Right or wrong? How feedback content and source influence children's mathematics performance and persistence.

The current study examined how different features of corrective feedback influenced children's performance and motivational outcomes on a mathematics task. Elementary school-aged children from the United States (N = 130; Mage = 7.61 years; 35% female; 60% White) participated in a Zoom session with a trained researcher. During the learning activity, children solved a series of mathematical equivalence problems and were assigned to different feedback conditions that varied in feedback content (correct answer alone vs. correct answer with verification) and feedback source (computer alone vs. computer with person). In terms of content, feedback with verification cues led to decreased persistence, decreased strategy variability, and higher reliance on entrenched strategies relative to feedback that contained the correct answer alone. In terms of source, feedback from the computer alone enhanced children's accuracy; however, the most resilient children received feedback from the computer and a person. Findings are discussed in light of existing feedback theories.

High variability in learning materials benefits children's pattern practice.

Concrete materials (e.g., pictures, objects) are believed to be helpful with learning, but not in all circumstances. Variability in these materials (i.e., using different materials vs. the same materials) could be an important factor. We compared how variability in concrete images influenced children's learning about repeating patterns (e.g., ABBABBABB). A total of 87 children aged 4 to 6 years from the United States (75% White; 44% female) completed an experiment via Zoom in which they received brief pattern training. Children were randomly assigned into Low, Medium, and High Variability training conditions, which differed in terms of whether the same materials were used over and over or they varied in their perceptual features. Children in the Low Variability condition performed better at the beginning of training, but this trend ultimately reversed. Children in the High Variability condition performed best by the end of training and on the posttest. Using variable materials may allow children to extract common structures across instances.

Feelings on feedback: Children's emotional responses during mathematics problem solving.

Theories of learning emphasize the importance of both the cognitive and affective state of the learner. The current study focused on children's affective reactions to corrective feedback during mathematics problem solving. Eighty-seven elementary school children (M age = 7.6 years, 41% female, 68% White) solved mathematical equivalence problems during an online video call and received trial-by-trial feedback on their answers. Trained researchers used children's facial expressions, tone of voice, and verbal statements to quantify their positive and negative affect on each trial. Overall, children tended to express more positive affect than negative affect. However, negative affect was more prominent when the child was incorrect and received negative feedback, and higher negative affect was associated with lower accuracy and lower persistence on the task. These results provide novel empirical evidence for the role of emotions during children's STEM learning in a non-evaluative context.

A developmental perspective on feedback: How corrective feedback influences children's literacy, mathematics, and problem solving.

Research in psychology and education indicates that corrective feedback can be a powerful learning tool. We provide a developmental perspective to focus specifically on how corrective feedback influences learning in childhood (~ages 3-11). Based on a systematic search, we review 44 empirical papers published between 1990 and 2022 examining the effects of corrective feedback on children's performance in the domains of literacy (n=18), mathematics (n=14), and problem solving (n=12). Across these domains, we synthesize research on how children respond to lessons and practice with, versus without, corrective feedback to provide theoretical and practical insights into (1) the effectiveness of corrective feedback in early childhood, (2) the features of effective feedback messages at different ages, and (3) the role of individual learner differences. We make several novel recommendations with some focused on future research questions and others focused on ways teachers can provide effective feedback to children.

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; Mage  = 79.5 months; 55% female; 88% White) and adults (N = 93; Mage  = 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.

Children's confidence using incorrect strategies on mathematical equivalence problems.

Children often struggle to solve mathematical equivalence problems correctly. The change-resistance theory offers an explanation for children's difficulties and suggests that some incorrect strategies represent the overgeneralization of children's narrow arithmetic experience. The current research considered children's metacognitive abilities to test a tacit assumption of the change-resistance theory by providing a novel empirical examination of children's strategy use and certainty ratings. Children were recruited from U.S. elementary school classrooms serving predominantly White students between the ages of 6 and 9. In Study 1 (n = 52) and Study 2 (n = 147), children were more certain that they were correct when they employed arithmetic-specific incorrect strategies relative to other incorrect strategies. These findings are consistent with the change-resistance theory and have implications for the development of children's metacognition.

Generalizability, transferability, and the practice-to-practice gap.

Emphasizing the predictive success and practical utility of psychological science is an admirable goal but it will require a substantive shift in how we design research. Applied research often assumes that findings are transferable to all practices, insensitive to variation between implementations. We describe efforts to quantify and close this practice-to-practice gap in education research.

The benefits of a metacognitive lesson on children's understanding of mathematical equivalence, arithmetic, and place value.

Metacognition is theorized to play a central role in children's mathematics learning. The primary goal of the current study was to provide experimental evidence in support of this role with elementary school students learning about mathematical equivalence. The final sample included 135 children (59 first-graders and 76 second-graders) who participated in the study across three sessions in their classrooms. They completed a pretest during session one, a lesson and posttest during session two, and a two-week delayed retention test during session three. For session two, children were randomly assigned to receive a lesson on mathematical equivalence with or without integrated metacognitive questions. Relative to the control lesson, children who received the metacognitive lesson demonstrated higher accuracy and higher metacognitive monitoring scores on the posttest and retention test. Further, these benefits sometimes extended to uninstructed items targeting arithmetic and place value. No condition effects were observed for children's metacognitive control skills within any of the topics. These findings suggest a brief metacognitive lesson can improve children's mathematics understanding.

ABBABB or 1212: Abstract language facilitates children's early patterning skills.

