Sowey, E. R. (2001) "Striking Demonstrations in Teaching Statistics" Journal of Statistics Education, 9(1). (www.amstat.org/publications/jse/v9n1/sowey.html)
I read with great interest "Striking Demonstrations in Teaching Statistics" in the March
2001 JSE. The article is timely, as there seems to be a keen interest in such
demonstrations by teachers of mathematics and statistics (consider, for example, the
current NSF-funded "Demos with Positive Impact" project at Georgia Southern University
or the paradoxes with "teacher's notes" collected by Movshovitz-Hadar and Webb (1998)). Striking demonstrations can certainly add
vitality to the university level courses described by Shaughnessy (1992, p. 466) that "continue to be either rule-bound recipe-type courses for calculating
statistics, or overly mathematized introductions to statistical probability that were the norm a decade ago. Thus, college level students ... rarely get
the opportunity to improve their statistical intuition ...
We know students in mathematical statistics courses often respond differently to striking examples than do students in introductory courses, but it is nice to see examples for both types of courses in Sowey’s list. Suggestions for additional examples include: the fact that the mean class size on a per-class basis never exceeds the mean class size on a per-student basis (Hemenway 1982), the fact that defining a "line of best fit" by minimizing the sum of the absolute residuals does not work (see, for example, Lesser 1999b), the fact that neither practical nor statistical significance need imply the other, the fact that scale type is not just an attribute of data (Velleman and Wilkinson 1993), the fact that most positive results will be false positives when testing for a sufficiently rare condition even with a fairly accurate test, the fact that a skewed unimodal distribution need not have unequal mean, median, and mode (Eisenhauer 2002), the fact that "in a long coin-tossing game one of the players remains practically the whole time on the winning side, the other on the losing side" (Feller 1968, pp. 78, 80), and so on. There is also room to add striking demonstrations based on specific data sets, such as how a bimodal set of data can appear unimodal by the choice of number of bins in the histogram, as we learn from inter-eruption times for the Old Faithful geyser in Yellowstone National Park (West and Ogden 1998).
My main goal here, though, is not to "complete" the list of striking examples (indeed, there will always be more to add, and perhaps Sowey or JSE might maintain an ongoing Web page of additional contributions), but rather: (1) to distinguish between "intuitive" and "counterintuitive" striking examples, and (2) to discuss the effort that has been made to identify such demonstrations for use in teaching.
Lesser (1994, pp. 23-24) defines a counterintuitive example as requiring "both that [most students would] have an initial expectation or primary intuition (a directional hypothesis, so to speak) and that that primary
intuition with respect to a result contradicts and is, at least initially, very resistant to the normative view." While the great value of counterintuitive examples (for getting student attention and facilitating critical thinking) may seem obvious even to those who may be unaware of the scholarship cited below, some educators may not be aware
of the possible pitfall or conflict with the view of former National Council of Teachers
of Mathematics president Burrill (1990, p. 13),
repeated by the ASA (1994): "The emphasis in
teaching statistics should be on good examples and building intuition, not on probability
paradoxes or using statistics to deceive." Falk and
Konold (1992, p. 161) express a similar concern: "It is tempting to bring some of the
more devious problems to the classroom to demonstrate to students their erroneous tendencies
and perhaps enlighten them. However, if a teacher persists in pointing out to students how
prone they are to inferential errors, they may become so convinced of their incapacities that
they despair of ever mastering more appropriate techniques." On a related note,
Borasi (1994, p.170) cautions: "To fully
appreciate the radical nature of approaching errors as springboards for inquiry in
mathematics instruction, it is also important to realize that such an approach is at odds
with most teachers’ and students’ current views of errors ...
In any case, Sowey’s characterization of striking examples groups together "counterintuitive" examples (such as example 3 in his Section 4.2) and "intuitive" examples/analogies (such as example 6a in Section 4.6 or example 1 in Section 4.5). Lesser (1994) gives definitions, context and rationale for distinguishing between intuitive and counterintuitive (and "non-intuitive", for that matter) and then reconciles paradigmatic differences underlying the use of the counterintuitive type. On this topic, the emerging statistics education literature cannot (and should not) avoid overlap with the literature of mathematics education, science education and educational psychology, and there are several theories and models to take into account, including: heuristics, constructivism, the "origins" paradigm (Fischbein 1987) of intuition, bridging analogies and conceptual change (see, for example, Brown 1992). A lesser consequence of grouping intuitive and counterintuitive examples together is that the four characteristics in Section 2 may not all apply to all striking demonstrations (for example, a demonstration may provoke much reflection and curiosity and eventual insight, but not be immediately grasped).
Interestingly, there are instances when deeper understanding can be facilitated by juxtaposing a counterintuitive example with an intuitive analogy. For example, an instructor may challenge a typical student’s undue initial emphasis on sampling fraction (over sample size or sampling method) by asking (see, for example, Paulos 1994, p. 35) whether "a random sample of 500 people taken from the entire US population of 250 million is generally far more predictive than a random sample of 50 people out of a population of 2,500." After students have explored the scenario, the instructor could then offer this intuition-building analogy by Freedman, Pisani, Purves, and Adhikari (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 [that is, the sample] would reflect quite faithfully the composition of the whole bottle [that is, the population] and it really wouldn’t matter if the bottle was a test tube or a gallon jug."
Let us now discuss the efforts that have been made to identify and discuss striking demonstrations for use in teaching. Emphasizing examples that occur in real-life contexts and that are accessible without calculus, Lesser(1994, 1998) identifies and catalogues examples (a list which neither contains nor is contained by Sowey’s list) and also provides further rationale, criteria and a framework for their selection and premeditated use.
