Advances and trends in research on mathematical working spaces: A systematic review

This study addresses the research developed around the theory of mathematical working spaces (MWS). To this end, factors such as researchers ’ country of origin are considered, as well as methodological characteristics, proposed objectives in each study, specific theoretical factors, and the links between MWS and other theoretical perspectives. This systematic review follows the guidelines stated by reporting items for systematic reviews and meta-analyses for documenting each stage of the review. In total, 102 studies were considered from databases including Dialnet, SciElo, Scopus


INTRODUCTION
In recent decades, the field of mathematics education research has undergone the development of diverse theoretical and methodological positions, lines of research, and research communities that contribute to addressing and explaining issues from different points of view (Bikner-Ahsbahs et al., 2015;Laborde, 2007;Lerman, 2020).These contributions reflect the evolution and thematic vitality of the discipline, especially its development in recent years (Gaona & Arévalo-Meneses, 2023).From this context, a growing interest has emerged in carrying out analyses of literature centered on the state of the art of research focused on specific mathematical domains or theoretical approaches (e.g., Cevikbas et al., 2022;Schoenherr & Schukajlow, 2023;Vásquez et al., 2023).The present study considers research developed around a theory known as mathematical working spaces (MWS), whose results are presented in this article.The aim of this analysis is to contribute to a better understanding of the current research landscape in MWS and to inspire new studies.
The theory of MWS is currently in a period of advancement, which leads to the consideration of both theoretical and methodological aspects of research that is underpinned by this framework (Henríquez-Rivas et al., 2021;Kuzniak y Nechache, 2021;Nechache & Gómez-Chacón, 2022).Its main purpose within mathematics education is to describe, understand, and transform mathematical work in educational contexts (Kuzniak et al., 2022).
The beginnings of this theory trace back to the 1990s, with the first works emerging from researchers in France and centered on geometry teaching among primary school teachers (Houdement & Kuzniak, 1996).To this effect, the authors define a conceptual framework that organizes geometry around what they term geometric paradigms (Houdement & Kuzniak, 1999).With the theoretical expansion of Geometric working space to MWS (Kuzniak, 2011), research spread to other domains (e.g., analysis, probability, and algebra), reaching a notable level of collaboration and development by researchers from diverse countries, mainly in Europe and the Americas (Derouet & Parzysz, 2016;Montoya-Delgadillo & Vivier, 2014, 2016).
A particularly important venue for the development of MWS has been Mathematical Working Spaces Symposium, held in various countries beginning in 2009 Panqueban et al. / Advances and trends in research on mathematical working spaces 2 / 12 (Cyprus 2009, France 2010, Canada 2012, Spain 2014, Greece 2016, Chile 2018, France 2022), which has provided space for promoting the role of research in mathematics learning and teaching (Gómez-Chacón et al., 2015, 2017).The two most recent conferences have been organized around four thematic lines: (1) mathematical work and MWS, (2) specificity of tools and signs in mathematical work, (3) genesis and development of mathematical work: the role of the teacher and interactions, and (4) role and use of tasks in mathematical work (Derouet et al., 2023;Montoya-Delgadillo et al., 2019).

