Statistical Edutainment with Confidence

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Title: Statistical Edutainment with Confidence
Language: English
Authors: Pearl, Dennis K. (ORCID 0000-0003-1981-1826), Lesser, Lawrence M. (ORCID 0000-0001-5762-3987)
Source: Teaching Statistics: An International Journal for Teachers. Spr 2020 42(1):23-27.
Availability: Wiley-Blackwell. 350 Main Street, Malden, MA 02148. Tel: 800-835-6770; Tel: 781-388-8598; Fax: 781-388-8232; e-mail: cs-journals@wiley.com; Web site: http://www.wiley.com/WileyCDA
Peer Reviewed: Y
Page Count: 5
Publication Date: 2020
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 1544237
1544426
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Cartoons, Singing, Statistics, Teaching Methods, Discussion (Teaching Technique), Concept Formation, Mathematical Concepts, Mathematics Instruction
DOI: 10.1111/test.12213
ISSN: 0141-982X
Abstract: The use of cartoons, songs, and quotes can be a useful way to engage students in discussion and learning key concepts about quantifying the uncertainty in statistical estimates.
Abstractor: As Provided
Entry Date: 2020
Accession Number: EJ1241470
Database: ERIC
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  Value: <anid>AN0141416285;d8y01mar.20;2020Jan29.04:08;v2.2.500</anid> <title id="AN0141416285-1">Statistical edutainment with confidence </title> <p>The use of cartoons, songs, and quotes can be a useful way to engage students in discussion and learning key concepts about quantifying the uncertainty in statistical estimates.</p> <p>Keywords: cartoons; confidence intervals; margin of error; quotes; songs; teaching statistics</p> <hd id="AN0141416285-2">INTRODUCTION</hd> <p>The importance of secondary school students being able to infer information about parameters from sampled data and to develop and interpret a Margin of Error (MOE) as part of their studies of Statistics and Probability is seen in many national reports and standards (see, for example, HSS.IC.B.4 and HSS.IC.B.6 of the US High School Common Core Standards[<reflink idref="bib1" id="ref1">1</reflink>], the <emph>GAISE Pre‐K‐12 Report</emph>[<reflink idref="bib2" id="ref2">2</reflink>], and the UK Stage 4 National curriculum[<reflink idref="bib3" id="ref3">3</reflink>]). Similarly, at the introductory tertiary level, for example, the US college level, the quantification of uncertainty is often at the core of the statistics curriculum[<reflink idref="bib4" id="ref4">4</reflink>] and the use of confidence intervals and the ability to interpret the MOE and what affects its size are important learning goals. Teachers may wish to connect their students to these concepts in a memorable way by using educational fun or provocative materials within students' learning experiences.</p> <hd id="AN0141416285-3">ESTIMATING WITH CONFIDENCE: QUOTES</hd> <p>Quotes can be excellent starters of discussion. Some are amusing in highlighting absurdities, such as many about averages, while others can provoke more serious discussion. If students have already been introduced to both confidence intervals and hypothesis testing, the following quote[<reflink idref="bib5" id="ref5">5</reflink>] by British statistician Michael Oakes could be used in a classroom discussion comparing and contrasting the purpose of estimation vs hypothesis testing:</p> <p>"The researcher armed with a confidence interval, but deprived of the false respectability of statistical significance, must work harder to convince himself and others of the importance of his findings. This can only be good."</p> <p>The following quote[<reflink idref="bib6" id="ref6">6</reflink>] by the recently deceased American researcher and author Judith Bardwick can provoke considerable discussion about estimating proportions:</p> <p>"... motivation is highest when the probability of success is 50 percent: We don't get involved if the task is too easy or too hard."</p> <p>An instructor might ask: "What would prompt such a statement?" or "What about situations such as estimating proportions of defectives or the incidence of rarer diseases?" Such discussions can include a number of important points for students, including that uncertainty—and thus MOE—are highest in situations with population proportions closest to 0.5, as well as re‐emphasizing the approximate nature of the standard form of the MOE.</p> <p>One book[<reflink idref="bib7" id="ref7">7</reflink>] presents a very large (more than 2000 items) collection of statistics‐related quotes, not limited to just those valuable for educational use. The two examples above can be found in the 232‐quote collection at CAUSEweb.org/fun that is designed for educational use and provides a picture of the quote author along with pedagogical, bibliographic, and biographical annotations for the quote. It is also of some interest that this quote collection has more than 30% of its entries by female authors.