Concept–procedure interactions in children’s addition and subtraction

Cognitive predictors of children's arithmetic principle understanding

The understanding of arithmetic principles (APs) is an important component of our conceptual mathematical knowledge, but we have limited knowledge about how children acquire APs. The current study examined this issue through a longitudinal study with 273 Chinese fourth graders. These children were assessed on various cognitive capacities (e.g., verbal and visuospatial working memory, processing speed, inhibition skill, numerical magnitude representation) in Grade 4 as well as on their AP understanding using multifaceted assessment and their arithmetic competence in Grade 5. Results from structural equation modeling suggested that verbal memory and inhibition skill directly predicted AP understanding, which in turn predicted arithmetic competence. Visuospatial working memory predicted AP understanding through numerical magnitude representation. The findings allow researchers to better account for the individual differences in AP understanding.

Links between repeating and growing pattern knowledge and math outcomes in children and adults

This study examined repeating and growing pattern knowledge and their associations with procedural and conceptual arithmetic knowledge in a sample of U.S. children (N = 185; M_age  = 79.5 months; 55% female; 88% White) and adults (N = 93; M_age  = 19.5 years; 62% female; 66% White) from 2019 to 2020. Three key findings emerged: (1) repeating pattern tasks were easier than growing pattern tasks, (2) repeating pattern knowledge robustly predicted procedural calculation skills over and above growing pattern knowledge and covariates, and (3) growing pattern knowledge modestly predicted procedural and conceptual math outcomes over and above repeating pattern knowledge and covariates. We expand existing theoretical models to incorporate these specific links and discuss implications for supporting math knowledge.

The influence of sex on the relations among spatial ability, math anxiety and math performance

BACKGROUND

A large body of research has found stronger math anxiety in females and suggests that inferior spatial abilities (or attributes towards spatial abilities) in females compared to males are the origin of sex differences in math anxiety.

PURPOSE

To fully explore the complex relationship among math anxiety, spatial abilities, math performance and sex differences, the current study examined spatial skills, working memory skills, math anxiety, and self-efficacy as predictors of math performance.

BASIC PROCEDURES

Participating in the study were 89 undergraduate Israeli students (44 males and 45 females).

MAIN FINDINGS

The result showed sex differences in a few domains: math anxiety was higher in females compared to males, males outperformed females in number line performance and spatial skills. The relationships among spatial abilities, math performance, and math anxiety were stronger in males than in females. By contrast, the relationship between math self-efficacy and performance was stronger in females compared to males.

CONCLUSIONS

This finding demonstrated fundamental differences between the sexes, even with similar performances in curriculum-based assessments.

Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect "online" effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.

Teaching Congruences in Connection with Diophantine Equations

The presented paper is devoted to the new teaching model of congruences of computer science students within the subject of discrete mathematics at universities. The main goal was to create a new model of teaching congruences on the basis of their connection with Diophantine equations and subsequently to verify the effectiveness and efficiency of the proposed model experimentally. The teaching of congruences was realized in two phases: acquisition of procedural knowledge and use of Diophantine equations to obtain conceptual knowledge of congruences. Experiments confirmed that conceptual understanding of congruences is positively related to increasing the procedural fluidity of congruence resolution. Research also demonstrated the suitability of using Diophantine equations to link congruences and equations. Among other things, the presented research has confirmed the justification of teaching mathematics in computer-oriented study programs.

Reconceptualizing Symbolic Magnitude Estimation Training Using Non-declarative Learning Techniques

It is well-documented that mathematics achievement is an important predictor of many positive life outcomes like college graduation, career opportunities, salary, and even citizenship. As such, it is important for researchers and educators to help students succeed in mathematics. Although there are undoubtedly many factors that contribute to students' success in mathematics, much of the research and intervention development has focused on variations in instructional techniques. Indeed, even a cursory glance at many educational journals and granting agencies reveals that there is a large amount of time, energy, and resources being spent on determining the best way to convey information through direct, declarative instruction. The proposed project is motivated by recent calls to expand the focus of research in mathematics education beyond direct, declarative instruction. The overarching goal of the presented experiment is to evaluate the efficacy of a novel mathematics intervention designed using principles taken from the literature on non-declarative learning. The intervention combines errorless learning and structured cue fading to help second grade students improve their understanding of symbolic magnitude. Results indicate that students who learned about symbolic magnitude using the novel intervention did better than students who were provided with extensive declarative support. These findings offer preliminary evidence in favor of using learning combination of errorless learning and cue fading techniques in the mathematics classroom.

