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This conceptual paper proposes a model to describe the quality of student dialogue during participative collaborative problem solving. Drawing on the participation metaphor of learning, we argue that the construct of mathematical sophistication is useful to describe the quality. We then present two frameworks, mathematical competencies and mathematical practices, as ways to operationalise the construct in the collaborative problem-solving setting. We argue that by using a networking theories approach, the two frameworks will provide nuances of levels of mathematical sophistication that can be observed in student interaction. In addition, they could provide an analysis of both individual and group contributions to mathematical sophistication in a collaborative task setting. Implications of using two approaches for conceptualizing mathematical sophistication for future mathematics education research and teaching practices are provided.

Students in secondary school often find mathematics abstract and irrelevant, frequently questioning its usefulness and purpose. The discourse around STEM education has encouraged secondary teachers to attempt connecting mathematics with other STEM disciplines, most commonly with science. By making connections between content and skills through applications, it is anticipated that secondary school students will engage more in class that may result in improved mathematics achievement. This paper explores the rationale for making connections with science, examines the impact on student achievement in mathematics from previous research and discusses challenges for schools and teachers.

This paper reports on a study of choices about the use of a computer algebra system (CAS) or pen-and-paper (p&p) by a class of seven Year 11 Mathematical Methods (CAS) students as they completed a calculus worksheet. Factors that influenced students' choices are highlighted by comparing and contrasting the use of CAS and p&p between students. Teacher expectation of students' use of CAS and p&p reveals that, even in a small class, the students' use of CAS and p&p sometimes differed from what was expected. The analysis here indicates that there are a variety of factors that influence students' decisions, including speed of calculation and accuracy of p&p work.

From 2011-2013 the VCAA conducted a trial aligning the use of computers in curriculum, pedagogy and assessment culminating in a group of 62 volunteer students sitting their end of Year 12 technology-active Mathematical Methods (CAS) Examination 2 as a computer-based examination. This paper reports on statistical modelling undertaken to compare the distribution of results for this group with the standard cohort, and any differences in student response between the two groups at the item level.

Place value is a foundational competency for primary school mathematics and for this reason we have sought to investigate what the recent and current academic conversations are around this important concept. In this paper we present a survey of literature presented in the Australasian, European and Southern African contexts through a review of purposively selected conference proceedings and journals to establish what the conversations have been about the teaching and learning of place value in these research communities from 2013 to 2022.

This paper reports on a section of a larger study, investigating the mathematics experiences of Down syndrome (DS) learners in Australian Primary Schools. Developing the numeracy skills to experience independence in post school settings is crucial for individuals with DS. The aim of this paper is to share the school mathematics experiences of six DS adults and their parents/carers to ascertain if their experiences (DS adults) with mathematics when they were at school have had consequences for how they engage and participate in society as adults. Initial findings suggest that DS adults' mathematics experiences have impacted on their quality of life in a post-school setting.

In this position paper we highlight language as a perennial factor contributing to compromised meaning-making in multilingual primary school mathematics classrooms. We note use of the term 'translanguaging' in discussions around mitigating this meaning-making challenge. The paper argues that, while much work remains to be done towards clarifying the pedagogical insights, skills, and resources needed to ensure that translanguaging practices achieve their intended goals, potentially important parallels may be found between horizontal and vertical forms of translanguaging and horizontal and vertical mathematisation.

This paper reports changes in the extent to which a group of Year 11 students used a Computer Algebra System (CAS), pen-and-paper (P&P), or a combination of both, when solving routine problems across seven months in four different topics. Comparing the frequency of CAS use across topics shows students made greater use of CAS in the topics of "Linear, quadratic, and cubic functions" and "Exponential and logarithmic functions" than in the topics of "Trigonometry and circular functions" and "Calculus." These differences suggest that students either made different choices about the use of CAS in these topics or used CAS less frequently as they gained experience with CAS.

Research has shown that many primary students experience transition barriers between additive and multiplicative thinking. This paper analysed responses from 253 Years 3 to 6 students to a diagnostic assessment which consists of whole number multiplication and division problems involving equal groups, arrays, multiplicative comparison and Cartesian product situations. Based on the Rasch analysis, item responses were differentiated into five developmental Stages indicating a wide range of understanding and pointing to different transition barriers that students experience. The reasons for these are discussed in the paper and some advice is presented for teachers.

