Report

Welcome to the March 18, 2015 JSE webinar: The Median has a Balance Representation, too! presented by Larry Lesser & Amy Wagler Mathematical Sciences Dept. faculty at The University of Texas at El Paso and lead authors of the Nov. 2014 JSE paper http://www.amstat.org/publications/jse/v21n1/lesser.pdf “Finding a Happy Median: Another Balance Representation for Measures of Center” Outline Nov. 2014 JSE paper Today’s webinar 1. Introduction 2. Balance Representations: Mean & Median 3. Pilot Assessment of Intervention 4. Classroom Exploration of Model 5. Results of Assessment 6. Discussion Appendix: In-class Protocol Appendix: Pre-post Test Questions References • Brief motivation and illustration of Model • Some pedagogical considerations • Assessment of Model • Q&A Students’ procedural understanding does not imply conceptual understanding Groth and Bergner (2006) report that: • on a NAEP item, only 4% of Grade 12 students correctly chose (with satisfactory explanation) the median as the measure of center to summarize a particular data set • only 3 of 45 pre-service ES teachers gave descriptions of the median that addressed both ordering as well as when the median is not a value in the dataset balance model of the median (from Figure 1 of our paper) • 4” diameter clothesline pulley wheel • 6.5’ piece of twine tied into a loop, marked at 2” intervals • several 2” office (colored) binder clips (Lynch used spring-loaded clothespin-type mini-clamps instead) mean & median balance models are similar • Can be built inexpensively (< $10) • Support prediction questions • Show that particular measure of center need not be a value in the dataset, but must be between minimum and maximum balance models differ MEAN • Find balance point by trial and error • Board weight usually matters: our 48” stick will balance at 24 for {12,30,30}, but won’t balance at 18 for {12,12,30}; solution: have weights much heavier than the board or use a virtual applet MEDIAN • Find (a) median automatically • Weight of loop won’t matter 1-min demo video of median loop model (storyboarded by Lesser; the below URLs are on p. 7 of our paper) • • amstat.org/publications/jse/v22n3/pulley_loop_physical_model_of_median.html youtube.com/watch?v=-dDbWTPL0vE • Questions posed so video can be paused while your students make predictions. Q1: When I lift my index finger so the string is free to move, what will be at the bottom of the loop for this set of 3 unequally-spaced weights? 1-min demo video Q2: Now, what happens when I lift my finger so the string is free to move for an even number of values? Inspired by the O’Dell (2012) task sequence for the mean, Tables 2 & 3 use a more specific dataset sequence (price in $ of new-release DVDs) DATASET COMMENT {20,22,24,26,28} odd number of equally-spaced values {10,22,24,26,28} one value changed to being an outlier {12,24,26,28,30} add 2 to each value in previous dataset {12,24,24,30,30} dataset includes some repetition {10,11,12,24,24,30,30} dataset includes some repetition that affects median {12,24,24,30} even number of values with middle two equal {12,24,30,35} even number of values with middle two not equal, so median not unique Assessment (Fall 2013) • Population Description • Method: In-class Assessment and Pre-post test • Instrument: multiple choice and open ended questions • Results • Discussion and Extensions Fall 2013 Research • Setting: Southwestern US research institution with predominantly Latino/a student population • Population: Pre-service middle school teachers • Sample Size: 10 completed pre and post tests Method • Pretest about median given in prior class with 100% informed consent • Students participated in median model activity for approximately 45 minutes in subsequent class period • Median activity included in-class voting questions • Completed posttest directly after activity • Data deidentified and analyzed by statistics graduate student researcher Excerpt of In-Class Assessment Protocol Dataset #1 Assessment Instructor Prompts Based on the sample size of the data set, how many clips will we need? [Place clips as described in the above the first row of Table 3] Student Engagement Ask students questions to verify that they understand how to “read” the number loop. Targeted Observations * There is an odd