As observed by the National Mathematics Panel, “many students are woefully unprepared for algebra” ( Department of Education, 2008: p. 208 ) and struggle to solve algebraic equations. Equations such as 4x + 3 = 3x + 9 and 2(2x + 1) = 3x + 12 require to students have an understanding of the symbolic notation, the relational meaning of the equal sign, and the ability to work with the unknown—all of which are areas of deficiency cited by the Math Panel. Some of these deficiencies occur early in the student’s career. For example, elementary school students may have difficulty understanding the meaning of a term such as “4x”. They may wonder whether it represents a two-digit number written in base 10 ( Herscovics & Linchevski, 1994 ), in which case if x = 3, 4x would be 43. Alternatively, the student may think that 4x means four plus x ( Kieran, 1985 ), in which case 4x would be 7 when x is 3. Expressions such as 2(2x + 1) can also be confusing. Many students who try and work with such expressions by memorizing the distributive property forget to distribute to each term inside the parentheses and will incorrectly expand to 4x + 1 ( Booth, 1988 ; Ncube, 2016 ).

The equal sign is also a source of confusion for many students. Whereas almost every elementary school student is familiar with the operational meaning of the equal sign, that is, as an indicator that the result is coming next, such as in 3 + 4 = 7, many are not familiar with the relational meaning of the equal sign. For example, in the problem 5 + 3 = __ + 4, many students will enter an 8 since that is the sum of the numbers on the left side ( Falkner et al., 1999 ). This limited understanding of the meaning of the equal sign may persist into middle school, high school, and college ( Knuth et al., 2008 ). Although the operational meaning is a legitimate and essential one ( Ginsburg, 1996 ), students who only know this meaning will not realize that the equal sign in an equation such as 4x + 3 = 3x + 6 indicates that the total value of each side of the equal sign is the same ( Kieran, 1981 ). Since the meaning of a sign is arbitrary ( Chandler, 2007 ), students who do not have a meaningful experience with the use of the equal sign in its relational sense will not acquire that understanding ( Borenson, 2013 ; Hornburg et al., 2021 ; Sherman & Bisanz, 2009 ).

To solve equations such as 4x + 3 = 3x + 9 containing the unknown on both sides of the equal sign, students must be able to work with—or on—the unknown ( Filloy & Rojano, 1989 ; Herscovics & Linchevski, 1994 ). Two possible approaches to solving this equation are transposition and the subtraction property of equality ( Hall, 2002 ; Otten et al., 2019 ). According to Hall (2002) , many students who attempt to solve equations using symbolic manipulation find algorithmic work to be “daunting” (p. 17). For example, the “change side-change signs” rule is open to “oversimplification and abuse” (p. 57), leading to many errors. Hall found nine errors that many students make when solving simple linear equations with symbolic notation (switching addends, deletion errors, combining terms that cannot be combined, etc.). Indeed, when students do not “construct meaning for the new symbolism, they are reduced to performing meaningless operations on symbols they do not understand” ( Herscovics & Linchevski, 1994: p. 60 ), resulting in many mistakes ( De Lima & Tall, 2008 ; Hall, 2002 ; Kieran, 1985 ).

An approach often used by educators to assist students in understanding linear equations is the balance model ( Otten et al., 2019 ). There are physical, virtual, and drawn versions of a two-pan balance scale. In their systematic review of the literature on the balance model, Otten et al. (2019) found that in Grades 3 - 6, it is most often used to teach the relational meaning of the equal sign and help students find the missing number in problems such as 8 = ___ + 3 and 4 + 3 = ___ + 2. In Grades 7 and 8, the balance model is used to introduce students to equations such as ax + b = cx + d, having the unknown on both sides of the equal sign, where the coefficients, constants, and solutions are non-negative whole numbers ( Araya et al., 2010 ; Boulton-Lewis et al., 1997 ; Vlassis, 2002 ).

Boulton-Lewis et al. (1997) used a cups and discs balance model with above-average eighth-grade students. In this model, the unknown is represented by a cup, and a disc represents a unit. Hence, to represent the equation 2x + 5 = 17, the student places two cups and five discs on the left side of a line serving as a partition and seventeen discs on the right side. The objective is to determine the number of discs in each cup so that both sides have the same number. To solve the equation, students physically remove discs from each side. The students in the Bolton study preferred solving the above equation mentally rather than use the cups and discs. Although the authors attributed student hesitancy to use the materials to cognitive load, there are other explanations. First, since the students could solve this equation mentally, they saw no need to use these materials; secondly, the cups and discs balance model is very cumbersome. For example, the above equation would require 24 objects to represent the problem. This would be an operational, rather than a cognitive load, issue.

