Patterns of knowledge in children's addition

Reconsidering conceptual knowledge: Heterogeneity of its components

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.

Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research

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.

Understanding arithmetic concepts: Does operation matter?

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.

Children's understanding of additive concepts

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.

Cognition-emotion interactions: patterns of change and implications for math problem solving

Surprisingly little is known about whether relationships between cognitive and emotional states remain stable or change over time, or how different patterns of stability and/or change in the relationships affect problem solving abilities. Nevertheless, cross-sectional studies show that anxiety/worry may reduce working memory (WM) resources, and the ability to minimize the effects anxiety/worry is higher in individuals with greater WM capacity. To investigate the patterns of stability and/or change in cognition-emotion relations over time and their implications for problem solving, 126 14-year-olds’ algebraic WM and worry levels were assessed twice in a single day before completing an algebraic math problem solving test. We used latent transition analysis to identify stability/change in cognition-emotion relations, which yielded a six subgroup solution. Subgroups varied in WM capacity, worry, and stability/change relationships. Among the subgroups, we identified a high WM/low worry subgroup that remained stable over time and a high WM/high worry, and a moderate WM/low worry subgroup that changed to low WM subgroups over time. Patterns of stability/change in subgroup membership predicted algebraic test results. The stable high WM/low worry subgroup performed best and the low WM capacity-high worry “unstable across time” subgroup performed worst. The findings highlight the importance of assessing variations in cognition-emotion relationships over time (rather than assessing cognition or emotion states alone) to account for differences in problem solving abilities.

Spontaneously spotting and applying shortcuts in arithmetic-a primary school perspective on expertise

One crucial feature of expertise is the ability to spontaneously recognize where and when knowledge can be applied to simplify task processing. Mental arithmetic is one domain in which people should start to develop such expert knowledge in primary school by integrating conceptual knowledge about mathematical principles and procedural knowledge about shortcuts. If successful, knowledge integration should lead to transfer between procedurally different shortcuts that are based on the same mathematical principle and therefore likely are both associated to the respective conceptual knowledge. Taking commutativity principle as a model case, we tested this conjecture in two experiments with primary school children. In Experiment 1, we obtained eye tracking data suggesting that students indeed engaged in search processes when confronted with mental arithmetic problems to which a formerly feasible shortcut no longer applied. In Experiment 2, children who were first provided material allowing for one commutativity-based shortcut later profited from material allowing for a different shortcut based on the same principle. This was not the case for a control group, who had first worked on material that allowed for a shortcut not based on commutativity. The results suggest that spontaneous shortcut usage triggers knowledge about different shortcuts based on the same principle. This is in line with the notion of adaptive expertise linking conceptual and procedural knowledge.

Young children's use of derived fact strategies for addition and subtraction

Forty-four children between 6;0 and 7;11 took part in a study of derived fact strategy use. They were assigned to addition and subtraction levels on the basis of calculation pretests. They were then given Dowker's (1998) test of derived fact strategies in addition, involving strategies based on the Identity, Commutativity, Addend +1, Addend −1, and addition/subtraction Inverse principles; and test of derived fact strategies in subtraction, involving strategies based on the Identity, Minuend +1, Minuend −1, Subtrahend +1, Subtrahend −1, Complement and addition/subtraction Inverse principles. The exact arithmetic problems given varied according to the child's previously assessed calculation level and were selected to be just a little too difficult for the child to solve unaided. Children were given the answer to a problem and then asked to solve another problem that could be solved quickly by using this answer, together with the principle being assessed. The children also took the WISC Arithmetic subtest. Strategies differed greatly in difficulty, with Identity being the easiest, and the Inverse and Complement principles being most difficult. The Subtrahend +1 and Subtrahend −1 problems often elicited incorrect strategies based on an overextension of the principles of addition to subtraction. It was concluded that children may have difficulty with understanding and applying the relationships between addition and subtraction. Derived fact strategy use was significantly related to both calculation level and to WISC Arithmetic scaled score.

Spontaneous usage of different shortcuts based on the commutativity principle

Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied – even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paper-and-pencil-based as well as an eyetracking-based approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved three-addends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativity-based shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies.

Addressing the struggle to link form and understanding in fractions instruction

BACKGROUND

Although making explicit links between procedures and concepts during instruction in mathematics is important, it is still unclear the precise moments during instruction when such links are best made.

