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Thevenot C, Tazouti Y, Billard C, Dewi J, Fayol M. Acquisition of new arithmetic skills based on prior arithmetic skills: A cross-sectional study in primary school from grade 2 to grade 5. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2023; 93:727-741. [PMID: 36740227 DOI: 10.1111/bjep.12588] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Revised: 01/06/2023] [Accepted: 01/18/2023] [Indexed: 02/07/2023]
Abstract
BACKGROUND In several countries, children's math skills have been declining at an alarming rate in recent years and decades, and one of the explanations for this alarming situation is that children have difficulties in establishing the relations between arithmetical operations. AIM In order to address this question, our goal was to determine the predictive power of previously taught operations on newly taught ones above general cognitive skills and basic numerical skills. SAMPLES More than one hundred children in each school level from Grades 2 to 5 from various socio-cultural environments (N = 435, 229 girls) were tested. METHODS Children were assessed on their abilities to solve the four basic arithmetic operations. They were also tested on their general cognitive abilities, including working memory, executive functions (i.e., inhibition and flexibility), visual attention and language. Finally, their basic numerical skills were measured through a matching task between symbolic and nonsymbolic numerosity representations. Additions and subtractions were presented to children from Grade 2, multiplications from Grade 3 and divisions from Grade 4. RESULTS AND CONCLUSIONS We show that addition predicts subtraction and multiplication performance in all grades. Moreover, multiplication predicts division performance in both Grades 4 and 5. Finally, addition predicts division in Grade 4 but not in Grade 5 and subtraction and division are not related whatever the school grade. These results are examined considering the existing literature, and their implications in terms of instruction are discussed.
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Affiliation(s)
- Catherine Thevenot
- Institut de Psychologie, Batiment Géopolis, University of Lausanne, Lausanne, Switzerland
| | | | - Catherine Billard
- Centre de Référence sur les Troubles des Apprentissages, Bicêtre Hospital, Le Kremlin-Bicêtre, Paris, France
| | - Jasinta Dewi
- Institut de Psychologie, Batiment Géopolis, University of Lausanne, Lausanne, Switzerland
| | - Michel Fayol
- Université de Clermont Auvergne (LAPSCO, UMR 6024 UCA-CNRS), Clermont-Ferrand, France
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2
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Reconsidering conceptual knowledge: Heterogeneity of its components. J Exp Child Psychol 2023; 227:105587. [PMID: 36512922 DOI: 10.1016/j.jecp.2022.105587] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2021] [Revised: 10/24/2022] [Accepted: 10/26/2022] [Indexed: 12/14/2022]
Abstract
Cognitive arithmetic classically distinguishes procedural and conceptual knowledge as two determinants of the acquisition of flexible expertise. Whereas procedural knowledge relates to algorithmic routines, conceptual knowledge is defined as the knowledge of core principles, referred to as fundamental structures of arithmetic. To date, there is no consensus regarding their number, list, or even their definition, partly because they are difficult to measure. Recent findings suggest that among the most complex of these principles, some might not be "fundamental structures" but rather may articulate several components of conceptual knowledge, each specific to the arithmetic operation involved. Here, we argue that most of the arithmetic principles similarly may rather articulate several core concepts specific to the operation involved. Data were collected during a national mathematics contest based on an arithmetic game involving a large sample of 9- to 11-year-old students (N = 11,243; 53.1% boys) over several weeks. The purpose of the game was to solve complex arithmetic problems using five numbers and the four operations. A principal component analysis (PCA) was performed. The results show that both conceptual and procedural knowledge were used by children. Moreover, the PCA sorted conceptual and procedural knowledge together, with dimensions being defined by the operation rather than by the concept. This implies that "fundamental structures" rather regroup different concepts that are learned separately. This opens the way to reconsider the very nature of conceptual knowledge and has direct pedagogical implications.
