Abstract
How does a maths teacher rated as outstanding according to the ‘DocenteMás’ Performance Assessment System of Chile semiotically mediate her seventh-grade students’ learning? With the goal of answering this question, this article used a qualitative, ethnographic case study method to describe the pedagogical strategies of the teacher and her students when covering the thematic unit on numbers and operations in primary school. The results show that the pedagogical strategies emerge from the communicative interaction in the classroom via a local and a global dynamic. With regard to the local dynamic, the pedagogical strategies discovered include exposition, experiential articulation and arithmetic understanding. In the global dynamic, pedagogical strategies of semiotic mediation like metaphors, question series and joint understanding of knowledge stood out.
How is the communicative interaction dynamic between an outstanding maths teacher and her seventh-grade students organized? What types of pedagogical strategies emerge from the communicative interaction dynamic between an outstanding maths teacher and her students that favour learning? Much of the international research on learning interactions has been based on Vygotsky’s (1934) historical-cultural approach to learning and development, in which the focus of study is the communicative interaction between teachers and their students (Howe & Mercer, 2007; Mercer, 1997, 2008, 2010). The focus of these studies was how teachers deploy their discourse to make students learn, which is essential in all educational establishments that aspire to achieve academic success that lasts over time. Many of students’ successes and failures depend not solely on their own efforts or discoveries but also on the forms of culturally situated communicative interaction in the classroom (Edwards & Mercer, 2012). In the historical-cultural approach to addressing classroom learning, it is important to consider how the teacher generates an environment that fosters communicative interaction, connects the subject-matter content with the students’ idiosyncratic experiences and uses pedagogical strategies based on semiotic mediation to foster the construction of meaning (Rogoff, 1998). These new strategies can be understood within the framework of semiotic mediation as sensorimotor reconfigurations of dynamic coordination from enactive and ecological approaches that expand the understanding of educational interaction from the socio-constructivist perspective (Shvarts & Abrahamson, 2023).
During the past 10 years in Chile, the evidence shows that teachers rated as outstanding in the Chilean National Teaching Professional Performance Assessment System (DocenteMás) operate with a wider variety of ways of presenting the content to the students (Cornejo et al., 2011). It is also known that the traditional discourse of the maths teacher is geared at training skills and doing routine processes, which inhibits the reconstruction of the knowledge and concepts generated by the students, as well as the opportunity to ground, question and link different strategies that guarantee meaningful learning (Araya & Dartnell, 2009; Preiss et al., 2011). However, the majority of these studies have been conducted by analysing brief class segments (between 10 and 15 minutes) in which many of the interpretations and inferences made reflect only part of the class session. Considering this evidence, in this research we studied one teacher rated as outstanding in maths (according to the Professional Teaching Assessment System) in interaction with a group of seventh-grade students during eight complete class sessions. The purpose of the study was to understand the pedagogical strategies that emerge from the classroom interaction while covering content on numbers and operations. The relevancy of this study lies in the fact that it analyses a longer time segment, which allows for more regular observation and foregrounds the classroom ecology in order to understand the pedagogical strategies of an outstanding teacher within her context.
Theoretical framework
Dialogic learning interaction between teachers and students
The study of learning based on classroom dialogue, meant as a discursive web, is extraordinarily important for education, given that it enables us to understand how teachers use their discourse in contexts of learning interactions to co-construct learning with their students (Mercer & Howe, 2012). Language as a cultural tool is also very interesting for researchers who study communication in classroom settings, who have described how teachers can mediate their students’ learning to help them understand and perform their tasks using linguistic scaffolding tools such as questions, feedback and explanations of the structure of the task (Pratt, 2006).
At school, this semiotic mediation process carried out by the teacher and students involves a set of psychological processes in constant transformation, which vary according to the type of discourse with others, the communication with oneself and the sociocultural context (Fossa et al., 2016). This semiotic mediation process is prompted during communicative interaction in the classroom between teachers and students via the use of pedagogical strategies that articulate the relationship between the technical language of the curriculum and students’ experiential language. Understanding this recursive cycle from public to private, from discourse to thinking, is essential in achieving academic success in the educational process (Mercer, 2010). Within this framework, the educational trends based on understanding school knowledge instead of merely memorizing it suggest that teachers include pedagogical strategies that foster students’ meaningful learning through the use of semiotic resources like metaphors and analogies.
Mercer (1997) is a pioneer in studying classroom interactions and pedagogical strategies, as he began by analysing how teachers guide the construction of their students’ knowledge. Within the Vygostkian tradition, Mercer and Littleton (2007) have researched classroom communication and distinguished three different levels of analysis in the real communication that occurs in any interactive educational activity. The first level is linguistic, where what is being studied is the speech acts of students or teachers. The second level is psychological, referring to how the teachers interact with the students or the students with their peers during class. The third level is cultural, emphasizing the context in which the communicative interaction takes place (Samuelsson, 2010).