THEORETICAL FRAMEWORK Mathematical Working Spaces
In this section, the theoretical framework that underlies the research reviewed in this study is presented, followed by the research questions formulated thereupon.
MWS is a didactic theory that is unique from other approaches currently used in mathematics education (Radford, 2017); it relates mathematical content by intricately combining epistemological aspects of mathematics and cognitive processes of subjects.The richness of this theoretical corpus has facilitated its progress and deeper development, both in theoretical and methodological aspects, which is especially highlighted by the research undertaken in the past several years.MWS theory initially emerged for geometry teaching (Houdement & Kuzniak, 1999); subsequently, it has been further developed in distinct mathematical domains (Kuzniak, 2011).This theory addresses the description, understanding, and educational formation of mathematical work on the part of educational actors to enable working with these phenomena (Kuzniak, 2022).Thus, the objective of the theory is didactic study of the mathematical work in which students and teachers participate to contribute to the understanding of the work done in solving tasks, as well as allowing for the characterization of the routes taken in their resolution (Kuzniak et al., 2016b).
In MWS theory, tasks occupy an important place and, therefore, have important implications for associated research, as they are understood as the medium for solving problems (Kuzniak, 2022).While tasks are not an explicit component of the model, they are understood as activators of the work of the individual in a specific institution (Kuzniak, 2011).For this reason, the study of tasks, their design, and their implementation is a topic that has gained increasing relevance in MWS research.
MWS theory considers the epistemological principles of the objects that are studied within a mathematical domain (e.g., geometry and probability) (Kuzniak, 2011;Montoya-Delgadillo & Vivier, 2016).Likewise, the theory integrates the human (or social) component, which entails considering a cognitive dimension related to the epistemological dimension (Kuzniak, 2022).These two dimensions, termed epistemological and cognitive planes, intend to capture the mathematical content of the domain being studied and the cognitive activity of the individual when they acquire, develop, or utilize those mathematical contents (Kuzniak, 2011).
The epistemological plane has three interacting components: the representamen, associated with a concrete and tangible group of symbols based on the interpretations and relations constructed by the individual; artifacts, such as drawing tools, software, or a symbolic system, utilized as an instrument for action; and the referential, a theoretical reference system corresponding to definitions, properties, and theorems.The cognitive plane is organized around three processes: visualization, related to the deciphering and interpretation of signs; construction, a process based on the actions triggered by the artifacts utilized and the associated usage techniques; and proof, a process understood as all discursive reasoning that allows the formulation of deductively-organized arguments, definitions, hypotheses, and conjectures, along with enunciating counterexamples, with the support of the referential (Kuzniak, 2022;Kuzniak et al., 2016b).The connection between these planes occurs through semiotic, instrumental, and discursive geneses, which enable coordinating and specifying the nature of mathematical work in diverse educational and institutional contexts (Kuzniak, 2011).
Semiotic genesis represents the relation between the mathematical object and the cognitive process of visualization that gives it meaning.In instrumental

Contribution to the literature
• The results of this review reveal how MWS is utilized for the study of mathematical work of students, future teachers, in-service teachers, and teacher educators in distinct school levels and contexts.• One contribution of this research is the identification of research trends, productive ideas, and future lines of research from the perspective of MWS.• Another contribution of this study is related to expanding the possibilities of proposing research that considers theoretical connections between MWS and other perspectives.
EURASIA J Math Sci Tech Ed, 2024, 20(6), em2450 3 / 12 genesis, artifacts become operational through the construction carried out by the individual.Discursive genesis relates the referential and the proof process.Thus, in the resolution of a given task, these three geneses and the relations among the components of the planes can interact to give meaning to the mathematical work of the individual.Coutat and Richard (2011) and Kuzniak and Richard (2014) recognize the concept of vertical planes, understood as the interactions between two geneses and the associated components.In these interactions, three different vertical planes are identified: using artifacts in the construction of results under certain conditions or in the exploration of semiotic representations (Sem-Ins vertical plane); the process of proof or the validation of a construction based on experimentation using an artifact (Ins-Dis vertical plane); and coordinating the process of visualization of represented objects with validation logic (Sem-Dis vertical plane) (Kuzniak et al., 2016a).The relations among the planes, components, geneses, and vertical planes are illustrated in the following diagram (Figure 1).
In this manner, MWS research is based on studying and understanding the dynamic of mathematical work through the role of each of these geneses and their interactions (Kuzniak, 2018), which is termed circulation in MWS (Montoya-Delgadillo et al., 2014).Thus, MWS, with its planes, components, and geneses, is a methodological and analytical tool used to identify the different phases of the process of problem-solving and to describe the evolution of mathematical work (Kuzniak & Nechache, 2021;Nechache & Gómez-Chacón, 2022).In the analysis of circulations, changes between two mathematical domains can be identified (Montoya-Delgadillo & Vivier, 2014), as well as barriers, difficulties, and restraints (Henríquez-Rivas & Montoya-Delgadillo, 2016;Kuzniak, 2022) and other cases in which varied interactions exist, which is known as complete mathematical work (Kuzniak et al., 2016a).
Finally, three types of MWS are distinguished in the theory depending on users, their position in a school institution, and their role in the implementation of the school curriculum (Gómez-Chacón et al., 2016a, 2016b, 2016c).These include the referential MWS, related to people or institutions responsible for the school institution in accordance with mathematical criteria (Montoya-Delgadillo & Reyes-Avendaño, 2022); suitable MWS, understood as the mode in which mathematical contents developed by a teacher or researcher are designed, adapted, and presented for teaching in a given place and context (Henríquez-Rivas et al., 2021, 2022); and personal MWS, linked to the reality of students' work when they appropriate and maneuver problemsolving (Menares- Espinoza & Vivier, 2022).