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3
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Variables versus numbers: Effects of symbols and algebraic knowledge on students’ problem-solving strategies. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2022. [DOI: 10.1016/j.cedpsych.2022.102114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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4
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Chan JYC, Ottmar ER, Lee JE. Slow down to speed up: Longer pause time before solving problems relates to higher strategy efficiency. LEARNING AND INDIVIDUAL DIFFERENCES 2022. [DOI: 10.1016/j.lindif.2021.102109] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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5
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Lubin A, Kana L, Ducloy N, Delteil F, Perdry H, Mikaeloff Y. Do children with mathematical learning disabilities use the inversion principle to solve three-term arithmetic problems?: The impact of presentation mode. J Exp Child Psychol 2021; 216:105343. [PMID: 34968744 DOI: 10.1016/j.jecp.2021.105343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2021] [Revised: 10/28/2021] [Accepted: 11/22/2021] [Indexed: 10/19/2022]
Abstract
Numerical inversion is the ability to understand that addition is the opposite of subtraction and vice versa. Three-term arithmetic problems can be solved without calculation using this conceptual shortcut. To verify that this principle is used, inverse problems (a + b - b) can be compared with standard problems (a + b - c). If this principle is used, performance on inverse problems will be higher than performance on standard problems because no calculation is required. To our knowledge, this principle has not been previously studied in children with mathematical learning disabilities (MLD). Our objectives were (a) to study whether 10-year-olds with MLD are able to use this conceptual principle in three-term arithmetic problems and (b) to evaluate the impact of the presentation mode. A total of 64 children with or without MLD solved three-term arithmetic problems (inverse and standard) in two presentation modes (symbolic and picture). The results showed that even though children with MLD have difficulties in performing arithmetic problems, they can do so when the inverse problem is presented with pictures. The picture presentation mode allowed children with MLD to efficiently identify and use the conceptual inversion shortcut and thus to achieve a similar performance to that of typically developing children. These results provide interesting perspectives for the care of children with MLD.
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Affiliation(s)
- Amélie Lubin
- Normandie Univ, UNICAEN, Laboratoire de Psychologie Caen Normandie, F-14000 Caen, France.
| | - Laetitia Kana
- Institut de Psychologie, Université de Paris, F-75006 Paris, France
| | - Nelly Ducloy
- Child Neurological Rehabilitation Unit and Learning Disorders Reference Centre, Assistance Publique-Hôpitaux de Paris, Centre Hospitalo-Universitaire de Bicêtre, Pôle Neurosciences Tête et Cou, F-94270 Le Kremlin-Bicêtre, France
| | - Florence Delteil
- Child Neurological Rehabilitation Unit and Learning Disorders Reference Centre, Assistance Publique-Hôpitaux de Paris, Centre Hospitalo-Universitaire de Bicêtre, Pôle Neurosciences Tête et Cou, F-94270 Le Kremlin-Bicêtre, France; Centre de Recherche en Epidémiologie et Santé des Populations (CESP), INSERM U1018, UFR Médecine, Université Paris-Saclay, F-91190 Gif-sur-Yvette, France
| | - Hervé Perdry
- Centre de Recherche en Epidémiologie et Santé des Populations (CESP), INSERM U1018, UFR Médecine, Université Paris-Saclay, F-91190 Gif-sur-Yvette, France
| | - Yann Mikaeloff
- Centre de Recherche en Epidémiologie et Santé des Populations (CESP), INSERM U1018, UFR Médecine, Université Paris-Saclay, F-91190 Gif-sur-Yvette, France; Child and adolescent protection Centre, Assistance Publique-Hôpitaux de Paris, Centre Hospitalo-Universitaire de Bicêtre, Département médico-universitaire santé de l'enfant et de l'adolescent, F-94270 Le Kremlin-Bicêtre, France
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6
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Eaves J, Gilmore C, Attridge N. Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research. Trends Neurosci Educ 2021; 23:100152. [PMID: 34006361 DOI: 10.1016/j.tine.2021.100152] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Revised: 02/18/2021] [Accepted: 02/24/2021] [Indexed: 11/29/2022]
Abstract
Individuals use diverse strategies to solve mathematical problems, which can reflect their knowledge of arithmetic principles and predict mathematical expertise. For example, '6 + 38 - 35' can be solved via '38 - 35 = 3' and then '3 + 6 = 9', which is a shortcut-strategy derived from the associativity principle. The shortcut may be critical for understanding algebra, however approximately 50% of adults fail to use it. We review the research to consider why the associativity principle is challenging and highlight an important distinction between shortcut identification and execution. We also discuss how domain-specific skills and domain-general skills might play an important role in shortcut identification and execution, and provide an agenda for future research.
