$16. Using QR in Writing Assignments AALAC Workshop on Enhacing QR Across the Curriculum Carleton College October 11, 2014

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1 Using QR in Writing Assignments AALAC Workshop on Enhacing QR Across the Curriculum Carleton College October 11, The MegaPenny Project 1 billion pennies $16 1 trillion pennies Why are people so bad at thinking about large numbers? How do people think about numbers? 5 6

2 Subitization (of small quantities) Subitization Dehaene, S. (1997). The Number Sense. Oxford University Press. 7 8 Comparison Is this number larger or smaller than 65? Comparison Is this number larger or smaller than 65? Numerical comparisons Estimation (of large quantities) Distance effect 65 vs vs. 41 Magnitude effect 5 vs vs. 68 Dehaene, S. (1997). The Number Sense. Oxford University Press

3 Mental Number Line Counting Linguistic skill Longo, M Heuristics: Space Heuristics: Physical Space Piaget's conservation of number task 4 7 Cognitive Constraints on Cooperation was held constant ($3 to 300b; Study 1), as well as when both THE PRESENT STUDIES numeric and economic values increased (3b to 300b and $3 to To examine how representations of numeric value influence $300; Study 2). The linear model predicts no change in coopthe effect of economic rewards on cooperative behavior, we erative behavior with manipulations of numeric magnitude (e.g., manipulated numeric value, both independently of economic $3 vs. 300b); our model, however, predicts more cooperation for value (Studies 1 and 2) and in combination with it (Studies 2 and numerically larger rewards (300) than for numerically smaller 3), and observed four indices of IPD strategies: individual coop- rewards (3), regardless of economic value (3b, $3, $300). To eration, mutual cooperation, mutual defection, and forgiveness. directly test our underlying theory, Study 3 examined cooperathe IPD is defined by relations between payoffs two players earn tion under five conditions varying numeric and economic by cooperating or defecting (Fig. 1). This structure creates a value over several orders of magnitude. We predicted that dilemma in which individuals do best on any given iteration by cooperation would be better predicted by ratios of logarithmidefecting, yet overall both earn most by cooperating (Axelrod & cally compressed numeric values ln(r)/ln(t) than by ratios Hamilton, 1981). Specifically, the reward for unilateral defection of uncompressed values (R/T). (T) is greater than the reward for mutual cooperation (R), which is greater than the reward for mutual defection (P), which is in turn STUDY 1: TEMPTATION OF $3 VERSUS 300b greater than the reward for unilateral cooperation (S; see Fig. 1). Thus, the reward structure present in the IPD in which rewards Method for unilateral defection are greater than rewards for mutual Thirty-one pairs of undergraduates were randomly assigned to cooperation (i.e., when R/T < 1; Rapoport & Chammah, 1965) one of two economically equivalent payoff matrices, one earning can explain irrationally low rates of cooperation. dollars (R 5 $3; S 5 $0; T 5 $5; P 5 $1) and one earning cents Against this classical model, we hypothesized that cooperation (R 5 300b; S 5 0b; T 5 500b; P 5 100b). Pairs were initially depends on numeric structure of rewards and that manipulating separated; one was chosen as Subject and one as confederate. only numeric values of R and T would affect cooperation in the Confederates played Tit-for-Tat (TFT), initially cooperating IPD. That is, because payoffs for cooperating versus defecting and thereafter copying the Subject s behavior on the preceding are compared by brains representing numeric values logarithtrial. Subjects received no instruction on strategy but were mically (Dehaene, 1997), and because logarithmic coding fails to introduced to payoff matrices and practiced 10 IPD trials with preserve ratio information (Stevens, 1961), we expected that the Experimenter before playing the confederate; practice trials increasing numeric values of payoffs would make them less were not analyzed and served to introduce Subjects to procediscriminable, thereby reducing players temptation to defect. dures. Pairs were instructed to maximize earnings, posted after We tested our hypothesis by examining changes in cooperaellen E. Furlong andof John Opfer each 80E.trials. tive behavior when numeric value increased but economic value Heuristics: Magnitude Prisoner's Dilemma 80 Subject s Choice $1 Condition Cooperate (C) 80 Discussion Results and 3 Condition 300 Condition Although dollars $3 and cents conditions presented equivalent $300 Condition Condition economic rewards, the cents condition elicited more individual 2 cooperation, 60 F(1, 21) , prep 5.94, Zp ¼ :25, and mutual cooperation, F(1, 21) , prep 5.91, Zp 2 ¼ :21, than the dollars condition did. Similarly, the dollars condition elicited greater mutual defection, F(1, 21) , prep 5.97, Zp 2 ¼:31, 40 and a longer latency to forgive the confederate, or to cooperate after the confederate s first defection, F(1, 21) , prep 5.89, Zp 2 ¼ :18, than the cents condition did (Fig. 2). Defect (D) $3 (R) $5 (T) $0 0 (S) $1 (P) 40 Number of Trials 60 Number of Trials Cooperate (C) Defect (D) Partner s Choice 100 Condition 20 How to convey numbers to an audience? 20 Individual Mutual Mutual Cooperation Cooperation Defection STUDY 2: EFFECT OF NUMERIC VERSUS ECONOMIC VALUE ON COOPERATION Forgiveness 0 Individual Mutual Mutual Forgiveness Cooperation Cooperation Defection To test whether higher cooperation rates for 300b rewards than Fig. 2. Results from Study 1: cooperative behavior (measured by the number of trials showing individual and mutual cooperation), competitive Fig. 3. Results from Study 2: individual cooperation, cooperafor $3 rewards resulted from numeric values ofmutual rewards (300 vs. Fig. 1. Typical matrix values in the prisoner s dilemma game.defected), The game behavior (number of trials in which both players and fortion, and mutual defection rates and forgiveness latencies when values giveness latency (number of trialspayoff until a values player cooperated is defined by a mathematical relation between such that again the after 3) rather than from preference for dollars or cents,high Study 2 were numerically lowa(3b and $3 conditions) versus numerically the opponent defected) in the the prisoner s dilemma game when rewards temptation to defect (T) is greater than reward for mutual coopera(300b and $300 conditions). presented subjects with numerically equivalent payoffs of both were in dollars versus when rewards had equivalent monetary value but tion (R), which is greater than the punishment for mutual defection (P), were in cents. dollars and cents ($3, 3b; $300, 300b). Additionally, subjects in which is in turn greater than the sucker s reward (S), received when p 5.98, Z 2 ¼ :20; there was no effect of economic value, rep StudyF(2, 2 played ap computer, thereby removing social feedback. one has cooperated and one s partner has defected. 45) , prep Further, numerically large rewards Furlong, E., and Method Opfer, J. (2009). Cognitive constraints on how economic Forty-eight students were randomlyscience, assigned to20, play one of four elicited less mutual defection than numerically small ones did, rewards affect cooperation. Psychological 12 IPD games; two were identical to the games in Study 1 ( $1, n 5 12; 100b, n 5 12) and two were numerically identical to the games in Study 1 but had different units and therefore different economic values ( 1b, n 5 12: R 5 3b, S 5 0b, T 5 5b, P 5 1b; $100, n 5 12: R 5 $300, S 5 $0; T 5 $500; P 5 $100). Subjects in Study 2 played against computers that were programmed with TFT, thus behaving like the student confederates 17 F(1, 46) , prep 5.93, Zp 2 ¼ :11, with mutual defection showing no effect of economic value, F(2, 45) , prep Volume 20 Number 1 Intriguingly, players were very quick to forgive defections by the computer (M trials, SD 5 3.9), and neither numeric nor economic value influenced forgiveness number: F(1, 46) , prep 5.79; value: F(2, 45) , prep Finally, given the history of social motives producing effects 18