Many of the findings in studies of communicative interaction have reported that the pedagogical strategy that teachers use the most to make their students learn is systematic interrogation. This strategy is considered an important semiotic resource to favour higher psychological processes, in which intra-mental action becomes public and, once shared, becomes private once again (Howe, 2009). Additionally, Scott (2008) claims that when they encourage students to discuss their responses, teachers are promoting reflective actions that guide them to creating questions, which in turn guide the construction of new knowledge on the learning goal at hand, as well as monitoring of their own comprehension processes. Constant interrogation about how and why students use either strategy is essential so that they visualize their own perspective on the knowledge target being discussed and thus are able to effectively guide their future actions to fulfil the objectives of the task (Mercer & Dawes, 2008).
In a similar vein, Mercer and Howe (2012) found that teachers who use more interaction strategies based on open-ended questions contributed more to their students’ participation in class and got significantly better learning outcomes over time, unlike teachers who do not use them. These researchers primarily showed that teachers who use interrogation to encourage their students to provide reasons and explanations for their answers did more to favour the construction of their learning. And unlike other teachers who were only concerned with checking correct answers, they did this without encouraging an exchange of impressions about the target of study (Howe, 2010).
Semiotic mediation in the study of learning interactions in mathematics classes
Bearing in mind that mathematics is a cultural product and therefore emerges from interaction and the need for a social group to meet the demands of its environment, teaching maths implies a process of didactic transposition in which the teacher has to use strategies to transform technical knowledge of the discipline into teachable knowledge in the curriculum (Chevallard, 1998). However, transforming mathematics into teachable knowledge is not a guarantee that it will be learned, due to the fact that the teaching–learning process must also consider ‘how interpsychological functioning may be structured such that it maximises the growth of intrapsychological functioning’ (Wertsch, 1985, p. 87). To do so, a set of pedagogical strategies must be used that goes beyond developing quantitative reasoning, which evolves in individuals in the guise of schematic cognitive structures geared at objectives (Piaget, 1968). Instead, an entire series of functions situated in the zone of proximal development must be activated, which are socially in line with historical more than natural characteristics. This is justified in that the development is achieved and completed as individuals engage with cultural artefacts as a means of participation in the disciplinary practice (Vygotsky). Teachers are institutionally appointed social agents in charge of promoting students’ development through a variety of pedagogical strategies of semiotic mediation.
Inquiry into pedagogical strategies of semiotic mediation is important for research into teaching and learning mathematics, as it entails understanding the educational processes that facilitate the co-construction of mathematical meanings during class. The early interaction of mathematical dialogue referring to the initiation into logic, like the use of analogies between children and their parents and educators, solidifies the development of the literacy that is essential for school learning (Ryve et al., 2013). Literacy is considered fertile ground for learning mathematics in that it entails a socialization process in which the students are immersed in a world where the use of the different sign systems presented by the teacher becomes the fundamental basis for solving the mathematical problems and exercises that frequently appear in maths classes (Mercer, 2008).
This socialization process initiated at both home and school emerges from the reconfiguration of the modes of knowledge learned in formal and informal educational interactions. Preiss (2009) claims that learning mathematics at school consists of a process of representational re-description, which emerges from the transformation of action schemes into new sign systems that are learned in the classroom. This means that the sensory-motor experiences used for manipulation and visualization goals are enlisted as a foundation of new forms of abstract thinking.
This is consistent with the cognitive approach proposed in the curricular foundations of maths education corresponding to Bruner’s (1984) Concrete-Pictorial-Abstract method. In turn, Videla et al. (2018) claim that the configuration of mathematical meaning emerges at an early age and is distinct from representational re-description, given that representations obey contentious receptacles of experiences instead of being the result of dynamic ecological interactions between the organism and the environment. One interesting example of practical pedagogical strategies from this perspective is enactive metaphors, which operate as cognitive vehicles that articulate the thematic subject of the curriculum and students’ experience (Soto-Andrade, 2014). Pedagogical strategies of semiotic mediation based on metaphors are not merely rhetorical analogies but are grounded in our perceptive-motor system and the way our bodies inhabit the natural and sociocultural world (Díaz-Rojas et al., 2021). Something similar occurs with exposition, which uses nonverbal resources like gestures and signals to provide signs that maximize the meaning of what is being presented.
In Chile, the evidence from the study of pedagogical strategies in maths classes has focused more on mathematical discourse than on communicative interaction (Araya & Dartnell, 2009). Studies on discourse and mathematical thinking in Chilean classrooms have revealed that for most of the class time, the maths teacher’s discourse is geared at training skills and doing routine processes, thus inhibiting the mobilization of knowledge and concepts which create the opportunity for students to ground, question and link different strategies that guarantee meaningful learning (Preiss et al., 2011). If the maths teacher’s goal is for students to understand mathematics through a joint construction of meaning, they have to promote communicative processes in which the discourse acts and guides students’ meaning (Radovic & Preiss, 2010).
