Nicole M. McNeil

ANS acuity and mathematics ability in preschoolers from low‐income homes: contributions of inhibitory control

Recent findings by Libertus, Feigenson, and Halberda (2011) suggest that there is an association between the acuity of young childrens approximate number system (ANS) and their mathematics ability before exposure to instruction in formal schooling. The present study examined the generalizability and validity of these findings in a sample of preschoolers from low-income homes. Children attending Head Start (N = 103) completed measures to assess ANS acuity, mathematics ability, receptive vocabulary, and inhibitory control. Results showed only a weak association between ANS acuity and mathematics ability that was reduced to non-significance when controlling for a direct measure of receptive vocabulary. Results also revealed that inhibitory control plays an important role in the relation between ANS acuity and mathematics ability. Specifically, ANS acuity accounted for significant variance in mathematics ability over and above receptive vocabulary, but only for ANS acuity trials in which surface area conflicted with numerosity. Moreover, this association became non-significant when controlling for inhibitory control. These results suggest that early mathematical experiences prior to formal schooling may influence the strength of the association between ANS acuity and mathematics ability and that inhibitory control may drive that association in young children.

When Theories Don't Add Up: Disentangling he Manipulatives Debate

The use of manipulatives in the classroom has been advocated for decades. However, the theoretical and empirical support for this practice is mixed. Some researchers suggest that manipulatives facilitate learning by (a) providing an additional channel for conveying information, (b) activating real-world knowledge, and/or (c) improving memory through physical action. However, there are at least two reasons to question the efficacy of manipulative use. First, manipulatives might lead students to focus on having fun at the expense of deep learning. Second, manipulatives might make learning more difficult because they require dual representation. Although these two criticisms are disparate in terms of their underlying rationale, both converge on the idea that teachers should reduce their use of manipulatives that are highly familiar and/or perceptually interesting. More generally, the manipulatives debate highlights the need for teachers and researchers to work together to evaluate the costs and benefits of various classroom practices.

THE ROLE OF GESTURE IN CHILDREN'S COMPREHENSION OF SPOKEN LANGUAGE: NOW THEY NEED IT, NOW THEY DON'T

Two experiments investigated gesture as a form of external support for spoken language comprehension. In both experiments, children selected blocks according to a set of videotaped instructions. Across trials, the instructions were given using no gesture, gestures that reinforced speech, and gestures that conflicted with speech. Experiment 1 used spoken messages that were complex for preschool children but not for kindergarten children. Reinforcing gestures facilitated speech comprehension for preschool children but not for kindergarten children, and conflicting gestures hindered comprehension for kindergarten children but not for preschool children. Experiment 2 tested preschool children with simpler spoken messages. Unlike Experiment 1, preschool childrens comprehension was not facilitated by reinforcing gestures. However, childrens comprehension also was not hindered by conflicting gestures. Thus, the effects of gesture on speech comprehension depend both on the relation of gesture to speech, and on the complexity of the spoken message.

Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable

Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.

U-Shaped Development in Math: 7-Year-Olds Outperform 9-Year-Olds on Equivalence Problems

What is the nature of the association between age (7-11 years) and performance on mathematical equivalence problems (e.g., 7+4+5+7+_)? Many prevailing theories suggest that there should be a positive association. However, change-resistance accounts (e.g., N. M. McNeil & M. W. Alibali, 2005b) predict a U-shaped association. The purpose of the present research was to test these differing predictions. Results from two studies supported a change-resistance account. In the first study (N=87), performance on equivalence problems declined between the ages of 7 and 9 and improved between the ages of 9 and 11. The decrements in performance between the ages of 7 and 9 were then replicated in a second study (N=35). Results suggest that the association between age and performance on equivalence problems is U-shaped.

