This workshop introduces users to the Spirals Excel file which was developed to encourage exploration of the relation between geometric and numeric patterns for young learners. This can create a firmer understanding of multiplication, division, and fractions even before these concepts are formally introduced in a highly interactive and visual fashion. Such explorations may feel more like creating art than learning math. Although developed for independent exploration, the Spirals file could readily be employed in classroom settings for students with Excelenabled tablets, or via smart-boards to amplify a variety of topics in elementary and early secondary education. Introduction and Motivation Mathematics is the study of patterns, some of which are numeric and some of which are geometric. Learning is enhanced if one is able to see a connection between these two ways of knowing. This is particularly true for those who are transitioning from play to more formal learning situations. Research suggests that technology can enhance learning in the early elementary years, especially if learning geometric shapes [6]. When the images created are artistic in nature, young learners may view this exploration as play. One could argue that purposeful play creates situations where the individual becomes primed to learn from that play. This is consistent with Stein’s view of teaching mathematics based on The Triex: Explore, Extract, Explain [5]. He argues that we should encourage our students to explore and gather data, seek to extract some sort of pattern from those explorations and ultimately seek to explain what has been found. He notes that even if an individual is unable to achieve the level of “Explain,” they have been primed to appreciate the explanation provided by an instructor or another student [5]. Such play encourages learning and deepens understanding and can certainly occur prior to formally learning the concepts of multiplication and division in the middle elementary years [3]. The Spirals Excel file allows users to create polygons, stars, and spirals via three click-controlled parameters. The file begins by using the strategy proposed in [1] of teaching whole number patterns in the early elementary grades using regular polygons and stars based on two parameters: n, the number of vertices (and sides) in a regular polygon; and J, the number of jumps between vertices. The file adds one additional whole number parameter, the radius reduction factor, r. Figure 1 shows a bit of the scope of images one can create using Spirals. These images range from traditional spirals such as 1(a) to the swirling “twisted polygons” shown in 1(b) – 1(d). In the following sections we will lay out how workshop participants will explore creating images such as these, as well as more complex visual images, using just three parameters. (a) (b) (c) (d) Figure 1: Sample images created with the Spirals Excel file. Bridges 2021 Conference Proceedings

This paper describes a novel approach for exploring counting and pattern recognition in early elementary classrooms that ties numerical patterns to geometric patterns. Active-learning exercises that start out face-to-face (or using the virtual Class Circle sheet of the Polygons and Stars Excel file) can be explored quickly and more fully using the Polygons and Stars sheets. This file can be used by the teacher in the classroom, as well as by students undertaking independent explorations. Such explorations help energize young learners; as they explore, extract, and explain what they have found, they begin to recognize the beauty inherent in mathematical patterns.

This paper describes a novel approach for exploring counting and pattern recognition in early elementary classrooms that ties geometric patterns to numeric patterns. Active-learning exercises that start out face-to-face (or using the virtual Class Circle sheet of the Polygons and Stars Excel file) can be explored quickly and more fully using the Polygons and Stars sheets. This file can be used by the teacher in the classroom, as well as by students undertaking independent explorations. Such explorations help energize young learners; as they explore, extract, and explain what they have found, they begin to recognize the beauty inherent in mathematical patterns.

OBJECTIVE:Education abroad can be perceived as riskier for student health/safety than domestic study, but little data exists to test this assumption. We investigated one facet of student safety by comparing college student mortality rates occurring during educational programs abroad with rates occurring on U.S. campuses. Methods: Two insurers provided information about the duration of education abroad experiences for 1,342,405 individuals insured over seven calendar years and details of repatriation of remains claims during that period. Mortality rates for students abroad were calculated and compared with existing data from students on U.S. campuses. Results: Students are less likely to die during study abroad experiences than on U.S. campuses. This is driven by reduced likelihood of death among male students studying abroad. Conclusions: Results can guide staff and student training and policies by providing professionals involved in study abroad with objective information about the relative risk of overseas study and causes of death.