AIMS

The objective was to test the effectiveness of a 3-week classroom intervention on the fractions knowledge of grade 5/6 students. The instruction was based on a theory that specifies three sites during the learning process where concepts and symbols can be connected (Hiebert, 1984): symbol interpretation, procedural execution, and solution evaluation. Sample. Seventy students from one grade 5/6 split and two grade 6 classrooms in two public elementary schools participated.

METHOD

The students were randomly assigned to treatment and control. The treatment (Sites group) received instruction that incorporated specific connections between fractions concepts and procedures at each of the three sites specified by the Sites theory. Before and after the intervention, the students' knowledge of concepts and procedures was assessed, and a random subsample of 30 students from both conditions were individually interviewed to measure their ability to make specific connections between concepts and symbols at each of the three sites.

RESULTS

While all students gained procedural skill (p < .001), the students in the Sites condition acquired significantly more knowledge of concepts than the control group (p < .01) and were also better able to connect fractions symbols to conceptual referents (p < .025).

CONCLUSIONS

The current study contributes to the literature because it describes when it might be important to link concepts and procedures during fractions instruction.

Implicit learning of arithmetic regularities is facilitated by proximal contrast

Natural number arithmetic is a simple, powerful and important symbolic system. Despite intense focus on learning in cognitive development and educational research many adults have weak knowledge of the system. In current study participants learn arithmetic principles via an implicit learning paradigm. Participants learn not by solving arithmetic equations, but through viewing and evaluating example equations, similar to the implicit learning of artificial grammars. We expand this to the symbolic arithmetic system. Specifically we find that exposure to principle-inconsistent examples facilitates the acquisition of arithmetic principle knowledge if the equations are presented to the learning in a temporally proximate fashion. The results expand on research of the implicit learning of regularities and suggest that contrasting cases, show to facilitate explicit arithmetic learning, is also relevant to implicit learning of arithmetic.

Five- to 7-year-olds' finger gnosia and calculation abilities

The research examined the relationship between 65 5- to 7-year-olds’ finger gnosia, visuo-spatial working memory, and finger-use in solving single-digit addition problems. Their non-verbal IQ and basic reaction time were also assessed. Previous research has found significant changes in children’s representational abilities between 5 and 7 years. One aim of the research was to determine whether changes in finger representational abilities (finger gnosia) occur across these ages and whether they are associated with finger-use in computation. A second aim was to determine whether visuo-spatial working memory is associated with finger gnosia and computation abilities. We used latent class profile analysis to identify patterns of similarities and differences in finger gnosia and computation/finger-use abilities. The analysis yielded four finger gnosia subgroups that differed in finger representation ability. It also yielded four finger/computation subgroups that differed in the relationship between finger-use and computation success. Analysis revealed associations between computation finger-use/success subgroups, finger gnosia subgroups, and visuo-spatial working memory. A multinomial logistic regression analysis showed that finger gnosia subgroup membership and visuo-spatial working memory uniquely contribute to a model predicting finger-use in computation group membership. The results show that finger gnosia abilities change in the early school years, and that these changes are associated with the ability to use fingers to aid computation.

Individual differences in children's understanding of inversion and arithmetical skill

Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills.

SAMPLE

A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6-7 and 8-9 years).

METHODS

Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non-canonical forms was measured. Tests of general arithmetic ability were also administered.

RESULTS

Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the 'high ability' and 'low ability' groups showed conceptual understanding that was in-line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill.

CONCLUSIONS

The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem-solving skills.

Knowledge of counting principles: how relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.

The degree of abstraction in solving addition and subtraction problems

In this study, the incidence of the degree of abstraction in solving addition and subtraction problems with the unknown in the first term and in the result is analyzed. Ninety-six students from first grade to fourth grade in Primary Education (24 students per grade) solved arithmetic problems with objects, drawings, algorithms, and verbal problems. The participants were interviewed individually and all sessions were video-taped. The results indicate a different developmental pattern in achievement for each school grade depending on the levels of abstraction. The influence of the level of abstraction was significant, especially in first graders, and even more so in second graders, that is, at the developmental stage in which they start to learn these arithmetic tasks. Direct modeling strategies are observed more frequently at the concrete and pictorial level, counting strategies occur at all levels of abstraction, whereas numerical fact strategies are found at higher levels of abstraction.

What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.