Research Questions
To evince how the accumulated progress and deepening of research on MWS has been achieved during the last few decades, the present study is oriented by the following research questions: 1. How are MWS studies distributed in terms of questions of methodology, such as authors' country of origin, year of publication, and type of document?
2. In the same sense as question one, how are MWS studies distributed in terms of research design, participants' education level, and data collection techniques?
Subsequently, the following questions are intended to delve deeper into contents of MWS-based research: 1. What research objectives are addressed in MWSbased studies?
2. Which theoretical and empirical aspects have been considered by researchers in their analyses?

METHODOLOGY OF SYSTEMATIC REVIEW Search Strategies & Selection Criteria
To determine the research trends in MWS theory, a systematic literature review was conducted, which is understood as "a review of existing research that uses rigorous, explicit, and responsible research methods" (Gough et al., 2012, p. 6).This literature review followed the guidelines stated by reporting items for systematic reviews and meta-analyses (PRISMA) protocol (Page et al., 2021) for documenting each stage of the review.
The final search for information was carried out on 9 August 2023 on Dialnet, SciElo, Scopus, and Web of Science databases.To identify relevant studies within the field of mathematics education, we employed the following search strings (for English): "mathematic working space" OR "mathematical working" OR "geometry working space" OR "probability working space".For French, search string was translated, and a query was executed across all four databases.For Spanish, searches were conducted in Dialnet and SciElo.
The search encompassed all documents with research on MWS theory, and restrictions were not established regarding education level or year of publication.Likewise, no automatic filters were applied in the corresponding databases.The inclusion and exclusion criteria are described in Table 1.
The selection process was carried out in three stages following the approach proposed by Page et al. (2021): identification, selection, and inclusion.In the identification stage, the search strings were utilized, returning a total of 453 articles.The results obtained from each database were imported to a Microsoft Excel spreadsheet and grouped to eliminate duplicates, leading to the exclusion of 158 duplicate results.In the selection phase, the titles, and abstracts of the remaining studies (295 studies) were reviewed, which led to 148 potentially relevant results being chosen (147 were not related to MWS).To finalize the selection stage, the inclusion and exclusion criteria were applied to the 148 studies in question.Lastly, in the final stage, 105 studies were included; however, it was not possible to obtain three of these documents, so ultimately the analysis included 102 studies.Figure 2 shows a flowchart that summarizes the study selection process.

Data Analysis
A complete analysis of 102 studies was undertaken using a coding procedure established based on the   During this process, methodological and theoretical elements characteristic of MWS theory were considered.In terms of methodological elements, the categories defined in the systematic review by Cevikbas et al. (2022), which is related to mathematical modeling skills, were examined; these include year of publication, type of document, geographical distribution of authors, research type and design, participants' education level, sample size, and data collection and analysis methods.Regarding theoretical categories, the factors analyzed included mathematical domain, types of MWS studied, the type of task addressed, and any linkage with other theories.This analysis provided an understanding of research on MWS theory and its diverse methodological and theoretical aspects.

RESULTS OF SYSTEMATIC REVIEW
Below, the results of the review of MWS theory in mathematics education are presented.They are organized into two groups in relation to the five research questions posed in the second subsection of the theoretical framework.The first group covers question 1 and question 2, which address methodological issues.The second group considers questions 3, 4, and 5, which address questions and objectives in MWS research, aspects related to the theoretical framework, and connections with other theoretical perspectives.

