Introduction

In their recent survey about future theme scholars in Mathematics Education (ME) consider important to face in the next decade, Bakker et al. (2021) state that many researchers mention “so-called didactical, topic-specific research” (p. 12) as not less important though “perhaps less fashionable” (p. 12). In the survey, scholars also underline that research dealing with topic-specific issues and research with global issues, usually considered as innovative and ground-breaking, run along separate tracks with little or no crossover, at least from the beginning of what Stinson and Bullock (2012) call the socio-political-turn in ME. Supposing that innovative ground-breaking research involves some kind of new theoretical or methodological outcome, research focused on teaching/learning of specific mathematical topics (e.g., fractions, derivatives, proof, etc.) has a narrower theoretical range than context-specific research focused on questions like why and where Mathematics is taught/learned and who teaches/learns it. According to Assude et al. (2008), in ME, three types of theories can be distinguished: “local theories (e.g., what levels of development exist in students’ learning of fractions?), middle range theories (e.g., what is classroom mathematics instruction?), or grand theories (e.g., what is the mathematics education field?)” (p. 7). In this sense, research with main focus on topic-specificity produces local theoretical outcomes, while research with focus on context develops mainly grand and sometimes middle range-theoretical outcomes, where teaching/learning of specific mathematical topics has at best an illustrative role. Indeed, according to Stinson and Bullock, within the context-specific socio-political perspective, researchers explore “how socio-cultural and historical discourses have constructed and continuously shape teachers, students, and mathematics as subjects of inquiry” (Stinson & Bullock, 2012, p. 44). Therefore, this global focus needs to “zoom out,” as these authors would say, from topic-specificity. A closer connection between the two lines of research that here are called respectively topic-specific (TS) and context-specific (CS), would be desirable for the discipline, as Bakker et al. (2021) state, quoting, among others, Lyn English (Australia) in the survey mentioned above. English complains an excessive distancing of recent research from the original intent to improve teaching and learning mathematical contents: “literature has been moving away from the original goals of mathematics education. We seem to have been investigating everything but the actual learning of important mathematics topics” (Bakker et al., 2021, p. 12). Furthermore, a closer connection between TS- and CS-research could help to shed light on how the standard ways to carry out TS-research could impact on the social and cultural marginalization of particular groups such as students with learning disabilities (Niral Shah, USA, as cited by Bakker et al., 2021). But connecting the two lines of research could be useful also from a theoretical point of view. Indeed, it could help to evidence hidden variables in CS-research. The work focuses mainly on this theoretical aspect.

In the paper, firstly two possible reasons for the distance between TS- and CS-research are examined and then a way to reduce it is discussed. The main idea is that an investigation of the scientific production in ME research that has constructed and continuously shapes mathematical objects in the educational domain, could be helpful in this sense. It is shown that ME research practice produces specific mathematical objects, different from the mathematical ones that usually have the same name, and that they could be considered both, as a technical link between TS- and CS-research, as well as a factor at stake when aspects related to the social, political, and ethical implications of research in ME are discussed (Ernest, 2018, 2021; Stinson & Bullock, 2012). Indeed, such objects connect students’ learning of mathematical objects to the ways knowledge about their learning is acquired within ME as research domain (Asenova, 2021, 2022, 2023).

Two possible obstacles and the characterization of an epistemological approach in ME research

There are at least two reasons that hinder the connection between TS- and CS-research. Firstly, as stated in the introduction, the global evolution of ME as field of research toward a socio-political turn led in the last decades to a shift of the innovation focus from topic-specificity to context-specificity (e.g., Radford, 2008; Stinson & Bullock, 2012; Valero, 2010). Indeed, in their classification of different turning-moments concerning the evolution of ME as research field, Stinson and Bullock distinguish four moments, where the socio-political is the last one. The first, called process–product-moment, is characterized by the aim to predict objectively students’ outcomes by investigating teacher’s actions. The second one, called the interpretivist-constructivist moment, aims no more to only predict, but also to understand these phenomena. The third one, called the social-turn moment, is characterized by the necessity to socially contextualize teaching–learning phenomena. This contextualization is then extended by the socio-political turn in order to consider also the wider social and political context of the actors involved in the teaching–learning scenario.Footnote 1 This reconstruction of knowledge evolution in ME research is useful for the purposes of the present investigation because it helps to understand why TS-research could appear as outdated: it is limited to students’ and teacher’s relation to knowledge, that disregards, or barely touches, the constraints of the context, related to the cultural, social, and socio-political aspects of teaching and learning, while recent CS-research often substantially modifies the very idea of what learning, teaching and knowledge is about (e.g., Godino et al., 2007; Lavie et al., 2019; Radford, 2019; Sfard, 2008). This is in line with the local-theory and grand-theory scenario mentioned above: local theories have a narrower range than grand or middle-range theories and their outcomes are thus considered not so ground breaking. Secondly, discussion about knowledge acquisition of mathematical objects puts in the forefront the necessity of a reference to students’ epistemology, intended as theoretical assumptions about knowledge acquisition. In this way, TS-research is usually discussed within some middle-range or grand theory in ME that theorizes learning in a specific way. Thus, it seems that an epistemological approach that focuses on students’ epistemology makes it impossible to formulate general statements about knowledge acquisition related to mathematical objects in ME as research domain. Indeed, as Ernest (2012) states, citing von Glaserfeld: “To introduce epistemological considerations into a discussion of [mathematics] education has always been dynamite” (von Glaserfeld, 1983, quoted in Ernest, 2012, p. 1). This statement dates 30 years back, but it is still useful to illustrate how a reference to epistemology, intended as students’ knowledge acquisition, splits general discourses because of the multiplicity of approaches to learning in our discipline. At the same time, CS-research, where “context is moved from the margins to the center, as concepts of empowerment, class struggle, asymmetrical relations of power, and so forth are critically explored and uncovered” (Stinson & Bullock, 2015, p. 10), rests on general considerations on the discipline, as Stinson and Bullock’s interpretation exposed above shows. A general epistemological approach, able to face TS-knowledge acquisition in ME as research domain, could allow us to go beyond students’ epistemology and thus overcome the split due to different learning theories.

