The hierarchical relations among mathematical competencies: from fundamental numeracy to complex mathematical skills

Mathematical competencies can be conceptualized as layers of knowledge, with numeracy skills as the foundational core and more complex mathematical skills as the additional layers over the core. In this study, we tested an expanded hierarchical symbol integration (HSI) model by examining the hierarchical relations among mathematical skills. Undergraduate students (N = 236) completed order judgement, simple arithmetic, fraction arithmetic, algebra, and verbal working memory tasks. In a series of hierarchical multiple regressions, we found support for the hierarchical model: Additive skills (i.e., addition and subtraction) predicted unique variance in multiplicative skills (i.e., multiplication and division); multiplicative skills predicted unique variance in fraction arithmetic; and fraction skills predicted unique variance in algebra. These results support the framework of the HSI model in which mathematical competencies are related hierarchically, capturing the increasing complexity of symbolic mathematical skills

Refuting Misconceptions: Computer Tutors for Fraction Arithmetic

Fractions, known to be difficult for both children and adults, are especially prone to misconceptions and erroneous strategy selection. The present study investigated whether a computer tutor improves fraction arithmetic performance in adults and if supplementing problem solving with erroneous examples is more beneficial than problem solving alone. Seventy-five undergraduates solved fraction arithmetic problems using a computer tutoring system we designed. In a between-subjects design, 39 participants worked with a problem-solving tutor that was supplemented with erroneous examples and 36 participants worked with a traditional problem-solving tutor. Both tutors provided hints and feedback. Overall, participants improved after the tutoring interventions, but there were no significant differences in gains made by the two conditions. For students with low prior knowledge about fraction arithmetic, the numerical gains were higher in the erroneous-example group than the problem-solving group, but this effect was not significant. Thus, computer tutors are useful tools for improving fraction knowledge. While erroneous examples may be particularly beneficial for students with low prior knowledge who may hold more misconceptions, more research is needed to make this conclusion

Confidence is key: Unlocking the relations between ADHD symptoms and math performance

Are ADHD symptoms uniquely related to affect about mathematics in university students? Undergraduates (n = 425) completed three math performance measures (i.e., arithmetic, computational skills, and word problem solving) and self-report measures of ADHD symptoms, state anxiety, and confidence about math and literacy (i.e., affect and perceived self-efficacy). Students who reported more ADHD symptoms were less confident in both their mathematics and literacy skills. ADHD symptoms were indirectly linked to math performance through math confidence. People who reported more ADHD symptoms were anxious before starting the study, but this anxiety level remained constant. In contrast, people with lower math confidence had higher levels of state anxiety, and this level of anxiety increased during the study. In summary, ADHD symptoms were related to academic confidence in general; links with math performance were mediated through math confidence. There was no evidence for a specific link between attentional symptoms and mathematics

When Does the Story Matter? No Evidence for the Foregrounding Hypothesis in Math Story Problems

Math story problems are difficult for many solvers because comprehension of mathematical and linguistic content must occur simultaneously. Across two studies, we attempted to conceptually replicate and extend findings reported by Mattarella-Micke and Beilock (2010, https://doi.org/10.3758/PBR.17.1.106) and Jarosz and Jaeger (2019, https://doi.org/10.1002/acp.3471). Mattarella-Micke and Beilock found that multiplication word problems in which an irrelevant number was associated with the protagonist of the problem (i.e., foregrounded in the text) were solved less accurately than problems in other conditions. Jarosz and Jaeger used similar materials but tested the more general inconsistent-operations hypothesis that association with the protagonist would interfere with multiplication whereas dissociation would interfere with division. They found partial support: When division problems were primed with dissociative scenarios, solvers made more errors, but they failed to replicate the associative findings for multiplication. In the present research, we conducted two studies (Ns = 205 and 359), in which we similarly manipulated whether irrelevant content was associated with or dissociated from the story protagonist. In these studies, we did not find support for either the foregrounding or inconsistent-operations hypotheses. Exploratory error analyses suggested that solvers’ errors were most often the result of calculation difficulties or inappropriate operation choices and were unrelated to the presence of associative or dissociative story elements. Our careful implementation of this manipulation and much greater power to detect effects suggests that the association manipulation in irrelevant text does not influence adults’ performance on simple math story problems

A Longitudinal Investigation of the Co-Development and Bidirectional Relations Among Whole Number Arithmetic and Conceptual and Procedural Fraction Knowledge

Longitudinal data collected from 2021-202

We cannot ignore the signs: the development of equivalence and arithmetic for students from Grades 3 to 4

Students' understanding of the meaning of the equal sign develops slowly over the primary grades. In addition to updating their representations of equations to recognize that the equal sign represents an equivalence relation rather than signaling an operation, students need to move beyond full computation to efficiently solve equivalence problems. In this study, we examined the longitudinal relation between arithmetic and equivalence for students who were capable of accurately solving arithmetic problems in different formats. Chinese students (N = 612; M age = 9.0 years in Grade 3, 57% boys) completed measures of arithmetic fluency and equivalence fluency in Grade 3 and again in Grade 4. They also completed a non-verbal reasoning task in Grade 3. We tested a cross-lagged structural equation model to examine the reciprocal relations between arithmetic and equivalence fluency. We found reciprocal relations between the development of arithmetic and equivalence fluency from Grades 3 to 4, with a greater influence of arithmetic on the development of equivalence than the reverse. Furthermore, non-verbal reasoning predicted the development of equivalence, but not the development of arithmetic. Based on our findings, we conclude that for Chinese students with prior basic understanding of equivalence, flexible access to arithmetic facts supports their development of equivalence fluency

Controlling-Supportive Homework Help Partially Explains the Relation Between Parents' Math Anxiety and Children's Math Achievement

Researchers do not yet know why the math homework help of higher-math-anxious parents impedes children’s math learning and facilitates the development of math anxiety, as previously demonstrated by Maloney et al. (2015). In the present study, we explored a possible explanation for this phenomenon by examining the relations between parents’ math anxiety, their math homework-helping styles (i.e., autonomy- and controlling-supportive), and their child’s math achievement. Parents of children ages 11 to 14 completed an online survey. Using path analysis, we examined the relations among parental factors (i.e., math anxiety, math ability, and homework-helping styles) and child math achievement. Parents’ math anxiety was positively related to both autonomy-supportive and controlling-supportive math homework-helping styles. Notably, controlling-supportive style partially mediated the relation between parents’ math anxiety and their children’s math achievement. Thus, it is possible that the use of a controlling-supportive math homework-helping style may explain why the homework help offered by higher-math-anxious parents is detrimental to their children’s math learning. Identifying negative relations between parent factors and children’s math outcomes is crucial for developing evidence-based math learning interventions