4 2 million plastic bottles, the number used every 5 minutes in the US 106,000 cans, the number used every 30 seconds in the US. Chris Jordan, Running the Numbers Chris Jordan, Running the Numbers

5 How long ago? How far away? Seven Basic Principles Jane Miller The Chicago Guide to Writing About Numbers Writing about numbers is an essential skill, an important tool in the repertoire of expository writers in many disciplines. - Jane Miller The Chicago Guide to Writing About Numbers There were 25 million deaths. Establish contexts for your facts: the who what why and where During the fourteenth century, 25 million people died in Europe. When the Black Plague hit Europe in the latter half of the fourteenth century, it took the lives of 25 million people, young and old, city dwellers and those living in the countryside. The disease killed about onequarter of Europe s total population at the time (Mack, n.d.) 29 30

6 Pick simple plausible examples In 2001, the average temperature in the New York City area was 56.3 degrees Fahrenheit. In 2001, the average temperature in the New York City area was 56.3 degrees Fahrenheit, 1.5 degrees above normal. In 2001, the average temperature in the New York City area was 56.3 degrees Fahrenheit, 1.5 degrees above normal, making it the seventh warmest year on record for the area Select the right tool: tables, charts, prose Define your terms The relative hazard of failure for material C was Under simulated conditions, the bestperforming material (material C) lasted 1.28 times as long as the next best choice (material B). In conditions that mimic the weather and volume and weight of traffic in Littletown and Midville, the best-performing material (material C) has an average expected lifetime of 64 years, compared with 50 years for the next best choice (material B). Report and interpret 35 36

7 In 1998, total expenditures on health care in the United States were estimated to be more than $1.1 trillion (Centers for Medicare and Medicaid Services 2004). In 1998, total expenditures on health care in the United States were estimated to be more than $1.1 trillion, equivalent to $1,178 for every man, woman, and child in the nation (Centers for Medicare and Medicaid Services 2004). Between 1990 and 1998, the total costs of health care in the United States rose to $1,150 billion from $699 billion - an increase of 65%. Over that same period, the share of gross domestic product (GDP) spent for health care increased to 13.1% from 12.0% (Centers for Medicare and Medicaid Services 2004). Specify direction and magnitude In the United States, per capita health expenditures averaged $4,108 in the 1990s, equivalent to 13.0% of gross domestic product (GDP) - a higher share of GDP than in any other country in the same period. In comparison, Switzerland - the country with the second highest per capita health expenditures - spent approximately $3,835 per person, or 10.4% of GDP. No other country exceeded $3,000 per capita on health expenditures (World Bank 2001b) Mortality and age are correlated. As age increases, mortality increases. Among the elderly, mortality roughly doubles for each successive five-year age group. Summarize patterns: Generalization, Example, Exception 39 40