The Importance of Mathematical Relationship Formation in Math Word Problem Solving : An Eye Movement Analysis

An investigation into the importance of mathematical relationship formation in word problem solving sought to identify ways in which mathematical reasoning and achievement can be improved. The investigation broke problem solving into a series of steps with the hopes of facilitating solution accuracy though a proposed cognitive model which emphasized the importance of relative set size identification and relationship formation in the problem solving process. Participants were presented with a series of one, two and three part word problems aimed at examining the importance of each step, while their eye movements were monitored in order to identify patterns in attentional allocation during the process. It was hypothesized that following the steps in the proposed cognitive model would result in higher problem solving accuracy and that high accuracy problem solvers would devote more attention to the relational statement of the problem than low accuracy problem solvers. Results supported the proposed cognitive model, suggesting that problem solving accuracy is facilitated through model application. Finally, eye movement analyses revealed that high and low accuracy problem solvers focus their attention in different places within the problem text; specifically high accuracy problem solvers attend more to the question than low accuracy problem solvers. Trends identified in this study can be applied to education in order to improve the math problem solving skill of students. Mathematical Relationship Formation 3 The Importance of Mathematical Relationship Formation in Math Word Problem Solving: An Eye Movement Analysis Knowledge of mathematics and an ability to integrate mathematical skills into everyday life is becoming increasingly important in our industrialized society (Roman, 2004). Math education is critical in preparing young people for the mathematical and problem solving challenges that they will face as adults. Compared to many industrialized nations around the world, students in the US are lagging behind in mathematics achievement (Mayer, Tajika & Stanley, 1991). This indicates a real need for examining the math education system in the US. Mathematics education can be thought of as the teaching of two important topics: the execution of procedural skills and the solving of word problems (Nesher, 1986). A word problem sets up a situation in which one or more unknown variable(s) are defined in terms of a known variable(s). Often, word problems take the form of a short story, such as the example in Figure 4. Math educators are responsible for teaching students how to perform arithmetic operations as well as teaching them how to apply their arithmetic skill to real world quantitative problems. Interestingly, students who excel in arithmetic skill might not be successful problem solvers (De Corte, Verschaffel & Pauwels, 1990), indicating a need for students to develop a way of thinking about mathematics that is not simply a rote memorization of arithmetic facts. One aspect of problem solving is the progression that one must go through in order to arrive at a solution. For example, when presented with a problem, one must first decide what is being asked, then the appropriate information must be selected from the Mathematical Relationship Formation 4 problem in order to create a solvable mathematical relationship, and finally a series of computations must be performed in order to arrive at the numerical answer. Often, children are told what types of problem solving skills to use in given situations, making it difficult for them to make problem solving decisions when they are unprompted (Swan, 2004). One focus of math education should be to teach the skills necessary to facilitate independence and success in problem solving. This study will investigate the components of problem solving as they relate to mathematics performance in order to determine sources of difficulty in the problem solving process. In particular, the process of mathematical relationship formation will be examined in order to understand its contribution to problem solving accuracy. Additionally, an examination of the processes that facilitate mathematical relationship formation will be conducted. Through eye movement analyses, the attention and focus of participants will be monitored as they attempt to solve various math word problems. Eye movement analyses operate on the premise that eye fixations correspond to attention allocation. In this study, it is assumed that people will spend a greater proportion of time looking at the components of the word problems to which they are attending. By identifying common areas of attention and inattention associated with problem solving difficulty, we will be better equipped to understand the process of problem solving and will eventually be able to use this knowledge to improve math education. This paper will focus on the processes involved in mathematical relationship formation. Mathematical relationship formation is the process through which the text of a problem is transformed into a numerical mathematical relationship or equation. This relationship generally establishes a link between different variables in the text, often Mathematical Relationship Formation 5 defining the value of one variable in terms of the value of another. Because equations are the primary way to represent mathematical text as numbers (Hinsley, Hayes & Simon, 1977), the process of mathematical relationship formation can be used to monitor comprehension and to trace the solution process involved in math word problem solving. Cognitive Components of Mathematical Problem Solving According to Nesher (1986) two key topics exist in math education. Students must learn to execute procedural skills and they must learn to solve word problems. The difference here lies in the task itself: procedural skills are generally used to solve computational problems while a different type of thinking must be employed when students are trying to solve a word problem. (For the purposes of this paper, we will refer to procedural knowledge as computational skill.) The solving of word problems draws on both problem solving skill – how well a problem solver can interpret and set up the text of a word problem for solution computations – and procedural skill, as the process generally requires some sort of computation in order to arrive at a solution. In this way, problem solving accuracy is strongly dependant on correct procedural knowledge, but requires an additional ability to understand and interpret the logical structure of a given text. Because both computational skill and problem solving skill can influence the accuracy of solutions, it is important to differentiate between the processes involved in each in order to better understand the problem solving process as a whole. For the purposes of this paper, problem solving success will be defined as the arrival at an accurate solution. Computation vs. Problem Solving. The computational skills introduced by Nesher are necessary to perform arithmetic computations. These types of skills can be Mathematical Relationship Formation 6 further divided into procedural and propositional (or declarative) representations of knowledge (Nesher, 1986). A propositional representation of knowledge is a factual statement that is readily retrieved from memory. For example, it is easy for most adults to know, without performing any computation, that 3+2 = 5 because this is a math fact that has been practiced and memorized, and can be easily retrieved from memory. However, when presented with (3+2) (7-4) = ? , adults must draw on some sort of procedural knowledge that tells them to first perform the operations within the parentheses and then to multiply the two results. In this case, the values inside the parentheses are relatively simple and will most likely be computed using a propositional representation of knowledge, but the process which must be followed in order to perform the entire computation accurately requires a procedural representation of knowledge. Regardless of the processes required in computation, it is important to consider that arithmetic ability is only one component of solving word problems, and therefore only contribute so much to solving word problems successfully. Consequently, is possible for a person to be very good at computation but perform quite poorly on tests of problem solving ability (DeCorte, et al., 1990) resulting in inaccurate solutions. This is consistent with the notion that computational skill and problem solving ability constitute distinctly different domains, and skill with one does not necessarily imply skill with the other. A study conducted by Mayer, et al. (1991) specifically discriminates between problem solving ability and computational skill in order to uncover differences between the math abilities of American students and those of