The distinction of the two issues highlighted above allows us to divide the initial problem related to the connection of TS- and CS-research into two smaller problems. The first one is closer to the issue expressed in the quotations of ME scholars cited in the introduction, but it does not offer a possibility for a direct questioning because it is strongly related to scholars’ convictions. Thus, to face the problem, the second issue, related to the epistemological concern, is questioned. As it is argued in the “Conclusions” section, this approach allows us to question indirectly also the first issue.

To address the epistemological question highlighted above, a switch from students’ epistemology to epistemology intended as a branch of philosophy of science is accomplished. Thus, the focus in this context is on the epistemology of ME as field of research practice (Valero, 2010, p. XVII) that is investigated by examining pieces of research outcomes in this discipline. As stated above, the reference to particular learning theories needs to be overcome in this context because it entails a pragmatic approach that spans the divide between epistemology as foundational knowledge and hermeneutics as interpretive understanding. Indeed, epistemology is not intended here as normative science (e.g., Bachelard, 1938), in opposition to hermeneutics (e.g., Gadamer, 1960/2019; Rorty, 2004). The epistemological discourse in this paper does not claim for normative reconstruction of knowledge acquisition in research in ME but refers to epistemology as a special kind of hermeneutics (Rockmore, 1990). Rockmore highlights that the identification of epistemology with foundationalism, and thus with epistemology as normative science, can be overcome by the shift from foundationalism to anti-foundationalism in philosophy of science. An anti-foundationalist approach is compatible with the refusal of reductionism and absolute truth in a critical-postmodern approach and with uncertainty in ME as research field (Stinson & Bullock, 2012, 2015). In this article, the anti-foundationalist position relativizes foundational discourses, without refusing scientific knowledge acquisition, once accepted their relativeness and the impossibility to pose knowledge acquisition outside of presuppositions based on former knowledge (Rockmore, 1990). Following this concept of epistemology, the author’s position in this article is a pragmatical one (Peirce, 1960; Wittgenstein, 1953/2003). The research design is led by the research questions, as this is the case in a pragmatist approach (Johnson & Onwuegbuzie, 2004): according to the needs of the inquiry, conceptual webs, intended as locally legitimated language games (Lyotard, 1988), are produced, searching for global characteristics in TS-knowledge acquisition in ME research.

Some terminological clarifications

The shift from the students’ epistemology to the epistemology of the research discipline needs a distinction between two specific fields of knowledge: the one of educational practice and the one of research practice in ME (Valero, 2010), just mentioned above. Herein, a distinction between ME as praxeology and ME as research practice is made by extending Chevallard’s (1992) concept of praxeology. According to this author, praxeology is composed by a type of tasks, a technique to perform them, the technology needed to explain and justify the technique, and the theory that should be able to justify and “complete” the technology (Chevallard & Sensevy, 2014). For instance, in Mathematics there are praxeologies for calculating integrals or solving quadratic equations.

ME as praxeology is intended here as focused on TS teaching–learning phenomena that have their own praxeology. For instance, it could focus on the teaching–learning praxeology of calculating integrals in the Mathematics classroom; it is usually, although not exclusively, carried out by practitioners. ME as research practice, intended in TS-way, refers instead to the practice carried out by scholars in ME, adopting specific theoretical and methodological lenses, in theorizing and questioning TS-educational issues.

In the context of the present paper, practice differs from praxeology firstly because the former is located at a meta-level with respect to the latter, and secondly, because in the practice one or more of the four components that define a praxeology are missed, not explicitly defined, or under definition during the research practice itself. For instance, there is a praxeology for calculating integrals in the classroom and there is also a quite well-defined praxeology of teaching this mathematical content, but there could exist nearby countless practices for theorizing research on teaching/learning integral calculus and most of them are defined during the research practice itself.Footnote 2

In the forthcoming of the paper, the acronym MER is used to refer to Mathematics Education as research practice, while the acronym ME is used to refer to Mathematics Education as praxeology. Sometimes also the acronym ME(R) is used when referring to articles where the distinction between ME and MER is not explicitly faced and thus remains blurred.

The research questions

To focus the narrow research issue about the “empirical proof” of existence of Mathematical Objects specific to MER (MOsMER) and to investigate its potential to reduce the gap between TS- and CS-research, the following research questions are formulated:

RQ1. Assuming an epistemological viewpoint on MER, is there any evidence of the existence of MOsMER in the scientific production of the discipline, that allows us to make general epistemological TS-considerations, independent from specific learning theories?

