Unpacking the challenges and predictors of elementary-middle school students' use of the distributive property.

The distributive property plays a pivotal role in advancing students' understanding of multiplication, enabling the decomposition of problems and the acquisition of new facts. However, this property of multiplication is difficult for students to understand. We used two unique data sets to explore middle school students' use of the distributive property. Study 1 involved data from 1:1 structured interviews of students (N = 24) discussing worked examples and solving associated practice problems. We examined whether or not students used the distributive property to solve the problems and whether or not interviewers followed the recommended distributive property prompts or defaulted to more conventional methods. Despite exposure to worked examples using the distributive property and a protocol calling for attention to it, students and interviewers favored methods like PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) or long multiplication. Study 2 used a data set with middle school students' (N = 131) item-level responses on Kirkland's (2022; doctoral dissertation, University of Notre Dame) Brief Assessment of Mature Number Sense along with several related measures of domain-general and domain-specific skills. We extracted problems involving the distributive property for analysis. Surprisingly, there was no evidence that students' use of the distributive property improved from sixth grade to eighth grade. However, both grade-level mathematics achievement and cognitive reflection uniquely predicted the correct use of the distributive property. Results suggest that middle school students who exhibit stronger reflective thinking tend to perform better on distributive property problems. Findings highlight cognitive reflection as a potentially important construct involved in the understanding and use of the distributive property.

A bird's-eye view of research practices in mathematical cognition, learning, and instruction: Reimagining the status quo.

Research on mathematical cognition, learning, and instruction (MCLI) often takes cognition as its point of departure and considers instruction at a later point in the research cycle. In this article, we call for psychologists who study MCLI to reflect on the "status quo" of their research practices and to consider making instruction an earlier and more central aspect of their work. We encourage scholars of MCLI (a) to consider the needs of educators and schools when selecting research questions and developing interventions; (b) to compose research teams that are diverse in the personal, disciplinary, and occupational backgrounds of team members; (c) to make efforts to broaden participation in research and to conduct research in authentic settings; and (d) to communicate research in ways that are accessible to practitioners and to the general public. We argue that a more central consideration of instruction will lead to shifts that make research on MCLI more theoretically valuable, more actionable for educators, and more relevant to pressing societal challenges.

Features in children's counting books that lead dyads to both count and label sets during shared book reading.

This study examined how book features influence talk during shared book reading. We used data from a study in which parent-child dyads (n = 157; child's Mage  = 43.99 months; 88 girls, 69 boys; 91.72% of parents self-reported as white) were randomly assigned to read two number books. The focus was comparison talk (i.e., talk in which dyads count a set and also label its total), as this type of talk has been shown to promote children's understanding of cardinality. Replicating previous findings, dyads produced relatively low levels of comparison talk. However, book features influenced the talk. Books containing a greater number of numerical representations (e.g., number word, numeral, and non-symbolic set) and a greater word count elicited more comparison talk.

Partial knowledge in the development of number word understanding.

A common measure of number word understanding is the give-N task. Traditionally, to receive credit for understanding a number, N, children must understand that N does not apply to other set sizes (e.g. a child who gives three when asked for 'three' but also when asked for 'four' would not be credited with knowing 'three'). However, it is possible that children who correctly provide the set size directly above their knower level but also provide that number for other number words ('N + 1 givers') may be in a partial, transitional knowledge state. In an integrative analysis including 191 preschoolers, subset knowers who correctly gave N + 1 at pretest performed better at posttest than did those who did not correctly give N + 1. This performance was not reflective of 'full' knowledge of N + 1, as N + 1 givers performed worse than traditionally coded knowers of that set size on separate measures of number word understanding within a given timepoint. Results support the idea of graded representations (Munakata, Trends in Cognitive Sciences, 5, 309-315, 2001.) in number word development and suggest traditional approaches to coding the give-N task may not completely capture children's knowledge.

Improved set-size labeling mediates the effect of a counting intervention on children's understanding of cardinality.

How does improving children's ability to label set sizes without counting affect the development of understanding of the cardinality principle? It may accelerate development by facilitating subsequent alignment and comparison of the cardinal label for a given set and the last word counted when counting that set (Mix et al., 2012). Alternatively, it may delay development by decreasing the need for a comprehensive abstract principle to understand and label exact numerosities (Piantadosi et al., 2012). In this study, preschoolers (N = 106, Mage  = 4;8) were randomly assigned to one of three conditions: (a) count-and-label, wherein children spent 6 weeks both counting and labeling sets arranged in canonical patterns like pips on a die; (b) label-first,wherein children spent the first 3 weeks learning to label the set sizes without counting before spending 3 weeks identical to the count-and-label condition; (c) print referencing control. Both counting conditions improved understanding of cardinality through increases in children's ability to label set sizes without counting. In addition to this indirect effect, there was a direct effect of the count-and-label condition on progress toward understanding of cardinality. Results highlight the roles of set labeling and equifinality in the development of children's understanding of number concepts.