I agree with Sowey that there has been little discussion or consistency on this in textbooks. Lesser (1994) notes how some well-known introductory textbooks (see, for example, Devore and Peck 1990) do not mention Simpson’s Paradox at all, some discuss it in a section marked "optional" (see, for example, Cryer and Miller 1991), while Moore and McCabe (1993, section 2.5) involve it in the only in-text three-way table example as well as in every three-way table exercise (interestingly, some, but not all of these exercises "telegraph" the presence of the paradox in the data set) following that section.
JSE readers may be interested to know that in the mathematics/statistics education literature beyond textbooks, there are numerous scholarly articles (see, for example, Lesser 1999a, Lesser 2001) on the use of particular counterintuitive examples, and several studies that explore the power of using paradox to motivate student learning (for example, Shaughnessy (1977), Movshovitz-Hadar and Hadass (1990), Wilensky (1995)). An example of a pedagogical implication from Shaughnessy (1977), consistent with prior psychology research (see Lewin 1952), is the effectiveness 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. Such explorations would appear to have strong support from NCTM (2000, pp. 18-19): "Well-chosen tasks can pique students’ curiosity and draw them into mathematics ... Regardless of the context, worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work. Such tasks often can be approached in more than one way ... which makes the tasks accessible to students with varied prior knowledge and experience. Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students ... and how to support students without ... eliminating the challenge."
American Statistical Association (1994), G. Burrill (ed.), Teaching Statistics: Guidelines for Elementary to High School, Palo Alto, CA: Dale Seymour.
Borasi, R. (1994), "Capitalizing on Errors as ‘Springboards for Inquiry’: A Teaching Experiment," Journal for Research in Mathematics Education, 25 (2), 166-208.
Brown, D. E. (1992), "Using Examples and Analogies to Remediate Misconceptions in Physics: Factors Influencing Conceptual Change," Journal for Research in Science Teaching, 29 (1), 17-34.
Burrill, G. (1990), "Implementing the Standards: Statistics and Probability," Mathematics Teacher, 83 (2), 113-118.
Cryer, J. D., and Miller, R. B. (1991), Statistics for Business: Data Analysis and Modelling, Boston: PWS-KENT.
Devore, J., and Peck, R. (1990), Introductory Statistics, St. Paul, MN: West.
Eisenhauer, J. G. (2002), "Symmetric or Skewed?" College Mathematics Journal, 33 (1), 48-51
Falk, R., and Konold, C. (1992), "The Psychology of Learning Probability," in Statistics for the Twenty-First Century, eds. F. Gordon and S. Gordon, Washington, DC: Mathematical Association of America, 151-164.
Feller, W. (1968), An Introduction to Probability Theory and its Applications (Vol. 1; 3rd ed.), New York: John Wiley and Sons, Inc.
Fischbein, E. (1987), Intuition in Science and Mathematics, Dordrecht, Holland: D. Reidel.
Freedman, D., Pisani, R., Purves, R., and Adhikari, A. (1991), Statistics (2nd ed.), New York: W.W. Norton.
Hemenway, D. (1982), "Why Your Classes are Larger Than ‘Average’," Mathematics Magazine, 55 (3), 162-164.
Lesser, L. (1994), "The Role of Counterintuitive Examples in Statistics Education," Doctoral dissertation, University of Texas at Austin, Dissertation Abstracts International, 55, 10A, 3126-A.
----- (1998), "Countering Indifference Using Counterintuitive Examples," Teaching Statistics, 20(1), 10-12.
----- (1999a), "Exploring the Birthday Problem Using Spreadsheets," Mathematics Teacher, 92(5), 407-411.
----- (1999b), "The ‘Ys’ and ‘Why Nots’ of Line of Best Fit," Teaching Statistics, 21(2), 54-55.
----- (2001), "Representations of Reversal: Exploring Simpson’s Paradox," in The Roles of Representation in School Mathematics, eds. A. Cuoco and F. Curcio, Reston, VA: National Council of Teachers of Mathematics, 129-145.
Lewin, K. (1952), "Group Decision and Social Change," in Readings in Social Psychology, eds. G. Swanson, T. Newcomb, and E. Hartley, New York: Henry Holt, 459-473.
Moore, D. S., and McCabe, G. P. (1993), Introduction to the Practice of Statistics (2nd ed.), New York: W. H. Freeman.
Movshovitz-Hadar, N., and Hadass, R. (1990), "Preservice Education of Math Teachers Using Paradoxes," Educational Studies in Mathematics, 21, 265-287.
Movshovitz-Hadar, N., and Webb, J. (1998), One Equals Zero and Other Mathematical Surprises: Paradoxes, Fallacies, and Mind Bogglers, Berkeley, CA: Key Curriculum Press.
NCTM (2000), Principles and Standards for School Mathematics, Reston, VA: National Council of Teachers of Mathematics.
Paulos, J. A. (1994), "Counting on Dyscalculia," Discover, 15(3), 30-36.
Shaughnessy, 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, New York: Macmillan, 465-494.
Velleman, P. F., and Wilkinson, L. (1993), "Nominal, Ordinal, Interval and Ratio Typologies are Misleading," American Statistician, 47(1), 65-72.
West, R. W., and Ogden, R. T. (1998), "Interactive Demonstrations for Statistics Education on the World Wide Web," Journal of Statistics Education [Online], 6(3). (www.amstat.org/publications/jse/v6n3/applets/Histogram.html)
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
Armstrong Atlantic State University
11935 Abercorn Street
Savannah, GA 31419-1997
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