This paper reports on the development of complex unfamiliar mathematics questions using conventional mathematics questions. The autoethnographic research data included retrospective reflections of two educators and review of literature. Data analysis resulted in a three-phase development process: identifying conventional questions and subject matter to be modified to enhance levels of understandings and skills, modifying conventional questions and subject matter to complex unfamiliar questions, and identifying enhanced understandings and skills in modified questions. The paper then discusses the educators' insights developing complex unfamiliar mathematics questions for the practice of teaching and learning.

In this study, the evaluation of mathematical proficiency (MP) in one NSW Year 12 standardised exam paper was investigated. By considering the multi-strand model of MP, a literature-based criteria for scoring the assessment of MP in individual exam questions was first developed. The criteria were then applied to quantitatively evaluate the 2023 HSC Advanced Mathematics exam paper. The findings revealed a generally balanced assessment of all MP strands in the short-response exam questions. This study has methodological implications for evaluating the assessment of MP in external mathematics assessments, and the designing mathematics exams.

This paper reports on a key representation, a triple number line, designed as part of the first author's doctoral study. The study sought ways to represent multiple constructs of fractions in the context of merging music and mathematics to support learners' understanding of fractions. A problem scenario was designed guided by Realistic Mathematics Education principles. Findings shared in this paper are based on the process of designing and implementing the tasks around the triple number line. Data for this qualitative, participatory dual-design experiment in task design were collected via formal and informal interviews in two micro-Communities of Practice. We conclude that the key representation of the triple number line can be a powerful tool for supporting learners in their fraction understanding.

The study reported in this paper is on the professional development (PD) of primary school mathematics teachers. Teachers from two primary schools participated in the PD for two years. High-stakes mathematics examination tasks were used to kick start awareness and thinking about teaching for big ideas. Teachers did the tasks and discussed their solutions focusing on how their instruction could facilitate the acquisition of mathematical ideas as a body of connected knowledge. Data presented in this paper show that the tasks teachers worked with at the start of the PD did impact their understanding and instructional practice specific to big ideas in mathematics. Some challenges the teachers faced during the PD are also noted.

In this paper we report on an aspect of the findings of a larger three-phase study exploring the factors that influence teachers implementing pedagogies that cultivate students' STEM capabilities and 21st century skills. Data were collected through an online questionnaire, semi-structured interviews, focus groups and case studies. This paper will focus on the findings from the first phase of this study and initial analysis of focus groups and semi-structured interviews data. Preliminary findings show that participants hold mixed beliefs concerning student proficiency in mathematics and there are common factors that influence decisions concerning the use of pedagogical practices that support students' mathematical proficiency and the development of students' 21st century skills. These factors include teachers' personal beliefs and attitudes, perceived time and curriculum constraints, student behaviour and students' academic ability.

The interactional role of language use in the mathematics classroom is explored in the last few decades. This paper adds to the knowledge base by exploring different ways of talking that children and their teacher use in a New Zealand primary geometry English-medium classroom. Bakhtin's concept of speech genres is used for an analysing transcript of one audiovisually recorded whole-class interaction episode from a geometry lesson in a Year 5/6 classroom. The analysis suggests that the teacher and children use various genres to participate in classroom interactions. The use of several genres provides insights into what children and the teacher construct as mathematical in the real sense. The paper finished with a few implications for teaching and research.

The Queensland Certificate of Education (QCE) system is Queensland's senior school qualification. To support the introduction of the system in 2019, existing senior syllabuses were redeveloped and a new senior assessment model was established, this included the implementation of a mandatory high-stakes assessment task, the Problem-solving and Modelling Task (PSMT) in all four mathematics syllabi. The PSMT required new skills from both students and teachers to manage complex, open-ended investigations. In this paper, we reflect on our school's approach to build teacher capability in designing PSMTs and supporting student engagement with PSMTs.

Determining the most effective ways to support student-to-student talk requires some negotiating from a responsive teacher. This paper reports on a case study of two emergent multilingual students in Grade 3 (seven to eight-year-olds) who explained their strategies to each other. The transcript of their conversation was analysed using Chan and Sfard's (2020) "participation profile framework." Findings indicate that the two students learned from each other during the interaction because they revised their work to be more precise. One implication of this study is that strategic pairing may be a useful practice to eliminate inequitable power dynamics.

