Elementary school students often experience misconceptions (i.e., limited conceptions or alternative conceptions) in algebra that their teachers fail to recognize or understand (Smith, diSessa, & Roschelle, 1994).  Misconceptions have been defined as occurring when students use incorrectly learned strategies to solve new problems (Russell, O’Dwyer, & Miranda, 2009).  These wrong answers are not simple errors but instead are systematic in nature and come from student past experiences with and misunderstandings of such problems (Russell et al., 2009).  Revealing these student misconceptions is crucial, as students not provided with opportunities to experience algebra may continue to struggle throughout school (Smith, et al.).  Although Carpenter and colleagues (2005) have investigated these misconceptions in the area of equivalence, a paucity of research was found investigating misconceptions across the entire strand of algebra.  Because of this research gap, the following research question will be addressed: What algebra knowledge do elementary students have and what misconceptions are they experiencing? 

Perspectives

Although algebra has been a constant as a required course in America’s school system since prior to the 1980’s, it primarily existed as a specific subject to be studied only in the upper levels of middle and high schools (Wagner & Kieran, 1989; Smith & Thompson, 2008).  Many researchers would disagree with this mentality, however, instead describing algebra as a set of skills that should be utilized across all mathematical topics, across all grades kindergarten through twelfth grade, and not as a stand-alone topic strand.  Kaput and Blanton (2005), for example, clarified this: “Algebraic reasoning needs to develop over a long period of time in students’ mathematical experience, beginning in the early grades and engaging most mathematical topics” (p. 100).  Depriving students of such opportunities in the elementary grades may make learning algebra later (e.g., in middle and high school) more difficult.  This is especially the case considering that in the United States, algebra is often an important determinant in students’ mathematical lives; in that, often, it is after their experience with algebra that they determine if they would like to continue to pursue math (Mason, 2008). 

Algebra and algebraic reasoning are often considered abstract concepts, with a variety of potential definitions.  Some algebra researchers have defined algebra as containing three distinct strands: 1) modeling, 2) functions, and 3) generalized arithmetic (Kaput, 2008).  Modeling knowledge includes solving open number sentences, understanding equivalence, and using variables.  Functions in the elementary grades include possessing the ability to recognize, describe, extend, and create linear and nonlinear patterns.  Generalized arithmetic includes simplifying (i.e., simplify calculations using number relations and compensation strategies) and generalizing (i.e., utilizing mathematical properties like the commutative property, property of zero, etc.) (Wagner & Kieran, 1989).   

Data Sources and Methods

In this qualitative multiple case study, I analyzed the experiences of three in-service teachers and their students.  The teachers teach first, second, and fifth grade, respectively, at two high-poverty, low-achieving schools in a large urban school district in the Pacific Northwest. The student participants included 17 first grade students, 15 second grade students, and 20 fifth grade students, for a total of 52 students.   

To confirm findings and ensure data quality, data collection was triangulated through three different methods: multiple teacher interviews with each teacher participant, student interviews with each student participant, and classroom observations with each classroom (Miles & Huberman, 1994).  Teachers were asked about and students were asked to solve items in all three areas of algebra: modeling (i.e., equivalence problems such as 8+4=__+5), generalized arithmetic (i.e., simplification problems such as 15+7-6=__), and functions (i.e., linear patterns such as 3, 5, 7, 9, 11, __).  Data collection efforts began in January of 2011; data was collected over a period of four months. 

Results and Conclusions

The findings revealed that the students held a number of algebraic misconceptions that occurred throughout the three strands of algebra: modeling, generalized arithmetic, and functions.  In the area of modeling, for example, students across the three grade levels were able to solve open-number sentences such as 8+__=15 fairly easily, with rates of success for first, second, and fifth grade students of 65%, 87%, and 100%, respectively.  When this task was made conceptually more difficult, however, by introducing an equivalence task such as 6+7=__+4, these rates of success for first, second, and fifth grade students decreased to 6%, 13%, and 60%, respectively.  Misconceptions occurring here included adding the first two numbers, adding all three numbers, or ignoring the first number.  When asked to make this concept still more complex by introducing variables into the problem such as 6+B=9, the rates of success for first, second, and fifth grade student remained low at 29%, 47%, and 80%, respectively.  The primary misconception occurring with this problem involved visualizing B numerically (i.e., A=1 so B=2).  This type of analysis continued for the other two areas of algebra (i.e., functions and generalized arithmetic). 

Importance

The importance of this work lies in the concept that students who hold misconceptions surrounding algebraic reasoning in elementary school may continue to struggle with algebra in the later grades (National Mathematics Advisory Panel, 2008).  Carpenter and colleagues described this problem as: “If not addressed, misconceptions about equality and variables persist and provide serious impediments for learning high school algebra and other advanced mathematics” (p. 96).  This research supports Carpenter and colleagues’ (2005) work indicating that elementary students may be experiencing misconceptions in algebra that may need attention.  Further, this research supports the idea that teachers may need professional development in the area to improve student algebraic thinking skills.  Several researchers (Jacobs, Franke, Carpenter, Levi, & Battery, 2007; Carpenter, et al., 2005) have seen improvement of student algebraic reasoning skills through conducting professional development with teachers.  Despite several limitations, these analyses strive to gather an initial description of student misconceptions in algebra. 

This session will review the literature, discuss the misconceptions, and engage the audience in a discussion regarding potential solutions to remedy algebra misconceptions.  This session will address diagnostic assessment tools (i.e., to discover misconceptions) and professional learning (i.e., to inform teachers how to remedy such misconceptions).