</p> <p>As Kashin[<reflink idref="bib8" id="ref8">8</reflink>] relates, quotes can be a powerful pedagogical tool to generate discussion, to connect theory to application, and to provoke deeper thinking, especially when having students reflect on the quote as an individual before discussing in small groups and then in the whole room. The conciseness of quotes lends themselves to being readily shared (eg, via social media) to further expand the conversation. She mentions the free resource at https://quotescover.com for attractively displaying quotes (eg, for a wall poster or a social media post) in the classroom.</p> <p>Like Kashin, we believe that it is important to examine the pedagogical value of a quote for teaching and learning. For example, a quote that invokes negative stereotypes or attitudes about statistical concepts, practice, or practitioners would not be constructive. Hence, it is recommended to avoid such items and also to identify associated learning objectives.</p> <p>For estimating with confidence, some of the key concepts forming the learning objectives are:</p> <p>[concept 1] The size of the MOE is affected by the sample size (through <ephtml> <math display="inline" overflow="scroll" altimg="urn:x-wiley:0141982X:media:test12213:test12213-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.25em" /><msqrt><mi>n</mi></msqrt></math> </ephtml> ) (but not by the size of a total but finite population as long as the sample size is small in comparison);</p> <p>[concept 2] There is a trade‐off between confidence and precision so that smaller confidence levels imply a smaller MOE; a simple point estimate without an interval does not quantify uncertainty at all.</p> <p>[concept 3] The level of theoretical or population variation (eg, SD) also affects the MOE—with more variation associated with more uncertainty; and</p> <p>[concept 4] The interpretation of a confidence interval as the values compatible with the data under a probability model so that, for example, 95% of all samples would produce an interval covering the parameter.</p> <p>These concepts are consistent with the literature—for example, our concept 1 aligns with measured learning goal #11 from the GOALS assessment[<reflink idref="bib9" id="ref9">9</reflink>] and our concept 4 aligns with the framework in Pre‐K‐12 GAISE [<reflink idref="bib2" id="ref10">2</reflink>, see page 12)]. Let us see how these four concepts can all be illustrated with some other fun items.</p> <hd id="AN0141416285-4">ESTIMATING WITH CONFIDENCE: CARTOONS</hd> <p>Figure is a cartoon drawn by British cartoonist John Landers[<reflink idref="bib10" id="ref11">10</reflink>] that helps explain concept 4 by imagining many pollsters using the same methods on the same topic at the same time. Of course, the absurdity of them all interviewing the same person can be used to generate classroom discussion about how unlikely it is for a specific person to be contacted even once by a pollster when they are part of a large population. The interpretation of the confidence level as the percentage of intervals covering the true value among all samples might then be reinforced with a follow‐up activity. For example, giving each student in class a small package of plain M&M's candies and asking them each to make a confidence interval for the percentage of all such candies that are brown. You can demonstrate the concept in a hands‐on manner by then revealing the true percentage from an online source and checking what proportion of the class got intervals covering the answer.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01mar20/test12213-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12213-fig-0001.jpg" title="Cartoon[10] for addressing concept 4 [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <p>Figure is a Landers cartoon[<reflink idref="bib11" id="ref12">11</reflink>] that sets up a conversation about the value of an interval estimate over a point estimate; the value of giving some idea of variation of any estimate being a fundamental learning objective (concept 2).</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01mar20/test12213-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12213-fig-0002.jpg" title="Cartoon[11] for addressing concept 2 [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <p>The cartoon can be made more concrete in this context by saying the estimated home improvement cost, such as remodeling a bathroom, might be $5000, plus or minus $1000, which would be very helpful for planning a project contingency fund if you are on a tight budget. However, students should immediately be asking where the $1000 comes from, or is it just a "fudge factor." This can lead to a discussion on what is best: no indication of variation, a "fudge factor," or a quantification of uncertainty based on clearly identified assumptions and of how much uncertainty. The last requires more understanding and thought so students can discuss how this can be justified and explained to a layperson, including what is a good balance between correctness and approximation in communicating statistics?