BACKGROUNDThe Pediatrics Milestone Project uses behavioral anchors, narrative descriptions of observable behaviors, to describe learner progression through the Accreditation Council for Graduate Medical Education competencies. Starting June 2014, pediatrics programs were required to submit milestone reports for their trainees semiannually. Likert-type scale assessment tools were not designed to inform milestone reporting, creating a challenge for Clinical Competency Committees.OBJECTIVETo determine if milestone-based assessments better stratify trainees by training level compared to Likert-type assessments.METHODSWe compared assessment results for 3 subcompetencies after changing from a 5-point Likert scale to milestone-based behavioral anchors in July 2013. Program leadership evaluated the new system by (1) comparing PGY-1 mean scores on Likert-type versus milestone-based assessments; and (2) comparing mean scores on the Likert-type versus milestone-based assessments across PGY levels.RESULTSMean scores for PGY-1 residents were significantly higher on the prior year's Likert-type assessments than milestone-based assessments for all 3 subcompetencies (P < .01). Stratification by PGY level was not observed with Likert-type assessments (eg, interpersonal and communication skills 1 [ICS1] mean score for PGY-1, 3.99 versus PGY-3, 3.98; P  =  .98). In contrast, milestone-based assessments demonstrated stratification by PGY level (eg, the ICS1 mean score was 3.06 for PGY-1, 3.83 for PGY-2, and 3.99 for PGY-3; P < .01 for PGY-1 versus PGY-3). Significantly different means by trainee level were noted across 21 subcompetencies on milestone-based assessments (P < .01 for PGY-1 versus PGY-3).CONCLUSIONSInitial results indicate milestone-based assessments stratify trainee performance by level better than Likert-type assessments. Average PGY-level scores from milestone-based assessments may ultimately provide guidance for determining whether trainees are progressing at the expected pace.

BACKGROUND:Program evaluation is important for assessing the effectiveness of the residency curriculum. Limited resources are available, however, and curriculum evaluation processes must be sustainable and well integrated into program improvement efforts.INTERVENTION:We describe the pediatric Clinical Skills Fair, an innovative method for evaluating the effectiveness of residency curriculum through assessment of trainees in 2 domains: medical knowledge/patient care and procedure. Each year from 2008 to 2011, interns completed the Clinical Skills Fair as rising interns in postgraduate year (PGY)-1 (R1s) and again at the end of the year, as rising residents in PGY-2 (R2s). Trainees completed the Clinical Skills Fair at the beginning and end of the intern year for each cohort to assess how well the curriculum prepared them to meet the intern goals and objectives.RESULTS:Participants were 48 R1s and 47 R2s. In the medical knowledge/patient care domain, intern scores improved from 48% to 65% correct (P < .001). Significant improvement was demonstrated in the following subdomains: jaundice (41% to 65% correct; P < .001), fever (67% to 94% correct; P < .001), and asthma (43% to 62% correct; P  =  .002). No significant change was noted within the arrhythmia subdomain. There was significant improvement in the procedure domain for all interns (χ(2)  =  32.82, P < .001).CONCLUSIONS:The Clinical Skills Fair is a readily implemented and sustainable method for our residency program curriculum assessment. Its feasibility may allow other programs to assess their curriculum and track the impact of programmatic changes; it may be particularly useful for program evaluation committees.