Fractions are among the most problematic concepts that children encounter in their primary school years because of the many different conceptions of fractions. Textbook analyses have tried to provide insights into how fractions are introduced, focusing on the different concepts and representations of fractions. In this paper, we contribute to these efforts by investigating the way fractions are first introduced using the notion of potential concept images as afforded by the textbooks. Analyses of two textbooks, one from Northern Ireland and the other from Singapore, will be presented to highlight these potential concept images and their implications for practice.

Mathematical thinking is a complex, multi-faceted process that has been described as messy and difficult but can also show growth and insights into conceptual understanding and knowing. This paper explores the application of Edward de Bono's practical thinking, in particular, the occurrence of porridge words to examine the mathematical thinking of primary school students. The research employed qualitative research techniques using participant's drawings, their written descriptions, and interviews about their drawings. Employing discourse analysis uncovered patterns in how students used porridge words to communicate their mathematical thinking.

This paper explores factors that contribute to increased self-efficacy in preservice teachers as they practise teaching mathematics in a primary classroom. Thirteen preservice teachers completing a Master of Primary Teaching contributed to the research, four of whom became the focus of intense examination during two professional practices conducted over a 12-month period. Findings revealed six factors. When conceptualised into a framework the factors contributed to a better understanding of ways to support preservice teachers' self-efficacy for teaching mathematics.

In this paper we explored primary school teachers preference for different game modes to support mathematics teaching and learning. Eighty-four teachers played digital and non-digital addition and subtraction games that were functionally equivalent during professional learning workshops. Most teachers indicated that they would be more likely to use the non-digital mode; despite more mixed views around perceived effectiveness for supporting learning and anticipated student preferences. Key reasons as to why teachers tended to prefer non-digital or digital games are examined.

The current paper overviews a nine-month PhD study that investigated the impact of learning mathematics through sequences of challenging tasks on the mathematical competence and attitudes of Year 2 students (n = 59). Adopting a Self-Determination Theory lens, a pragmatist paradigm and a mixed-method design, the study found that at all levels of investigation and analysis, the experience of learning through sequences of connected, cumulative, and challenging tasks had positive benefits for Year 2 students.

The revised Australian curriculum presents a new emphasis in primary schools on the process of Mathematical Modelling. A modelling focus brings close attention to confidence and capability with a potentially new problem-solving process, associated language, and pedagogical processes. This paper presents a Year 4 classroom modelling experience that arose as students planned their end-of-year party through Guided Mathematical Inquiry. Classroom video data captured two students working on vertical whiteboards as they formulated and solved a problem involving carrot sticks and dip. Findings reflect the students as doers of mathematics, engaged in productive struggle.

This paper provides evidence that a short, fully online, well-constructed diagnostic test based on research literature can give teachers information about their students' thinking and strategies that is sufficiently accurate to use for formative assessment purposes. The example is a test for students beginning to learn to solve equations. The main goal is to inform their teacher of the solving strategies each can use. Accuracy data from 3010 students is analysed to judge how well the overall performance of each group matches predictions. Alongside a previous analysis of misconceptions and common errors, we show this test gives teachers a good picture of most students to plan their lessons.

This paper presents an analysis of how resources were designed to support implementation of the new Queensland senior secondary mathematics syllabuses. The analysis draws on the concept of educative curriculum materials that build teachers' subject matter knowledge and pedagogical content knowledge. Such resources are intended to help teachers "learn how to learn" to teach mathematical content that is new or unfamiliar to them.

Concern regarding the prevalence of out-of-field teaching continues to grow, as does the body of literature calling for a better understanding of the nuanced complexities associated with out-of-field teaching. In this paper, we extend our previous analysis of data from a survey of the profession teaching Years 7-10 mathematics in South Australia. Analysing the data through an identity lens indicates that out-of-field teachers who self-identify as mathematics teachers and express a preference for teaching mathematics are more likely to exhibit levels of interest, enjoyment, confidence, and commitment that align with those of in-field teachers.

Progressing from additive to multiplicative thinking is a key outcome of school mathematics, making ratios an essential topic of study in junior secondary. In this study, 15 Australian Year 8 students were administered a ratio test followed by semi-structured interviews to explore their conceptions of ratio prior to formal instruction. In this paper, students' responses to one of the ratio questions are analysed in detail. Analysis of incorrect responses was conducted using a modified version of Radatz's (1979) framework. Analysis of correct responses revealed that some students worked proficiently with ratio without formal instruction.

