Swee Fong Ng

The cognitive underpinnings of emerging mathematical skills: Executive functioning, patterns, numeracy, and arithmetic

BACKGROUND Exposure to mathematical pattern tasks is often deemed important for developing childrens algebraic thinking skills. Yet, there is a dearth of evidence on the cognitive underpinnings of pattern tasks and how early competencies on these tasks are related to later development. AIMS We examined the domain-specific and domain-general determinants of performances on pattern tasks by using (a) a standardized test of numerical and arithmetic proficiency and (b) measures of executive functioning, respectively. SAMPLE Participants were 163 6-year-olds enrolled in primary schools that typically serve families from low to middle socioeconomic backgrounds. METHOD Children were administered a battery of executive functioning (inhibitory, switching, updating), numerical and arithmetic proficiency (the Numerical Operations task from the Wechsler Individual Achievement Test-II), and three types of pattern tasks. RESULTS Contrary to findings from the adult literature, we found all the executive functioning measures coalesced into two factors: updating and an inhibition/switch factor. Only the updating factor predicted performances on the pattern tasks. Although performance on the pattern tasks were correlated with numerical and arithmetic proficiency, findings from structural equation modelling showed that there were no direct or independent relationships between them. CONCLUSIONS The findings suggest that the bivariate relationships between pattern, numeracy, and arithmetic tasks are likely due to their shared demands on updating resources. Unlike older children, these findings suggest that for 6-year-olds, better numerical and arithmetic proficiency, without accompanying advantages in updating capacities, will no more likely lead to better performance on the pattern tasks.

Strategic differences in algebraic problem solving : Neuroanatomical correlates

In this study, we built on previous neuroimaging studies of mathematical cognition and examined whether the same cognitive processes are engaged by two strategies used in algebraic problem solving. We focused on symbolic algebra, which uses alphanumeric equations to represent problems, and the model method, which uses pictorial representation. Eighteen adults, matched on academic proficiency and competency in the two methods, transformed algebraic word problems into equations or models, and validated presented solutions. Both strategies were associated with activation of areas linked to working memory and quantitative processing. These included the left frontal gyri, and bilateral activation of the intraparietal sulci. Contrasting the two strategies, the symbolic method activated the posterior superior parietal lobules and the precuneus. These findings suggest that the two strategies are effected using similar processes but impose different attentional demands.

Mathematics education : the Singapore journey

This comprehensive book is a state-of-the-art review of research and practices of mathematics education in Singapore. It traces the fascinating journey from the original development of the Singapore mathematics curriculum in the 1950s to the present day, and reports on diverse findings about the Singapore experience that are not readily available in print. All of the authors are active mathematics educators or senior mathematics teachers in Singapore, thus adding authenticity and distinctiveness to the stories covered in this book. The issues they so earnestly explore in this book will undoubtedly be of interest to graduate students, mathematics educators, and the international mathematics education community.

Survey of the State of the Art

This survey of the state of the art on research in early algebra traces the evolution of a relatively new field of research and teaching practice. With its focus on the younger student, aged from about 6 years up to 12 years, this document reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth of the field. The survey, in presenting examples drawn from the steadily growing research base, highlights both the nature of algebraic thinking and the ways in which this thinking is being developed in the primary and early middle school student. Mathematical relations, patterns, and arithmetical structures lie at the heart of early algebraic activity, with processes such as noticing, conjecturing, generalizing, representing, justifying, and communicating being central to students’ engagement.

Neuroscience and the Teaching of Mathematics

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the imperatives that led us to develop two neuroimaging studies that examined questions of curricular concerns. One of the challenges was to condense issues from classrooms into tasks suitable for neuroimaging studies. Another challenge, not particular to the neuroimaging method, was to draw suitable inferences from the findings and translate them into pedagogical practices. We describe our efforts and outline some continuing challenges.

Summary and Looking Ahead

This survey of the state of the art on research in early algebra traces the evolution of a relatively new field of research and teaching practice. With its focus on the younger student, aged from about 6 years up to 12 years, this document reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth of the field. The survey, in presenting examples drawn from the steadily growing research base, highlights both the nature of algebraic thinking and the ways in which this thinking is being developed in the primary and early middle school student. Mathematical relations, patterns, and arithmetical structures lie at the heart of early algebraic activity, with processes such as noticing, conjecturing, generalizing, representing, justifying, and communicating being central to students’ engagement.