</p> <p>The College GAISE report [<reflink idref="bib4" id="ref13">4</reflink>, see page 69] suggests having students create a cartoon or video as a mini‐project idea to use in teaching about confidence intervals.</p> <hd id="AN0141416285-7">ESTIMATING WITH CONFIDENCE: SONGS</hd> <p>The interactive song "Height of Confidence"[<reflink idref="bib12" id="ref14">12</reflink>] by Larry Lesser and Dominic Dousa develops, assesses, and re‐enforces student knowledge of the impact of sample size, SD, and confidence level on the width of a confidence interval (concepts 1, 2, and 3). This original song is interactive in the sense that students must answer questions about these learning objectives before being able to play the music. The recording has the student's own responses inserted into the song using a synthetic voice. In this way, the lyric created from the pre‐song prompts not only validates the students' answers but also provides them with rhymed couplets to help engage them, as seen from this excerpt regarding estimating a mean (the underlined words being examples of student inputs):</p> <p>When we seek that interval, it's always good to know</p> <p>What would make it shrink and what would make it grow.</p> <p>The interval gets tighter with increasing <emph>n</emph>:</p> <p>The added information helps us narrow in!</p> <p>The interval gets bigger with a larger <emph>s</emph>,</p> <p>That has come from a sample that's a more uncertain mess!</p> <p>The interval gets wider for higher confidence</p> <p>'Cause you have to cover lots more of your bets!</p> <p>If estimating proportions has been correctly covered in terms of estimating the mean of a binary (Bernoulli) variable, with the variance of the proportion being <emph>p</emph>(1 − <emph>p</emph>)/<emph>n</emph>, and the sample variance as its estimate, then the lessons from the song might be followed by a short activity using an R‐Shiny applet like the one at:</p> <p>https://psu-eberly.shinyapps.io/Inference_for_Proportions</p> <p>An interactive song on confidence intervals for proportions is "Super Bowl Poll"[<reflink idref="bib13" id="ref15">13</reflink>], which may be timely because of the 2020 Super Bowl game in American football (the National Football League) that was recently televised in 130 countries in more than 30 languages. This song helps students apply the margin of error concept in the context of a poll question, including the idea that variability decreases with the square root of the sample size (concept 1).</p> <hd id="AN0141416285-8">ESTIMATING WITH CONFIDENCE: GAMES</hd> <p>Another type of edutainment is games[<reflink idref="bib14" id="ref16">14</reflink>] and there is an excellent classroom trivia game[<reflink idref="bib15" id="ref17">15</reflink>] to further integrate and explore the concepts of confidence intervals (concept 4). In this activity, students are asked to respond to trivia questions that have a known numerical answer and to provide their answer in terms of a 90% confidence interval, rather than just a point estimate. After the last question, the answers are revealed and the class can compare what percentage of their intervals covered the true answer. For example, an instructor might ask "How tall was Napoleon in centimeters?" (true value: 169). See[<reflink idref="bib15" id="ref18">15</reflink>] for implementation guidance as well as sample questions.</p> <p>An extension of this activity might be to ask for an interval that covers the height of the next person to walk into the room. Students should be able to recognize that there is more variability in the result of that process, leading to a discussion of the difference between a confidence interval and a prediction interval.</p> <hd id="AN0141416285-9">CONCLUSION</hd> <p>As we have seen, edutainment items can be aligned with important concepts in the teaching and learning of confidence intervals. Can you bring edutainment into your classroom? We're <emph>confident</emph> you can!</p> <hd id="AN0141416285-10">ACKNOWLEDGMENTS</hd> <p>This work was supported by Project SMILES, NSF/EHR/DUE 1544426 (PSU), 1544237 (UTEP). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.