Consequences of Individual Differences in Children's Formal Understanding of Mathematical Equivalence.

Experts claim that individual differences in children's formal understanding of mathematical equivalence have consequences for mathematics achievement; however, evidence is lacking. A prospective, longitudinal study was conducted with a diverse sample of 112 children from a midsized city in the Midwestern United States (Mage [second grade] = 8:1). As hypothesized, understanding of mathematical equivalence in second grade predicted mathematics achievement in third grade, even after controlling for second-grade mathematics achievement, IQ, gender, and socioeconomic status. Most children exhibited poor understanding of mathematical equivalence, but results provide clues about which children are on the path to constructing an understanding and which may need extra support to overcome their misconceptions. Findings suggest that mathematical equivalence may deserve more attention from educators.

Comparing Meta-analysis and Individual Person Data Analysis Using Raw Data on Children's Understanding of Equivalence.

A prevailing theory of mathematical problem solving predicts that children will be less accurate solving a + b = c + __ problems versus a + b = __ + c. However, this has never been tested directly. Because of low base rates, information combined from multiple studies can help improve estimation accuracy and precision. This study compared meta-analysis and individual person data (IPD) analysis using raw data from 14 studies (N = 1,414; ns = 30-232; Mage reported = 8;7). Substantive results challenge the prevailing theory. Methodological results demonstrate the advantages of using meta-analysis and IPD over single-study analysis. Moreover, IPD can be more powerful than meta-regression for detecting between-study moderation effects.

Novel mouse model recapitulates genome and transcriptome alterations in human colorectal carcinomas.

Human colorectal carcinomas are defined by a nonrandom distribution of genomic imbalances that are characteristic for this disease. Often, these imbalances affect entire chromosomes. Understanding the role of these aneuploidies for carcinogenesis is of utmost importance. Currently, established transgenic mice do not recapitulate the pathognonomic genome aberration profile of human colorectal carcinomas. We have developed a novel model based on the spontaneous transformation of murine colon epithelial cells. During this process, cells progress through stages of pre-immortalization, immortalization and, finally, transformation, and result in tumors when injected into immunocompromised mice. We analyzed our model for genome and transcriptome alterations using ArrayCGH, spectral karyotyping (SKY), and array based gene expression profiling. ArrayCGH revealed a recurrent pattern of genomic imbalances. These results were confirmed by SKY. Comparing these imbalances with orthologous maps of human chromosomes revealed a remarkable overlap. We observed focal deletions of the tumor suppressor genes Trp53 and Cdkn2a/p16. High-level focal genomic amplification included the locus harboring the oncogene Mdm2, which was confirmed by FISH in the form of double minute chromosomes. Array-based global gene expression revealed distinct differences between the sequential steps of spontaneous transformation. Gene expression changes showed significant similarities with human colorectal carcinomas. Pathways most prominently affected included genes involved in chromosomal instability and in epithelial to mesenchymal transition. Our novel mouse model therefore recapitulates the most prominent genome and transcriptome alterations in human colorectal cancer, and might serve as a valuable tool for understanding the dynamic process of tumorigenesis, and for preclinical drug testing. © 2016 Wiley Periodicals, Inc.

An integrative data analysis of gender differences in children's understanding of mathematical equivalence.

This study examined gender as a potential source of variation in children's formal understanding of mathematical equivalence. The hypothesis was that girls would perform more poorly than boys. An integrative data analysis was conducted with 960 second and third graders across 14 previously conducted studies of children's understanding of mathematical equivalence. Measures included problem solving, problem encoding, and equal sign definition. Overall, children performed poorly on all measures. As predicted, girls were less likely than boys to solve mathematical equivalence problems correctly, even though there were no gender differences in calculation accuracy. In addition, girls were more likely than boys to use the "add-all" strategy, an incorrect strategy that has been shown to be more resistant to change than other incorrect strategies. There were not statistically significant differences for encoding or defining the equal sign, suggesting that deficits may reflect girls' tendency to follow taught algorithms.

Enhanced Glutathione Peroxidase Activity of Water-Soluble and Polyethylene Glycol-Supported Selenides, Related Spirodioxyselenuranes, and Pincer Selenuranes.

