What is the meaning of equivalence? Is equality the same?

I am on a quest to investigate the history of “equivalence” in mathematics, mathematics education, and personal journeys through mathematics learning. During my qualifiers, I focused largely on Sfard’s (2008) take on communication and cognition in her birthing of the phrase “commognition.” The logical consistency, coherence, and compatibility of Sfard’s theory, both conceptually and methodologically, were and still are extremely attractive to my research interests. Her notions of saming, word use, realizations, and signifiers were the primary focus of my research design and analysis. Now that I am moving further into my dissertation, I am diving into a broader investigation of sameness across theoretical paradigms and mathematical domains.

Unraveling the Complexities of the Famous “=” Symbol: Perspectives from Kaput and Sfard

I returned to the year 2008 and to the spirit of the Kaput Center at my university, on a mission to democratize education. Kaput (2008) opens his book with a chapter delineating that “the basic idea that one can replace one expression by an equivalent one” (p. 12) is at the soul of algebra, inherently elevating arithmetic to a structure that looks beyond determining values of computation and focusing more on the form and presentation of symbols. Kaput claims that a major jump from arithmetic to algebra involves the shift in meaning behind the equals sign symbol “=.” The debate in mathematics education literature around the vast world of interpretations around “=” has permeated the math education literature. A single conception has yet to be agreed upon as to why this set of little parallel lines has caused so much of a challenge for students and teachers.

Using Sfard’s commognition framework, the “=” may be a signifier of many realizations for a teacher and a student separately. Any interactions between the two participants in a mathematics classroom may generate conflict in that the two interpretations of the equals sign may differ, especially when the symbol goes beyond stating the result of a computation towards a general, universal statement of sameness. Sfard (2008) offers a symbolic example, which I attempt to visualize for the reader in Figure 1.1 below, to view possible distinctions in the interpretation of this symbol over the development of a notion from a process to an object, or, as she terms, “reification” (pp. 53-54). However, Sfard’s realization-signifier notions do not seem far off from Kaput’s symbol-referent ideas.

Using Kaput and colleagues’ (2008) points of view of algebra from a symbolization perspective conveys that relating two objects with the “=” is a constrained permission for participants to act upon statements to transform them via substitution. Kaput et al. (2008) claim, “because the syntax arises from previously accepted equivalences in the reference system for the symbols, it is guaranteed to yield correct results when used appropriately” (p. 19). Referencing the same equality framed in Figure 1.1, the expression 2 + 7 is transformed into 9 so that substituting one expression for the other in any circumstance within conventional constraints is permissible and may make visible vital components of a story of reference. The purpose of transformation and substitution is to generate new objects via symbols that continue to represent an initial reference story in a visibly distinct form.

Exploring the Epistemological Landscape: Symbolizing Mathematics and Conceptualizing Equivalence

Pedagogically, do these conceptualizations share some features of objectifying processes to represent ideas or stories through human-drawn symbols that represent some form of equivalence and leverage the symbol “=”? Does any of it matter?

How could these epistemological nuances influence the classroom and democratize education for all students? Kaput et al. (2008) utilize an analogy to a windshield in that seeing raindrops on a windshield in focus requires a blurring of the glass itself, and vice versa, likening this idea to the iteration necessary when symbolizing mathematics in that it is “a deep epistemological distinction between mathematics as an object of study in its own right versus mathematics as an intellectual tool” (p. 25). Sfard (2008) references the challenge in ontological-epistemological conflicts that are often invisible in such cases as manipulating and transforming symbols for mathematical purposes. What exists? Does an expression 2 + 7 actually exist? Or is it referencing objects? Or is it a special case lending itself to exemplify the property of commutativity algebraicly in that a + b = b + a?

A Symbol That Means More Than Just Equivalent

Whether the equals sign symbol refers to or signifies, any one person could interpret a distinct meaning from another. Are transformations equivalent versions of a single object that can serve as a substitution for the other? On the other hand, is a result equivalent to working out an arithmetic computation to be a determined value? Students and teachers spend little time making such distinctions in their use of these mini parallel lines, “=”. Through discourse and participation, teachers can convey a change in the meaning of the “=” symbol. A student may view an entire equation as an object in its own right, or they may feel an urge to compute a value. Or do these assumptions remain tacit, causing conflict and possibly disciplinary consequences (such as a lower grade or lunch detention to make up a quiz)?

My investigation of the notion of equivalence is vital to the field. Many students have little exposure to the word, let alone the vast interpretations of the word “equivalence” outside of school algebra. From the moment a child is born, they perceive equivalence or sameness alongside difference. However, at what point do we give children the opportunity to view our implicit thoughts that govern our actions, especially in a field as abstract as mathematics? When does a baby receive comfort from a sibling’s hug versus comfort from a soft blanket equivalent? When does the use of a line or a line segment seem equivalent?

References

Kaput, J. J. (2008). What Is Algebra? What Is Algebraic Reasoning? In J. J. Kaput, D. William Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). Lawrence Erlbaum Associates/National Council of Teachers of Mathematics.

Kaput, J. J., Blanton, B. L., and Moreno, L. (2008). Algebra From a Symbolization Point of View. In J. J. Kaput, D. William Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19-55). Lawrence Erlbaum Associates/National Council of Teachers of Mathematics.

Sfard, A. (2008). Thinking as communicating: Human development, development of discourses, and mathematizing. Cambridge University Press.