# Krzysztof CiporaLoughborough University | Lough · Centre for Mathematical Cognition

Krzysztof Cipora

PhD in Psychology

## About

85

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## Publications

Publications (85)

A short guidebook developed by UKRN Local Network Leads (LNLs) to enable new and current LNLs to understand their role, plan objectives and activities, and be aware of the many benefits of being an LNL.

Spatial-Numerical Associations (SNAs) refer to the demonstrations of spatial processing of numbers. The Mental Number Line (MNL) is a representation model describing numbers as aligning left-to-right (LR) was suggested to account for directional biases in participants’ responses during numerical tasks. One common behavioral demonstration of this is...

Although math anxiety has been extensively studied, its interplay with other emotional and attitude constructs is still unclear. The present dataset includes math anxiety and arithmetic performance alongside different types of anxiety (i.e., state, test, general anxiety and neuroticism) and attitudes towards math (i.e., math self-concept, math self...

Many studies have demonstrated spatial-numerical associations, but the debate about their origin is still ongoing. Some approaches consider cardinality representations in long-term memory, such as a Mental Number Line, while others suggest ordinality representations, for both numerical and non-numerical stimuli, originating in working or long-term...

A domain/field of research can often benefit from a consensus. Agreement regarding terminology and tasks used to measure specific constructs may be particularly beneficial. Our group authored such a consensus paper on measuring the Approximate Number System in young children (Krajcsi et al., 2024). In preparing that paper, we followed the procedure...

Two implicit generalizations are often made from group-level studies in cognitive experimental psychology and their common statistical analysis in the general linear model: (1) Group-level phenomena are assumed to be present in every participant with variations between participants being often treated as random error in data analyses; (2) phenomena...

The distance effect (comparing objects becomes easier with increasing differences in their magnitude) is observed in tasks ranging across domains, and its existence has been interpreted as evidence for analogue magnitude representation. Similarly, associations between response side and magnitude (faster left/right-sided responses to small/large obj...

Numbers are highly relevant in our everyday lives. Besides intentionally processing number magnitude when necessary, we often automatically process it even when not required. The SNARC (Spatial-Numerical Association of Response Codes; Dehaene et al., 1993) effect, describing faster left-/right-sided responses to smaller/larger numbers, respectively...

The approximate number system (ANS) is a hypothesized mechanism responsible for the representation and processing of numerical information in an imprecise fashion. According to the predominant theory, the ANS is essential in solving simple numerical tasks such as comparing which of two quantities is numerically larger, and some research has indicat...

Neuromodulation with transcranial direct current stimulation (tDCS) is an increasingly popular research tool to experimentally manipulate cortical areas and probe their causal involvements in behavior, but its replicability and regional specificity are not clear. This registered report investigated cathodal tDCS effects on spatial–numerical associa...

The approximate number system (ANS) is a hypothesized mechanism responsible for the representation and processing of numerical information in an imprecise fashion. According to the predominant theory, the ANS is essential in solving simple numerical tasks such as comparing which of two quantities is numerically larger, and some research has indicat...

Mathematics anxiety (MA), general and test anxieties affect mathematics performance. However, little is known about how different anxiety profiles (i.e. individual configurations of anxiety forms) influence the relationship between MA and mathematics performance in university students. To the best of our knowledge, studies that have categorized par...

Numerical cognition might be embodied, that is, grounded in bodily actions. This claim is supported by the observation that, potentially due to our shared biology, finger counting is prevalent among a variety of cultures. Differences in finger counting are apparent even within Western cultures. Relatively few indigenous cultures have been systemati...

The distance effect (comparing objects becomes easier with increasing difference in their magnitude) can be observed in tasks ranging across different domains and its existence has been interpreted as evidence for analog magnitude representation. Similarly, associations between response side and magnitude (faster left/right-sided responses to small...

In response to a point raised by Fischer et al. (2020), we discuss the theoretical implications of both the original Attentional SNARC (Att-SNARC) and its recent failed multi-laboratory replication. In our view, the theoretical importance of the original Att-SNARC can be summarized in two points: (1) there is a conceptual link between numbers and s...

Two implicit generalizations are often made from group-level studies in cognitive experimental psychology and their common statistical analysis in the general linear model: (1) Group-level phenomena are assumed to be present in every participant with variations between participants being often treated as random error in data analyses; (2) phenomena...