Within this context, the study conducted by Cornejo et al. (2011) explores the way maths teachers rated as outstanding use a wider variety of ways of presenting the content to the students. Specifically, they reported three ways of presenting a thematic subject: exposition, metaphor and ascription. Exposition consists of presenting a thematic subject via a case or particular situation that enables students to understand what the teacher wants them to grasp, in this case, by exemplifying the subject through different representation formats. Metaphor consists of presenting the thematic subject in a veiled fashion through a formulation which objectively pertains to another subject but addresses the subject at hand figuratively. Ascription seeks to present the subject via a linguistic formulation like definitions formulated clearly and precisely.
The results of that study revealed that semiotic resources like exposition, metaphors and ascription are used as pedagogical strategies to expand meaning and are significantly correlated to the second-grade maths teachers’ performance levels. The teachers who used exposition are the ones who were given an outstanding performance rating in the Chilean teacher assessment system, and they stood out from their peers because they had more than three ways of explaining the same content in a class. Metaphor as a cognitive vehicle is seldom used in general by the teachers rated with competent and basic performance. In contrast, ascription is the way of presenting content that is used the most among the teachers rated with basic and unsatisfactory performance.
These results are interesting, because they reveal the importance of pedagogical strategies sustained on semiotic resources applied in the classroom, particularly the way the content is presented. Even though these studies may provide a general and a specific view of the way teachers, according to their performance assessment, use semiotic resources so that their students can co-construct meanings in mathematics classes, the majority of these studies analyse time segments 10 to 15 minutes long to report on what is happening in the classrooms. Although this is justified by the power of the sample size, these brief segments of analysis often lose the depth of the context in which these pedagogical strategies arise and disconnect the continuous communicative process of brief, partial dialogues from the much more global and articulated whole that stems from the course of interaction in the class. Thus, the sociocultural and ecological richness of these studies is diluted, as they place the emphasis on the pedagogical strategies per se, while ignoring the plasticity of their use according to contexts and instances during classroom interaction. Likewise, the pedagogical strategies based on semiotic mediation analysed for brief periods are taken outside the broader global dynamic that enables their strategic uses in pedagogical practice to be understood, that is, the how and when any given strategy is used based on the dynamic that emerges during the communicative interaction in the classroom.
Given the above, our study’s general objective is to understand the pedagogical strategies of semiotic mediation used by a teacher rated as outstanding in maths education to get her seventh-grade students to learn the content of numbers and operations. To do so, we set three specific objectives: (a) to describe the communicative interaction dynamic between an outstanding teacher and her seventh-grade students which makes the emergence of pedagogical strategies to mediate learning possible; (b) to interpret these pedagogical strategies; and (c) to classify the pedagogical strategies of semiotic mediation between this teacher and her seventh-grade students that contribute to understanding and learning in the classroom.
Methodology
The methodology is a qualitative study that seeks to describe, interpret and classify the pedagogical strategies used in the classroom within a specific school cultural framework, in which a teacher rated as outstanding in mathematics semiotically mediates her students’ learning. The research process involved an inductively constructed analysis based on particular to general themes, in which the researcher interprets the significance of the data (Creswell, 2007).
Type of design
This research is an ethnographic case study. In this type of design, research cases are often delimited by time and activity factors (Creswell, 2007). According to Stake (1999), in many case studies there is no possibility of choice and the case is given. However, in this study in particular, the choice of the case reflected the characteristic of the sample, namely the only teacher who ended up being rated as outstanding among all the maths teachers assessed. This gives the research a fundamental importance, which is not to learn about other cases or general problems but to learn from this particular case.
For this case study, we chose descriptive recording of systematic class observation as a data-collection tool. We are thus affiliated with an ethnographic approach to producing/collecting information. The choice of this methodological design is coherent with what Mercer (2010) posited for studying communication in the classroom, because ethnographic analysis yields a detailed description of the events observed through the researchers’ continuous close participation in the social milieu that they are studying.
The richness of ethnographic observation was achieved by non-participant observation for four months, with visits to two seventh-grade mathematics classes per week, each lasting one hour and 30 minutes. During the observation process, the classes in the thematic units of numbers and operations were recorded. The choice of this thematic unit stems from the fact that it is the most abstract area taught, and therefore many interaction processes are limited to mechanical problem-solving procedures which often push for the syntactic automation of the content over meaningful comprehension of it (Preiss et al., 2011). These pedagogical strategies are generally part of the repertoire of the teachers rated at the basic and competent levels, so observing this scenario with an outstanding teacher may provide relevant information on other pedagogical strategies that foster her students’ learning.