Function Tasks, Input, Output, and the Predictive Rule: How Some Singapore Primary Children Construct the Rule

Function-machine tasks are not part of the formal Singapore Primary Mathematics curriculum and hence not taught formally. The corpus of data shows that provision of the expressions input, output, and ‘the rule is’ aided primary children, particularly those in the upper primary grades, to construct the predictive rule underpinning function-machine tasks. Children’s annotations showed that many were willing to write the literal form of input ± a = output, while others were open to the symmetric equivalence construct of the non-literal form of output = input ± a. Primary children’s knowledge reflected the spiral structure of the Singapore Primary Mathematics curriculum, where number facts and processes are introduced in bite sizes. Children at all upper grades found implicit functions challenging.

Learning and solving algebra word problems: The roles of relational skills, arithmetic, and executive functioning.

Although algebra is a prerequisite for higher mathematics, few studies have examined the mathematical and cognitive capabilities that contribute to the development of algebra word problems solving skills. We examined changes in these relations from second to ninth grades. Using a cross-sequential design that spanned 4 years, children from 3 cohorts (Mage = 7.85, 10.05, and 12.32) were administered annual tests of algebra word problems, mathematical skills (mathematical relational tasks, arithmetic word problems), and cognitive capabilities (working memory, updating, inhibitory, task switching, and performance intelligence). The cross-sectional findings showed that ability to solve mathematical relational problems was associated strongly with performance in algebra word problems. Working memory and updating explained variance in the relational, but not the algebra problems. Using an autoregressive cross-lagged model with structured residuals to analyze the longitudinal data, we found relational and arithmetic performance predicted independently algebra performance from one year to the next. The strength of these relations was consistent across grades. These findings point to the importance of developing skills in relational problems as one of the tools for improving algebra performance.

Summary of Findings of Chapters in Part II and Introduction to Part III

The two studies, the Singapore and the Hong Kong study, provide contrasting evidence of how two groups of teachers choose to teach factorisation. Teacher-HK in the Hong Kong study placed great emphasis on the choice of tasks used to teach factorization. By selecting suitable tasks which are related but with increasing complexity, Teacher-HK presented factorization as a reverse of expansion and how a set of unique factors could be extracted from a given expression. In fact Teacher-HK wanted to ensure that the students in the class understood why each expression had a unique set of factors and why it was necessary to extract all the factors. He used specific mathematical concepts to highlight why a given set of factors is correct. For example, although \( n\left(2a+2b\right),2n\left(a+b\right) \) and \( 2\left(na+nb\right) \) are three equivalent sets of factors for the expression \( 2na+2nb, \) only \( 2n\left(a+b\right) \) is the acceptable form. Teacher-HK helped students relate factorization of algebraic expressions to the factorization of whole numbers. Although it is correct to express twelve as the product of two factors \( 4\times 3, \) it was possible to factorise 12 completely to ‘\( 2\times 2\times 3 \)’. Teacher-HK tried to ensure that the students knew when they had the complete set of factors.

Summary of Part I and Introduction to Part II

The findings reported in the five chapters in Part I broadly support the use of videos in professional development of teachers. One message emerging from the work of the various authors suggests that effective leadership provided by senior management is crucial to the professional growth of teachers. Senior management must value and hence be willing to invest in the professional development of teachers. When professional growth of teachers is seen as an important contributing factor to how students learn, senior management would be willing to invest time and space for teachers to meet and discuss how best to conduct their professional growth. Furthermore they are open to listen to what mathematics educators have to share with teachers in such professional development courses. The role senior management plays in the professional development of teachers is shown clearly in Lim and Kor’s work with the Malaysian teachers. When senior management offered their support to the mathematics educators, those teachers who attended the in-service courses learnt about the nature of lesson study and also the good practices of their peers. By participating in the in-service courses, the teachers have some grounded images of how lesson study could be conducted. However when such support was not forthcoming, it was not possible for teachers to have any form of grounded images of lesson study.