Teacher: What do you think the equal sign means?

Student 1: I think it means like what's the next number of the addition problem.

Student 2: It means the sum.

Student 3: It tells what the number is in an addition or subtraction problem.

Stop and think for a moment. If these students were given the equation 5 + 6 = -- + 4, would they put a 7 in the blank? Probably not. Their comments demonstrate what is referred to as an operational view of the equal sign (Carpenter, Franke, & Levi, 2003). Students who view the equal sign operationally believe that the symbol indicates that they should be performing an operation. Thus, these students would most likely fill in the blank with an 11 (the sum of 5 and 6) or a 15 (the sum of all numbers in the equation). In contrast, students with a relational view of the equal sign recognize the relationship that the symbol represents between the two sides of the equation (Carpenter et al., 2003). Students with a relational understanding will state that the equal sign indicates that one side of the equation is the same as the other side. A student with a relational view would know to put a 7 in the blank in order to achieve the equal relationship, resulting in the same amount on each side of the symbol.

It has been well-documented that many students do not understand the meaning of the equal sign (Carpenter et al., 2003; Faulkner, Levi, & Carpenter, 1999; Knuth, Alibali, Hattikudur, McNeil, & Stephens, 2008; Knuth, Stephens, McNeil, & Alibali, 2006; Molina & Ambrose, 2006). Thus, researchers have called for instruction that specifically addresses such misconceptions (Carpenter et al., 2003; Faulkner et al., 1999), and have indicated that such work must start at the elementary level (Knuth, Alibali, et al., 2008).

In response to these recommendations, some state curriculums require that students gain a full understanding of the equal sign. For example, the state of Mississippi has the following objective included in its 3rd-grade curriculum: "Create models for the concept of equality, recognizing that the equal sign (=) denotes equivalent terms such that 4 + 3 = 7, 4 + 3 = 6 + 1 or 7 = 5 + 2" (Mississippi Department of Education, 2007, p. 26). Similarly, Connecticut's 2nd-grade curriculum states, "Demonstrate an understanding of equivalence or balance of sets using objects, models, diagrams, numbers whole number [sic] relationships (operations) and the equals sign, e.g. 2 + 3 = 5 is the same as 5 = 2 + 3 and the same as 4 + 1 = 5" (Connecticut State Department of Education, 2007, p. 20). In addition, the Common Core Standards explicitly state that students in 1st grade need to gain an understanding of the equal sign (Common Core State Standards Initiative, 2010).

In response to curriculum requirements and the research recommendations, mathematics educators have sought to design tasks or experiences that will aid students in developing an accurate understanding of the equal sign. As we were planning instruction for our classroom, a review of the available resources revealed tasks that fell into two distinct categories. In the first category, the teacher utilizes equations to engage students in discussions about the meaning of the equal sign (Faulkner et al., 1999; Knuth, Alibali, et al., 2008; Mann, 2004; Molina & Ambrose, 2006). The teacher might ask, for example, if the equation 17 = 17 is a true or false equation. Tasks in the second category engage students in balance-scale activities (Cuevas & Yeatts, 2001; Mann, 2004). Figure 1 provides an example of a balance-scale activity.

We did not find in our review of resources, however, word problems that utilize a context for thinking about equality. Recognizing the importance, in general, of including word problems when developing students' understanding, we created a set of problems to draw upon in our instructional unit on equality. The purpose of this article is to share word problems that provide a context with which to facilitate students' thinking about equality. After a brief description of the importance of teaching through problem solving, the equality problems will be shared, along with supporting student work.

TEACHING THROUGH PROBLEM SOLVING

Prior to teaching any mathematical topic, teachers must recognize what mathematics the students need to know and then make instructional decisions about how to facilitate student engagement in thinking about that topic for understanding. As Lambdin (2003) states, "Understanding takes place in the students' minds as they connect new information with previously developed ideas, and teaching through problem solving is a powerful way to promote this kind of thinking" (p. 11). Teaching through problem solving, however, goes beyond simply providing students with problem-solving tasks or teaching problem-solving skills once a week at the end of a lesson. Teaching through problem solving begins with developing tasks or problems that engage students in thinking about a problem that is accessible, but for which they have no immediate route for solving. Once engaged in the problem, the students typically represent the problem with pictures or manipulatives, given that they do not have a previously learned procedure or formula to apply to the problem. Utilizing pictures and manipulatives allows students of all ability levels to represent and solve the problem in multiple ways. After the problem has been worked, these multiple representations allow for student-led discussions of the various representations and solution strategies. These discussions then provide the teacher with an avenue for introducing mathematical topics in a meaningful way.