Sample
The sampling procedure was non-probabilistic. Unlike probabilistic criteria, in this case the samples are not random. The type of sampling used is the extreme case criterion, which consists of knowledge of a phenomenon of interest based on unusual manifestations (Miles & Huberman, 1994). The extreme case used here is the outcome of the teaching quality assessment according to the performance category conducted by the Professional Teaching Assessment System of Chile, which establishes four performance classification criteria: unsatisfactory, basic, competent and outstanding. The professional teaching performance assessment system is a compulsory assessment for teachers in public schools and consists of four instruments: portfolio, self-assessment, peer assessment and reference reports by third parties. The outstanding performance category corresponds to ‘professional performance that clearly and consistently exceeds expectations. It tends to be manifested by a broad repertoire of behaviours regarding what is being assessed, or by pedagogical richness above and beyond fulfilment of the aspect assessed’ (DocenteMás, 2018, p. 1). Given that the focus of this study was on outstanding teachers, we considered all the teachers that met this requirement in the teaching assessment from the previous year. Out of a total of 44 maths education teachers assessed in the región de Coquimbo, Norte Chico zone of Chile, only one outstanding maths teacher was found.
The school where the research was conducted belongs to the local education service and depends on the government. The socioeconomic level of the students is lower-middle class. The year observed is seventh grade, and the class had 24 students, 11 females and 13 males. The students’ mean age was 12 years and five months. The teacher studied was 30 years old and had six years of professional experience, three in a private subsidized school and three at her current school. Her university degree was in primary education with a minor in maths education.
Information-collection techniques
The classes were recorded using a recorder and an observation log that provided us with a primary document on the communicative interaction between the teacher and her students. Nonparticipant observation was used to collect the information. To carry out the observation, no observation template that specified what to observe a priori was used, but instead an open-ended observation log focused on communicative interaction between the teacher and her students, particularly on the use of pedagogical strategies and the way the students understood and learned. The same ethnographer recorded all the classes twice a week. The corpus chosen for the analysis is the thematic unit on numbers and operations, where the learning objective pursued by the teacher in eight class sessions consisted of: (a) multiplying natural numbers by decimals; (b) multiplying decimals by decimals; and (c) solving problems which involve multiplying decimals by decimals.
Analysis procedure
The analysis adopted an inductive model of fragmentation, reduction and classification. We began by fragmenting the complete class observation corpuses via inductive codes, which we later related to each other to draw general conclusions via reduction. This means that first a set of codes was created, and after a recodification based on a systematic revision, the most relevant codes were chosen according to the objectives of the study. Finally, the classification was created, which consisted of grouping all the codes into families of codes or categories. However, it is important to note that the inductive model which allowed the aforementioned objectives to be reached corresponds to the designation of categories inherent in any emerging model (Strauss & Corbin, 2002). In this regard, the categories chosen focused on the phenomenon of pedagogical strategies of semiotic mediation in context. That is, they focused on describing and classifying not only the pedagogical strategies per se but also the contextual conditions that made it possible for them to appear during the classroom interaction. To do so, emphasis was placed on the organization of the different class times and the communicative interactions between the teacher and students.
Criteria of scientific rigour
Two criteria scientific rigour were used in this study, namely credibility and saturation. Credibility was achieved inasmuch as the ethnographer collected information through systematic observations and conversations with the teachers and students, which were later recognized by the informants as a credible approximation of what they said and did. Thus, the data recorded which had any doubts or additions were re-examined and clarified after class with the key informants. With regard to saturation, this enabled us to determine first the number of observations needed to avoid reiteration, and secondly whether the codes remained steady over time and could be grouped into more abstract categories. To analyse the corpus, the qualitative software Polimnia was used, which is a CAQDAS based on a model of argumentative complexity.
Results
Below we present the results organized around three questions that enabled us to address the objectives of the study.
How does the context of communicative interaction between the outstanding mathematics teacher and her seventh-grade students foster the emergence of pedagogical strategies?
Based on the corpus of class observations, at the initial level of analysis we were able to identify the pedagogical strategies used the most by the teacher. One of these strategies is stimulating prior knowledge, which consists of eliciting knowledge associated with the students’ experience in the thematic subject at hand with the goal of situating the thematic context of the class. Other important codes were identifying the start, development and end of class, the three main acts which comprise structure of the class which guarantee its organization and continuity, according to the framework of good teaching in Chile. At the start, the content to be studied is contextualized, while the development of the class reflects the operational scenario in which knowledge is put into practice, in this case by solving exercises and problems associated with the content. Likewise, the end of the class was considered one of the main acts in which the teacher ascertained whether her students had learned what they had discussed in class and developed the skills stated in the achievement indicators, as well as glimpsing obstacles to learning and new questions which urged them to open up alternative ways of understanding.
Some of the strategies used the most often at the start of class are metaphor and logical reasoning. Metaphor is meant here as a cognitive vehicle that enables meaning to be transferred from one source or specific domain to an elevated or abstract domain. Logical reasoning, in turn, can be viewed as a series of actions that link together causal relationships and enable conclusions to be drawn based on antecedents. Another important code is formalizing the rule through induction, which stems from logical reasoning and means reaching a general conclusion, like the formula of an operation, based on the partial elements comprising it. Both metaphor and the formalization of the rule exist on different linguistic levels; metaphor refers to the semantic plane while formalization of the rule is associated with the syntactic plane. The dialectical interplay between both planes that the teacher establishes is effective in capturing her students’ interest and helping them to understand the task. To illustrate this, below is an excerpt from the start of class where metaphor and the formalization of the rule emerge as the outcome of a specific presentation related to the students’ experience.
