The first part of this Discussion focuses on the language that students relied upon, both in their written work and in their oral contributions to the group discussions, and thereby serves to synthesize our main results. In that the structurally-oriented equivalence transformations that students described were characterized by both invisibility and visibility, yet in a manner that tended to collapse both in terms of hidden numerical values, the second part of the Discussion looks more deeply at this aspect—one which suggests not only the tight connection between the numerical and the structural in students’ algebraic thinking but also some implications for transitioning to letter-symbolic algebra. The third and last part of the Discussion returns to an issue that arose in the literature review and, in view of our results, casts new light on the question regarding the separation of the exchanging and sameness components of equality in students’ approaches to generating and justifying equivalences.
The language that students used to express sameness, properties, and equivalence
We asked ourselves several questions at the outset of our analysis, questions focusing on the kind of language that students might use to express sameness, properties, and equivalence and the role of this language in developing their thinking. Our approach in this regard is in line with Sfard (2008) who argues that “linguistic communication is the primary source of sustainable, accumulable changes in human forms of doing” (p. 123), and where thinking and communicating, whether the communicating be intra- or inter-personal, are united. In our study, both the student-sharing of their written work within the group and of their justifications of the approaches taken, as well as the interviewer-probing of students’ responses or his hinting at alternative ways of tackling a given task, were all conducive to expanding students’ mathematical discourse.
With respect to the kinds of language the students used to justify the equivalence transformations that they carried out in order to indicate the truth-value of their rewritten numerical equalities, it was seen that they initially computed the total of each side of the equalities and justified the truth-value with a language that centered on the results they had obtained: “We get the same result.” Even after rewriting those given equalities with a different decomposition of each side, their written and oral justifications continued with the same language.
When subsequently presented with a new set of equalities and being asked explicitly to not calculate the total of each side before rewriting the equalities, their thinking began to evolve. Instead of first calculating the total for the left side, they began by decomposing it. Their language shifted to explaining where each of the numbers of the decomposed left side of their partially rewritten equality came from with respect to the initial equality. It is noted, however, that the search for appropriate transformations for the numbers of the initial left side sometimes required a few tries, and several back-and-forth comparisons with the initial right side, before arriving at a decomposition that would work equally well for the right side. As seen in the schematic representation of the process of transforming an initial equality, illustrated in Fig. 6, the students came to reason that, since the various numbers of A can be decomposed in a top–down manner into A1 and since B can be equivalently decomposed in the same way, the resulting equality A1 = A1 which is obviously true demonstrates that the initial equality A = B is also true.
After completing the rewriting of the decomposed equality, the students’ oral justification of its truth-value was expressed as: “It will give the same”—suggesting that they no longer needed to compute the result of each side, but that it would surely be the same. In fact, the language they used to justify both the numerical transformation of a number into a decomposed, equivalent numerical subexpression, and the final form of the rewritten equality with its two identical sides, involved the word same in both cases, but the former top–down transformation had a qualification attached to it: “It looks different but gives the same result.” Thus, the term same applied to both the value of the decomposed numerical subexpressions in relation to the corresponding numbers of the initial sides and the appearance of the final expressions on both sides of the resulting equality.
With respect to the properties that the students mentioned, it was noted that, in transforming a number into its decomposed form and in comparing the decomposed side with the not-yet-decomposed side of the initial equality, there was an implicit reference to the order and addition structures for whole numbers and a more explicit reference to the decomposability of these numbers. There was also a written, but implicit, reference to the reflexive property of equality when describing the final decomposed equality: “Because if I look and compare each number, they are the same and obviously it is the same.” However, the students’ lack of precise names for these properties of number and of equality led to their using more informal, everyday language to describe them. This was seen in their manner of referring to decomposing in the last question of Task-set B—a question on describing in a general way their approaches to rewriting the equalities so as to show their truth-value—with terms such as simplifying, making the numbers smaller, and converting the numbers. This was, however, not unexpected. These students had not had any prior experience with explicit statements of properties, or structure-based approaches to equivalence, in their earlier classroom mathematical experience. Even that part of the question that included the phrase, “regardless of the numbers involved in the equality,” seemed somewhat unusual to them. Nevertheless, their ability to transplant familiar words to unfamiliar situations was a first step for them in the building of new mathematical discourses related to equivalence. We hasten to add however that, while students never used the term equivalence (and only very rarely, the term equal), the language that they did use provided evidence of their growing sense of how equalities could be transformed to show equivalence (i.e., sameness), and this by means of top–down structural decompositions that relied on number properties related to the addition structure, such as place-value, halving, doubling, and so on.