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Accepting that teaching through problem solving is vital for understanding, the challenge for teachers is in identifying the proper resources to help facilitate this type of instruction. When students are actively engaged in problem solving, they are making connections to the real world and building on their prior knowledge. The context of the problem allows this to happen for all students. Unfortunately, in planning instruction on the meaning of the equal sign, we had difficulty finding contextual problems.

EQUALITY PROBLEMS

Recognizing the role that problem solving plays in understanding mathematics, we used the problem-creating framework (Barlow, 2010) to create word problems specifically designed to engage students in thinking about equality. When implemented in the classroom, the problem contexts enabled students to represent and solve the problems with pictures and/or manipulatives. By engaging students in representing the mathematics, the problems provided students with the opportunity to think about equivalence and balance without rushing toward representing the relationships with symbols.

In the following paragraphs, we will share three of the problems. When reading the problems, the reader may note that the problem contexts seem contrived or unrealistic. According to Van de Walle (2003), the realism of the problem context is not as important as the opportunity that the problem provides for students to think about important mathematics functions.

Ants Problem

The Ants Problem (see Figure 2) utilizes small numbers so that students can easily represent the problem on a pan balance. Figure 3 shows a student solving the problem with a pan balance. This student accurately represented the problem by placing the correct number of unifix cubes (the "ants") on each side of the pan balance. The majority of the class employed this solution process. Some students, however, failed to understand that ants needed to be added to the pan balance. Instead, they moved ants from one side of the balance to the other. These contrasting solutions allowed us to facilitate a rich discussion of not only the problem context, but also the mathematical ideas. One assumption in this problem is that the red and blue ants all weigh the same amount, an assumption that the students typically do not question. This is an important assumption to examine, however, and therefore we posed the different-sized ants problem that follows.

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Different-sized Ants Problem

The Different-sized Ants Problem (see Figure 4), like the Ants Problem, utilizes small numbers and can be easily represented on a pan balance. Unlike the Ants Problem, the red ants and the blue ants do not weigh the same amount. To represent the ants for this problem, students will have to account in some way for the different weights. Most of the students in our class used two red unifix cubes snapped together to represent a red ant and used one blue unifix cube to represent a blue ant. This allowed them to represent the problem with the pan balance and correctly solve it. Some students initially did not understand the impact of the different weights, but they self-corrected as they collaborated. We found it interesting to note that those students who solved the problem from the day before by moving ants now demonstrated, with this problem, that they recognized the need to add ants.

Seesaw Problem

Unlike the previous two problems, the Seesaw Problem (1) (see Figure 5) contains larger numbers and is not easily represented on a balance. In this problem, most of our students drew a picture to aid them in thinking about the relationships expressed in the problem. The differences in their work, however, were revealed through their solution processes. In order to find the friend's weight, the students either utilized the hundreds chart, symbolically represented the problem with the missing addend (see Figure 6a), or counted by using tally marks (see Figure 6b). The work represented in Figure 6a is representative of the majority of the class.

In Figure 6b, the student began by incorrectly working with a weight of 49 rather than 47. Although she is able to find a solution for this weight, she recognized that the weight was actually 47. Next, she correctly added 45 and 52. It appears as though she then became confused, as she added 45, 52, and 40. At some point, she began tallying to try to compute the correct answer. Although her work is incomplete, it does indicate her means for thinking about the problem.

You will notice that neither student has written an equation to represent the problem. At the time this problem was given to the class, the focus was on the idea of balance and not on representing that idea symbolically. Once the equal sign had been provided as a means of representing balance, students were able to represent this problem with the equation 45 + 52 = 40 + 57.

