Meanings given to algebraic symbolism in problem-posing

Abstract

Some errors in the learning of algebra suggest students have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing in order to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process depending on the characteristics of the algebraic statements given. We designed a written questionnaire composed of eight closed algebraic statements expressed symbolically, which was administered to 55 students who had finished their compulsory education and that had some previous experience in problem posing. In our analysis of the data, we examine both syntactic and semantic structures of the problem posed. We note that in most cases students posed problems with syntactic structures different to those given. They did not include computations within variables, and changed the kinds of relationships connecting variables. Students easily posed problems for statements with additive structures. Other differences in the type of problems posed depend on the characteristics of the given statements.

Development of the Sci-math Sensemaking Framework: categorizing sensemaking of mathematical equations in science

“…These steps included (1) borrowing from the tens column, (2) subtracting 4 from the ones column, (3) borrowing from the hundreds column, (4) taking away 6 from the tens column, and (5) taking away 1 from the hundreds column. The focus on procedural knowledge in mathematics classes has been criticized as one reason for students' difficulty in understanding the meaning of equations or adopting efficient problemsolving strategies for new or complex problems (Cañadas, Molina, & del Río, 2018;Jacobs et al, 2007;Peled & Segalis, 2005). Similarly, in science classrooms, researchers have found that students tend to rely on algorithms without conceptual understanding of the science in solving problems and teachers tend to focus instruction on using mathematical procedures to do calculation (Bain, Rodriguez, & Towns, 2019b;Bing & Redish, 2007;Hansson et al, 2015;Hu & Rebello, 2013;Kuo et al, 2013;Lehavi et al, 2017;Redish, 2017;Redish & Gupta, 2010;Redish & Kuo, 2015;Schuchardt, 2016;Schuchardt & Schunn, 2016;Tuminaro & Redish, 2007;Wink & Ryan, 2019).…”

“…In the Math-Structure sensemaking category, the focus is on understanding the arrangement of variables (symbols) and operations of the mathematical equations (Bain, Rodriguez, & Towns, 2019a, 2019bBecker & Towns, 2012;Bing & Redish, 2007;Brahmia et al, 2016;Cañadas et al, 2018;Hestenes, 2010;J. Hiebert & Lefevre, 1986;Hu & Rebello, 2013;Izsák, 2004;Jacobs et al, 2007;Karam, 2014;Karam & Krey, 2015;Kirshner, 1989;Kuo et al, 2013;Moss & Case, 1999;Pospiech, 2019;Redish, 2017;Redish & Kuo, 2015;Rodriguez et al, 2019;Rodriguez et al, 2018;Schuchardt & Schunn, 2016;Sherin, 2001).…”

“…Math-Relation sensemaking refers to understanding the quantitative relationships expressed in the equation (Bain, Rodriguez, & Towns, 2019a;Baroody et al, 2007;Becker et al, 2017;Becker & Towns, 2012;Cañadas et al, 2018;Carlson et al, 2002;Dixon et al, 2001;Hestenes, 2010;Izsák, 2004;Izsák & Jacobson, 2017;Jacobs et al, 2007;Karam, 2014;Kuo et al, 2013;Lehavi et al, 2017;Levy & Wilensky, 2009;Moss & Case, 1999;Pietrocola, 2009;Pospiech, 2019;Redish, 2017;Redish & Kuo, 2015;Rodriguez et al, 2018;Rodriguez et al, 2019;Schuchardt, 2016;Sherin, 2001;Smidt & Weiser, 1995;Thompson & Carlson, 2017;Tuminaro & Redish, 2007;Uhden et al, 2012;Von Korff & Sanjay Rebello, 2014;Wink & Ryan, 2019). Carlson et al (2002) defined covariational reasoning as attending to the way in which two variables change with respect to one another.…”

“…The categories identified in the mathematics sensemaking dimension of the Sci-Math Sensemaking Framework provide a nuanced description of the multiple types of mathematics sensemaking that can occur when working with mathematical equations in science. The teaching and learning of mathematical procedures, rules, structure, relation, and concepts have been widely discussed in mathematics education studies (Baroody, 2003;Cañadas et al, 2018;Hiebert & Lefevre, 1986;Moss & Case, 1999;Peled & Segalis, 2005;Star, 2005). The use of mathematics in science offers the opportunity for students to practice mathematics as mathematicians, thus developing their mathematical thinking (Schoenfeld, 1992).…”

Improving mathematical learning in Scotland's Curriculum for Excellence through problem posing: an integrative review

“…Although such conditions can spawn a diversity of topics, a specific mathematical theme can similarly be employed. For instance, Canadas et al (2018) present situations using algebraic statements to extend mathematical thinking as opposed to performing meaningless operations on equations. Table 1 displays posed problems from the previous research.…”