Concept Mapping in Mathematics: Research into Practice

Karoline Afamasaga-Fuata'i (ed.)Concept Mapping in Mathematics1Research into Practice10.1007/978-0-387-89194-1_1© Springer Science+Business Media, LLC 2009

1. The Development and Evolution of the Concept Mapping Tool Leading to a New Model for Mathematics Education

Joseph D. Novak¹, ²   and Alberto J. Cañas²  

(1)

Cornell University Ithaca, New York, USA

(2)

Florida Institute for Human & Machine Cognition, Pensacola, FL, USA

Joseph D. Novak (Corresponding author)

Email: [email protected]

Alberto J. Cañas

Email: [email protected]

A research program at Cornell University that sought to study the ability of first and second grade children to acquire basic science concepts and the affect of this learning on later schooling led to the need for a new tool to describe explicit changes in children’s conceptual understanding. Concept mapping was invented in 1972 to meet this need, and subsequently numerous other uses have been found for this tool. Underlying the research program and the development of the concept mapping tool was an explicit cognitive psychology of learning and an explicit constructivist epistemology, described briefly in this paper.

A research program at Cornell University that sought to study the ability of first and second grade children to acquire basic science concepts and the affect of this learning on later schooling led to the need for a new tool to describe explicit changes in children’s conceptual understanding. Concept mapping was invented in 1972 to meet this need, and subsequently numerous other uses have been found for this tool. Underlying the research program and the development of the concept mapping tool was an explicit cognitive psychology of learning and an explicit constructivist epistemology, described briefly in this paper.

In 1987, collaboration began between Novak and Cañas and others at the Institute for Human and Machine Cognition, then part of the University of West Florida. This led to the development of software to facilitate concept mapping, evolving into the current version of CmapTools, now widely used in schools, universities, corporations, and governmental and non-governmental agencies.

CmapTools allows for selective use of Internet and other digital resources that can be attached to concept nodes and accessed via icons on a concept, providing a kind of knowledge portfolio or knowledge model. This capability permits a new kind of learning environment wherein learners build their own knowledge models, individually or collaboratively, and these can serve as a basis for life-long meaningful learning. Combined with other educational practices, use of CmapTools permits a New Model for Education, described briefly. Preliminary studies are underway to assess the possibilities of this New Model.

1.1 Introduction: The Invention of Concept Mapping

During the 1960s, Novak’s research group first at Purdue University and then at Cornell University sought to develop a coherent theory of learning and theory of knowledge that would form a basis for more systematic research in education and a scientific basis for school curriculum design. We found that Ausubel’s assimilation theory of learning presented in his Psychology of Meaningful Verbal Learning (1963) spoke to what we were most interested in, namely, how do learners grasp the meanings of concepts in a way that permits them to use these concepts to facilitate future learning and creative problem solving? Ausubel stressed the distinction between learning by rote and learning meaningfully. Rote learning or memorizing information may permit short-term recall of this information, but since it does not involve the learner in actively integrating new knowledge with concepts and propositions already known, it does not lead to an improved organization of knowledge in the learner’s cognitive structure. In contrast, meaningful learning requires that the learner chooses actively to seek integration of new concept and propositional meanings into her/his cognitive structure, thereby enhancing and enriching her/his cognitive structure. Since the 1960s, many studies have shown that what distinguishes the naive or novice learner from the expert is the extent to which the person has a highly organized cognitive structure and metacognitive strategies to employ this knowledge in new learning or novel problems solving (Bransford, Brown, & Cocking, 1999). In short, one builds expertise in any discipline by building powerful knowledge structures that characterize the key intellectual achievements in that discipline, as well as strategies to use this knowledge.

Also occurring in the 1960s was a philosophical movement away from positivism where knowledge creation was seen as a search for truths unfettered by prior ideas or emotion. Kuhn’s (1962) book, The Structure of Scientific Revolutions marked a turning point toward constructivist ideas that saw knowledge creation as a human endeavor that involved changing methodologies and paradigms and an evolving set of ideas and methodologies leading to useful but evolving paradigms and ideas. We saw that this constructivist epistemology and cognitive psychology was equally applicable to mathematics and mathematics teaching. The challenge was: how do we get educators and the school contexts to change to enhance the utilization of these new insights (Novak, 1986, p. 184). As we proceeded in our mathematics education studies, we found we could work with a theory of learning that explained how new concepts are acquired and used that complemented a theory of knowledge that focused on the evolving creation of new concepts and problem solving approaches.