The collapsing of invisible and visible sameness in justifying their structure-based equivalence transformations
In our conceptual framework, we introduced a distinction between invisible and visible sameness, and used this distinction in analyzing students’ talk about the vertical transformations carried out in the generating of equivalent numerical subexpressions and the resulting horizontally-displayed common form for each side of the equality. We expected that the students would differentiate between the invisible and visible aspects of sameness in their justifications, but they did not use these exact terms. As highlighted above, they referred to the vertical, invisible sameness as: “it looks different but gives the same result”; and for the horizontal, visible sameness as: “Because if I look and compare each number, they are the same and obviously it is the same.” While the term, same, was used for both cases of “it”, for the former they were referring to particular numerical values that were safeguarded by the top–down, vertical transformations; and for the latter, to the invisible total of each side as represented by the same numbers on each side. So, in fact, there were really two types of invisible sameness inherent to the students’ statements—both of them referring to unseen computational totals. In other words, both the initial untransformed numbers and their corresponding transformed equivalents, vertically speaking, and the identical transformed numbers on each side of the equal sign, horizontally speaking, shared something in common: some same, but hidden, numerical results. This brings us back to an earlier statement made by Melhuish and Czocher (2020) that “we often mean that the two objects are the same in some (meaningful) respect but are not the same in every respect” (p. 39, italics in original). The form into which the numbers of the initial equality had been transformed was clearly different from their original version; however, their numerical values remained the same, albeit hidden from view.
What is of particular interest is that it was the aspect of the “same numerical result”—a computational underpinning to their structural approach—that allowed the students to collapse the vertical invisibility and horizontal visibility of equivalence into a single type of sameness justification. This suggests that computational underpinnings may be more central to structure-based transformational work than previously acknowledged, not only at primary school, but also later on in secondary school algebra. Examples to support this point of view can be found in several past research studies as, for instance, in a study with eighth-grade students (13- and 14-year-olds) on the introduction of various transformations involved in algebraic equation-solving (Kieran, 1987). When the students were presented at the end of that eight-session study with questions related to the equivalence of pairs of equations, numerical thinking clearly provided a basis for their decisions about sameness in what was patently an algebraic context. In fact, the kind of control afforded by the numerical in algebraic activity has recently led some researchers to conceive of algebraic thinking more broadly and to rename it as numerico-algebraic (Pilet & Grugeon-Allys, 2021) and as arithmetico-algebraic thinking (Fernando Hitt, personal communication, May 2021).
We emphasize, moreover, that the numerical is not only useful but also necessary for certain kinds of algebraic activity. When students encounter algebraic expressions that represent rational functions, a property-based structural perspective will not be sufficient. A numeric perspective will also be required—that is, the domain needs to be accounted for when considering equivalence between, for example, the expression (x − 2)/(x2 − 2x) and its structurally-transformed version 1/x (Solares & Kieran, 2013; see also Kieran et al., 2013). It may seem ironic that the development of a structural view of equivalence at the primary school level tends to involve some dampening down of the usage of the computational, but that both the computational and the structural dimensions of equivalence will be needed within the later study of algebra.
