Children's Mathematical Thinking Research Articles

An exploratory mixed methods study about teacher candidates’ descriptions of children’s confusion, productive struggle, and mistakes in an elementary mathematics methods course

Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres’ rights of the learner

The math talk learning environment: Testing an early childhood math intervention

Given the importance attributed to math talk, this study sought to test the effects of a Math Talk Learning Environment for exposing children to math talk and whether this effort would improve children's early math skills. The intervention used guided play in small group math games focused on numeracy skills. The analytic sample comprised 95 racially and ethnically diverse children enrolled in Head Start classrooms randomly assigned to one of two treatment groups or the counterfactual. Students in one group engaged in math games while being encouraged to use math language, whereas the other group engaged in the same math games without any explicit math language encouragement. Linear mixed modeling was used to regress children's residualized math gains on condition. Results showed a significant difference in the general math skills and the specific numeracy skills gained by children who participated in the two treatment groups compared to the counterfactual. No significant differences could be detected between the treatment groups; yet, there are indicators that the intervention warrants further investigation. This research parallels prior research and contributes new findings that will inform early childhood practitioners, developmental psychologists, and others interested in applying cognitive science to the classroom. The discussion includes descriptions of the supports required and challenges to fostering young children's mathematical thinking.

“I did it wrong, but i know it”: Young children's metacognitive knowledge expressions during peer interactions in math activities

Previous studies reveal that children's metacognitive skills make massive progress during the early childhood period. We believe that examining how children use metacognitive skills in the learning process is crucial for shaping future learning experiences. This case study explores how children's metacognitive knowledge emerges through peer interactions in mathematical measurement activities. Sixteen activities based on the dimensions of mathematical measurement skills of length, area, weight, and volume were applied and video recorded. We systematically observed two 5-year-old children in these activities for 10 weeks. A framework of analysis was developed from the results of previous research on children's metacognition. Children's metacognitive knowledge was analyzed in mathematical statements and other variables were also extracted. Using qualitative analysis, this study indicates how children's mathematical thinking skills are reflected in their expressions of metacognitive knowledge during peer interactions. Difficulties in assessing and measuring children's metacognition are also discussed.

Development and Practice of Children's Mathematical Thinking Ability in Board Games

Development and Practice of Children's Mathematical Thinking Ability in Board Games

Dynamic assessment and teachers' knowledge of children's mathematical thinking: a case study in children's mathematics

This paper considers the kind of pedagogical knowledge and principles involved in the operationalisation of knowledge of children's mathematical thinking as a process of dynamic assessment. Using a case study of a particular child, this paper explores planning and instruction for a child determined by a detailed and informed interpretation of the child's conceptual understanding through a dynamic process. It presents as a case study the observations of a teacher who had undertaken professional development in children's mathematical thinking, theoretically informed by Cognitively Guided Instruction and Maths Recovery. The observations revealed the child's mathematical understanding and how the teacher used this knowledge dynamically to inform teaching. The paper outlines the kind of knowledge required of teachers to enact this dynamic process in mathematics teaching and argues for the centrality of this to the development of inclusive practice.

Early mathematical education in narrative contexts and the local environment

The article describes the implementation of the project that took the form of a teaching experiment. The “Myśleć jak architect” [en. “Think like an Architect”] project was aimed at developing children's spatial, mathematical and creative skills, integrating the social environment: academic, educational, family and cultural, and promoting science. Its essence was to design and conduct a mathematical and creative didactic cycle in the university space and in the local cultural space. The essence of educational meetings were: play, tasks, games and mathematical problems undertaken and solved by children in active and creative action, also using innovative teaching materials. The workshop didactic program was designed as a cycle of 12 hours of didactic classes, in which there was a deliberate use of rhymes, poems, fairy tales and mathematical narratives. The project lasted from November 2018 to May 2019. The beneficiaries of the project were the oldest pre-schoolers and younger students from grades I-III, in 9 groups, each consisting of 12 people. The study confirmed the effectiveness and attractiveness of integrating literature with mathematics and learning mathematics in various areas. The study has shown that the strategies for teacher management of construction play using narration are an important and effective ways of supporting the development of a child's mathematical thinking.

Professional Noticing into Practice: An Examination of Inservice Teachers’ Conceptions and Enactment

ABSTRACTTeacher noticing and the related construct of professional noticing of children's mathematical thinking have proven to be fertile ground for education researchers. Professional noticing is a framework for a teaching practice consisting of three component parts: attending, interpreting, and deciding. The current study investigates the conceptions and enactment of professional noticing of 24 elementary and middle grades teachers participating in professional learning programs that incorporated professional noticing. These teachers demonstrated a wide range of interpretations of professional noticing which varied in consistency with respect to the literature in this area. This diversity of conceptions is seen as a consequence of teachers having different definitions and scopes of application for professional noticing. This study adds to current discussions about the meaning and role of professional noticing by considering the perspective of practitioners, a group whose input is often secondary to education researchers but whose conceptions and enactment of such noticing is critical for student success.

