by David
Mathews and Julie
M. Clark
This paper is one in an emerging series of studies by members of the Research in Undergraduate Mathematics Education Community, or RUMEC ,concerning the nature of and development of college students'mathematical knowledge. We present analyses of audio-taped clinical interviews with college freshmen immediately after they completed an elementary statistics course with a grade of "A". The point of these interviews was not to see how quickly isolated facts could be recalled, nor was the point to see how little students remember. Rather, the goal was to determine as precisely as possible the conceptions of mean, standard deviation and the Central Limit Theorem which the most successful students had shortly after having completed a statistics course. Be it in statistics or any other mathematical content area, teachers want their students to learn and retain as much mathematics as possible. But as Schlomo Vinner put it in "The function concept as a prototype for problems in mathematical learning" (1992):
There is one overall question which bothers mathematics teachers as well as all other teachers: What will remain in our students' minds after the end of the course and the final exam?
Since experienced teachers generally know (and are dissatisfied with) the answer to this question, one might well be tempted to experiment with pedagogical innovations to increase retention of knowledge. But such an approach assumes that we know what constitutes statistical knowledge. As early as the late 1970's, leaders within the mathematics education community were urging a moratorium on such studies of performance and argued for an increased research effort to help us to understand the cognitive development in the learners of mathematics (Kilpatrick, 1978, Schoenfeld, 1994). Considerable progress has been made in this area with regard to functions (see Breidenbach et al, 1992, Harel & Dubinsky, 1994, Vinner, 1992 for examples of this). Moreover, a number of general epistemological frameworks have been developed (see Asiala et al, 1996, Gray & Tall, 1991, Sfard, 1987, Tall & Vinner, 1981) and applied to a variety of mathematical content areas. Very little work, however, has been done relative to studying the cognitive development of college students with regard to statistical concepts. In an excellent survey entitled Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research, Garfield and Ahlgren (1988) summarized the situation and noted:
Although many articles in the education literature recommend how to teach statistics better, there is little published literature on how students actually learn statistics concepts.