OK, children. Look at the rule. See that the size of the ant is almost halfway between 0 and 1.
Ah, that’s true.
So what quantity is that?
0.4, Miss.
Good. Now let’s assume that we look at this ant with a magnifying glass. What would happen with its size?
It would get bigger.
OK, and if this magnifying glass enlarges the size, how much would an ant that originally measured 0.4 and now is doubled in size measure?
0.8 cm.
Why 0.8?
Because 0.8 is 0.4 times 2.
So, how did you do that?
I multiplied 0.4 × 2.
In the excerpt above, we see how the teacher starts the class by activating prior knowledge using metaphor. Here, the metaphor alludes to the presentation of multiplication as an enlargement of the ant’s size with a magnifying glass. Based on the visual representation of an ant with a magnifying glass and a ruler, the teacher seeks to connect the symbolic and syntactic nature of the operation of multiplying natural numbers by decimals with an element of specific experience that allows for possibilities of meaning. To do so, the teacher asks the child to associate the action of the magnifying glass increasing the ant’s size with multiplication, which could establish the elements comprising the operation according to the position of the ant on the ruler, in this case, the decimal factor corresponding to the initial size of the ant and the numerical factor corresponding to the increase in size. The metaphorical pedagogical strategy implies a process of semiotic mediation that is essential for student learning and is also consistent with the didactic guidelines of the mathematics curricular foundations proposed by the COPISI methodological approach (MINEDUC, 2020).
Metaphor as a semiotic language resource used in mathematics works as a cognitive vehicle that leads to the construction of knowledge from a fertile, experiential realm to a more abstract one (Núñez & Lakoff, 2005). Through the use of metaphor, the children are able to imagine something that is intangible or beyond their direct experience, such as numbers and operations. The teacher's goal is for the students to understand what they are automatizing so they can realize why they are doing what they are doing. This action comes from subsequent interviews and is considered fundamental, given that much of mathematical activity at school when dealing with numbers and operations is disassociated from a contextual framework that makes educational processes relevant to everyday life. In this sense, the teacher makes sure that numbers and operations are not disassociated from the physical objects of the world with which the students interact daily. Because of this, the use of metaphors captures the students’ attention, because it connects with their experiential background and ultimately expands learning opportunities.
The class structure proposed by the Ministry of Education of Chile (MINEDUC), namely the beginning, development and end, is carried out by the teacher to contextualize what the students are learning within what they have done in previous classes. This is evidenced when the teacher starts the class by stimulating prior knowledge through metaphor, a pedagogical strategy of semiotic mediation that fosters a connection between students’ unique experiences and the technical content of the curriculum, as it consists of seeing one thing in terms of another (Soto-Andrade, 2014). In this case, they see the augmented size of the ant through the magnifying glass on a ruler with decimals that are multiplied. Subsequently, the teacher communicates that the class has begun using an induction exercise to put the knowledge into practice. To do so, she uses the trope of interrogation, where she urges the students to formalize the rule of multiplying decimals by natural numbers. After several exercises, she brings students up to the board and asks their seated classmates to justify aloud the decisions that the child at the board is taking, thus facilitating cooperative learning. This latter can be evidenced and understood as the execution of coordinated activities with joint participation to respond to different challenges. At the end of the class, the teacher ensures that both the mechanics of the operation’s procedure and understanding the meaning of the operation have been learned. To do so, she summarizes what they have discussed and asks metacognitive questions that foster students’ self-monitoring of their own learning processes in relation to the learning objective publicly presented at the beginning of class.
How do the pedagogical strategies that the outstanding maths teacher uses foster understanding and learning of numbers and operations in her seventh-grade students?
The pedagogical strategies that were fundamental are thematic recapitulation, the use of teaching resources, metaphorical semiotic mediation, arithmetic conceptualization and experiential articulation. Thematic recapitulation consists of weaving a web of learning by connecting information dealt with in the previous class and linking it with the new information in order to achieve the proposed learning objective. In the case of the teaching resources used, we found the specific, pictorial and symbolic material suggested by MINEDUC’s methodological guidelines for mathematics teaching. The importance of using different materials is that it promotes learning by doing, which is a feature of the new contemporary perspectives on cognition, which replace the sensory-motor development of abstract thinking through sensorial multimodality. In the teacher’s classes, we observed the use of pictorial representations, specific materials, videos and guides corresponding to the mathematics study programme that enrich understanding. Bearing in mind that mathematics is by nature symbolic, it requires the use of specific, pictorial material that serves as a bridge towards abstraction. Especially in the unit on numbers and operations, it is essential to use pedagogical strategies of semiotic mediation that provide the scaffolding needed for the gradual co-construction of knowledge so that the students can gain increasing autonomy resulting from consolidating abstract thinking and sequential reasoning in the domain of operations.
