1 Introduction

In the twenty-first century, the use of mind maps in education has had an upturn with the rise of multimodality and technology. Trends related to 21st Century Competencies often present technology, collaboration, and visualization as mutually related components. For example, the Ontario Government (2016) in their document Towards Defining 21st Century Competencies for Ontario defines clear connections between digital tools and resources, key transformational learning practices, and competency development. The document mentions graphing tools and concept mapping tools as technologies that can foster, amongst other competencies, coordination, communication, metacognition, analysis, problem-solving, and reasoning.

Mathematics education does not escape these trends in multimodality and technology. As Hoyles and Noss (2009) explained, “the very need for remote communication of mathematical ideas—either synchronous or asynchronous—provides a motivation to produce explicit formal expression of mathematical ideas” (p. 141). In teacher education, Gadanidis and Namukasa (2013) noted that the affordances of new media help preservice teachers to communicate mathematics in multimodal ways and to see mathematics as a collaborative enterprise.

In this frame of ideas, we have included the use of collaborative mind mapping activities as an alternative to threaded forums in the elementary mathematics education program at Western University. We have done this for over three years in two different courses: a computational thinking in mathematics education course, and a mathematics teaching methods course. In this paper, we present a grounded theory developed from these experiences of collaborative mind mapping. The emerging theory responds to the question How do preservice mathematics teachers construct mathematical and pedagogical knowledge while they interact through three different tools for online collaborative mind mapping (i.e., Popplet, Mindmeister, and Mindomo)?

2 Literature review

To respond to our question, we categorized our literature into two areas: (1) models of online interaction and knowledge construction, and (2) literature about the online learning of mathematical and pedagogical knowledge. Under the first category, many authors have documented the ways in which students construct knowledge in online settings. Particularly, Harasim (2017) and Garrison (2017) developed influential theories that help researchers understand the processes that allow knowledge construction through online interaction. In collaborativism learning theory (Harasim 2017) learning is defined as Intellectual Convergence, the higher stage of a collective cognitive process. This theory involves three stages or phases that students need to go through to achieve learning, namely, Idea Generating, Idea Organizing, and Intellectual Convergence. Additionally, the originator of the Community of Inquiry model (Garrison 2017) describes how teacher and student participants develop roles while engaging in online discussion through three key elements or presences, namely, the cognitive, the social, and the teaching presences.

Particularly applicable to mathematics teacher education, Clay et al. (2012) proposed a model to support the development of mathematical knowledge for teaching (MKT) that consists of a sequence of six activities: (1) reviewing an expert model, (2) creating initial responses to the task, (3) listening to/viewing others’ responses, (4) reviewing and commenting on others’ responses, (5) discussing, and (6) revising initial responses (See also Silverman and Clay 2009).

The aforementioned models have important contributions to understanding and describing interactions in mind maps in terms of variables to observe and a general frame of reference. However, these models do not yield tools for the in-depth analysis of knowledge construction during collaborative mind mapping, which has semiotic possibilities in addition to conversation and writing (arranging items, sizing, highlighting, linking or separating ideas). Based on these semiotic possibilities, we carried out the review of literature under the umbrella of a second category.

In the category of online construction of mathematical and pedagogical knowledge, there is plenty of research on how the use of online technologies and multiple modes of representation is beneficial for mathematics learning. Hoyles and Noss (2009) outlined four categories of digital tools which have the potential to shift the way in which mathematics is taught and learned: “(1) dynamic and graphical tools, (2) tools that outsource processing power, (3) new representational infrastructures, and (4) the implications of high-bandwidth connectivity on the nature of mathematics activity” (p. 129). Documented research from all these categories can be found in volumes such as those by Martinovic et al. (2013) and Heid and Blume (2008).

Particularly, in mathematics teacher education, researchers have explored the potential of particular multimedia applications, such as online videos (LeSage 2013; Llinares and Valls 2009) to improve learning about mathematics and mathematics pedagogy. In the research of Gadanidis et al. (2008), an environment that included multimodal communication and collaboration proved to be helpful in the development of mathematical ideas for preservice teachers. In our own work designing blended learning experiences for mathematics teacher candidates, we have documented the positive impact of online media to help preservice teachers learn new approaches to mathematics pedagogy (Gadanidis and Namukasa 2013), and we have determined the important role of online interaction in the development of mathematics pedagogy ideas such as the low floor, high ceiling approach, sharing mathematics experiences with audiences outside the classroom, aesthetic experiences in mathematics learning, among other approaches (Gadanidis and Cendros Araujo 2017). We have also studied the positive impact of online activities, including mind maps, in the development of computational thinking for mathematics teacher candidates, and determined that in addition to being a source for teacher learning, they also offer models for classroom teaching (Gadanidis et al. 2017a, b).

