Abstract

Teachers of mathematics often use diagrams to explain concepts related to quantity. Students of mathematics often have difficulty, however, understanding how the diagrams represent the intended concepts (Uttal, Liu, & DeLoache, 2006). This is consistent with the research on students’ difficulties with mathematical symbols and notation (Hiebert, 1992), and previous studies have demonstrated that teachers need to make the connections between symbols and their conceptual referents explicit (Osana & Pitsolantis, 2013). This study examined the impact of instruction that explicitly teaches preservice teachers the conceptual meaning of a mathematical symbol called the “bar diagram.” Fifty undergraduate students (N = 50) were assigned to one of three conditions: Bar Diagram with Links (BDL), Bar Diagram with No Links (BDNL), or Comparison. The students in the BDL condition were explicitly shown, with the use of concrete materials, that Bar Diagrams are mathematical symbols used to represent quantity and relations among quantities. The BDNL condition was exposed to the Bar Diagram symbol without any explicit connection to referents. Students in the third condition served as a comparison group. Objectives of this study were to determine whether it is critical to make explicit connections to the conceptual meaning of a mathematical symbol prior to appropriately applying the symbol to (a) solve word problems presented using text in a story format, (b) view the bar diagrams as pictures that represent quantities for the purpose of solving problems, (c) understand the quantitative meaning of bar diagrams, and (d) use algebraic symbols to solve analogous problems.