Developing a more conceptual understanding of matrices & systems of linear equations through concept mapping and vee diagrams.

Abstract

The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating and monitoring students' developing conceptual understanding of matrices and systems of linear equations in an undergraduate mathematics course. The study also explored the tools' role in scaffolding and facilitating students' critical and conceptual analyses of problems in order to identify potential methods of solutions. Data collected included students' progressive maps/diagrams, journals of reflections and justifications of revisions, and final reports and researchers' annotated comments on students' maps/diagrams and anecdotal notes from presentations. Findings showed that students developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. Findings suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these metacognitive tools into students' repertoire of effective learning strategies.

Introduction

Current emphases in national and state curricular frameworks urge the promotion of deep knowledge and deep conceptual understanding of students as well as enhancing students' abilities and skills in working, thinking and communicating mathematically. To achieve these content and process outcomes, mathematics teachers are encouraged to be innovative, investigative and explorative in their pedagogical approaches to designing and developing learning activities (NCTM, 2000; NSW 2002). External examination reports (MANSW, 2005) indicate that a high proportion of students have difficulties comprehending the meanings of key concepts in the context of problems, justifying solutions, and presenting coherent mathematical arguments. Furthermore, first year university students' mathematical performances (Mays, 2005) in diagnostic tests show that most have mathematical misconceptions with fractions, percentages and multi-digit subtraction. Similarly, national surveys in Samoa confirm that learning by rote-memorization is quite prevalent in most schools (DOE, 1995). Such findings resonate with recurring comments in examiners' reports concerning students' obvious inabilities to effectively apply existing knowledge to successfully answer exam questions (Afamasaga-Fuata'I, 2001, 2002a, 2002b, 2002c, 2003, 2005a, 2005b).

In foundation and undergraduate mathematics classes in Samoa, students find it difficult to explain and justify their answers mathematically in terms of the conceptual structure of relevant topics. Instead their verifications are often in terms of sequences of steps of procedures. Whilst this may work for familiar problems, this procedural view constrains them when solving qualitatively and structurally different problems (i.e., novel problems). According to Richards (1991), this manifestation is typically a communication problem resulting from students' inability to understand the meaning of a language (i.e., concepts, principles, theorems and theories) used in mathematical discussions and dialogues of more mathematically literate others. Subsequently, less mathematically literate students are unable to make sense of such conversations, offer conjectures or evaluate mathematical assumptions. When doubtful, students tend to use any procedure to get an answer without really checking whether an algorithm is suitable to the problem (Schoenfeld, 1996). Such behavior typically reflects classroom practices wherein students focus more on completing sets of exercises that practice applications of procedures with little opportunity to reflect upon the processes that lead to the construction and generation of solutions. In such settings, students are not likely to question, challenge or influence the teaching of mathematics in the classroom (Knuth & Peressini, 2001). Thinking reflectively and communicating effectively are critical skills that students could be enculturated into as a routine part of classroom practices.

In Samoa, problems associated with learning mathematics are indicative of an educational system that is traditional and examination-driven where there is little to no time for mathematical discussions and dialogues. Instead, there is an urgency to complete the syllabus in time before external examinations. As a result over the many years of secondary schooling, problem-solving skills students acquire may not necessarily be situated "within a wider understanding of overall concepts" and would probably not be "long-lasting" (Barton, 2001). Furthermore, students may have the knowledge of relevant content areas but are often unable to, independently, apply what they know to problems unless substantial guidance is provided (Afamasaga-Fuata'I, 2003; Afamasaga-Fuata'I & Retzlaff, 2003). These tendencies are characteristic of students who have learnt mathematics by memorizing collection of facts and procedures in compartments to be recalled when necessary with little effort in making interconnections and linkages between topics. From analyses of national examination results and reports (Afamasaga-Fuata'I, 2001, 2002a, 2002b, 2002c, 2003), students' problems in mathematics seem to be centered around the interactions of four broad factors namely students': (i) narrow perceptions of mathematics, (ii) lack of critical skills to transfer existing knowledge to new situations, (iii) inability to communicate mathematically to others, and (iv) lack of critical thinking, reflection and analysis. The factors are mutually interactive but the separation shall facilitate analysis and discussion later on. Finally, in an effort to partially redress student difficulties and to explore innovative ways in which mathematics learning could be enhanced and guided by findings of a study with secondary students at the local government secondary school (Afamasaga-Fuata'I, 1998), the author undertook a series of concept map and vee diagram studies (mapping studies) at the National University of Samoa (NUS) (Afamasaga-Fuata'I, 2000, 2002a, 2002d, 2004b, 2004c, 2005a, 2005b). The case study reported here was part of these mapping studies.

Aims of the Mapping Studies

The researcher is particularly interested in examining and exploring the ways students' understanding could be influenced by constructing and using concept maps and vee diagrams to learn mathematics. Therefore, the aims of the studies are to investigate the impact of constructing and using concept maps and vee diagrams on students' understanding of mathematics in terms of their:

(1) critical ability to analyze, illustrate and justify their knowledge of the hierarchical interconnections between main and relevant concepts, principles and procedures of a topic,

(2) critical ability to communicate their mathematical understanding efficiently to others, and

(3) developing skills and competence in thinking reflectively and communicating effectively.

More specifically, the researcher examines students' developing ability to critically analyze a topic, fluently with the usage of their mathematics knowledge (i.e. language, concepts and principles) to justify hierarchical interconnections and solutions, effectiveness in articulating and publicly communicating their understanding, and identifying ways students' mathematics perceptions may have been influenced as a result of using the metacognitive tools.