A study by Araya et al. (2010) with seventh-grade students did not use any physical props. Instead, half of the group saw a 15-minute video demonstration of the cups and discs balance model employed to solve algebraic equations. In this instance, the cups and discs are illustrated as being placed on the bins of a two-pan stationary balance scale, and the cup’s weight is assumed to be zero. The objective is to find the weight of each disc. The other half of the group was presented with a video showing the traditional abstract solution. Both groups were given a posttest shown on a computer screen. All calculations were performed mentally. The group presented with the cups and disc balance model video scored significantly higher than the other group. Furthermore, students with a below-average GPA who had been presented with the balance model videos did as well as the above-average GPA students who were presented with the symbolic notation videos. In considering why the analogies model could have such an immediate and significant positive impact, the authors attributed it to a) the ease with which the mapping could be understood and b) the intuitive nature of the two-pan balance and the principles for maintaining equilibrium. They suggested that the latter may be part of our biological primary knowledge. However, the 15-minute video exposure to the model did not result in any improvement for the lowest-achieving mathematics students.

Vlassis (2002) used a drawn balance model in her study with lower-achieving eighth-grade students. Her objective was to have students transfer the procedure used in simplifying the pictorial equation to the traditional written notation for solving equations. For example, she presents the students with a drawing showing a balance scale with two bins containing images representing weights. On one bin, there is a drawing of two squares with an x inside each of them and a circled 14; on the other, there is a drawing of three squares, each one containing an x, and a circled 8. The students used arrows or cross-outs to remove the same weight from each side. Thereafter, they were presented with equations written in the traditional symbolic notation. Her study showed that the students could transfer the procedures learned with the drawing. Vlassis concluded that the isomorphism between the representation and the equation enabled the students to form an operative mental image that they could readily access, even months after instruction. In particular, the model enabled the students to understand the equality between the two sides of the equation and that removing the same value maintains the balance between the two sides. Whereas the model helped the students apply their learning to equations with positive values, it did not do so with those involving negative values, such as 8x − 5 = 2x + 7 or −6x = 24.

In summary, the studies by Araya et al. (2010) and Vlassis (2002) show that the balance model concept can enable seventh- and eighth-grade students to understand equivalence and the subtraction property of equality and apply those concepts to the solution of linear equations with the unknown on both sides of the equal sign. As noted earlier, students in Grades 3 - 6 can use the balance model to understand equivalence and find the missing number in simple addend problems, such as 8 = __ + 3 ( Otten et al., 2019 ). Students in these grades also understand that removing the same weight from each side of a two-pan balanced system maintains the balance of the system ( Brizuela & Schliemann, 2004 ; Mann, 2004 ). Taylor-Cox (2003) demonstrated that even 5-year-old children understand the concept of maintaining balance using a two-pan seesaw. As one child noted, “If Alex wants to get off the seesaw, then Angela has two get off too since she is the one who weighs the same” (p. 18).

Consequently, it makes sense to inquire whether upper elementary school students can learn to solve linear equations with the unknown on both sides of the equal sign if they are provided with a concrete version of the balance model that they can easily manipulate. In solving those equations, the students would be learning essential algebraic concepts. Further, it would be of interest to explore the effect of the concrete model on the achievement of students in Grade eight since, according to the National Math Panel, many of those students have difficulty with such equations.

As a mathematics supervisor in the 1980s, the author learned that many algebra students had difficulty in solving algebraic linear equations having the unknown on both sides of the equal sign. He wondered if a hands-on approach could be developed to enable elementary school students to experience success with such equations, thereby enhancing their self-perception as learners and introducing them to powerful algebraic concepts that may pay dividends later on. At the time, the author was aware of the existence of algebra tiles. These are manipulatives intended to model operations with polynomials ( Howden, 1985 ). From attending NCTM conferences, however, the author realized that teachers also used them to model linear equations in Grades 6 - 8 and high school. Algebra tiles are an area-based model: A long green rectangular bar is considered to have a length of x and a width of 1, thereby having a surface area of x; a small 1 × 1 yellow square has a surface area of 1. Using algebra tiles, the equation 3x + 2 = 2x + 3 is illustrated as shown in Figure 1, with the two sides of the equation representation separated by a vertical line.

The student removes two green bars and two yellow blocks from each side to solve. The remaining setup shows one green bar on the left side of the partition and one yellow square on the right. Notwithstanding the difference in areas between the long green bar and the small yellow square, the student is expected to conclude that in this example, the green bar (or x) has the same value as the yellow square—that is, x = 1. The author of this paper did not think that upper elementary-grade students would have an easy time grasping and conceptualizing why two entities of obviously different sizes were equal to each other. Hence, he concluded that algebra tiles would be inappropriate for elementary school students.

