How does a Standards-based curriculum really differ from traditional curricula? So far, only a few studies have been done to identify those important features of Standards-based curricula that distinguish them from traditional curricula and fewer studies focus their research on the algebraic strand within those curricula.

The study presented here is part of a larger research project, titled “Longitudinal Investigation of the Effect of Curriculum on Algebra Learning” (LieCal Project). The LieCal Project is designed to longitudinally compare the effects of the Connected Mathematics Program (CMP) to the effects of more traditional middle school curricula (hereafter called non-CMP curricula) on students' learning of algebra. One of the purposes of the LieCal Project is to provide a profile of the intended treatment of algebra in the CMP curriculum with a contrasting profile of the intended treatment of algebra in non-CMP curricula.

Background of the Study

In the 1990s, with extensive support from the National Science Foundation (NSF), a number of school mathematics curricula were developed and implemented to align with the recommendations of the Standards (Senk & Thompson, 2003). There have been heated discussions about the benefits of using these so-called Standards-based mathematics curricula in the United States (e.g., Herman et al., 2006; Schoenfeld, 2006; Wu, 1997).

Even though there has been a major shift in the landscape of school mathematics in recent years (Chazan, 2008), learning to solve equations is still an essential element in the study of algebra (Mathematical Sciences Education Board, 1998). Unfortunately, many students still have a difficult time learning algebra, particularly learning the concepts and skills related to equations and equation solving (Kieran, 2007; Loveless, 2008; National Mathematics Advisory Panel, 2008). How can we teach students the fundamental ideas related to equations and equation solving that will provide a solid foundation from which to satisfy the quantitative demands of their future endeavors (Kaput, 1999; RAND, 2003).

Methods

The Curriculum Materials

The purpose of this study is to present a fine-grained analysis that compares the treatment of equation solving in the NSF-funded Connected Mathematics Program (CMP) [1] (e.g., Lappan et al., 2002) with that of the more traditionally-based curricula (non-CMP): Glencoe Mathematics: Concepts and Applications (e.g., Bailey et al., 2006), McDougal Little School Math (Larson et al., 2005), Prentice Hall Mathematics (Charles et al., 2004), Saxon Math (Hake and Saxon, 2002, 2004) .

Research Questions

 In this study, our analysis focuses on the following research questions:

1. How is the concept of equation introduced in the five curricula? 

2. How is equation solving introduced in the five curricula? 

3. What are differences in the types of problems that involve equations in the five curricula?

Brief Summary of the Results

Due to space limitations, we only briefly summarize the results in part.

Defining Variables

To understand how equations are defined, we need to first understand how variables are defined in the CMP and non-CMP curricula. CMP defines variables as quantities that change or vary, and it uses them to represent relationships. The non-CMP curricula formally define a variable as a symbol (or letter) used to represent one or more numbers, treat variables predominantly as placeholders, and use them mostly to represent unknowns in expressions and equations.

Defining Equations

In CMP, the functional approach to equation is a natural extension of its development of the concept of a variable as a changeable quantity used to represent relationships. Rather than seeing equations simply as objects to manipulate, students learn that equations often describe relationships between varying quantities that arise from meaningful, contextualized situations (Bednarz, Kieran, & Lee, 1996). In CMP equations are formally defined as rules that are expressed with mathematical symbols, and that are often used for describing the relationship between two variables.

In the non-CMP curricula, the definition of a variable as a symbol develops naturally into two iconic hallmarks of a structural focus: the use of decontextualized (or “naked”) equations and an emphasis on procedures for solving them. For example, immediately after defining an equation as “…a sentence that contains an equals sign, =,” Glencoe Mathematics provides examples like 2 + x = 9, 4 = k – 6, and 5 – m = 4 (Bailey, et al., 2006, p. 34). Students are told that the way to solve an equation is to replace the variable with a value that results in a true sentence.

Introducing Equation Solving

In the CMP curriculum, equation solving is introduced within the context of discussing linear relationships. Thus, linear equation solving is introduced by making visual sense of what it means to find a solution using a graph.

After CMP introduces equation solving graphically, the symbolic method is introduced within a single contextualized example. That is, CMP uses real-life contexts to help students understand the meaning of each step of the symbolic method of equation solving, including why inverse operations are used

In contrast, the non-CMP curricula first introduce equation solving as the process of finding a number to make an equation a true statement. Specifically, solving an equation is described as replacing a variable with a value (called the solution) that makes the sentence true. The process of equation solving is introduced in the non-CMP curricula symbolically by using the additive property of equality and the multiplicative property of equality.

Cognitive Demand of Algebraic Problems in the Five Curricula

Using a scheme developed by Stein et al. (1996), we classified the Algebraic tasks in the CMP and the four non-CMP curricula into four increasingly demanding categories of cognition: memorization, procedures without connections, procedures with connections, and doing mathematics. As Figure 1 shows, significantly more tasks in the CMP curriculum than in the non-CMP curriculum are higher-level tasks (procedures with connections and doing mathematics) [X²(3, N = 10766) = 1620.08, p < .0001].

Figure 1. Distribution of cognitive demand of algebraic problems in the five curricula.

In the full paper, we will also present achievement data to show how the nature of CMP and non-CMP curricula impact on students' learning.

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