RQ2. If RQ1 can be answered positively, in which sense does the existence of MOsMER connect general TS-considerations with issues related to CS-research and how does this reduce the distance between these two lines of research?

To answer RQ1 it will be necessary and sufficient to bring some examples of such objects (empirical proof) examining and interpreting a piece of scientific production in MER (i.e., assuming an epistemological viewpoint on it). This means that what we are looking for here is not a definition of MOsMER but evidence of their existence. The answer to RQ1 is propaedeutic to, and aimed at, the answer to RQ2. The answers to the research questions are discussed in the “Discussion” section.

An empirical proof of existence of mathematical objects specific to MER as answer to RQ1

In this section, the existence of mathematical objects different from the mathematical objects in Mathematics and specific to MER is discussed. An “empirical proof of existence” is provided by exhibiting some examples of such objects drawn from MER. The arguments are derived from the consideration that mathematical objects are ontologically different in MER with respect to Mathematics because of the “didactical context” within they are conceived and studied by the scholars in MER. The existence of MOsMER would give a proof for the ontological creativity of MER that could be considered as a variable to be investigated in CS-research.

A double analogy

Since mathematical objects in Mathematics and MOsMER usually have the same name, it is important to show why it is argued that they are different. To this purpose a double analogy is used.Footnote 3

Just as Mathematics studies mathematical objects that have their specific existence not in empirical practice but in mathematical research practice, MER studies mathematical objects that have a specific existence not in the praxeology of the classroom, but in ME as research practice.

Therefore, empirical reality (a1) is to mathematical research practice (a2), as the praxeology of the classroom (a1’) is to research practice in MER (a2’).

Just as in Mathematics, mathematical objects firstly arise as a methodological tool and then become an object of study and research in Mathematics, in MER the mathematical objects firstly emerge as a tool which usually has the same name as the one in Mathematics, and then turn into specific objects of study in MER.

Therefore, the methodological tool for or of Mathematics (b1) is to the correspondent mathematical object of Mathematics (b2), as the mathematical object as tool in MER (b1’) is to the correspondent mathematical object specific to MER (b2’).

The first analogy (a1 is to a2 as a1’ is to a2’) rests on the similarity of the two relations established between the disciplines and their domains of research, but it highlights also the difference between mathematical objects in Mathematics as research discipline and in MER, due to an ontological dimension, based on their different domains of existence (there are two different relations that are linked together by the analogy). The second analogy (b1 is to b2 as b1’ is to b2’) stands for the similarity of emergence of the two kinds of mathematical objects within the practice of the respective discipline. The two analogies are related to each other as follows: the ontological aspect expressed by the first analogy invokes the epistemological viewpoint required by the second analogy; this viewpoint provides in turn an appropriate methodology. It suggests investigating MER-scholars’ scientific production, looking for shifts from mathematical objects as tools in MER to MOsMER. The first analogy is based on a pragmatist viewpoint on meaning (D’Amore, 2001; Kutschera, 1979), widely accepted and used in MER (e.g., Dörfler, 2005, 2016; Ernest, 1991/2004; Godino et al., 2007; Lavie et al., 2019; Sriraman & English, 2010). According to the pragmatist viewpoint, mathematical objects have a subjective, interpretation-depending meaning. They can be seen for instance as determined by the use within a language game (Wittgenstein, 1953/2003) and by the semiotic resources the subject is able to activate in a context where the object emerges as final logical interpretant (Peirce, 1960).

Why can the two analogies exposed above be considered as valid or appropriate for the scope of this article? The first analogy is considered as a valid one because once the pragmatist viewpoint is accepted, (1) the contexts of the working mathematician’s research practice and the context of the students’ learning activity should be considered as different domains and thus the mathematical objects they produce cannot be the same; (2) the scholar in MER works in an even different context, where the mathematical objects are the students’ epistemic objects, that means objects-to-be known or known objects, that emerge into this specific research context. Thus, mathematical objects in Mathematics and MOsMER are ontologically different in a pragmatical sense because of their different domains of emergence.Footnote 4

The second analogy assumes that this different kind of mathematical objects, the MOsMER, evolve in a way as mathematical objects in Mathematics: they firstly are used in an operational way, and then become objects of study and reflection, turning into a structural form, as highlighted by Sfard (1991). Sfard shows how in the epistemological evolution of Mathematics as well as in students’ conceptual evolution, mathematical notions firstly emerge in an operational sense, as processes, and then are reified in a structural sense, turning into objects of study and reflection. Referring to Sfard’s discussion, which is analyzed in more detail in the “Theoretical framework and methodology of research in reference to RQ1” section, the present investigation argues in favor of the possibility to add a third kind of switch from an operational to a structural type of mathematical notions: beside the one in the practice of the working mathematician and the one in the learning students’ activity, a shift in the practice of the working researcher in MER. While the first two shifts are detected assuming an epistemological viewpoint on the practice of the working mathematician and on the learning processes of the learning subject, the third one could be characterized by assuming an epistemological viewpoint on the practice of the working researcher in MER, as suggested by the second analogy. The second analogy is used as a basis for the methodology of the investigation about process-object shifts in MER-scholars TS production, with the aim to answer RQ1; thus, its validity, intended as appropriateness, is confirmed in an experimental way.