Research Commentary: Curricular Noticing: A Framework to Describe Teachers' Interactions With Curriculum Materials

Building on the work of Professional Noticing of Children's Mathematical Thinking, we introduce the Curricular Noticing Framework to describe how teachers recognize opportunities within curriculum materials, understand their affordances and limitations, and use strategies to act on them. This framework builds on Remillard's (2005) notion of participation with curriculum materials, connects with and broadens existing research on the relationship between teachers and written curriculum, and highlights new areas for research. We argue that once mathematics educators better understand the strategic curricular practices that support ambitious teaching, which we refer to as professional curricular noticing, such knowledge could lead to recommendations for how to support the curricular work of teachers and novice teachers in particular.

Translating Professional Development for Teachers Into Professional Development for Instructional Leaders

In recent years, teacher noticing of children's mathematical thinking has emerged as an important and generative construct in mathematics education (Sherin, Jacobs, & Philipp, 2011). Less is known about ways instructional leaders notice teachers' learning. Between 2011 and 2015, we facilitated professional development (PD) in which coaches, principals, and teachers studied mathematics teaching and learning together. Our initial focus on teacher decision-making was inadequate in meeting instructional leaders' learning needs. We adapted the PD to focus instructional leaders' attention on the work of learning teaching. Analysis of leaders' discourse revealed shifts from noticing teacher characteristics to noticing dilemmas and decision-making within teaching and coaching. Findings suggest new roles for teacher educators and new forms of PD for instructional leaders.

Thoughtful practice: Taking professional development a step further

This paper reports on a follow-up study of one teacher, two years after she participated in a numeracy staff development contract based on the Cognitively Guided Instruction (CGI) philosophy. The discussion is based on classroom observations and a semi-structured interview focusing on Ms Smith's perspective on the experience. She explains how she uses the research-based knowledge about the development of children's mathematical thinking gained from professional development to inform the teaching and learning of mathematics in her classroom. This single case study is used to give some insight into reflective practice, self-sustaining and generative change in the teaching of mathematics, and professional development.

Between the Big Trees: A Project-based Approach to Investigating Shape and Spatial Thinking in a Kindergarten Program

SUPPORT FOR CHILDREN'S EMERGING mathematical thinking is a characteristic of high-quality early childhood education. Young children's spatial thinking, an important component of mathematical thinking, is both innate and influenced by experience. Since spatial thinking contributes to children's mathematical thinking, it is important for children to engage in activities that support this learning. Early childhood educators are calling for guidance in how to support children's mathematical thinking in the context of an informal curriculum. In this paper, we describe how a project-based approach to mathematics teaching and learning provided a range of opportunities for children to investigate and rehearse understandings of two- and three-dimensional (2D and 3D) shapes and spatial thinking within the context of a project that was of ‘real world’ interest to the children. By intentionally embedding multiple opportunities for children to explore shapes and spatial thinking in a sequence of core learning experiences and complementary experiences, educators provided children with opportunities to rehearse shape and spatial concepts and related language in differing ways. Opportunities for formative assessment of children's learning are also discussed.

Connecting Children's Mathematical Thinking with Family and Community Knowledge in Mathematics Instruction

This department publishes brief news articles, announcements, and guest editorials on current mathematics education issues that stimulate the interest of TCM readers and cause them to think about an issue or consider a specific viewpoint about some aspect of mathematics education.

Learning to Leverage Children's Multiple Mathematical Knowledge Bases in Mathematics Instruction

In this article, the authors explore prospective elementary teachers' engagement with and reflection on activities they conducted to learn about a single child from their practicum classroom. Through these activities, prospective teachers learned about their child's mathematical thinking and the interests, competencies, and resources she or he brought to the mathematics classroom, and then wrote reports that included instructional suggestions as to next steps to further the child's growth in mathematics. The authors' analyses of these reports indicate that there were a variety of ways which prospective teachers made connections to one or more of their child's knowledge bases. In a high percentage of cases, prospective teachers attended to one of these knowledge bases, indicating that they were attending to particularities about their child and developing the dispositions to continue to do so. Implications for research and practice are discussed.

Young Children Dancing Mathematical Thinking

EARLY CHILDHOOD EDUCATORS ARE required to recognise multiple sources of evidence of children's mathematical thinking. This paper encourages early childhood educators to consider how young children's spontaneous dance improvisations provide evidence of their exploration of the mathematical concepts of spatial orientation and spatial visualisation. In this research, examples of four-year-old children's improvisational dance are drawn from a larger corpus of children's dance vignettes and analysed, highlighting children's demonstrations of embodied spatial orientation and spatial visualisation. Recognising children's spatial thinking provides opportunities for educators to develop contingent learning experiences that facilitate children's exploration of concepts of lines, angles, direction and two- and three-dimensional shapes through other forms of symbolic expression.