The Spatial-Numerical Association of Response Codes (SNARC) effect-i.e., faster responses to small numbers with the left compared to the right side and to large numbers with the right compared to the left side-suggests that numbers are associated with space. However, it remains unclear whether the SNARC effect evolves from a number's magnitude or t...

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiti...

The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively) is considered to be strong evidence for the link between numbers and space. Studies have shown considerable variation in this effect. Among the factors determining individual differences in the...

In everyday life, adults need to manipulate numbers in several ways. A large range of measures evaluating arithmetic fluency is currently used, differing in definitions and operationalizations of the underlying construct. The basic arithmetic operations include addition, subtraction, multiplication, and division. Here, we introduce an instrument fo...

Mathematics anxiety (MA) is negatively associated with mathematics performance. Although some aspects, such as mathematics self‐concept (M self‐concept), seem to modulate this association, the underlying mechanism is still unclear. In addition, the false gender stereotype that women are worse than men in mathematics can have a detrimental effect on...

Maths anxiety is common and refers to feelings of anxiety, fear and other negative emotions and thoughts in individuals when confronted with mathematical tasks or numerical information. Self-report measures of maths anxiety have been created, but the majority are in English and are not culturally relevant to all countries. This study aimed to trans...

In this paper, we discuss several largely undisputed claims about mathematics anxiety (MA) and propose where MA research should focus, including theoretical clarifications on what MA is and what constitutes its opposite pole; discussion of construct validity, specifically relations between self‐descriptive, neurophysiological, and cognitive measure...

Working memory (WM) is one of the most important cognitive functions that may play a role in the relation between math anxiety (MA) and math performance. The processing efficiency theory proposes that rumination and worrisome thoughts (induced by MA) result in less available WM resources (which are needed to solve math problems). At the same time,...

Psychological science could greatly benefit from research synthesis, especially when it comes to overcoming limitations of too small samples tested in single studies. However, initiatives aimed at sharing large, curated trial-level data from cognitive tasks remain scarce. To address this problem, we initiated a collective effort allowing to assembl...

The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster left/right side responses to small/large magnitude numbers, respectively) is considered as strong evidence for the link between numbers and space. The studies have shown considerable variation in this effect. Among the factors determining individual differences in the...

Despite variety of cultures, our shared biology and the universality of finger counting suggests that numbers are embodied. Another lines of research show that numerical cognition might be bound to what our bodies are able to do. Differences in finger counting are apparent even within Western cultures. Relatively few indigenous cultures have been s...

One of the most fundamental effects used to investigate number representations is the Spatial-Numerical Association of Response Codes (SNARC) effect showing that responses to small/large numbers are faster with the left/right hand, respectively. However, in recent years, it is hotly debated whether the SNARC effect is based upon cardinal representa...

Teachers are strong role models for their pupils, especially at the beginning of education. This also holds true for math: If teachers feel anxious about math, the consequences on the mathematical education of their pupils is detrimental. Previous studies have shown that (future) elementary school teachers have higher levels of math anxiety than mo...

In this paper we list ten fairly undisputed claims about Mathematics Anxiety (MA) and propose where MA research should focus on. The areas future MA research should focus on comprise (a) theoretical clarifications on what MA is, and what constitutes its opposite pole, (b) construct validity – specifically relations between self-descriptive, (neuro)...

Maths anxiety is common and refers to feelings of anxiety, fear and other negative emotions and thoughts in individuals when confronted with mathematical tasks or numerical information. Self-report measures of maths anxiety have been created, but the majority are in English and are not culturally relevant to all countries. This study aimed to trans...

Working memory (WM) is one of the most important cognitive functions that may play a role in the relation between math anxiety (MA) and math performance. The processing efficiency theory proposes that the rumination and worrisome thoughts (induced by MA) result in less available WM resources (which are needed to solve math problems). At the same ti...

This is a draft of the Editorial of a Special Issue "Direct and Conceptual Replication in Numerical Cognition".

Mathematics anxiety (MA) is negatively associated with mathematics performance. Although some aspects, such as mathematics self-concept (M-self-concept), seem to modulate this association, the underlying mechanism is still unclear. In addition, the false gender-stereotype according to which women are worse than men in mathematics, can have a detrim...

