Silke Melanie Göbel

Peer-reviewed journal articles

• Kaufmann, L., von Aster, M., Göbel, S.M., & Klein, E. (2020). Developmental dyscalculia in adults: Current issues and open questions for future research. Lernen und Lernstörungen 9(2), 1-12. doi: 10/1024/2235-0977/a000294 Review
• Lin, C.-J., & Göbel, S.M. (2019). Arabic digits and spoken number words: Timing modulates the cross-modal numerical distance effect. Quarterly Journal of Experimental Psychology, 72(11), 2632-2646. doi:10.1177/1747021819854444
•  Göbel, S.M., McCrink, K., Fischer, M.H., & Shaki, S. (2018). Observation of directional storybook reading influences young children’s counting direction. Journal of Experimental Child Psychology 166, 49-66. doi: 10.1016/j.jecp.2017.08.001
• Moll, K., Göbel, S.M., & Snowling, M. J. (2015). Basic number processing in Children with Specific Learning Disorders: Co-morbidity of Reading and Mathematics Disorders. Child Neuropsychology 21(3), 399-417. doi:10.1080/09297049.2014.899570
• Göbel, S.M.,Watson, S.E., Lervåg, A., & Hulme, C. (2014). Children’s arithmetic development: it is number knowledge, not the approximate number sense, that counts. Psychological Science, 25(3), 789-798. doi:10.1177/0956797613516471

Textbook

• Gilmore, C., Göbel, S.M., & Inglis, M. (2018). An Introduction to Mathematical Cognition. International Texts in Developmental Psychology. Hove: Routledge

Book chapters

• Göbel, S.M. (2018). How Culturally Predominant Reading Direction and Number Word Construction Influence Numerical Cognition and Mathematics Performance. In Daniel B. Berch, David C. Geary and Kathleen Mann Koepke (Eds). Language and Culture in Mathematical Cognition, Vol 4, MCL, Burlington, UK: Academic Press. pp. 229-256
• Göbel, S.M. (2015). Reading difficulties and mathematics. In Dowker, A. & Cohen Kadosh, R. (Eds)..Oxford Handbook of Numerical Cognition, 680-704. Oxford: OUP. doi: 10.1093/oxfordhb/9780 199642342.013.044

• Sixtus, Elena; Lonnemann, Jan; Yan, Song; Li, Peng & Göbel, Silke Melanie (2026). Language-dependency of the left-digit effect in number line estimation and the role of number word inversion. Scientific Reports. 16(1). doi: 10.1038/s41598-026-54579-w. Full text in Research Archive Show summary

  Abstract Language plays a critical role in shaping cognitive processes. The present study focuses on the impact of language-dependent number representations, specifically examining the effects of number word inversion and the left-digit effect across different linguistic groups. Number word inversion is a phenomenon where in certain languages, such as in German, spoken two-digit numbers start with the unit preceding the decade (e.g., for 91, literally “one-and-ninety” rather than the Arabic numeral order “ninety-one”). Previous studies employing various numerical tasks indicate that the inversion effect—where the unit digit exerts a disproportionate impact on how the total magnitude of a two-digit number is perceived—influences children’s performance. This study addresses the inversion effect in adults by comparing German-speakers to English- and Mandarin-speakers in a number line estimation task. Linear regression slope analyses of estimated positions conducted across decades identified the inversion effect, demonstrating that German-speakers provided comparatively higher estimates for numbers with larger unit digits compared to the other language groups. Additionally, the left-digit effect, which conversely is characterised by the smaller impact of the unit digit on perceived number magnitudes, became evident in all three language groups. This left-digit effect is further modulated by membership of a specific language group. The structure of Mandarin number words is discussed as a significant factor contributing to the pronounced amplification of the effect. Our findings suggest that language plays a pivotal role in persistently shaping the cognitive processing of numbers.