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Affiliation(s)
- Joanne Eaves
- University of Nottingham: University Park Campus, Psychology, Pharmacy, Life Sciences, East Drive, Nottingham, NG7 2RD, United Kingdom.
| | - Camilla Gilmore
- Centre for Mathematical Cognition, Loughborough University, LE11 3TU, United Kingdom
| | - Nina Attridge
- Department of Psychology, Portsmouth University, PO1 2UP, United Kingdom
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Wong TTY, Leung COY, Kwan KT. Multifaceted assessment of children's inversion understanding. J Exp Child Psychol 2021; 207:105121. [PMID: 33756277 DOI: 10.1016/j.jecp.2021.105121] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2020] [Revised: 12/03/2020] [Accepted: 01/21/2021] [Indexed: 11/19/2022]
Abstract
The current study was aimed at examining various theoretical issues concerning children's inversion understanding (i.e., its factor structure, development, and relation with mathematics achievement) using a multifaceted assessment. A sample of 110 fourth to sixth graders was evaluated in three different measures of inversion understanding: evaluation of examples, explicit recognition, and application of procedures. The participants were also evaluated on their mathematics achievement. A one-factor structure best explains inversion understanding involving different arithmetic operations. Grade-related improvements were observed in some facets of inversion understanding. Latent profile analysis using the three inversion measures revealed seven classes of children with different inversion profiles. Furthermore, classes with better inversion understanding also had higher mathematics achievers. The current findings provide evidence to support the multifaceted nature of inversion understanding, grade-related improvements in children's inversion understanding as well as the relation between inversion understanding and mathematics achievement.
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Affiliation(s)
- Terry Tin-Yau Wong
- Department of Psychology, The University of Hong Kong, Pok Fu Lam, Hong Kong.
| | - Chloe Oi-Ying Leung
- Department of Psychology, The University of Hong Kong, Pok Fu Lam, Hong Kong
| | - Kam-Tai Kwan
- Department of Psychology, The Education University of Hong Kong, Tai Po, New Territories, Hong Kong
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9
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Understanding arithmetic concepts: The role of domain-specific and domain-general skills. PLoS One 2018; 13:e0201724. [PMID: 30252852 PMCID: PMC6155447 DOI: 10.1371/journal.pone.0201724] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Accepted: 07/20/2018] [Indexed: 11/19/2022] Open
Abstract
A large body of research has identified cognitive skills associated with overall mathematics achievement, focusing primarily on identifying associates of procedural skills. Conceptual understanding, however, has received less attention, despite its importance for the development of mathematics proficiency. Consequently, we know little about the quantitative and domain-general skills associated with conceptual understanding. Here we investigated 8–10-year-old children’s conceptual understanding of arithmetic, as well as a wide range of basic quantitative skills, numerical representations and domain-general skills. We found that conceptual understanding was most strongly associated with performance on a number line task. This relationship was not explained by the use of particular strategies on the number line task, and may instead reflect children’s knowledge of the structure of the number system. Understanding the skills involved in conceptual learning is important to support efforts by educators to improve children’s conceptual understanding of mathematics.
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Dubé AK, Robinson KM. Children's understanding of multiplication and division: Insights from a pooled analysis of seven studies conducted across 7 years. BRITISH JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2017; 36:206-219. [PMID: 29134666 DOI: 10.1111/bjdp.12217] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2016] [Revised: 09/13/2017] [Indexed: 11/28/2022]
Abstract
Research suggests that children's conceptual understanding of multiplication and division is weak and that it remains poor well into the later elementary school years. Further, children's understanding of fundamental concepts such as inversion and associativity does not improve as they progress from grades 6 to 8. Instead, some children simply possess strong understanding while others do not. Other studies have identified an increase across these grades. The present investigation analyses data from seven studies of Grade 6 (n = 226), Grade 7 (n = 221), and Grade 8 (n = 216) children's three-term problem-solving (e.g., 3 × 24 ÷ 24 and 3 × 24 ÷ 6) and provides a unified account of multiplication and division understanding, one in which grade differences and individual variability coexist and are moderated by sex. Statement of contribution What is already known on this subject? Children's conceptual understanding of multiplication and division is weak and it is unclear whether it increases across the key grades of 6-8. Understanding of the inversion and associativity concepts is characterized by high individual variability, but grade and sex have never been found to be a contributing factor. What does this study add? A meta-analysis of seven data sets (n = 643) indicates that grade differences and individual variability coexist and are moderated by sex. Understanding increases across grade only for boys, but an equal number of boys and girls are in the top 10% of conceptual problem-solvers.