Diaryl selenides containing o-hydroxymethylene substituents function as peroxide-destroying mimetics of the antioxidant selenoenzyme glutathione peroxidase (GPx), via oxidation to the corresponding spirodioxyselenuranes with hydrogen peroxide and subsequent reduction back to the original selenides with glutathione. Parent selenides with 3-hydroxypropyl or 2,3-dihydroxypropyl groups produced the novel compounds 10 and 11, respectively, with greatly improved aqueous solubility and catalytic activity. The phenolic derivative 28 displayed similarly ameliorated properties and also modest radical-inhibiting antioxidant activity, as evidenced by an assay based on phenolic hydrogen atom transfer to the stable free radical DPPH. In contrast, several selenides that afford pincer selenuranes (e.g., 20 and 21) instead of spiroselenuranes upon oxidation showed inferior catalytic activity. Several selenide analogues were attached to polyethylene glycol (PEG) oligomers, as PEG substituents can improve water solubility and bioavailability, while retarding clearance. Again, the PEG derivatives afforded remarkable activity when oxidation generated spirodioxyselenuranes and diminished activity when pincer compounds were produced. Several such compounds proved to be ca. 10- to 100-fold catalytically superior to the diaryl selenides and their spirodioxyselenurane counterparts investigated previously. Finally, an NMR-based assay employing glutathione in D2O was designed to accommodate the faster reacting water-soluble mimetics and to more closely duplicate in vivo conditions.

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.

Oxidation of Disulfides to Thiolsulfinates with Hydrogen Peroxide and a Cyclic Seleninate Ester Catalyst.

Cyclic seleninate esters function as mimetics of the antioxidant selenoenzyme glutathione peroxidase. They catalyze the reduction of harmful peroxides with thiols, which are converted to disulfides in the process. The possibility that the seleninate esters could also catalyze the further oxidation of disulfides to thiolsulfinates and other overoxidation products under these conditions was investigated. This has ramifications in potential medicinal applications of seleninate esters because of the possibility of catalyzing the unwanted oxidation of disulfide-containing spectator peptides and proteins. A variety of aryl and alkyl disulfides underwent facile oxidation with hydrogen peroxide in the presence of catalytic benzo-1,2-oxaselenolane Se-oxide affording the corresponding thiolsulfinates as the principal products. Unsymmetrical disulfides typically afforded mixtures of regioisomers. Lipoic acid and N,N'-dibenzoylcystine dimethyl ester were oxidized readily under similar conditions. Although isolated yields of the product thiolsulfinates were generally modest, these experiments demonstrate that the method nevertheless has preparative value because of its mild conditions. The results also confirm the possibility that cyclic seleninate esters could catalyze the further undesired oxidation of disulfides in vivo.

LGR5 positivity defines stem-like cells in colorectal cancer.

Like normal colorectal epithelium, colorectal carcinomas (CRCs) are organized hierarchically and include populations of cells with stem-like properties. Leucine-rich-repeat-containing G-protein-coupled receptor 5 (LGR5) is associated with these stem cells in normal colorectal epithelium; however, the precise function of LGR5 in CRC remains largely unknown. Here, we analyzed the functional and molecular consequences of short hairpin RNA-mediated silencing of LGR5 in CRC cell lines SW480 and HT-29. Additionally, we exposed Lgr5-EGFP-IRES-CreERT2 mice to azoxymethane/dextrane sodium sulfate (AOM/DSS), which induces inflammation-driven colon tumors. Tumors were then flow-sorted into fractions of epithelial cells that expressed high or low levels of Lgr5 and were molecularly characterized using gene expression profiling and array comparative genomic hybridization. Silencing of LGR5 in SW480 CRC cells resulted in a depletion of spheres but did not affect adherently growing cells. Spheres expressed higher levels of several stem cell-associated genes than adherent cells, including LGR5. Silencing of LGR5 reduced proliferation, migration and colony formation in vitro and tumorigenicity in vivo. In accordance with these results, NOTCH signaling was downregulated upon LGR5 silencing. In AOM/DSS-induced colon tumors, Lgr5 high cells showed higher levels of several stem cell-associated genes and higher Wnt signaling than Lgr5 low tumor cells and Lgr5 high normal colon cells. Array comparative genomic hybridization revealed no genomic imbalances in either tumor cell fraction. Our data elucidate mechanisms that define the role of LGR5 as a marker for stem-like cells in CRC.

Effects of methoxy substituents on the glutathione peroxidase-like activity of cyclic seleninate esters.

Cyclic seleninate esters function as mimetics of the antioxidant enzyme glutathione peroxidase and catalyze the reduction of hydrogen peroxide with a stoichiometric thiol. While a single electron-donating methoxy substituent para to the selenium atom enhances the catalytic activity, m-methoxy groups have little effect and o-methoxy substituents suppress activity. The effects of multiple methoxy groups are not cumulative. This behavior can be rationalized by opposing mesomeric and steric effects. Oxidation of the product disulfide via its thiolsulfinate was also observed.

