Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.
more »
« less
Arithmetic operations without symbols are unimpaired in adults with math anxiety
This study characterises a previously unstudied facet of a major causal model of math anxiety. The model posits that impaired “basic number abilities” can lead to math anxiety, but what constitutes a basic number ability remains underdefined. Previous work has raised the idea that our perceptual ability to represent quantities approximately without using symbols constitutes one of the basic number abilities. Indeed, several recent studies tested how participants with math anxiety estimate and compare non-symbolic quantities. However, little is known about how participants with math anxiety perform arithmetic operations (addition and subtraction) on non-symbolic quantities. This is an important question because poor arithmetic performance on symbolic numbers is one of the primary signatures of high math anxiety. To test the question, we recruited 92 participants and asked them to complete a math anxiety survey, two measures of working memory, a timed symbolic arithmetic test, and a non-symbolic “approximate arithmetic” task. We hypothesised that if impaired ability to perform operations was a potential causal factor to math anxiety, we should see relationships between math anxiety and both symbolic and approximate arithmetic. However, if math anxiety relates to precise or symbolic representation, only a relationship between math anxiety and symbolic arithmetic should appear. Our results show no relationship between math anxiety and the ability to perform operations with approximate quantities, suggesting that difficulties performing perceptually based arithmetic operations do not constitute a basic number ability linked to math anxiety.
more »
« less
- Award ID(s):
- 1654089
- NSF-PAR ID:
- 10400281
- Date Published:
- Journal Name:
- Quarterly Journal of Experimental Psychology
- ISSN:
- 1747-0218
- Page Range / eLocation ID:
- 174702182211135
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can per-form a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruc-tion. We presented 5- to 8-year-olds with nonsymbolic multipli-cands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a vis-ible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multipli-cation, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit symbolic multiplication equations (e.g., 2 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between chil-dren’s Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children’s emerging symbolic math competence. These findings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.more » « less
-
more » « less
Abstract Mathematical knowledge is constructed hierarchically from basic understanding of quantities and the symbols that denote them. Discrimination of numerical quantity in both symbolic and non‐symbolic formats has been linked to mathematical problem‐solving abilities. However, little is known of the extent to which overlap in quantity representations between symbolic and non‐symbolic formats is related to individual differences in numerical problem solving and whether this relation changes with different stages of development and skill acquisition. Here we investigate the association between neural representational similarity (NRS) across symbolic and non‐symbolic quantity discrimination and arithmetic problem‐solving skills in early and late developmental stages: elementary school children (ages 7–10 years) and adolescents and young adults (AYA, ages 14–21 years). In children, cross‐format NRS in distributed brain regions, including parietal and frontal cortices and the hippocampus, was positively correlated with arithmetic skills. In contrast, no brain region showed a significant association between cross‐format NRS and arithmetic skills in the AYA group. Our findings suggest that the relationship between symbolic‐non‐symbolic NRS and arithmetic skills depends on developmental stage. Taken together, our study provides evidence for both mapping and estrangement hypotheses in the context of numerical problem solving, albeit over different cognitive developmental stages.
-
more » « less
Abstract Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic‐like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function‐like structure, like symbolic arithmetic. Children (
n = 74 4‐ to ‐8‐year‐olds in Experiment 1;n = 52 7‐ to 8‐year‐olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them “about the same.” We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge. -
null (Ed.)The Academy of Engineering Success (AcES) program, established in 2012 and supported by NSF S-STEM award number 1644119 throughout 2016-2021, employs literature-based, best practices to support and retain underprepared and underrepresented students in engineering through graduation with the ultimate goal of diversifying the engineering workforce. A total of 71 students, including 21 students supported by S-STEM scholarships, participated in the AcES program between 2016-2019 at a large R1 institution in the mid-Atlantic region. All AcES students participate in a common program during their first year, comprised of: a one-week summer bridge experience, a common fall professional development course and spring “Engineering in History” course, and a common academic advisor. These students also have opportunities for: (1) faculty-student, student-student, and industry mentor-student interaction, (2) academic support and student success education, and (3) major and career exploration – all designed to help students develop feelings of institutional inclusion, engineering self-efficacy and identity, and academic and professional success skills. They also participate in the GRIT, Longitudinal Assessment of Engineering Self-Efficacy (LAESE), and the Motivated Strategies for Learning Questionnaire (MSLQ) surveys plus individual and focus group interviews at the start, midpoint, and end of each fall semester and at the end of the spring semester. The surveys provide a measure of students’ GRIT, their beliefs related to the intrinsic value of engineering and learning, their feelings of inclusion and test anxiety, and their self-efficacy related to engineering, math, and coping skills. The interviews provide information related to the student experience, feelings of inclusion, and program impact. Institutional data, combined with the survey and interview responses, are used to examine four research questions designed to examine the relationship of the elements of the AcES program to participants’ academic success and retention in engineering. Early analyses of the student retention data and survey responses from the 2017 and 2018 cohorts indicated students who ultimately left engineering before the start of their second year initially scored higher in areas of engineering self-efficacy and test anxiety, than those who stayed in engineering, while those who retained to the second year began their engineering education with lower self-efficacy scores, but higher scores related to the belief in the intrinsic value of engineering, learning strategy use, and coping self-efficacy. These results suggest that students who start with unrealistically high expectations of their performance leave engineering at higher rates than students who start with lower personal performance expectations, but have stronger value of the field and strategies for meeting challenges. These data appear to support the Kruger-Dunning effect in which students with limited knowledge of a specific field overestimate their abilities to perform in that area or underestimate the level of effort success may require. This paper will add an analysis of the academic success and retention data from 2019 cohort to this research, discuss the impact of COVID-19 to this program and research, as well as illuminate the quantitative results with the qualitative data from individual and focus group interviews regarding the aspects of the AcES program that impact student success, their expectations and methods for overcoming academic challenges, and their feelings of motivation and inclusion.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1177/17470218221113555