The author considered the notation proposed by Sawyer (1960) to be more sensible for introducing algebraic notation to elementary school students. Sawyer suggested a method for teaching algebraic number puzzles to fifth-graders. In this approach, the unknown number is represented by a drawing of a sack containing an unknown number of stones; the drawing of a stone represents each additional unit. With this model, the instruction, “Think of a number. Add 3”, would be expressed pictorially as shown in Figure 2. Although the author did not find this approach appealing for working with equations, it did demonstrate that young students could understand the concept of an unknown.

The author came across an encouraging statement by Barbel Inhelder, a student of Piaget: “Advanced notions of mathematics are perfectly accessible to children of seven to ten years of age, provided they are divorced from their mathematical expression and studied through materials that the child can handle

himself” (as cited in Bruner, 1960: p. 43 , emphasis added). Inhelder was referring to a hands-on isomorphic system that would be the counterpart of the abstract mathematical system. The power of an isomorphic system to enable young students to experience advanced mathematical concepts was further confirmed by Post (1981: p. 112) : “An isomorphism is an extremely important concept in mathematics, for if any two systems can be shown to be isomorphic to one another, it becomes possible to work in the simpler and more available system and transfer all conclusions to the less accessible one”.

With this perspective in mind, the author undertook a two-year research and experimentation process seeking to develop an isomorphic manipulative system enabling students as early as the third grade to solve equations with the unknown on both sides of the equal sign. He wanted the system to work with equations containing the terms x and/or −x and positive and/or negative constants. Upon completing the instructional system, the author applied for and was granted a patent ( Borenson, 1986a ). The system, known as Hands-On Equations ( Borenson, 1986b ), consists of a series of sequential lessons and the accompanying manipulatives, as described below.

In this instructional system, concrete objects and physical actions are the counterparts of abstract symbols and mathematical processes. A game piece, namely a blue pawn, represents the unknown x, and another game piece, namely a white pawn, represents (−x). The system includes eight blue pawns and eight white pawns. Red-numbered cubes represent positive constants, and green-numbered cubes represent negative constants. The system has two red cubes numbered 0 - 5 and two numbered 5 - 10; the green cubes are similarly numbered. This innovation for representing the constants makes it possible, for example, to have the constant of 9 represented by just one game piece, namely the red cube displaying the number 9, thereby simplifying the representation of the constant compared to the Sawyer or algebra tile model. Furthermore, the author wanted to use a balance to model the two sides of the equation, but he did not want students to rely on a moving balance to determine whether their solutions were correct. Hence, he decided to use a flat laminated scale for the student and a three-dimensional stationary balance scale for the teacher. Figure 3 shows the physical representation of equation 4x + 5 = 2x + 13.

In this mapping, the “4x” is translated to the placement of four blue pawns on the left side of the balance scale. The plus sign followed by the 5-constant is an

instruction to place on the same side of the balance a red cube displaying the number 5. The equal sign is an instruction to continue the setup on the other side of the scale. Once the setup is completed, to solve the equation, the student performs “legal moves”, that is, moves that maintain the theoretical balance of the system. In this example, the student simultaneously removes one pawn from each side of the balance scale, as shown in Figure 4. The student does so again, and then removes a 5-value from the cubes on each side. This leaves two pawns on the left side of the scale and a value of 8 on the cube(s) on the right. At this point, the student realizes that the value of the pawn is 4, since 4 + 4 = 8. The student writes the solution as x = 4. The check value of 21 = 21 is obtained by evaluating the original physical setup shown in Figure 3 when the pawn has the value of 4.

The first six lessons are presented with manipulatives. The seventh lesson transitions to a pictorial representation of the concrete solution using only paper and pencil. Whether using the hands-on or the pictorial solution (see Appendix), the isomorphic solution process is the same: the student translates or maps the given abstract equation into its concrete or pictorial representation. Next, legal moves are made to simplify the setup and thereby solve for the value of the pawn or

shaded triangle that will make both sides “balance”. That value will be the value of x that solves the given abstract equation. The process is shown schematically in Figure 5.

This study with fourth- and eighth-graders aimed to achieve three research objectives: first, it aimed to determine the extent to which the seven-lesson instructional treatment improved student performance from the pretest to the post-tests with and without manipulatives; secondly, it aimed to compare the performance of each grade group at the three test points; and lastly, it aimed to assess the pre- and posttest performance of the fourth-grade students using the eighth-grade pretest as a point of reference or benchmark. These objectives were accomplished by addressing the research questions outlined in Table 1.

Since the author of this paper is the author/inventor of this early algebra