Technology- based algebra learning: Epistemological discontinuities and curricular implications
In general, an important goal of the Visual Math curriculum is to help students develop strong algebra skills and to learn to perform a variety of standard algebraic manipulations with understanding rather than by rote memorization. The early parts of the Visual Math curriculum focus on one-variable functions and equations conceptualized as the comparison of two one-variable functions. Thinking in this way about equations or inequalities, students acquire, in addition to algebraic procedures, alternative methods of solving equations. Before using symbolic manipulations, and in the process of doing so, students are encouraged to use systematic guessing and intuitive numeric and graphic analysis strategies, and to conjecture about the visual effects of symbol manipulations. For example, function graphing tools support the students’ attempts to explore questions of equivalence as they learn to manipulate equations algebraically. Conjecturing, demonstrating and reasoning whether an operation on an equation or on an inequality would result in an equivalent equation (or an equivalent inequality) is a central activity in this curriculum. Legitimate manipulations are those involving the same operations on both sides of the equation or inequality. Such manipulation, as shown in Figure 3.1, changes each of the compared functions, each of the graphs, as well as the points of intersection of the two graphs, but preserves the solution set, so that changes should not affect the x values of the intersection points. Understanding equivalence equips learners with the tools they need to discuss questions such as: Why can’t one always multiply each side of an equation by x? What happens when an inequality is multiplied by a negative number that
causes the inequality sign to change direction? Chazan and Yerushalmy (2003) described these questions as those that students seldom have opportunities to ask. Another central activity in the Visual Math curriculum that takes advantage of early learning of functions is solving problems in context. Figure 3.2 describes a common algebra problem and a solution that appears commonly in the work of Visual Math algebra beginners (seventh and eighth graders). The solution identifies two processes that need to be compared in the story-problem. It consists of a sketch describing the structure of the situation in the problem (two intersecting graphs, each describing the change in position over time of one vehicle relative to point A), two algebraic expressions that match the functions in the graph (g(x)=56x and f(x)= 476−80x), visual and numeric scripts of the rate of change on each graph (an annotated ‘step’ indicating its size and direction), and an equation (476−80x=56x) that represents the comparison of the two functions. The numeric solution is the x value of the intersection point. In general, solutions included a graph and algebraic expressions of two functions that matched the graph. Graphs did not replace the algebra but rather served as a visual aid to formulate the algebraic equation. Viewing equations as models analogous to graphs and situations; viewing an equation as a comparison of two functions, most often graphically; and viewing algebraic letters as variables of functions served as powerful resources, leading to an exceptional success rate in comparative studies (Gilead & Yerushalmy 2006). Using these resources, students were able to solve problems for which they had not yet studied an algorithmic solution method, and although they were still algebra beginners they exhibited profound understanding of advanced calculus ideas related to rate of change of non-linear processes (Shternberg & Yerushalmy 2003). They demon-
Figure 3.1 Graphs of two equivalent inequalities x2 – 4 > – 2 – x and 3(x2 – 4) > 3(– 2 – x).