Algebraic reasoning in and through quantitative conceptions
Abstract
In a 4-month constructivist teaching experiment, I investigated how one 7th grade child and two 8th grade children constructed algebraic reasoning as a transformation in their multiplicative schemes. Algebraic reasoning is the purposeful functioning of a child’s schemes and operations in contexts I defined as algebraic. The algebraic contexts explored in my study required a child to create equality between two multiplicative situations (compilations) by operating on/with the individual units (1s) and composite units that constitute the compilations. Algebraic reasoning emerges in two ways from these contexts. First, when a child operates on and with the structure of his or her schemes, the child is reasoning algebraically. Second, if a child’s schemes form the underpinnings for formal algebraic conceptions, the child is reasoning algebraically. I identified three specific schemes of equality, a unidirectional scheme of equality, a relational scheme of equality, and a quantitative relational scheme (QRE) of equality, that were constructed by the children in my study as they produced equality between two compilations. A QRE scheme of equality is comprised of operations on composite units across two compilations. The QRE scheme constitutes algebraic reasoning because a child operates on the structure of their multiplicative schemes with the structure of their additive schemes. A child also operates on the structure of their equality scheme with the structure of their multiplicative schemes. Moreover, because the QRE scheme is comprised of operations on composite units across two compilations, it forms the cognitive roots for the algebraic concepts of the distributive property, quantitative conservation, and solving linear equations. The construction of the QRE scheme by a child Joe was crucial to my study because it provided evidence of the existence of the hypothesized QRE scheme. The second scheme of equality, a relational scheme, consists of balancing the total 1s from each compilation via operations on 1s. This scheme also constitutes algebraic reasoning because a child increases the smaller total via addition and decreases the larger total via subtraction until they produce equality. They operate with the structure of their additive scheme on the structure of their equality scheme. A child with the third scheme of equality, a unidirectional scheme, creates equality by transforming the total from one of the given compilations to produce the second total. A child who operates with this scheme is not reasoning algebraically. I suggest a trajectory for the development of these three equality schemes. A child begins with a unidirectional scheme of equality. Next, they construct a relational scheme of equality as their scheme progresses to include operating on both totals with 1s to create balance. A second progression occurs to the most advanced equality scheme, a QRE scheme, when a scheme is constructed that moves past operations on 1s to include operations on composite units. My research goes beyond a description of the three schemes as it provides evidence as to how a child transitions from not having a QRE scheme to having one. Moreover, my research informs the teaching of mathematics as the UDS and QRE tasks offer elementary school teachers a way to give children opportunities to form the foundations for the formal algebraic concepts of the distributive property, solving linear equations, and quantitative conservation.
Degree
Ph.D.
Advisors
Kastberg, Purdue University.
Subject Area
Mathematics education
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