Theoretical framework and methodology of research in reference to RQ1

As stressed before, Sfard (1991) highlights that there are two moments in the historical development of mathematical concepts: an initial one related to an “operational” conception (the concept conceived as a mere tool) and a subsequent one related to a structural conception (the same concept, however, thought as an object in itself, and therefore, as a specific object of possible theoretical study). Sfard (1991) emphasizes an analogy between the occurrences in the history of mathematical thinking and the ones in the cognitive construction of the individual. For instance, the concept of function in history of Mathematics firstly emerges as a process (e.g., function as computational process) and later becomes an object of study in and of itself, presenting a structural dimension (e.g., function as set of ordered pairs, but also functions as solutions of partial differential equations, starting from the vibrating string problem in the eighteenth century). Sfard shows that the first acquisition of the concept of function by a learning subject goes through analogous stages.

Sfard (1991) furthermore characterizes three steps in the transition from the operational to the structural phase: interiorization, condensation, and reification. Following Piaget, she stresses that a process has been interiorized if it can be carried out through mental representations and “in order to be considered, analyzed and compared it needs no longer to be actually performed” (Sfard, 1991, p. 18). Subsequently, during the condensation phase, the subject becomes “more and more capable of thinking about a given process as a whole, without feeling an urge to go into details” (Sfard, 1991, p. 19). The last stage, the reification, is somehow different from the previous two. While interiorization and condensation require a gradual change, reification means that a qualitative change occurs; it is defined “as an ontological shift–a sudden ability to see something familiar in a totally new light (…) an instantaneous quantum leap: a process solidifies into object, into a static structure” (Sfard, 1991, pp. 19–20).Footnote 5

Sfard’s investigation is extended here to what was called above the epistemological viewpoint in MER. Since, as Ernest (2012) states, our research methods and methodologies “are all underpinned by epistemology” (p. 10), it is important to show that the basic assumptions that underpin the adopted methodology are not arbitrary because they inform the research outcomes. Firstly, the possibility to consider an epistemology of MER as autonomous science is supposed; secondly, the applicability of Sfard’s approach in MER is assumed. Concerning the first issue, Assude et al. (2008) show that although an absolute autonomy does not appear sustainable for ME(R), a relative autonomy can be assumed. Indeed, in MER, on the one hand, we adapt and develop tools taken “from outside,” on the other hand, we also carry out autonomous research and theory production. Concerning the second issue, one can state that the transposition of Sfard's approach to MER is based on what Sfard (2009) calls “the objectifying metaphor” (p. 43) which is not discipline-specific: for reasons of language efficiency and to facilitate reflective activity without overloading memory, people tend to objectify the contents of language that are familiar to them. An example in this sense is the substantiation of verbs or adjectives (e.g., to add—addition; continuous function—the continuity of functions). To transpose Sfard’s approach for the present purposes, an explanation and adaptation of the terms “operational” and “structural” to MER is needed. Indeed, in MER they cannot be the same as the ones characterized by Sfard (1991) in Mathematics and in ME. Furthermore, an explanation of the criteria adopted to establish that condensation/reification occurred is needed.

The “operational” dimension of objects specific to MER

While the operational dimension of mathematical notions in Mathematics has relatively intuitive characteristics and does not require explanations (think for example of solution-algorithms or of the counting procedure), in the case of MER, the meaning with which this term can be used does not seem so immediate. Indeed, in MER there are no calculation algorithms to refer to. However, if instead of operational one uses the term instrumental,Footnote 6 it becomes also possible to identify in MER characteristics analogous to the operational ones in Mathematics.

To characterize what we call here “the instrumental aspect” we do not use definitions but a pragmatical characterization as an outcome of its possible instantiations. One can talk about an instrumental aspect of the mathematical notions in MER, referring for example to cases where mathematical concepts have their origin directly in problems related to the teaching–learning processes. These kinds of notions are drawn directly from the classroom praxeology. In this sense, the researcher in MER resorts to the instrumental aspect of mathematical notions specific to MER:

  1. 1.

    To allow or facilitate the interpretation of the students’ or the teacher’s conceptions, mental models or mental actions;

  2. 2.

    To make it possible to represent a specific mathematical content or to communicate about it in classroom interactions;

  3. 3.

    To foster the analysis of students’ oral or written production or of teacher’s interventions.

The previous list does not claim to be exhaustive and can be enriched with other instrumental aspects that may emerge during the examination of specific cases.

The structural dimension of mathematical objects specific to MER

The sense in which it is possible to understand the term “structural” in MER is very different from the one considered in Mathematics and therefore also in Sfard (1991). In fact, for Sfard the structural version of mathematical notions is primarily linked to Bourbaki-style set definitions. Such a reference is not possible in MER because its object language is not a set-theoretical formalizable but a discursive one. The role assumed by the formal structure in Sfard’s approach can be replaced in the case of MER by a conceptual structure that is called here a semantical complex. A semantical complex can be thought of as a conceptual structure, in which two or more concepts are linked together by semantic relationships, for example by a semantical relation of complementarity (the characteristics represent partial views on the same concept which complement each other) or inclusion (one is subsumed by the other, that means the meaning of one is englobed by the meaning of the other) or partial overlap (two or more concepts can be partially subsumed by the same concept). The semantical complexes represent the structural dimension of the MOsMER. They differ from the corresponding mathematical objects because they are not monolithic objects defined in themselves but are conceptual webs. Not all the aspects that characterize a semantical complex can be found in the correspondent mathematical object. There is always a gap, some discrepancy, between them, due to their different domain of emergence and the needs they accomplish (research on knowledge acquisition in MER versus research in Mathematics).

