# Lieven VerschaffelKU Leuven | ku leuven

Lieven Verschaffel

## About

596

Publications

175,252

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

16,634

Citations

## Publications

Publications (596)

Early detection of and relevant information on children’s mathematical difficulties is important to initiate targeted teaching and intervention. This study investigated the extent to which strategy use in single-digit addition provides additional predictive information about 61 grade one children’s (6-year-old) mathematical achievement 3 years late...

Research showed that the capacity of making simple estimations begins to develop already at the age of five, but little is known about the early development of this estimation capacity and the strategies that underly it. The current study longitudinally followed the estimation capacity and strategies of 332 children from first to second grade of pr...

This study examined 91 preservice mathematics teachers’ strategy repertoire that they referred to when solving one direct and one inverse proportion missing-value word problems. When encouraged to provide multiple solutions, the preservice teachers exhibited the ability to solve the two problems using more than one strategy. However, they used a si...

Not only children but also adolescents and adults encounter great difficulties in learning to reason proportionally. Despite these difficulties, research increasingly shows that proportional reasoning emerges early, before it is being instructed in school. There have however been very few attempts to stimulate this early emerging ability. The aim o...

In this article we review the research on the flexible or adaptive use of solution strategies in school mathematics, with a focus on the most recent work in the field. After a short introduction, we provide an overview of the various ways in which strategy flexibility has been conceptualized and investigated in the research literature. Then we revi...

The present study investigated the effect of the physical estimation environment and the size of the to-be-estimated (TBE) objects on the use of measurement units and length estimation performance. Following a between-subjects design, forty young adults were asked to estimate the length of nine pieces of ribbon presented in two different conditions...

While symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achieveme...

In 2016, we started a 6-year-longitudinal research project on the development of 4- to 9-year-olds’ competencies in four early core mathematical domains, i.e., (1) mathematical patterns and structures, (2) computational estimation, (3) proportional reasoning, and (4) probabilistic reasoning. These four competencies are conceived of as a combination...

When solving subtraction problems such as 83–46, children use the direct subtraction (DS) strategy (e.g., 83 – 40 = 43, 43 – 6 = 37) or the subtraction by addition (SBA) strategy (e.g., 46 + 4 = 50, 50 + 30 = 80, 80 + 3 = 83, so the answer is 4 + 30 + 3 = 37). This study is the first to use the choice/no-choice method to examine DS and SBA use in t...

Children start preschool with large individual differences in their early numerical abilities. Little is known about the importance of heterogeneous patterns that exist within these individual differences. A person-centered analytic approach might be helpful to unravel these patterns and the cognitive and environmental factors that are associated w...

Recent studies demonstrated that the adult-preschooler interaction during shared book reading (SBR) contributes to its effectiveness (Mol et al., 2008). The level of abstraction, or complexity, of the mathematical questions adults formulate during SBR serves as an indicator of the interaction quality. We aimed to investigate the chance of spontaneo...

For the education systems of Singapore and Spain, problem solving is the cornerstone of the primary mathematics curriculum. However, TIMSS results indicate that Singaporean students can solve more difficult word problems than Spanish students. This may be due, to some extent, to the way in which each country's curriculum is concretized through its...

This study examined adults’ frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA)....

The current paper presents an introduction to a special issue focusing on mathematical flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious, outcome of mathematics education. Mathematical flexibility involves the flexible, creative, meaningful, and innovative use of mathematical concepts, relations, rep...

The results of different international mathematics achievement assessments (e.g., TIMSS-19) show very important differences in the ability of Spanish and Singaporean students to solve PAVs. In this sense, higher levels of mathematical proficiency seem to be associated with the ability to solve more difficult PAVs (Mullis et al., 2020). While much o...

The success or failure of education systems in promoting the problem-solving skills of students 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 asses...

Research Findings: Shared-picture book reading can stimulate children’s mathematical development. Evidence of learning-supportive characteristics in picture books is limited in this domain. A first step is systematically analyzing the occurrence of domain-specific features in publicly available picture books. We analyzed the occurrence of general b...

Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional explana...

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study...

The present study longitudinally investigated proportional reasoning abilities in early elementary school before the start of its instruction. Three aims were put forward: (a) distinguishing the different developmental states in young children’s understanding of missing-value proportional situations, (b) investigating how children transition throug...