• Göbel, Silke M.; Shiron, Veniamin; Tucker, Miriam & Bruin, Angela De (2026). Number comparison in bilinguals is not affected by their second language. Scientific Reports. 16(1). doi: 10.1038/s41598-026-52236-w. Full text in Research Archive Show summary

  Abstract Language background can show subtle effects on number comparison even with Arabic digits. Number pairs can be decade-unit compatible, i.e. the decision about number size is the same for unit and decade (e.g. 24 and 57, 2 < 5 and 4 < 7), or decade-unit incompatible (e.g. 29 and 57, 2 < 5 but 9 > 7). Incompatible pairs often lead to longer response times. This compatibility effect is larger in languages with number word inversion (e.g. German). We aimed to investigate the influence of language on the compatibility effect in bilinguals. In Experiment 1, we compared the compatibility effect in German monolinguals and German-English bilinguals. We found a significant compatibility effect, but no significant differences between the mono- and bilinguals. In Experiment 2, we manipulated language activation within participants by asking German-English bilinguals to describe scenes either in English or German before completing number comparison trials. There was a significant cost of switching languages, showing that participants did activate German and English. However, the size of the compatibility effect was not significantly influenced by participants activating German or English. Overall, we did not find an influence of a bilingual’s language on number comparison with Arabic digits.

• Göbel, Silke Melanie; Landerl, Karin & Lervåg, Arne Ola (2026). Longitudinal Predictors of Conceptual Understanding of Arithmetic Principles. Journal of Numerical Cognition. 12. doi: 10.5964/jnc.19523. Full text in Research Archive
• Murray, Ewan; Horner, Aidan J. & Göbel, Silke Melanie (2025). A Meta-analytic Review of the Effectiveness of Spacing and Retrieval Practice for Mathematics Learning. Educational Psychology Review. ISSN 1040-726X. 37(3). doi: 10.1007/s10648-025-10035-1. Full text in Research Archive Show summary

  Spaced retrieval practice harnesses two well-studied phenomena: the spacing effect, where spacing out practice over several sessions leads to a gain in retention compared to massed practice in one session; and the testing effect, where material that is tested is better retained than material that is restudied. This meta-analysis investigates if, and under what circumstances, spaced and retrieval practice can benefit mathematics learning. We found a robust small to medium effect of spaced versus massed practice overall (g = 0.28, 27 studies, 53 effect sizes). Those studies can be split into two subsets based on their experimental design, where material was either taught in isolation (10 studies, 27 effect sizes) or as part of a course (17 studies, 26 effect sizes). We found a larger, yet less robust, effect for the isolated learning (g = 0.43) than for course-embedded (g = 0.24). Our search also revealed seven studies, 32 effect sizes, which manipulated testing versus restudy. The weighted mean effect of testing versus restudy was g = 0.18. However, the 95% confidence interval crossed zero, suggesting the testing effect is not robust. Overall, our results suggest that spaced practice can improve mathematics learning for material in isolation and within a course. However, the effect may be smaller than in other domains. Additionally, the current literature does not provide conclusive evidence for a consistent effect of retrieval practice for mathematics learning, possibly due to the smaller number of studies available. Many decisions go into the design of a program of learning. What to learn and how to learn it are thought about in painstaking detail to maximize the chances of success for the learners. A critical question to ask when designing a learning program is when should revision of prior material take place? Precisely when a session takes place during a day is often out of the hands of instructional designers and up to timetablers, but once time has been allotted, there is often freedom to choose what specific questions to ask and when. When an instructor decides which questions to ask during the start of a lesson, during the main tasks or as homework, it is here that there is the opportunity to improve learning through the use of retrieval practice and spaced repetition. There has been increased interest in harnessing retrieval practice, actively retrieving information rather than restudying it, and spaced practice, spreading out practice over multiple sessions rather than a single session, to improve mathematics learning, both in research and in applied settings. Given this recent spike in interest, it is valuable to synthesize the current research and see if these techniques are as effective for mathematics learning as they are in other domains. The goal of this meta-analysis is to review and synthesize the current literature surrounding the spacing and testing effect in mathematics learning.