In this paper we reflect on our combined work in some of the most marginalised educational contexts in the Southern Hemisphere. We draw on the work of Bourdieu to frame the paper. We propose the working in marginalised education settings requires a particular habitus or way of being to be able to play the research game. Underpinning our approach is the South African construct of Ubuntu, which is very much about collaboration--I am because we are--so that there is a move away from doing research "on" participants and contexts to one which is very much about doing research "with" participants and contexts. We find Bourdieu's notion of game as a powerful construct to theorise ways of thinking about the field of educational research.

School Mathematics Leaders seek to improve the mathematical learning outcomes of the students in their schools. Recognising that improved learning is dependent on high quality teaching, leaders are often keen to support teachers' knowledge of mathematics content and pedagogy. This paper reports ways in which School Mathematics Leaders support and lead the professional development of teachers in team meetings. Results from survey data and case study research are reported in this paper to describe and highlight the supportive actions undertaken by School Mathematics Leaders as they provide professional learning opportunities for teachers.

This paper reports on the potential for engaging students in an activity that considers the interconnectedness of the five Big Ideas of Statistics in the context of conducting random trials. In the study, two classes of Year 6 students (aged 11-12 years) used "TinkerPlots" to determine the sample space of a "Mystery Spinner." Analysed for this paper were data collected from entries made in completed workbooks while engaged in the learning activity and responses to relevant questions in an end-of-year questionnaire from 27 students. The results indicated using an activity that included a probability model contributed to students working mathematically with percentages and frequencies and supported the development of intuitions about randomness and informal inference. This was promoted by students analysing the variation in the distribution of data and describing their expectations about an unknown sample space.

This paper examines the types of teacher questions that assist young students to interpret and explain their critical mathematical thinking (CMT). Focusing on students who enter their first year of formal schooling (aged 5-6 years), this paper draws on data from a one-on-one task based clinical interview conducted with 16 students. Teacher questioning data were analysed for question type (probing, factual, guiding) and further analysed against the conceptualized critical mathematical thinking framework for young students. Findings indicated that when teachers used clarifying, noting relationships, and offering opinions style probing questions, young students were supported to interpret and explain their critical mathematical thinking.

This paper describes the authors' journey in designing a linear representation that aligns mathematics and music to support fractional reasoning. The three authors, guided by the theoretical framing of realistic mathematics education, engaged in a task design process over a 12-month period to develop integrated resources. Data were collected in the form of Zoom meeting recordings, among other sources. This paper describes some key findings from a narrative analysis of this process. We posit that the contribution is twofold--methodological insight of a task design journey and findings around the obstacles and resolutions of aligning representations across mathematics and music, despite the obvious confluence.

In Australia, all teachers are expected to explicitly attend to numeracy in the subjects they teach. Pre-service teachers, therefore, need to begin to develop effective numeracy teaching strategies but there is a limited research base to inform the design of courses that address this need. This paper reports on findings from a study investigating the impact of one such course. The paper draws on data from two pre-service teachers: two course tasks completed at the beginning of the course and interviews conducted at the end of the course. Data were analysed using the 21st Century Numeracy Model. Findings suggest there were changes to the pre-service teachers' understanding of numeracy and their capacity to embed numeracy in the subjects they will teach.

This paper emerges from a broader study that investigated the strategies employed by teachers to continue mathematics teaching and learning during South Africa's COVID-19 lockdowns and through subsequent phased and partial re-opening of schools. In this paper, we focus on teacher views of the role of parents in these efforts gathered through two questionnaires administered from 2020 to 2021. Twenty-five Grade 4-7 mathematics teachers from schools in the Eastern Cape province took part in the study. We address the question: What were teacher perspectives on the role of parents in the continuation of mathematics teaching and learning during COVID-19 and the gradual reopening of schools? We explore the teacher responses and show how the pandemic context provided a stimulus to forge stronger teacher-parent relationships and opened opportunities for productive ways of extending mathematics learning almost exclusively undertaken in the classroom into homes.

