The Genesis and Development of Layers of Algebraic Generality in Young Students

Funding agency:

Summary

Algebra has always been a difficult topic to learn. Many students get lost in a jungle of formulas and equations. And they get lost forever. Recent research, however, suggests that a progressive introduction to algebra in the early grades may facilitate the students’ access to more advanced algebraic concepts later on. This suggestion rests on two innovative claims:

First, a mature knowledge of arithmetic is not the inescapable prerequisite for starting to learn some elementary algebraic concepts.

Second, letters are neither a necessary nor a sufficient condition for thinking algebraically. Algebra, in fact, is not about using letters. It is rather about using letters or other signs to think in a certain way. This is why students can legitimately think algebraically through spoken words, gestures and, as our current research reveals, rhythm (Radford, Bardini, & Sabena, 2006, in press).

These claims do not challenge the power of symbolic algebra. What they bring forward is:

(1) the recognition of other genuine forms of practicing algebra and

(2) the contention that these forms may be useful in creating the conditions that allow students to gain access to other ways of thinking about unknowns, variables, equations, formulas and so on.

This research program rests on the aforementioned claims and the key idea that the students’ algebraic activity can be carried out through different, interrelated, layers of generality.

Layers of generality depend, in particular, on the signs or semiotic systems that the students use to deal with unknowns, variables, parameters, etc. and the ensuing conceptualizations that these semiotic systems afford. In some of these layers, the students deal with algebraic generality in a contextual and embodied way; in other layers, in a rather detached or disembodied manner. Although our previous research with Junior and Senior High School students has made evident some layers of generality (Radford, 2006c), almost nothing is known about the genesis, development and transformation of the embodied and other layers of generality in young students. To successfully introduce the students to algebra in the early grades, to know how to create the conditions that will allow them to go as far as possible within each layer of algebraic generality and to have the students gain access to the algebraic symbolism, further research needs to be conducted.

Focusing on the generalization of patterns, in this research program, we want to investigate in detail the emergence, consolidation and evolution of layers of algebraic generality. Since the introduction to algebraic symbolism occurs in Grade 7 in Ontario, we will conduct two simultaneous longitudinal studies. The first study will focus on a Grade 2 class of students which we will follow as the students move to Grades 3 and 4. The other study will focus on a Grade 5 class of students which we will follow as the students move to Grades 6 and 7. The first study will allow us to investigate the formation of the earlier layers of algebraic generality. The second study will allow us to investigate the transition to, and emergence of, the symbolic or alphanumeric layer of generality. These studies will provide us with large longitudinal and detailed data with which to study the transformation of, and relationship between, layers of algebraic generality.

The importance and originality of this research is both theoretical and practical. On the practical level, this research will inform curriculum designers and educational policy makers about the potential for introducing algebra in the early years. It will also inform teachers about young students’ limits and their capacity to deal with algebra. On the theoretical level, we are proposing a non-reductive approach that draws on epistemology, semiotics, philosophy, anthropology and psychology to understand the central problem of algebraic generality. The results can be of interest to teachers, mathematics educators, and cultural and developmental psychologists.

The Genesis and Development of Layers of Algebraic Generality in Young Students

Funding agency:

Summary

Algebra has always been a difficult topic to learn. Many students get lost in a jungle of formulas and equations. And they get lost forever. Recent research, however, suggests that a progressive introduction to algebra in the early grades may facilitate the students’ access to more advanced algebraic concepts later on. This suggestion rests on two innovative claims:

First, a mature knowledge of arithmetic is not the inescapable prerequisite for starting to learn some elementary algebraic concepts.

Second, letters are neither a necessary nor a sufficient condition for thinking algebraically. Algebra, in fact, is not about using letters. It is rather about using letters or other signs to think in a certain way. This is why students can legitimately think algebraically through spoken words, gestures and, as our current research reveals, rhythm (Radford, Bardini, & Sabena, 2006, in press).

These claims do not challenge the power of symbolic algebra. What they bring forward is:

(1) the recognition of other genuine forms of practicing algebra and

(2) the contention that these forms may be useful in creating the conditions that allow students to gain access to other ways of thinking about unknowns, variables, equations, formulas and so on.

This research program rests on the aforementioned claims and the key idea that the students’ algebraic activity can be carried out through different, interrelated, layers of generality.

Layers of generality depend, in particular, on the signs or semiotic systems that the students use to deal with unknowns, variables, parameters, etc. and the ensuing conceptualizations that these semiotic systems afford. In some of these layers, the students deal with algebraic generality in a contextual and embodied way; in other layers, in a rather detached or disembodied manner. Although our previous research with Junior and Senior High School students has made evident some layers of generality (Radford, 2006c), almost nothing is known about the genesis, development and transformation of the embodied and other layers of generality in young students. To successfully introduce the students to algebra in the early grades, to know how to create the conditions that will allow them to go as far as possible within each layer of algebraic generality and to have the students gain access to the algebraic symbolism, further research needs to be conducted.

Focusing on the generalization of patterns, in this research program, we want to investigate in detail the emergence, consolidation and evolution of layers of algebraic generality. Since the introduction to algebraic symbolism occurs in Grade 7 in Ontario, we will conduct two simultaneous longitudinal studies. The first study will focus on a Grade 2 class of students which we will follow as the students move to Grades 3 and 4. The other study will focus on a Grade 5 class of students which we will follow as the students move to Grades 6 and 7. The first study will allow us to investigate the formation of the earlier layers of algebraic generality. The second study will allow us to investigate the transition to, and emergence of, the symbolic or alphanumeric layer of generality. These studies will provide us with large longitudinal and detailed data with which to study the transformation of, and relationship between, layers of algebraic generality.

The importance and originality of this research is both theoretical and practical. On the practical level, this research will inform curriculum designers and educational policy makers about the potential for introducing algebra in the early years. It will also inform teachers about young students’ limits and their capacity to deal with algebra. On the theoretical level, we are proposing a non-reductive approach that draws on epistemology, semiotics, philosophy, anthropology and psychology to understand the central problem of algebraic generality. The results can be of interest to teachers, mathematics educators, and cultural and developmental psychologists.