Investigating the Conceptual Basis for the Early Introduction of Negative Numbers
Laura Bofferding

About the research


NAEd/Spencer Postdoctoral Fellowship

Award Year



Purdue University

Primary Discipline

Mathematics Education
In an effort to improve STEM teaching and learning, there has been a renewed focus on what mathematics is taught in school, how it is taught, and when. The adoption of the Common Core State Standards has contributed to one of STEM advocates’ main goals: reducing the number of concepts taught at each grade with the goal of focusing on them in more depth (Board on Science Education and Board on Testing and Assessment, 2011). The same group also pushes for content that is “logically sequenced over time to develop more comprehensive understanding” (p. 2).Based on the belief that they are too abstract for young children to learn, negative numbers are usually not taught until middle school. Because children mostly learn whole numbers during elementary school, many learn that they cannot subtract a larger number from a smaller one (Ball & Wilson, 1990). This belief frequently results in the smaller-from-larger bug where students incorrectly reverse the numbers, solving 4-9 as 9-4 (Murray, 1985; VanLehn, 1982) or solving two-digit subtraction problems by subtracting the smaller number from the larger number instead of regrouping (e.g., solving 62-48=26 instead of 14; Fuson, 2003). These conceptions later contribute to students’ difficulties with solving problems with negative numbers (Murray, 1985).My proposed research project extends the work of Siegler and Ramani(2009) to the domain of integers; over two years, I will investigate first-graders’ (year 1) and kindergartners’ (year 2) developing understanding of negative numbers as they play a linear board game (modified from Siegler & Ramani). Results will add to and clarify aspects of Case’s (1996) central conceptual structures theory regarding the development of numerical concepts. Contrasted with control groups, results of the game playing group will highlight if children can learn negative number concepts while learning positive number concepts or only after developing positive number knowledge. Further, children’s ability to solve problems such as 3-9 will clarify whether early introduction of negatives could help lessen the prevalence of subtraction errors (e.g., solving 3-9 as 9-3).Specifically, this research project addresses the following three goals: Evaluate the extent to which playing linear board games can contribute to the development of a mental number line for integers. Determine if students must have formed their mental number line for positive numbers before they can make sense of negative integer order and values or whether they can learn these concepts as they are developing an understanding of the correspondence between positive number order and values. Explore the extent to which young children can use their mental integer number line to solve simple integer addition and subtraction problems.
About Laura Bofferding
Laura Bofferding is an assistant professor at Purdue University in the department of curriculum and instruction and specializes in elementary mathematics education. She received her Ph.D. in Curriculum Studies and Teacher Education (Mathematics Education) from Stanford University in 2011. Her primary research area is on elementary-school students’ learning of integer concepts and integer learning trajectories. Her dissertation work leveraged a mixed methods design to investigate elementary students’ mental models of negative number concepts and evaluate instruction that can support students’ transition from reasoning about whole number to integer concepts. Most recently, Dr. Bofferding received a grant from the Purdue Research Foundation to examine the processes and knowledge involved in pre-service teachers’ solutions to integer problems. In addition to her research, Dr. Bofferding has also served as a mathematics consultant for ACE consulting through the University of Notre Dame to provide schools with feedback on ways to improve their mathematics teaching.

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