Informal Mathematical Knowledge Research Articles

Dimensionality of preschoolers’ informal mathematical abilities

Recent research examining children’s early mathematical abilities has focused primarily on number and operations (e.g., counting, addition) with considerably less attention directed to the role of other possible dimensions of early mathematical abilities, such as, measurement, geometry, and patterning. The current study examined the dimensionality of informal mathematical abilities by conducting categorical confirmatory factor analysis (CCFA) using data from a large sample of preschool children from low-income families (N = 1630; Mean age = 4.46 years, SD = .37) using the Child Math Assessment (CMA; Klein & Starkey, 2004). The best fitting model consisted of four factors that include Number and Operations, Measurement, Geometry, and Patterning, with the Number and Operations factor explaining common variance in three first-order factors of Numbering, Operations, and Relations. These findings support the view that informal mathematical knowledge is a multi-dimensional construct that includes of these separable dimensions. Additionally, a Multiple Indicators Multiple Causes model was used to determine if mathematical ability differed for male and female preschoolers on each of the four factors or on each of the 35 items of the CMA. Results showed no differences for mathematical abilities between males and females at this age. Future research and curricular implications are discussed.

PRECISE MATHEMATICS COMMUNICATION: THE USE OF FORMAL AND INFORMAL LANGUAGE

INTRODUCTION. When explaining their reasoning, students should communicate their mathematical thinking precisely, however, it is unclear if formal terminology is necessary or if students can explain or describe mathematical concepts using everyday language. This paper reports the results of two studies. The first explores the relation between students’ use of formal and informal mathematical language and procedural knowledge in mathematics; the second replicates and extends these findings using a longitudinal design over the course of an entire school year. METHOD. Study 1 uses a pre-test, intervention, post-test method across one unit of study. Study 2 uses the same pre-test, intervention, post-test method, however, it implements a longitudinal design across an entire school year using growth curve analysis. RESULTS. Findings show that students benefit most when they attempt to describe the targeted mathematical concepts, regardless of the type of language used. This finding is consistent with prior work showing that having students go through the process of self-explaining, independent of the quality of those explanations, yields benefits to learning (i.e. Chi, 2000). DISCUSSION. Although teachers should still use formal language in their classrooms; they should not be discouraged if students are initially unable to use formal language correctly. We suggest that teachers allow students to explain their reasoning using either formal or informal terms, especially while students are in the midst of developing an understanding the mathematical concepts.

Comparing Turkish Early Childhood Education Curriculum with Respect to Common Core State Standards for Mathematics

Increasing recognition of mathematics in society and its effects in development of children takes attention to early childhood education and as a result early childhood curriculum. Although children have informal mathematical knowledge before they begin formal schooling, a qualified mathematics education during early childhood will enrich and prevent learning difficulties in further years. For this purpose, the aim of the study was to compare Turkish Early Childhood Education Curriculum (TECEC) objectives with respect to kindergarten level Common Core State Standards for Mathematics (CCSSM). The study carried out as a document analysis method. Results are presented under five domains of CCSSM; counting and cardinality, operations and algebraic thinking, number and operations in base ten, measurement and data, and geometry. Results of the study indicate that TECEC and CCSSM for kindergarten level have both similarities and differences. While CCSSM dedicate more time numbers than other topics, TECEC gives significance to many mathematical skills not specifically the numbers like geometry, measurement, etc. About numbers topic in CCSSM have in depth expectations from children. Some other differences are also seen in patterning topic in TECEC and in place value topic in CCSSM. More similarities are found in geometry domain in both TECEC and CCSSM.

Strategi Pengembangan Kemampuan Numerik Anak Usia Dini

Child at the age of zero to five years is seen as a golden age (golden age) to learn because in this age range their growth and development takes place quickly and rapidly. One is the Son has the informal mathematical knowledge is incredible. On this basis, educators need to provide the right stimuli that optimal child growth and development. This research is literature. The research results indicate that the child has an informal mathematical knowledge gained extraordinary and studied in the context of everyday life. It is important for children to understand the concept that teaches mathematics first is through the use of mathematical intuition contextual and then to mathematical symbols. Symbols are then abstracted by the children in his brain, this is called a visual representation (visual imagery). Mental imagery (mental imagery) as an object mentally for reflective thinking processes that will help and guide individuals to solve problems as well as helping individuals to reorganize (construct) a wide range of knowledge and reconstruct the concept of the problem. Therefore, context familiarity, ease of substance, the attractiveness of the object, and fun activities become a keyword in teaching mathematics in early childhood.