The SNARC (Spatial-Numerical Association of Response Codes) effect (i.e., a tendency to associate small/large magnitude numbers with the left/right hand side) is prevalent across the whole lifespan. Because the ability to relate numbers to space has been viewed as a cornerstone in the development of mathematical skills, the relationship between the...

Research on multi-digit number processing suggests that, in Arabic numerals, their place-value magnitude is automatically activated, whenever a magnitude-relevant task was employed. However, so far, it is unknown, whether place-value is also activated when the target task is magnitude-irrelevant. The current study examines this question using the p...

The symbolic number comparison task has been widely used to investigate the cognitive representation and underlying processes of multi-digit number processing. The standard procedure to establish numerical distance and compatibility effects in such number comparison paradigms usually entails asking participants to indicate the larger of two present...

A recent study based on the SPoARC effect (spatial position association response codes) showed that culture heavily shapes cognition and more specifically the way thought is organized; when Western adults are asked to keep in mind a sequence of colors, they mentally organize them from left to right, whereas right‐to‐left reading/writing adults spat...

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiti...

The symbolic number comparison task has been widely used to investigate the cognitive representation and underlying processes of multi-digit number processing. The standard procedure to establish numerical distance and compatibility effects in such number comparison paradigms usually entails asking participants to indicate the larger of two present...

Teachers are strong role models for their pupils, especially at the very beginning of education, and as such, pupils tend to share their teacher’s attitudes. This also holds true for math: If teachers feel anxious about math, the consequences on the mathematical education of their pupils is detrimental.Previous studies have shown that (future) elem...

Doing basic maths seems to be a pretty common thing. 2 + 2 equals 4, both in France and in China. 7 × 8 equals 56, both in the United States of America and in Germany. Although most of us use the same symbols to write down numbers (1, 2, 3, 4, …), we use very different words for these numbers simply because we speak different languages. In this art...

Numerous studies revealed effects of some linguistic properties like inversion or reading/writing direction on number processing. However, it remains more controversial, whether influences at a syntactic level, such as singular vs. plural form associated with certain numbers, can also influence magnitude and place-value processing and vice versa. I...

In response to a recent point raised by Fischer at al. (2020), we discuss the theoretical implications of both the original Attentional SNARC (Att-SNARC) and its recent failed multi-laboratory replication. In our view, the theoretical importance of the original Att-SNARC can be summarized in two points: (1) there is a conceptual link between number...

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relat...

The attentional spatial-numerical association of response codes (Att-SNARC) effect (Fischer, Castel, Dodd, & Pratt, 2003)—the finding that participants are quicker to detect left-side targets when the targets are preceded by small numbers and quicker to detect right-side targets when they are preceded by large numbers—has been used as evidence for...

Evidence from multiple studies conducted in the past few decades converges on the conclusion that numerical properties can be associated with specific directions in space. Such spatial–numerical associations (SNAs), as a signature of elementary number processing, seem to be a likely correlate of math skills. Nevertheless, almost three decades of re...

An important part of any math curriculum is solving word problems. Such problems are particularly difficult for many students. It has been shown that the difficulty of word problems arises from the characteristics of the problems themselves (e.g., arithmetic and linguistic characteristics) and the solver (e.g., mathematical and reading ability as w...

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize the analogue number magnitude representation. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Pre...

Evidence from multiple studies conducted in few past decades converge to the conclusion that numerical properties can be associated with specific directions in space. Such Spatial-Numerical Associations (SNAs) as a signature of elementary number processing seem to be a likely candidate for a correlate of math skill. Nevertheless, almost three decad...

Based on a theory of impulsive and reflective human behavior, we test the effects of transcranial direct current stimulation (tDCS) targeting either prefrontal or parietal cortex in either hemisphere. In a confirmatory registered report, cathodal tDCS is administered to conceptually reproduce tDCS modulations of implicit spatial-numerical associati...

In everyday situations, numbers appear in a linguistic context: we communicate and perceive numbers within sentences. However, languages differ in how they refer to numerosities. One such difference is grammatical number: the way singular and plural grammatical forms are used when referring to certain numbers. We know that differences in grammatica...

[This corrects the article DOI: 10.3389/fpsyg.2018.01081.].

The aim of cognitive psychology is to obtain insights into human cognition in general. For this purpose, group-studies are typically conducted on representative samples so that the results can be generalized to the population. Using this approach, individual differences in such group-level cognitive phenomena are typically neglected and not much is...