Working with elementary school children, we sought to design new instruction in such a way that meaningful learning would be enhanced, and to demonstrate that such learning could facilitate future learning and problem solving. To do this we found that we needed an assessment method that could monitor the evolving knowledge frameworks of our learners. Moreover, we were interested in demonstrating that young children (ages 6–8) could acquire significant science concepts and that this learning would facilitate later learning. The advances in learning theory and epistemology permitted Novak to construct a theory of education, first presented in 1977 (Novak, 1977) and modified and elaborated in 1998 (Novak, 1998). This theory has been guiding our work for some three decades.

Given our interest in teaching young children basic science concepts such as the nature of matter, energy, and energy transformations and related ideas, we were faced with the reality that most primary grade teachers did not posses the knowledge to teach these ideas. Therefore, we developed a series of audio-tutorial lessons where children were guided by audio instruction in the manipulation of pertinent materials and presented the vocabulary needed to code the concepts they were learning. Figure 1.1 shows an example of one of these audio-tutorial lessons and the carrel unit in which they were presented.

Fig. 1.1

An audio-tutorial carrel unit showing a 7 year old student learning about energy transformations

Lessons proceeded on a schedule of one new lesson every two weeks placed in classrooms in Ithaca Public schools. Earlier studies had shown that a variety of paper and pencil tests were inadequate for monitoring the growth in the children’s understanding of concept meanings, so we used interviews to probe the children’s knowledge. This, however created another problem in that it was difficult to see in the interview transcripts just how the children’s cognitive structure was changing and how new concepts were being integrated into the child’s cognitive structure. After struggling with the problem for some weeks, Novak’s research group came up with the idea of transforming the interview transcripts into a hierarchically arranged picture showing the concepts and proposition revealed in the interview. We called the resulting drawing a concept map. We found that we could now see explicitly what concept and propositions were being integrated into a child’s mind as they progressed through the audio-tutorial lessons, and also in later years as they encountered school science studies.

Figure 1.2 shows an example of a concept map drawn from an interview with one child at the end of grade two and another for the same child in grade 12.

Fig. 1.2

Two concept maps drawn from interviews with children in grade 2, (top) and grade 12 (bottom). These show the enormous growth in conceptual understanding for this child over 10 years of schooling

This child was obviously a meaningful learner and not only were some misconceptions remediated, but he developed an excellent knowledge structure for this area of science. The results from this 12-year longitudinal study demonstrated several things: concept maps could be a powerful knowledge representation and assessment tool; young children can acquire significant understanding of basic science concepts (widely disputed in the 1960s and 70s); technology can be used to deliver meaningful instruction to students; early meaningful learning of science concepts highly influenced learning in later science studies (Novak & Musonda, 1991).

Shortly after we developed the concept map tool for the assessing of changes in learner’s cognitive structure, we found that our staff and others were reporting that concept maps were very helpful as a study tool for virtually any subject matter. This led Novak to develop a course called Learning To Learn, and he taught this course at Cornell for some 20 years. One of the outcomes from the course was the book Learning How to Learn (Novak & Gowin, 1984). The book has been published in 9 languages and remains as popular as when it was first published. Concept maps are also powerful metacognitive tools helping students to understand the nature of knowledge and the nature of meaningful learning (Novak, 1990). More recently we have found concept maps to be an excellent tool for capturing expert knowledge for archiving and training in schools and corporations, and also for team problem solving. Beginning some 10 years ago, the Institute for Human and Machine Cognition has developed CmapTools, software that not only facilitates building concept maps but also offers new opportunities for learning, creating, and using knowledge, as will be discussed further.

1.2 The Use of Concept Maps in Mathematics

Our early work using concept maps in mathematics was focused on demonstrating how mathematical ideas could be represented in this form. Cardemone (1975) showed how the key ideas in a remedial college math course could be represented using concept maps. He found that the use of concept maps could help teachers design a better sequence of topics and helped students see relationships between topics. Minemier (1983) found that when students made concept maps for the topics they studied they not only performed better on problems solving tests but they also gained increased confidence in their ability to do mathematics. Fuata’i (1985, 1998) used concept maps along with vee diagrams with Form Five students in Western Samoa. She found that students became more autonomous learners and better at solving novel problems as compared with students not using these tools. Figure 1.3 shows an example of a concept map produced by one of her students.

