Logical Aspects of Equations and Equation Solving
Specialerapport. IND's studenterserie nr. 52. Matematik.
Tanja Rosenberg Nielsen, september 2016.
Abstract
This thesis concerns danish upper secondary school students' practices with equations. Data for this thesis have been collected through questionnaires to which 161 students spread over eight classes from two different upper secondary schools responded.
The purpose of the thesis is to study how the students define the concept of equations, which difficulties they meet when solving equations and how they perceive the equality sign in equations. Moreover, the students' conceptions of equations are compared to the scholarly concept of equations, and some textbooks for upper secondary school are analyzed in terms of their presentation of equations and compared to the students conceptions.
Of particular interest to this study is the four features: equality sign, variable, something to solve and whether the equation is always true, always false or conditionally true (ie true for some but not all possible values of the unknown).
The collected data show that students value the presence of a variable most important, the presence of an equality sign second most important, the fact that there is something to solve/rearrange third most import and the expression being conditionally true least important in order for something to be an equation.
Moreover, we have looked at students' mastery of operations applied to equations in order to solve them that cause an expansion or a reduction of the domain of the equation to be solved. Data show that students are not aware of these types of mistakes at all. Moreover, they are likely to make calculation errors or get stuck before they even reach the point where the mistake could possibly be made.
Stephens et al. (2013) describe three different ways in which one can perceive the equality sign. This study is especially interested in the relational-operational understanding and the relational-structural understand. A person holding a relational-operational understanding is aware that the equality sign expresses a relation between the left hand side and the right hand side and confirms the relation by calculating. A person holding a relational-structural understanding views the equality sign as a symbol expressing equivalence between two expressions rather than between two calculations. The data show that one fifth show a relational-structural understanding, and about three fifth show a
relational-operational understanding at some point. But it is also found that the shown understanding differs from task to task, and very few students show a consistent understanding.