Problem posing via “what if not?” strategy in solid geometry — a case study
Introduction
Problem posing involves creating new problems for investigating a given situation as well as reformulating a problem during its solving process (Silver, 1994). The term “problem posing” in the literature usually refers to an activity in which the problem posing itself is the focus of attention and not a problem-solving tool. The activity of problem posing can take place before, during or after solving a given problem. The present study focuses on problem posing before and after solving a given problem.
The Israeli mathematics matriculation examination is built in a manner that there are optional questions from which one may choose to do some and ignore others. These questions are each related to a certain topic so that, for example, a student who did not study solid geometry can choose an alternative question in another topic and still have the opportunity to get the maximum score in the exam. As mathematics teachers’ educators we had many informal talks with in-service Israeli mathematics teachers who told us that they prefer not to teach solid geometry in high schools since the subject is difficult to teach and to learn. Moreover, Israel matriculation examinations results indicate that the percentage of students choosing to answer questions in solid geometry is lower than the percentage of students who choose to answer questions in any other mathematical area (Latner & Movshovitz-Hadar, 1999).
Shifting responsibility for problem posing from teachers to students “can have clear affective gains [and] necessarily involves some review of material used” (Watson & Mason, 2002). We supposed that by the activity of creating solid geometry problems instead of just solving them, we would reduce, even for a little, the fear and the restraint from teaching this subject. We decided to adopt the “what if not?” strategy (Brown & Walter, 1993) to make the process of problem posing more systematic, structured and traceable. According to “what if not?” strategy, we examine each component of the problem and manipulate it through the process of asking “what if not?” We will discuss this strategy more broadly later.
The study participants were familiar with the “what if not?” strategy. They have used this strategy in previous courses through different mathematical topics such as function analysis and algebra. In the present study, we wanted to see how they implement the “what if not?” strategy in solid geometry problems.
Our study focuses on issues referring to the intersection of three topics: solid geometry concepts, posing problems related to mathematical thinking, and pre-service teachers’ education.
Section snippets
The role of problem posing in students’ mathematics education
During their study of school mathematics, students experience problem solving. Usually they get the problems from their math teacher or from text books. Only rarely are they asked to pose problems similar to, or different from, problems they have seen before. However, mathematics education researchers have long emphasized the educational value of problem posing by students and suggested incorporation of activities of problem posing within the mathematics sessions in school (Brown & Walter, 1983a
The role of problem posing in the education of mathematics teachers
Developing skills of mathematical problem posing is especially important for mathematics teachers (Silver et al., 1996, Southwell, 1998). Southwell (1998) found that posing problems based on given problems could be a useful strategy for developing the problem solving ability of pre-service mathematics teachers. Integrating problem posing activities in their mathematics lessons enabled them to get to know their students’ mathematical knowledge and understanding better. Since teachers have an
The “what if not?” strategy
Posing new problems can be based on free, semi-structured and structured situations (Southwell, 1998). A free problem posing situation refers to the case in which the learner is given an open situation and has a free hand formulating new problems. A semi-structured problem posing situation relates to an open situation in which the learner is requested first to explore its structure and complete it and then to pose new problems. A structured problem posing situation refers to the case in which
The study
Twenty-eight pre-service teachers participated in the study. The participants were on their second or third year of studies (out of four years) towards graduating as high school mathematics teachers. The participants were students in a “teaching methods for secondary school mathematics” course. Prerequisites for this course included mathematics courses, such as linear algebra, and group theory, and a course of general methods and teaching practices.
As students in the “teaching methods for
Research findings
A total of 108 new problems were posed by the participants. All research participants from both groups posed problems. Analysis of the posed problems revealed a wide range of problems, from problems including a change of one of the numerical data in the problem to another specific one, to a proof problem. Several classifications of the posed problems were made and will be broadly described later. Table 1 displays the data components of the given problems to facilitate the tracing of the
The kinds of posed problems
The first classification refers to the type of change participants made of one of the given problem components. The posed problems were divided into two main categories: changing one of the problem data components; changing the problem question. Each one of the above categories was refined into subcategories. The first category was divided into three subcategories: 1. Changing the numerical value of the data; 2. Changing the kind of data. 3. Eliminating one of the problem data.
The second
The educational value of posing problems using the “what if not?” strategy
In this section, we discuss different aspects of the educational value of problem posing activity using “what if not?” strategy through two dimensions: the pre-service teachers as learners; and the pre-service teachers as future teachers. We refer to the following aspects: rethinking of the involved geometrical concepts; deepening students’ understanding of geometrical concepts; developing thinking skills.
Additional mathematical observations
Although the focus of our study was the activity of posing problems rather than solving them, in this section we will refer to two supplementary mathematical observations referring to the solvability of the posed problems and the influence of the data change on the number of data components in the posed problem. Discussing the above issues while engaging in problem posing activity using the “what if not?” strategy might be regarded as educational benefits of the activity. Questions connected to
Concluding remarks
In the present study, we characterized the different kinds of posed problems on the basis of a given one, using the “what if not?” strategy. We observed two main basic ways for changing a given problem into a new one: changing one of the data components and changing the problem question. The most common change the research’s participants applied was the change of a specific data (numerical or data kind) into another specific one. Unlike problem solving, the activity of problem posing is rarely
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Problem formulation: insights from student generated questions
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