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## 1. Introduction

Over the past decades mathematical education has been the subject of many controversial discussions relating as much to its goals and content as to the nature of processes of learning and teaching mathematics. For instance, the so-called ‘new math reform’ of the 1960s and 1970s, which resulted in substantial changes in the mathematics curriculum in many countries, afterwards often elicited heated debates, and the same holds true for issues relating to teaching and learning mathematics, such as the role of discovery learning, and the use of technological devices in general, and the calculator in particular, in the mathematics classroom. This research paper does not allow us to give a complete overview of this vast domain of inquiry. Therefore, we will focus selectively on a few major aspects of mathematical education: (a) a dispositional view of the goals of mathematics education; (b) mathematics learning as the construction of knowledge and skills in a sociocultural context; (c) designing powerful teaching–learning environments for mathematics; and (d) constructing innovative forms of assessment instruments tailored to the new goals of mathematics education. For a more complete overview, we refer to the following volumes in which the vast body of studies has been excellently reviewed: Handbook of Research on Mathematics Teaching and Learning (Grouws 1992); International Handbook of Mathematics Education (Bishop et al. 1996); Theories of Mathematical Learning (Steﬀe et al. 1996). These volumes also show that since the 1970s mathematical education has evolved into an interdisciplinary ﬁeld of study in which instructional psychologists, mathematics educators, mathematicians, and anthropologists are major participants. Within this community has developed an enriched conception of mathematics learning as involving the construction of meaning, understanding, and problem-solving skills based on the modeling of reality. During the same period, several important shifts have taken place in both conceptual and methodological approaches to mathematical education: from a concentration on the individual to a concern for social and cultural factors; from ‘cold’ to ‘hot’ cognition; from the laboratory to the classroom as the primary setting for research; and from a mainly quantitative experimental approach to a more diversiﬁed methodological repertoire including qualitative and interpretative techniques.

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## 2. A Dispositional View of Mathematics Learning

This section focuses on what students should learn in order to acquire competence in mathematics. In this respect, the more or less implicit view which often still prevails in today’s educational practice is that computational and procedural skills are the essential requirements. This contrasts sharply with the view that has emerged from the research referred to above. Indeed, there is nowadays a broad consensus that the major characteristics underlying mathematical cognition and thinking are the following (see De Corte et al. 1996, Schoenfeld 1992):

(a) A well-organized and ﬂexibly accessible domainspeciﬁc knowledge base involving facts, symbols, conventions, deﬁnitions, formulas, algorithms, concepts, and rules that constitute the contents of mathematics as a subject-matter ﬁeld.

(b) Heuristic methods, that is search strategies for problem solving that do not guarantee that one will ﬁnd the solution, but substantially increase the probability of success because they induce a systematic approach to the task (e.g., thinking of an analogous problem; decomposing a problem into subgoals; visualizing the problem using a diagram or a drawing).

(c) Metacognition, involving knowledge and beliefs concerning one’s own cognitive functioning on the one hand (e.g., believing that one’s mathematical ability is strong), and skills relating to the self-regulation of one’s cognitive processes on the other (e.g., planning a mathematical solution process; monitoring an ongoing solution process; evaluating and, if necessary, debugging a solution; reﬂecting on one’s mathematics learning and problem-solving activities).

(d) Aﬀective components involving beliefs about mathematics (e.g., believing that solving a mathematics problem requires eﬀort versus believing that it is a matter of luck), attitudes (e.g., liking versus disliking story problems), and emotions (e.g., satisfaction when one ﬁnds the solution to a diﬃcult problem).

It is certainly useful to distinguish these four categories of components, but it is also important to realize that in expert mathematical cognition they are applied integratively and interactively. For example, discovering the applicability of a heuristic to solve a geometry problem is generally based, at least partially, on one’s conceptual knowledge about the geometrical ﬁgures involved. A negative illustration is that some beliefs observed in students, such as ‘solving a math problem should not last more than a few minutes’, will inhibit a thorough heuristic and metacognitive approach to a diﬃcult mathematics problem. But, while this integration of the diﬀerent components is necessary, it is not yet a suﬃcient condition to overcome the phenomenon of inert knowledge observed in many students: although the relevant knowledge is often available and can even be recalled on request, students do not spontaneously apply it in situations where it is appropriate to solve a new math problem. In other words, acquiring competence in mathematics involves more than the mere sum of the four components listed above. As a further elaboration of this view, the notion of a ‘mathematical disposition’ introduced in the Curriculum and E aluation Standards for School Mathematics (National Council for Teachers of Mathematics 1989, p. 233) in the USA, points to the integrated availability and application of the diﬀerent components:

Learning mathematics extends beyond learning concepts, procedures, and their application. It also includes developing a disposition toward mathematics and seeing mathematics as a powerful way for looking at situations. Disposition refers not simply to attitudes but to a tendency to think and to act in positive ways. Students’ mathematical dispositions are manifested in the way they approach tasks—whether with conﬁdence, willingness to explore alternatives, perseverance, and interest—and in their tendency to reﬂect on their own thinking.

According to Perkins et al. (1993), the notion of disposition involves, besides ability, inclination and sensitivity; the latter two aspects are essential in view of overcoming the phenomenon of inert knowledge. Inclination is the tendency to engage in a given behavior due to motivation and habits; sensitivity refers to the feeling for, and alertness to, opportunities for implementing the appropriate behavior. Ability, then, combines both the knowledge and the skill—in other words, most of the components mentioned above—to deploy that behavior. The acquisition of a disposition—especially the sensitivity and inclination aspects—requires extensive experience with the different categories of mathematical knowledge and skills in a large variety of situations. As such, a mathematical disposition cannot be directly taught, but has to develop over an extensive period of time.

## 3. Mathematics Learning as the Construction of Meaning in Sociocultural Contexts

The question arises, then, as to what kind of learning processes are conducive to the attainment of the intended mathematical disposition in students. The negative answer seems to be that this disposition cannot be achieved through learning as it occurs predominantly in today’s classrooms. Indeed, the international literature bulges with ﬁndings indicating that students in our schools are not equipped with the necessary knowledge, skills, beliefs, and motivation to approach new mathematical problems and learning tasks in an eﬃcient and successful way (see, e.g., De Corte 1992). This can largely be accounted for by the prevailing learning activities in today’s schools, consisting mainly of listening, watching, and imitating the teacher and the textbook. In other words, the dominating view of learning in the practice of mathematical education is still the information-transmission model: the mathematical knowledge acquired and institutionalized by past generations has to be transmitted as accurately as possible to the next generation (Romberg and Carpenter 1986).

An additional shortcoming of current mathematical education, which is related to the inappropriate view of learning as information absorption, is that knowledge is often acquired independently from the social and physical contexts from which it derives its meaning and usefulness. This has become very obvious through a substantial amount of research carried out since the mid-1980s on the inﬂuence of cultural and situational factors on mathematics learning, and commonly classiﬁed under the heading ‘ethnomathematics and everyday mathematical cognition’ (Nunes 1992). For example, a series of investigations on so-called ‘street mathematics’ has shown that a gap often exists between formal school mathematics and the informal mathematics applied to solve everyday, real-life problems.

The preceding description of the learner as an absorber and consumer of decontextualized mathematical knowledge contrasts sharply with the conception supported by a substantial amount of evidence in the literature showing that learning is an active and constructive process. Learners are not passive recipients of information; rather, they actively construct their mathematical knowledge and skills through interaction in meaningful contexts with their environment, and through reorganization of their prior mental structures.

Although there are conceptual diﬀerences along the continuum from radical to realistic constructivism, the idea is broadly shared that learning is also a social process through which students construct mathematical knowledge and skills cooperatively; opportunities for learning mathematics occur during social interaction through collaborative dialog, explanation and justiﬁcation, and negotiation of meaning (Cobb and Bauersfeld 1995). Research on small-group learning supports this social constructi ist perspective: cooperative learning can yield positive learning eﬀects in both cognitive and social-emotional respects. However, it has also become obvious that simply putting students in small groups and telling them to work together is not a panacea; it is only under appropriate conditions that small-group learning can be expected to be productive. Moreover, stressing the social dimension of the construction of knowledge does not exclude the possibility that students also develop new knowledge and skills individually. In addition, most scholars share the assumption of the so-called cultural constructi ist perspective that active and constructive learning can be mediated through appropriate guidance by teachers, peers, and cultural artifacts such as educational media.