In a summarizing work by Borba and Llinares (2012) it was determined that online learning environments supported mathematics education in three ways, namely, (1) providing asynchronous collaborative communication interfaces that allow participants to spend more time building their arguments, (2) providing sharing and co-creating tools that allow teachers to compare and share their ideas, and (3) providing student teachers with a knowledge base that help them justify and evaluate their arguments. Another important finding of Borba and Llinares (2012) was that in the revised literature, three factors influenced the way in which participants in interactive learning environments produced concepts related to mathematics teaching and mathematical knowledge, as follows: (1) the topics under discussion, (2) the goals of the learning environment (cognitive scaffolding), and (3) the assigned reading.

Mind maps, which offer a flexible and creative layout, have rarely been explored in previous research in mathematics learning. We set out to develop a theory that describes the abundance of multimodal information contained in online collaborative mind mapping and that interprets the elements of meaning that have significance for knowledge construction in mathematics education. The grounded theory developed in the present study seeks to fill a gap in the understanding of how preservice mathematics teachers construct knowledge when they interact through online collaborative mind mapping.

3 Theoretical framework

The theoretical underpinnings that guided this research are framed in Borba and Villarreal’s (2005) Humans-with-Media concept, in which researchers understand that any kind of new media introduced in the mathematics teaching and learning process, such as a mind map tool, reorganizes human thinking. Reorganization refers to the fact that instead of substituting humans or merely serving their purposes, media act and interact in knowing. Humans and media form part of a collective that thinks and constructs knowledge. In this framework, visualization is a way of accessing mathematical knowledge, and the role of media in visualization goes beyond simply showing an image. “In the case of visualization, what we see is always shaped by the technologies of intelligence that form part of a given collective of humans with-media, and what is seen shapes our cognition” (Borba and Villarreal 2005). Our way of defining mind maps in our study is as visualizations of mathematics education knowledge, constructed in a collective of teacher candidates interacting with one another, and with a mind mapping tool that reshapes the way the collective thinks. The knowledge constructed in these mind maps is distinct from the one developed through other online discussion tools such as threaded forums, in the sense that it incorporates asynchronous possibilities different from those in classroom settings, but also introduces powerful visualizations of collective knowledge.

4 Context and participants

In this research, we implemented a multiple case study (Stake 2005) of collaborative mind mapping, carried out in the undergraduate elementary mathematics teacher education program at Western University. Participants were enrolled in blended courses (using online activities as a support for face-to-face learning), where some of the online activities used collaborative mind mapping for knowledge construction. Students used different mind mapping tools and received different scaffolding techniques in terms of prompts and number of participants per group. Each one of these courses is treated as a case and they are described below. Table 1 shows a summary of the cases and their characteristics.

Table 1 The three cases and their characteristics
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4.1 Case 1

We studied a computational thinking in mathematics education course, which had a duration of nine weeks, 2 h per week, where the five odd-numbered sessions were face-to-face, and the four even-numbered sessions were online. The online component included the collaborative knowledge construction and reflection in small groups (4–7 participants) of mind-maps through the online tool Popplet (popplet.com). Below is the weekly outline of topics for the online weeks of the course: Week 2: Algorithms, coding, and CT in the context of geometry.

  • Week 4: CT in the context of probability.

  • Week 6: CT in the context of patterning and algebra.

  • Week 8: CT and mathematics pedagogy in the context of measurement and number sense.

There were a total of 31 small groups across the five sections. The total number of mind-maps created was 93, distributed as follows:

  • Week 2: 31 mind-maps.

  • Week 4: 31 mind-maps.

  • Weeks 6 and 8: 31 mind-maps.

Prior to each online week, teacher candidates received a link with access to their group’s mind-map, which was initially blank. For weeks 6 and 8 each group used only one canvas, so for week 8 students connected ideas and topics within the mind-map they used in week 6. The prompts used by the instructor to guide participants in developing the mind-maps included an explanation on the use of Popplet, a list of suggested—not mandatory—topics to address in the mind-map, and a video on how to use the tool. The teacher presence in the mind maps was minimal, with only short responses in the mind maps for week 2. Figure 1 shows a sample mind map developed by a group of students in Case 1.

Fig. 1
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Sample mind map created by participants in Case 1 (https://bit.ly/case1-sample)

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4.2 Case 2

We studied the work of a new cohort of students in the computational thinking in mathematics teacher education course. Characteristics of this case were the same as in Case 1 in terms of duration, mode of delivery, and contents. In regard to mind map construction, this case included groups of 8 participants (groups were larger than in Case 1) through the online tool Mindmeister (https://www.mindmeister.com). In this case, only weeks 2 and 4 required mind map construction. As a result, a total of 60 mind maps were created (30 from week 2, and 30 from week 4).

