter because the way in which teachers respond affects student learning. Although studies have provided important insights into the nature of teacher responses, little is known about the extent to which these responses take into account the potential of the instance of SMT to support learning. This study investigated teachers’ responses to a common set of instances of SMT with varied potential to support students’ mathematical learning, as well as the productivity of such responses. To examine variations in responses in relation to the mathematical potential of the SMT to which they are responding, we coded teacher responses to instances of SMT in a scenario-based interview. We did so using a scheme that analyzes who interacts with the thinking (Actor), what they are given the opportunity to do in those interactions (Action), and how the teacher response relates to the actions and ideas in the contributed SMT (Recognition). The study found that teachers tended to direct responses to the student who had shared the thinking, use a small subset of actions, and explicitly incorporate students’ actions and ideas. To assess the productivity of teacher responses, we first theorized the alignment of different aspects of teacher responses with our vision of responsive teaching. We then used the data to analyze the extent to which specific aspects of teacher responses were more or less productive in particular circumstances. We discuss these circumstances and the implications of the findings for teachers, professional developers, and researchers. PubDate: 2022-03-01
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Abstract: Abstract The connection between spatial reasoning and mathematics learning and pedagogy in primary school children has been the subject of an increasing number of studies in recent years. There has been no comprehensive analysis, however, of how studies based on spatial reasoning interventions may lead to improvements in students’ mathematics learning in school classroom environments. This article considers 18 studies selected from a combined systematic literature review of 133 studies, from Scopus and Education Research Complete (ERC) using PRISMA, and 23 studies recommended by the research team from bibliographies of major international research centres with a spatial reasoning dedication. This combination approach has allowed a synthesis of research and practice in an analytical way, assisting construction of a framework for spatial reasoning interventions for consideration in developing core knowledge and skills within the primary school mathematics curriculum. The findings highlight the importance of designing and evaluating spatial reasoning programs for primary school children in order to improve students’ mathematics classroom learning, including evidence from standardized tests, as they progress through the school system. The article supports the need for further research on interventions that provide sustainable school-based spatial reasoning programs. PubDate: 2022-03-01
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Abstract: Abstract The purpose of this study is to investigate how triadic dialog promotes generalization of growing figural patterns during classroom talk in multilingual classrooms and crucial linguistic terms that the teacher and students draw on in their home language, as they engage in pattern generalization. Ten sessions in two classrooms in grade 7 were videotaped. A mixed quantitative-qualitative approach was adopted to analyze the videotaped sessions. At a macro-level analysis, the sessions were mostly hybrid including various generalization level episodes. At a micro-level analysis, triadic dialog was the dominant mode of interaction during classroom talk. Nuances in the discourse structure of triadic dialog were associated with shifts between generalization level episodes in each session. Classroom talk involved deployment of colloquial Arabic to refer to linguistic means that influenced the processes of generalization. The study presents and analyzes representative excerpts of classroom talk during various generalization level episodes to demonstrate the findings. The selected episodes show the details of classroom interaction and provide evidence how triadic dialog involved the use of crucial words in home language to promote various levels of generalization. PubDate: 2022-03-01
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Abstract: Abstract Studies in mathematics education have shown interactions between a variety of emotions and the quality of learning. Research has found that positive emotions are among the main factors that engender a sense of well-being. In the current study, we focused on emotions that may be aroused while coping with mathematical challenges, and sought to evaluate their intensity and frequency in that context. The research was conducted among a unique group: competitors of the Israel International Math Competition for Girls (IIMCG), a competition involving problem solving in mathematical thinking, which targets female high school students. The participants were 403 of the competitors (304 Israeli, 99 American) who had reached the second round of the competition. The research instrument was a Math Emotions Measuring Instrument, which assesses positive and negative emotions expressed while coping with mathematical challenges. Results emphasized the numerical prominence of positive as opposed to negative emotions that were reported by participants. The ratio of positive to negative emotions (i.e., the positivity ratio) was high – 4.2:1. In addition, a statistically significant cultural difference was found between the participants from Israel and those from the USA in terms of the intensity of 10 out of 28 emotions experienced. Our findings shed new light on the connection between mathematics and emotions and prove the importance of generating strategies to enhance a positive climate in teaching and learning mathematics. PubDate: 2022-03-01
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Abstract: Abstract Explorations of the national NAPLAN numeracy data consistently reveal a strong relationship between achievement on these tests and students’ socio-economic background. A small but persistent pattern of gender difference favouring males in mean NAPLAN numeracy scores is also reported. Whether these patterns are replicated for students attending single-sex schools is examined in this paper. PubDate: 2022-03-01