## 4. Designing Powerful Teaching–Learning Environments

Taking into account the view of mathematical learning as the construction of meaning and understanding, and the goal of mathematics education as the acquisition of a mathematical disposition involving the mastery of diﬀerent categories of knowledge and skills, a challenging task has to be addressed. It consists of elaborating a coherent framework of research-based principles for the design of powerful teaching–learning environments, i.e., situations and contexts which can elicit in students the learning activities and processes that are conducive to the intended mathematical disposition.

A variety of projects attempting the theory-based design of powerful mathematics learning environments has already been carried out (see De Corte et al. 1996 for a selective overview), reﬂecting the methodological shifts toward the application of teaching experiments in real classrooms and toward the use of a diversity of techniques for data collection and analysis, including qualitative and interpretative methods.

For example, Lampert (1986) has designed a learning environment that aims at promoting meaning construction and understanding of multiplication in fourth graders by connecting and integrating principled conceptual knowledge (e.g., the principles of additive and multiplicative composition, associativity, commutativity, and the distributive property of multiplication over addition) with their computational skills. She starts from familiar and realistic problems to allow children to use and explore their informal prior knowledge, and practices collaborative instruction whereby she engages in cooperative work and discussion with the whole class. Students are solicited to propose and invent alternative solutions to problems that are then discussed, including their explanation and justiﬁcation. It is obvious that this learning environment also embodies a classroom climate and culture that diﬀers fundamentally from what is typical of traditional mathematics lessons.

A second and more comprehensive example is Realistic Mathematics Education (RME) developed in the Netherlands. RME, already initiated by Freudenthal in the 1970s, conceives mathematics learning essentially as doing mathematics starting from the study of phenomena in the real world as topics for mathematical modeling, and resulting in the reinvention of mathematical knowledge. Based on this fundamental conception of doing mathematics the design of ‘realistic’ learning environments is guided by a set of ﬁve inter-related principles: (a) learning mathematics is a constructive activity; (b) progressing toward higher levels of abstraction; (c) encouraging students’ free production and reﬂection; (d) learning through social interaction and cooperation; and (e) interconnecting knowledge components and skills (Treﬀers 1987).

These and other representative examples (see, for instance, Cobb and Yackel 1998, Cognition and Technology Group at Vanderbilt 1997, Fennema and Romberg 1999, Verschaﬀel et al. 1999) illustrate eﬀorts within the domain of research on mathematics learning and teaching to implement innovative educational settings embodying to some degree ideas that have emerged from theoretical and empirical studies, such as the constructivist view of learning, the conception of mathematics as human activity, the crucial role of students’ prior—informal as well as formal— knowledge, the orientation toward understanding and problem solving, the importance of social interaction and collaboration in doing and learning mathematics, and the need to embed mathematics learning into authentic and meaningful contexts, as well as to create a new mathematics classroom culture. The results of those projects reported so far are promising as they demonstrate that this kind of learning environment can lead to fundamental changes in the sort of mathematics knowledge, skills, and beliefs that children acquire, and to making them more autonomous learners and problem solvers. However, these projects also raise questions for future research. For example, there is a strong need for additional theoretical and empirical work aiming at a better understanding and ﬁne-grained analysis of the acquisition processes that this type of learning environment elicits in students, of the precise nature of the knowledge and beliefs they acquire, and of the critical dimensions that can account for the power of such environments.

## 5. Constructing Innovati e Forms of Assessment Instruments

To support the implementation in educational practice of the innovative approach to mathematics learning and teaching, and especially to evaluate the degree of attainment of the new goals of mathematics education, appropriate forms of assessment instruments are required. In this respect, traditional techniques of educational testing, predominantly based on the multiple-choice item format, have been severely criticized as being inappropriate for evaluating students’ achievement of the intended objectives of mathematics education. As a consequence, an important line of research has been initiated since the early 1990s aiming at the development of alternative forms of assessment which are tailored to the new conception of the goals and nature of mathematics learning and teaching, and which reﬂect more complex, real-life or so-called authentic performances (Romberg 1995). At the same time the need for a better integration of assessment with teaching and learning, as well as the importance of assessment instruments to yield information to guide further learning and instruction have been emphasized.

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