Concept Mapping as an Assessment Tool for Conceptual Understanding in Mathematics

Haiyue JIN

• Verlag: Routledge
• Jahr: 2022
• ISBN:
  9781000585629
• Auflage: 1
• Thema: Pedagogía
• Thema: Educación general

This book investigates the practicability and effectiveness of the concept map as a tool for assessing students' conceptual understanding in mathematics.

The author first introduces concept mapping and then employs it to investigate students' conceptual understanding of four different mathematical topics. Alongside traditional scoring methods, she adopts Social Network Analysis, a new technique, to interpret student-constructed concept maps, which reveals fresh insights into the graphic features of the concept map and into how students connect mathematical concepts. By comparing two traditional school tests with the concept map, she examines its concurrent validity and discusses its strengths and drawbacks from the viewpoint of assessing conceptual understanding. With self-designed questionnaires, interviews, and open-ended writing tasks, she also investigates students and teachers' attitudes toward concept mapping and describes the implications these findings may have for concept mapping's use in school and for further research on the topic.

Scholars and postgraduate students of mathematics education and teachers interested in concept mapping or assessing conceptual understanding in classroom settings will find this book an informative, inspiring, and overall valuable addition to their libraries.

1IntroductionConcept Mapping for Mathematics

DOI: 10.4324/9781003269373-1

Background

Over recent decades, psychological and educational research on mathematics learning has firmly established the importance of conceptual understanding in terms of our ability to use knowledge flexibly and apply what we learn appropriately in different settings (e.g. Bransford et al., 1999; National Council of Teachers of Mathematics (NCTM), 2000). Educators, researchers, and curriculum designers alike (Afamasaga-Fuata’I, 2008; Kilpatrick et al., 2001; Ministry of Education, Singapore, 2019; NCTM, 2000) have acknowledged this importance and contributed to explorations of the field. Moreover, substantial efforts have been made to measure levels of conceptual understanding (e.g. Niemi, 1996a; Webb and Romberg, 1992; White and Gunstone, 1992; Jones et al., 2019; Ceran and Ates, 2020). However, researchers have long known that, as Niemi (1996a, p. 351) puts it, ‘commonly used achievement tests provide, at best, only indirect and highly limited information on students’ conceptual understanding’. Accordingly, exploration of more direct and informative means of measuring learners’ conceptual understanding in mathematics is needed.

Conceptual understanding is by nature a cognitive construct grounded in various and multiple theories, such as Schema Theory (Rumelhart, 1980; Rumelhart and Ortony, 1977) and Piaget’s (1977) Equilibration of Cognitive Structure Theory. Hiebert and Lefevre’s (1986) description of conceptual knowledge remains highly influential and forms a basis for most contemporary work:

Conceptual knowledge is characterized most clearly as knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions so that all pieces of information are linked to some network.

(pp. 3–4)

Accordingly, the extent of not only our knowledge of relevant concepts but also of the relationships between them determines the extent of our conceptual knowledge. A technique that appears to be relevant to the features of conceptual knowledge is Novak and Gowin’s (1984) concept mapping.

Concept map is defined as a two-dimensional graphical representation of knowledge. Concepts, referred to as ‘nodes’, are linked with labelled arrows to denote relations between nodes in a pair (Novak and Gowin, 1984). Although concept mapping has been used extensively in assessing conceptual understanding in science education (Novak, 1990; Wallace and Mintzes, 1990) and is considered effective, the practice remains comparatively rare in mathematics education. As mathematics learning and science learning involve many of the same psychological and epistemological properties, it is worth exploring whether concept mapping can be likewise effective adapted to mathematics (Novak, 2006) and thereby recommended for more extensive applications. This book aims to introduce the application of concept mapping as an assessment of students’ conceptual understanding of mathematics.