This hands-on multi-class exercise allows students to physically confront the marginal trade-offs that are inherent in many economic situations. It helps students with limited mathematical training better visualize how economists represent 3-dimensional surfaces in two dimensions as indifference curves, isoquants and isoprofit contours using a balloon. This is a PowerPoint presentation (provided as a pdf), optimized for computer projection. Many students study microeconomics with limited formal mathematical training in analyzing multivariate functions. Despite this weakness, the marginal analysis that is at the core of microeconomics requires that we teach students to think in terms of marginal trade-offs and to see how these trade-offs must be balanced to achieve an optimal outcome. Although developed for managerial economics and intermediate microeconomics classes, one could use this exercise in introductory level classes for those without a background in calculus due to the geometric approach developed here.. This paper provides detailed instructions for building the balloon as well as how to use the surface of the balloon to model economic concepts such as marginal product, the economic region of production, and marginal rate of technical substitution (and their consumer side counterparts). Instructions show students how to find and interpret a point where x is more valuable than y and to compare that with the reverse situation. This protocol concludes with instructions allowing students to analyze constrained optimization visually including what the tangency condition implies regarding the marginal trade-offs inherent in the constrained optimal solution. Algebraic counterparts are provided in appendices.

URL: http://users.dickinson.edu/∼erfle/ExcelGraphics/Multicollinearity.xlsxInstructors teaching empirical techniques in managerial economics or economic and business statistics courses often face a...

In introductory and intermediate microeconomics courses, students are taught about production functions and cost functions but rarely are they shown a direct link between production and cost. If we assume production is Cobb-Douglas, then the CDProductionToCost.xlsx Excel file shows the link between short-run, SR, and long-run, LR, production and cost in geometric detail. This geometric approach is particularly useful in classes without a calculus prerequisite. The file has three interactive figure sheets and four information sheets. The figure sheets do not require a knowledge of Excel to operate; they can be manipulated simply using sliders and click-boxes. To focus attention on the graphical material presented, the equations that created each figure are hidden and the interactive figure sheets are write-protected. The information sheets are not write-protected so instructors can alter them as they wish. The accompanying paper provides additional graphical and algebraic detail and suggests alternative ways to use the file to deliver material to students. Although the main purpose of this file is to show the geometric link between production and cost, the supplementary sheets allow instructors (and students) to explore the underlying production function and costminimizing solutions that allow one to link production to cost.

This paper extends the strategy espoused by Holmgren (2017) to employ discrete choice utility tables to teach the consumer choice model in introductory microeconomics classes.These extensions show instructors how to explain the equal marginal utility per dollar spent rule using discrete choice tables.One can also examine the more basic issue of the meaning of marginal utility and the ordinal nature of utility using these tables.This paper also provides additional tables that allow instructors to distinguish between goods that are substitutes or complements and to show why an individual may choose to not purchase a good once its price increases sufficiently.Finally, this hyperlink provides an Excel file that allows instructors to create new tables as well as graphical overlays that instructors may use to bridge between tabular display and graphical image.

Price points are an important consideration in the analysis of optimal pricing. We contribute to this literature by proposing that consumers have total expenditure barriers when purchasing a bundle of items from a seller. We suggest a rationale for, and test the proposition that, individuals set reservation expenditure levels for meal purchases using salient points in multiples of $5. We test this hypothesis using a sample of online orders from 257 restaurants clustered into five restaurant types. Single meal orders were identified by removing transactions having more than one entrée leaving 52,583 transactions for individual level analysis. Orders were analyzed by Payment Type (cash versus credit) × Meal Category (weekday lunch, weekday dinner, weekend lunch, weekend dinner) for each restaurant cluster. Two distributions are examined, final ticket amount and proportion of individuals with a particular base cost that added an additional item of $3 or less to the order. The data demonstrate that final ticket amounts peak just below a multiple of $5 and order adding behavior is greater just above than just below a multiple of $5. Both distributions exhibit strong evidence of expenditure barrier purchasing for dinner purchases and mixed evidence of such purchasing for lunch purchases.