Engagement in mathematics education is an important factor in a successful student experience. This paper reports findings from a study of Year 5 student perceptions of engagement during a two-week, inquiry-based learning (IBL), problem-posing investigation. The study triangulated data from semistructured interviews, video observations and student work samples to understand the student's perspectives of IBL; however, this paper reports the interview data. The findings indicate that most of the students were behaviourally, emotionally, and cognitively engaged during the IBL investigation, and that the investigation provided an opportunity for students to experience levels of competence, relatedness, and autonomy need satisfaction.

Mathematics self-efficacy is considered an important variable in mathematics education because of its links to mathematics achievement. This paper reports on the development of the "Self-efficacy-effort in Mathematics Scale" (SEEMS), an instrument that has strong theoretical and psychometric properties. Based on a sample of n = 224 Australian primary students and n =133 secondary students, the instrument demonstrated positive correlations with related measures of mathematics achievement and mathematics self-concept. Girls had higher mathematics achievement within the primary and secondary school data. There were no significant gender difference in mathematic self-concept scores for the primary and the secondary school students. For mathematics self-efficacy, a gender difference was only identified in the high school data. The implications of this research to mathematics education practices are reviewed in the paper.

Mathematics education research has focused on developing teachers' knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher's thinking visible and enhancing a teacher's pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a teacher's pedagogical reasoning was made visible and highlight aspects of his thinking in relation to his instructional decisions during a series of lessons on division. Implications for professional learning are discussed.

Early years mathematics textbooks are support material for teachers and learners as they contain visual representations which communicate and clarify mathematical concepts. In this paper, we report on the types and functions of visual representations of multiplication and division in South African early years mathematics textbooks. There seems to be a reliance on textbooks amongst teachers in South Africa as they assist in providing pedagogic content knowledge. The research from which this paper emerged was a document analysis of multiplication and division visual representations in nine textbooks (Grade 1-3). A Visual Representation Framework was used to analyse the textbooks. The findings indicated that the most dominant type of visual representations across all three textbook series and across the three grades were equal groups and the majority of visual representations had an exemplifying function.

This theoretical paper examines views about the role of language and mathematical discourse in learning mathematics. Current research is still addressing what constitutes a mathematical discourse. As new conceptions of the purpose of language use in mathematics are explored, and associated ontological and epistemic positions are revealed, one might ask: how are we able to reframe our view of language to support a social participation perspective? This paper proposes the consideration of Wittgenstein's philosophy of language games to shift our conception of classroom language use in mathematics to encompass broader contextual features such as participation, patterns of exchange and social norms.

The "Let's Count" Community Professional Pilot 2019 took place in six sites across three states (New South Wales, Queensland and South Australia). The aim of the pilot program was to implement the "Let's Count" face-to-face program for the first time with a group of people who work with young children and their families but who are not trained early years educators working within early childhood education and care centres. This paper reports on the evaluation of the pilot program with specific emphasis on expanding the reach of "Let's Count" whilst maintaining its integrity and outcomes. [This paper is the third in a symposium of three papers. For the first paper, "'Let's Count' in an Online Learning Environment," see ED616193. For the second paper, "'Let's Count' in Early Childhood Teacher Education," see ED616194.]

"Let's Count Online" is a new e-learning approach to delivering "Let's Count" professional learning. It was evaluated in 2018. The findings suggest that the e-learning platform was successful, and that the outcomes for educators were similar to those achieved by participants using the face-to-face workshop professional learning model. Several key differences in outcomes were noted, and these inform recommendations for refining "Let's Count Online." [This paper is the first in a symposium of three papers. For the second paper, "'Let's Count' in Early Childhood Teacher Education," see ED616194. For the third paper, "'Let's Count' and Community Professionals," see ED616191.]

This paper reports on one aspect of a bigger project: teachers' design of instructional materials. We found a number of design moves used by the teachers in our study. In this paper, we report three of them: Making things explicit, making connections, and resequencing practice examples. [This paper is the fourth in a symposium of four papers. For the first paper "Models of Mathematics Teaching Practice in Singapore Secondary Schools," see ED616186. For the second paper "An Experienced and Competent Teacher's Instructional Practice for Normal Technical Students: A Case Study," see ED616178. For the third paper "Imbuement of Desired Attitudes by Experienced and Competent Singapore Secondary Mathematics Teachers," see ED616187.]