As in Case 1, participants received a link with access to their group’s mind-map and the prompt used to guide the construction was a list of suggested—not mandatory—topics, but this time we included a live presentation including questions and answers on how to use the tool (one for each section), with reinforcement videos made available for students. In this case, the instructor did not participate in the mind maps. Figure 2 shows a mind map created by a group of students in Case 2.

Fig. 2
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Sample mind map created by participants in Case 2 (https://bit.ly/case2-sample)

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4.3 Case 3

In Case 3 instructors decided to include the mind map activity in the mathematics methods course for teacher candidates. This cohort was the same set of students who participated in Case 2. The course had a total duration of 17 weeks, using mainly a face-to-face delivery mode, coupled with an online learning component which included discussion groups and mind maps. The mind map activity was used in weeks 9–12, which covered the following contents:

  • Week 9: Paying Attention to Fractions.

  • Week 10: Fencing the Dog (area and perimeter).

  • Week 11: Paying Attention to Spatial Reasoning.

As in Case 2, this activity included groups of 6–8 participants through the online tool Mindomo (https://www.mindomo.com). A total of 96 mind maps were obtained from this case (32 for each week). Figure 3 shows a mind map created by a group of students in Case 3. Since this group of participants had already used collaborative mind maps, instructors only shared a video about using Mindomo and allowed TCs to create their own mind maps, asking to be invited to view them (rather than creating the blank canvas and sending a link to TCs). The instructors did not participate in the mind maps. In this case, instructors decided to use questions as prompts to guide the activity because the discussion was intended to be more focused on single weekly topics, such as the following:

(a) What are some ways you can classify fractions? What are some fractions that would be part of each classification? (b) A fraction can be defined as a portion or division. Which definition means more to you? Explain. (c) Which is more important: fractions or decimals? How would you convince someone that you are correct? (d) What are some ways fractions and decimals are used in other areas of mathematics?

Fig. 3
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Sample mind map created by the participants in Case 3 (https://bit.ly/case3-sample)

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4.4 The three tools and their characteristics

Generally speaking, all three tools afforded the enhancement of discussion in three ways: (1) allowing participants to create visual connections between mathematics and mathematics pedagogy topics, (2) allowing participants to organize topics and comments visually, and (3) allowing participants easily to include images and videos as a part of the discussion.

However, there were some technical differences among the three tools that affected the knowledge constructed in mind maps. First, the connections of mathematics and mathematics pedagogy between topics were made in different ways depending on the tool. In general, Popplet was the simplest tool, allowing only one type of connection between topics, and only one way to display such connections (a gray line). On the other hand, Mindmeister and Mindomo allowed different types of lines and colour for the connectors, and the possibility of adding text to connectors, which generated more intricate networks of concepts. Second, although the three tools allowed students to add notes or comments to concepts, they were more easily viewed and organized in Mindmeister and Mindomo. Table 2 summarizes some technical differences between the three mind mapping tools.

Table 2 The three mind mapping tools and their characteristics
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5 Method

5.1 Sources of data and sample

The data used in this study consisted of two elements. The first was the set of artifacts (mind maps) created by the students as a final product, which included the students’ texts, images, videos and layouts they used to represent knowledge (examples in Figs. 1, 2, 3). The second source of data was the online records of students’ interaction during collaborative mind mapping, obtained through Mindomo’s, Mindmeister’s and Popplet’s history feature, which allowed researchers to look at the whole process of mind-map construction. Additionally, a visual version of the interaction was obtained by recording the process of mind map construction in the history feature. Examples of this process can be seen in the following URLs:

Of the total number of mind maps constructed in each case, for this study we selected only those for which students had given consent to participate. We were required to delete all comments and participation logs of students who did not give consent to participate. As a result, we used only the mind maps for which all students gave consent to participate, a total of 47 mind maps (out of 93) for Case 1, 25 mind maps (out of 60) for Case 2, and 33 mind maps (out of 96) for Case 3.

5.2 Grounded theory

According to Charmaz (2014), grounded theory methods are a set of “systematic, yet flexible guidelines for collecting and analyzing qualitative data to construct theory from the data themselves. Thus, researchers construct a theory ‘grounded’ in their data” (p. 1). These data are constructed through observations, interactions, and gathered materials, which are systematically examined, coded, and categorized to generate an “analytical product rather than a purely descriptive account. Theory development is the goal” (Hood 2007, p. 154).

