Since 1988, I've taught statistics to a variety of undergraduate and graduate student audiences, including business majors, mathematics majors, science majors, preservice elementary teachers, and in-service secondary teachers. One common theme has been the search for ways to bring intuition to concepts that were initially nonintuitive or even counterintuitive to students. I personally saw very little consistency in how textbooks and teachers addressed this, and I dedicated my dissertation (Lesser 1994) and much of my subsequent research (e.g., Lesser 1998) to exploring this.
Using Multiple Representations
Sometimes it is simply a matter of finding an additional representation (e.g., verbal, numerical, graphical/geometrical, algebraic) for the concept. For example, Simpson's Paradox is usually presented only in tabular form, which shows that a comparison can be reversed upon aggregation, but that form gives no insight about why or when such a reversal happens. In my articles "Illumination Through Representation: An Exploration Across the Grades" (Lesser 2005) and "Representations of Reversal: An Exploration of Simpson's Paradox" (Lesser 2001), I explore alternative representations that can supply such insight.
Students with less mathematically sophisticated backgrounds often benefit the most when an algebraic formula is not the first representation they see of a concept. Utts (2005) is one example of a textbook with this approach, where each chapter is meant to be fully read and understood before the Greek letters and algebraic notation appear at the very end of each chapter.
While students may not be required to memorize many formulas on exams, they can still benefit greatly by gaining an intuitive feel for why the formulas make sense, where they come from, and what else they connect to. For example, the correlation formula becomes very intuitive when it is seen to be an average product of z-scores (which produce positive contributions to the formula for standardized points in the first and third quadrants). For another example, the probability formulas for Pr(A or B) and for Pr(A|B) can be made quite intuitive with Venn diagram representations. And Martin (2003) uses a seesaw analogy to give intuition to the formula for expected value.
Using Intuitive Analogies
Another method of bringing engagement to the classroom has been to collect and compile a repertoire of intuitive bridging analogies, metaphors, visualizations, anchors, heuristics, and so on until you have at least one for each major statistics concept. Martin (2003), with the exchange of published letters attached, is a good resource to jump-start your own collection. Some of these (e.g., the judicial analogy for hypothesis testing or the target analogy for an estimator's bias and variance) will likely be familiar to most teachers, while others may be new. Martin (2003) advises us to note whether similarities are superficial or structural and to explore where the analogy breaks down.
Using (and Resolving) Counterintuitive Examples
Some fundamental and/or famous results in probability and statistics are counterintuitive. For example, inferential statistics ultimately rests on the not-immediately-intuitive Central Limit Theorem, and the mathematical theory of probability was launched by the paradox of Chevalier de Méré. The birthday problem ("How many people do you need to have at least a 50 percent chance of at least one match of birthdays?") is perhaps the most famous instance of a counterintuitive example. By considering the "number of opportunities" for matches, I was successful in helping make this result intuitive for my students (Lesser 1999).
Many counterintuitive results can be used effectively to capture students' attention and force them to engage with concepts in a more sustained way and with a deeper understanding. Shaughnessy (1977) illustrates the importance and benefit of group activities in which members commit or buy into the task by making a guess at the outcome before the activity, carrying out the activity, noting the results of the activity, and comparing those empirical results with their preconceptions. Several studies illustrate the power of using paradox to motivate student learning, for example, Shaughnessy (1977), Movshovitz-Hadar and Hadass (1990), and Wilensky (1995). In the Journal of Statistics Education, Sowey (2001) and my subsequent letter to the editor (2002) are good resources to jump-start your own collection.
Conditional probability scenarios are another source of results that do not become intuitive before an explicit sample space diagram is made. The Monty Hall Let's Make a Deal three-doors problem is a famous one (e.g., Barbeau 1993). Another is given by Shaughnessy (1992, p. 474): "There are three cards in a bag. One card has both sides green, one card has both sides blue, and the third card has a green side and a blue side. You pull a card out, and see that one side is blue. What is the probability that the other side is also blue?"
Besides losing the inherent benefits that come from challenges to sharpen one's conceptual knowledge, teachers (or textbook writers) who might be tempted to avoid all counterintuitive examples would be doing their students a further disservice, for several counterintuitive phenomena actually occur in the real world, and awareness of them is therefore needed for statistical literacy. For example, Simpson's Paradox is listed as essential for citizenship by the National Council on Education and the Disciplines (2001).
Another major source of counterintuitive examples involves principles of sampling. For example, students can be asked which is more predictive: a random sample of 500 people taken from a population of 250,000,000 or a random sample of 50 from a population of 2,500 (Paulos 1994, p. 35). Students tend to choose the latter because they have an initial reliance on sampling fraction over sample size. Intuition can be brought in with the analogy of Freedman (1991, p. 339): "Suppose you took a drop of liquid from a bottle, for chemical analysis. If the liquid is well mixed, the chemical composition of the drop [i.e., the sample] would reflect quite faithfully the composition of the whole bottle [i.e., the population], and it really wouldn't matter if the bottle was a test tube or a gallon jug."
