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Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that: (1) analogical reasoning predicts students' abilities only in generalized arithmetic; (2) serial reasoning predicts students' abilities only in generalized arithmetic; (3) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (4) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed.

In this paper, we present a way of describing variation in young children's learning of elementary arithmetic within the number range 1-10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.

When a number sentence includes more than one operation, students are taught to follow the rules for the order of operations to get the correct result. In this context, brackets are used to determine the operations that should be calculated first. However, it seems that the written format of an arithmetical expression has an impact on the way students evaluate this expression. It also seems that a connection exists between this way of evaluation and an understanding of structure. Both issues are examined in this paper. A number of arithmetical expressions in a rational form were given to primary school students from Greece and Sweden. The collected findings strengthen our hypothesis that this rational form of the arithmetical expressions was of critical importance for the students' decision on how to evaluate these expressions. They temporarily put aside their knowledge about the rules for the order of operations. Instead, the way they evaluated the expressions indicates an implicit use of what we call in this paper "mental" brackets. It is very likely that the use of these "mental" brackets is closely connected with students' structure sense.

A key aim of mathematics teaching is for children to develop appropriate and efficient strategies for solving tasks. The analysis presented in this paper moves beyond the exploration of changes in the strategies that children employ to solve tasks and extends to observation and exploration of changes that occur when their overall solving approach remains seemingly stable. We present an analysis of data from two qualitative studies, each of which combined a microgenetic design with task-based interviews, to examine changes that occur in 5-6-year-old children's verbal reports when solving an additive task, and in 9-10-year-old children's verbal reports when solving fraction word problems. Children's verbal reports were analysed through the lens of the theory of shifts of attention. We found that phases of stability are underlain by dynamic changes in how the same strategy is communicated and conceptualised over a number of sessions and these changes appear to be accounted for by changes related to shifts in the object and structure of children's attention, i.e. "what" children attend to and "how," when reporting on their solving approach. The paper extends the theory by revealing and studying microqualities that underlie different learners' structure of attention during phases of stability in arithmetic tasks. The findings provide new, significant insights for understanding qualitative dynamics of change in learning. Sensitivity to differences and changes in learners' shifts of attention is essential for teachers to make sense of what learners experience and identify opportunities conducive to further learning.

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved.

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform--which was eventually itself replaced in the longer term--provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms "numeration units" (ones, tens, hundreds, thousands, etc.) and "powers-of-ten written in figures" (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention group n = 65 and control group n = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks.

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the undergraduate level attended by 12 participants during the fall of 2013. The goal of the intervention was to enhance participants' CAC in the case of Elementary Arithmetic Theorems by using the rationality construct derived from Habermas' work. Qualitative analyses of participants' discussions showed how the construct was gradually appropriated by them and contributed to their CAC. Trends emerging from quantitative comparisons with previous implementations of the same course provided some evidence regarding the effects of the intervention on the quality of participants' CAC performances at the end of the course. The method chosen for participants' assessment and their cultural background will be discussed as conditions for the success of the intervention.

The semiotic approach to mathematics education introduces the notion of "semiotic system" as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts.

In this paper, we focus on Finnish pre-service elementary teachers' (N = 269) and upper secondary students' (N = 1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: "We know that 498:6 = 83. How could you conclude from this relationship (without using long-division algorithm) what 491:6 = ? is?" This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners' reasoning strategies in particular play a central role when teachers try to improve learners' proficiency.

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of "unknown" interferes with the interpretation of the variable "in functional relation" and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16-19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16-17 years.

History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context. But there is a problem because genuine historical studies necessitate ever greater particularity whereas mathematics and philosophy require generality and abstraction. The paper offers some methodological reflections about these matters together with two case studies from nineteenth century history of arithmetic and integration theory, respectively, which try to strike a balance between the directly opposed requirements.

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of "division with decimals" in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to conceive multiplication and division as a system of operations.