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ASSESSING STUDENT UNDERSTANDING

After solving and discussing these problems, the students in our classroom were better equipped for engaging in discussions about the equal sign, as well as for completing balance activities. At the conclusion of the unit, we assessed students' understandings by having them describe in their journals what the equal sign means. One student's journal entry is provided as a sample response (see Figure 7). An edited transcript of this entry is below:

The equal sign means to balance on a math scale and in order for it to balance you have to have the same amount of blocks or manipulatives in each side of the scale and now I know that is because if I get one [balance] and put 12 [cubes] in one side and put 14 [cubes] in the other side and that if I took 2from [the side with] 14 it would, it would balance or I could add 2 to the 12 and it would balance.

From this student's writing, we see that she has connected the meaning of the equal sign to the balancing of a "math scale" and has therefore moved toward a relational view of the equal sign. While it is not clear from this writing sample that the student developed her understanding as a result of the problem contexts used in class, one can assume, based on the previously cited literature, that her understanding was enhanced by the development of this topic through problem solving.

CONCLUSION

Research has indicated that students often have difficulties understanding the meaning of the equal sign (Knuth, Alibali, et al., 2008). When exploring the causes of these difficulties, researchers found that curriculum materials may be one of the largest contributors to the difficulties (Knuth, Alibali, et al., 2008; Knuth, Stephens, et al., 2006). Therefore, providing teachers with alternative tasks and problems is essential if they are to help their students work with this critical concept.

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Our review of available resources revealed tasks that fell into one of two categories, namely class discussions of equations and balance activities. However, we did not find problems that utilized a context to which students could relate and connect the meaning of the equal sign. As a result, we felt it was important to share the problems we created.

Our work with students demonstrated that these problem contexts enabled them to link the idea of balance to the equal sign in a meaningful way. After solving and discussing the problems, the connections made by the students provided them with a means for accessing class discussions about equations that lacked a context. In addition, the problem contexts helped students to make meaning of the balance activities. As was the case with our students, we believe that teachers who engage their students in solving these or similar problems, in addition to equation discussions and balance activities, will successfully engage students in thinking about equality and help them develop a strong understanding of the equal sign.

References

Barlow, A.T. (2010). Building word problems: What does it take? Teaching Children Mathematics, 17, 140-148.

Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann.

Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved June 25, 2010, from www.corestandards.org/the-standards/mathematics

Connecticut State Department of Education. (2007). Connecticut prekindergarten-grade 8 mathematics curriculum standards. Hartford, CT: Author.

Cuevas, G. J., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.

Faulkner, K. R, Levi, L., & Carpenter, T. P., (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 56-60.

Kahan, J. A., & Wyberg, T. R. (2003). Mathematics as sense making. In H. L. Schoen (Ed.), Teaching mathematics through problem solving: Grades 6-12 (pp. 15-26). Reston, VA: National Council of Teachers of Mathematics.

Knuth, E. J., Mibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514-519.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M.W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 36, 297-312.

Lambdin, D.V. (2003). Benefits of teaching through problem solving. In F.K. Lester, Jr. (Ed.), Teaching mathematics through problem solving: Prekindergarten-grade 6 (pp. 3-14). Reston, VA: National Council of Teachers of Mathematics.

Mann, R. L. (2004). Balancing act: The truth behind the equals sign. Teaching Children Mathematics, 11, 65-69.

Mississippi Department of Education. (2007). 2007 Mississippi mathematics framework revised. Jackson, MS: Author.

Molina, M., & Ambrose, R. C. (2006). Fostering relational thinking while negotiating the meaning of the equals sign. Teaching Children Mathematics, 13, 111-117.

Van de Walle, J. A. (2003). Designing and selecting problem-based tasks. In F.K. Lester, Jr. (Ed.), Teaching mathematics through problem solving: Prekindergarten-grade 6 (pp. 67-80). Reston, VA: National Council of Teachers of Mathematics.

Footnote

Although this problem deals with what can sometimes be a sensitive subject (people's weight), the students in this classroom viewed it as just another problem to solve. There was no discussion or concern expressed regarding the weight of the children in the problem, the weight of the students in our class, or the fact that the heaviest child on the seesaw was Mary's friend.

Angela T. Barlow is Professor, Mathematics Education, Middle Tennessee State University Murfreesboro, TN. Shannon E. Harmon is Visiting Assistant Professor, Mathematics Education, Department of Curriculum and Instruction, University of Mississippi, University, MS.
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