</p> <ref id="AN0141416285-11"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Funding information Division of Undergraduate Education, Grant/Award Numbers: 1544237, 1544426; National Science Foundation</bibtext> </blist> </ref> <ref id="AN0141416285-12"> <title> REFERENCES </title> <blist> <bibtext> National Governors Association Center for Best Practices & Council of Chief State School Officers, Common Core State Standards for Mathematics, Authors, Washington, DC, 2010.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> C. Franklin et al., Guidelines for assessment and instruction in statistics education (GAISE) report: A pre‐k‐12 curriculum framework, American Statistical Association, Alexandria, VA, 2007.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> Department for Education. The national curriculum in England: Key stages 3 and 4 framework document, 2014, available at <ulink href="http://www.gov.uk/dfe/nationalcurriculum">www.gov.uk/dfe/nationalcurriculum</ulink>.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> GAISE College Report ASA Revision Committee, Guidelines for assessment and instruction in statistics education college report, 2016, available at <ulink href="http://www.amstat.org/education/gaise">http://www.amstat.org/education/gaise</ulink>.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> M. Oakes, Statistical Inference: A Commentary for the Social and Behavioural Sciences, John Wiley & Sons, Hoboken, NJ, 1986.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref6" type="bt">6</bibl> <bibtext> J. M. Bardwick, Danger in the Comfort Zone: From Boardroom to Mailroom – How to Break the Entitlement Habit That's Killing American Business, 2nd ed., AMACOM, New York, 1995.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref7" type="bt">7</bibl> <bibtext> C. C. Gaither and A. E. Cavazos‐Gaither, Statistically Speaking: A Dictionary of Quotations, CRC Press, Boca Raton, FL, 1996.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref8" type="bt">8</bibl> <bibtext> D. Kashin, Quotes to provoke Reggio‐inspired teaching and learning, available at https://tecribresearch.wordpress.com/2016/03/26/quotes-to-provoke-reggio-inspired-teaching-and-learning/</bibtext> </blist> <blist> <bibl id="bib9" idref="ref9" type="bt">9</bibl> <bibtext> Sabbag, A. G., Garfield, J., and Zieffler, A., Quality Assessments in Statistics Education: A Focus on the GOALS Instrument, Advances in Statistics Education: Developments, Experiences, and Assessments. Proceedings of the Satellite Conference of the International Association for Statistical Education (IASE) (M.A. Sorto, ed.), Rio de Janeiro, Brazil, 2015, available at <ulink href="http://iase-web.org/documents/papers/sat2015/IASE2015%20Satellite%2041%5fSABBAG.pdf">http://iase-web.org/documents/papers/sat2015/IASE2015%20Satellite%2041%5fSABBAG.pdf</ulink></bibtext> </blist> <blist> <bibtext> J. Landers (2006a). The Pollsters, available at https://<ulink href="http://www.CAUSEweb.org/cause/resources/fun/cartoons/pollsters">www.CAUSEweb.org/cause/resources/fun/cartoons/pollsters</ulink></bibtext> </blist> <blist> <bibtext> J. Landers (2006b). Home Improvement, available at https://<ulink href="http://www.CAUSEweb.org/cause/resources/fun/cartoons/home-improvement">www.CAUSEweb.org/cause/resources/fun/cartoons/home-improvement</ulink></bibtext> </blist> <blist> <bibtext> L. Lesser and D. Dousa (2015) Height of Confidence, available at https://<ulink href="http://www.causeweb.org/smiles/songs/height%5fof%5fconfidence">www.causeweb.org/smiles/songs/height%5fof%5fconfidence</ulink></bibtext> </blist> <blist> <bibtext> L. Lesser (2015) Super Bowl Poll, available at https://<ulink href="http://www.causeweb.org/smiles/songs/super%5fbowl%5fpoll">www.causeweb.org/smiles/songs/super%5fbowl%5fpoll</ulink></bibtext> </blist> <blist> <bibtext> S. Kuiper and R. Sturdivant, Games as a Locus of Self‐empowered Collaborative Learning. Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9, July, 2014), (K. Makar, B. de Sousa, and R. Gould, eds.), International Statistical Institute, Flagstaff, AZ, and Voorburg, available at <ulink href="http://icots.info/9/proceedings/pdfs/ICOTS9%5fC117%5fKUIPER.pdf">http://icots.info/9/proceedings/pdfs/ICOTS9%5fC117%5fKUIPER.pdf</ulink></bibtext> </blist> <blist> <bibtext> X. Wang, N. G. Reich, and N. J. Horton, Enriching students' conceptual understanding of confidence intervals: An interactive trivia‐based classroom activity, The American Statistician 73 (1) (2019), 50 – 55.</bibtext> </blist> </ref> <aug> <p>By Dennis K. Pearl and Lawrence M. Lesser</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref11"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref12"></nolink> <nolink nlid="nl3" bibid="bib12" firstref="ref14"></nolink> <nolink nlid="nl4" bibid="bib13" firstref="ref15"></nolink> <nolink nlid="nl5" bibid="bib14" firstref="ref16"></nolink> <nolink nlid="nl6" bibid="bib15" firstref="ref17"></nolink>
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