Divergence of verbal expression and embodied knowledge: Evidence from speech and gesture in children with specific language impairment

It has been suggested that phonological working memory serves to link speech comprehension to production. We suggest further that impairments in phonological working memory may influence the way in which children represent and express their knowledge about the world around them. In particular, children with severe phonological working memory deficits may have difficulty retaining stable representations of phonological forms, which results in weak links with meaning representations; however, nonverbal meaning representations might develop appropriately due to input from other modalities (e.g., vision, action). Typically developing children often express emerging knowledge in gesture before they are able to express this knowledge explicitly in their speech. In this study we explore the extent to which children with specific language impairment (SLI) with severe phonological working memory deficits express knowledge uniquely in gesture as compared to speech. Using a paradigm in which gesture-speech relationships have been studied extensively, children with SLI and conservation judgement-matched, typically developing controls were asked to solve and explain a set of Piagetian conservation tasks. When gestures accompanied their explanations, the children with SLI expressed information uniquely in gesture more often than did the typically developing children. Further, the children with SLI often expressed more sophisticated knowledge about conservation in gesture (and in some cases, distributed across speech and gesture) than in speech. The data suggest that for the children with SLI, their embodied, perceptually-based knowledge about conservation was rich, but they were not always able to express this knowledge verbally. We argue that this pattern of gesture-speech mismatch may be due to poor links between phonological representations and embodied meanings for children with phonological working memory deficits.

Limitations to Teaching Children 2 + 2 = 4: Typical Arithmetic Problems Can Hinder Learning of Mathematical Equivalence

Do typical arithmetic problems hinder learning of mathematical equivalence? Second and third graders (7-9 years old; N= 80) received lessons on mathematical equivalence either with or without typical arithmetic problems (e.g., 15 + 13 = 28 vs. 28 = 28, respectively). Children then solved math equivalence problems (e.g., 3 + 9 + 5 = 6 + _), switched lesson conditions, and solved math equivalence problems again. Correct solutions were less common following instruction with typical arithmetic problems. In a supplemental experiment, fifth graders (10-11 years old; N= 19) gave fewer correct solutions after a brief intervention on mathematical equivalence that included typical arithmetic problems. Results suggest that learning is hindered when lessons activate inappropriate existing knowledge.

Effects of Perceptually Rich Manipulatives on Preschoolers' Counting Performance: Established Knowledge Counts

Educators often use concrete objects to help children understand mathematics concepts. However, findings on the effectiveness of concrete objects are mixed. The present study examined how two factors-perceptual richness and established knowledge of the objects-combine to influence childrens counting performance. In two experiments, preschoolers (N = 133; Mage = 3;10) were randomly assigned to counting tasks that used one of four types of objects in a 2 (perceptual richness: high or low) × 2 (established knowledge: high or low) factorial design. Findings suggest that perceptually rich objects facilitate childrens performance when children have low knowledge of the objects but hinder performance when children have high knowledge of the objects.

Challenges in Mathematical Cognition: A Collaboratively-Derived Research Agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition.

An eye for relations: eye-tracking indicates long-term negative effects of operational thinking on understanding of math equivalence.

Prior knowledge in the domain of mathematics can sometimes interfere with learning and performance in that domain. One of the best examples of this phenomenon is in students’ difficulties solving equations with operations on both sides of the equal sign. Elementary school children in the U.S. typically acquire incorrect, operational schemata rather than correct, relational schemata for interpreting equations. Researchers have argued that these operational schemata are never unlearned and can continue to affect performance for years to come, even after relational schemata are learned. In the present study, we investigated whether and how operational schemata negatively affect undergraduates’ performance on equations. We monitored the eye movements of 64 undergraduate students while they solved a set of equations that are typically used to assess children’s adherence to operational schemata (e.g., 3 + 4 + 5 = 3 + __). Participants did not perform at ceiling on these equations, particularly when under time pressure. Converging evidence from performance and eye movements showed that operational schemata are sometimes activated instead of relational schemata. Eye movement patterns reflective of the activation of relational schemata were specifically lacking when participants solved equations by adding up all the numbers or adding the numbers before the equal sign, but not when they used other types of incorrect strategies. These findings demonstrate that the negative effects of acquiring operational schemata extend far beyond elementary school.