The process of analyzing data using grounded theory methods involves coding and categorizing the data to find patterns, using memo writing and theoretical sampling as a part of the process. To help with this endeavor, we used the QSR NVivo 11 software, which was selected from a variety of qualitative data analysis software (QDAS) packages, because it allows a researcher to import and code multimodal sources of data such as the mind maps and interaction videos from our study. The following subsections describe the steps taken to perform the initial, focused, and theoretical coding.

5.2.1 Initial coding

The initial coding in grounded theory helps researchers start to make sense of the data (Charmaz 2014). In this research, coding was started by looking for actions rather than themes, thus, using gerunds to label observable actions performed by participants in the activity. Using the comparative method of grounded theory, researchers looked at each instance in mind map construction, paying attention to newly emerging actions, as well as patterns that repeated from instance to instance. It is important at this point to remark that the initial coding stage was completed using data only from Cases 1 and 2. The reason behind this selection was to allow researchers to later conduct theoretical sampling using data from a new case (Case 3). It is important to note that in constructivist grounded theory, data collection and analysis are conducted simultaneously in an iterative process (Charmaz 2014). This process involves going back to the field to seek pertinent data after an emerging theory has been drafted (theoretical sampling). This procedure allows researchers to elaborate and refine the categories that constitute the theory. When using digital data, which is previously stored and readily available, going back to the field for pertinent data is not possible, so the theoretical sampling needs to adjust to this circumstance (Whiteside et al. 2012). We achieved this criterion by coding and analyzing a subset of the data (Cases 1 and 2), in order to gain a sense of the emerging theory, and then theoretically to sample using the rest of the data available (Case 3). As a limitation of this process, we note that it did not allow us to highlight differences between the three cases, but rather diminish them in our attempt to observe overarching patterns. However, when differences were noticeable, we explained them in our results.

As a trail of evidence, “Appendix” shows all initial codes and frequencies, as generated in the initial coding stage. For a thorough description of each code, the reader may refer to the document in the following URL: https://bit.ly/codebook2019. As can be implied, the capabilities of the three different mind mapping tools supposed an additional challenge for the process of initial coding. For example, the code Using the chat feature could be observable only in Case 2 when gaining a sense of the emerging theory, and in Case 3 for theoretical sampling, but the tool used in Case 1 (Popplet) does not have a chat feature. The codes Coding with colours and Making aesthetic decisions were underrepresented in Case 1 due to the limited range of editing possibilities in Popplet. In these opportunities, we strove to generalize results based on all the editing possibilities of Mindmeister and Mindomo.

5.2.2 Focused coding

During and after the initial coding, the researchers engaged in some focused coding. It is possible to note in “Appendix” that some codes were grouped into some early categories. This grouping helped us define more relevant codes and facilitated subsequent coding. Once the initial coding was finished, we looked more deeply into our codes, code descriptions, and data references to further group our initial codes into focused ones. This process is known as focused coding (Charmaz 2014) or selective coding (Glaser and Strauss 1967). At the end of this stage, four broad categories emerged: building concepts, developing discourse, developing leadership, and expression variations. Table 3 shows the four categories and their composition in initial codes.

Table 3 Focused and initial codes
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5.2.3 Theoretical coding

Theoretical coding is the stage where the grounded theory is built in terms of relating the codes and categories and generating (or raising) a core theme or category (Charmaz 2014). In this process, we used theoretical memos and integrative diagrams to establish relationships between our codes. Theoretical memos were written during the initial and focused coding stages, and in the theoretical coding stage they were analyzed and related to our focused codes. They also helped merge some codes and identify which of our categories (Building Concepts) could relate to all other codes, raising it to core category.

The second tool that helped us determine relationships among codes was the integrative diagram. We created integrative diagrams to relate nodes in each category, and then a larger diagram relating all four categories. Figure 4 shows our main integrative diagram created using Nvivo. Finally, with an emerging theory taking shape, we developed theoretical sampling, a process through which we tested our codes and categories with new data (Case 3), further refining their conceptualization and integration.

Fig. 4
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Integrative diagram showing the relationships between our codes and categories. Created using Nvivo

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We solidified the theory by naming it Knowledge building through mind mapping, which originated from the core category Building Concepts. Using the theoretical memos and integrative diagrams developed through the coding process, we created a narrative account of the theory, which includes the following constructs: (1) Stages of knowledge building through mind mapping, (2) Results of knowledge building through mind mapping, and (3) Expression variations in knowledge building through mind mapping.

6 Results: a grounded theory of knowledge building through mind mapping

The following theory describes specific stages, results, and variations of online collaborative mind mapping in mathematics teacher education. In other words, it describes how preservice mathematics teachers construct knowledge when they interact through online collaborative mind mapping. It is intended as a model to guide educators and researchers through the implementation and observation of visual and collaborative online learning experiences.