We investigated elementary school children's use of direct subtraction (DS) and subtraction by addition (SBA) when mentally solving multi-digit subtractions. Fourth- to sixth-grade children of varying mathematical achievement levels were offered subtractions in one choice condition (choice between DS or SBA) and two no-choice conditions (mandatory...

This commentary addresses the five contributions to Part IV, dealing with counting, additive reasoning, multiplicative reasoning, fractional reasoning, and proportional reasoning. The aim of the commentary is twofold. First, we point to elements and aspects of the authors’ work that we think are the most important contributions for pre-service and...

El propósito del estudio fue examinar si los niños preescolares se enfocan espontáneamente en estructuras matemáticas –patrones y clasificación– y si dicho enfoque se asocia con sus habilidades para trabajar patrones. La metodología fue cuantitativa de tipo descriptivo y comparativo. Participaron en el estudio 60 preescolares que fueron evaluados a...

Recent literature suggests proportional reasoning develops at an earlier age than traditionally assumed. It also indicates that in tasks where both additive and multiplicative reasoning are appropriate, some children have a preference for additive relations, while others have a preference for multiplicative relations. In this study we investigated...

Ellis et al. (2021, this issue; in the following abbreviated as EEA) conducted a conceptual replication and extension of a recent meta-analysis relating number line estimation to broader mathematical competence (Schneider et al., 2018; in the following: SEA). EEA pooled and analyzed data from seven new studies and compared the results to SEA’s find...

This study presents a patterning intervention for 5-year-olds, who were in their third year of preschool in Flanders (Belgium), with two key elements: A focus on the structure of patterns, and the inclusion of growing patterns (e.g., ABAABAAAB) in addition to repeating patterns (e.g., ABABAB). We evaluated the impact of this intervention on pattern...

This longitudinal cross-lagged panel study investigated the development of the structure of young children’s spontaneous number focusing tendencies and their longitudinal associations with numerical abilities and mathematics achievement over the course of a 3-year period, that is, in the second and third year of kindergarten and in first grade of p...

Research distinguishes three types of arithmetic: exact arithmetic, computational estimation and approximate arithmetic. Little is, however, known about the interrelationship among these three arithmetic skills and the general cognitive and early numeracy skills that underlie these arithmetic skills. The current study investigates this interrelatio...

The current study examined upper elementary school children’s frequent, efficient and adaptive use of direct subtraction ( DS ) and subtraction by addition ( SBA ) when mentally solving multi-digit subtractions, replicating and expanding previous research by Torbeyns et al. (2018). First, children were offered subtractions in two choice conditions...

Computational estimation is seen as an important mathematical competence. Little is known, however, about the mathematical skills that are predictive of early computational estimation development. The current study longitudinally followed a group of about 350 children at four time points: second (K2, 4-year-olds) and third grades of kindergarten (K...

The nature of the arithmetic word problems found in maths textbooks influences the way students develop their ability to solve them, as teachers use the books in their classes quite frequently. Given that students are better able to reason through and solve authentic problems that are contextualized in situations familiar to them, and that differen...

The present study aimed to analyze the direction of the associations between repeating patterning, growing patterning, and numerical ability. Participants were 410 children who were annually assessed on their repeating patterning, growing patterning, and numerical ability, at ages 4, 5, and 6 years (i.e., spring 2017, 2018, and 2019). A cross-lagge...

In this chapter, we review a long-standing and comprehensive line of research on the use of the subtraction by addition (SBA) strategy when solving subtractions, in which our own research center has played an important role. We document this phenomenon in a variety of subdomains of the elementary arithmetic curriculum, ranging from single-digit men...

Proportional reasoning is one part of mathematics that is particularly hard to apprehend for children. At the same time, research suggests that already at an early age children develop an ability to make sense of some proportional situations. Previous research has shown that language-be it language in general or language related to mathematics-play...

Selecting a large and diverse sample of 5-6-year-old preschool children (179 boys and 174 girls; M
age
= 70.03 months, SD
age
= 3.43), we aimed to extend previous findings on variability in children's home math environment (i.e., home math activities, parental expectations, and attitudes) and its association with children's mathematical skills. We...

Children’s spontaneous focusing on Arabic number symbols (SFONS) has been identified as a relevant component of their early mathematical development. This study investigated whether SFONS is a separate construct from spontaneous focusing on numerosity (SFON) and examined whether it is uniquely related to numerical abilities and mathematics achievem...

