Arithmetical Thinking Research Articles

Revisiting the Relationship of Arithmetical Thinking and Letter-Symbolic Algebra

Revisiting the Relationship of Arithmetical Thinking and Letter-Symbolic Algebra

An Analysis of 3rd Graders’ Problem Solving Strategies on Early Algebra

The purpose of this study was to investigate the understanding of and the preparation for third graders’ learning of algebra, and to derive didactic implications for helping students’ transition from arithmetical thinking to algebraic thinking. Twenty third graders in elementary school were selected as subjects of this study from I elementary school in Seoul. Since the students have learned the four fundamental arithmetic operations on whole numbers, they were able to understand equations, additions, subtractions, multiplications, and divisions and perform them.
 Questionnaires were developed by adapting and reconstructing the problems for preparing algebraic learning, as proposed by Blanton et al. (2011) and Lim & Chang (2019), to be available to elementary school students. The students’ problem solving strategies were analyzed by means of the questionnaires. Implications for developing essential understandings of algebraic thinking and supporting elementary students’ transition from arithmetical thinking to algebraic thinking were identified through the results of the analysis.

Distinct and Convergent Beneficial Effects of Estrogen and Insulin on Cognitive Function in Healthy Young Men.

ContextSystematic investigations into the cognitive impact of estradiol and insulin in male individuals are sparse, and it is unclear whether the 2 hormones interact to benefit specific cognitive functions in humans.ObjectiveWe investigated the acute effect of estradiol and insulin and of their combined administration on divergent (creative) and convergent (arithmetical) thinking as well as short-term and working verbal memory in healthy young men.MethodsAccording to a 2 × 2 design, 2 groups of men (each n = 16) received a 3-day transdermal estradiol (100 µg/24 h) or placebo pretreatment and on 2 separate mornings were intranasally administered 160 IU regular human insulin and, respectively, placebo before completing a battery of cognitive tests; we also determined relevant blood parameters.ResultsEstrogen compared with placebo treatment induced a 3.5-fold increase in serum estradiol and suppressed serum testosterone concentrations by 70%. Estrogen in comparison to placebo improved creative performance, that is, verbal fluency and flexibility, but not arithmetical thinking, as well as verbal short-term memory, but not visuospatial memory. The combination of estrogen and insulin enhanced recognition discriminability at delayed verbal memory recall; insulin alone remained without effect.ConclusionEstrogen specifically enhances core aspects of creativity and verbal memory in young male individuals; delayed recognition memory benefits from the combined administration of estradiol and insulin. Our results indicate that insulin’s acute cognitive impact in young men is limited and not robustly potentiated by estradiol. Estradiol per se exerts a beneficial acute effect on creative and verbal performance in healthy young men.

초등학교 6학년의 근/원 일반화 문제 해결 과정 분석을 통한 대수적 일반화 가능성 탐구

Students who have learned arithmetic in elementary mathematics would learn algebra in middle schools in earnest. So, the transition from arithmetical thinking to algebraic thinking is an important success factor for algebra learning. Therefore, in this study, 64 sixth graders solved the problems of near generalization and far generalization concerning the generalization of some figure patterns. Subsequently we analyzed their problem solving. As a result of the analysis, several cases of students having difficulties in performing algebraic generalization were identified. Through the discussion of the research results, several pedagogical implications were drawn to support the transition from arithmetical thinking to algebraic thinking in elementary school students.

Realistic Mathematics Education Principles for Designing a Learning Sequence on Number Patterns

The number pattern is one of mathematics topics taught for junior high school students that relate between arithmetic and algebra domain. This topic bridges arithmetical and algebraic thinking. Therefore, the learning for this topic should be designed meaningfully. This research aims to design a learning sequence on the number patterns using principles of Realistic Mathematics Education (RME). To do this, we used design research method, particularly the preliminary design phase, with the following three steps. First, literature study was conducted to collect student difficulties in the learning of number patterns, relevant studies, and the theory of RME. Second, we observed Indonesian mathematics textbooks addressing the number patterns to see a learning sequence for this topic. Finally, we designed a learning sequence for the number patterns using the RME principles, particularly the reality principle, level principle, and intertwinement principle. The result of this research includes the learning sequence for the number patterns according to the RME principles, which consists of three activities: relationship between patterns and numbers; exploration of numbers; and generalization of number patterns. We conclude that the three principles of RME are fruitful for designing a meaningful learning sequence for the topic of number patterns.

초등학교 2학년의 초기 대수 관련 문제 해결 전략 분석

The purpose of this study was to investigate the preparation for learning algebra of 2nd graders and to derive some didactic implications of transition from arithmetical thinking to algebraic thinking. 100 2nd graders in elementary school were selected as subjects of this study from H, J, K, W elementary schools in Seoul, and B elementary school in Gyeonggi-do. These students were found to be less than 1st level of arithmetical thinking as a result of the preliminary test according to the arithmetical thinking level test developed by Lim & Chang(2017). Questionnaires were prepared by adapting and reconstructing four word-problems for preparing algebraic learning which are proposed to be available to 1st and 2nd graders in elementary school by Caldwell et al.(2011), and Caldwell et al.(2014). Using the questionnaires, the students’ problem solving strategies were analyzed. From the results of the analysis, some meaningful implications for helping early algebraic learning and the elementary students’ transition from arithmetical thinking to algebraic thinking were found.