Effects of Tutorial Interventions in Mathematics and Attention for Low-Performing Preschool Children

ABSTRACTTwo intervention approaches designed to address the multifaceted academic and cognitive difficulties of low-income children who enter pre-K with very low math knowledge were tested in a randomized experiment. Blocking on classroom, children who met screening criteria were assigned to a Math + Attention condition in which the Pre-Kindergarten Mathematics Tutorial (PKMT) intervention was implemented (4 days/week for 24 weeks) in addition to 16 adaptive attention training sessions, a Math-Only condition using the PKMT intervention, or a business-as-usual condition. Five hundred eighteen children were assessed at pretest and posttest. There was a significant effect of the PKMT intervention on a broad measure of informal mathematical knowledge and a small but significant effect on a measure of numerical knowledge. Attention training was associated with small effects on attention, but did not provide additional benefit for mathematics. A main effect of state on math outcomes was associated with a stronger, numeracy-focused Tier 1 mathematics curriculum in one state. Findings are discussed with respect to increasing intensity of math-specific and domain-general interventions for young children at risk for mathematical learning difficulties.

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of non-trivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics – a book on Tarski’s geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37 % of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42 %. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge.

Effect of Perceptual Functions Level on the Development of Pre-mathematical Concepts in Czech Pre-school Children

The aspects affecting the academic success of students in mathematics include the level of informal mathematical knowledge and communication skills. Therefore this study analyses the influence of the level of perceptual (visual and auditory) functions on the development of pre-mathematical concepts in 71 Czech pre-school children. The survey was carried out using a standardised foreign test of early mathematical skills and a test of reading and writing risks. The results confirmed the tightness of the relationship between the two variables, which can be described as “medium” addiction.

Empowering educationally disadvantaged mathematics students through a strategies-based problem solving approach

A major impediment to problem solving in mathematics in the great majority of South African schools is that disadvantaged students from seriously impoverished learning environments are lacking in the necessary informal mathematical knowledge to develop their own strategies for solving non-routine problems. A randomized pretest–posttest control group design was used to empirically investigate the effectiveness of a strategies-based problem solving approach on the problem solving performance of 9th grade disadvantaged students. In this approach students receive explicit instruction on a wide repertoire of problem solving strategies. The results reported in this study show a significant improvement in problem solving performance when a strategies-based approach to problem solving was being implemented. Quantitative and qualitative analyses of the responses to the items showed how the treatment group students had internalized as part of their problem solving repertoire the strategies on which they had been explicitly instructed on. The findings of this study make a case for the adoption of this approach so that the gap between the student’s existing problem solving competence, and the cognitive demands of a problem solving task can be bridged.

Skolem, the Skolem ‘Paradox’ and Informal Mathematics

I discuss Skolem's own ideas on his ‘paradox’, some classical disputes between Skolemites and Antiskolemites, and the underlying notion of ‘informal mathematics’, from a point of view which I hope to be rather unusual. I argue that the Skolemite cannot maintain that from an absolute point of view everything is in fact denumerable; on the other hand, the Antiskolemite is left with the onus of explaining the notion of informal mathematical knowledge of the intended model of set theory. 1 conclude that one must take seriously the embodiment of the uncountable into countable languages, as a symptom of some specific features which characterize mathematical languages with respect to other languages.

Achievement in Mathematics in Grades 9 and 11 in Limpopo Province of South Africa: Introduction of a Problem-Based Approach

800 students in Grades 9 and 11 of schools in the Central Region of the Limpopo Province of South Africa completed the Study Orientation Questionnaire in Mathematics. Mean age in Grade 11 was 17.5 yr. (SD = 1.4) and in Grade 9 15.1 yr. (SD = 1.2). Intervention was aimed at teachers and students in this group. Teachers in the trained group received training in a problem-based approach to teaching and learning in mathematics and introduced these principles into their classes. Analysis of variance on the differences between post- and pretest scores of the six subscales and the marks in mathematics and English yielded no effects for grade, sex, or grade after 6 mo. Pearson correlations for students in Grade 11 were positive between study orientation and achievement in mathematics. Improving teachers' training and expertise, transforming disadvantaged learning environments, and developing necessary formal and informal mathematical knowledge seem both essential and difficult.