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Abstract: Abstract Despite research highlighting the negative impact of ability grouping on student outcomes in mathematics, the practice seems entrenched in New Zealand schools. The Ministry of Education launched a professional development initiative with the aim that mathematics lead teachers would drive change towards teaching in mixed-ability groups using a problem-solving pedagogy. Lead teachers were seen as uniquely positioned to promote change within schools. This study examined lead teachers’ changes in their own practice and that affected in their schools following the initiative. The study used a problem-based methodology. Through careful, in-depth interviews with lead teachers from six primary schools, the researchers examined lead teachers’ theories-of-action and the extent to which they were able to affect change. Findings showed different levels of change in lead teachers’ own and their school’s practice. Only two lead teachers made considerable changes in their own practice, and only one could fully incorporate mixed-ability grouping into school-wide practices. Analyses across the six cases revealed four factors impacting lead teachers’ ability to affect change: positional authority, senior leadership support, expertise in content and pedagogy, and changing teacher beliefs to bring about change. The study concludes that while the notion of lead teachers implementing change through working closely with their colleagues seems exciting and promising, the conditions for teacher leadership to be effective need to be met. Furthermore, the case of this national initiative shows that shortcuts in professional development are ineffective, especially for a matter so disputed and entrenched as ability grouping in mathematics. PubDate: 2022-03-01
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Abstract: Abstract Teachers are encouraged to connect mathematic instruction to the real-world by posing tasks that are situated in rich, relevant contexts, but research has found that many teachers integrate contextual problems (CPs) as motivators rather than as supports for conceptual development. To provide insight into how teachers’ conceptions about CPs shift as they teach through contextual problem solving, we interviewed six teachers before and after they taught from a unit designed from principles of realistic mathematics education, an instructional design theory which positions realistic contexts as learning supports. Our findings indicate that teachers initially viewed CPs primarily as affective or motivating enhancements, but after teaching the RME unit with university-based support, the teachers articulated integrated understandings of how CPs can function as supports for conceptual development. The teachers articulated how CPs provide initial access, sites for progressive representational formalization, and references to which students can fall back in order to interpret subsequent tasks. The authors identify connections between these ideas and the support provided by the university team and the teachers’ guides. PubDate: 2022-03-01
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Abstract: Abstract This paper makes a case for placing knowledge at the centre of the school mathematics curriculum, and for knowledge building and knowledge differentiation as critical for both equity and excellence, emphasising that knowledge is much more than a set of descriptions of content as might typically be found in a curriculum document or textbook. The paper commences by discussing the implications of the traditional epistemological view of knowledge as justified true belief for mathematics education and uses this to build a preliminary description of knowledge building. Ideas from critical realism are then used to show that it is not so much the content of knowledge that matters but the production of knowledge and to build an enhanced conception of knowledge building in school mathematics. A distinction is made between knowledge and knowing that provides a non-relativist yet fallible view of knowledge, recognising its emergent but directed nature through its production and legitimation within established fields. The importance of knowledge building as a democratic right is then discussed, highlighting the importance of specialised knowledge and arguing that knowledge differentiation provides a basis for a conception of school mathematics curriculum that is dynamic and empowering. The paper concludes by discussing a range of potential theoretical and empirical research projects arising from a focus on knowledge and knowledge building in school mathematics. PubDate: 2022-03-01
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Abstract: Abstract The aim of this article is to carry out a work of networking theories which combines two perspectives on the mathematical activity involved in a modelling process, in order to answer the following question: To what extent does the application of the onto-semiotic tools complement the analysis from a cognitive perspective of a mathematical modelling process' To this end, we considered two theoretical frameworks: on the one hand, the onto-semiotic approach, which provides tools for the analysis of any mathematical activity and which here we applied to the activity of modelling; on the other hand, the modelling cycle from a cognitive perspective, which is a reflection on the specific mathematical activity of modelling. Then, we took a modelling problem that we applied to prospective mathematics teachers (at undergraduate and postgraduate level), and we analysed it from the perspective of both frameworks, in order to identify concordances and complementarities between these two ways of analysing the mathematical activity involved in the modelling process. The main conclusion is that both frameworks complement each other for a more detailed analysis of the mathematical activity that underlies the modelling process. Specifically, the analysis with the tools provided by the onto-semiotic approach reveals the phases or transitions of the modelling cycle as a conglomerate of mathematical practices, processes, and the primary objects activated in these practices. PubDate: 2022-02-16 DOI: 10.1007/s13394-022-00411-3
