2014 — 2015 
Matthews, Percival Grant 
R03Activity Code Description: To provide research support specifically limited in time and amount for studies in categorical program areas. Small grants provide flexibility for initiating studies which are generally for preliminary shortterm projects and are nonrenewable. 
Delimiting and Leveraging Children's Natural Sense of Proportion @ University of WisconsinMadison
DESCRIPTION (provided by applicant): Mathematical competence is an important determinant of life chances. Recent research suggests that understanding fractions particularly understanding their relative sizes is critical for the development of mathematical competence. Unfortunately, children and adults often encounter considerable difficulties understanding fractions. To explain these widespread difficulties, many researchers have argued for an innate constraints account of fraction cognition. On this account, fractions are difficult to understand because they lack an intuitive basis, whereas whole number understanding can be grounded in our perceptual abilities to process numerosities (i.e., collections of countable objects). Thus, innate constraint theorists argue that fraction learning is challenging because it does not benefit from existing cognitive abilities similar to those that facilitate whole number learning and that fractions must instead be learned through adapting whole number understanding. In short, they argue that fractions are somehow less natural than whole numbers. The proposed research will investigate a competing hypothesis, the cognitive primitives' account, which integrates previously unrelated findings from neuroscience, developmental psychology and education. This hypothesis contends that cognitive systems tuned to the processing of nonsymbolic fractions (such as the relative lengths of two lines or the relative areas of two figures) are present before children begin formal instruction. The existence of these primitive nonsymbolic fraction processing abilities suggests that they might serve as a foundation for understanding the magnitudes of symbolic fractions. On this view, children may be equipped with cognitive mechanisms that support fraction concepts prior to formal education in the same way that the ability to process numerosities equips them to learn about whole numbers. If substantiated, this hypothesis can inform interventions designed to improve fraction learning and may contribute to the detection and treatment of math learning difficulties. To test the predictions of the cognitive primitives account, the project will use behavioral tasks to investigate children's (6yearolds an 10 to 11yearolds) abilities to perceive the magnitudes of nonsymbolic fractions. It will also aim to develop a training program that pairs nonsymbolic fractions with symbolic fractions to teach children about the magnitudes of symbolic fractions. These findings will have important implications for our understanding of number processing and for designing interventions that are optimal for promoting fraction learning. If perceptual sensitivity to nonsymbolic fractions can provide a foundation for the acquisition of symbolic fraction knowledge, then instruction should attempt to exploit these primitive abilities. If deficits in these nonsymbolic abilities contribut to math learning difficulties, then screening should include measures of nonsymbolic abilities and interventions should be designed to strengthen these abilities.

1 
2014 — 2017 
Hubbard, Edward (coPI) [⬀] Matthews, Percival 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Using Nonsymbolic Ratios to Promote Fraction Knowledge: a Neurocognitive Approach @ University of WisconsinMadison
This project, to be conducted by researchers at the University of Wisconsin, Madison, will study knowledge of fractions in college students and eighthgraders. The project will use behavioral and brain imaging measures. The main hypothesis is that some kinds of visual perception ability (such as sensitivity to the ratio formed by the lengths of two lines) are related to fraction understanding. The project will develop a training method aimed at improving knowledge about fractions. This project will advance the work of the REAL (Research on Education and Learning) program in studying the cognitive and neural basis of STEM (science, technology, engineering, and mathematics) learning.
The project will use fMRI (functional magnetic resonance imaging) to study brain activity, focusing on brain regions including the PFC (prefrontal cortex) and IPS (intraparietal sulcus). In particular, the research will use an adaptation method, looking at activation recovery when a novel ratio is presented in a series of ratios. The project addresses a cognitive primitives account of fraction magnitude processing, which posits that perceptual sensitivity to nonsymbolic ratio magnitudes plays a major role in supporting the developing understanding of symbolic fraction magnitudes. This account implies that neurocognitive architectures tuned to the processing of nonsymbolic ratiosuch as the relative lengths of two lines or the relative areas of two figuresare present even before learners receive fractions instruction.