Emerging research demonstrates a central role of early patterning skills in supporting cognitive development. This study focused on the labels used to describe patterns. Children (N = 90; Mage = 5.4 years) solved and explained 10 pattern abstraction tasks (i.e., recreated a model pattern using novel materials). Using a between-participants design, children were taught using one of four labels: letters (AAB, AAB), numbers (112, 112), quantitative grouping labels (two one, two one), or no labels (this part, this part). All three forms of abstract language were beneficial relative to no labels. Grouping labels, which conveyed information about quantity, also aided performance on posttest items. Children's speech and gesture provided further insights into how abstract language may support early patterning skills and attention to structure.

Mathematical thinking in children with developmental language disorder: The roles of pattern skills and verbal working memory.

Previous research suggests that children with language disorders often have difficulties in mathematical tasks. In the current study, we investigated two relevant factors - working memory and pattern skills - that may underlie children's poor mathematics performance. Children with developmental language disorder (DLD, n = 18, ages 6-13) and age-matched typically-developing children (n = 18) completed three math tasks that tapped calculation skill and knowledge of concepts. Children also completed a visual pattern extension task and a verbal working memory task. There were four key findings: (1) children with DLD exhibited poorer mathematical knowledge than typically-developing children, both in calculation and on key math concepts, (2) children with DLD performed similarly to typically-developing children on the visual pattern extension task, (3) children with DLD had lower verbal working memory scores than typically-developing children, and these differences in working memory accounted in part for their poorer calculation performance, and (4) children's pattern extension scores predicted their arithmetic calculation scores, but not their concept scores.

Diagrams benefit symbolic problem-solving.

BACKGROUND: The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic equations can benefit problem-solving performance as well. AIMS: We tested the impact of diagram presence on students' performance on algebra equation problems to determine whether diagrams increase problem-solving success. We also examined the influence of item- and student-level factors to test the robustness of the diagram effect. SAMPLE: We worked with 61 seventh-grade students who had received 2 months of pre-algebra instruction. METHOD: Students participated in an experimenter-led classroom session. Using a within-subjects design, students solved algebra problems in two matched formats (equation and equation-with-diagram). RESULTS: The presence of diagrams increased equation-solving accuracy and the use of informal strategies. This diagram benefit was independent of student ability and item complexity. CONCLUSIONS: The benefits of diagrams found previously for story problems generalized to symbolic problems. The findings are consistent with cognitive models of problem-solving and suggest that diagrams may be a useful additional representation of symbolic problems.

Relations between patterning skill and differing aspects of early mathematics knowledge.

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.

Early Math Trajectories: Low-Income Children's Mathematics Knowledge From Ages 4 to 11.

Early mathematics knowledge is a strong predictor of later academic achievement, but children from low-income families enter school with weak mathematics knowledge. An early math trajectories model is proposed and evaluated within a longitudinal study of 517 low-income American children from ages 4 to 11. This model includes a broad range of math topics, as well as potential pathways from preschool to middle grades mathematics achievement. In preschool, nonsymbolic quantity, counting, and patterning knowledge predicted fifth-grade mathematics achievement. By the end of first grade, symbolic mapping, calculation, and patterning knowledge were the important predictors. Furthermore, the first-grade predictors mediated the relation between preschool math knowledge and fifth-grade mathematics achievement. Findings support the early math trajectories model among low-income children.

Improving conceptual and procedural knowledge: The impact of instructional content within a mathematics lesson.

BACKGROUND: Students, parents, teachers, and theorists often advocate for direct instruction on both concepts and procedures, but some theorists suggest that including instruction on procedures in combination with concepts may limit learning opportunities and student understanding. AIMS: This study evaluated the effect of instruction on a math concept and procedure within the same lesson relative to a comparable amount of instruction on the concept alone. Direct instruction was provided before or after solving problems to evaluate whether the type of instruction interacted with the timing of instruction within a lesson. SAMPLE: We worked with 180 second-grade children in the United States. METHODS: In a randomized experiment, children received a classroom lesson on mathematical equivalence in one of four conditions that varied in instruction type (conceptual or combined conceptual and procedural) and in instruction order (instruction before or after solving problems). RESULTS: Children who received two iterations of conceptual instruction had better retention of conceptual and procedural knowledge than children who received both conceptual and procedural instruction in the same lesson. Order of instruction did not impact outcomes. CONCLUSIONS: Findings suggest that within a single lesson, spending more time on conceptual instruction may be more beneficial than time spent teaching a procedure when the goal is to promote more robust understanding of target concepts and procedures.

The benefits of computer-generated feedback for mathematics problem solving.

The goal of the current research was to better understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In a randomized experiment, second-grade children received instruction on a correct problem-solving strategy and then solved a set of relevant problems. Children were assigned to receive no feedback, immediate feedback, or summative feedback from the computer. On a posttest the following day, feedback resulted in higher scores relative to no feedback for children who started with low prior knowledge. Immediate feedback was particularly effective, facilitating mastery of the material for children with both low and high prior knowledge. Results suggest that minimal computer-generated feedback can be a powerful form of guidance during problem solving.

Easy as ABCABC: Abstract Language Facilitates Performance on a Concrete Patterning Task.

The labels used to describe patterns and relations can influence children's relational reasoning. In this study, 62 preschoolers (Mage  = 4.4 years) solved and described eight pattern abstraction problems (i.e., recreated the relation in a model pattern using novel materials). Some children were exposed to concrete labels (e.g., blue-red-blue-red) and others were exposed to abstract labels (e.g., A-B-A-B). Children exposed to abstract labels solved more problems correctly than children exposed to concrete labels. Children's correct adoption of the abstract language into their own descriptions was particularly beneficial. Thus, using concrete learning materials in combination with abstract representations can enhance their utility for children's performance. Furthermore, abstract language may play a key role in the development of relational thinking.

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.

Benefits of practicing 4 = 2 + 2: nontraditional problem formats facilitate children's understanding of mathematical equivalence.

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional "operations on left side" format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.