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Abstract: Abstract The purpose of the current study was to explore pre-service middle school mathematics teachers’ personal concept definitions of a trapezoid, parallelogram, rectangle, rhombus, square, and kite. The data were collected by a self-report instrument through which the pre-service teachers provided their answers in written form. The participants’ definitions were coded by using Zazkis and Leikin’s (Educational Studies in Mathematics, 69(2), 131–148, 2008) framework, which includes the following four main categories for determining mathematical correctness: necessary and sufficient, necessary but not sufficient, sufficient but not necessary, and neither necessary nor sufficient. The findings revealed that about half of the all definitions were correct. More specifically, the participants generated considerably higher proportion of correct definitions for a parallelogram and rhombus, while they displayed a very low performance in defining a kite. The possible reasons of the participants’ varied performance levels in defining the six basic quadrilaterals are discussed based on the linguistic (syntactic, semantic, and lexical) structure of the Turkish names given to the these quadrilaterals. The current study may provide some feedback to teacher education programs regarding the extent of knowledge that should be possessed by the pre-service teachers about definitions of special quadrilaterals before they leave their programs. Such feedback may also help mathematics teacher educators ponder on more fruitful approaches that may promote the development of pre-service teachers’ knowledge and understanding of definitions of special quadrilaterals. PubDate: 2022-02-05 DOI: 10.1007/s13394-022-00412-2
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Abstract: Abstract The current study is part of a comprehensive research on linking visualization, students’ construction of geometrical concepts and their definitions, and students’ ability to prove. The aim of the current study is to investigate the effect of learners’ understanding of definitions of geometrical concepts on their understanding of the essence of geometrical proofs and their ways of proving. By “understanding” (geometrical definitions and/or proofs), we mean knowing the definition’s and/or proof’s meaning and their role within the logic structure of geometry and also the ability to define and/or prove in “geometrically correct” ways. Ninety grade 11 students from an Arab high school in Israel participated in the comprehensive study in geometry of which the current study forms a part. Research tasks for the investigation of the current study’s aim were designed, constructed, and used in a questionnaire to the whole research population and in interviews with about 10% of the population. The findings point clearly to effects of elements within the students’ understanding of definitions on their understanding of proofs and on their ability to prove. These elements are as follows: (a) The difficulty to internalize that an incomplete definition is an incorrect definition and may lead to an incomplete proof. (b) The difficulty to deal with non-economical definitions forms the basis to non-economical proofs. (c) Students have difficulties to accept equivalent definitions to the same geometrical concept. (d) The students’ lack of understanding the origin of a constructive definition of geometrical concept. PubDate: 2022-01-29 DOI: 10.1007/s13394-021-00406-6
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Abstract: Abstract The purpose of this study is to observe the changes on pre-service teachers’ cognitive (i.e. fluency, flexibility, and originality) and affective outcomes (e.g. perceptions towards creativity-directed tasks, beliefs about the nature of mathematics) related to mathematical creativity after they participated in a creativity-focused mathematics method course. The pre-service teachers (n = 40) were randomly assigned into two groups as the intervention (n = 21) and the control (n = 19). The results revealed that the pre-service teachers who received the intervention developed all of the cognitive and affective outcomes more than the pre-service teachers in the control group (p < .05). The implication of these findings is that integrating creativity-directed tasks into mathematics education college courses can better equip pre-service teachers to develop mathematical creative abilities in their future students. PubDate: 2022-01-24 DOI: 10.1007/s13394-022-00409-x
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Abstract: Abstract Originating from the learning sciences community, design-based research (DBR) is attracting interest from many educational researchers including those focused on mathematics. Beyond its research role, it is being seen as a collaborative way to engage teachers in deep professional development leading not only to changes in skill but also in purpose. Previous reviews of the approach, however, have offered only cautious optimism for the capacity of DBR to support widespread or scalable change. This paper will use methods drawn from the digital humanities and social sciences to explore patterns relating to regional differences, theoretical underpinnings, practical implementation and methodological choice in recent DBR research in the domain of mathematics education. The findings suggest that much of the work presented as DBR might be better characterized as ‘implementation studies’ as they contain only limited commitment to theoretical development capable of supporting the scaling of innovation. The exceptions appear to occur in settings with well-developed research capacity which recognizes the need for ontological innovation of theory whilst iteratively and comprehensively exploring the complexities of authentic learning interventions. An example of a rich mathematical research study using DBR is provided to point to the potential for this methodology to achieve its overarching aims more fully. PubDate: 2022-01-23 DOI: 10.1007/s13394-021-00407-5
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