We view understanding of mathematical material as a function of (1) connections of text concepts and formulas to real-world referents; (2) integration of concepts and formulas within the text; and (3) explanation of formulas. This view has provided a basis for constructing three written treatments of elementary probability, presumably varying in the degree to which they convey understanding. Our view of the processes involved in solving problems has led us to use both formula and story problems, and to emphasize analyses of error protocols. A research study is described involving 48 undergraduate students, randomly assigned to three text conditions. Results indicated that very different patterns of knowledge appear to be present in subjects in the non-explanatory (standard and low-explanatory texts) and explanatory conditions. Subjects in the two non-explanatory groups performed considerably less well on story than on formula problems, and often used the correct formula for a problem (that is, met the lenient criterion) but failed to solve it. A closer look at answers to story problems revealed that subjects in the non-explanatory conditions often required the explicit presence of key words which unambiguously pointed to certain operations, tended to misclassify problems in the presence of irrelevant or redundant information, and made many errors when the values in the story required modification before insertion into the formula. In contrast, subjects in the highly explanatory condition performed equally well on story and formula problems, tended to solve whenever they showed evidence of knowing the appropriate formula, and were considerably less hindered by absence of key words, the presence of irrelevant information, and the need to translate values in the story.
The CAUSE Research Group is supported in part by a member initiative grant from the American Statistical Association’s Section on Statistics and Data Science Education