In the “Empirical evidence of mathematical objects specific to MER: some examples” section, two examples of such semantical complexes are provided.

Reification/condensation in MER

Concerning indices of condensation or reification, Sfard states:

There is one thing, however, which is much too essential to be passed over in silence. It is the potential role of names, symbols, graphs and other representations in condensation and reification. Judging from the history, the importance of this factor can hardly be overestimated. (Sfard, 1991, p. 21)

For the present purpose, to be able to state that reification, or at least condensation, occurs, it would be important to detect, beside specific representations, some kind of substantiation of the terms that denote the semantical complex or its elements. Indeed, this kind of shift in language can be seen as an index of an objectifying metaphor (Sfard, 2009), as explained above. In MER, whose language is discursive, the substantiation acquires an important role in the objectification of specific (mathematical) notions. Following Sfard (2009), reification/condensation is not intended as the creation of some kind of platonic objects in MER but as a sort of linguistic reification that can be detected in the discursive language used by scholars in MER.Footnote 7

Empirical evidence of mathematical objects specific to MER: some examples

In this section, two examples of mathematical objects specific to MER are provided, arguing empirically for the conjecture that it is possible for MER to consider a kind of mathematical objects that share the same name with the corresponding ones in Mathematics, but specific to MER.

Division as mathematical object specific to MER

Fischbein et al. (1985) inquire into the role of tacit modelsFootnote 8 to which students in primary school resort when they are solving problems concerning multiplication or division. In this context, referring to division, Fischbein and co-authors give the following definitions:

Division (…) The structure of the problem determines which model is activated. Partitive division. The first model, which might also be termed sharing division, an object or collection of objects is divided into a number of equal fragments or subcollections. The dividend must be larger than the divisor; the divisor (operator) must be a whole number; the quotient must be smaller than the dividend (operand). Quotative division. In the second model, which might also be termed measurement division, one seeks to determine how many times a given quantity is contained in a larger quantity. In this case, the only constraint is that the dividend must be larger than the divisor. If the quotient is a whole number, the model can be seen as repeated substruction. (Fischbein et al., 1985, p. 7)

The quotation shows that the authors give a definition of two different types of intuitive or tacit models of division and use them to describe and interpret students’ behavior during the problem-solving processes in classroom praxeology. This seems to be a clear recourse to the terms partitive division and quotative division in an instrumental sense, as highlighted in the “The “operational” dimension of objects specific to MER” section. Indeed, these terms are used to allow or facilitate the interpretation of the students’ mental models or mental actions (according to point (1) in the “The “operational” dimension of objects specific to MER” section) and also to foster the analysis of students’ oral or written production (according to point (3) in the “The “operational” dimension of objects specific to MER” section).

The “didactical division” characterized above is widespread both in the textbooks and in the scientific production of scholars in MER. In the latter context, it is used not only in the instrumental sense exposed at the beginning of the section but also in a more structural way. Indeed, Downton (2008) writes as follows:

Fischbein et al. (1985) proposed two aspects of division: partitive and quotitive. In partition division (commonly referred to as the sharing aspect) the number of subsets is known and the size of the subset is unknown, whereas in quotation division (otherwise known as measurement division), the size of the subset is known and the number of subsets is unknown. (Downton, 2008, p. 171)

We can notice that this author does not talk any more about models used to interpret students’ mental actions but talks about two aspects of division: partitive and quotative. Furthermore, the two adjectives “partitive” and “quotative” in the first sentence of the quotation eventually become nouns in the second sentence (“partition” and “quotation”), creating two kinds of division, named partition division and quotation division.

In the example there are both: a recourse to a specific name and a recourse to substantiation (partition division and quotation division). We conclude that while Fischbein et al. (1985) use the notion of partitive and quotative division in an instrumental way, where this notion is considered still a mental model used to describe and analyze students’ behavior, in the case of Downton (2008) an objectification that satisfies the requests of reification highlighted above, occurs.

One can state that the “didactical object” division characterized here is a kind of conceptual complex, where the two kinds of division (partition division and quotation division) are linked by a semantical relation of complementarity (Fig. 1).

Fig. 1
figure 1

Representation of the semantical complex “division” that characterizes a mathematical object “division” specific to MER: the two kinds of division are complementary and represent together “division” from MER viewpoint

Full size image

Considering this kind of “didactical division,” we can notice that it possesses characteristics that have no counterpart in the mathematical object division. For instance, in the latter, there is never a recourse to a description of a figural representation that involves a spatial arrangement of a collection of objects and there are no restrictions due to the need that the divisor must be smaller than the dividend. Division as mathematical object is defined as the inverse operation to multiplication, a binary operation closed in reference to a (number) set. In this sense, division is defined in reference to (numerical) sets where each element has an inverse multiplicative element, that means not in ℕ, the set of natural numbers, but only starting from ℚ+, the set of positive rational numbers. Division as mathematical object cannot be characterized in terms of partitive or quotative division because neither the divisor nor the dividend is necessarily a whole positive number, that is, numbers belonging to the set of natural numbers. For the present purpose it does not matter that in Mathematics there are other kinds of division (e.g., Euclidean division) or that one can consider also rings with units and invertible elements, instead as fields that guarantee the closure of the operation. What matters here is that partition division and quotation division does not have a counterpart in a mathematically well-defined object and that this discrepancy is due to the different domain of emergence of such objects. Indeed, the mathematical context is irrelevant from the cognitive point of view, while the one specific to MER rests on it.