Also on the topic of sampling, students initially rely on sample size over sampling method. This is brought out when students are asked what is more predictive: a random sample of 3,000 from a population of 10 million or 2.3 million responses from surveys mailed to those 10 million. (Teachers may recognize these numbers from the successful challenge of George Gallup to Literary Digest in predicting the 1936 presidential election.) An analogy by Davis (2005) is to imagine making a large jar's worth of salad dressing with oil, vinegar, spices, and chunks of garlic. If the dressing is well mixed or shaken, a spoonful off the top of the jar is accurate. If the dressing is not well mixed, then ingredients may separate, and that same spoonful off the top may be quite misleading.
I present these examples and references to further examples in hopes that they will give teachers a richer repertoire for those times when students need further motivation or scaffolding to attain deeper conceptual understanding. Readers who encounter additional examples are invited to relay them to me via my Web site.
Barbeau, E. 1993. "The Problem of the Car and Goats." College Mathematics Journal 24 (2): 149-154.
Davis, L. 2005. Personal communication with the author.
Freedman, D., R. Pisani, R. Purves, and A. Adhikari. 1991. Statistics. 2nd ed. New York: W. W. Norton.
Lesser, L. 2005. "Illumination Through Representation: An Exploration Across the Grades." Statistics Teacher Network, no. 66: 3-5. Also at www.amstat.org/education/stn/pdfs/STN66.pdf.
Lesser, L. 2002. Letter to the Editor. Journal of Statistics Education 10 (1). www.amstat.org/publications/jse/v10n1/lesser_letter.html.
Lesser, L. 2001. "Representations of Reversal: An Exploration of Simpson's Paradox." In The Roles of Representation in School Mathematics, ed. A. A. Cuoco and F. R. Curcio, 129-145. Reston, Virginia: National Council of Teachers of Mathematics.
Lesser, L. 1999. "Exploring the Birthday Problem with Spreadsheets." Mathematics Teacher 92 (5): 407-411.
Lesser, L. 1998. "Countering Indifference Using Counterintuitive Examples." Teaching Statistics 20 (1): 10-12. Also available as a pdf at psu.edu.
Lesser, L. 1994. "The Role of Counterintuitive Examples in Statistics Education" (doctoral dissertation, University of Texas at Austin). In Dissertation Abstracts International 55 (10A), 3126-A. University Microforms Inc. #DA9506033.
Martin, M. 2003. "It's Like . . . You Know: The Use of Analogies and Heuristics in Teaching Introductory Statistical Methods." Journal of Statistics Education 11 (2). www.amstat.org/publications/jse/v11n2/martin.html.
Movshovitz-Hadar, N., and R. Hadass. 1990. "Preservice Education of Math Teachers Using Paradoxes." Educational Studies in Mathematics 21: 265-287.
National Council on Education and the Disciplines. 2001. Mathematics and Democracy: The Case for Quantitative Literacy. Princeton, New Jersey: Woodrow Wilson National Fellowship Foundation.
Paulos, J. A. 1994. "Counting on Dyscalculia." Discover 15 (3): 30-36.
Shaughnessy, J. M. 1977. "Misconceptions of Probability: An Experiment with a Small-Group, Activity-Based, Model Building Approach to Introductory Probability at the College Level." Educational Studies in Mathematics 8: 285-316.
Shaughnessy, J. M. 1992. "Research in Probability and Statistics: Reflections and Directions." In Handbook of Research on Mathematics Teaching and Learning, ed. D. A. Grouws, 465-494. Reston, Virginia: National Council of Teachers of Mathematics.
Sowey, E. 2001. "Striking Demonstrations in Teaching Statistics." Journal of Statistics Education 9 (1). www.amstat.org/publications/jse/v9n1/sowey.html.
Utts, J. M. 2005. Seeing Through Statistics. 3rd ed. Belmont, California: Thomson.
Wilensky, U. 1995. "Paradox, Programming and Learning Probability: A Case Study in a Connected Mathematics Framework." Journal of Mathematical Behavior 14: 253-280.
Lawrence M. Lesser is a mathematics education professor at the University of Texas at El Paso. He began university teaching in 1988 and has also worked for two years as a state agency statistician and two years as a high school teacher and department head. Statistics education is a major focus of his teaching, book chapters, journal articles, and conference presentations. He has connected statistics to such diverse areas as ethics, music, and the lottery. He served a three-year term on the Journal of Statistics Education editorial board and in 2005 began serving on the Research Advisory Board of the Consortium for the Advancement of Undergraduate Statistics Education (CAUSE).
Lawrence M. Lesser
University of Texas at El Paso
El Paso, Texas