Concomitantly, we also found the pedagogical strategy of arithmetic conceptualization, which the teacher used based on the different definitions proposed by the students in order to co-construct the formal definition of the arithmetic procedures carried out during the classes. This could be viewed as epistemological vigilance in the sense of a heuristic reconfiguration of the students’ interpretations to inform the meaning of the operation’s formal definition based on their own experiences. This action is very important because the teacher uses the pedagogical strategies to generate a sociocultural reconfiguration of the mathematical meaning in which all the students participate in exchanging their own historically co-constructed conceptions to serve the common learning objective. In this sense, the teacher incorporates the students’ experiences through her experiential language, respecting all the definitions suggested yet also mediating each of them in order to co-construct the one that is the most compatible with the formal definition. Another pedagogical strategy found in this study is experiential articulation, which refers to the way the teacher connects the content to be discussed with the students’ experiences around the thematic subject being presented. This is due to the fact that experiential articulation allows the meaning that each student communicates to be woven together with what the teacher says. The teacher does not detract importance from the proposed ideas if there are mistakes in them, as she reminds all the students of what they already know and makes this same knowledge transform into a new meaning. Here, the thematic subject emerges in the constant transformation of the social fabric of the content of the class.
The configuration of the communicative interaction based on pedagogical strategies that the outstanding teacher uses in the mathematics classrooms inverts the traditional logic of classes, where the automation of arithmetic procedures takes precedence over comprehension of the content. This is fundamental in achieving meaningful learning and a significant feature of the pedagogical performance of an outstanding teacher. This communicative interaction also promotes an atmosphere that is propitious for class learning in which the teacher remains alert to her students’ questions or needs, offers equitable opportunities for her students to participate and promotes collaboration among them. It is important to highlight the co-construction of knowledge based on the students’ responses and results. Below is an excerpt from the corpus that illustrates this.
How much is 0.15 × 0.2?
0.30
How do you know it’s 0.30?
I multiplied it like we just did. 2 × 0.15.
What about the 0 before the 2?
Hmm, that’s true. We’re multiplying a decimal by a decimal. How do you do that?
What do you think? Think about what we did last class.
I know. You multiply number times number, and at the end you bring down the zero and move the point depending on the number of decimals on either side.
OK, so how did you come up with that rule?
I remember last class when you told us about it.
Do your classmates remember this? OK! Let’s solve it together.
Regarding this excerpt, we can establish that the use of the strategy of thematic recapitulation is woven in with the semiotic resource of interrogation. This is clear in the constant communication between the teacher and her students, as well as among the students, which is essential to standardize the criteria of heterogeneous access to the content. It also allows the negotiation of meanings to highlight the ecological nature of classroom communication, in which active, deliberate participation is transformed into a crucial tool to ensure the continuity of meaning. The teacher’s use of successive questions prevents the knowledge presented from being reduced to mere information stripped of meaning. When the teacher uses the expressions ‘What do you think?’ and ‘Think about what we did last class’, she is asking the students to draw from their own learning experiences while also laying out the road map that establishes the thematic continuity between the available knowledge and subsequent knowledge. To do so, it is essential for the students to draw from their long-term memory to dredge up the previous content on the specific use of the arithmetic procedures. Because it is generated by the use of continuous questions, experiential articulation facilitates reflection and self-monitoring of the processes that each student creates based on their sociocultural background. This is revealed in the way the teacher allows the students to participate so that they can use their own experiences to weave and reconfigure meaning in a new discursive fabric, thus allowing their prior experiences with the content to be used and then made public and placed at the service of new knowledge, which then reaffirms or expands upon what they have already learned.
How do pedagogical strategies of semiotic mediation reconfigure the organization of the communicative interaction between an outstanding teacher and her seventh-grade students in maths classes?
Based on the observations made, it is possible to state that the dynamic of the organization of the classroom interaction reveals key two aspects: local interaction and global interaction. Both interactions occur at different points in the class structure: the beginning, development or end. Local interaction refers to particular aspects of communicative interaction and corresponds to dialogues between the teacher and one student based on the questions she asks to clarify doubts, find out individual concerns or promote metacognition, like monitoring their own comprehension processes. Global interaction corresponds to the dialogue that emerges from collective student participation and is usually promoted based on a trope of metaphor and interrogation, which entails a series of consecutive responses and questions from a group of diverse students that lead to the joint construction of learning.
The presence of these types of interactions was dominant during the eight sessions observed; however, the one that is the most predominant is global interaction over local. This is due to the fact that in her classes, the teacher prioritizes the public nature of learning over the private, such that her intention is continuous reflection that leads to the emergence of new answers and new questions. The emergence of answers based on those already established between the teacher and some of the students encourages each child to seek approaches based on their own sociocultural experiences that best link the knowledge already shared, such as with metaphor.