Function as mathematical object specific to MER

Balacheff and Gaudin (2009) analyze the literature related to the topic “function” in ME(R) and dwell on eight different conceptions of function in students aged 15–19 detected by Vinner (1992). These conceptions are formulated in a descriptive mode which aims to highlight how students conceive the idea of function: e.g., “A function must be an algebraic term”; “A function is identified with one of its graphical or symbolic representations”; “A function should be given by one rule…” (Vinner, 1992, quoted in Balcheff & Gaudin, 2009, p. 24). Starting from this analysis of students’ conceptions, Balacheff and Gaudin (2009) frame their research on students’ conceptions of function within the cK¢ model (Balacheff, 1995, 2017). In this model, a conception is a quadruple (P, R, L, Ʃ), where P represents a set of paradigmatic problems able to bring out the conception; R represents a set of operators (algorithmic, theoretical etc.), that can be used to act in the context of the solution to the paradigmatic problems; L represents one or more representational systems which can be used in this context; Ʃ represents a control system that allows to evaluate the solutions in order to establish if they can be effectively considered as such. Starting from Vinner’s empirically determined categories and from a careful historical and epistemological inquiry, Balacheff and Gaudin (2009) work out theoretically the following four possible conceptions of functionFootnote 9: (1) table conception: CT = (PT, RT, Table, ƩT) “has essentially empirical foundations; the validity of a table depends on the precision of measurement and of the related computations against the requirements of a given experimental context” (p. 21); (2) curve conception: CC = (PC, RC, Curve, ƩC): “whose corresponding sphere of practice (PC)—in the beginning of the eighteenth century—was constituted by the important problem of long-distance navigation where coasts were out of sight. (…) A curve was not yet the graphical representation that we acknowledge nowadays as being the graph of a function considered as a relationship between entities (numbers or even quantities)” (Balacheff & Gaudin, 2009, p. 22); (3) analytic conception: CA = (PA, RA, Algebra, ƩA) “introduces a rupture in the epistemology of functions. A function defined by an analytical expression does not need to refer to an experimental field (either of natural phenomena nor of mechanical drawings)” (Balacheff & Gaudin, 2009, p. 23); (4) relation conception: CR = (PR, RR, LR, ƩR) that starts with the vibrating string controversy in the eighteenth century and the consequent need to study functions as objects in themselves, that means as solutions of partial differential equations.

Starting from this classification, Balacheff and Gaudin, using the tool Cabri Geometre, carry out an experimentation within a paradigmatic situation, able to allow students’ specific conceptions to emerge. The authors identify, examining students’ protocols, two types of conceptions: Curve-Algebraic and Algebraic-Graph conception, which are based on a strong connection between the conceptions CC e CA within the context of analytical geometry.Footnote 10

In the Curve-Algebraic conception, the diagram that represents the function is seen as a geometrical object, situated in a reference system; there is a so-called rigid link between the curve (in this case a parabola) and its equation, as a relationship of the curve with the remaining elements of the reference system is not established. In this case there is no true coordination between the graphic and the algebraic register. The use of mathematical tools belonging to infinitesimal analysis in solving the problem is reduced to mere symbolic manipulation.

In the Algebraic-Graph conception, the diagram is an object of the reference system, and the graphical representation system is considered in close coordination with the algebraic one; the function is, at the same time, both the equation and its graph, and the Cartesian plane is the place where this link is highlighted. The Curve-Algebraic conception bridges the distance between synthetic and analytic geometry, while the Algebraic-Graph conception bridges the distance between Analytic geometry and Calculus. In Balacheff (2017), the author discusses a further elaboration of the conceptions of function discussed above, referred to preservice teachers. In this case, the problem-solving-activity-environment is not Cabri Geometre, but the tool Maple. The paradigmatic situation is determined by the requirement of the choice of the best approximation among five possible ones in a point of a 3-grade-polynomial function when the coordinates of 20 points of its curve are known with a range of error of 10%. The experimentation shows that the problem-solving activity allows the emergence of another conception of function: the conception of function-as-object, that unifies both, the Curve-Algebraic and the Algebraic-Graph conceptions.

In the beginning of the section, we presented Vinner’s classification (Vinner, 1992) from which Balacheff and Gaudin (2009) take inspiration for their subsequent modeling of high school students’ conceptions of function. It provides a range of categories useful for describing the conceptions that have emerged from an empirical investigation and so it is possible to state that they are used in an instrumental manner: to allow or facilitate the interpretation of the students’ conceptions (in accordance with point (1) in the “The “operational” dimension of objects specific to MER” section) and to represent a specific mathematical content or to communicate about it in classroom interactions (in accordance with point (2) in the “The “operational” dimension of objects specific to MER” section).

On the other hand, the characterization of the function-as-object concept provided by Balacheff (2017) can be seen as a reification of a mathematical object “function” specific to MER, for which Balacheff and Gaudin’s (2009) characterizations represent the condensation phase.