Toward Deeper Assessment of Conceptual Understanding

In many countries, school education is exam-oriented (e.g. Tan, 2006; Zhang et al., 2004), and although contemporary mathematics curricula emphasise development of learners’ conceptual understanding, commonly used school measures primarily emphasise problem-solving proficiency. Lack of relevant and effective assessment tools and strategies that directly address students’ conceptual understanding makes implementing such curricular demands difficult and hinders timely intervention when students’ understanding is in need of adjustment. This book is a response in particular to the need for effective assessment techniques and tool for measuring students’ conceptual understanding in mathematics.

When exploring novel assessment methods, researchers should keep in mind fundamental elements of assessment that ensure that meaningful inferences are drawn. Pellegrino et al. (2001) identified three elements of effective assessment, namely, cognition, observation, and interpretation, also known as the Assessment Triangle. Ruiz-Primo and Shavelson (1996) characterised assessments, particularly concept mapping assessments, in terms of assessment task, students’ responses, and scoring of responses. As these two models are complementary, in the present study they are combined to form the following categories, assessment task, student response, and interpretation. The assessment task element involves checking that the task assigned is backed by solid theories or beliefs indicating that it can represent student knowledge in a subject domain. The student response element involves whether the examiner can observe and infer students’ performance of the construct being assessed from the responses provided. Interpretation of responses replaces Ruiz-Primo and Shavelson’s (1996) scoring because interpreting a concept map is much more than scoring. This element involves checking that the interpretation method selected is valid and reliable for drawing inferences from the students’ responses obtained.

For the first element, assessment task, broad theories support concept mapping as an effective means of capturing attributes of conceptual understanding. In cognitive psychology, it is generally agreed that human knowledge is stored in the memory in information packets comprising schemas (Jonassen et al., 1993). When learning occurs, we tend to incorporate new information into our schemas. If the information is contrary to what we already know, we may need to adapt our schemas to better accommodate it. These incorporating and adapting processes are known as assimilation and accommodation, respectively (Piaget, 1977). The balance between the two processes reflects how knowledge develops in the mind. Explorations into concept formation, concept acquisition, and conceptual learning in mathematics (e.g. Sfard, 1991; Skemp, 1987) support the pattern of relations among mathematical concepts and equilibration processes. Skemp argues that to understand a concept is ‘to assimilate it into an appropriate schema’ (1987, p. 29). From this representational view of understanding (Perkins, 1998), conceptual understanding, which emphasises the understanding of concepts, can be defined as the structured interrelatedness among concepts in a domain. Once conceptual understanding is represented externally, it can be assessed by others. A concept map, with its specific attributes, i.e. nodes, links, linking phrases, and structure, can explicitly represent conceptual understanding. It can serve as a ‘window into the mind’ (Shavelson et al., 2005, p. 1).

The second element is student responses. The two main areas affecting students’ responses in concept mapping tasks (CM tasks) are their concept map drawing skills and the quality of their conceptual understanding. Studies on the use of the concept map as an assessment tool usually begin with an introduction to concept map and training on how to construct one (e.g. Ruiz-Primo, Schultz, et al., 2001; Williams, 1994). Concept mapping refers to the process of drawing a concept map. Training students on concept mapping is necessary, just as training them on the use of software computer in computer-based test assessment is necessary. Most researchers have found that students can quickly learn to construct concept maps with limited practice (e.g. Freeman, 2004; Ruiz-Primo, 2004; Ruiz-Primo, Schultz, et al., 2001). However, others argue that concept mapping imposes high cognitive demands on students by requiring them to identify important concepts, relationships, and structures within a given domain of knowledge (e.g. Novak and Gowin, 1984; Schau and Mattern, 1997). As such, students may experience initial difficulty constructing concept maps.