0.915 
2016 — 2020 
Hubbard, Edward Michael [⬀] Matthews, Percival Grant 
R01Activity Code Description: To support a discrete, specified, circumscribed project to be performed by the named investigator(s) in an area representing his or her specific interest and competencies. 
Perceptual and Cognitive Mechanisms of Developing Fractions Knowledge: a CrossSequential Approach @ University of WisconsinMadison
Project Summary Mathematical competence is an important determinant of life chances in modern society, and knowledge of fractions is a foundational skill for establishing mathematical competence. Despite the importance of fraction knowledge, children and adults often encounter considerable difficulties understanding fractions. To explain these widespread difficulties, many researchers have argued for an innate constraints account. They propose that fractions are difficult because they do not correspond to any preexisting categories in our brain, unlike whole numbers, which correspond to sets of countable things. Thus, they argue fraction concepts are challenging because they do not benefit from existing cognitive abilities and instead must be learned through adapting children's whole number understanding. The study team proposes a competing hypothesis, the cognitive primitives account, which integrates previously unrelated findings from neuroscience, developmental psychology and education. We argue that a primitive ability that we dub the ratio processing system (RPS) is tuned to the processing of nonsymbolic fractions?such as the relative length of two lines or the relative area of two figures?and is present even before formal instruction. On this view, children are equipped with cognitive mechanisms that support fraction concepts in the same way that the ability to process countable sets equips them to learn about whole numbers. To test the predictions of these competing hypotheses, this project will follow two cohorts of children (2nd graders until 5th grade and 5th graders until 8th grade) using behavioral and brain imaging methods to (a) trace the development of nonsymbolic fraction processing abilities, (b) determine how symbolic fraction knowledge builds on these abilities and (c) investigate whether individual differences in the RPS predict later math achievement. To test whether the acuity or recruitment of these nonsymbolic architectures plays a role in fraction difficulties as well as general math learning difficulties, the study team will compare the behavioral performance and neural activity on a battery of cognitive tasks. This research has important implications for our understanding of number processing and for designing educational practices that are optimal for fraction learning. Improving fractions understanding would help children to clear a critical hurdle in the acquisition of higherorder mathematical competencies that impact educational, employment, and even health outcomes. If cognitive primitives for nonsymbolic fractions can provide a foundation for the acquisition of symbolic fraction ability, then instruction should attempt to recruit these primitives. If deficits in these primitives contribute to math learning difficulties, then screening should include measures of nonsymbolic abilities and interventions should be designed to address these abilities.

1 
2018 — 2021 
Alibali, Martha (coPI) [⬀] Stephens, Ana (coPI) [⬀] Matthews, Percival 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Cultivating Knowledge of Mathematical Equivalence @ University of WisconsinMadison
This project will develop and test the effectiveness of a semesterlong conceptuallybased instruction for promoting understanding of mathematical equivalence and associated gains in algebraic thinking. Participants in the research will be elementary and middle school students. Although many brief, singlesession laboratory studies have suggested effective ways to promote equal sign knowledge in the short term, these studies have generally failed to produce practical guidelines for use by regular classroom teachers. This project will apply findings from laboratory studies over a longer time period with the goal of packaging a practical approach to developing students' knowledge of the equal sign for classroom teachers. The hypothesis is that improved equal sign knowledge will lead to improved access to algebra, an important pathway into higher mathematics and science that provides access to Science, Technology, Engineering, and Mathematics (STEM) fields that help power modern society. This project will explore the effectiveness of spaced, conceptuallybased instruction for promoting understanding of mathematical equivalence and associated gains in algebraic thinking. Multiple experimental studies have shown that brief, conceptuallybased instructional interventions can lead to improvements in children's equal sign knowledge. The proposed research will test whether spacing such interventions over time can lead to more substantial and longterm gains in equal sign knowledge, and whether such knowledge, in turn, fosters algebraic reasoning. One component of the research will aim to optimize the conceptually based intervention. The other component will be a study that investigates the effects of the intervention over time, using a crossover design. The research will employ a measure of equal sign knowledge that is more sensitive than most commonly used measures, allowing for detection of relatively finegrained gains in response to the instructional intervention. In total, the work will contribute to the field's understanding of how to improve equal sign knowledge and understanding of the causal impact of equal sign instruction on student competence in algebra.
This project is cofunded by the Discovery Research preK12 program (DRK12) that seeks to significantly enhance the learning and teaching of science, technology, engineering, and mathematics (STEM) by preK12 students and teachers, through research and development of innovative resources, models, and tools.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

0.915 