Young children’s early repeating patterning abilities are important foundations for their later mathematical development. Prior studies on young children’s repeating patterning abilities have been conducted exclusively in developed countries differing in economic, societal, and educational characteristics from developing countries. In this study, w...

The transition from informal to formal mathematics is an important episode in children’s mathematical development. The current study investigated how young children’s computational estimation performance and strategy use develops in this transitional period. The computational estimation performance of 350 children was assessed before the start of f...

Children’s spontaneous focusing on Arabic number symbols (SFONS) has been identified as a relevant component of their early mathematical development. This study investigated whether SFONS is a separate construct from spontaneous focusing on numerosity (SFON) and examined whether it is uniquely related to numerical abilities and mathematics achievem...

Children’s own spontaneous mathematical activities are crucial for their mathematical development. Mathematical thinking and learning does not only occur in explicitly mathematical situations, such as the classroom. Those children with higher tendencies to recognize and use mathematical aspects of their everyday surroundings, both within the classr...

Recently, responding with eye fixations in the number line estimation (NLE) task was introduced. To determine if this new mode of responding replicates typical mouse cursor NLE results, 33 adults completed a 0-1000 NLE task and responded with their eyes (eye condition) or the mouse (mouse condition), while their eyes were tracked. In the mouse cond...

Previous research demonstrated that some children inappropriately solve multiplicative missing-value word problems additively, while others inappropriately solve additive missing-value word problems multiplicatively. Besides lacking skills, children’s preference for additive or multiplicative relations has been shown to contribute to those errors....

Although a good rational number understanding is very important, many learners struggle to understand fractions. Recent research attributes many of these difficulties to the natural number bias – the tendency to apply natural number features in rational number tasks where this is inappropriate. Previous correlational dual process studies found evid...

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and de...

In this chapter we review and discuss a relatively new line of research in the domain of early mathematical development that addresses the dispositional rather than the ability aspect of children’s early mathematical competence, namely their tendencies to attend to and focus on various mathematical elements in their environment and its relation wit...

This paper provides a survey of the research on the relationship between teachers’ professional knowledge and their instructional quality in the domain of mathematics education. First, an overview is provided of the way in which mathematics professional knowledge has been conceptualized. Second, the relation between mathematics teachers’ profession...

Young children show large individual differences in their tendency to focus spontaneously on numerical aspects (e.g., numerosities or Arabic number symbols) of their everyday environment. The origins of these individual differences are unclear. Given the role of the home numeracy environment (HNE) in children’s early mathematical development and th...

We sketch the turbulent history of primary mathematics education in Belgium during the last (half) century. The outline starts with traditional mathematics in the period before and shortly after World War II, an approach that is often, but partly unjustly, labelled as ‘mechanistic’. Then we focus on the rise of New Math or ‘modern mathematics’ in t...

The home numeracy questionnaire (HNQ) developed by LeFevre et al. (2009) is a self-report instrument to assess the home numeracy of young children. In this questionnaire, young children’s parents have to indicate how often their child participates both in direct (e.g., counting down: 10, 9, 8, …) and indirect numeracy activities (e.g., playing boar...

Singaporean children are the best performers on international achievement tests in mathematics (i.e. the TIMSS). Their excellent results could be due at least partly to certain characteristics of the textbooks used there (Oates 2014). Therefore, these materials could be taken as a good benchmark to describe and evaluate aspects of the textbooks of...

Background:
Early patterning competence has recently been identified as an important precursor of mathematical development. Whereas the focus of this research has been on children's ability regarding repeating patterns, children might also differ in their spontaneous attention to patterns.
Aims:
The present study aimed to explore 4- to 5-year ol...

El objetivo del presente estudio fue examinar la influencia del nivel socioeconómico de los niños de jardín infantil en el desarrollo de sus habilidades numéricas tempranas y su enfoque espontáneo en el número (SFON), después de controlar por variables cognitivas de dominio general, como inteligencia y memoria de trabajo, y por edad. Al iniciar el...

This article encompasses a systematic review of the research on ICT-based learning environments for metacognitively oriented K-12 mathematics education. This review begins with a brief overview of the research on metacognition and mathematics education and on ICT and mathematics education. Based on a systematic screening of the databases Web of Sci...

