Children’s reasoning about data sets Amy M. Masnick, Bradley J. Morris, & Christopher Was
Introduction Individual numbers represented as both approximate, relative magnitudes with error variance and as precise, verbal categories (Dehaene, 2009) Number sets have unique properties, beyond individual numbers, because there are summary characteristics: • Mean • Variance Some past work shows means and variance guide reasoning and eyetracking behavior in adults, suggesting summary representations (Morris & Masnick, 2015) Less systematic work (Masnick & Morris, 2008) showed some ability to rely on these characteristics in third and sixth graders, though before Common Core U.S. Common Core standards in math emphasize teaching of basic statistics (including central tendency and variance) by sixth grade, but not by 4th (Common Core, 2010) Research questions: Which data characteristics do children use? (Mean, variance, set size) Are there age difference in use? Which strategies do children report using, and do their reported strategies match their behavior?
Method Participants 44 4th grade students Mean age = 9.78 years, SD = 0.60 38 6th grade students Mean age = 12.14 years, SD = 0.45
Stimuli Number sets varied by: Sample size (4, 6, or 8 observations per set) Ratio of means (4:5 or 9:10) Coefficient of variation (.10 or .20 of mean)
Twelve possible combinations Three trials each combination Thirty-six total trials
Strategies
Procedure: You will be shown the results from a series of golf drives (a single golf shot to achieve maximum distance). Each slide will show how far a golfer hit a series of balls from one of two tees (LEFT or RIGHT). Your job is to tell me which golfer, on average, hit the ball FARTHER (all drives were measured in feet). Table 1. Example data sets Example A Golfer 1 Golfer 2 155 130 175 160 180 220 190 180 130 175 Mean:
Example B Golfer 3 Golfer 4 150 170 220 200 150 210 175 160 178 170
166 173 174.6 182 Note. Participants given such a task do not see the mean.
After you make your choice, I will ask you how sure you are using the scale in front of you. The scale goes from 1 to 4. A 1 indicates that you were NOT SO SURE about which golfer hit the ball farther and a “4” indicates that you were TOTALLY SURE that one golfer hit the ball farther. I will ask you: How sure are you that this golfer hit this ball farther? Please say a number.
Results Age: 6th graders more accurate (90%) than 4th graders (82%) Set size: Sets of 8 more difficult (82% accuracy) than sets of 4 or 6 (86% and 89% accuracy) No difference between sets of 4 and 6, but a difference between 6 & 8 Mean difference: Larger mean difference leads to greater accuracy (92%) than smaller mean difference (82%) Variance Larger variance leads to lower accuracy (84%) than smaller variance (88%) Eye tracking column switches: Fewer switches with larger mean difference and lower variance
Comparison Comparisons are made from scanning a subset of the total set (e.g., first two, last two).
High-Low Comparisons are made between the highest (or lowest) values in each set.
Gist Comparisons are based on estimates of set properties (e.g., mean, variance)
Sixth graders used the gist strategy most (74% of the time), whereas fourth graders were much more mixed, with 30% comparison, 45% gist, 25% high-low). Sixth graders also were more accurate when using the gist strategy (94%) than the fourth graders (72%).
Discussion Evidence suggests children have intuitions about summary characteristics of data In particular, even the majority of fourth graders who haven’t been formally taught about such characteristics reason based on them Sixth graders use these characteristics more consistently, particularly at larger set sizes, by using strategies more efficiently
References . Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Common core state standards (college-and career-readiness standards and K–12 standards in English language arts and math). Dehaene (2009). Origins of Mathematical Intuitions: The Case of Arithmetic. The Year in Cognitive Neuroscience 2009: Annual New York Academy of Science, 1156, 232–259. doi: 10.1111/j.1749-6632.2009.04469 Masnick, A. M., & Morris, B. J. (2008). Investigating the development of data evaluation: The role of data characteristics. Child Development, 79(4), 1032-1048. doi:10.1111/j.1467-8624.2008.01174.x Morris, B. J., & Masnick, A. M. (2015). Comparing data sets: Implicit summaries of the statistical properties of number sets. Cognitive Science, 39(1), 156-170. doi:10.1111/cogs.12141