These two positions seem somewhat conflicting, and the research has not adequately addressed the optimal amount of training required, taking into account different levels of students. These issues are of concern for the further development and application of concept mapping in assessment. In addition, there is a gap in most studies between students’ concept mapping training and the mapping performance criteria used by examiners to gauge their conceptual understanding; that is, examiners (e.g. Edwards and Fraser, 1983; Jin, 2007; Williams, 1998) seldom evaluate students’ mapping skills after training but instead proceed directly to assessing students’ constructed concept maps, thus assuming that such maps are faithful representations of students’ conceptual understanding. These studies assume that participating students have equivalent mapping skills after the same training and that those who draw better concept maps have higher levels of conceptual understanding. These assumptions are questionable because it is quite possible for a student who is experienced in concept mapping but who has a low level of conceptual understanding to construct a higher-level concept map than a student with a high level of conceptual understanding but who is not clear on the concept map construction process. In such cases, it is unfair to assess students’ understanding by comparing their concept maps. Skill at mapping and its relationship with the resulting maps should be clarified before going deeper into any study treating students’ concept maps as representations of their conceptual understanding.

Once a concept map is completed, interpreting the information it contains becomes the main concern. This is the third element of the Assessment Triangle. Researchers (e.g. McClure et al.’s, 1999) often build scoring systems for concept maps based on features, such as valid nodes, meaningful propositions, and structure. The scores allow examiners to assess conceptual development by directly comparing different students’ concept maps or comparing concept maps constructed by the same student at different times. Other researchers (e.g. Afamasaga-Fuata’I’s, 2009a, 2009b; Liu and Hinchey, 1996) employ qualitative methods to focus directly on the details of students’ performance, e.g. the connections they make that show insight into or misconceptions about the conceptual relationships. Both scoring methods and qualitative methods are valuable for extracting information embedded in student-constructed concept maps.

This though leads naturally to the question of whether a given interpretation of a concept map is valid; that is, whether the information obtained and the inferences drawn fairly represent the quality of students’ conceptual understanding. The existing literature, however, does not provide satisfactory answers to such questions, and researchers have found that information collected from concept maps and other measures, e.g. interviews, writing tasks, multiple-choice tests, and the Science Achievement Test (e.g. Edwards and Fraser, 1983; Hoz et al., 1990; Novak, 2005) yields inconsistent conclusions about students’ achievement. The inconsistent results may be partially due to students’ lack of familiarity with concept mapping (Snead and Snead, 2004), which would leave them inadequately prepared to effectively map their knowledge of a topic. Ruiz-Primo, Schultz, et al. (2001) have also reported that different mapping tasks, e.g. ‘fill-in-the-map’ and ‘construct-a-map’ tasks, are not equivalent in their capacity for representing students’ knowledge. In addition, evidence to date suggests that concept maps measure different aspects of achievement, compared to other instruments (Anderson and Huang, 1989; Markham et al., 1994; Novak et al., 1983). Different scoring methods may also result in differing conclusions about students’ mapping performance (Ruiz-Primo, 2004). Given these inconsistencies, whether a concept map is a valid representation of conceptual understanding should be further examined, with attention to mapping skills and mapping formats.

Based on these considerations, this book is guided by following research questions:

1. Do students need training to construct informative concept maps? What level of training is appropriate and sufficient?
2. What attributes of students’ conceptual understanding can be assessed from student-constructed concept maps?
3. What differences among mathematical topics, if any, should be considered in the use of concept mapping as a technique for assessment of students’ conceptual understanding?
4. Is concept mapping a valid technique for assessing students’ conceptual understanding in mathematics?
5. What are students’ and teachers’ attitudes toward concept mapping in mathematics?

The main study described in this book began with training on concept mapping to prepare the participating students with basic knowledge about what a concept map is and how to construct one. The effectiveness of the training is examined to ensure that they were prepared with the ...

Inhaltsverzeichnis

• Cover
• Half Title
• Title
• Copyright
• Contents
• List of figures
• List of tables
• Preface
• 1 Introduction: Concept Mapping for Mathematics
• 2 Conceptual Understanding in Mathematics and Its Measures
• 3 Why Concept Mapping?
• 4 Developing a Training Programme for Concept Mapping in Mathematics
• 5 Assessing Conceptual Understanding in Mathematics With Concept Mapping
• 6 Comparing Concept Mapping and School Tests
• 7 Attitudes Toward Concept Mapping
• 8 Summary and Implications
• Appendices
• References
• Index