• Göbel, Silke M.; Inglis, Matthew & Bahnmueller, Julia (2025). The cognitive processing of numerical bases: a review of empirical evidence. Philosophical Transactions of the Royal Society of London. Biological Sciences. ISSN 0962-8436. 380(1937). doi: 10.1098/rstb.2024.0221. Full text in Research Archive Show summary

  We review the empirical research on numerical bases in the context of spoken and written numerical expressions, visual number symbols and the mappings between these expressions and symbols. Most of this research has been carried out with speakers of Indo-European languages, which have decimal systems. Children typically acquire spoken numerical expressions before they learn associated visual number symbols, but numerical expressions are language-specific. The structure of the numerical expressions influences number word learning, with evidence that an explicit base-10 structure in the spoken numerical expressions (i.e. base transparency ) is advantageous. Later, when visual base-10 number symbols are acquired, inconsistencies in the mapping between those and the numerical expressions affect numerical and mathematical learning. We also review research on base-20 numeration systems and on inversion in number words: when the order of constituents in the spoken number word is inversed in relation to their order in the visual number symbols, e.g. ‘fourteen’ versus ‘14’. In conclusion, we discuss evidence about how using number symbols with a non-decimal base affects the processing of numbers and make suggestions for future research. This article is part of the theme issue ‘A solid base for scaling up: the structure of numeration systems’.

• Simpson, Heather G.; Henderson, Lisa M. & Göbel, Silke Melanie (2025). An ERP study on multiplication and its relationship to phonological processing in children and adults. Psychological Research. ISSN 0340-0727. 89(1). doi: 10.1007/s00426-024-02036-6. Full text in Research Archive
• Mononen, Riikka; Korhonen, Johan; Hægeland, Karoline; Younesi, Matin; Göbel, Silke Melanie & Niemivirta, Markku (2025). Domain-specific and domain-general skills as predictors of arithmetic fluency development. Learning and Individual Differences. ISSN 1041-6080. 117. doi: 10.1016/j.lindif.2024.102585. Full text in Research Archive Show summary

  We investigated Norwegian children's (n = 262) development in arithmetic fluency from first to third grade. Children's arithmetic fluency was measured at four time points, domain-specific (i.e., symbolic magnitude processing and number sequences) and domain-general skills (i.e., working memory, rapid naming, non-verbal reasoning, and sustained attention) once in the first grade. Based on a series of growth mixture models, one developmental trajectory best described the data. Multigroup latent growth curve models showed that girls and boys developed similarly in their arithmetic fluency over time. Symbolic magnitude processing and number sequence skills predicted both initial level and growth in arithmetic fluency, and working memory predicted only initial level, similarly for boys and girls. Mother's education level predicted the initial level of arithmetic fluency for boys, and rapid naming predicted growth for girls. Our findings highlight the role of domain-specific skills in the development of arithmetic fluency.

• Goebel, Silke Melanie; Terry, Rebecca; Klein, Elise; Hymers, Mark & Kaufmann, Liane (2022). Impaired Arithmetic Fact Retrieval in an Adult with Developmental Dyscalculia: Evidence from Behavioral and Functional Brain Imaging Data. Brain Sciences. 12(6). doi: 10.3390/brainsci12060735. Full text in Research Archive Show summary