Histograms of push-up and curl-up performances of middle school students show that students stop at multiples of five more often than random processes suggest. The natural question arises: do students who stop at these focal outcomes settle for them or did they strive to achieve them? Erfle and Gelbaugh (2013) and Erfle (2014b) argue that the latter is true and that these students are more motivated than their less focal peers. This motivation to perform transfers across activities, especially for focal push-up performers. The present chapter examines whether students implicitly use other counting bases in their physical activity performances. The relative performance of those who count by other bases are also examined. A performance ending in a multiple of k may not be the result of an intentional strategy of skip counting by ks. In the absence of student survey information regarding skip counting strategies we must infer intentionality to perform using a specific base from actual performance outcomes. As a result, one task is to define intentionality to attain or avoid a specific outcome. If N students complete performances and k is a counting base, then having more than N/k students who end their performance in multiples of k provides prima fascia evidence of counting by k. Similarly, having fewer than N/k students who end their performance just below multiples of k offers further evidence of counting by k. Evidence of both behaviors exists for a number of bases for both activities. Average performance profiles for those who end in multiples of k as well as those achieving a remainder of k-1 are examined as well.

Since muscle is more dense than fat, athletes tend to have greater mass and BMI than similarly sized non-athletes. Comparing direct adiposity measures and BMI confirms that BMI is a biased proxy for adiposity for elite athletes. A similar bias should exist for non-elite athletes as well as fit individuals. This paper provides a methodology for indirectly estimating the size of the fitness bias in BMI using median physical activity performances. Approximately 30% of females and 33% of males are fit using this definition. Using data from 9 062 students, regressions suggest 3.1, 95% CI [0.9, 5.3], of a female’s BMI percentile of 85, and 3.6% of her weight, CI [1.6%, 5.6%], is due to being fit, but 5.6, CI [3.3, 7.9], of a male’s BMI percentile of 85, and 5.9% of his weight, CI [3.9%, 7.9%], is due to being fit. These increases in weight are smaller than, but consistent with, the bias of more than 20% for elite athletes. Strong performance on individual physical activities decreases BMI percentile and BMI, but doing well on multiple physical activities has the reverse effect. This provides evidence of a fitness bias. BMI report cards should include the caveat that BMI may overstate the adiposity status of fit children.

"On Using a Barter Edgeworth Box to Discuss Efficiency Early in the Semester." The Journal of Economic Education, 46(4), pp. 440–441

BACKGROUND:In 2009, the Pennsylvania Department of Health developed the Active Schools Program (ASP) which required 30 minutes of daily physical education (PE) in middle schools to reduce childhood obesity. This investigation evaluated the ASP effects on physical fitness and weight status in middle school adolescents throughout 1 academic year.METHODS:A quasi-experimental design was used to recruit middle schools into an intervention group (N = 30) or control group (N = 9).RESULTS:Physical fitness outcomes had larger intervention effects than weight status outcomes. These effects were most profound among at-risk students. Multiple linear regression analysis provided a best-guess effect of daily PE on body mass index (BMI) percentile of -1.2, 95% confidence interval (CI) (-1.9, -0.5) for at-risk females and -0.8, 95% CI (-1.5, -0.1) for at-risk males. Much of this benefit is attributable to the differential increase in physical fitness achieved by students with the benefit of having daily PE.CONCLUSIONS:Thirty minutes of daily PE can be considered a scientific approach to ameliorate health outcomes in at-risk middle school adolescents, particularly among females. Improvements on BMI percentile among at-risk youth are presaged by greater improvements in physical fitness. This investigation supports a school-based approach aimed to improve behavioral risk factors as a means to reduce childhood obesity.

Background. (Erfle & Gelbaugh, 2013) and (Erfle, 2014) examined a regular irregularity in physical activity performance histograms for curl-ups and push-ups from a sample of more than 9,000 middle-school students. These histograms showed periodic spikes at 5 and 10 unit intervals. They showed that students who used focal counting on one event were more likely to do so on another event, or on the same event at a later assessment. They also found that students who ended at these focal endings outperformed their non-focal peers on these fitness tests. They found that males were more likely to be focal than females. Methods. This chapter examines focal proclivity and performance using two Male U.S. Service Academy Cadet convenience sample datasets. One dataset is of 520 Cadets who did at least one pull-up, push-up, and sit-up; the other is of 301 Cadets who did at least one cadence pull-up, (CPU), and one 175-pound bench press repetition, (BP rep). Results. Neither dataset exhibits the pronounced regular spikes seen in the middle-school data. We therefore expand our analysis to counting by bases other than 5 and 10. The BP-CPU dataset exhibits focal counting by 2, 4, 6 and 8 for BP reps and by 5s for CPU. There is limited evidence of focal counting in the PullPush-Sit dataset with sit-ups being the only event with significant focal counting (by base 5). Pull-up performance by those who count by 4s have higher pull-up performance than those who do not count by 4s (∆M = 0.98, p = .01). Those who count BP reps by 2s and 4s significantly outperform those who do not on both BP reps and CPUs. Regression models suggest the counting BP reps by 6s leads to 2.91 more BP reps, p < .001, all else held constant.