Aneuploidy, oncogene amplification and epithelial to mesenchymal transition define spontaneous transformation of murine epithelial cells.

Human epithelial cancers are defined by a recurrent distribution of specific chromosomal aneuploidies, a trait less typical for murine cancer models induced by an oncogenic stimulus. After prolonged culture, mouse epithelial cells spontaneously immortalize, transform and become tumorigenic. We assessed genome and transcriptome alterations in cultures derived from bladder and kidney utilizing spectral karyotyping, array CGH, FISH and gene expression profiling. The results show widespread aneuploidy, yet a recurrent and tissue-specific distribution of genomic imbalances, just as in human cancers. Losses of chromosome 4 and gains of chromosome 15 are common and occur early during the transformation process. Global gene expression profiling revealed early and significant transcriptional deregulation. Chromosomal aneuploidy resulted in expression changes of resident genes and consequently in a massive deregulation of the cellular transcriptome. Pathway interrogation of expression changes during the sequential steps of transformation revealed enrichment of genes associated with DNA repair, centrosome regulation, stem cell characteristics and aneuploidy. Genes that modulate the epithelial to mesenchymal transition and genes that define the chromosomal instability phenotype played a dominant role and were changed in a directionality consistent with loss of cell adhesion, invasiveness and proliferation. Comparison with gene expression changes during human bladder and kidney tumorigenesis revealed remarkable overlap with changes observed in the spontaneously transformed murine cultures. Therefore, our novel mouse models faithfully recapitulate the sequence of genomic and transcriptomic events that define human tumorigenesis, hence validating them for both basic and preclinical research.

Fluxional cyclic seleninate ester: NMR and computational studies, glutathione peroxidase-like behavior, and unexpected rearrangement.

The oxidation of allyl selenide 12 with hydrogen peroxide produced the corresponding allyl selenurane 14, the cyclic seleninate ester 4, or the rearranged O-allyl seleninate ester 18, dependng on the conditions. Crossover experiments with selenide 12 and its deuterated crotyl analogue 27 indicated an intramolecular rearrangement that proceeds by an intramolecular pathway where the allyl or crotyl group is translocated via its distal carbon atom to the hydroxymethyl functionality. Variable-temperature NMR experiments with cyclic seleninate ester 4 revealed fluxional behavior at room temperature that was catalyzed by trifluoroacetic acid. Computational studies indicated an activation energy of 12.3 kcal mol(-1) for hydroxyl interchange at selenium, comparable to the value of 15.5 kcal mol(-1) derived from the NMR experiments. The glutathione peroxidase-like activity of 4 was measured in an assay where the catalysis of the reduction of hydrogen peroxide with benzyl thiol was monitored by the appearance of dibenzyl disulfide. The catalytic activity of 4 was double that observed with the unsubstituted seleninate ester 2 but was limited by the competing accumulation of the relatively inert selenenyl sulfide 32, resulting in a deactivation pathway that competes with the primary catalytic cycle.

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 children's 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.

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 children's 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 children's performance when children have low knowledge of the objects but hinder performance when children have high knowledge of the objects.

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.

Spontaneous transformation of murine epithelial cells requires the early acquisition of specific chromosomal aneuploidies and genomic imbalances.

Human carcinomas are defined by recurrent chromosomal aneuploidies, which result in a tissue-specific distribution of genomic imbalances. In order to develop models for these genome mutations and to determine their role in tumorigenesis, we generated 45 spontaneously transformed murine cell lines from normal epithelial cells derived from bladder, cervix, colon, kidney, lung, and mammary gland. Phenotypic changes, chromosomal aberrations, centrosome number, and telomerase activity were assayed in control uncultured cells and in three subsequent stages of transformation. Supernumerary centrosomes, binucleate cells, and tetraploidy were observed as early as 48 hr after explantation. In addition, telomerase activity increased throughout progression. Live-cell imaging revealed that failure of cytokinesis, not cell fusion, promoted genome duplication. Spectral karyotyping demonstrated that aneuploidy preceded immortalization, consisting predominantly of whole chromosome losses (4, 9, 12, 13, 16, and Y) and gains (1, 10, 15, and 19). After transformation, focal amplifications of the oncogenes Myc and Mdm2 were frequently detected. Fifty percent of the transformed lines resulted in tumors on injection into immunocompromised mice. The phenotypic and genomic alterations observed in spontaneously transformed murine epithelial cells recapitulated the aberration pattern observed during human carcinogenesis. The dominant aberration of these cell lines was the presence of specific chromosomal aneuploidies. We propose that our newly derived cancer models will be useful tools to dissect the sequential steps of genome mutations during malignant transformation, and also to identify cancer-specific genes, signaling pathways, and the role of chromosomal instability in this process.