Relevant Literature Review & Conceptual Framework 

Mathematics knowledge is increasingly being viewed as knowledge that is socially constructed. It is knowledge that has evolved and developed over time to its current status through a social process of conjecturing, refutations, proof and warranting by the community of mathematicians (Ernest, 1994a, 1994b, 1998; Hersh, 1994). Ernest in his review of research in the philosophy of mathematics, points to a convergence of shared acceptance of mathematics as an essentially social phenomenon, something long agreed by historians (and sociologists) of mathematics, but long denied by traditional philosophers of mathematics. In this emerging perspective, the role of proof in mathematics is social; it serves to persuade the appropriate mathematical community to accept knowledge as warranted. As one well-known mathematician said: "A proof becomes a proof after the social act of 'accepting it as a proof. This is true of mathematics as it is of physics, linguistics and biology" (Manin, 1977, p.48). Learning mathematics, therefore, may be viewed as knowledge construction involving both individual and social processes. In schools, most students view mathematics learning as learning how to solve certain types of problems. This view is nonproblematic to students most of the time and as long as the problems are familiar to them. However, students begin to experience a sense of frustration when well-memorized procedures and algorithms could not be effectively applied to solving novel problems. Without deep conceptual understanding of relevant topics, students will struggle to identify potential methods. The prevalent practice of emphasizing and requiring only correct answers (encouraged by assessment using mostly multiple-choice questions) without explaining the processes that generate the answers is a practice that does not nurture and sustain a deeper understanding of the mathematics. Hence, any attempt to improve mathematics teaching and learning must address not only individual students and their immediate learning environment, but must also consider the social processes that govern educational institutions, and in turn, how these processes legitimize and generate certain societal (and individual) expectations of what it means to learn and succeed in mathematics. These expectations tend to influence the nature and dynamics of actual classroom practices to the point that students learn procedures and methods rotely. Usually in these classrooms, there is little to no time set aside for students to engage in reflective or metacognitive strategies such as reflecting on how problems are solved, why methods work, how previous knowledge is applied in current situations, planning variable approaches and identifying potential future directions of learning based on recently constructed knowledge.

To encourage focused reflection and critical analysis for connections between prior and current knowledge, and between principles and procedures, concept maps and vee diagrams may be used to facilitate illustration, communication and negotiation of shared meanings in social settings. Concept maps and vee diagrams can also be used to assess knowledge acquisition, organization, and application in solving problems. A concept map is defined as a two-dimensional hierarchical map of interconnecting concepts (Novak & Gowin, 1984; Baroody & Bartels, 2000; Novak, 2002, 2004a, 2004b; Novak & Canas, 2004, or a graph consisting of nodes representing concepts and labeled lines denoting the relation between a pair of nodes. A student's concept map is interpreted as representing important aspects of the organization of concepts in his or her memory (cognitive structure) (Novak & Gowin, 1984; Ruiz-Primo & Shavelson, 1996; Ruiz-Primo, 2004). A concept map aims to show how a student perceives the structure and links between things, ideas, or people. Student-constructed concept maps show how students link ideas, and their view of the structure of a topic. Once students understand the process of the task, concept maps are quicker and more direct and considerably less verbal than essays.

The use of hierarchical concept maps in the mapping studies is grounded in Ausubel's cognitive theory of meaningful learning. It views the process of meaningful learning taking place if the student relates and/or links what he or she knows to new knowledge. The linking of knowledge may take place through the process of integrative reconciliation and/or progressive differentiation as less inclusive and less general concepts are subsumed under more inclusive and more general concepts, and/or more general concepts subsume less general concepts. Subsequently, existing knowledge is modified to accommodate and/or assimilate new knowledge (Ausubel, 2000; Ausubel, Novak & Hanesian, 1981; Novak, 1998, 2004a, 2004b). Ausubel's theory, therefore, provides guidance as to what constitutes a legitimate concept map. Consequently, concept maps invented and pioneered by Novak and Gowin (1984) are hierarchical with superordinate concepts at the top with progressively less inclusive and more specific concepts towards the bottom with examples where appropriate, labeled with linking words, and crosslinked so that relations between subbranches of the hierarchy are identified. The hierarchy is expanded when new concepts and new links are added to the hierarchy either by creating new branches or by differentiating new ones even further. Meaning increases for students when they can identify new links between sets of concepts or propositions at the same hierarchical level or other subbranches of the map at other levels. These crosslinks represent the integrative connection among different subdomains of the structure (Novak & Gowin, 1984). Concept maps may show linear chains (no integration), or evidence of crosslinkages and give evidence of a much more integrated structure. Also a concept map allows students to display knowledge they have acquired elsewhere. The quality of a concept map may be characterized by the number of crosslinks, showing more integration (in contrast to a linear one). Students may add concepts voluntarily to make the map more meaningful. Invalid linkages can show common difficulties in understanding particular concepts, which can cause substantial problems in learning those ideas.