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Affiliation(s)
- Adam K Dubé
- McGill University and University of Regina, Regina, Saskatchewan, Canada
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11
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Robinson KM, Price JAB, Demyen B. Understanding arithmetic concepts: Does operation matter? J Exp Child Psychol 2017; 166:421-436. [PMID: 29049950 DOI: 10.1016/j.jecp.2017.09.003] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2016] [Revised: 08/24/2017] [Accepted: 09/01/2017] [Indexed: 11/26/2022]
Abstract
Most research on children's arithmetic concepts is based on (a) additive concepts and (b) a single concept leading to possible limitations in current understanding about how children's knowledge of arithmetic concepts develops. In this study, both additive and multiplicative versions of six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and associativity) were investigated in Grades 5, 6, and 7. The multiplicative versions of the concepts were more weakly understood. No grade-related differences were found in conceptual knowledge, but older children were more accurate problem solvers. Individual differences were examined through cluster analyses. All children had a solid understanding of identity and negation. Some children had a strong understanding of all the concepts, both additive and multiplicative; some had a good understanding of equivalence or commutativity; and others had a weak understanding of commutativity, equivalence, inversion, and associativity. Associativity was the most difficult concept for all clusters. Grade did not predict cluster membership. Overall, these results demonstrate the breadth of individual variability in conceptual knowledge of arithmetic as well as the complexity in how children's understanding of arithmetic concepts develops.
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Affiliation(s)
- Katherine M Robinson
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada.
| | - Jill A B Price
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
| | - Brendan Demyen
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
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12
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Fyfe ER, Evans JL, Matz LE, Hunt KM, Alibali MW. Relations between patterning skill and differing aspects of early mathematics knowledge. COGNITIVE DEVELOPMENT 2017; 44:1-11. [PMID: 36777545 PMCID: PMC9914987 DOI: 10.1016/j.cogdev.2017.07.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Abstract
Patterns are often considered central to early mathematics learning; yet, the empirical evidence linking early pattern knowledge to mathematics performance is sparse. In the current study, 36 children ranging in age from 5 to 13 years old (M = 9.1 years) completed a pattern extension task with three pattern types that varied in difficulty. They also completed three math tasks that tapped calculation skill and knowledge of concepts. Children were successful on the pattern extension task, though older children fared better than younger children, potentially due in part to their explanations that considered both dimensions of the pattern (shape and size). Importantly, success on the pattern extension task was related to mathematics performance. After controlling for age and verbal working memory, patterning skill predicted calculation skill; however, patterning skill was not associated with knowledge of concepts. Results suggest that patterning may play a key role in the development of some aspects of early mathematics knowledge.
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Affiliation(s)
- Emily R Fyfe
- Wisconsin Center for Education Research, University of Wisconsin-Madison
| | - Julia L Evans
- School of Behavioral and Brain Sciences, University of Texas at Dallas
| | | | - Kayla M Hunt
- Department of Psychology, University of Wisconsin-Madison
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McCrink K, Shafto P, Barth H. The relationship between non-symbolic multiplication and division in childhood. Q J Exp Psychol (Hove) 2017; 70:686-702. [PMID: 26880261 PMCID: PMC5228418 DOI: 10.1080/17470218.2016.1151060] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
Children without formal education in addition and subtraction are able to perform multi-step operations over an approximate number of objects. Further, their performance improves when solving approximate (but not exact) addition and subtraction problems that allow for inversion as a shortcut (e.g., a + b - b = a). The current study examines children's ability to perform multi-step operations, and the potential for an inversion benefit, for the operations of approximate, non-symbolic multiplication and division. Children were trained to compute a multiplication and division scaling factor (*2 or /2, *4 or /4), and were then tested on problems that combined two of these factors in a way that either allowed for an inversion shortcut (e.g., 8*4/4) or did not (e.g., 8*4/2). Children's performance was significantly better than chance for all scaling factors during training, and they successfully computed the outcomes of the multi-step testing problems. They did not exhibit a performance benefit for problems with the a*b/b structure, suggesting that they did not draw upon inversion reasoning as a logical shortcut to help them solve the multi-step test problems.