The SNARC (Spatial-Numerical Association of Response Codes) effect (i.e., a tendency to associate small / large magnitude numbers with left / right hand side), an instance of implicit, directional Spatial-Numerical Association (SNA), is prevalent across whole lifespan, including children and adolescents. However, there are considerable interindivid...

The Spatial–Numerical Association of Response Codes (SNARC) effect (i.e., faster reactions to small/large numbers on the left-/right-hand side) is usually observed along with the linguistic Markedness of Response Codes (MARC) effect—that is, faster left-/right-hand responses to odd/even numbers. The SNARC effect is one of the most thoroughly invest...

Tight cognitive links between space and number processing exist. Usually, Spatial-Numerical Associations (SNAs) are interpreted causally: spatial capabilities are a cornerstone of math skill. We question this seemingly ubiquitous assumption. After presenting SNA taxonomy, we show that only some SNAs correlate with math skill. These correlations are...

Battista et al. rightly point out the importance of the relation between spatial and geometric reasoning. They are also working towards more fine-grained analyses considering different aspects of both spatial and mathematical reasoning, and postulate that analyses should examine particular skills, rather than these very general constructs. The auth...

A strong link between bodily activity and number processing has been established in recent years. Although numerous observations indicate that adults use finger counting (FC) in various contexts of everyday life for different purposes, existing knowledge of FC routines and their use is still limited. In particular, it remains unknown how stable the...

A commentary being part of the book edited by K.S. Mix and M. Battista.

Undoubtedly, tight links between space and number processing exist. Usually, findings of Spatial-Numerical Associations (SNA) are interpreted causally, i.e., that spatial capabilities aid or are even a fundamental cornerstone of mathematical skill. In this book chapter, we question this seemingly ubiquitous assumption.To start with, there is no rob...

Research on multi-digit number processing suggests that, in Arabic numerals, the magnitude of a number is automatically processed based on the place-value system. The current study examined whether this automatic processing extends to magnitude-irrelevant parity processing by using the parity congruency effect. The parity status of a decade digit i...

Numerical categories such as parity, i.e., being odd or even, have frequently been shown to influence how particular numbers are processed. Mathematically, number parity is defined categorically. So far, cognitive, and psychological accounts have followed the mathematical definition and defined parity as a categorical psychological representation a...

The Abbreviated Math Anxiety Scale (AMAS) is one of the most popular instruments measuring math anxiety (MA). It has been validated across several linguistic and cultural contexts. In this study, we investigated the extent of administration method invariance of the AMAS by comparing results (average scores, reliabilities, factorial structure) obtai...

The Spatial Numerical Association of Response Codes (SNARC) effect describes a stimulus-response association of left with small magnitude and right with large magnitude. Usually, it is estimated by means of regression slopes, where the independent variable only has a limited number of levels. Inspection of the literature reveals that it is not diff...

In the field of numerical cognition it is often highlighted that the domain-specific systems, referred to as “Approximate Number System” (ANS), or “The Number Sense” (NS)1, constitute the basis for mathematical skills (Feigenson et al., 2004; Dehaene, 2011). However, recently, Leibovich et al. (2016) stressed the role of domain-general factors, esp...

Maths anxiety is considered to be one of the important predictors of maths achievements. Numerous studies indicate that a teacher’s maths anxiety as well as teaching methods may be one of the causes of maths anxiety in children. Data obtained from American college students indicate that students choosing majors in pre-service education are characte...

While mathematically impaired individuals have been shown to have deficits in all kinds of basic numerical representations, among them spatial-numerical associations, little is known about individuals with exceptionally high math expertise. They might have a more abstract magnitude representation or more flexible spatial associations, so that no au...

The mental number line metaphor describes how numbers are associated with space. These spatial-numerical associations (SNA) are subserved by parietal structures (mainly intraparietal sulcus [IPS] and posterior superior parietal lobule [PSPL]). Generally, it is assumed that this association is a basic cornerstone for arithmetic skills. In this revie...

Math anxiety has an important impact on mathematical development and performance. However, although math anxiety is supposed to be a transcultural trait, assessment instruments are scarce and are validated mainly for Western cultures so far. Therefore, we aimed at examining the transcultural generality of math anxiety by a thorough investigation of...