Recharging Rational Number Understanding

In 1978, only 24% of 8th grade students in the United States correctly answered whether 12/13+7/8 was closest to 1, 2, 19, or 21 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980). In 2014, only 27% of 8th grade students selected the correct answer to the same problem, despite the ensuing forty years of effort to improve students’ conceptual understanding (Lortie-Forgues, Tian, & Siegler, 2015). This is troubling, given that 5th grade students’ fraction knowledge predicts mathematics achievement in secondary school (Siegler et al, 2012) and that achievement in math is linked to greater life outcomes (Murnane, Willett, & Levy, 1995). General rational number knowledge (fractions, decimals, percentages) has proven problematic for both children and adults in the U.S. (Siegler & Lortie-Forgues, 2017). Though there is debate about which type of rational number instruction should occur first, it seems it would be beneficial to use an integrated approach to numerical development consisting of all rational numbers (Siegler, Thompson, & Schneider, 2011). Despite numerous studies on specific types of rational numbers, there is limited information about how students translate one rational number notation to another (Tian & Siegler, 2018). The present study seeks to investigate middle school students’ understanding of the relations among fraction, decimal, and percent notations and the influence of a daily, brief numerical magnitude translation intervention on fraction arithmetic estimation. Specifically, it explores the benefits of Simultaneous presentation of fraction, decimal, and percent equivalencies on number lines versus Sequential presentation of fractions, decimals, and percentages on number lines. It further explores whether rational number review using either Simultaneous or Sequential representation of numerical magnitude is more beneficial for improving fraction arithmetic estimation than Rote practice with fraction arithmetic. Finally, it seeks to make a scholarly contribution to the field in an attempt to understand students’ conceptions of the relations among fractions, decimals, and percentages as predictors of estimation ability. Chapter 1 outlines the background that motivates this dissertation and the theories of numerical development that provide the framework for this dissertation. In particular, many middle school students exhibit difficulties connecting magnitude and space with rational numbers, resulting in implausible errors (e.g., 12/13+7/8=1, 19, or 21, 87% of 10>10, 6+0.32=0.38). An integrated approach to numerical development suggests students’ difficulty in rational number understanding stems from how students incorporate rational numbers into their numerical development (Siegler, Thompson, & Schneider, 2011). In this view, students must make accommodations in their whole number schemes when encountering fractions, such that they appropriately incorporate fractions into their mental number line. Thus, Chapter 1 highlights number line interventions that have proven helpful for improving understanding of fractions, decimals, and percentages. In Chapter 2, I hypothesize that current instructional practices leave middle school students with limited understanding of the relations among rational numbers and promote impulsive calculation, the act of taking action with digits without considering the magnitudes before or after calculation. Students who impulsively calculate are more likely to render implausible answers on problems such as estimating 12/13+7/8 as they do not think about the magnitudes (12/13 is about equal to one and 7/8 is about equal to one) before deciding on a calculation strategy, and they do not stop to judge the reasonableness of an answer relative to an estimate after performing the calculation. I hypothesize that impulsive calculation likely stems from separate, sequential instructional approaches that do not provide students with the appropriate desirable difficulties (Bjork & Bjork, 2011) to solidify their understanding of individual notations and their relations. Additionally, in Chapter 2, I hypothesize that many middle school students are unable to view equivalent rational numbers as being equivalent. This hypothesis is based on the documented tendency of many students to focus on the operational rather than relational view of equivalence (McNeil et al., 2006). In other words, students typically focus on the equal sign as signal to perform an operation and provide an answer (e.g., 3+4=7) rather than the equal sign as a relational indicator (e.g., 3+4=2+5). Moreover, this hypothesis is based on the documented whole number bias exhibited by over a quarter of students in 8th grade, such that students perceived equivalent fractions with larger parts as larger than those with smaller parts (Braithwaite & Siegler, 2018b). If middle school students are unable to perceive equivalent values within the same notation as equivalent in size, it seems probable that they might also struggle perceiving equivalent rational numbers as equivalent across notations. This is especially true in light of evidence that many teachers often do not use equal signs to describe equivalent values expressed as fractions, decimals, and percentages (Muzheve & Capraro, 2012). Chapter 2 underscores the importance of highlighting the connections among notations by discussing the pivotal role of notation connections in prior research (Moss & Case, 1999) and the benefit of interleaved practice in math (Rohrer & Taylor, 2007). Finally, I propose a plan for improving students’ understanding of rational numbers through linking notations with number line instruction, as an integrated theory of numerical development (Siegler et al, 2011) suggests that all rational numbers are incorporated into one’s mental number line. Chapter 3 details two experiments that yielded empirical evidence consistent with the hypotheses that students do not perceive equivalent rational numbers as equivalent in size and that this lack of integrated number sense