The hypothesis that in Balacheff (2017), there is a reification of a mathematical object “function” specific to MER seems to be confirmed both by the schematic nature of the representation that facilitates the reference to the different conceptions (Balacheff proposes a table representation of the three conceptions, highlighting their characteristics), as well as by the substantification of the terms used to refer to them. In fact, while Balacheff and Gaudin (2009) still talk about Curve-Algebraic conception and Algebraic-Graph conception, Balacheff (2017) talks about Curve-algebra conception, Algebra-graph conception, and Function-as-object conception. In the last case, the terms that characterize the noun ‘conception’ are themselves all nouns (Curve-algebra, Algebra-graph, Function-as-object), while in the previous case some of them are adjectives (Curve-Algebraic, Algebraic-Graph). The conception of Function-as-object is a more complete semantical complex that includes the other two conceptions (Curve-algebra conception and Algebra-graph conception), which interact with each other according to a semantical relation of complementarity, integrating the algebraic and the graphic registers related to them (Fig. 2).

Fig. 2
figure 2

Representation of the semantical complex “function” that characterizes a mathematical object “function” specific to RME: the Curve-algebra conception and the Algebra-graph conception are complementary to each other, and the Function-as-object conception represents an integration of the algebraic and the graphic registers related to them

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Furthermore, one can state that the specific characteristics of the object function, dependent on the way it is introduced to students, do not belong to the features of the mathematical object function as it is conceived in Mathematics. Indeed, in Mathematics, a function is defined as a relation between two sets and as a subset of the Cartesian product of two sets, independently from the way it is introduced in the classroom. Neither of these definitions is related to algebra nor analysis in particular, as in Balacheff’s categorization. Thus, also in this case there is a discrepancy due to the different domains of emergence of the two kinds of mathematical objects, the one in Mathematics and the one in MER.

Discussion

In the “An empirical proof of existence of mathematical objects specific to MER as answer to RQ1” section, two examples of MOsMER were analyzed and discussed and the corresponding semantical complexes which give rise to the structural dimension of their dual nature as process and object were sketched. To detect these objects, an epistemological viewpoint on pieces of scientific production in MER was assumed. In this way, RQ1 can be answered positively, providing an “empirical proof” of existence of such objects.

As stated in the “Two possible obstacles and the characterization of an epistemological approach in ME research” section, in a pragmatical and anti-reductionist perspective, the viewpoint of the epistemologist is necessarily an interpretative (or hermeneutical) one. According to D’Amore (2015), a hermeneutical viewpoint should create a “conceptual scheme designed above the real, not a second real” (p. 154). In this sense, the conceptual complexes that were worked out to exemplify the MOsMER have no claim for ontological uniqueness. Semantical complexes can be formed in different ways related to the same mathematical object. This allows us to consider different ways to make TS-research in ME and to reify MOsMER, without refusing the multiplicity of school Mathematics (Ernest, 2000). Moreover, the shift from the instrumental to the structural meaning of MOsMER is not a one-way-path and once reification is accomplished, the instrumental and the structural meaning co-exist and are used pragmatically, as needed in research. The possibility of such a characterization in terms of specificity in MER does not have to be the case for every mathematical object, but this does not affect the present inquiry because it could simply mean that there is still a lack of intensive TS-research on these objects in MER.

Furthermore, the reified objects are not fixed but could evolve over time, creating more complex webs that could require a new organization of previously established semantical complexes. Thus, they are characterized by uncertainty and are concepts “under construction” (Skovsmose, 2016, p. 12) or locally legitimated language games (Lyotard, 1988).

Anyway, what emerges from the inquiry in the “An empirical proof of existence of mathematical objects specific to MER as answer to RQ1” section is a sort of “relational formalism,” applicable to MER that could allow us to connect mathematical objects in broader semantical webs specific to MER. Thus, if some kind of normativity should be recognized to the objects emerging in the present investigation, it is related to these relational aspects that connect MOsMER in webs similar to Lakatos’ informal theories (Lakatos, 1978) in Mathematics. Although this last interpretation of TS epistemology in MER could be considered as a metanarrative (Lyotard, 1988), refused by a postmodern approach (Stinson & Bullock, 2012), it could be considered as a useful abstract metanarrative “under construction” (Skovsmose, 2016, p. 12).

To answer RQ2, we start from the affirmative answer to RQ1 and argue in what sense the ontological creativity of MER could be considered as technical link and as factor at stake in bringing closer TS- and CS-research.

A first aspect to be highlighted is that the consideration of MOsMER allows us to look at topic-specificity in MER “as a whole.” This allows us to avoid specific theories of knowledge and makes TS-research “epistemologically suitable” to be considered as a variable not only in local, but also in middle-range and grand theories, typical of CS research. In this sense, the emergence of MOsMER allows us to start a discourse in MER that can be considered a topic suitable for middle range and grand theories. Indeed, a discourse about MOsMER allows us to hypothesize research questions suitable for middle range theories (e.g., How does the reification of MOsMER shape Mathematics instruction?) and also for grand theories (e.g., How does the emergence of MOsMER affect the identity of ME as research domain and its relation to Mathematics?). Thus, MOsMER can be considered as a technical link between the two lines of research because they allow topic-specificity to access more general and abstract realms in MER.

Moving away from these more technical considerations, it is interesting to show in what sense MOsMER allows us to consider TS-research as factor at stake in bringing closer to each other TS- and CS-research.