In this paper, we will describe the challenges and insights gained from conducting professional development workshops aimed at helping faculty prepare materials to support the development of students' quantitative skills in different disciplinary contexts. We will examine some of the mistakes we made, and misconceptions we had, in conducting the workshops and describe how we adapted in successive years to improve the curriculum materials developed during the workshops. In particular, we will discuss how we encouraged faculty to focus on clarifying student learning goals and developing effective assessment.Much of the creative intellectual work currently be- ing conducted is occurring at the borders between disciplines and across disciplines, and there is a growing interest in bringing more interdisciplinary inquiry into the undergraduate curriculum. Yet our lack of familiarity and comfort with disciplinary frameworks outside the ones in which we were educated and work can impede that progress, limiting our ability to introduce new ideas and methods into our teaching.. In this paper, we will describe the challenges and insights gained from conducting profes- sional development workshops aimed at helping faculty prepare materials to support the development of students' quantitative skills in different disciplinary contexts. These disciplines included nursing, geography, education, soci- ology and history, in addition to the natural sciences and economics (Vacher & Lardner, 2010). In this report which summarizes our project evaluation findings, we examine some of the mistakes we made, and misconceptions we had, in conducting the workshops and describe how we adapted in successive years to improve the curriculum materials developed during the workshops. The faculty development project was entitled Spreadsheets Across the and was funded from 2005 - 2008 by the National Science Foundation (NSF DUE # 0442629).In particular, we overestimated participating teach- ers' understanding of the extent of their students' quantita- tive knowledge and understanding. Many of the modules written early in the project had unrealistic expectations of what students knew and how much could be learned from a particular module. We also underestimated the extent to which the participating teachers' interest in learning more about the technical aspects of Power Point and Ex- cel would divert the focus on student learning. We will explore these issues in more depth below and discuss the interventions we made to address these problems.BackgroundThe idea of Writing Across the Curriculum (WAC) was promoted in the United Kingdom in the 1960's and was first introduced in the early 1970's in the US. By the 1990's the idea was widely accepted and was being enthusiastically promoted by high-ranking college and university administrators (Bazerman et al., 2005). The idea of Math Across the Curriculum had a later start and to date has not yet gotten the kind of attention that WAC has. Related to, but distinct from, Math Across the Cur- riculum, the notion of quantitative literacy (also known as numeracy) has, in the last 10 years, captured the interest of many people concerned about undergraduate education. This interest goes beyond mathematics educators and extends to faculty working in many different fields.In 2001, Lynn Steen (National Council on Education and the Disciplines, 2001) edited a volume entitled Math- ematics and Democracy: The Case for Quantitative Literacy in which he states:Quantitatively literate citizens need to know more than formulas and equations. They need a predisposition to look at the world through mathematical eyes, to see the benefits (and risks) of thinking quantitatively about com- monplace issues, and to approach complex problems with confidence in the value of careful reasoning. Quantitative literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. …

The authors examined how undergraduate students negotiated roles and developed a shared understanding of mathematics while working together on computer-based modules. The subjects were videotaped while working on these modules and their computer output was simultaneously collected on a separate videotape. Examination of the data tentatively suggests ways pairs of students approach academic work and ways they interact as they process and organize knowledge. This study builds on an exploratory study the authors conducted that generated a set of research questions addressing the nature of learning in interactive technological environments.

We identify and analyze some widespread problems with the implementation of studentcentered instruction in introductory college science and mathematics laboratory courses. Specifically, we observe potential problems with interactions between the instructor and individual students, interactions between the instructor and small groups of students, and the instructor’s ability to monitor the learning environment. We describe our underlying assumptions regarding the purpose and nature of student-centered introductory college laboratory course, and analyze the problems that we identify using these assumptions. We provide practical suggestions for dealing with each category of problems.

One of the chief obstacles for student retention in the Science, Technology, Engineering, or Math (STEM) disciplines is the difficulty in transitioning from high school to college mathematics and sciences. This is especially true for first generation college students and other disadvantaged populations. Building on the work of David Bressoud and others, mathematics educators and support staff at Duke University, a highly selective private university in North Carolina, aim to build support structures to enable these students to successfully navigate the transition. Bressoud writes in the Chronicle of Higher Education that the pressure to offer calculus in high school is "pushing underprepared students prematurely into the subject." He suggests that if we are to meet the challenges created by the growth of high-school calculus we must re-examine first-year college mathematics. His observations are consistent with the views of many mathematics faculty, especially at selective colleges and universities where students entering college with aspirations to enter the STEM disciplines often become discouraged during first-year mathematics courses. Students entering selective universities with less preparation than the average student at the university are less likely to graduate with degrees in the STEM disciplines. At Duke, such students with aspirations of majoring in the STEM disciplines start their math sequence in Math 25L, Laboratory Calculus with Functions I. The class is recommended for students with a Math SAT score below 680. This score puts students in the class well below the average score at the university. Students in Math 25L are far more likely than the average Duke student to have come from lower-income families or be first-generation college students. Thus they often have not had the same educational opportunities as many of their peers. This paper examines a support system created through collaboration between mathematics educators and the Academic Resource Center at Duke. Hastily organized support sessions in Fall 2008 had very mixed results. In subsequent semesters, having examined Scholarship of Teaching and Learning (SoTL) literature, the collaborators decided to use Marilyn Carlson's Pre-Calculus Concept Assessment (PCA) as a diagnostic. Gaps were diagnosed in the students' understanding of functions as processes, and in their metacognitive skills. Support structures were adapted to take these issues into account, resulting in far greater success in 2010/11 and 2011/12.

