Representations of reversal: An exploration of Simpson's Paradox

To support NCTM's newest process standard, the potential of multiple representations for teaching repertoire is explored through a real-world phenomenon for which full understanding is elusive using only the most common representation (a table of numbers). The phenomenon of "reversal of a comparison when data are grouped" is explored in surprisingly many ways, each with their own insights: table, circle graph, slope & correlation coefficients, platform scale, trapezoidal representation, unit square model, probability (balls in urns), matrix determinants, linear transformations, vector addition, and verbal form. For such a mathematically-rich phenomenon, the number of distinct representations may be too large to expect a teacher to have time to use all of them. Therefore, it is necessary to learn which representations might be more effective than others, and then form a sequence from those selected. Pilot studies were done with pre-service secondary teachers (n1 = 7 at a public research university and n2 = 3 at a public comprehensive university) on exploring a sequence of 7 different representations of Simpson's Paradox. Students tended to want to stay with the most concrete and visual representations (note: a concrete-visual-analytic progression may not be expected to apply in the usual manner in the particular case of Simpson's Paradox).