The pedagogical strategies of semiotic mediation which emerged during the local interaction in the eight class sessions were exposition, experiential articulation, explanations and problem-solving strategies. Exposition consists of presenting the content in different formats, accompanied by gestures and movements, which may be concrete, iconic, auditory and symbolic, with the goal of getting the students to somehow understand the thematic subject. This is evidenced during class when the teacher uses resources of semiotic mediation with different ways of representing the content, such as graphic material at the board, exercises and problem-solving from the textbook. Stemming from exposition, explanation resembles exposition or an examination of the why, what for, how and what of the thematic subject. Both exposition and explanation draw from a dynamic of presentational respectivity, when the teacher tries to make the content visible and understandable for all students, especially those who are expressing doubts and making mistakes.
Considering that arithmetic understanding implies the meaning of the content and the proceduralization of the formula, problem-solving is a teacher resource to promote the use of both aspects. When solving problems, students use different strategies during the classes, with the predomination of rote solution of the algorithm, and to a lesser extent heuristics, referring to inventiveness when proposing new ways of solving problems.
Regarding global interaction, the pedagogical strategies used the most are question series, assessing different strategies, joint understanding of knowledge and metaphor. Question series consists of asking questions about the questions that the teacher asks her students so that they all understand what is being presented in a cultural setting which is organized based on the students’ own knowledge. This semiotic mediation resource was the most predominant in the observations made and is also the most important one in a dialogic approach to classroom interaction. Private elaboration and public sharing of knowledge help to consolidate understanding of the content being covered.
Assessing the use of different strategies could be viewed as a positive expression of the diverse ways students respond to the learning activities. This assessment of diversity generates confidence in students who learn more quickly and asks those with greater difficulties to push themselves further, given that the teacher generates an affective environment of respect and tolerance for the different expressions that emerge during students’ collective participation. Regarding the pedagogical strategies of joint understanding of knowledge and question series, they are extremely useful for understanding the conceptual meaning of the operation. This strategy was evidenced in the promotion of an understanding completed by the students who, by the teacher asking them questions, proposed a clarifying and deeper approach to the content being studied through turn-taking.
Joint understanding of knowledge by the teacher promotes constant reflection and deliberate public participation regarding the content being studied, which allows students to understand how they influence their own answers and monitor their own arguments by contrasting them with their peers’ theories. Once this has been done, the teacher provides specific, useful feedback so that the students can improve their learning according to the aspects that they have both mastered and not mastered. To better illustrate the organizational dynamic of communicative classroom interaction, below is an excerpt from the class, the first part of which is a local interaction while the second is a global interaction:
OK, children, let’s pay attention. Look closely: what’s happening with the product? What makes the remainder?
The point is the remainder!
Right. So why? Look at each product [points to the board with the products of 4.2 and 5.76, reaffirming the number of decimals in each of them in relation to the multiplications that led to these products].
Because the numbers after the point have to be considered.
OK. If I have 1.12 × 4.5, how many decimals will the product have?
Three!
Great! Now, to make sure you don’t forget it, let’s come up with the definition together. Let’s do it all together and write it in our notebooks.
You get the product of a decimal multiplied by a natural number by multiplying the factors as natural numbers. The decimal point should be placed in the product so it has the same number of decimals as the factors.
This excerpt illustrates two important aspects of the organization of the classroom learning interaction, as mentioned above. The first refers to the local interaction, in which interrogation operates as a pedagogical strategy of semiotic mediation that facilitates precision in the use of the rule of multiplying decimals by decimals. To do so, the teacher engages in dialogue with the students, who express their perspectives on what they consider essential for finding the product of the multiplication. The second aspect comes in the global interaction, where the teacher ensures the students share their perspectives and automate the arithmetic procedures through a continuous review of each student’s contribution. Here, the teacher pays attention to what each student says, generating a communicative interaction characterized by the flow of an exchange of impressions. Once the teacher collects the different impressions on precisely how the arithmetic procedure is done and makes sure that most of the students understand it, in this case the use of the point in the result, we find that one major feature that was not reported in studies of Chilean classrooms with outstanding teachers is a dialectic play between the semantic and syntactic planes of comprehension of the arithmetic operation, which are used in a complementary fashion through the co-construction of mathematical meaning in conjunction with the students.
Conclusions and discussion
To respond to the overarching objective, the aspects of semiotic mediation that contribute to understanding the pedagogical strategies that foster the organization of communicative classroom interaction for the development of seventh-grade student learning were classified. Generally speaking, we found that the classroom organization dynamic was marked by two types of interaction: local and global. Both were dominant during the eight sessions observed. Local interaction consists of local dialogues between the teacher and a student, while global interaction corresponds to collective interactions. This study’s fundamental findings in terms of local interaction correspond to the pedagogical strategies of semiotic mediation like exposition, explanation and problem-solving strategies. Exposition was carried out by the teacher inasmuch as she used different presentation formats of this thematic subject to ensure that all the students were able to understand it. Stemming from exposition, explanation is another resource of semiotic mediation that serves as discourse while the content is being presented. In the case of global interaction, the pedagogical strategies of semiotic mediation are metaphor, arithmetic understanding, experiential articulation and question series. The latter is an important finding, as it enabled us to establish the way the classroom learning interaction between an outstanding mathematics teacher and her seventh-grade students is organized, relying on an interaction dynamic that goes from the local to the global and vice versa, which was dominant during the eight sessions observed and throughout the entire class structure proposed by MINEDUC. Finally, the emerging aspects that mark the presence of these types of local–global interaction in the classroom are pedagogical strategies of arithmetic conceptualization and joint understanding of knowledge for mathematical meaning.