While engaging in online collaborative mind mapping, participants follow a straight sequence (Construct 1: Stages of Knowledge Building). First, they introduce topics, which are built upon by adding subtopics. When participants are done adding subtopics, they proceed to make connections to other participants’ topics, and sometimes make small contributions to others’ topics.

There are two byproducts of this process (Construct 2: Results of Knowledge Building): firstly, participants develop discourse by adding comments, asking questions, sharing life experiences, and referring to class activities and resources. And secondly, they also develop leadership by engaging in behaviors such as building a base, making aesthetic decisions, resolving technical issues, giving directions, grouping topics, highlighting, filling or leaving blanks, and by overcoming leadership obstacles.

Participants engage in collaborative mind mapping in varied forms, according to their preference or the nature of the content. We refer to these varied forms of participation as expression variations (Construct 3: Expression Variations in Knowledge Building) and they consist of using the chat, descriptions, images, or videos, all of which may or may not include collaborating in real time. The integration of these three constructs can be visualized in Fig. 4. Leadership and discourse emerge as results of the mind mapping process, while the expression variations constitute a property. Below, we elaborate on the constructs of our theory and discuss how these constructs relate to existing literature.

6.1 Stages of knowledge building through mind mapping

6.1.1 Introducing a topic

Each participant began their session by introducing a new topic related to mathematics or mathematics pedagogy (e.g. types of fractions, area and perimeter, computational thinking, an approach to mathematics education) to the mind map, unless a previous base of main topics was already built by a student leader (which happened in most mind maps). There were some variations in the kinds of topics introduced, which were greatly dependent on the prompt used by the instructor. When using a topics list (Cases 1 and 2) participants were more likely to introduce a topic related to something that stood out from class or from readings (e.g., the growth mind set, mathematical thinking, gaming), or to introduce a question (e.g., what is your previous experience with coding?). However, when using a question prompt, the topics introduced were more likely to come directly from the readings, in some cases even labeling a topic by the page numbers of the reading. Another common way of introducing a topic was labeling with the participant’s name and adding many different topics to it (building an individual mind map inside the larger mind map).

6.1.2 Building a concept

The process of building a concept is generally done by a single student (there is little collaboration at this stage). One participant adds all subtopics on a previously created topic, or on one they just created, and rarely adds to others’ topics. Note in Appendix A that adding or building on (others’) topics was observed only 2.95 times per mind map, which is lower than adding to their own topic (7.53 times per mind map) or adding a comment to other’s post (13.24 times per mind map). So, the most common way of collaborating was not by adding sub-topics, but by adding comments. We discuss this in more detail when we describe Developing Discourse as a result of knowledge building.

6.1.3 Making sense of the whole picture

This stage is where most of the collaboration happens. As stated before, when building a concept, participants tend to stay in their own topic and contributions to other topics are minimal. But when participants finish building their initial concepts, they move to find ways to connect their ideas with others. We named this process ‘making sense of the whole picture’ as this was the purpose of all the connections made. These connections result in a cohesive mind map that shows how collective ideas relate to each other. Other ways in which participants made sense of the whole picture were by arranging the layout, highlighting main topics, or separating sections. However, this was done by only one or two students, if any, for the whole mind map.

Case 3 showed an addition to the process of making connections. Participants would extend the colour they used for topics, to the connections. This means that participants would use their assigned (or selected) colour, which they had used to develop their concept, to make connections. The purpose behind this was visually to demonstrate that their participation was not limited to adding topics and comments, but that connections were also part of their work. An example of this aspect can be seen in Fig. 3. Finally, Cases 2 and 3 had the possibility of adding connecting words to connectors (a feature not available in Popplet—Case 1). However, connecting words were rarely used by participants.

6.2 Results of knowledge building through mind mapping

6.2.1 Developing discourse on mathematics and mathematics pedagogy

In terms of content, the most common themes in participants’ discussions were class activities and resources, and life experiences related to mathematics and coding. The mathematical and pedagogical insights that teacher candidates shared and discussed in mind maps were closely aligned to course goals, readings, and class activities, as well to the philosophical orientation of our mathematics education program in relation to the following: computational modelling in mathematics education, a low floor, high ceiling approach, real world contexts for mathematics learning, aesthetic experiences in mathematics learning, and sharing mathematics with audiences outside the classroom. While the discussion about readings and class activities was expected—since that was the purpose of the mind maps—participants also communicated about life experiences such as experiences and feelings towards coding, past (usually negative) experiences with mathematics pedagogy, compared to the new pedagogical approaches they were learning, experiences from placement, and code developed as class activity or personal practice.