  Developmental dyscalculia (DD) is a developmental disorder characterized by arithmetic difficulties. Recently, it has been suggested that the neural networks supporting procedure-based calculation (e.g., in subtraction) and left-hemispheric verbal arithmetic fact retrieval (e.g., in multiplication) are partially distinct. Here we compared the neurofunctional correlates of subtraction and multiplication in a 19-year-old student (RM) with DD to 18 age-matched controls. Behaviorally, RM performed significantly worse than controls in multiplication, while subtraction was unaffected. Neurofunctional differences were most pronounced regarding multiplication: RM showed significantly stronger activation than controls not only in left angular gyrus but also in a fronto-parietal network (including left intraparietal sulcus and inferior frontal gyrus) typically activated during procedure-based calculation. Region-of-interest analyses indicated group differences in multiplication only, which, however, did not survive correction for multiple comparisons. Our results are consistent with dissociable and processing-specific, but not operation-specific neurofunctional networks. Procedure-based calculation is not only associated with subtraction but also with (untrained) multiplication facts. Only after rote learning, facts can be retrieved quasi automatically from memory. We suggest that this learning process and the associated shift in activation patterns has not fully occurred in RM, as reflected in her need to resort to procedure-based strategies to solve multiplication facts.

• Finke, Sabrina; Vogel, Stephan E.; Freudenthaler, H. Harald; Banfi, Chiara; Steiner, Anna F. & Kemény, Ferenc [Show all 8 contributors for this article] (2022). Developmental trajectories of symbolic magnitude and order processing and their relation with arithmetic development. Cognitive development. ISSN 0885-2014. 64. doi: 10.1016/j.cogdev.2022.101266. Full text in Research Archive Show summary

  Efficient processing of absolute magnitude and relative order of numbers is key for arithmetic development. This longitudinal study tested 1) whether there is a developmental shift in the contribution of symbolic magnitude and order processing to arithmetic between Grades 1 and 2, and 2) whether the development of symbolic numerical abilities is characterized by reciprocal predictive relations. In two independent samples (UK: N = 195, Austria: N = 161), order processing did not predict early developmental change in arithmetic, but emerged as a predictor in Grade 2. Symbolic magnitude processing in Grade 1 predicted subsequent developmental change in arithmetic, but not later on. Moreover, we observed cross-lagged relations between symbolic magnitude and order processing. Our findings confirm that the contributions of symbolic magnitude and order processing to arithmetic development are interactive and change across the first years of schooling. This may be driven by a developmental shift from procedural strategies to retrieval of arithmetic facts.

• Banfi, Chiara; Clayton, Francina J.; Steiner, Anna F.; Finke, Sabrina; Kemény, Ferenc & Landerl, Karin [Show all 7 contributors for this article] (2022). Transcoding counts: Longitudinal contribution of number writing to arithmetic in different languages. Journal of Experimental Child Psychology. ISSN 0022-0965. 223. doi: 10.1016/j.jecp.2022.105482. Full text in Research Archive Show summary

  Number writing involves transcoding from number words (e.g., “thirty-two”) to written digit strings (32) and is an important unique predictor of arithmetic. The existing longitudinal evidence about the relation between transcoding and arithmetic is mostly language specific. In languages with number word inversion (e.g., German), the order of tens and units is transposed in spoken number words compared with Arabic numbers. This makes transcoding more challenging than in languages without number word inversion (e.g., English). In the current study, we aimed to understand whether the contribution of number writing to the development of arithmetic is similar in languages with and without number word inversion. German-speaking children (n = 166) and English-speaking children (n = 201) were followed over the first 3 years of primary school. In a series of multiple linear regressions, we tested whether number writing of multi-digit numbers was a significant unique predictor of arithmetic after controlling for well-known non-numerical predictors (nonverbal reasoning and working memory) and numerical predictors (symbolic and nonsymbolic magnitude comparison). Number writing in Grade 1 predicted arithmetic in Grades 1, 2, and 3 over and above the other predictors. Crucially, number writing performance was of comparable importance for arithmetic development in German- and English-speaking children. Our findings extend previous evidence by showing that transcoding predicts the development of arithmetic skills during the first 3 years of primary school in languages with and without number word inversion.