Concept maps can also show low level of understanding such as in a star shaped map where all concept maps are linked to one central concept. Linkages may be vague or show erroneous conception. Leaving links unlabelled may indicate vagueness, oversight or misconception. Maps may have more linkages but can represent inadequate understanding, and can reveal a key problem in the student's ideas. Also the links may not fully reveal the student's conception (Novak, 1998, 2004b). Alternatively, vee diagrams may be used to seek further evidence of students' conceptions of connections between concepts and methods. A vee diagram is an epistemological tool to analyze and illustrate the interplay between conceptual (or thinking) elements (Theories, Principles, Concepts) and methodological (or doing) elements (Records, Transformations, Knowledge Claims, Value Claims) of a problem or activity (Object/Event, Focus Question). Figure 1 shows an adaptation of Gowin's epistemological vee (Gowin, 1981) for analyzing a mathematics problem including the guiding questions for each vee element. In contrast to meaningful learning, rote-learning is when students tend to accumulate isolated propositions (i.e. linear sequence of links that are unconnected to any other subdomain) rather than developing integrated hierarchical networks of concepts (i.e., structurally complex progressive differentiation and integrative reconciliation between subdomains). With vee diagrams, students may be able to complete the methodological (or doing) elements (Records, Transformations and Knowledge Claims) for familiar problems but may not be able to identify all or most of the conceptual (or thinking) elements (underlying principles and main concepts). With novel problems, students may not be able to complete any of the elements on the conceptual and methodological sides.

When concept maps are used in conjunction with vee diagrams, the illustrated information complement and reinforce each other as well as provide explicit material for evaluating students' critical analytical skills in integratively reconciling methods of solutions and mathematical principles and vice versa. The literature on metacognition (Novak, 2002, 2004a, 2004b; Schoenfeld, 1987) shows that the task of metacognitively reflecting upon one's existing conceptions can foster one's critical, analytical, and reflective thinking. One can evaluate these levels of understanding by discussing, interacting and negotiating meanings with students centered on appropriate learning activities. Accordingly, the research reported here contributes to the metacognition idea but goes beyond normal activities of a mathematics classroom by requesting that, in addition, students illustrate the state of their existing understanding by mapping it out on concept maps and vee diagrams and through these maps/diagrams present, justify and/or communicate their mathematical understanding in social settings (group and/or one-on-one). A number of research studies have been conducted in the sciences (Mintzes, Wandersee & Novak, 1998, 2000; Novak & Canas, 2004; Novak, 2002, 2004a, 2004b), and mathematics (Afamasaga-Fuata'I, 1998, 2000, 2002d, 2004b, 2004c 2005a; Baroody & Bartels, 2000; Liyanage & Thomas, 2002; Schmittau, 2004; Vagliardo, 2004; Williams, 1998) to investigate the value of concept maps and/or vee diagrams as metacognitive tools.

[FIGURE 1 OMITTED]

Methodology

The mapping studies are qualitative, exploratory teaching experiments conducted over a semester of 14 weeks with different cohorts of second and third year mathematics university students. They construct concept maps of mathematics topics and vee diagrams of problems to illustrate and highlight the structure of mathematics knowledge, in terms of hierarchical interconnections between concepts, principles, formulas and methods relevant to the selected topic. These individually constructed concept maps and vee diagrams are presented in class for peer critiques interspersed with one-on-one presentations to the researcher. There are at least two interactions of this process before a final one-on-one presentation at the end of the semester.

Students begin the semester by learning how to construct maps of relatively easy topics such as functions and types of functions before undertaking their project work; for this cohort, their topic was matrices and systems of equations. Students also practice negotiating meaning during group presentations; they argue, explain, justify and revise work appropriately and when necessary as they strive to reach a consensus. The new socio-mathematical norms (in contrast to the traditional transmission model of teaching and learning) expect students to present their work publicly, be prepared to justify and address critical comments from peers and researcher, and then later on critique their peers' presented work. This paper reports the work of one student (Dora) who was a practicing teacher of secondary mathematics and science at the time of the study.

Data and Analysis

Data collected include students' progressive maps/diagrams, journals of reflections and justifications for revisions, and final reports and researcher's annotated comments on students' maps/diagrams and anecdotal notes during presentations. Concept map data is presented first followed by those for vee diagrams.

Progressive Concept Maps

Concept maps are analyzed in terms of structural complexity, nature of node contents and valid propositions using counts of occurrences of each criterion to provide a comparative basis for changes between versions of concept maps. Structural complexity is described in terms of hierarchical levels, multiple-branching nodes, crosslinks, uplinks and subbranches. In this paper, multiple-branching nodes are those with at least two progressive differentiation links, crosslinks connect horizontally to nodes in adjacent sub-domains or same subdomain at the same/higher/lower hierarchical level whilst uplinks cross vertically from lower hierarchical levels to nodes in higher levels. In comparison, valid propositions indicate valid (concept [linking words.[right arrow]] concept) triads that singly or combine to form meaningful propositions. In contrast, nature of node labels are analyzed in terms of number of concepts, cumulative overlaps and new concepts. A content analysis of node labels reveals more information about evolving improvements in clarity and succinctness in naming key/relevant concepts and ideas. Whilst number of concepts is total nodes in current map, cumulative overlaps indicate number of common labels that appeared in previous versions. The difference of the two counts represents new additional concepts appearing for the first time in the current concept map.

Dora's progressive maps (maps A to J) were constructed and revised during the semester primarily as a result of her own evolving conceptual and integrated understanding of matrices and systems of equations and based on feedback from critiques. Revised maps were subsequently subjected to critiques from peers and/or researcher during group one-on-one presentations. Maps A, B, C, F and J have been selected for presentation and detailed analysis to benchmark the developmental trend of Dora's evolving conceptual understanding of systems of equations up to the final version for order 2 systems (i.e., J) and before switching back to order n. Versions after map J are not included in this paper. Minimal references will be made occasionally to other intermediary versions (i.e., D, E, G, H, and I not shown) when necessary to provide relevant background information. Figures 2 and 3 show data for structural complexity, valid propositions and nature of node contents for the selected maps. Maps A, B, C, F and J are shown in Figures 4, 5, 6, 7, and 8 respectively.

