This workbook works to address the following three common misconceptions for students learning algebra: 1) Understanding the equals sign operationally to mean “the answer is,” “add up the numbers,” or “multiplies to,” rather than a relational concept of the equals sign that means to “is equivalent to” 2) Interpreting a negative sign as subtraction rather than a negative number or variable 3) Viewing a variable as a label (B is books) rather than as a number (B is the number of books, or B is the cost of a book) Bar modeling techniques can help address each of these misconceptions, although we found that emphasis from the instructor is also crucial to develop correct conceptions. In this workbook, we have tried to correct these misconceptions in the following ways: 1) The equals sign in bar modeling is shown when the two bars are of equal size. When a student is working through the problems, it is imperative that they consistently show equality by drawing the bars as equal size. Additionally, attending to precision also develops the spatial reasoning necessary to solve mathematical problems. 2) A negative in bar modeling is always part of a specific number block or variable block. The right side of a negative block gets attached to a bar, rather than the left side as with positive blocks. The parallel to the number line is clear with this technique. 3) For the word problems in this book, the student is asked to define the variable in the upper-right hand corner. They may choose which letter they wish to use, although the most important part of this exercise is that they define the variable clearly. In particular, their definition must refer to a number - usually, the question in the word problem contains the definition of the variable. For a comprehensive review of common misconceptions for middle school students in algebra, see Sarah Busha and Karen S. Karp, “Prerequisite algebra skills and associated misconceptions of middle grade students: A review,” in The Journal of Mathematical Behavior 32, no. 3 (2013): 613–632, http://dx.doi.org/10.1016/j.jmathb.2013.07.002. For more discussion about how misconceptions are addressed in our workbook, as well as more sources, see our Literature Review. |
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