This paper is based on a long-term didactic engineering about division problems (only in a numerical setting) at primary school. Situations and students' work are analyzed by means of a double theoretical framework: the theory of situations and the theory of conceptual fields (Vergnaud 1991). The analysis focuses mainly on classroom interactions and on the didactic memory from both the teacher perspective and the learner perspective: in particular, it not only investigates how didactic memory is managed by the teacher, but also how students recall past events or reread those events in a-didactic situations.

The problem of innumeracy in general and at the primary school level in particular is attributed to a structuralist design of instruction emphasizing an algorithmic approach to arithmetic. Offered is an alternative learning approach developing arithmetic as an informal context-bound activity tied to mental arithmetic and estimation. (MDH)

A key aim of mathematics teaching is for children to develop appropriate and efficient strategies for solving tasks. The analysis presented in this paper moves beyond the exploration of changes in the strategies that children employ to solve tasks and extends to observation and exploration of changes that occur when their overall solving approach remains seemingly stable. We present an analysis of data from two qualitative studies, each of which combined a microgenetic design with task-based interviews, to examine changes that occur in 5–6-year-old children's verbal reports when solving an additive task, and in 9–10-year-old children's verbal reports when solving fraction word problems. Children's verbal reports were analysed through the lens of the theory of shifts of attention. We found that phases of stability are underlain by dynamic changes in how the same strategy is communicated and conceptualised over a number of sessions and these changes appear to be accounted for by changes related to shifts in the object and structure of children's attention, i.e. what children attend to and how, when reporting on their solving approach. The paper extends the theory by revealing and studying microqualities that underlie different learners' structure of attention during phases of stability in arithmetic tasks. The findings provide new, significant insights for understanding qualitative dynamics of change in learning. Sensitivity to differences and changes in learners' shifts of attention is essential for teachers to make sense of what learners experience and identify opportunities conducive to further learning. [ABSTRACT FROM AUTHOR]

An attempt is made to establish a link between ordinary arithmetical situations and relevant mathematical concepts by the analyzing of complexity of concepts. (MP)

Empirical research has documented how children's early counting develops into an increasingly abstract process, and initial counting procedures are reified as children develop and use more sophisticated counting. In this development, the learning of different oral counting sequences that allow children to count in steps bigger than one is seen as an essential skill that supports children's mental calculation strategies. This paper proposes that the reification or refinement of the counting process that results to increased-in-sophistication use of counting is underlaid by the process of knowledge explicitation that the model of representational redescription postulates. The paper uses a case study to provide insight into the pathway that a 6-year-old child followed from learning how to verbally count in twos and tens to being able to use this knowledge for calculation purposes. The proposal is that knowledge of oral sequences is redescribed in more explicit and accessible formats before children are able to connect their knowledge of the verbal counting with the goal of using the sequence for calculation. The discussion presented here queries the notion of spontaneity as an inherent element of the theory and discusses the role that social interaction may play in supporting knowledge redescription. If it is the case that children's knowledge of oral counting sequences is redescribed into increasingly explicit formats before it can be applied for calculation, then children need to be provided early in their education with structured activities that trigger knowledge redescription and support the necessary connections between counting, number structure and calculation. [ABSTRACT FROM AUTHOR]

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved. [ABSTRACT FROM AUTHOR]

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention group n = 65 and control group n = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks. [ABSTRACT FROM AUTHOR]

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based surveys and interviews. We document what attributes of binary operation group theory students perceive as critical and what types of metaphors students use to convey these attributes. We found that many students treat superficial features as critical (such as element-operator-element formatting) and do not always perceive critical features as essential (such as the binary attribute). Further, these attributes were communicated across three metaphor categories: arithmetic-related, function-related, and organization-related. [ABSTRACT FROM AUTHOR]

At the start of mathematics education children are often presented with addition and subtraction problems in the form of pictures. They are asked to solve the problems by filling in corresponding number sentences. One type of problem concerns the representation of an increase or a decrease in a depicted amount. A decrease is, however, more difficult to represent in paper pictorially than an increase. Within the Cognitive Load Theory framework, we expected that commonly used decrease problems would be harder and slower to solve than increase problems because of higher intrinsic cognitive load. We also hypothesised that combining the pictorial information with auditory information would reduce this load and, as a result, would lead to improved performance. We further expected that these effects would be most prominent in children who score below average on a general mathematics test. We conducted an experiment with sixty children attending the first grade of primary school, who were divided into a higher and lower mathematics-achieving group. As expected, children performed worse on the decrease problems compared to the increase problems. Also, the combination of pictorial and auditory information reduced the accuracy lag of the decrease problems compared to the increase problems. With response time this effect occurred only in the group of lower mathematics achieving children. These results are in line with the cognitive load theory framework, but we also offer an alternative explanation regarding attention. [ABSTRACT FROM AUTHOR]