Structural Complexity and Valid Propositions

Shown in Figure 2 are the counts of hierarchical levels (H/Levels), multiple-branching nodes (M/B nodes), subbranches or subdomains (S/Branches), crosslinks, uplinks and valid propositions (ValidProps) for maps A, B, C, F and J for comparison. Each criterion shows more or less increasing trends over the semester except for uplinks. Collectively, they illustrate that the final map was relatively more complex in terms of hierarchical levels, multiple branching nodes (i.e., progressive differentiation links), subdomains, integrative crosslinks and uplinks. Valid propositions have also increased substantially by map J. Invalid propositions were substantially lower at (2, 2, 0, 7, and 4} for maps {A, B, C, F and J} respectively. Invalidity was mainly due to missing (blank), vague or incorrect linking words, and errors in labels such as notational and algebraic errors in derivation of general values. The next sections describe the nature of node contents.

[FIGURE 3 OMITTED]

Nature of Node Contents

Figure 3 shows that total concepts increase steadily over time from more or less the same number (i.e., 20, 22, and 20) with the first three maps (A, B and C), up to 38 for map F and 59 by map J. Similarly, cumulative overlaps started low at 5, then 8 and increased steadily to 29 by map F and rose to 40 (at map H) and then stabilized at 39 in the last two versions I and J. The highest number (i.e., 20) of new additional concepts for maps A to J occurs with map J.

The following sections describe each of the five maps in more details before summarizing the main points. (1)

Concept Map A -- Map A in Figure 4 has 8 hierarchical levels, 5 multi-branching nodes, 3 subbranches, 10 cross-links and 3 uplinks. Dora arranged her concepts hierarchically with maxtrix (2) as the most general but evidently INVERSE seems to be the central idea, with concepts matrix, I=[AA.sup.-1], EROS, adjoining, [A.sup.-1] and formula centrally linked to it. Progressive differentiating links from EROS forms a sub-branch that includes the two methods: Gauss-Jordan and Gauss Elimination, Reduced Row Echelon Form and Row Echelon Form and terminating with concept: back substitution. In contrast, an adjacent branch is formed by integrative reconciliation links from formula and [A.sup.-1] to X=[A.sup.-1]B with a single link from the latter to end node variables. A second link from [A.sup.-1] connects to B with a subsequent single link from the latter to vector from which two progressive differentiating links connect to end nodes constants and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Within the same formula-[A.sup.-1] sub-branch is a more general concept solutions, which subsumes variables and constants. A crosslink from back substitution of the EROS branch to variables of the formula-[A.sup.-1] branch in-tegratively reconciles the two subdomains. At the top of the map, the more general matrix is linked to its neighboring nodes identity matrix and I=[AA.sup.-1] by a linear sequence of single links. A crosslink from identity matrix connects the matrix sub-domain to the EROS branch. Similarly, a crosslink from I=[AA.sup.-1] integrates it to the centrally located INVERSE node. In contrast, an uplink from adjoining connects it to matrix thus forming one of the most significant propositions that integratively reconciles the smallest subdomain (adjoining) and matrix branch. That is, "INVERSE can be calculated by adjoining the given matrix to an identity matrix." Other valid propositions comprise a mixture of single or combined triads such as, "INVERSE if multiplied with original matrix produce the identity matrix then applying EROS produce Reduced Row Echelon Form;" INVERSE plays an important role in formula of X=[A.sup.-1]B;" and "X=[A.sup.-1]B is the set of variables." Invalid propositions are due to missing linking words such as:

[FIGURE 4 OMITTED]

"INVERSE [blank.[right arrow]] EROS" and "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]." This might be due to oversight or uncertainty about the most effective description to create meaningful propositions.

[FIGURE 5 OMITTED]

Concept Map B -- Map B in Figure 5 has 8 hierarchical levels, 4 multi-branching nodes, 3 subbranches, 6 crosslinks and 5 uplinks. The map focuses on Cramer's Rule and its use of determinants to solve a system of n equations. Progressive differentiation links from Cramer's Rule result in two branches. The system branch describes the propositions: "system has a unique solution given by a quotient of determinants i.e. |[A.sub.1]|/|A|, |[A.sub.2]|/|A|,... ,... where A is the square matrix" and "system of n equations in n unknowns or a square matrix." Cross-links from determinants and quotient form propositions: "determinants which can be labeled det [A.sub.1], det [A.sub.2] becomes the numerator" and "quotient which is numerator/denominator." An uplink forms the proposition, "determinants of the square matrix", and a link from the latter forms the proposition, "square matrix is the denominator" collectively forming at least 3 closed cycles of links within this subdomain.

In contrast to the system branch, the AX=B branch has three progressive differentiation links from AX=B to form 3-linear chains of single links, each describing the meanings of matrices A, X and B. The linear sequence for matrix A ends without linking words between the last two nodes. However, an uplink from the end node columns integratively reconciles it with the linear chain for matrix B resulting in the proposition "columns are replaced by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]." A method is implied by this proposition but not explicitly elaborated upon in this map. The three linear propositions in this subdomain include, "AX=B where A is the coefficient matrix i.e. "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]" "AX=B where X is the column vector i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]", and "AX=B where B is the vector i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]."

The key crosslinks integratively reconcile the two subdomains. The first one connects system to AX=B at Level 2 describing the proposition "system can be written as AX=B" with the second one (an uplink) from |[A.sub.1]|/|A|, |[A.sub.2]|/|A|,... at Level 7 of the system branch to Level 5 of the AX=B branch, forming the proposition "|[A.sub.1]|/|A|, |[A.sub.2]|/|A|,... gives values for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]". The two invalid propositions, "Cramer's Rule[blank.[right arrow]]system" and "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]", are due to missing linking words.