Fig. 1.3

A concept map made by a Western Samoan student in a study by Fuata’i (1998, p. 65)

1.3 CmapTools and the Internet

CmapTools goes beyond facilitating the construction of concept maps through an easy-to-use map editor, leveraging on the power of technology and particularly the Internet and WWW to enable students to collaborate locally or remotely in the construction of their maps, search for information that is relevant to their maps, link all types of resources to their maps, and publish their concept maps, (Cañas et al., 2004)¹. CmapTools facilitates the linking of digital resources (e.g. images, videos, text, Web pages, or other digital concept maps) to further explain concepts through a simple drag-and-drop operation. The linked resources are depicted by an icon under the concept that represents the type of resource linked. By storing the concept maps and linked resources on a CmapServer (Cañas, Hill, Granados, C. Pérez, & J. D. Pérez, 2003), the concept maps are converted to Web pages and can be browsed through a Web browser. The search feature in CmapTools takes advantage of the context provided by a concept map to perform Web searches related to the map, producing more relevant results that may include Web pages and resources, or concept maps stored in the CmapTools network (Carvalho, Hewett, & Cañas, 2001). Thus, a small initial map can be used to search for relevant information which the student can investigate, which leads to an improved map, to another search, and so on. The student can link relevant resources found to the map, create other related maps, and organize these into what we call a knowledge model (Cañas, Hill, & Lott, 2003). Figure 1.4 illustrates a concept map made with CmapTools and the insets show some of the resources attached to this map that can be opened by clicking on the icon for the resource. Other examples will be given in subsequent chapters of this book.

Fig. 1.4

A concept map about numbers showing some resources that have been opened by clicking on the various icons under the concepts

1.4 A New Model for Education

Given the new technological capabilities available now, and combined with new ideas for applying the latest thinking about teaching and learning, it is possible to propose a New Model for Education (Novak & Cañas, 2004; Cañas & Novak, 2005). The New Model involves these activities that will be further elaborated:

1.

Use of expert skeleton concept maps to scaffold learning.

2.

Use of CmapTools to build upon and expand the expert skeleton concept maps by drawing on resources available on the CmapServers, on the Web, plus texts, image, videos and other resources.

3.

Collaboration among students to build knowledge models.

4.

Explorations with real world problems providing data and other information to add to developing knowledge models.

5.

Written, oral, and video reports and developing knowledge models.

6.

Sharing and assessing team knowledge models.

1.4.1 Expert Skeleton Concept Maps

The idea behind the use of expert skeleton concept maps is that for most students (and many teachers) it is difficult to begin with a blank sheet and begin to build a concept map for some topic of interest. By providing a concept map prepared by an expert with 10–15 concepts on a given topic, this skeleton concept map can help the learner get started by providing a scaffold for building a more elaborate concept map. Vygotsky (1934), Berk and Winsler (1995) and others point out that the apprentice learner is often very insecure in their knowledge and needs both cognitive and affective encouragement. While the teacher can best provide the latter, the skeleton concept map can provide the cognitive encouragement to get on with the learning task. Moreover, students (and often teachers) may have misconceptions or faulty ideas about a topic that would impede their learning if they were to begin with a blank sheet. The scaffolding provided by the expert map can get the learner off to a good start, and as they begin to research relevant resources and to add concepts and resources to their map, there is a good chance that their misconceptions will also be remediated (Novak, 2002). Figure 1.5 shows an example of an expert skeleton concept map.

Fig. 1.5

An example of an expert skeleton concept map that can serve as a starting point for building a knowledge portfolio about number ideas. Figure 1.4 above shows an example of how this skeleton map could be elaborated using CmapTools and resources drawn from the Web. In general, the number of concepts expected to be added by the student is proportional to (e.g. two or three times) the number of concepts originally in the skeleton map