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students’ development of arithmetic skills. [ABSTRACT FROM AUTHOR]

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the undergraduate level attended by 12 participants during the fall of 2013. The goal of the intervention was to enhance participants' CAC in the case of Elementary Arithmetic Theorems by using the rationality construct derived from Habermas' work. Qualitative analyses of participants' discussions showed how the construct was gradually appropriated by them and contributed to their CAC. Trends emerging from quantitative comparisons with previous implementations of the same course provided some evidence regarding the effects of the intervention on the quality of participants' CAC performances at the end of the course. The method chosen for participants' assessment and their cultural background will be discussed as conditions for the success of the intervention. [ABSTRACT FROM AUTHOR]

In this paper, we present a cognitive analysis of the relationship between the argumentation process leading to the construction of a conjecture and its algebraic proof in solving Calendar Algebra problems. To solve this kind of problem, students encounter two sources of potential difficulties: the shift from using arithmetic in the argumentation to using algebra in the proof and the shift from an inductive argument towards a deductive proof. Thus, the aims of this article are to describe these cognitive difficulties and to show how students overcome them. Methodologically, we compare students' problem solving process corresponding to three problems presented in the first four lessons of a teaching experiment. The analysis and comparison between these three resolution processes is performed using Toulmin's model. [ABSTRACT FROM AUTHOR]

History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context. But there is a problem because genuine historical studies necessitate ever greater particularity whereas mathematics and philosophy require generality and abstraction. The paper offers some methodological reflections about these matters together with two case studies from nineteenth century history of arithmetic and integration theory, respectively, which try to strike a balance between the directly opposed requirements. [ABSTRACT FROM AUTHOR]

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. “+”, “=”) made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra. [ABSTRACT FROM AUTHOR]

The ability to handle proof is the focus of a number of well-documented complaints regarding students' difficulties in encountering degree-level mathematics. However, in addition to observing that proof is currently marginalised in the UK pre-university mathematics curriculum with a consequent skills deficit for the new undergraduate mathematics student, we need to look more closely at the nature of the gap between expert practice and the student experience in order to gain a full explanation. The paper presents a discussion of first-year undergraduate students' personal epistemologies of mathematics and mathematics learning with illustrative examples from 12 student interviews. Their perceptions of the mathematics community of practice and their own position in it with respect to its values, assumptions and norms support the view that undergraduate interactions with proof are more completely understood as a function of institutional practices which foreground particular epistemological frameworks while obscuring others. It is argued that enabling students to access the academic proof procedure in the transition from pre-university to undergraduate mathematics is a question of fostering an epistemic fluency which allows them to recognise and engage in the process of creating and validating mathematical knowledge. [ABSTRACT FROM AUTHOR]

This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly notable for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties. [ABSTRACT FROM AUTHOR]

The use of the algebraic method for solving word problems is a challenging topic for secondary school students. Students' difficulties are usually associated with extracting the problem's network of relationships between quantities and with formalizing these relationships into algebraic language in a problem model. Both sources can coexist and interact; thus, it is usually not possible to determine which source of difficulty is more relevant. In addition, there are specific errors, such as the error by multiple referents for the unknown, which are directly linked to the wording of the problem text, and in which the same two sources of error coexist. In this work, we present the results of an experiment conducted with 255 secondary school students assessing the effect of two common difficulties on the accuracy of problem models and on the rate of multiple referents for the unknown. The first difficulty is the use of algebraic language in the construction of the problem model; the second is the use of the same expression to designate different quantities within the problem text. We used a 2 × 2 between-between design, with one factor related to the symbolic language (algebraic or arithmetic) in which the problem model is constructed, and the other factor related to the actual language features of the text problem. Our results indicate that overall, the main source of difficulty for students is the use of algebraic language to formalize a problem model, representing a large effect size. [ABSTRACT FROM AUTHOR]