Analysis Group on Methodological Issues
First, general aspects are addressed, such as authors' country of origin, year or publication, and document type, along with certain methodological questions that aid in the understanding of how MWS theory has been researched and utilized in different contexts and education levels.

Document types of years of publication
This analysis included 102 studies of which 76 were empirical and 26 theoretical.These studies were distributed across different document types, as follows: 70 are journal articles, 29 are conference proceedings, two are book chapters, and one is a thesis.Among the articles, 53 are published in journals specialized in mathematics education, while five are published in science and mathematics education journals, six in education journals, five in interdisciplinary journals, and one in a technology journal.In terms of conference proceedings, all studies come from conferences on mathematics education.

Geographical distribution of authors
Authors who have utilized MWS theory are located in different parts of the world, and their institutional affiliations reflect a wide geographic distribution (Figure 4).Examining the affiliations of all authors recorded in the publications analyzed reveals that they originate from a total of 15 countries, eight of which are from the Americas and seven from Europe.The countries with greatest representation in the publications reviewed are Chile (23.0%),Spain (21.0%),France (16.7%), and Mexico (14.3%).The other countries recorded comprise less than 7.0% of the total.It should be noted that the results show that 81.0% of the articles analyzed have less than three authors, while the maximum number of authors was six.

Research design & education level of study participants
The analysis showed that, of the 76 empirical studies, 69 correspond to qualitative research, and three to quantitative research, while four combine both qualitative and quantitative methods, but without explicitly being mixed-methods research.Meanwhile, 26 of the studies reviewed are theoretical studies.In terms of the methods utilized most frequently in empirical studies, 24.0% employed case study research, while 6.0% used didactic engineering as a methodological approach.
In terms of participants in the studies analyzed, four studies involve primary school students, 25 involve secondary school students, eight involve university students from non-teaching-focused programs (e.g., mathematics or engineering majors), 15 involve preservice mathematics teachers (four primary and 11 secondary), 16 involve in-service mathematics teachers (four primary and 12 secondary), and eight utilize mixed samples of primary and secondary school students, university students, and pre-service mathematics teachers.This distribution is illustrated in Figure 5.
Table 2 shows the sample sizes (or numbers of participants) utilized in the studies analyzed.Half of these studies (n=52) use samples with fewer than 50 participants, 11 studies do not mention sample size, and five studies use sample sizes that range from 51 to 100 participants; meanwhile, five studies use samples ranging from 201 to 500 participants.

Data collection methods
The analysis revealed that 79 of the 102 studies utilize a range of one to seven distinct methods of data collection.More specifically, 83.5% of the studies employ between one and three data collection methods.The most used method, employed in 40.2% (n=41) of studies, is the application of tasks or problems, while 27.5% (n=28) rely on video recordings, and 21.6% (n=22) utilize interviews.The remainder of methods reported are used in less than 10.0% of studies, as shown in Figure 6.

Analysis Groups on Theoretical Topics
Below, results are presented in relation to the objectives proposed by the studies, theoretical aspects specific to MWS, and connections between MWS and other theoretical perspectives.In particular, in terms of theoretical topics, attention is focused on mathematical domains, tasks, and types of MWS.

Research objective addressed by studies
The findings demonstrate that the most frequent research objectives include the following purposes: analyzing (n=21), characterizing (n=14), exploring (n=6), and understanding (n=6).
The studies centered on analyzing tend to address diverse forms of solving a task or problem, examine the work of an expert solving a problem in the context of referential MWS, or analyze MWS when students or teachers solve a task.Meanwhile, those studies that seek to characterize specifically describe the mathematical work, either of teachers or students, when solving a task or problem.Regarding the studies with objectives focused on understanding, the majority seek to understand a mathematical concept through MWS theory.A synthesis of the objectives across all studies analyzed is presented in Table 3.