The Impact of Junior Kindergarten on Math Skills in Elementary School

Using data from the first cycle of the National Longitudinal Survey of Children and Youth, this study examines the impact of junior kindergarten on children's later skills in math, above and beyond regional differences and individual/ household factors. It was hypothesized that earlier schooling would better prepare children for first formal learning of arithmetic, resulting in a more developed knowledge-base for optimal performance at the middle and end of elementary school. We also hypothesized that junior kindergarten attendance would reduce the performance gap between children from economically disadvantaged and those from advantaged families. Our results suggest that earlier schooling did not provide the cognitive boost that would lead to better performance in arithmetic, especially for girls. These results are above and beyond a number of controls (sex, age, region, SES, family functioning, family configuration, education, and family size). Junior kindergarten attendance did not help bridge the performance gap based on economic advantage/disadvantage. Primary school children from poor families who attended junior kindergarten did not perform on a par with their middle-class peers. This study makes an important case for enrichment in the central conceptual structures related to intuitive and informal mathematical knowledge through games and activities potentially featured in the junior kindergarten curriculum.

SES, ethnic, and gender differences in young children's informal addition and subtraction: A clinical interview investigation

The main goal of this study was to examine possible socioeconomic status (SES) differences in 4- and 5-year-old children's informal mathematical knowledge. One hundred and two children, 32 from lower, 39 from middle, and 31 from upper SES families participated in the study. Each participant was given a clinical interview involving several addition and subtraction problems and a related representation task. With and without scaffolding, the children from the upper SES group showed higher rates of success on many problems than did the children from the lower and middle SES groups. At the same time, the results fail to reveal significant differences in the performance levels of children from lower and middle SES families. Furthermore, all of the SES groups used very similar strategies. Although the children from the upper SES group showed higher levels of performance, their underlying knowledge (strategies) differed little from that of children from the lower and middle SES groups. These results are consistent with the hypotheses that both common biological inheritance and a common environment rich in opportunities for mathematical learning insure the development of basic mathematical competence in virtually all children. The study also found almost no gender or ethnic differences in performance and underlying knowledge. The results suggest that evaluators and educators should employ flexible methods, like the clinical interview, to assess mathematical knowledge and that preschoolers can benefit from a rich mathematics education.

Babies overboard! The complexities of incorporation culturally relevant teaching into mathematics instruction

A collective case study guided by reconceptualized critical theory explored the teaching of four elementary teachers as they incorporated a model of culturally relevant teaching into their mathematics instruction at four predominantly Black schools in Bermuda. The study examined how teachers sought to (a) foster critical mathematical thinking and critical consciousness, (b) build on informal mathematical and cultural knowledge; and (c) utilize empowerment orientations toward students' culture and experience. The teachers embraced culturally relevant teaching but varied in their efforts to incorporate these perspectives into instruction. Four facets of their teaching are illuminated. These complexities included situations were teaching focused on sustaining empowered relationships;building on or to students' cultural knowledge; and fostering critical thinking as framed in the model, or personal definitions.

Fostering Parental Support for Children's Mathematical Development: An Intervention with Head Start Families

The first national education goal, school readiness, recognizes a need for young children to be better prepared for entry into elementary school. Many low-income children exhibit a pattern of underachievement in school mathematics. Research has revealed a developmental gap between low-income preschool children and their middle-class peers with respect to the extent of their numerical knowledge. Research has also found that many low-income children do not receive a broad base of support for mathematical development at home or in preschool. In each of two studies, we conducted a bi-generation (parent and child) mathematics intervention with Head Start families. The intervention was designed to enhance parental support for pre-kindergarten children's mathematical development. It was found that low-income parents were willing and able to support this area of their children's development once they were provided with the training to do so. The support that parents provided to their children through the intervention was clearly effective in enhancing the development of children's informal mathematical knowledge. Intervention children developed more extensive mathematical knowledge than a comparison group of low-income children. Thus, an important step toward achieving the school readiness goal can be taken by fostering low- income parents' support for young children's mathematical development.

Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals by Abstraction" program differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis.

Culturally Relevant Mathematics Teaching in a Mexican American Context

This article examines mathematics instruction and its intersection with culturally relevant teaching in an elementary/middle school in a Mexican American community. The findings are based on a collaborative-research and school-change project involving university researchers, teachers, and the school's principal. On the basis of ethnographic data and an interdisciplinary theoretical framework, we propose a three-part model of culturally relevant mathematics instruction. The 3 components are (a) building on students' informal mathematical knowledge and building on students' cultural and experiential knowledge, (b) developing tools of critical mathematical thinking and critical thinking about knowledge in general, and (c) orientations to students' culture and experience.

The Features of Informal Mathematical Knowledge in Childhood.

The Features of Informal Mathematical Knowledge in Childhood.

Connecting Formal and Informal Mathematical Knowledge

Connecting Formal and Informal Mathematical Knowledge