influences estimation ability. The findings identify a discrepancy in performance in magnitude comparison across different rational number notations, in which students were more accurate when presented with problems where percentages were larger than fractions and decimals than when they were presented with problems where percentages were smaller than fractions and decimals. Superficially, this finding of a percentages-are-larger bias suggests students have a bias towards perceiving percentages as larger than fractions and decimals; however, it appears this interpretation is not true on all tasks. If students always perceive percentages as larger than fractions and decimals, then their placement of percentages on the number line should be larger than the equivalent fractions or decimals. However, this was not the case. The experiments revealed that students’ number line estimation was most accurate for percentages rather than the equivalent fraction and decimal values, demonstrating that students who are influenced by the percentages-are-larger bias are most likely not integrating understanding of fractions, decimals, and percentages on a single mental number line. Furthermore, empirical evidence provided support for the theory of impulsive calculation defined earlier, such that many students perform worse when presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies than in situations without such lures. Importantly, integrated number sense, as measured by the composite score of all cross-notation magnitude comparison trials, was shown to be an important predictor of estimation ability in the presence of distracting information on number lines and fraction arithmetic estimation tasks, often above and beyond number line estimation ability and general math ability. The experiments reported in Chapter 3 also evaluated whether Simultaneous, integrated instruction of all notations improved integration of rational number notations more than Sequential instruction of the three notations or a control condition with Rote practice in fraction arithmetic. The experiments also evaluated whether the instructional condition influenced fraction arithmetic estimation ability. The findings supported the hypothesis that a Simultaneous approach to reviewing rational numbers provides greater benefit for improving integrated number sense, as measured by more improvement in the composite score of magnitude comparison across notations. However, there was no difference among conditions in fraction arithmetic estimation ability at posttest. The experiments point to potential areas for improvement in future work, which are described subsequently. Chapter 4 attempts to explore further students’ understanding of the relations among notations. For this analysis, a number of data sources were examined, including student performance on assessments, interview data, analysis of student work, and classroom observations. Three themes emerged: (1) students are employing a flawed translation strategy, where students concatenate digits from the numerator and denominator to translate the fraction to a decimal such that a/b=0.ab (e.g., 3/5=0.35). (2) percentages can serve as a useful tool for students to judge magnitude, and (3) students equate math with calculation rather than estimation (e.g., in response to being asked to estimate addition of fractions answers, a student responded, “I can’t do math, right?”). Moreover, case studies investigated the differential effect of condition (Simultaneous, Sequential, or Control) on students’ strategy use. The findings suggest that the Simultaneous approach facilitated a more developed schema for magnitude, which is crucial given that a student’s degree of mathematical understanding is determined by the strength and accuracy of connections among related concepts (Hiebert & Carpenter, 1992). Chapter 5 concludes the dissertation by discussing the contributions of this work, avenues for future research, and educational implications. Ultimately, this dissertation advances the field of numerical cognition in three important ways: (1) by documenting a newly discovered bias of middle school students perceiving percentages as larger than fractions and decimals in magnitude comparisons across notations and positing that a lack of integrating notations on the same mental number line is a likely mechanism for this bias; (2) by demonstrating that students exhibit impulsive calculation, as measured by the difference in performance between situations where students are presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies and situations that do not involve lures; and (3) by finding that integrated number sense, as measured by the composite score for magnitude comparison across notations, is a unique predictor of estimation ability, often above and beyond general mathematical ability and number line estimation. In particular, students with higher integrated number sense are more than twice as likely to correctly answer the aforementioned 12/13+7/8 estimation problem than their peers with the same number line estimation ability and general math ability. This finding suggests that integrated number sense is an important inhibitor for impulsive calculation, above estimation ability for individual fractions and a general standardized test of math achievement. Finally, this dissertation advances the field of mathematics education by suggesting instruction that connects equivalent values with varied notations might provide superior benefits over a sequential approach to teaching rational numbers. At a minimum, this dissertation suggests that more careful attention must be paid to relating rational number notations. Future work might examine the origins of impulsive calculation and the observed percentages-are-larger bias. Future research might also examine whether integrated number sense is predictive of estimation ability beyond general number sense within notations. From these investigations, it might be possible to design a more impactful intervention to improve rational number outcomes