Cumulative Overlaps: Maps A to B--Figure 3 shows that only 5 concepts are common between maps A and B (see Figures 4 and 5) indicating that the rest of the concepts in both maps are qualitatively different from each other. Of the 5 common ones, two have remained exactly the same (B and vector).

Whilst B and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from map A combine into one label: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in map B, solutions changes to unique solution and X = [[A.sup.-1]B] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These changes reflect an attempt to be more precise and succinct in expressing concept labels and hence enhance the meaningfulness of propositions and compactness of subdomain hierarchies.

Concept Map C -- Map C shown in Figure 6 has relatively fewer hierarchical levels (6) than earlier maps (A and B), increasing number of multi-branching nodes (7), fewer subbranches (2), 7 crosslinks and 3 uplinks. Dora's focusing concept is Elementary Row Operations (abbreviated here to EROS) with outgoing links to matrix, rows, system of linear equations and tools. Two progressive differentiating links from system of linear equations to Reduced Row Echelon form (RREf) and Row Echelon form (Ref) with subsequent branching forming an overall main branch which terminates with example matrices "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]" and "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]" illustrating the two methods of Gauss-Jordan Elimination (G-JE) and Gauss Elimination (GE) respectively. Adjacent to this main branch is that for the sub-domain matrix at the top level of clarity the acceptable operations (EROS) on rows and columns. It is worth noting that the treatment of matrix in this map is more focused in detailing the acceptable EROS in direct contrast to the more general treatment of matrix in relation to the identity matrix, I=[AA.sup.-1] previously encountered in map A.

[FIGURE 6 OMITTED]

An uplink from constant at the bottom of the matrix branch to Row Echelon form of the main EROS branch integratively reconciles the two subdomains. Another crosslink from EROS and an uplink from system of linear equations both to matrix, integratively reconcile the two branches resulting in the propositions "EROS can be used on a matrix" and system of linear equations can be represented in the form of a matrix".

Figure 3 indicates that map C has only 8 common concepts with previous versions (A and B) and 12 new additional concepts of its own. Seven of the 8 common concepts (i.e., matrix, EROS, constant, G-JE, GE, RREf and Ref) are from map A with only one (i.e., system of linear equations) that combine two separate labels, system and n equation, from map B. Another significant change is the shifting of EROS from a lower hierarchical level in map A (Level 4) to become one of two main organizing ideas alongside matrix (see Figure 6).

Map C shows only 14 valid propositions and no invalid ones. Some of the significant propositions include "EROS can solve a system of linear equations by reducing to either Reduced Row Echelon form or Row Echelon form", "Reduced Row Echelon form have zeros off the main diagonal," "Row Echelon form may have leading Is on the main diagonal," "EROs deals with rows and columns," "rows and columns can be interchanged or multiplied by a constant" and "rows and columns can be interchanged and then added or subtracted', to name a few.

Concept Maps D to E -- Concept hierarchies are notably more complex according to increasing counts in Figure 3 (i.e., 26 to 31 total nodes and 15 to 19 cumulative overlaps) with concepts matrices and system(s) of equations still maintaining top position in both maps. It is also at map D that Dora chooses to change from a more general system of equations with n unknowns to focus on systems of 2 linear questions with two unknowns. The high number of cumulative overlaps (i.e., 15 out of 26 in map D) indicates minimal effect of this change (in terms of labels) on the overall map. Subsequent peer critiques and feedback from consultations led to further revisions of the maps up to and including map F. The latter is as shown in Figure 7. Particular and significant additions and changes from progressive versions up to map F are discussed next.

Concept Map F -- Map F in Figure 7 has 9 hierarchical levels, 7 multi-branching nodes, 4 subbranches, 8 crosslinks, one uplink and 25 valid propositions. Figure 3 indicates that of the 38 total nodes, 29 are cumulative overlaps with previous versions and 9 new concepts. Of the 9 new concepts, 3 of them have incorrect notations (i.e., [A.sub.1] and [A.sub.2] instead of [X.sub.1] and [X.sub.2] respectively in the end nodes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which might be due to oversight and the third error is in the value of [X.sup.1] in the node [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first two corrected notations should be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the Cramer's Rule branch. For the third error, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] should be either [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (I) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (II) where the major error may have been in the incomplete derivation (I) or derivation (II) of the [X.sub.1] value whilst it was probably an oversight with the omission of the [-a.sub.2][a.sub.3] term from the denominator of the first term [[a.sub.1][b.sub.2]]/[[a.sub.1][a.sub.4]] for the [X.sub.2] value.

[FIGURE 7 OMITTED]

The most significant concept additions were the more detailed elaboration of Cramer's Rule last viewed in map B (Figure 4) in more general terms and only a single-node mention in map E. The subdomain is developed to clarify how [X.sub.1] and [X.sub.2] values are determined using determinants. In Figure 7, the crosslink between Cramer's Rule and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] integrates the two ideas with the latter branch illustrating a more developed network of interconnections than the former. Another significant addition is the inclusion of the integrative representation [A : I[EROS.[right arrow]]I : [A.sup.-1]] (initially appeared in map E as the merging together of different concepts which appeared as single labels in previous versions; for example, identity matrix (AEF) (3), inverse (ADEF), adjoint matrix A:I (ADEF) and EROS (ACDEF). Other concepts that appeared first in map E and remained unchanged in map F are those detailing invertibility of matrices namely: det A, det=0, det [not equal to] 0, not invertible, invertible and augmented matrix. In comparison, concepts EROS (ACDEF), constants, and matrices appeared in all versions (ABCDEF). The two concepts systems and n equations first appeared separately in map B but later were combined to form system of linear equations in map C, were revised to system(s) of equations in maps D and E but were changed to system of equations in map F. This progressive refinement of labels indicates a developmental trend in being more concise and succinct in naming key concepts hence enhancing the meaningfulness of propositional links with adjacent nodes particularly from maps B to C to F.