• Finke, Sabrina; Kemény, Ferenc; Clayton, Francina J.; Banfi, Chiara; Steiner, Anna F. & Perchtold, Corinna M. [Show all 9 contributors for this article] (2021). Cross-Format Integration of Auditory Number Words and Visual-Arabic Digits: An ERP Study. Frontiers in Psychology. 12. doi: 10.3389/fpsyg.2021.765709. Full text in Research Archive Show summary

  Converting visual-Arabic digits to auditory number words and vice versa is seemingly effortless for adults. However, it is still unclear whether this process takes place automatically and whether accessing the underlying magnitude representation is necessary during this process. In two event-related potential (ERP) experiments, adults were presented with identical (e.g., “one” and 1) or non-identical (e.g., “one” and 9) number pairs, either unimodally (two visual-Arabic digits) or cross-format (an auditory number word and a visual-Arabic digit). In Experiment 1 (N=17), active task demands required numerical judgments, whereas this was not the case in Experiment 2 (N=19). We found pronounced early ERP markers of numerical identity unimodally in both experiments. In the cross-format conditions, however, we only observed late neural correlates of identity and only if the task required semantic number processing (Experiment 1). These findings suggest that unimodal pairs of digits are automatically integrated, whereas cross-format integration of numerical information occurs more slowly and involves semantic access.

• Finke, Sabrina; Banfi, Chiara; Freudenthaler, H. Harald; Steiner, Anna F.; Vogel, Stephan E. & Goebel, Silke Melanie [Show all 7 contributors for this article] (2021). Common and distinct predictors of nonsymbolic and symbolic ordinal number processing across the early primary school years. PLOS ONE. 16(10), p. 1–23. doi: 10.1371/journal.pone.0258847. Full text in Research Archive Show summary

  What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing.

• Bahnmueller, Julia; Cipora, Krzysztof; Goebel, Silke Melanie; Nuerk, Hans-Christoph & Soltanlou, Mojtaba (2021). Pick the smaller number: No influence of linguistic markedness on threedigit number processing. Journal of Numerical Cognition. 7(3), p. 295–307. doi: 10.5964/jnc.6057. Full text in Research Archive Show summary

  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 presented multi-digit Arabic numbers rather than to indicate the smaller number. In terms of linguistic markedness, this procedure includes the unmarked/base form in the task instruction (i.e., large). Here we evaluate distance and compatibility effects in a three-digit number comparison task observed in Bahnmueller et al. (2015, https://doi.org/10.3389/fpsyg.2015.01216) using a marked task instruction (i.e., ‘pick the smaller number’). Moreover, we aimed at clarifying whether the markedness of task instruction influences common numerical effects and especially componential processing as indexed by compatibility effects. We instructed German- and English-speaking adults (N = 52) to indicate the smaller number in a three-digit number comparison task as opposed to indicating the larger number in Bahnmueller et al. (2015). We replicated standard effects of distance and compatibility in the new pick the smaller number experiment. Moreover, when comparing our findings to Bahnmueller et al. (2015), numerical effects did not differ significantly between the two studies as indicated by both frequentist and Bayesian analysis. Taken together our data suggest that distance and compatibility effects alongside componential processing of multi-digit numbers are rather robust against variations of linguistic markedness of task instructions.

• Cipora, Krzysztof; Soltanlou, Mojtaba; Smaczny, Stefan; Goebel, Silke Melanie & Nuerk, Hans-Christoph (2021). Automatic place-value activation in magnitude-irrelevant parity judgement. Psychological Research. ISSN 0340-0727. 85(2), p. 777–792. doi: 10.1007/s00426-019-01268-1. Full text in Research Archive
• Steiner, Anna F.; Banfi, Chiara; Finke, Sabrina; Kemény, Ferenc; Clayton, Francina J. & Goebel, Silke Melanie [Show all 7 contributors for this article] (2021). Twenty-four or four-and-twenty: Language modulates cross-modal matching for multidigit numbers in children and adults. Journal of Experimental Child Psychology. ISSN 0022-0965. 202, p. 1–18. doi: 10.1016/j.jecp.2020.104970. Full text in Research Archive Show summary