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Affiliation(s)
- Koleen McCrink
- a Psychology Department , Barnard College, Columbia University , New York , NY , USA
| | - Patrick Shafto
- b Department of Mathematics and Computer Science , Rutgers University , Newark , NJ , USA
| | - Hilary Barth
- c Psychology Department , Wesleyan University , Middletown , CT , USA
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Liu J, Zhang H, Chen C, Chen H, Cui J, Zhou X. The neural circuits for arithmetic principles. Neuroimage 2017; 147:432-446. [DOI: 10.1016/j.neuroimage.2016.12.035] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2016] [Revised: 12/10/2016] [Accepted: 12/13/2016] [Indexed: 10/20/2022] Open
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Robinson KM, Dubé AK, Beatch JA. Children's understanding of additive concepts. J Exp Child Psychol 2016; 156:16-28. [PMID: 28024177 DOI: 10.1016/j.jecp.2016.11.009] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2016] [Revised: 11/15/2016] [Accepted: 11/18/2016] [Indexed: 11/24/2022]
Abstract
Most research on children's arithmetic concepts is based on one concept at a time, limiting the conclusions that can be made about how children's conceptual knowledge of arithmetic develops. This study examined six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and addition and subtraction associativity) in Grades 3, 4, and 5. Identity (a-0=a) and negation (a-a=0) were well understood, followed by moderate understanding of commutativity (a+b=b+a) and inversion (a+b-b=a), with weak understanding of equivalence (a+b+c=a+[b+c]) and associativity (a+b-c=[b-c]+a). Understanding increased across grade only for commutativity and equivalence. Four clusters were found: The Weak Concept cluster understood only identity and negation; the Two-Term Concept cluster also understood commutativity; the Inversion Concept cluster understood identity, negation, and inversion; and the Strong Concept cluster had the strongest understanding of all of the concepts. Grade 3 students tended to be in the Weak and Inversion Concept clusters, Grade 4 students were equally likely to be in any of the clusters, and Grade 5 students were most likely to be in the Two-Term and Strong Concept clusters. The findings of this study highlight that conclusions about the development of arithmetic concepts are highly dependent on which concepts are being assessed and underscore the need for multiple concepts to be investigated at the same time.
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Affiliation(s)
- Katherine M Robinson
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada.
| | - Adam K Dubé
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
| | - Jacqueline-Ann Beatch
- Department of Psychology, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
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Robinson KM, Dubé AK, Beatch JA. Children's multiplication and division shortcuts: Increasing shortcut use depends on how the shortcuts are evaluated. LEARNING AND INDIVIDUAL DIFFERENCES 2016. [DOI: 10.1016/j.lindif.2016.06.014] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Baroody AJ, Purpura DJ, Eiland MD, Reid EE. Fostering First Graders’ Fluency With Basic Subtraction and Larger Addition Combinations Via Computer-Assisted Instruction. COGNITION AND INSTRUCTION 2014. [DOI: 10.1080/07370008.2014.887084] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Watchorn RPD, Bisanz J, Fast L, LeFevre JA, Skwarchuk SL, Smith-Chant BL. Development of Mathematical Knowledge in Young Children: Attentional Skill and the Use of Inversion. JOURNAL OF COGNITION AND DEVELOPMENT 2013. [DOI: 10.1080/15248372.2012.742899] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Robinson KM, Dubé AK. Children's additive concepts: Promoting understanding and the role of inhibition. LEARNING AND INDIVIDUAL DIFFERENCES 2013. [DOI: 10.1016/j.lindif.2012.07.016] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Gilmore CK, Bryant P. Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill. BRITISH JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2010. [DOI: 10.1348/026151007x236007] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Robinson KM, Dubé AK. Children’s understanding of addition and subtraction concepts. J Exp Child Psychol 2009; 103:532-45. [PMID: 19167015 DOI: 10.1016/j.jecp.2008.12.002] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2008] [Revised: 12/03/2008] [Accepted: 12/05/2008] [Indexed: 10/21/2022]
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25
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Canobi KH. Concept–procedure interactions in children’s addition and subtraction. J Exp Child Psychol 2009; 102:131-49. [PMID: 18809182 DOI: 10.1016/j.jecp.2008.07.008] [Citation(s) in RCA: 62] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2007] [Revised: 07/29/2008] [Accepted: 07/29/2008] [Indexed: 11/24/2022]
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