1.4.2 Adding Concepts and Resources Using CmapTools

It is well known today that meaningful learning requires that the learner chooses to interact with the learning materials in an active way and that he/she seeks to integrate new knowledge into her/his existing knowledge frameworks (Novak, 1998; Bransford et al., 1999). Through the drag-and-drop feature of CmapTools that allows for linking supplemental resources to a concept map by simply dragging and dropping the icon for an URL, an image, video, another concept map or any other digital resource on a concept, a learner can build an increasingly complex knowledge model for any domain of knowledge, which can serve as a starting point for later related learning. Moreover, CmapTools provide for easy collaboration between learners either locally or remotely, and either synchronously or asynchronously. When the recorder option of CmapTools is turned on, it will record step-by-step the history of the creation of a concept map, indicating the sequence of building steps and who did what at each step. Obviously this also provides a new tool for cognitive learning studies, but such work is just beginning. For example, Miller, Cañas, & Novak (2008) using the Recorder tool has shown that in the process of learning how to construct concept maps, patterns in map changes for teachers in Proyecto Conéctate al Conocimiento in Panama (Tarté, 2006) were similar for teachers who had no previous experience using computers when compared with those who reported previous experience.

A major problem in mathematics studies is that there is rarely clear focus on the concepts underlying the mathematical operations learners are asked to do. It is even more rare to have an explicit record of the conceptual thinking of the learners as they progress in their studies, and a record the learner can turn to when related materials are studied. Another important advantage is that concept maps can be easily related to one another, for example as sub-concept maps in a more general, more encompassing concept map. Examples of this are shown in other chapters.

1.4.3 Collaboration Among Students

With the rediscovery of the studies of Vygotsky (1934) in the past 20 years, educators are increasingly recognizing the importance of social exchange in the building of cognitive structure, as well as for motivation for meaningful learning. Although the work of the Johnson brothers (1988) and others have shown some of the merits of cooperative learning, most of these studies could not take advantage of the facilitation offered by CmapTools for cooperative learning. Often the advantages of cooperative learning were found to be small at best. We need new research studies showing the effect of collaboration on learning using CmapTools.

1.4.4 Exploration with Real World Problems

One of the important conclusions from many recent studies on situated cognition is the importance of placing learning into a meaningful, real world context. We recognize this value in our New Model and urge that whenever possible, new ideas in mathematics should be introduced within the framework of some real world problem. While math teachers have been using for decades the idea that ratio and proportion problems, for example, should be introduced with tangible activities, such as using objects of different densities for comparison, most of these activities have not made explicit the mathematics concepts involved in the problems given, and the focus has been mostly on procedures to get the correct answer. Of course, this has also carried over into physics teaching and teaching in other subjects.

When real world activities are tied in with the use of CmapTools and the creation of knowledge models for the domains studied, research is beginning to show the resulting improvement in learning (Cañas, Novak, & González, 2004; Cañas & Novak, 2006; Cañas et al., 2008).

In the Proyecto Conéctate al Conocimiento (Tarté, 2006) effort in Panama, we are finding that the collaboration possibilities available with CmapTools are leading not only to sharing in knowledge building but also to a variety of social exchanges. During the training programs for teachers, teachers are invited to prepare a concept map in the form of biography referred to as Who am I?. This has led to teachers and principals also building concept maps about their schools, communities and a variety of related exchanges (Sánchez et al., 2008). This personal engagement has had strong motivating effects for pursuing other collaborations and we expect this will increase over time as the social network grows. Figure 1.6 shows a sample montage of some of the work done by teachers and school principals in Panama. With thousands of teachers and students involved in this project, we are learning many new possibilities for ways to use concept maps to facilitate meaningful learning.

Fig. 1.6

Concept maps drawn by Panamanian teachers illustrating Who am I? concept maps posted on Web sites and serving to provide motivation and recognition, as well as a basis for collaborations

1.4.5 Written, Oral, and Video Reports and Developing Knowledge Models

While we will continue to see an increase in the use of electronic communications in the future, there will always be an important place for written and oral reports. Whether in schools or corporate settings, our students need to become effective written and oral communicators. In one of our early studies, we found that fifth grade children who prepared concept maps prior to attempting to write out their ideas not only wrote better stories but they were also better able to tell their stories (Ben-Amar, 1990). In fact, they wrote a play derived from their stories and it was so well received they were invited to present it at other elementary schools!

The full range of capabilities for organizing knowledge available using CmapTools is too recent to have an empirical research base to document the value of the possibilities created, including what we call A New Model for Education. Hopefully, after the publication of this book, many empirical studies will be done to assess the value of CmapTools not only for improving instruction in mathematics, but also in improving students’ ability to communicate their mathematics ideas and a new excitement for learning mathematics.