수와 연산 영역에서 부진을 경험한 중학생의 산술적 사고 수준 변화 및 대수적 사고로의 이행에 관한 사례 연구

Arithmetic and algebra are considered to be the most basic and core contents in school mathematics. However, many students in elementary and middle school have difficulties in arithmetic and algebra learning, and immense studies have been carried out in the field of mathematics education to solve these problems. The present study is a follow-up study which was conducted following the previous study conducted on a student who had difficulties in number and operation domain. This study investigated the manner in which the level of arithmetical thinking changed and how the transition from arithmetical thinking to algebraic thinking occurred after that student spent a year studying mathematics in middle school. Based on the results of the analysis, we suggest several implications for teaching and learning of number and operation domain at elementary and middle school in terms of the development of arithmetical thinking.

Neuroscientific studies of mathematical thinking and learning: a critical look from a mathematics education viewpoint

In this commentary we take a critical look at the various studies being reported in this issue about the relationship between cognitive neuroscience and mathematics, from a mathematics education viewpoint. After a discussion of the individual contributions, which we have grouped into three categories—namely neuroscientific studies of (a) children’s numerical magnitude representation, (b) arithmetical thinking, and (c) more advanced mathematical thinking—and which nicely document the scientific progression being made within the domain of educational neuroscience applied to the domain of mathematics education during the last 5 years, we point to some general caveats that should be considered in future research.

Algebraic Models of Mental Number Axes: Part II

The paper presents a formal model of the system of number representations as a multiplicity of mental number axes with a hierarchical structure. The hierarchy is determined by the mind as it acquires successive types of mental number axes generated by virtue of some algebraic mechanisms. Three types of algebraic structures, responsible for functioning these mechanisms, are distinguished: BASAN-structures, CASAN-structures and CAPPAN-structures. A foundational order holds between these structures. CAPPAN-structures are derivative from CASAN-structures which are extensions of BASAN-structures. The constructed formal model unifies two competitive conceptions of cognitive arithmetic: namely, the conception of the mental number line and the conception of parallel individuation. The paper is the continuation of a paper entitled Representational structures of arithmetical thinking, in which rich empirical evidence supporting the model is presented. The main result achieved in the present paper may be philosophically interpreted as an attempt to formalize the Kantian conception of the pure idea of time, understood as the a priori form of human arithmetical thinking. In this way, our theory may be comprehended as a result of applying the hard method of logical reconstruction of fundamental epistemological categories.

Representational Structures of Arithmetical Thinking: Part I

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures.

Development of Arithmetical Thinking: Evaluation of Subject Matter Knowledge of Pre-Service Teachers in Order to Design the Appropriate Course

One of the key courses in the mathematics teacher education program in Israel is arithmetic, which engages in contents which these pre-service mathematics teachers (PMTs) will later teach at school. Teaching arithmetic involves knowledge about the essence of the concept of “number” and the development thereof, calculation methods and strategies. properties of operations on different sets of numbers, as well as the properties of the numbers themselves. Hence, the question arises: how to educate PMTs in order to supplement their mathematical knowledge with the required components? The present study explored the development of arithmetic thinking among pre-service teachers intending to teach mathematics at elementary school. This was done by matching the van Hiele theory of the development of geometric thinking to arithmetic. Analysis of findings obtained both in the present study and in many studies of geometry teaching indicates that this approach to considering the learners’ level of thinking development might lead to meaningful learning in arithmetic course for PMTs.

Arithmetical thinking in children attending special schools for the intellectually disabled

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children's own ways of operating, which implies a triadic teaching process.

Teachers' Mini-Case Studies of Children's Mathematics

This article reports findings from a five-day summer institute conducted with 27 practicing primary-grade teachers. The institute was part of a year-long professional development project that supported teachers' thinking about children's arithmetical understanding. From Quicktime movies that contained excerpts of six children's interviews, pairs of teachers developed mini-cases about the children's arithmetical understanding. At the end of the institute, the teachers presented their mini-case studies to the group. Teachers developed two types of hypotheses: hypotheses about children's thinking and hypotheses about what it means for children to know and do mathematics. After providing examples to clarify the two types of hypotheses, we address the challenges we faced in developing multimedia inquiry-based environments that facilitated teachers' thinking about children's arithmetical thinking.

Promising Research, Programs, and Projects: Using Technology to Foster Teachers’ Reflections about Children's Arithmetical Thinking

Twenty-eight primary-grade teachers, working in pairs, stared intently at the screens of their computers as they watched a CD-ROM containing QuickTime movie excerpts of students responding to mathematical tasks during an interview.

Affective Variables and Mathematics Education

to general feelings such as liking/disliking of mathematics, nor is it meant to exclude perceptions of the difficulty, usefulness, and appropriateness of mathematics as a school subject. There are several ways affective variables are related to mathematics learning. It is likely that a student who feels very positive about mathematics will achieve at a higher level than a student who has a negative attitude toward mathematics. It is also likely that a high achiever will enjoy mathematics more than a student who