Recent literature suggests that children develop proportional reasoning abilities at an earlier age than traditionally assumed. It also indicates that children's preference for additive and multiplicative relations plays a role in this development. In this study we investigated four-to nine-year olds' proportional reasoning abilities in missing-val...

The present study cross-sectionally investigated proportional reasoning abilities in 5-to 9-year-old children (n = 185) before they received instruction in proportional reasoning. This study addressed two important aspects of the development of proportional reasoning that remain unclear in the current literature: (1) the age range in which it devel...

In this study, we aimed to address two gaps in research on early mathematical patterning, namely the lack of attention (1) to growing patterns and (2) to the association between different aspects of patterning and numerical ability. Participants were 400 four-year olds from a wide range of socioeconomic backgrounds. Children's patterning and numeri...

For several decades computational estimation is seen as an important topic in primary mathematics education. While previous narrative literature reviews on computational estimation summarized the available research on computational estimation abilities in students and adults, the present systematic literature review focuses on the measurement, deve...

Previous research has shown that upper primary school children frequently erroneously solve additive word problems multiplicatively, while younger children frequently erroneously solve multiplicative word problems additively. It has been suggested that children's preference for additive or multiplicative relations explains these errors, besides the...

Activities with patterns (e.g., making a necklace with alternating blue and red pearls) are common in many preschool and kindergarten settings. These activities are assumed to play an important role in children’s mathematical development. Nevertheless, most research on early mathematical development focuses on numerical and arithmetical abilities,...

Many studies have investigated the association between children’s spontaneous attention to number and their early mathematical abilities. This work has presented number in a non-symbolic format by exclusively using numerosities as their stimuli. Therefore, little is known about children’s spontaneous attention for Arabic number symbols. We aimed to...

Mathematics-related affect comprises an individual’s attitudes, beliefs, emotions and motivations towards mathematics. These affective constructs have been widely studied in mathematics education and cognitive psychology and have been shown to be related to cognitive outcomes such as performance on a range of mathematical tasks. However, it is not...

The current study investigated developmental changes children's benchmark-based strategy use in number line estimation. Third and fifth graders solved a 0-1000 number line estimation task in one of three conditions. In the control condition only the origin and endpoint were specified, the midpoint condition included an additional benchmark at 50%,...

This study aims to look into the effect of L2 proficiency level on composing processes of L2 student writers in L2 writing. The research to date has produced conflicting evidence concerning the relationship between L2 proficiency and composing processes. A number of L2 studies found that L2 proficiency does not influence L2 writing performance and...

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symb...

The number line estimation task is widely used to investigate mathematical learning and development. The present meta-analysis statistically synthesized the extensive evidence on the correlation between number line estimation and broader mathematical competence. Averaged over 263 effect sizes with 10576 participants with sample mean ages from 4 to...

Numerical competencies acquired in preschool are foundational and predictive for children's later mathematical development. It remains to be determined whether there are gender differences in these early numerical competencies which could explain the often‐reported gender differences in later mathematics and STEM‐related abilities. Using a Bayesian...

More and more research suggests that proportional reasoning emerges already at a very early age in childhood. The present study aimed to investigate these abilities in four-to five-year-old children. A five-item proportional reasoning task involving discrete quantities was administered in 389 children. On average children could solve about one item...

Recent research suggests that individual differences in mathematics
achievement relate to basic number processing skills such as the ability to
process numerical magnitudes. A key issue in this recent field of research is
whether non-symbolic magnitude processing predicts mathematics achievement.
An alternative account posits that accessing non-sym...

We investigated the use of the subtraction by addition strategy, an important mental calculation strategy in children with different levels of mathematics achievement. In doing so we relied on Siegler’s cognitive psychological model of strategy change (Lemaire & Siegler, 1995), which defines strategy competencies in terms of four parameters (strate...

Recent research suggests that individual differences in mathematics achievement relate to basic number processing skills such as the ability to process numerical magnitudes. A key issue in this recent field of research is whether non-symbolic magnitude processing predicts mathematics achievement. An alternative account posits that accessing non-sym...

Research has shown that children have a strong tendency to exclude real world considerations when solving word problems. In this study, we investigated a novel way to try to change this tendency. We tested whether children would adapt their behavior when solving word problems in which realistic considerations are required (P-items) when these probl...