Discourse development is the construct of our theory directly related to mathematical knowledge. Teacher candidates were able to make connections among concepts related to mathematics and mathematics pedagogy using multimodal elements, as evidenced by the presence of connectors and colour groupings of many different concepts (e.g., in a particular mind map participants used different colours to separate specific tools such as Scratch, Arduino, Sphero, and others, from the approaches used to introduce them, such as play, tinkering, and others). However, even though mind maps have many of these visual features that would allow participants to develop a multimodal discourse, students mostly interacted through comments, by attaching notes to topics. Consequently, the prevalent mode of communicating ideas was written discourse. Sometimes, the structure of a mind map resembles a threaded forum, especially when participants construct their mind maps by name, rather than by topics (see Sect. 5.1.1).

In terms of the nature of interactions through comments, they were generally of agreement and support. Disagreement, conflict, or debate were rare (observe the frequency of Disagreeing in “Appendix”). Questioning was more often used as a conversation starter, such as a poll, or when a main topic was taken directly from the question prompt (Case 3). But participants rarely used questions to follow up conversations, and long conversation threads—of three or more back-and-forth comments—were not seen in the mind maps.

6.2.2 Developing leadership

One important characteristic of mind map interaction is that most groups appeared to have one or two student leaders. Each of these leaders seemed to be executing his or her vision of the mind map. In Cases 1 and 2, where the maps were created by the instructor and participants were invited, the first student to access the mind map was generally the student leader. In Case 3, where participants created their own mind maps and invited the instructors, the leader was often the same person who creates the mind map and invited others. Some of the most important tasks that the leader performed were as follows:

  • Developing a base of main topics on which the rest of participants build.

  • Choosing a colour and layout template for the mind map.

  • Deciding the colour that each participant will use for their contributions (in the instances where the group used a colour legend).

  • Developing an initial topic—the structure of which is followed by all other members.

  • Grouping topics together by proximity if they have many connections or common concepts.

  • Highlighting important parts by using shapes, fonts, or colours.

Besides these behaviors that contributed to the construction of a cohesive mind map, other behaviors were obstacles to this goal. In a high number of mind maps (93% as per “Appendix”), one or more participants in the group made their contributions on the day of the deadline. This allowed little time for their topics and comments to be integrated in the whole mind map. In other instances, participants would ignore the rules others were following, such as using the same colour for all contributions of a participant, or filling a topic introduced by the leader.

6.3 Expression variations in knowledge building through mind mapping

6.3.1 Collaborating in real time

Real time collaboration often happened near a deadline, and when it happened, participants worked in different parts of the mind map, staying within their own section. Chat communication in Mindmeister and Mindomo (Cases 2 and 3) showed that they preferred not to use the software at the same time because it would not allow them to undo changes while two or more participants were connected.

6.3.2 Using descriptions

Descriptions added to a topic were heavily used as they were the main vehicle for developing discourse (see Developing Discourse). Descriptions is the only area where students shared life experiences and emotions related to mathematics and coding. Even in the cases where students shared an image related to a life experience, it was accompanied by a description.

6.3.3 Using images

Images are included in mind maps for a variety of purposes:

  • As mathematical visual mediators, for example to include fractions and their graphic representations.

  • To discuss an image that stood out from the readings, i.e., screen captures, photos, or scans taken directly from a class resource.

  • To show a finding as a triggering element for discussion, e.g., an infographic found online about the growth mindset.

  • To share products of their work in class, e.g., a screen capture of a program developed in scratch.

  • To illustrate a concept, as accessory, e.g., a photo of children using a computer to discuss coding in mathematics education.

6.3.4 Using videos

Videos were used only as triggering agents, since participants used this feature only to share web videos that sparked their curiosity or showed something of interest related to mathematics education or computational thinking. This feature was used only in 40% of mind maps, as per “Appendix”.

6.3.5 Using the chat

The chat feature was used rarely (only in 6.67% of mind maps, as per “Appendix”), but when it was used, only organizational aspects were discussed, such as the following:

  • Requirements of the mind map assignment or other class assignments.

  • Task organization, i.e., who should contribute what and when, often with the intention of avoiding using the mind map at the same time.

  • Technical issues.

It is also important to note that the chat feature was available only in two of the case studies (Case 2 and 3) because of the tools’ capabilities: this aspect of interaction was not present when students interacted through Popplet. Even with this differentiation, participants did not transfer organizational aspects of the mind maps (described above) to any of the other variations of expression.

