Proportionality,Computataion and Equality - Prerequisite Knowledge for the Learning of Algebra

Proportionality

Fractions commonly appear in beginning algebra in the form of proportions, which provide wonderful examples of naturally occurring linear functions. Because of this, Post, Behr, and Lesh feel that proportionality has the ability to connect common numerical experiences and patterns, with which students are familiar, to more abstract relationships in algebra. Proportions can also be used to introduce students to algebraic representation and variable manipulation in a way that parallels their knowledge of arithmetic.

In fact, proportions are useful in a multitude of algebraic processes, including problem solving, graphing, translating and using tables, along with other modes of algebraic representation. Due to its vast utility, Post et al. (1988) consider proportionality to be an important contributor to students’ development of pre-algebraic understanding. Similarly, Readiness Indicator number 4 focuses on the importance of ratios, rates and proportions in the  study of algebra (Bottoms, 2003).

Proportional reasoning requires a solid understanding of several rational number concepts including order and equivalence, the relationship between a unit and its parts, the meaning and interpretation of ratio, and various division issues (Post et al., 1988). Therefore, these concepts could be considered, along with proportional reasoning, prerequisite knowledge for the learning of algebra.

Computations

In addition to understanding the properties of numbers, algebra students need to understand the rules behind numerical computations, as stated in Readiness Indicator number. Computational errors cause many mistakes for algebra students, especially when simplifying algebraic expressions. Booth (1984) claims elementary algebra students’ difficulties are caused by confusion surrounding computational ideas, including inverse operations, associativity, commutativity, distributivity, and the order of operations convention. These misconstrued ideas are among basic number rules essential for algebraic manipulation and equation solving (Watson, 1990). The misuse of the order of operations also surfaced within an example of an error made by collegiate algebra students that Pinchback (1991) categorized as result of lack of prerequisite knowledge. Other errors deemed prerequisite occurred while adding expressions with radical terms and within the structure of long division (while dividing a polynomial by a binomial) (Pinchback, 1991).

Mentioned by Rotman (1991) as a prerequisite arithmetic skill, the order of operations is also included in Readiness Indicator number 10 (Bottoms, 2003). In fact, this convention has been found to be commonly misunderstood among algebra students in junior high, high school, and even college (Kieran, 1979, 1988; Pinchback, 1991). The order of operations relies on bracket usage; however, algebra requires students to have a more flexible understanding of brackets than in arithmetic. Therefore, according to Linchevski (1995), prealgebra should be used as a time to expand students’ conceptions of brackets.

Kieran (1979) investigated reasons accounting for the common misconception of the order of operations and alarmingly concluded that students’ issues stem from a much deeper problem than forgetting or not learning the material properly in class. The junior high school students, with which Kieran worked, did not see a need for the rules presented within the order of operations. Kieran argues that students must develop an intuitive need for bracket application within the order of operations, before they can learn the surrounding rules. This could be accomplished by having students work with arithmetic identities, instead of open-ended expressions.

Although teachers see ambiguity in solving an open-ended string of arithmetic operations, such as 2 + 4 x 5, students do not. Students tend to solve expressions based on how the items are listed, in a left-to-right fashion, consistent with their cultural tradition of reading and writing English. Therefore, the rules underlying operation order actually contradict students’ natural way of thinking. However, Kieran suggests that if an equation such as 3 x 5 =15 were replaced by 3 x 3+ 2 =15, students would realize that bracket usage is necessary to keep the equation balanced (Kieran, 1979).

Equality

Kieran’s (1979) theory assumes that students have a solid understanding of equations and the notion of equality. Readiness Indicator number 10 suggests that students are familiar with the properties of equality before entering Algebra I (Bottoms, 2003). However, equality is commonly misunderstood by beginning algebra students (Falkner, Levi, & Carpenter, 1999; Herscovics & Kieran, 1980; Kieran, 1981, 1989). Beginning algebra students tend to see the equal sign as a procedural marking that tells them “to do something,” or as a symbol that separates a problem from its answer, rather than a symbol of equivalence (Behr, Erlwanger, & Nichols, 1976, 1980). Even college calculus students have misconceptions about the true meaning of the equal sign (Clement, Narode, & Rosnick, 1981).

Kieran (1981) reviewed research addressing how students interpret the equal sign and uncovered that students, at all levels of education, lack awareness of its equivalence role. Students in high school and college tend to be more accepting of the equal sign’s symbolism for equivalence, however they still described the sign in terms of an operator symbol, with an operation on the left side and a result on the right. Carpenter, Levi, and Farnsworth (2000) further support Kieran’s conclusions by noting that elementary students believe the number immediately to the right of an equal sign needs be the answer to the calculation on the left hand side. For example, students filled in the number sentence 8 + 4 =  __ +5 with 12 or 17.

According to Carpenter et al. (2000), correct interpretation of the equal sign is essential to the learning of algebra, because algebraic reasoning is based on students’ ability to fully understand equality and appropriately use the equal sign for expressing generalizations. For example, the ability to manipulate and solve equations requires students to understand that the two sides of an equation are equivalent expressions and that every equation can be replaced by an equivalent equation (Kieran, 1981). However, Steinberg, Sleeman, and Ktorza (1990) showed that eighth- and ninth-grade algebra students have a weak understanding of equivalent equations.

Kieran (1981) believes that in order to construct meaning while learning algebra, the notion of the equal sign needs to be expanded while working with arithmetic equalities prior to the introduction of algebra. If this notion were built from students’ arithmetic knowledge, the students could acquire an intuitive understanding of the meaning of an equation and gradually transform their understanding into that required for algebra. Similarly, Booth (1986) notes that in arithmetic the equal sign should not be read as “makes”, as in “2 plus 3 makes 5” (Booth, 1986), but instead as “2 plus 3 is equivalent to 5”, addressing set cardinality.