Regarding this study’s contribution, we would highlight the presence of a complete study of eight class sessions, unlike the vast majority of studies on learning interactions in the classroom, which have been conducted by analysing brief segments of class time lasting between 10 and 15 minutes, in which most of the interpretations and inferences made reflect only a single part of the class. In contrast, although this is a case study focused on two units (eight sessions) examining numbers and operations, it provides a more in-depth picture of particular and general descriptions about how the communicative interaction dynamic in the classroom is organized based on the use of pedagogical strategies of semiotic mediation.
As a way of furthering the richness of this longitudinal case study, in contrast to studies with videographic analyses of brief time segments, we can cite as evidence that the pedagogical strategies used by the outstanding teacher emerge in a particular context in which the communicative interaction dynamic in the classroom prompts strategies of semiotic mediation like metaphor at the beginning of class, exposition and interrogation during the development of the class and joint understanding at the end. These aspects are lost when studying brief time periods because only the strategies per se are described, but they are stripped of the scenario of exchange and sociocultural transformation that the teacher can have and use to mediate the co-construction of mathematical knowledge.
According to Ryve et al. (2013), this aspect is fundamental because it enables the students to reconfigure what they have learned based on their own experiences, assessing their contributions and promoting self-monitoring of their own actions to maximize the meaning of the task. Coupled with the above, the pedagogical strategy of semiotic mediation called experiential articulation was a relevant finding, as it highlights the way the teacher gets her students to weave their own meetings en route to a meaning that is co-constructed through global communicative interaction in the classroom. Hence, the students are capable of articulating the curricular content with its sociocultural underpinning. The relevance of this articulation process is that the teacher weaves a web in which the different contents form a whole geared at producing a communicative interaction between technical language and experiential language.
With regard to our findings about how the learning interaction is organized in the classroom, we found that the communicative interaction dynamic in the mathematics classroom displays two types of interactions, local and global, and that different pedagogical strategies which contribute to learning can be observed in them. In the case of local interaction, we have reported that exposition is a resource of semiotic mediation that appears within the pedagogical repertoire when covering the content. This finding is coherent with those reported by Cornejo et al. (2011) in a national teacher assessment study, showing that exposition is part of the repertoire of strategies of teachers rated as outstanding inasmuch as they seek to expand their students’ construction of meaning through different ways of presenting the content. Unlike that study and stressing the particular depth of this case study over generalizations, we found that exposition does not appear mechanically as a pedagogical strategy of semiotic mediation in outstanding teachers but also depends on the type of interaction, which generally features local interactions in which the goal is to expand on students’ understanding of the operation. This strategy, in turn, is not used in isolation from other strategies that complement it, like interrogation and arithmetic understanding. The way the communicative interaction dynamic promotes the emergence of exposition in the context is sequentially based on non-understanding of the arithmetic procedures.
Unlike the study by Cornejo et al. (2011), which states that metaphors are seldom used by outstanding teachers as a pedagogical strategy of semiotic mediation, in this study this strategy is dominant in much of the observation corpus. In addition to using figurative language, the teacher also incorporates concrete and iconic means of representation that contribute to the transformation of the sensory-motor schema into systems of signs that contribute to maximizing meaning.
Finally, noting this transition from a local dynamic to a global one and vice versa as an outstanding teacher interacts with her students contributes to the sociocultural comprehension of mathematical meaning in a systematic, scaffolded way. The synchronic passages described in the ethnography show the contextual conditions that make the differentiation and scope of the different pedagogical strategies possible when dealing with content related to numbers and operations. Unlike merely instrumental pedagogical practices that tend to be accompanied by the excessive use of formulas and mechanisms to automate procedures that the students have to reproduce, the outstanding teacher in this study emphasizes joint understanding through semiotic resources that highlight the situational nature of the experiences and systematic interrogation. These findings converge with the meta-analysis on pedagogical classroom practices in Chile by Godoy et al. (2016) by revealing key aspects of significant pedagogical practices in mathematics, like sensitivity to academic needs, organization of the class and constant monitoring of what the children are learning, including the teacher’s ability to promote the students’ engagement with their own learning during the course of dynamic interactions that promote the use of a variety of strategies.
In order to further explore and complement this study, we suggest that the number and breadth of similar sociocultural studies could be increased to contribute to taking a situated and non-representationalist approach to understanding the ways teachers get their students to learn different mathematical contents. We believe that future studies should further explore the contextual conditions, both historical and emerging, of teaching practice in order to provide clues for educational organizations to successfully guide improvements in their own change processes. Considering the educational challenges of the twenty-first century, expanding the capacities of teacher professional development through contextual studies is a key factor to consider in subsequent research in this field.