Mathematical domain
In this review, the mathematical domains studied based on MWS theory have also been examined.Table 4 shows that 65.0% of the studies analyzed are focused on one mathematical domain: 31.4% of studies are centered on the domain of geometry, nearly 5.9% on algebra, and approximately 9.8% on arithmetic and analysis, respectively, while fewer studies explore statistics, and probability.Meanwhile, approximately 19.0% of studies analyzed are concerned with two mathematical domains in tandem; the pairs of domains most studied are algebra and arithmetic (n=3), algebra and geometry (n=3), and arithmetic and geometry (n=4).Of the 19 studies that analyze two domains, only 12 explicitly address the change from one domain to another in their analysis.

Types of mathematical working spaces & tasks
Regarding types of MWS, 57 studies were found to focus on the personal MWS of the participants of which 54 are empirical studies and three are theoretical studies (Figure 7).Meanwhile, the suitable MWS is present in 18 studies, with 17 corresponding to empirical studies and one to a theoretical study.Some studies analyze two types of MWS, including personal-suitable (n=1) and referential-suitable (n=4).No theoretical and empirical studies were found that analyze all MWS types.Moreover, 22 of the studies are theoretical in nature and do not address any specific type of MWS.
Meanwhile, in terms of types of MWS and mathematical domains, the personal MWS is utilized most to analyze the geometric domain (n=19) and the arithmetic domain (n=10) in the studies reviewed.The other domains are represented in three studies or less in relation to this type of MWS.Suitable MWS has been studied in relation to the geometric domain (n=6) and in relation to other domains to a lesser extent, with three studies or fewer.Additionally, of the 12 studies that analyze a change of domain, 10 are of empirical nature.Two are centered on the suitable MWS and eight on the personal MWS, while two are theoretical studies.

Mathematical tasks
In this review, the task types utilized in studies related to MWS theory were also analyzed.48 studies employed unpublished tasks, which means that the authors designed their own tasks considering their specific studies and taking their research objective into account.In 13 studies, the tasks were proposed by the teacher in the context of teaching with examples, class activities, or evaluations.In eight studies, the tasks were drawn from existing research, which implies that tasks were utilized that had already been developed in other publications.In five studies, tasks from textbooks were used, both K-12 and university level.Furthermore, two studies combine the categories mentioned above.This data is exhibited in Table 5.

Links/connections with other theories or perspectives
During the review, several studies were found in which MWS theory is linked to other theories or perspectives in mathematics education research.Those, which were most often utilized in connection to MWS to analyze data included mathematical modeling (n=9), mathematics teachers' specialized knowledge model (MTSK) (n=8), activity theory (n=4), affect in mathematics education (n=3), and problem-solving (n=3).Some studies also address other theories of mathematics education, but in lesser measure.This data is summarized in Table 6.Regarding the first research question, the results indicate that the majority of authors of the studies reviewed are from Chile, Spain, France, or Mexico.In this sense, it should be mentioned that participation in joint international projects (e.g., French-Chilean ECOS-Sud C13H03 project) have contributed to developing and deepening understanding of important theoretical aspects of MWS (e.g., Montoya-Delgadillo & Vivier, 2016).Also, proceedings of MWS Symposium and issues dedicated to theory in prestigious mathematics education journals have had a strong impact on number of documents found and analyzed in this review.