1.4.6 Sharing and Assessing Team Knowledge Models

Already indicated above are ways in which the sharing of individual and team knowledge models can be facilitated using the collaboration tools of CmapTools. However, many teachers want to know how they can evaluate knowledge models. In our own teaching, we have used a variety of strategies including having teams post their knowledge models anonymously and then asking students to rank the models from lowest to highest, including criteria for their rankings. Using digital knowledge models created with CmapTools, these can be posted on the class server and provide easy access for assessment. Students can be very insightful, and often brutally honest, in their assessments. Furthermore, serious assessment is an educational experience, and students learn how they can improve their own knowledge models.

1.5 In Conclusion

Our objective in this chapter was to provide a brief history of the development of the concept mapping tool, including the development of the computer software, CmapTools, designed to facilitate concept map making and to provide new opportunities for individual and collaborative learning. Although the research to date supports the value of concept mapping to facilitate meaningful learning (Coffey et al., 2003), very little research has been done in the field of mathematics education. It is our hope this book will encourage such research. We also hope to see studies in mathematics learning that will utilize what we call a New Model for Education, and that libraries of expert skeleton concept maps in mathematics will be posted on web sites. We observed an increase in the number of papers dealing with mathematics education presented at international conferences on concept mapping from 2004 to 2008 (Cañas, Novak, et al., 2004; Cañas & Novak, 2006; Cañas et al., 2008) and we are hopeful that even more and improved studies will be presented at following conferences (see http://cmc.ihmc.us).

References

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Ben-Amar Baranga, C. (1990). Meaningful learning of creative writing in fourth grade with a word processing program integrated in the whole language curriculum. Unpublished M.S. thesis, Cornell University.

Berk, L. E., & Winsler, A. (1995). Scaffolding children’s learning: Vygotsky and early childhood education. Washington, DC: National Assn. for Education for the Education of Young Children.

Bransford, J., Brown, A. L., & Cocking, R. R. (Eds.). (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

Cardemone, P. F. (1975). Concept maps: A technique of analyzing a discipline and its use in the curriculum and instruction in a portion of a college level mathematics skills course. Unpublished M.S. thesis, Cornell University.

Cañas, A. J., Hill, G., & Lott, J. (2003). Support for constructing knowledge models in CmapTools (Technical Report No. IHMC CmapTools 2003-02). Pensacola, FL: Institute for Human and Machine Cognition.

Cañas, A. J., Hill, G., Granados, A., Pérez, C., & Pérez, J. D. (2003). The network architecture of CmapTools (Technical Report IHMC CmapTools 2003-01). Pensacola, FL: Institute for Human and Machine Cognition.

Cañas, A. J., Hill, G., Carff, R., Suri, N., Lott, J., Eskridge, T., et al. (2004). CmapTools: A knowledge modelling and sharing environment. In A. J. Cañas, J. D. Novak, & F. Gonázales (Eds.), Concept maps: Theory, methodology, technology. Proceedings of the first international conference on concept mapping (Vol. I, pp. 125–133). Pamplona, Spain: Universidad Pública de Navarra.

Cañas, A. J., Novak, J. D., & González, F. (Eds.) (2004). Concept maps: Theory, methodology, technology. Proceedings of the first international conference on concept mapping. Pamplona, Spain: Universidad Publica de Navarra.

Cañas, A. J., & Novak, J. D. (2005). A concept map-centered learning environment. Symposium at the 11th Biennial Conference of the European Association for Research in Learning and Instruction (EARLI), Cyprus.

Cañas, A. J., & Novak, J. D. (2006). Re-examining the foundations for effective use of concept maps. In A. Canãs & J. D. Novak (Eds.), Concept maps: theory, methodology, technology. Proceedings of the second international conference on concept mapping (Vol. 1, pp. 494–502). San Jose, Costa Rica: Universidad de Costa Rica.

Cañas, A. J., Reiska, P., Åhlberg, M. K., & Novak, J. D. (2008). Concept mapping: Connecting educators. Third international conference on concept mapping. Tallinn, Estonia: Tallinn University.