In 2011, the "Let's Count" professional learning program was developed into an elective distance education subject offered at Charles Sturt University. The resulting subject, EMC101: Let's Count, has been offered every year since 2012, and has to date been completed by 796 students. This paper details the subject design and provides enrolment and evaluation data that attest to the success of the subject. [This paper is the second in a symposium of three papers. For the first paper, "'Let's Count' in an Online Learning Environment", see ED616193. For the third paper, "'Let's Count' and Community Professionals", see ED616191.]

We present a model that employs "cultural-historical activity theory" (CHAT) concepts to inform research with designated Educational Leaders in early years settings. We theorise practice change in early years mathematics education in terms of motive objects of activity and mediation by cultural tools. We show how CHAT can be used to lead development of a strengths-based approach to support young children's early mathematics education through systematic professional learning activity. Our overarching aim is to understand how educational leadership in early learning spaces can be reimagined, drawing on CHAT to theorise this under-researched area of mathematics leadership in early learning settings. [This paper is the third in a symposium of three papers. For the first paper "An Introduction to the Strengths Approach," see ED616177. For the second paper "A Strengths Approach to Birth to 3 Mathematics Education: The Case of Banjo Childcare Centre," see ED616174.]

This paper reports how 30 experienced and competent Singapore secondary mathematics teachers attempted to imbue desired attitudes in their students and some possible factors that might have influenced the teachers' choice of instructional approaches. It was found from the analysis of lesson observations of these teachers that most of those teaching lower-ability students tended to build their students' confidence and perseverance, while those teaching higher-ability students were more inclined to help their students appreciate the relevance of mathematics. Only a minority of the teachers tried to make lessons fun by using mathematics-related resources or telling non-mathematics-related jokes. It was also discovered from the teacher interviews that two factors appeared to influence the teachers' choice of the types of positive attitudes to develop in their students: the abilities of their students and the beliefs of the teachers on what mathematics is. [This paper is the third in a symposium of four papers. For the first paper "Models of Mathematics Teaching Practice in Singapore Secondary Schools," see ED616186. For the second paper "An Experienced and Competent Teacher's Instructional Practice for Normal Technical Students: A Case Study," see ED616178. For the fourth paper "Singapore Mathematics Teachers' Design of Instructional Materials," see ED616188.]

A model of instruction is a set of strategies that guide teachers in their instructional practice. The purpose of this paper is to dispel the myth that mathematics teaching in Singapore schools is all about drill and practice, as perceived of many Asian systems. This paper draws on data of a large project that examined the enactment of school mathematics curriculum in Singapore secondary schools. Based on the teaching practices of 30 experienced and competent teachers, a survey was constructed and administered to 677 teachers. The data from the survey showed that teachers go well beyond traditional forms of instruction in their teaching practices in Singapore secondary schools. [This paper is the first in a symposium of four papers. For the second paper, "An Experienced and Competent Teacher's Instructional Practice for Normal Technical Students: A Case Study," see ED616178. For the third paper, "Imbuement of Desired Attitudes by Experienced and Competent Singapore Secondary Mathematics Teachers," see ED616187. For the fourth paper, "Singapore Mathematics Teachers' Design of Instructional Materials," see ED616188.]

This paper provides a foundation for the Research Symposium, "Strengths Approaches in Early Childhood Mathematics Education" by providing an overview of the development of strengths-based approaches in social work and education. A framework, adapted from the Strengths Approach (McCashen, 2017), for applying a strengths-based approach in early childhood mathematics education is introduced. [This paper is the first in a symposium of three papers. For the second paper "A Strengths Approach to Birth to 3 Mathematics Education: The Case of Banjo Childcare Centre," see ED616174. For the third paper "Leadership of Strengths-Based Approaches for Early Years Mathematics Education: Using CHAT as a Framework for Educational Leaders' Professional Learning Leadership," see ED616189.]

This paper contributes to the Research Symposium, "Strengths approaches in early childhood mathematics education" by providing an illustration of how an early childhood centre adopts a strengths approach to mathematics education for birth to three-year-old children. A case illustration is drawn from a current Australian Research Council-funded study focusing on mathematics education for children under three years of age. The case is analysed and described using a five-step strengths-based framework. [This paper is the second in a symposium of three papers. For the first paper, "An Introduction to the Strengths Approach," see ED616177. For the third paper, "Leadership of Strengths-Based Approaches for Early Years Mathematics Education: Using CHAT as a Framework for Educational Leaders' Professional Learning Leadership," see ED616189.]