Developing problem solving experiences in practical action projects

Abstract: Problem solving doubtless is an essential element of mathematical learning, so that mathematics educators often are satisfied when finding situations that lead their students to such activity. But in many cases, the chosen situations and the ways to guide students' works are not sufficiently analyzed from a didactic point of view. Our goal in the present analysis is to underline the possible ways for managing the situations, and to exhibit the parameters that educators have at their disposition within their role as mediator between students and mathematical knowledge and know-how.Keywords: Problem solving, problem for investigating, project based learning, a priori analysis, mediation.(ProQuest: ... denotes formula omitted.)1. Classical problem solvingProblem solving at its broad extent is clearly exposed by Alan Schoenfeld (Schoenfeld, 2006, p.41): "[Problem solving] includes a child's actions in interacting with its parents, a student working on a mathematics problem in class or in the laboratory, and a teacher's decision-making while teaching a mathematics (or other) lesson. " In this paper, we will only consider problem solving in mathematics instruction. Nevertheless, what we mostly found in mathematics educational literature and in learning material designed for students is a restricted use, with sometimes a reduction to drill for improving skills. In our opinion, this approach is too limited. Thus we first analyze some typical situations used in mathematics education with a problem solving purpose.Our goal in the following analysis is to underline the possible ways for managing the situations, and to exhibit the parameters that educators have at their disposition within their role as mediator between students and mathematical knowledge and know-how.1.1. Too easy... and how to do it betterThe following Math Problem Solving example comes from a worksheet of Rhl School. We may consider it as a representative of certain trend in exploiting problem solving in mathematics education."Ryan's ClassThere are exactly twelve children in Ryan's class. Only four of the children are boys. The following questions refer to a time when all the children are present in the class. There are no visitors in the class. There might be more than one correct answer to a question.1. Which of the lettered statements must be true?2. Which of the lettered statements cannot be true?3. Which of the lettered statements could be true or not true?a. There are twice as many girls as boys in Ryan's class.b. There are eight more girls than boys in Ryan's class.c. There are four more girls than boys in Ryan's class.d. If Ryan is sitting at a table with all the girls, there are exactly nine children at that table.e. If only three of the boys are standing on their heads, one of the boys is not standing on his head."Math Problem Solving, Vol. 8, No. 1, May 5,2003. Retrieved from http://www.rhlschool.com/math8nl .htmComments about the task: Giving an answer to the lettered statements a, b, c and e only supposes to understand them. The only true problem here comes from the fact that we don't know Ryan's gender (while mostly used for boys, Ryan has been used for girls in the United States since the 1970s). And for the lettered statement d, if Ryan is a boy, the statement is true, and if Ryan is a girl, the statement is false; then the correct answer for us is the answer 3, but the correct answer would be different for somebody who knows Ryan's gender. So we note that there is a subtle semantic distinction in that statement; nevertheless, in terms of mathematics knowledge, we will say that the proposed task is too easy. That means that the mathematical activity for answering is poor, even though understanding the statements may have a relatively high intellectual cost for the students. Compare with the situation that would result from the following questions, without changing the data (i. …