Carvalho, M., Hewett, R. R., & Canãs, A. J. (2001). Enhancing web searches from concept map-based knowledge models. In N. Callaos, F. G. Tinetti, J. M. Champarnaud, & J. K. Lee (Eds.), Proceedings of SCI 2001: Fifth multiconference on systems, cybernetics and informatics (pp. 69–73). Orlando, FL: International Institute of Informatics and Systemics.

Coffey, J. W., Carnot, M. J., Feltovich, P. J., Hoffman, R. R., Canãs, A. J., & Novak, J. D. (2003). A summary of literature pertaining to the use of concept mapping techniques and technologies for education and performance support. Technical Report submitted to the US Navy Chief of Naval Education and Training, Institute for Human and Machine Cognition, Pensacola, FL.

Fuata’i, K. A. (1985). The use of Gowin’s Vee and concept maps in the learning of form five mathematics in Samoa College, Western Samoa. Unpublished M.S Thesis, Cornell University.

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Miller, N. L., Cañas, A. J., & Novak, J. D. (2008). Use of the CmapTools recorder to explore acquisition of skill in concept mapping. In A. J. Cañas, P. Reiska, M. Åhlberg & J. D. Novak (Eds.), Concept mapping: Connecting educators. Proceedings of the third international conference on concept mapping (Vol. 2, pp. 674–681). Tallinn, Estonia: Tallinn University.

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Footnotes

1

CmapTools can be downloaded and used at no cost from: http://cmap.ihmc.us

Part 2

Part B: Primary Mathematics Teaching And Learning

Karoline Afamasaga-Fuata'i (ed.)Concept Mapping in Mathematics1Research into Practice10.1007/978-0-387-89194-1_2© Springer Science+Business Media, LLC 2009

2. Analysing the Measurement Strand Using Concept Maps and Vee Diagrams

Karoline Afamasaga-Fuata’i¹ 

(1)

School of Education, University of New England, Armidale, Australia

The chapter presents data from a case study, which investigated a primary student teacher’s developing proficiency with concept maps and vee diagrams as tools to guide the analyses of syllabus outcomes of the Measurement strand of a primary mathematics syllabus and subsequently using the results to design learning activities that promote working and communicating mathematically. The student teacher’s individually constructed concept maps of the sub-topics length, volume and capacity are presented here including some vee diagrams of related problems. Through concept mapping and vee diagramming, the student teacher’s understanding of the mapped topics evolved and deepened, empowering her to confidently provide mathematical justifications for strategies and procedures used in solving problems which are appropriate to the primary level, effectively communicate her understanding publicly, and developmentally sequence learning activities to ensure future students’ conceptual understanding of the sub-topics.

The chapter presents data from a case study, which investigated a primary student teacher’s developing proficiency with concept maps and vee diagrams as tools to guide the analyses of syllabus outcomes of the Measurement strand of a primary mathematics syllabus and subsequently using the results to design learning activities that promote working and communicating mathematically. The student teacher’s individually constructed concept maps of the sub-topics length, volume and capacity are presented here including some vee diagrams of related problems. Through concept mapping and vee diagramming, the student teacher’s understanding of the mapped topics evolved and deepened, empowering her to confidently provide mathematical justifications for strategies and procedures used in solving problems which are appropriate to the primary level, effectively communicate her understanding publicly, and developmentally sequence learning activities to ensure future students’ conceptual understanding of the sub-topics.

2.1 Introduction

Various Professional Teaching Standards point to the need for teachers of mathematics to have deep understanding of students’ learning, pedagogical content knowledge of the relevant syllabus and the ability to plan learning activities that develop students’ understanding, as essential to achieve excellence in teaching mathematics (AAMT, 2006). These Standards therefore imply that student teachers should develop deep knowledge and understanding of principles, concepts and methods they are expected to teach their future students. For example, the underlying theoretical principles of the New South Wales Board of Studies’ K-6 Mathematics Syllabus (NSWBOS, 2002) encourage the development of students’ conceptual understanding through an appropriate sequencing of learning activities and implementation of working and communicating mathematically strategies. To this end, this chapter proposes that the application of the metacognitive tools of hierarchical concept maps (maps) and vee diagrams (diagrams), and the innovative strategies of concept mapping and vee diagramming can influence (a) the development of students’ meaningful learning and conceptual understanding and (b) the dynamics of working and communicating mathematically within a social setting. Therefore, the focus question for this chapter is: "In