7 Discussion

7.1 Stages of knowledge building through mind mapping

Participants in our multiple case study engaged in a process of knowledge building in accordance with the ideas of Bereiter (2002), who, as a part of his connectionist model of the mind, defined knowledge as connections made through common goals, group discussion, and synthesis of ideas. The visual affordances of mind maps allowed for viewing, linking, and manipulating ideas, which are functions that contribute to collaborative knowledge building, in a way that threaded discussions cannot support (Scardamalia and Bereiter 2003).

According to Borba and Villarreal (2005) the affordances of new media not only change how students perform, but also reorganize how they think. In this way, mind maps were an agent that reorganized the ways in which students thought about mathematics, computational thinking, and mathematics pedagogy. Instead of describing or commenting on different aspects of readings and class activities, as is usually done in threaded forums, mind maps made students abstract the main topic or idea they wanted to communicate and build around that topic. While comments and descriptions were still part of mind maps, participants changed the ways they started discussions, and how they looked at concepts and the relationships among them.

Reorganization was evident in the way multimodal elements (e.g., the picture of a code in scratch and its corresponding URL) were used to introduce topics and to connect multiple ideas (e.g., the scratch code mentioned above appeared connected to concepts such as visualization tools, coding environments, and scratch).

The different technical characteristics of the three tools (Popplet, Mindmeister and Mindomo) also had an impact in the reorganization of knowledge. For example, when discussing types of fraction representation (volume model, area model, or number line), the possibility of adding words to connectors allowed participants to connect to specific cases where they could (or could not) be useful (e.g., Area model is useful to show equivalency. The words “is useful to show” were placed on a connector between the concepts Area model and equivalency). However, these connecting words were not largely used, and neither were other distinctive capabilities of Mindmeister and Mindomo such as adding boundaries or icons. For this reason, we strove for generalization of our constructs rather than a direct focus on the differences among the three tools. Also, we found that our emerging three stages of knowledge building through mind mapping—introducing a topic, building a concept, and making sense of the whole picture—can be compared with the Online Asynchronous Collaboration (OAC) model to support the development of mathematical knowledge for teaching (MKT) proposed by Clay et al. (2012). In the first stage of knowledge building through mind mapping, when the teacher candidates introduced a topic, they often did so in consideration of an expert model of mathematics teaching reviewed in class or in course readings, and created an initial response to the task (Stages 1 and 2 in OAC). When the prompt was a list of topics (Cases 1 and 2), teacher candidates openly included many themes related to class, while a question prompt generated mind maps that revolved around a single resource or issue. This also generated more consistent (similar-looking) mind maps among groups.

In the second stage of knowledge building, when the teacher candidates built a concept, they engaged in activities that included viewing, reviewing and commenting on others’ responses (Stages 3 and 4 in OAC). However, in our grounded theory, we found that in the final stage (making sense of the whole picture) participants did not engage in discussing and revising initial responses (Stages 5 and 6 in OAC). Instead, the affordances of the tools prompted them to connect ideas and relate concepts. For example, when discussing the low floor high ceiling approach to designing mathematics learning activities, they were able to connect it with coding and how it can make learning experiences more flexible and accessible for all students. These kinds of connections, which were not explicitly made in the readings and materials, were a great indicator of teacher candidates’ learning. We also noted a difference among the three tools in terms of connections, because Mindomo (Case 3) allowed participants to change the colour of the connection, so they could give additional meaning to it, more often to indicate which participant had made the connection.

However, in our multiple case study, there was still much space for improving collaboration throughout the whole process of mind map construction. While the final products (mind maps) were presented as collaborative work, the construction process in its two early stages—introducing a topic and building concepts—shows mainly signs of cooperation, understood as a process “in which each member contributes an independent piece to the whole in a form of a division of labour” (Harasim 2017, p. 121). This result could be due to an issue of authorship, where participants did not feel comfortable adding to or editing work authored by another person.

In Gadanidis et al.'s (2008) study using collaborative writing in a wiki, participants also had difficulty allowing themselves to edit the work of others. They attributed this aspect to a matter of ownership of ideas. “When a student’s [product] is edited by peers, is that [product] still the original student’s work?” (Gadanidis et al. 2008, p. 130). We believe that when a mind map is co-created by students, they feel the need to set boundaries to their own work and that of others, so that they can fulfill the purpose of demonstrating their knowledge and original ideas to the instructor.

7.2 Results of knowledge building through mind mapping

The constructs we called results of knowledge building through mind mapping—developing discourse and developing leadership—contain elements of the three roles that Borba and Llinares (2012) identified in mathematics education online learning environments. The first role refers to providing asynchronous collaborative communication interfaces that allow participants to spend more time building their arguments. While participants in mind maps relied heavily on language to describe concepts and express thoughts, the interface allowed them also to use connectors, shapes, and colours to relate and highlight ideas as part of their discourse development, hence spending more time building arguments than verbally explaining ideas.