CONCLUSIONS
From a methodological perspective, the results reveal that nearly 26.0% of studies analyzed focus on theoretical aspects of MWS, and 69.0% are carried out in qualitative research frameworks.It is important to highlight that a large quantity of these studies is based on case study research.Meanwhile, the majority of participants were secondary school students and inservice mathematics teachers.In terms of data collection techniques, the most common were the application of tasks or problems and video recordings.
The results suggest that an opportunity exists for future studies to consider qualitative research focused on education levels including early childhood, primary, and adult education, where mathematical work can be strengthened based on teacher education and the efforts of teacher educators.Meanwhile, quantitative research approaches have been less utilized from this theoretical perspective, and they could be considered, for example, in mixed-methods research or projects (Creswell & Creswell, 2023) in which MWS theory could contribute not only to describing or characterizing mathematical work, but also to understanding and transforming MWS of diverse actors (teachers, students, investigators) in an educational context (Kuzniak et al., 2022).The latter point in particular (transforming MWS) stands out as a key opportunity for future research.
In relation to types of research objectives addressed in the scope of MWS theory, analyzing and characterizing are the most frequent.This underscores the diversity of analysis that has been carried out sustained by MWS theory over time.Findings illustrate a wide panorama of the advances made in this field and offer orientation toward future research that seeks to delve deeper into the process of teaching and learning mathematics through MWS theory.Furthermore, this theory has allowed for analytical and methodological models to be proposed for analyzing mathematical work based on distinct perspectives, levels, contexts, and participants.
In terms of specific aspects of the studies reviewed, one important point is that the domains that have been favored thus far are geometry and arithmetic.These findings point to the necessity of considering studies in other mathematical domains, as well as greater focus on changes of domain (Montoya-Delgadillo & Vivier, 2014).Regarding types of MWS, it is evident that referential MWS has been the least studied, which opens possibilities for study and exploration from a theoretical and methodological point of view.In this sense, future research that focuses on the referential MWS of teachers at different education levels could contribute to the formulation of improvements in mathematics teacher education in terms of curriculum, teaching organization, and the design of signature tasks and materials available for students and teachers (Kuzniak, 2022), all of which could fall within the purview of researchers utilizing this perspective.As observed in Table 5, 31 of the 102 studies reviewed were developed in connection with other theories.Of these 31 studies, it is noteworthy that nine utilize mathematical modeling, while eight also consider MTSK model (Carrillo et al., 2015).Some of these examples are developed explicitly from the perspective of networking theories (e.g., Verdugo-Hernández et al., 2022).Without a doubt, this research perspective merits further exploration.
Concerning the limitations or restrictions of the present study, it is important to mention that the exclusion criteria of certain databases may have resulted in the omission of studies relevant to MWS theory; for example, only one doctoral thesis based on MWS theory was considered, and the proceedings of conferences such as RELME or CERME have not been accounted for.Moreover, the lack of access to some documents could have limited the inclusion of certain studies in the analysis.In future research, manual searches could be carried out that consider other search terms or the detection of studies published in journals whose indexation is distinct from that considered in the present review.But this systematic review largely reflects advances and developments that MWS theory has undergone both in recent years and since its beginnings.
Ultimately, this study presents a systematic review of the current state of existing literature built upon the framework of MWS theory, demonstrating an expansive body of research that could contribute to further research in this theoretical line.

Figure 2 .
Figure 2. Diagram of process of identification & selection of studies (Source: Authors' own elaboration) The majority originate from Symposium on Mathematical Work (n=26), while others are proceedings from International Group for the Psychology of Mathematics Education (PME) (n=2) and the conference organized by Sociedad Española de Investigación en Educación Matemática [Spanish Society for Research in Mathematics Education] (SEIEM) (n=1).Regarding the years of publication, the studies reviewed were published between 2006 and 2023, as shown in Figure 3.It is important to highlight that in 2014, Revista Latinoamericana de Investigación en Matemática [Latin American Journal of Research in Mathematics] (RELIME), dedicated two issues to research on MWS theory.Likewise, this occurred with journals Bolema (Boletim de Educação Matemática [Mathematics Education Bulletin]) and ZDM-Mathematic Education in 2016.Meanwhile, in 2015, proceedings of 4 th Symposium on MWS Theory were registered.

Figure 3 .
Figure 3. Distribution of studies based on year of publication (Source: Authors' own elaboration)

Table 1 .
Inclusion & exclusion criteria Inclusion criteriaExclusion criteria IC1.Studies at all levels of mathematics education.EC1.Studies in a discipline other than mathematics education.IC2.Studies addressing MWS in the theoretical framework or introduction.EC2.Studies that do not address MWS in the results.IC3.Studies published in English, French, and Spanish.EC3.Studies published in languages other than English, French, or Spanish.IC4.Document types: articles, conference proceedings, theses, and book chapters.

Table 2 .
Sample size/number of participants reported in studies Sample size/participants Frequency (n) Percentage (%) Figure 6.Data collection methods (Source: Authors' own elaboration)

Table 3 .
Objectives in MWS investigations

Table 4 .
Domains analyzed in studies

Table 5 .
Task types reported in studies

Table 6 .
Theories linked with MWS in studies