Other new concept additions are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and I:[A.sup.-1].

Some of the most substantive propositions include: "system of equations e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and AX=B can be split into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]"; "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be adjoined to form A:B"; "A:B uses EROS i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];" "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] called unknown vector"; "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] called Augmented matrix or the coefficient matrix"; "coefficient matrix and identity matrix can be adjoined to form Adjoint matrix A:I"; and "Adjoint matrix A:I and EROS result in [A : I[EROS.[right arrow]]I: [A.sup.-1]]".

Of the 7 invalid propositions, 3 are due to vague linking words at the progressive differentiation node formed by crosslinks between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and Cramer's Rule, 3 are due to errors in the notation (i.e. |[A.sub.1]| and |[A.sub.2]| instead of |[X.sub.1]| and |[X.sub.2]| respectively) and general values for [X.sub.1] and [X.sub.2] as mentioned earlier and one is due to missing linking words on the connection from [A : B[EROS.[right arrow]]I: X] to I:[A.sup.-1].

Concept Map J -- Map J in Figure 8 has 10 hierarchical levels, 11 multi-branching nodes, 6 subbranches, 23 crosslinks, 3 uplinks and 54 valid propositions. Of the 59 total nodes, 20 of them are new additional concepts. Compared to map F, the order branch has been extended to include illustrative examples of R x C with a multi-branching node and a crosslink to the adjacent [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] branch. Similarly, the symbols branch is also crosslinked to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The determinants branch has a new concept, NO INVERSE, to strengthen the connection with noninvertible with a crosslink from det A to new node: det [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Cramer's Rule branch thereby integratively reconciling the two subdomains. Additionally, Levels 7 to 9 of map F (Figure 7) within the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. -AX=B branch have been extended another level and substantively revised in map J with additional new concepts, Row of zeros, No rows of zero, [A.sup.-1] = inverse of A, Matrix I:X, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to enhance further the meanings of interconnections within the A:I-EROS-A:B subbranches. Furthermore, label [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of map F has been decomposed in map J into labels such as: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [x.sub.1] = [b.sub.1]/[a.sub.1], and [x.sub.2] = [[a.sub.1][a.sub.2][b.sub.2] - [a.sub.2][a.sub.3][b.sub.1]]/[[a.sub.2.sup.2][a.sub.4] - [a.sub.1][a.sub.2][a.sub.3]] in an attempt to generate more meaningful clarifications and propositions but for the errors in the application of EROS to the matrix A:B to generate I:X.

[FIGURE 8 OMITTED]

Appropriate corrections would show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [X.sub.1] = [[a.sub.4][b.sub.1] - [a.sub.2][b.sub.2]]/[[a.sub.1][a.sub.4] - [a.sub.2][a.sub.3]] and [X.sub.2] = [[a.sub.1][a.sub.2][b.sub.2] - [a.sub.2][a.sub.3][b.sub.1]]/[[a.sub.1][a.sub.2][a.sub.4] - [a.sub.2.sup.2][a.sub.3]].

Integrative reconciliation links from end nodes of the well-developed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] -AX=B branch include an uplink from the two end nodes for X1 and X2 to unknown matrix at level 5 and others from end nodes Rows of zeros and No rows of zero to NO INVERSE and INVERSE respectively, of the adjacent determinant branch.

More additions of and progressive differentiation at nodes occur at the Cramer's Rule branch with an extension to another level to provide definitions: det [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], det [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], det [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and det A = ([a.sub.1][a.sub.4] - [a.sub.2][a.sub.3]) hence correcting in map J the oversight noted in map F where |[A.sub.1]| and |[A.sub.2]| were used in the labels: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but should have been [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead.

There is also further evidence of refinement of labels to be more concise and succinct; for example, revising unknown vector to unknown matrix and from constants to constants matrix since map F. These label (and linking word) changes taken collectively result in the formation of substantive meaningful propositions in map J such as, "Reduced Row Echelon form if it has Row of zeros then it has NO INVERSE" and Reduced Row Echelon form if it has No rows of zero then it has an INVERSE." Correction of algebraic errors would result in substantive propositions such as "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [X.sub.1] = [[a.sub.4][b.sub.1] - [a.sub.2][b.sub.2]]/[[a.sub.1][a.sub.4] - [a.sub.2][a.sub.3]] and [X.sub.2] = [[a.sub.1][a.sub.2][b.sub.2] - [a.sub.2][a.sub.3][b.sub.1]]/[[a.sub.1][a.sub.2][a.sub.4] - [a.sub.2.sup.2][a.sub.3]] are values of the unknown matrix".