The second role refers to providing sharing and co-creating tools that allow teachers to compare and share their ideas. The mind maps showed that participants engaged in this comparison and sharing as they decided the topics to add, and which colours to use, taking into consideration the topics others had added. The role of the leader was also important in determining what topics were included and the general structure of the mind map.

The third role refers to the fact that the visual representation serves as a group memory of the work, where participants are reminded of previous ideas and their implications. In mind maps, the topics introduced by others are visually available at all points of the interaction, so participants did not include those topics again and instead added comments if they agreed or had something to say about a previously included topic.

Many authors have stressed the importance of discourse to develop thinking (Harasim 2017; Sfard 2008), build knowledge (Bruffee 1999; Harasim 2007; Scardamalia and Bereiter 2003) and develop identity (Scollon et al. 2012). Specifically for mathematical thinking, Sfard (2008) conceptualizes mathematics as a discourse, that is, communication that consists of word use, visual mediators, routines, and narratives. While participants in our case study did include a large number of mathematical visual mediators (such as symbols, diagrams, and graphs) in the form of images, comments added to nodes were the most frequent way of expression used by participants, and the largest source of meanings in our data. This result aligns with the view that “while discourse plays a key role in learning, text or writing is considered the most important type of conversation in knowledge building” (Harasim 2017, p. 131). Henceforth, while the affordances of the mind maps allowed for more direct ways of communicating relationships, highlighting central ideas, and illustrating mathematical concepts through images and videos, and our participants used these features conveniently, they still relied on the power of written speech to articulate and represent most of their thoughts.

Since the mind maps in our context were a case of self-directed learning environments—with minimal to no participation from the instructors—there were many opportunities for participants to develop leadership. In Garrison’s (2017) model of online interaction, instead of referring to a teacher presence, he refers to a teaching presence, since he observed that when the teacher withdraws from the discussion, participants develop the role of directing the cognitive and social processes. In our case study, the assumption of this role by some students was a natural response to the task, since the activity prompts did not include any role designations or instructions on how to start and organize the collaborative work. Scardamalia and Bereiter (2003) referred to this process as assuming epistemic agency and collective responsibility, by which students set goals and plan as they take responsibility for their own learning and the advancement of the group project.

7.3 Expression variations in knowledge building

The prompts in the mind map activity did not encourage one mode of expression over the others. However, as explained in previous sections, participants relied heavily on descriptions to express their knowledge. This could be due to a general preference of students towards the traditional forms of assessment (Furnham et al. 2011; Iannone and Simpson 2015), in this case, essay type or written evaluations. However, there is much to gain by encouraging that students use multiple ways to represent their knowledge.

For example, Gadanidis et al. (2008) in their study with preservice mathematics teachers, concluded that “multimodal communication does make a difference in an online learning environment. And, this difference is not only in terms of having more ways of communicating; it is also a qualitative difference in the ideas that are communicated” (p. 126). The different expression variations afforded by mind maps did make a qualitative difference in what participants expressed. For example, life experiences were shared only through descriptions attached to nodes, while videos and images were used mainly as triggering agents that started discussion, and chat communication was limited to organizational conversations. Also, the integration of visuals and text in mind maps facilitated the integration of graphical, narrative, and symbolic realizations, which is an important indicator of mathematics learning and mathematics discourse development (Sfard 2008).

Particularly, regarding videos, those included by participants in mind maps sparked discussions about computational thinking, mathematics, and mathematics pedagogy, which often related to past experiences with the subject matter. LeSage (2013) asserted that videos are a valuable instructional tool in mathematics for auditory or visual learners, and she continued to explain that the use of videos provides control to students and that “for elementary teachers with a history of negative mathematics experiences; being in control of mathematics is a novel yet welcome change” (p. 203).

8 Concluding remarks

We set out to find how preservice mathematics teachers interact and construct knowledge through collaborative mind mapping, which may resemble online discussions carried out through forums, but which have little in common with these forms of linear text. In our case study, mind maps enabled distinct semiotic possibilities that written conversations do not. These possibilities included more options in arranging items, sizing, highlighting, linking or separating ideas. We observed that teacher candidates followed a straight sequence when constructing knowledge through collaborative mind mapping. There are two byproducts of this process: participants develop a discourse on mathematics and mathematics pedagogy and leadership, while using varied forms of expression according to their preference or the nature of the content.

Finally, the theory generated in this study is valuable for expanding existing literature, especially concerning the use of visual tools for mathematics teacher education, and for informing practice and generating suggestions for teachers and developers to implement this type of learning experience in other courses and/or other education levels, as well as to set the stage for further research.