Previous studies have indicated that the presence of atypical endpoints (e.g., 1,639 and 2,897) on a number line has a negative effect on number line estimation (NLE) performance (Booth & Newton, 2012; Hurst et al., 2014). In the present study, we investigated whether this effect could be attributed to a disruption in the ease with which benchmarks...

Interactive whiteboard offers a high-potential innovative tool in mathematical educational environments, in which teachers’ modeling processes and students’ exploring activities can easily be executed. Nevertheless, these affordances are not self-evident. There is a gap between the potential claims of the tool and its actual use in the classrooms....

This research aims at exploring an effective set to stimulate mathematics understanding and learning in an Interactive Whiteboard (IWB) environment. IWB affordances appear to be best used when mathematical tasks engage students in mathematical reasoning and when all students are involved in the discussion. The intent of this research project was to...

This chapter focuses on the neuro-cognitive, cognitive and developmental analyses of whole number arithmetic (WNA) learning. It comprises five sections. The first section provides an overview of the working group discussion. Section 7.2 reviews neuro-cognitive perspectives of learning WNA but looks beyond these to explain the transcoding of numeral...

This chapter provides an overview of the ICME 23 Study panel on special needs in research and instruction in whole number arithmetic. It starts with a general introduction by Verschaffel about the state of affairs in and the major issues and challenges for research and educational practice in the field of mathematical learning difficulties (MLD). A...

The focus of this paper is on two definite areas: designing of powerful learning environments, and graphical representation of music. With respect to the former, technology-enhanced learning environments (TELEs) have revealed as powerful tools for fostering students’ learning (Azevedo & Cromley, 2004). In this respect, scaffolding (Collins, Brown,...

This study aimed at analyzing the development of Ecuadorian children’s early numerical abilities during the Kindergarten year in relation to their SES, the quality of their early mathematics education, and the attended school type. 179 Ecuadorian Kindergartners (18 classrooms, 6 classrooms per school type) were offered a standards-based early numer...

Young children’s spontaneous focusing on numerosity (SFON) predicts their later mathematical competencies. In this study we investigated the development of SFON in Ecuadorian kindergartners as well as the contribution of early numerical abilities and the quality of mathematics education to this development. The participants were 100 kindergartners...

Children’s Spontaneous Focusing On Numerosity (SFON) predicts later mathematics performance. This association is assumed to rely on children’s self-initiated practice in number recognition during everyday activities, which would enhance their further mathematical development. Consequently, SFON in experimental tasks should be associated with SFON d...

Students have a great deal of difficulties learning about rational number concepts, as they are confounded by misapplying reasoning about natural numbers to fractions and decimals, referred to as a natural number bias. For example, students often think that the number of digits of a decimal, or the size of the component numbers of fractions, is eno...

This study analyses whether the primary school mathematics textbooks from two Spanish publishers show a varied instructional diet of addition and multiplication problems at different levels of complexity. To do so, it analyses the problems in all the primary grades by the publishers Santillana and SM according to two levels of complexity: (a) proce...

In this study we longitudinally followed 201 upper elementary school learners in the crucial years of acquiring rational number understanding. Using latent transition analysis we investigated their conceptual change from an initial natural number based concept of a rational number towards a mathematically more correct one by characterizing the vari...

We investigated the effectiveness of the Building Blocks program for enhancing Ecuadorian kindergart-ners' early numerical abilities and spontaneous focusing on numerosity (SFON), after controlling for working memory, intelligence, age, and SES. Following a pretest-intervention-posttest design, 18 classes comprising 355 children from varied SES bac...

Dieser Tagungsband dokumentiert die Ergebnisse der Jahrestagung des Arbeitskreises Grundschule in der Gesellschaft für Didaktik der Mathematik (GDM), die in diesem Jahr wieder in Bad Salzdetfurth stattfand. Vom 09. bis 11. November 2018 widmete sich der Arbeitskreis dem Thema „Inhalte im Fokus – Mathematische Strategien entwickeln“. Mathematische B...

We aimed to investigate upper elementary children's strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross‐sectional sample of 119 Dutch and 122 Flemish fourth to sixth graders solved a varied set of multidigit division problems. With latent class analysis, three distinct str...

Young children’s spontaneous focusing on numerosity (SFON) as measured by experimental tasks is related to their mathematics achievement. This association is hypothetically explained by children’s self-initiated practice in number recognition during everyday activities. As such, experimentally measured SFON should be associated with SFON exhibited...