The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching–learning process in the classroom. [ABSTRACT FROM AUTHOR]

Mathematical word problem solving is influenced by various characteristics of the task and the person solving it. Yet, previous research has rarely related these characteristics to holistically answer which word problem requires which set of individual cognitive skills. In the present study, we conducted a secondary data analysis on a dataset of N = 1282 undergraduate students solving six mathematical word problems from the Programme for International Student Assessment (PISA). Previous results had indicated substantial variability in the contribution of individual cognitive skills to the correct solution of the different tasks. Here, we exploratively reanalyzed the data to investigate which task characteristics may account for this variability, considering verbal, arithmetic, spatial, and general reasoning skills simultaneously. Results indicate that verbal skills were the most consistent predictor of successful word problem solving in these tasks, arithmetic skills only predicted the correct solution of word problems containing calculations, spatial skills predicted solution rates in the presence of a visual representation, and general reasoning skills were more relevant in simpler problems that could be easily solved using heuristics. We discuss possible implications, emphasizing how word problems may differ with regard to the cognitive skills required to solve them correctly. [ABSTRACT FROM AUTHOR]

In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: proportional equivalence and unit equivalence. These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. Past research has largely overlooked the distinction between the two types of equivalence. However, this may provide an important foundation for central topics that build on equivalence, and a better understanding of these two types of equivalence may support a more flexible understanding of fractions. Last, we propose future directions for teaching equivalence in mathematics. [ABSTRACT FROM AUTHOR]

Eye tracking is an increasingly popular method in mathematics education. While the technology has greatly evolved in recent years, there is a debate about the specific benefits that eye tracking offers and about the kinds of insights it may allow. The aim of this review is to contribute to this discussion by providing a comprehensive overview of the use of eye tracking in mathematics education research. We reviewed 161 eye-tracking studies published between 1921 and 2018 to assess what domains and topics were addressed, how the method was used, and how eye movements were related to mathematical thinking and learning. The results show that most studies were in the domain of numbers and arithmetic, but that a large variety of other areas of mathematics education research was investigated as well. We identify a need to report more methodological details in eye-tracking studies and to be more critical about how to gather, analyze, and interpret eye-tracking data. In conclusion, eye tracking seemed particularly beneficial for studying processes rather than outcomes, for revealing mental representations, and for assessing subconscious aspects of mathematical thinking. [ABSTRACT FROM AUTHOR]

Most studies of children's attitudes to mathematics have dealt with children in second grade or later, and have suggested that attitudes deteriorate, and anxiety increases with age. The present study investigated attitudes to mathematics in 67 English and 49 Chinese children at the end of their first year of school. The participants were given Thomas and Dowker's (2000) Mathematics Attitude Questionnaire, which uses pictorial rating scales to assess primary school children's mathematics anxiety, liking for mathematics, unhappiness at poor performance in mathematics, and self-rating in mathematics. They were also given the British Abilities Scales Basic Number Skills test. Attitudes were generally positive, though not more so than previously found for older primary school children. The Chinese children performed better in the arithmetic test and also rated themselves higher than the English children, but did not differ in other attitudes. Self-rating in mathematics and lack of unhappiness at poor performance were associated with better performance in the English group. There were no significant relationships between attitudes and performance in the Chinese group. Implications of the findings are discussed. [ABSTRACT FROM AUTHOR]

Although students' invented strategies typically prove to be meaningful and effective in improving the students' mathematical understanding, much remains unexplored in the current literature. This study examined, through a teaching-scenario task, the nature of 80 preservice teachers' reasoning and responses to students' informal and formal strategies for whole number subtraction. This study also examined challenges reported by preservice teachers attempting to connect students' informal strategies to a traditional method. The broader implications of this study for the international community are discussed, and recommendations for teacher education programs are presented in accordance with the findings of the study. [ABSTRACT FROM AUTHOR]