No Child Overlooked: Mental Health Triage in the Schools

Mental health problems among children in schools are on the increase. To exercise due diligence in their responsibility to monitor and promote mental health among our nation’s children, school counselors may learn from triage systems employed in hospitals, clinics, and mental health centers. The School Counselor’s Triage Model provides school counselors with an easy-to-use, time efficient assessment tool to enable them to screen large groups of students to determine their mental health needs. By engaging in systematic mental health screening, school counselors can efficiently and effectively demonstrate their commitment to a core value of school counseling: addressing every child’s social-emotional needs

Critical behavior in the variation of GDR width at low temperature

We present the first experimental giant dipole resonance (GDR) width systematics, in the temperature region 0.8 $\sim$ 1.2 MeV for $^{201}$Tl, a near Pb nucleus, to investigate the evolution of the GDR width in shell effect & pairing dominated region. The extracted GDR widths are well below the predictions of shell effect corrected thermal shape fluctuation model (TSFM) and thermal pairing included phonon damping model. A similar behavior of the GDR width is also observed for $^{63}$Cu measured in the present work and $^{119}$Sb, measured earlier. This discrepancy is attributed to the GDR induced quadrupole moment leading to a critical point in the increase of the GDR width with temperature. We incorporate this novel idea in the phenomenological description based on the TSFM for a better understanding of the GDR width systematics for the entire range of mass, spin and temperature.Comment: Accepted for publication in Phys. Lett. B, 7 pages, 4 figure

Aluminum oxide barrier coatings on polymer films for food packaging applications

In the field of packaging, barrier layers are functional films, which can be applied to polymeric substrates with the objective of enhancing their end-use properties. For food packaging applications, the packaging material is required to preserve packaged food stuffs and protect them from a variety of environmental influences, particularly moisture and oxygen ingress and UV radiation. Aluminum metallized films are widely used for this purpose. More recently, transparent barrier coatings based on aluminum oxide or silicon oxide have been introduced in order to fulfill requirements such as product visibility, microwaveability or retortability. With the demand for transparent barrier films for low-cost packaging applications growing, the use of high-speed vacuum deposition techniques, such as roll-to-roll metallizers, has become a favorable and powerful tool. In this study, aluminum oxide barrier coatings have been deposited onto biaxially oriented polypropylene and polyethylene terephthalate film substrates via reactive evaporation using an industrial 'boat-type' roll-to-roll metallizer. The coated films have been investigated and compared to uncoated films in terms of barrier properties, surface topography, roughness and surface energy using scanning electron microscopy, atomic force microscopy and contact angle measurement. Coating to substrate adhesion and coating thickness have been examined via peel tests and transmission electron microscopy, respectively. © 2013 Elsevier B.V

Pupil responses associated with coloured afterimages are mediated by the magno-cellular pathway

Sustained fixation of a bright coloured stimulus will, on extinction of the stimulus and continued steady fixation, induce an afterimage whose colour is complementary to that of the initial stimulus; an effect thought to be caused by fatigue of cones and/or of cone-opponent processes to different colours. However, to date, very little is known about the specific pathway that causes the coloured afterimage. Using isoluminant coloured stimuli recent studies have shown that pupil constriction is induced by onset and offset of the stimulus, the latter being attributed specifically to the subsequent emergence of the coloured afterimage. The aim of the study was to investigate how the offset pupillary constriction is generated in terms of input signals from discrete functional elements of the magno- and/or parvo-cellular pathways, which are known principally to convey, respectively, luminance and colour signals. Changes in pupil size were monitored continuously by digital analysis of an infra-red image of the pupil while observers viewed isoluminant green pulsed, ramped or luminance masked stimuli presented on a computer monitor. It was found that the amplitude of the offset pupillary constriction decreases when a pulsed stimulus is replaced by a temporally ramped stimulus and is eliminated by a luminance mask. These findings indicate for the first time that pupillary constriction associated with a coloured afterimage is mediated by the magno-cellular pathway. © 2003 Elsevier Science Ltd. All rights reserved