  Does number–word structure have a long-lasting impact on transcoding? Contrary to English, German number words comprise decade–unit inversion (e.g., vierundzwanzig is literally translated as four-and-twenty). To investigate the mental representation of numbers, we tested the effect of visual and linguistic–morphological characteristics on the development of verbal–visual transcoding. In a longitudinal cross-linguistic design, response times (RTs) in a number-matching experiment were analyzed in Grade 2 (119 German-speaking and 179 English-speaking children) and in Grade 3 (131 German-speaking and 160 English-speaking children). To test for long-term effects, the same experiment was given to 38 German-speaking and 42 English-speaking adults. Participants needed to decide whether a spoken number matched a subsequent visual Arabic number. Systematic variation of digits in the nonmatching distractors allowed comparison of three different transcoding accounts (lexicalization, visual, and linguistic–morphological). German speakers were generally slower in rejecting inverted number distractors than English speakers. Across age groups, German speakers were more distracted by Arabic numbers that included the correct unit digit, whereas English speakers showed stronger distraction when the correct decade digit was included. These RT patterns reflect differences in number–word morphology. The individual cost of rejecting an inverted distractor (inversion effect) predicted arithmetic skills in German-speaking second-graders only. The moderate relationship between the efficiency to identify a matching number and arithmetic performance could be observed cross-linguistically in all age groups but was not significant in German-speaking adults. Thus, findings provide consistent evidence of a persistent impact of number–word structure on number processing, whereas the relationship with arithmetic performance was particularly pronounced in young children.

• Steiner, Anna F.; Finke, Sabrina; Clayton, Francina J.; Banfi, Chiara; Kemény, Ferenc & Goebel, Silke Melanie [Show all 7 contributors for this article] (2021). Language effects in early development of number writing and reading. Journal of Numerical Cognition. 7(3), p. 368–387. doi: 10.5964/jnc.6929. Full text in Research Archive Show summary

  Reading and writing multidigit numbers requires accurate switching between Arabic numbers and spoken number words. This is particularly challenging in languages with number-word inversion such as German (24 is pronounced as four-and-twenty), as reported by Zuber, Pixner, Moeller, and Nuerk (2009, https://doi.org/10.1016/j.jecp.2008.04.003). The current study aimed to replicate the qualitative error analysis by Zuber et al. and further extended their study: 1) A cross-linguistic (German, English) analysis enabled us to differentiate between language-dependent and more general transcoding challenges. 2) We investigated whether specific number structures influence accuracy rates. 3) To consider both transcoding directions (from Arabic numbers to number words and vice versa), we assessed performance for number reading in addition to number writing. 4) Our longitudinal design allowed us to investigate transcoding development between Grades 1 and 2. We assessed 170 German- and 264 English-speaking children. Children wrote and read the same set of 44 one-, two- and three-digit numbers, including the same number structures as Zuber et al. For German, we confirmed that a high amount of errors in number writing was inversion-related. For English, the percentage of inversion-related errors was very low. Accuracy rates were strongly related to number syntax. The impact of number structures was independent of transcoding direction or grade level and revealed cross-linguistic challenges of transcoding multidigit numbers. For instance, transcoding of three-digit numbers containing syntactic zeros (e.g., 109) was significantly more accurate than transcoding of items with lexical zeros (e.g., 190). Based on our findings, we suggest adaptations of current transcoding models.

• Clayton, Francina J.; Copper, Clare; Steiner, Anna F.; Banfi, Chiara; Finke, Sabrina & Landerl, Karin [Show all 7 contributors for this article] (2020). Two-digit number writing and arithmetic in Year 1 children: Does number word inversion matter? Cognitive development. ISSN 0885-2014. 56:100967, p. 1–14. doi: 10.1016/j.cogdev.2020.100967. Full text in Research Archive

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