This article examines the proclivity and performance attributes of focal students across time and activities using data from 9,345 students. Three systematic focal behavior partitions are examined: Across activities, across time, and across activities and time. A student’s performance is focal if it ends in 0 or 5 for push-ups and 0 for curl-ups. Chi-square tests confirm that individual focal outcomes and systematic focal outcomes occur more frequently than random processes would suggest. In each instance, the only cell that is less populated than random processes would suggest is the one that exhibits no systematic focal behavior and the cell that exhibits the greatest deviation from expected is the full focal cell. Focal students outperform their peers on three activities at two assessments. Students with two-systematically focal outcomes have superior performance to students with no systematic focal outcomes but inferior performance to those with three or four focal outcomes.

Histograms of push-ups and curl-ups from a sample of more than 9,000 students show periodic spikes at five and 10 unit intervals. This article argues that these spikes are related to focal points, a game theoretic concept popularized by Nobel Laureate Thomas Schelling. Being focal on one test makes one more likely to be focal on the other. Focal students (whose push-up score is a multiple of 5 and whose curl-up score is a multiple of 10) behave differently from their non-focal peers. They are more likely athletic, older, and male. Focal students, on average, did 2.2 more push-ups, 1.7 more curl-ups, and ran the mile 15 seconds faster than non-focal students, even controlling for these covariates of performance. By contrast, being focal on a single activity did not produce a statistically significant mile time difference. Students who systematically stop at focal outcomes appear differentially motivated toward physical activity performance.

This article presents a test of the argument that vulnerability to economic dislocation, primarily as it is represented by economic homogeneity in communities, exercises a constraint on levels of support for Quebec nationalism. The guiding hypothesis is that residents of economically vulnerable ridings will be reluctant to accept the risks attached with political moves toward independence. The authors employ data on the level of support for the Bloc Québécois in the federal elections of 1993 and 1997, and the proportion voting "Yes" in the 1995 sovereignty referendum in Quebec, in the province's 75 federal electoral districts as their measures of support for sovereignty. Results suggest that there is a relationship between the geographies of potential economic vulnerability and the level of nationalist support, particularly evident in the 1995 referendum voting and the 1997 election. Calculations involving perceived economic vulnerability and risk remain as powerful defenders of the political status quo in Quebec.

Managerial economics is one of the most applied areas of economics. Managerial economics teaches students how to use the tools of economics to make decisions. As a result, this class is most effective if it simulates what students face when working in a business setting. Students should therefore be taught using the same toolkit on which business managers rely. Today, that means using Excel as the platform on which to teach the course. Excel is integrated into the course content not only during computer labs and in homework but also during class. Such integration requires more than a set of spreadsheet exercises to accompany a text; it requires Excel programs that faculty can use to interactively introduce concepts during lectures. This article describes a series of exercises and programs that teach students how to use Excel, and how to use Excel to analyze data needed to make reasoned decisions.

The paper presents an ecological analysis evaluating two explanatory perspectives on third/minor party advances in recent English general elections. The impact of community cohesion on third/minor party mobilisation is complex and differentiated. Cohesion stemming from the socio-economic homogeneity of constituency environments tends to buttress support for the two main class-based parties. However, communal cohesion stemming from workplace-residence proximity tends to favour the channeling of collective protest into third/minor parties. Third/minor party advances are also shown to be positively related to prior situations of major party dominance in constituencies, particularly in the 1983 election.

Abstract.The paper presents an ecological analysis evaluating two explanatory perspectives on third/minor party advances in recent English general elections. The impact of community cohesion on third/minor party mobilisation is complex and differentiated. Cohesion stemming from the socio‐economic homogeneity of constituency environments tends to buttress support for the two main class‐based parties. However, communal cohesion stemming from workplace‐residence proximity tends to favour the channeling of collective protest into third/minor parties. Third/minor party advances are also shown to be positively related to prior situations of major party dominance in constituencies, particularly in the 1983 election.