An additional aspect that is suggested by our finding related to the students’ notion of the hidden numerical values that characterize both the invisible top–down equivalences of initial and final equalities and the visible left–right equivalence of the resulting transformed numerical equality concerns the question of making transitions to letter-symbolic algebra—a world that also comprises hidden numerical values. Herscovics and Kieran (1980) have described a didactical approach to introducing algebraic equations with 7th and 8th graders that begins with numerical equalities that are identities and that involves hiding one of the numbers with a literal symbol. While space constraints do not permit a detailed description of the approach and how it is extended to include the top–down equivalences entailed in equation-solving, suffice it to say that the meaning given by the students to equation-solving was that of uncovering the hidden number by means of successive transformations that were oriented toward maintaining the truth-value of each step. In view of the way in which our 6th graders came to justify the equivalence transformations that they carried out in order to indicate the truth-value of their rewritten numerical equalities, we suggest that an avenue for future research would be to design a study that brings together the didactic approaches of the prior research reported by Herscovics and Kieran and of the research presented in this paper within an integrated sequence that transitions from numerical to algebraic transformational equivalences.
The linking of exchanging and sameness in students’ structural approaches to generating and justifying equivalences
We now return to a point that arose earlier within the literature review: the claim by Jones (2009) that a distinction ought to be made between the “sameness” meaning for the equals sign, which promotes distinguishing statements by truthfulness, and the “can be exchanged for” meaning, and his follow-up argument that one meaning does not imply the other. Jones’s claim is based on his research involving the Sum Puzzles digital environment that promoted a substitutive meaning for the equals sign and where students were presented with tasks such as that of transforming 31 + 40 into a single result by selecting from among the various equality tiles that were offered (e.g., 31 = 30 + 1, 30 + 1 = 1 + 30, etc.). These tiles allowed the students to make the following repeated substitutions: 31 + 40, 30 + 1 + 40, 1 + 30 + 40, 1 + 70, leading to the desired single result of 71 (see also Jones & Pratt, 2012). With such tasks, some of the students made substitutive exchanges without considering truthfulness, which was deemed to provide support for the notion that sameness and exchanging/substitution are two distinct facets of equivalence. However, aspects of our study prompt an alternative interpretation with respect to the sameness versus exchanging debate.
If one takes into account the properties underlying those very substitutive acts—such as, when 30 + 1 is substituted in Sum Puzzles for 31—then it would seem reasonable to suggest that it is precisely because 30 + 1 is the “same” as 31 (i.e., by the addition structure, 30 + 1 = 31, and by symmetry 31 = 30 + 1) that the substitution is tenable. From this perspective, exchanging depends on the support of sameness. In all fairness, however, it should be stated that Jones’s tasks were quite different from the ones used in our study. Tasks requiring students to choose from among a set of given exchanging options in order to solve sum puzzles are markedly not the same as tasks that involve generating equivalent numerical equalities. As our results suggest, tasks that explicitly ask students to rewrite an equality in such a way that demonstrates its truth-value can lead them to become sensitive to value-preserving transformations/substitutions. In such tasks, possible distinctions between sameness and exchanging would seem to fade completely.
To the same extent that exchanging depends on the support of sameness, it is also suggested that sameness can be demonstrated by exchanging. Recall from the literature review the case of Miguel who signaled the truthfulness of the equality 7 + 7 + 9 = 14 + 9 by mentally exchanging the 7 + 7 for 14 (Molina et al., 2008). More generally, determining the truthfulness of simple equalities—even when done by means of computing each side and substituting the computed totals for the given addition operation(s)—can be said to involve exchanging. Thus, we suggest that determining truthfulness/sameness within equivalence tasks of the kind used in our study, as well as within some of the tasks of the earlier empirical research on young students’ views of equality, will usually always involve some exchanging action, just as exchanging usually always involves the support of sameness. And this argument extends beyond the numerical domain to that of the algebraic.
To conclude this discussion, we come back to our construct of invisible sameness and propose that this construct encapsulates an essential characteristic of the duality of exchanging and sameness within equivalence transformations. Even if the students participating in our study did not use the exact terminology of invisibility when referring to the sameness of the equivalent expressions they were generating, we would argue nonetheless that the notion of invisible sameness captures the spirit of the truth-maintaining exchange process whereby the result of an equivalence transformation is invisibly the same as its pre-transformed version—such sameness being justifiable by properties, broadly speaking. Taking a cue from our students who referred to both truth-value and exchanging with the singular term of “sameness,” we suggest that invisible sameness is integral to activity related to the top–down equivalence transformations of numerical equalities, and at times to left–right equivalence too.