Numbers - Prerequisite Knowledge for the Learning of Algebra

Researchers have focused on the value of understanding numbers prior to algebra introduction (Booth, 1984, 1986; Gallardo, 2002; Kieran, 1988; Rotman, 1991; Wu, 2001). According to Watson (1990), a better understanding of number basics would give students a stronger ability to handle algebraic operations and manipulations. What types of numbers need to be studied prior to learning algebra? The Readiness Indicator for algebra focuses on students’ ability to read, write, compare, order, and represent a variety of numbers, including integers, fractions, decimals, percents, and numbers in scientific notation and exponential form (Bottoms, 2003). Some of these forms have also been mentioned in research addressing prerequisite number knowledge for the learning of algebra.

Gallardo (2002) focused on the fact that the transition from arithmetic to algebra is where students are first presented with problems and equations that have negative numbers as coefficients, constants and/or solutions. Therefore, she believes that students must have a solid understanding of integers in order to comprehend algebra. Lack of this understanding will affect students’ abilities to solve algebraic word problems and equations. However, Gallardo’s research showed that 12- and 13-year-old students do not usually understand negative numbers to the fullest extent.

Misconceptions of negative numbers were identified in earlier research done by Gallardo and Rojano (1988; cited in Gallardo, 2002) while investigating how 12- and 13-year-old beginning algebra students acquire arithmetic and algebraic language. One major area of difficulty involved the nature of numbers. Specifically, students had troubles conceptualizing and operating with negative numbers in the context of prealgebra and algebra. Therefore, Gallardo (2002) argues that while students are learning the language of algebra, it is imperative that they understand how the numerical domain can be extended from the natural numbers to the integers.

Kieran (1988) also found misunderstandings regarding integers to affect the success of algebra students in grades 8-11. During interviews with Kieran, students who had taken at least one year of algebra made computational equation-solving errors involving the misuse of positive and negative numbers. Furthermore, when these students were required to use division as an inverse operation, they tended to divide the larger number by the smaller, regardless of the division that was actually required within the operation. Therefore, students’ errors extended into the division of integers, which implies a lack of understanding of fractions.

An opinion article regarding how to prepare students for algebra further supports the inclusion of fractions as prerequisite knowledge for the learning of algebra. According to Wu (2001), fraction understanding is vital to a student’s transformation from computing arithmetic calculations to comprehending algebra. Wu believes that K-12 teachers are not currently teaching fractions at a deep enough level to prepare students for algebra. In fact, she believes that the study of fractions could and should be used as a way of preparing students for studying generality and abstraction in algebra.

Fractions were also stressed when Rotman (1991) chose number knowledge as a prerequisite arithmetic skill for learning algebra. During a research project that mounted evidence against the assumption that arithmetic knowledge is prerequisite for successful algebra learning, Rotman constructed a list of arithmetic skills he considers as prerequisite to algebra. Based on his experiences as a teacher, Rotman argues that algebra students need to understand the structure behind solving applications, the meaning of symbols used in arithmetic, the order of operations and basic properties of numbers (especially fractions). Of course, in order to operate with fractions students are required to know basic number theory ideas including least common multiple. Therefore, the necessity of fraction knowledge partially supports Readiness Indicator, which states that students need to be able to determine the greatest common factor, least common multiple, and prime factorization of numbers (Bottoms, 2003).