number of values, equally spaced. Ask a student to place or guide your placement of the clips to correspond to the observations listed. Students predict using choices: A) 24 (9 votes) B) something else (1 vote) * Because the loop is formed from a number line, the values are already sorted. Why do you think it was 24? Is 24 a particular measure of center? Discussion Now, where should I put the fulcrum to make this “platform model” Discussion The value 24 is actually the value of several measures of center in this case (mean, median, and midrange) and so more exploration is needed to make a conjecture. The midpoint value 24 is the only point on the stick where it balances. Are the values already sorted? Why? When I let go of the string so the loop is free to move, what value will come to rest at the bottom of the loop? The number 24 quickly moves to the bottom of the loop. 10-question pre/post test • 1 created by the authors from scratch; • 9 adapted from “Measures of Center” ARTIST database (Garfield & Gal, 1999) • 5 multiple choice • 2 multiple choice followed by “Explain” • 3 open-ended Pretest Item Example 1) A test was given to 60 individuals. Half answered 80% of the questions correctly, the other half of the class answered 90% correctly. With regard to the mean and median of the class scores, which of the following statements is correct? a. mean > median b. mean = median c. mean < median. Explain. Pretest Item Example 10) NCAA wrestler weights are summarized in the table below. However, when determining these weights it was discovered that the scale topped out at 250 lbs. so that the exact weight of any player 250 lbs. or more could not be determined. The following is a sample of weights from a particular team: Which can we say about the median? a. It cannot be determined b. The median is 175 c. The median is 200 d. The median is 225 e. The median is 250 f. The median can be determined, but is not one of the choices Weight 175 200 225 250+ Frequency 4 8 15 9 Results • Overall test of proportion correct was conducted for all 7 multiple-choice questions (Ha: µpre-µpost < 0; Z = -1.71; p = 0.0438, permutation test p = 0.034) • Follow-up analysis of individual items using permutation tests Multiple-choice Item Results Item: Concept Proportion correct Pre Post Permut. p-value Adjusted p-value Proportion correct for in-class polls 2: Compares mean and median for symmetric data 3: Def. of median 0.1 0.1 0.234 1.000 0.900 (#1) 0.5 0.7 0.077 0.539 0.678 (#1-7) 4: Compares mean/median for skewed data 5: Find median 0.2 0.9 0.000 0.000 0.900 (#6) 0.6 0.7 0.290 1.000 0.800 (#5) 6: Find median 0.7 0.7 0.323 1.000 0.100 (#7) 7: Interpretation 0.3 0.6 0.031 0.217 1.000 (#2) 10: Find median from table 0.3 0.5 0.074 0.518 0.678 (#1-7) Open-ended Items 3 statistics graduate students independently graded open-ended responses using this rubric: 1 = virtually no relevance 2 = some relevance 3 = substantively relevant 4 = completely relevant Responses to Open-ended Items Description of Question Pre-test mean Likert rating (intraclass correlation coefficient) Post-test mean Likert rating (intraclass correlation coefficient) 1: Provides an example of skewed count data 2.13 (0.101) and asks for reason why the mean would be inappropriate as a measure of center. 2.60 (0.194) 2: Asks for comparison of mean and median 1.70 (0.291) for symmetric data, then requests explanation 1.73 (0.253) 4: Asks for comparison of mean and median 1.47 (0.594) for skewed data, then requests explanation 8: Asks for approximate value of median from 1.87 (0.147) a dotplot 9: Asks for comparison of mean and median 1.60 (0.290) for a dotplot 2.37 (0.408) 1.63 (0.607) 1.93 (0.187) Interpretation of Results • Biggest insight gain: how mean and median are affected differently by outliers • Median need not be unique • May not transfer well without physical model • Students gain a concrete basis to visualize the data as ordered when finding the median • Reinforces idea that mean and median are not always an observed value (De)limitations • Preservice middle school teachers provide “upper bound” benchmark – If they struggle with median concepts, middle school students will even more • Sample size small due to only one section of class available • Results are preliminary, not definitive