Summary

Dora's initial 3 maps were developed separately to have different emphasis. However within each subdomain, Dora attempted to integrate and depict interconnections within and across concept hierarchies. Dora found class critiques useful as they provided her with feedback and suggestions for expanding her maps. There was evidence of linear chains of connections initially but these were subsequently revised to illustrate more integrative reconciliations across hierarchies and more progressive differentiation from level to level within branches such as changes for the symbols and determinants branches from map F to map J. Revisions were also made to linking words and node labels to enhance the meanings of propositions and overall hierarchy for sub-domains such as the splitting of inverse into INVERSE and NO INVERSE. Additional progressive differentiating links from previous end nodes result in extensions of hierarchical levels as Dora tried to "clarify some of the labels by decomposing it so that they would be understood." An example of this extension is the addition of another hierarchical level and nodes such as rows of zeros, no rows of zero, Gauss-Jordan Elimination, [X.sub.1] = [b.sub.1]/[a.sub.1] and [X.sub.2] = [[a.sub.1][a.sub.2][b.sub.2] - [a.sub.2][a.sub.3][b.sub.1]]/[[a.sub.2.sup.2][a.sub.4] - [a.sub.1][a.sub.2][a.sub.3]] by map J to clarify end nodes at Level 9 of map F. Evidently, Dora's map J was extensively more structured, integrated and differentiated than earlier versions mainly as a result of her growing proficiency and confidence in using concept maps and her own increasing understanding of matrices and systems of equations over the semester. Additionally, thinking metacognitively, reflectively, and dialectically whilst revising and transforming current versions to accommodate feedback from previous critiques prompted the inclusion of additional concepts levels and links to extend maps and enhance the meaningfulness of subdomains. Her active participation in social interactions, communication, and negotiations of meanings with her peers and researcher also contributed significantly to the overall meaningfulness and comprehensiveness of her topic concept map.

Progressive Vee Diagrams

Dora's 4 vee diagrams illustrate four different methods of solving one word problem. The underpinning theoretical principles and main concepts relevant to each of these methods are displayed on the conceptual (thinking) side of the vee diagrams. The hierarchical conceptual interconnections between main and relevant subsidiary concepts of three of the four methods are illustrated in some of the maps from A to J. In particular, the guiding principles for Method 1 in Figure 9 are depicted in maps C, F and J, for Method 2 in Figure 10 are illustrated in maps B, F and J, for Method 3 in Figure 11 are found in maps A and J whilst principles for Method 4 (Figure 12) are not included in any of the maps A to J.

Basically, Dora interpreted and translated the word problem (shown in the element OBJECT in Figure 9) first by introducing variables x, y and z and then algebraically representing each type of can (regular, holiday and party) with its weights of each type of nuts (cashews, peanuts and walnuts) as linear equations hence generating a system of 3 linear equations in 3 unknowns (see the TRANSFORMATION element in Figure 9). From this system, Dora identified 4 methods; three of them depend on transforming the system of equations into a linear matrix equation AX=B as described by the first listed principle in Figure 9 and 11 and as illustrated by the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. -A:X=B branch of maps F and J (Figures 7 and 8). Specifically, Method 1 is conceptually illustrated in the subdomain leading up to [A : B[EROS.[right arrow]]I : X] in map J (Figure 8), generally illustrated in Figure 6 and similarly supported by principles numbered 1 and 2 in Figure 9.

[FIGURE 9 OMITTED]

In contrast to Method 1, Method 2 is an application of Cramer's Rule as supported by the principles listed in vdiagram 2 in Figure 10 and conceptually well-explicated in map J (Figure 8) and depicted generally in map B (Figure 5). The third method (Figure 11) is the application of the multiplicative inverse A-1 determined from [A : I[EROS.[right arrow]]I: [A.sup.-1]] and then using X=[A.sup.-1]B to determine values of unknown as described by principles 2, 3 and 4 of vdiagram 3 in Figure 11 and conceptually illustrated by the A:I-EROS sub-domain of map J and formula-[A.sup.-1] and EROS branches of map A. The rest of the transformations for vdiagram 3 to solve X=[A.sup.-1] was completed on Dora's worksheets to generate the knowledge claim shown. Whilst all three methods share a common base (linear matrix equation AX=B), the fourth method solves the system of equations simultaneously, supported by the listed principles on vdiagram 4 in Figure 12. An independent concept map illustrating interconnections between main concepts of the simultaneous method or integratively reconciling it to the concept of system of equations is not included in the maps A to J under discussion in this paper. It appeared that Dora was able to flexibly and fluidly shift within methods and between multiple methods as a consequence of mapping relevant conceptual interconnections and diagramming supporting principles specific to each method. The illustrated interplay and synthesis on vee diagrams between theoretical principles and methods (the latter are exemplars of applications of the former) routinely consolidates and reinforces for Dora this often de-emphasized theory-application connection

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The use of concept maps and vee diagrams to scaffold and guide metacognitive strategies and reflective thinking facilitated Dora's critical analyses and illustrations of conceptual interconnections of a mathematics topic and problem. Without the visual and explicit level of detail that is naturally part of a map/diagram, communication of the same ideas publicly in a social setting would not have been as effective and as efficient not only for Dora as the constructor and presenter but equally so for her audience of critics. The relative ease of referring to correct, faulty or vague propositions/principles when visually displayed on a concept map/vee diagram enhances the efficiency of communicating, exchanging and discussing mathematical ideas.