Multifaceted assessment of children's inversion understanding

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.

Understanding arithmetic concepts: The role of domain-specific and domain-general skills

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.

Effects of Instructional Guidance and Sequencing of Manipulatives and Written Symbols on Second Graders’ Numeration Knowledge

Concrete objects used to illustrate mathematical ideas are commonly known as manipulatives. Manipulatives are ubiquitous in North American elementary classrooms in the early years, and although they can be beneficial, they do not guarantee learning. In the present study, the authors examined two factors hypothesized to impact second-graders’ learning of place value and regrouping with manipulatives: (a) the sequencing of concrete (base-ten blocks) and abstract (written symbols) representations of the standard addition algorithm; and (b) the level of instructional guidance on the structural relations between the representations. Results from a classroom experiment with second-grade students (N = 87) indicated that place value knowledge increased from pre-test to post-test when the base-ten blocks were presented before the symbols, but only when no instructional guidance was offered. When guidance was given, only students in the symbols-first condition improved their place value knowledge. Students who received instruction increased their understanding of regrouping, irrespective of representational sequence. No effects were found for iterative sequencing of concrete and abstract representations. Practical implications for teaching mathematics with manipulatives are considered.

Children's understanding of the addition/subtraction complement principle

BACKGROUND

In the last decades, children's understanding of mathematical principles has become an important research topic. Different from the commutativity and inversion principles, only few studies have focused on children's understanding of the addition/subtraction complement principle (if a - b = c, then c + b = a), mainly relying on verbal techniques.

AIM

This contribution aimed at deepening our understanding of children's knowledge of the addition/subtraction complement principle, combining verbal and non-verbal techniques.

SAMPLE

Participants were 67 third and fourth graders (9- to 10-year-olds).

METHODS

Children solved two tasks in which verbal reports as well as accuracy and speed data were collected. These two tasks differed only in the order of the problems and the instructions. In the looking-back task, children were told that sometimes the preceding problem might help to answer the next problem. In the baseline task, no helpful preceding items were offered. The looking-back task included 10 trigger-target problem pairs on the complement relation.

RESULTS

Children verbally reported looking back on about 40% of all target problems in the looking-back task; the target problems were also solved faster and more accurately than in the baseline task. These results suggest that children used their understanding of the complement principle. The verbal and non-verbal data were highly correlated.

DISCUSSION

This study complements previous work on children's understanding of mathematical principles by highlighting interindividual differences in 9- to 10-year-olds' understanding of the complement principle and indicating the potential of combining verbal and non-verbal techniques to investigate (the acquisition of) this understanding.

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.

An alternative time for telling: when conceptual instruction prior to problem solving improves mathematical knowledge

BACKGROUND

The sequencing of learning materials greatly influences the knowledge that learners construct. Recently, learning theorists have focused on the sequencing of instruction in relation to solving related problems. The general consensus suggests explicit instruction should be provided; however, when to provide instruction remains unclear.

AIMS

We tested the impact of conceptual instruction preceding or following mathematics problem solving to determine when conceptual instruction should or should not be delayed. We also examined the learning processes supported to inform theories of learning more broadly.

SAMPLE

We worked with 122 second- and third-grade children.

METHOD

In a randomized experiment, children received instruction on the concept of math equivalence either before or after being asked to solve and explain challenging equivalence problems with feedback.

RESULTS

Providing conceptual instruction first resulted in greater procedural knowledge and conceptual knowledge of equation structures than delaying instruction until after problem solving. Prior conceptual instruction enhanced problem solving by increasing the quality of explanations and attempted procedures.

CONCLUSIONS

Providing conceptual instruction prior to problem solving was the more effective sequencing of activities than the reverse. We compare these results with previous, contrasting findings to outline a potential framework for understanding when instruction should or should not be delayed.

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.