Background: The volume of information that physicians must learn is increasing; yet, trainee educational time is limited. Many experts propose using trainees’ learning preferences to guide teaching. However, data regarding predominant learning preferences within pediatrics are limited. Aim: Identify predominant learning preferences among pediatric residents in a Residency Training Program. Methods: The Visual–Aural–Read/Write–Kinesthetic (VARK) questionnaire and Kolb Learning Style Inventory (LSI) were administered anonymously to 50 pediatric residents. Results: Learning style assessments were completed by 50 pediatric residents. Residents were significantly more likely to be accommodating on the Kolb LSI, which is consistent with an interactive learning preference (p < 0.01); 30% demonstrated a multimodal preference on the Kolb LSI (Figure 1). VARK assessments demonstrated that 45 (90%) respondents were kinesthetic, which is also consistent with a significant preference for interactive learning (p < 0.01). Forty (80%) were found to be multimodal on the VARK (Figure 1). There was no association between learning preference and the residents’ anticipated career choice or level of training. Conclusions: The predominant learning preferences among a cohort of pediatric residents from a single training program were consistent with a preference for interactive learning, suggesting that some trainees may benefit from supplementation of educational curricula with additional interactive experiences. Continued investigation is needed in this area to assess the effectiveness of adapting teaching techniques to individual learning preferences.

BACKGROUND Resuscitation education for pediatric residents may be limited due to the low frequency of actual codes in children. Mock codes represent an opportunity to increase trainee education in acute resuscitations, and we designed a unique multidisciplinary mock code curriculum that uses senior pediatric residents as teachers. METHODS A novel mock code curriculum was designed and integrated into an existing night-float rotation. Our 2-tiered curriculum not only focuses on improving teaching proficiency for resident educators (REs) but also includes separate goals to augment simulation-based resuscitation education for resident participants (RPs) and the multidisciplinary staff. RESULTS Seventy-six residents (17 REs, 59 RPs) and more than 75 nurses have participated in the curriculum. After participation, 100% of residents felt that this curriculum would improve the quality of actual resuscitations, and 94% of RPs reported receiving valuable feedback. Comfort with teaching and feedback increased for REs (P < .05), and comfort in resuscitation and crisis resource management improved for RPs (P < .05). The nursing staff also felt that communication, teamwork, and collaboration improved due to implementation of this curriculum. CONCLUSIONS A unique mock code curriculum can improve resident comfort with teaching, peer facilitation, feedback, and resuscitation. Curricular interventions of this nature may also be able to improve the balance between service and education within a residency training program. As we move toward a competency based training model within graduate medical education, further investigation is needed to link educational modifications of this nature to clinical outcomes and actual resident performance.