There are several cases of research where CS-aspects are investigated focusing on topic-specifity (e.g., D’Amore & Santi, 2021; Radford, in press), but this is usually done by considering particular TS-arguments in geometry, algebra, etc., as illustrative for the CS-investigations or by considering CS-theories as theoretical frameworks for particular TS-investigations. On the other hand, the socio-cultural and socio-political issues, which are at the core of CS-research, are investigated in reference to mathematical knowledge “as a whole” (e.g., Ernest, 2000, 2018; Radford, 2019; Stinson, 2013; Stinson & Bullock, 2012) and this fits the general epistemological viewpoint on topic-specificity assumed in this inquiry by investigating the emergence of MOsMER. But how can the emergence of MOsMER be concretely related to variables discussed in the socio-political, critical, and postmodern approaches? Discussing the technical aspect, the focus was on the formulation of suitable global research questions related toMOsMER. Now the focus is on suitable research problems that show how the emergence of MOsMER pushes topic-specificity into the field of CS research interests. To this purpose, five possible research problems are exemplified.

  1. 1.

    MOsMER represent reifications of modalities used to frame knowledge acquisition in the classroom and are at the same time tools that shape classroom Mathematics because research results influence textbooks, teacher training, and teaching practices (e.g., Parise, 2021). In this sense, such objects acquire a normative value with respect to the interpretation of the learning processes in Mathematics classrooms (Gascón & Nicolás, 2017). Thus, their role in the larger cultural and socio-political context of the praxeology of ME (Godino et al., 2019) and of the practice of MER should be considered as a variable within CS-research. The ontological creativity of MER influences (and implicitly shapes) the broader educational domain and contributes to reify some kind of MOsMER rather than others. This seems to require an in-depth-investigation as research problem within critical theory, with focus on the kind of ontological space that TS-MER supports or eventually represses.

  2. 2.

    Reification of MOsMER leads to a sort of self-referentiality in MER: usually, only scholars in MER possess the theoretical tools for really understanding and discussing the reasons that led to their emergence and thus they alone hold the knowledge about how to manage them, while practitioners (and mathematicians) become ‘laymen’ that need the mediation of the initiated experts. This seems to be unavoidable because of increasing science specialization, but the way TS-reification determines the autonomy of MER as scientific discipline should be considered an issue to be investigated within the context of deconstructive postmodern and critical ME(R) (Skovsmose, 2016; Stinson & Bullock, 2012).

  3. 3.

    In the Philosophy of ME the harmfulness of Mathematics (Ernest, 2018) is discussed. The emergence of education-specifical Mathematics suggests investigating also the “harmfulness” of this kind of Mathematics and its relations to Mathematics as research domain. It would be interesting to explore if there are cases in which education-specific Mathematics had an impact on research in Mathematics, suggesting new research-directions and the impact of such “contaminations” on both these domains of knowledge.

  4. 4.

    MOsMER are strictly related to what Ernest (2002) calls the epistemological empowerment that “concerns the individual’s growth of confidence not only in using Mathematics, but also a personal sense of power over the creation and validation of knowledge” (p. 1). But while Ernest considers students’ and teacher’s epistemological empowerment, MOsMER refer firstly to the Mathematics on which the researcher’s epistemological empowerment is based. In regard to Ethics as first Philosophy in ME(R) (Ernest, 2012, 2021), it would be interesting to investigate how the researcher’s empowerment, based on the reification of MOsMER, impacts on teachers’ empowerment.

  5. 5.

    The reification of MOsMER influences further TS-research practices in MER because the researcher him/herself is not a monolithic fixed being but a multiplicitous and fragmented dynamic entity that evolves over time and space (Stinson, 2013). A research problem within critical postmodern theory could be related to the need to understand if and to what extent the existence of reified MOsMER fosters, but maybe also limits, new approaches in TS-research because of possible cultural and normative hegemonies, and how this could impact on the marginalization of particular groups, such as students with special educational needs (e.g., Lambert & Tan, 2020).

The five examples show concretely in what sense, by considering the emergence of MOsMER, TS-research can be linked to CS-research, answering positively RQ2. Indeed, the impact of the ontological creativity of MER on textbooks, teacher-training, teaching practices, further TS-research-practices, as well as on the researcher’s epistemological empowerment and on the self-referentiality of ME research, moves TS-research from the periphery to the heart of CS-research. This shows in which sense TS-discourses in MER construct and “continuously shape teachers, students, and mathematics as subjects of inquiry” (Stinson & Bullock, 2012, p. 44).

Conclusions

The investigation presented in the article allowed us to detect at least some “puzzle pieces” of a possible way to bring closer to each other TS- and CS-research. The study highlights how topic-specificity, if investigated by “zooming out” of learning-theoretic specificities, is not so far away from the concerns discussed in innovative, ground-breaking (CS) research, developed within middle-range and grand theories. Thus, questioning the second issue from the “Two possible obstacles and the characterization of an epistemological approach in ME research” section, related to epistemological constrains that stem from the distance between TS- and CS-research, allows us to recognize potential ways to face also the first one, related to the lack of attractiveness of TS-research for ground-breaking innovation. Indeed, the five points discussed in the previous section are examples of “more fashionable” and innovative research that considers topic-specificity as the main variable.