Problem Posing in Consumer Mathematics Classes: Not Just for Future Mathematicians

Abstract

IntroductionEllerton (2013) points out, To state that problem posing is as fundamental to mathematics as problem solving should be to state the obvious-after all, one cannot solve a problem unless first, a problem has been posed. (p. 87) However, the importance of problem posing is much greater. Freire (2007) says problem-posing education involves a constant unveiling of reality (p. 72), and sees problem posing as a fundamental cornerstone of education. Similarly, Nicolaou and Xistouri (2011) among others, identify problem posing as lying at the heart of mathematical activity (p. 612). The importance of problem posing is identified in various national curricula including (a) The United States, in both the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), and the Common Core Standards (Flick & Kuchey, 2010); (b) China (Silver, 2013; Van Harpen & Sriraman, 2013); (c) Australia (Silver, 2013); and (d) Ontario (Ontario Ministry of Education, 2005, 2007). The journal Educational Studies in Mathematics (January, 2013) recognized the importance of problem posing by devoting a special issue exclusively to the topic.Problem posing can be either a goal of mathematics education or an instructional strategy (Stoyanova, 2005). However, in many curricula problem solving often dominates over problem posing (Stoyanova, 2003). This is despite research showing that problem posing (a) improves students' problem-solving skills, attitudes, and confidence, understanding of concepts, and mathematical thinking (Singer et ak, 2013); (b) reinforces basic mathematical skills, increases motivation, responsibility, and thinking flexibility (Ponte & Henriques, 2013); and (c) is useful for teachers to assess students' cognitive processes, identify misconceptions, and modify instruction (Ponte & Henriques, 2013). There is also extensive research relating problem posing to creativity. Voica and Singer (2013) point out that there are correlations between problem solving, problem posing, and creativity. Among the multiple descriptions of creativity are problem finding, problem solving, and problem posing, and problem posing is frequently used in assessing creativity (Bonotto, 2013; Pelczer & Rodriguez, 2011).Much of the research in problem posing has focused on elementary students (Barlow & Cates, 2006; Bonotto, 2013; Cai et ah, 2013; Kerekes, Diglio, & King, 2009; Nicolaou & Xistouri, 2011; Singer & Voica, 2013; Stoyanova, 2003; Voica & Singer, 2012, 2013); college and university students (Koichu & Kontorovich, 2013; Ponte & Henriques, 2013); gifted high school students (Singer et al., 2013; Stoyanova, 2003; Van Harpen & Sriraman, 2014; Voica & Singer, 2013); teachers and teacher education (Ellerton, 2013; Shriki & Lavy, 2012; Singer et al., 2013; Stoyanova, 2005; Ticha & Hospesova, 2013). There are relatively few references to problem posing in nonacademic streams even thoughPreparing students for life is seen by some educators as an ultimate goal of school education. Those students who will not become professional scientists will need to be able to apply mathematics in everyday life situations. It is therefore important that students' experiences in mathematics classrooms help them to become competent users of mathematics by being able to pose, analyze and solve real world problems. (Stoyanova, 2003, p.33)The current case study examines an instructional strategy with a focus on problem posing in six grade 11 consumer mathematics classes over a 3-year period.Problem Posing ModelsProblem posing refers to both the generation of new and the reformulation of given problems (Singer, 1994, as cited in Pelczer & Rodriguez, 2011). Problem posing and problem solving are interlinked. Silver (1994) identifies three temporal periods when problem posing can occur while engaging in problem solving: before problem solving (pre-solution), during (within-solution), and after (post-solution), as cited in Singer et al. …