(n + 1)th Proof of Stirling’s Formula, The, Reinhard Michel, 844 Another Continued Fraction for π , Thomas J. Pickett and Ann Coleman, 930 Another Simple Proof of the High Girth, High Chromatic Number Theorem, Simon Marshall, 68 Applied Linear Algebra. By Peter J. Olver and Chehrzad Shakiban, reviewed by Leslie Hogben and Wolfgang Kliemann, 373 Arithmetic in the Ring of Formal Power Series with Integer Coefficients, Daniel Birmajer and Juan B. Gil, 541 Arithmetic Progressions and Binary Quadratic Forms, Ayşe Alaca, Şaban Alaca, and Kenneth S. Williams, 252 Baer’s Result: The Infinite Product of the Integers Has No Basis, Stefan Schröer, 660 Borsuk-Ulam Implies Brouwer: A Direct Construction Revisited, Alexey Yu. Volovikov, 553 Bounds on the Zeros of the Derivative of a Polynomial with All Real Zeros, Aaron Melman, 145 Brunnian Spheres, Hugh Nelson Howards, 114 Carathéodory and Kobayashi Metrics and Applications in Complex Analysis, The, Steven G. Krantz, 304 Closed Curves on Spheres, W. Holsztyński, J. Mycielski, G. C. O’Brien, and S. Świerczkowski, 149 Complex Analysis as Catalyst, Steven G. Krantz, 775 Conjugate Phase Portraits of Linear Systems, Patrick D. McSwiggen and Kenneth R. Meyer, 596 Continued Fractions Approach to a Result of Feit, A, John P. Robertson and Keith R. Matthews, 346 Covers for Angleworms, Banyat Sroysang, John E. Wetzel, and Wacharin Wichiramala, 61 Curious Result Related to Kempner’s Series, A, Bakir Farhi, 933 Cycles in Graphs and Groups, William M. Kantor, 559 Derivatives and Eulerian Numbers, Grzegorz Rza̧dkowski, 458 Developing Assessment Methodologies for Quantitative Literacy: A Formative Study, Jack Bookman, Susan L. Ganter, and Rick Morgan, 911 Devil’s Platform, A, Washek F. Pfeffer, 943 Dynamical Systems and Irrational Angle Construction by Paper-Folding, Cayanne McFarlane and Wm. Douglas Withers, 355 Easy Proofs of Some Borwein Algorithms for π , Jesús Guillera, 850 Edge Detection Using Fourier Coefficients, Shlomo Engelberg, 499 Elementary Proof of Marden’s Theorem, An, Dan Kalman, 330 Elementary Proof of the Hitting Time Theorem, An, Remco van der Hofstad and Michael Keane, 753 Enumerative Algebraic Geometry of Conics, Andrew Bashelor, Amy Ksir, and Will Traves, 701 Euler Through Time: A New Look at Old Themes. By V. S. Varadarajan, reviewed by Ranjan Roy, 469 Extreme Palindromes, Kathy Q. Ji and Herbert S. Wilf, 447 Fair Majority Voting (or How to Eliminate Gerrymandering), Michel Balinski, 97 Fearless Symmetry: Exposing the Hidden Patterns of Numbers. By Avner Ash and Robert Gross, reviewed by Jeffrey Nunemacher, 79 Four Shots for a Convex Quadrilateral, Christian Blatter, 837 Freshman’s Approach to Conway’s Napkin Problem, The, Niklas Eriksen, 492 From Möbius to Gyrogroups, Abraham A. Ungar, 138

This paper compares the attitudes about mathematics of students from traditionally taught calculus classes and those from a “reformed” calculus course. The paper is based on three studies, which together present a consistent picture of student attitudes about calculus reform. The reformed course appeared to violate students' deeply held beliefs about the nature of mathematics and how it should be learned. Although during their first months in the reformed course most students disliked it, their attitudes gradually changed. One and 2 years after, reform students felt significantly more than the traditionally taught students that they better understood how math was used and that they had been required to understand math rather than memorize formulas.

ABSTRACT Duke University was the site of one of the first large scale calculus reform projects funded by the National Science Foundation (NSF) in the post-Tulane conference era. In the seven years during which Project CALC has first taught, it has undergone numerous revisions. In this paper, we will discuss these revisions and the reasons behind them. In particular, we will describe the changes we have made over the last seven years in the: mathematical content; text; software and hardware; emphasis on computational skill; amount and nature of student writing; amount and nature of student homework; and grading, testing and assessment of student learning. The reasons for these changes include: examination of the results of our formal evaluation; the attitudes of students; and the attitudes of faculty.

ABSTRACT The purpose of this study was to examine the differences between the metacognitive behaviors exhibited by experts and novices. Of particular interest was the degree to which subjects managed or controlled their behavior and the extent to which solutions, particularly by experts, were schema driven. Nine novices (college freshman) and six experts (first and second year graduate students in mathematics) were asked to think aloud while solving four mathematics problems: (1) a routine problem; (2) a problem with more than one obvious path; (3) a nonroutine problem that involved the use of the skills used in the routine problem; and (4) a problem with insufficient or contradictory information. The verbal protocols provide evidence that: experts in this study possessed and used schemas to solve problems but schema use did not fully or adequately characterize expertise; of the two aspects of metacognition - beliefs about cognition and control of cognition - beliefs played a more important role than control; and assessment of work is not in and of itself useful; assessments must be acted on.