[FIGURE 12 OMITTED]

Discussion of Concept Maps and Vee Diagrams

As mentioned in the introduction, Samoan students' recurring problems in learning mathematics seemed to be a mutual interaction between their narrow perceptions of mathematics, inability to critically transfer existing knowledge to new situations, difficulties in justifying and communicating mathematically, and lack of opportunity to practice metacognitive strategies to think critically, reflectively and analytically. Dora's data reported here attempted to explore in-depth the impact of constructing concept maps and vee diagrams on her developing conceptual and integrated understanding of the topic matrices-and-systems-of-equations (MASOE). The literature on metacognition shows that the task of metacognitively reflecting upon one's existing conceptions can foster one's critical, analytical, and reflective thinking (Schoenfeld, 1987). Whilst Dora's case study demonstrated this claim, it also went beyond it by explicitly having her visually map and explicate her understanding of conceptual interconnections on maps/diagrams over the semester. The cognitive and mental activities of identifying, organizing, linking, describing, and evaluating necessarily force Dora to critically reflect and think more deeply about her own existing understanding to enable the construction of maps/diagrams that are meaningful, coherent and make sense. Discussion of the data may be organized around three main points, namely Dora's critical ability to conceptually analyze the structure of knowledge within the topic MASOE, efficiency and effectiveness in communicating mathematically, and developing competence in thinking reflectively and metacognitively.

Dora's developing critical ability to conceptually analyze the MASOE topic was scaffolded and assisted by constructing concept maps and vee diagrams. Her expertise in (a) identifying key concepts and other subsidiary concepts, (b) interconnecting relevant ones and describing the nature of interrelationships in order to develop hierarchical networks of meaningful propositions and (c) identifying key principles to justify main steps of methods of solutions improved over the semester as a result of her own increasing engagement in thinking metacognitively, socially interacting and negotiating meanings during group and one-on-one critiques, and increasing proficiency in constructing maps/diagrams. Her initial maps had different emphasis but these were subsequently structured and integrated into one map with well-developed integrated and differentiated links to illustrate meaningful interconnections and propositions. Counts measuring structural complexity and valid propositions increased significantly by map J whilst her four vee diagrams illustrated conceptually-supported multiple methods each with its own set of key principles. Most of the listed principles on vee diagrams are also indicated and illustrated on the final concept map. The information depicted on both concept maps and vee diagrams mutually reinforce and complement each other thereby portraying and emphasizing both the conceptual and methodological elements on the MASOE topic, not just the procedural elements at the expense of the conceptual.

Over time, Dora found it easier to communicate her understanding and intentions to others by referring to visually illustrated hierarchical connections on her maps and identifying principles/concepts and methods on vee diagrams. The visual depiction of conceptual interconnections made the communication of her ideas more explicit and effective for her critics as well as making it easier for her critics to critique her work in general and in particular. Her fluency with, and usage of appropriate language, concepts and principles to justify hierarchical interconnections in maps and solutions in vee diagrams reflected her growing confidence in her own integrated and differentiated understanding of concepts and methods.

Dora noted in her final report that comments from peer and one-on-one critiques were constructive and insightful which in turn prompted her to be more critical and reflective in her revisions. As a consequence, her revisions generated additional nodes, progressive differentiation and integrative reconciliation links, succinct and concise labels, and extensions of hierarchical levels resulting in concept maps that were structurally more complex and cohesive. Because she had to defend and justify her conceptual interconnections and vee diagram analyses to her peers and researcher, Dora was relatively more thorough and reflective in planning and organizing her revisions to ensure that previous comments had been addressed appropriately and that she was able to justify the overall meaningfulness of maps/diagrams.

Conclusions and Recommendations

Dora's mathematical perceptions, critical application of existing knowledge and communicating mathematically have been influenced substantively as a result of having to construct progressive concept maps and vee diagrams.

Firstly, Dora developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Her end-of-study perceptions of mathematics had evidently expanded beyond being able to solve problems competently to include in addition a well-structured, integrated and differentiated knowledge of conceptual interconnections to illustrate the structure of knowledge within a topic and relevant principles pertinent to solving a problem.

Secondly, the construction of comprehensive topic concept maps and multiple-solution vee diagrams raised her critical awareness and practical understanding of the interconnections between mathematical concepts and procedures which is critical for effective and novel problem solving.

Thirdly, verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. The interconnections are visually displayed making it easier to publicly evaluate and assess mathematical correctness and validity of statements.

Conclusions suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these meta-cognitive tools into students' repertoire of effective learning strategies. The mental activities of critical, analytical and reflective thinking are sharpened and enhanced by the cognitive demands of: (a) organizing concepts, constructing and describing interconnections in a topic (or problem) and (b) identifying and articulating key principles that support and guide methods and procedures for solving problems. The presentation of completed maps/diagrams can focus and foster social interactions, negotiations of meanings, communication and exchange of ideas in classroom settings. Findings from this case study suggest that there is educational value in incorporating these metacognitive tools into normal classroom teaching and learning activities.

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Karoline Afamasaga-Fuata'I

University of New England, Australia 

(1) Concept maps have been carefully re-drawn for legibility but still maintaining their original hierarchical structure.

(2) Actual concept labels and linking words used by Dora on her concept maps will be italicised.

(3) AEF means that the relevant concept appeared in map A then map E and map F in that order. This notion will be used herceforth to indicate the sequence of appearances of concepts in the various versions A up to J.

Progressive Concept Maps--Structural Complexity & Valid Propositions

                   Counts
             A   B   C   F   J

H/Levels      8   8   6   9  10
M/B Nodes     5   4   7   7  11
S/Branches    3   3   2   4   6
Cross-Links  10   6   7   8  23
Uplinks       3   5   3   1   3
ValidProps   17  20  14  25  54

Structural Complexity & Valid Propositions
H/Levels -- Hierarchical Levels; M/B Nodes -- Multiple Branching Nodes
S/Branches -- sub-Branches; ValidProps -- Valid Propositions

Figure 2. Dora's Progressive concept Maps--Structural Complexity & Valid
Propositions.

Note: Table made from bar graph.