- From Alan Schoenfeld,
"What's all the fuss about metacognition", pp. 195-6, in
Cognitive Science and Mathematics Education, Alan
Schoenfeld, ed. (Lawrence Erlbaum, 1987):
One of the problems on the NAEP secondary mathematics exam, which was administered to a stratified sample of 45,000 students nationwide, was the following: An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?

Seventy percent of the students who took the exam set up the correct long division and performed it correctly. However, the following are the answers those students gave to the question of "how many buses are needed?": 29% said..."31 remainder 12"; 18% said..."31"; 23% said..."32", which is correct. (30% did not do the computation correctly).

It's frightening enough that fewer than one-fourth of the students got the right answer. More frightening is that

**almost one out of three students said that the number of buses needed is "31 remainder 12".**[our emphasis] - Abstract from Kurt Reusser and Rita Stebler, "Every Word Problem Has a
Solution--The Social Rationality of Mathematical Modelling in Schools,"
Learning & Instruction 7(4): 309-27 (Dec 1997):
Two experiments involving 67 elementary school and 439 high school students show that students

**present solutions to many unsolvable problems without showing realistic reactions.**Results are discussed with respect to the quality of word problems in teaching mathematics, the culture of teaching and learning, and the issue of social rationality in mathematics problem solving. [our emphasis] - From A.H. Schoenfeld, "On Mathematics as Sense-Making: An Informal
Attack On the Unfortunate Divorce of Formal and Informal
Mathematics," in Informal Reasoning and Education,
J.F. Voss, D.N.
Perkins, and J.W. Segal, eds. (Erlbaum, 1991), pp. 311-344:
Kurt Reusser asked 97 first and second grade students the following question: "There are 26 sheep and 10 goats on a ship. How old is the captain?" Seventy-six of the 97 students "solved" the problem, providing a numerical answer by adding 26 and 10.

The reference to Reusser is to Kurt Reusser, "Problem Solving beyond the Logic of Things: Contextual Effects on Understanding and Solving Word Problems," Instructional Science 17(4): 309-38 (1988). The abstract:

Discusses social and cognitive influences on understanding and solving word problems. Descriptions of studies with elementary school, secondary school, and college students highlight subject matter-related attitudes; the fact that

**students can solve problems without understanding them**; and contextual expectations, including the issue of authority. (41 references) [our emphasis]

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