How Intense Policy Demanders Shape Postreform Politics: Evidence from the Affordable Care Act

The implementation of the Affordable Care Act (ACA) has been a politically volatile process. The ACA\u27s institutional design and delayed feedback effects created a window of opportunity for its partisan opponents to launch challenges at both the federal and state level. Yet as recent research suggests, postreform politics depends on more than policy feedback alone; rather, it is shaped by the partisan and interest-group environment. We argue that “intense policy demanders” played an important role in defining the policy alternatives that comprised congressional Republicans\u27 efforts to repeal and replace the ACA. To test this argument, we drew on an original data set of bill introductions in the House of Representatives between 2011 and 2016. Our analysis suggests that business contributions and political ideology affected the likelihood that House Republicans would introduce measures repealing significant portions of the ACA. A secondary analysis shows that intense policy demanders also shaped the vote on House Republicans\u27 initial ACA replacement plan. These findings highlight the role intense policy demanders can play in shaping the postreform political agenda

Compilation of Giant Electric Dipole Resonances Built on Excited States

Giant Electric Dipole Resonance (GDR) parameters for gamma decay to excited states with finite spin and temperature are compiled. Over 100 original works have been reviewed and from some 70 of which more than 300 parameter sets of hot GDR parameters for different isotopes, excitation energies, and spin regions have been extracted. All parameter sets have been brought onto a common footing by calculating the equivalent Lorentzian parameters. The current compilation is complementary to an earlier compilation by Samuel S. Dietrich and Barry L. Berman (At. Data Nucl. Data Tables 38(1988)199-338) on ground-state photo-neutron and photo-absorption cross sections and their Lorentzian parameters. A comparison of the two may help shed light on the evolution of GDR parameters with temperature and spin. The present compilation is current as of January 2006.Comment: 31 pages including 1 tabl

The worlding of St. Petersburg and Shanghai: comparing cultures of communication in two cities before and after revolutions

In this article we propose an alternative model for comparative communication research. We first make the case for comparing cities, especially worlding cities outside what is traditionally called the “West.” We then explicate what we mean by comparing cultures of communication and why this offers an opportunity to reevaluate methodological nationalism and the cultural dynamics of worlding. We go on to use Shanghai and St. Petersburg as two historical examples to demonstrate how worlding cities (1) compel us to see cultural hybridization as a historical process; (2) offer good opportunities to observe contested elements of cultures; (3) make it possible to analyze cities as texts that are always connected with, but not necessarily contained by the nation

Self-consistent Green's function method for nuclei and nuclear matter

Recent results obtained by applying the method of self-consistent Green's functions to nuclei and nuclear matter are reviewed. Particular attention is given to the description of experimental data obtained from the (e,e'p) and (e,e'2N) reactions that determine one and two-nucleon removal probabilities in nuclei since the corresponding amplitudes are directly related to the imaginary parts of the single-particle and two-particle propagators. For this reason and the fact that these amplitudes can now be calculated with the inclusion of all the relevant physical processes, it is useful to explore the efficacy of the method of self-consistent Green's functions in describing these experimental data. Results for both finite nuclei and nuclear matter are discussed with particular emphasis on clarifying the role of short-range correlations in determining various experimental quantities. The important role of long-range correlations in determining the structure of low-energy correlations is also documented. For a complete understanding of nuclear phenomena it is therefore essential to include both types of physical correlations. We demonstrate that recent experimental results for these reactions combined with the reported theoretical calculations yield a very clear understanding of the properties of {\em all} protons in the nucleus. We propose that this knowledge of the properties of constituent fermions in a correlated many-body system is a unique feature of nuclear physics.Comment: 110 pages, accepted for publication on Prog. Part. Nucl. Phy