A Study of Pre-Kindergarten Teachers' Knowledge about Children's Mathematical Thinking

WITHIN THE CURRENT ERA of educational reform, greater numbers of early childhood teachers have become engaged in teaching mathematics curricula that require teachers to have a deep understanding of children's mathematical thinking. The purpose of this study is to examine how pre-kindergarten teachers in the United States (US) notice and interpret children's mathematical thinking. To address teacher knowledge of children's mathematical thinking, a qualitative case study was conducted. The sources of data include classroom observation and post-observation interviews with each teacher. The results indicate that these teachers varied in their level of noticing and attending to students' mathematical thinking. Although some teachers could successfully engage their students' mathematical thinking, the extent to which they interpreted that thinking was different.

The role of children's mathematical aesthetics: The case of tessellations

In order to understand the role of mathematical aesthetics in the classroom and how it promotes mathematical learning, we must first understand children's current aesthetic values. Six children, ages 8–10, were interviewed as they explored tessellations for the first time. The students repeatedly used certain aesthetic themes. Further interviews revealed mathematicians invoked the same themes, though in different ways. The students’ aesthetic criteria motivated their explorations, guided their creation of tessellations, and informed their evaluations. Rather than being an optional topic, mathematical aesthetics may be foundational for children's mathematical thinking, just as it is for mathematicians’.

Developing equitable elementary mathematics classrooms through teachers learning about children's mathematical thinking: Cognitively Guided Instruction as an inclusive pedagogy

This paper reports on a study carried out in Scotland which involved introducing the principles of Cognitively Guided Instruction (CGI) to 21 mainstream elementary teachers. It considers the effects of developing CGI in classrooms focussing on teacher learning and particularly their capacity to support all learners. The findings demonstrate teachers' awareness of their own learning and how increased understanding of children's mathematical thinking left them better placed to support all learners. The study highlights the importance of developing teachers' knowledge of children's mathematical thinking in order to promote inclusive practices with CGI providing a useful framework for this professional development.

Primary special school teachers' knowledge and beliefs about supporting learning in numeracy

This paper presents findings from a qualitative study of a group of 12 teachers in primary special schools in Scotland for children with moderate learning difficulties. It sets out an analysis of classroom observations and interviews that explored teachers' knowledge and beliefs about teaching and learning in mathematics with children with moderate learning difficulties. The teachers were interviewed pre- and post-intervention; this was a research-based professional development programme in children's mathematical thinking (Cognitively Guided Instruction) which teachers then developed in their classrooms. The findings showed that prior to the professional development, the teachers had a limited knowledge of children's mathematical development with teaching frequently informed by intuitive beliefs and dated and sometimes discredited practices. Most teachers had low expectations of children with learning difficulties. Post-intervention, the teachers reviewed this stance and affirmed that a deeper understanding of children's mathematical thinking provided a more secure knowledge base for instruction. They also recognised the extent to which learners were constrained by existing classroom practices. The paper argues for the commonality of this knowledge base and considers the problematic nature of viewing such knowledge as sector specific.

Young children's recognition of quantitative relations in mathematically unspecified settings

Children have been found to be able to reason about quantitative relations, such as non-symbolic proportions, already by the age of 5 years. However, these studies utilize settings in which children were explicitly guided to notice the mathematical nature of the tasks. This study investigates children's spontaneous recognition of quantitative relations on mathematically unspecified settings. Participants were 86 Finnish-speaking children, ages 5–8. Two video-recorded tasks, in which participants were not guided to notice the mathematical aspects, were used. The tasks could be completed in a number of ways, including by matching quantitative relations, numerosity, or other aspects. Participants’ matching strategies were analyzed with regard to the most mathematically advanced level utilized. There were substantial differences in participants’ use of quantitative relations, numerosity and other aspects in their matching strategies. The results of this novel experimental setting show that investigating children's spontaneous recognition of quantitative relations provides novel insight into children's mathematical thinking and furthers the understanding of how children recognize and utilize mathematical aspects when not explicitly guided to do so.

Investigating the Relationship Between Musical Training and Mathematical Thinking in Children

This study examines the potential for music education to enhance children's mathematical thinking. Specification of potential cognitive correlates between musical and mathematical components is sought and underpins the design (3 variables x 2 conditions each=6 groups). Nearly 200 children aged 7-8 years experienced weekly music lessons (duration=9 months). Lessons emphasized melody, rhythm or form; in half of the classes, the teacher made the musical-mathematical parallels explicit.Apart from the specific musical-mathematical foci, the lesson content was kept as constant as possible within primary school settings. Pre-tests and post-tests in musical, creative, spatial and mathematical thinking were administered. Statistical analyses will examine improvement over time while considering differences among three musical components and two conditions for each.This research addresses concerns that governments’ quests for higher standards in mathematics may result in impoverished curricula with limited access to the arts. If it is shown that musical training appears to benefit logical thinking, as hypothesized, it may add to a growing body of research suggesting that policy-makers and educationalists reconsider curriculum balance.