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This reflective paper aims to illuminate some of the challenges that mathematics educators, including myself, face in an attempt to implement new sets of ideas called for by the mathematics reform communities. The paper begins with a discussion of the epistemological and ontological position of constructivism. A discussion of the conceptual and pedagogical challenges of implementing constructivist instruction will follow. The paper concludes with a discussion of some suggestions the reform communities can embrace as they fashion a vision of constructivism that involves more than theories of learning or teaching. Day (2000) argues that in these uncertain times the aim of education should be to develop skills, processes and understandings that enable the learner to respond to, initiate rapid changes, pose and solve complex problems. To achieve this aim educators are being expected to initiate and teach in reform minded ways. The rapidly expanding socio-cultural and psychological knowledge base on learning coupled with the unique social, technological and economical contexts of learning today calls for a new and disciplined examination of the challenges of constructivist teaching.
Constructivist Epistemology and Ontological Position
Constructivism as opposed to objectivism derives from a philosophical position that human beings have no access to an objective reality i.e. a reality independent of our way of knowing it. Rather knowledge of our world is constructed from our perceptions and experiences, mediated through our previous knowledge. Constructivist ontological position (Guba and Lincoln, 1994) is that realities are relative and that we perceive them in form of multiple, intangible mental constructions, socially and experientially based, local and specific in nature, and dependent for their form and content on the individual persons or groups holding the constructions. In this position also the constructions made including their realities are alterable. Constructivist epistemology is subjective and transactional whereby the learner and the object of the investigation are assumed to be interactively connected. From this perspective, learning is a process by which human beings adapt to their world around them. Von Glasersfeld (1995) argues that we have no way of knowing whether a concept matches an objective reality, but rather our concern should be whether a given concept works i.e. fits within our experiential world. In clarifying the above perspective, Glasersfeld (1995) argues that a concept works or is viable to the extent that it does what we need it to do: to make sense of our perceptions or data, to make an accurate prediction, to solve a problem, or to accomplish a personal goal.
Steffe & Gale (1995) observed a diversity of epistemological perspectives due to the wide spread interest in constructivism among mathematics education researchers, theorist and practitioners leading to a plethora of different meanings for constructivism. Numerous literatures within the constructivism domain support various conceptions of learning and instruction. The various types of constructivism that have emerged include; radical/ cognitive, social, physical, postmodern constructivism and information processing constructivism to name but a few. Literature relative to mathematics educators can be prudently grouped into cognitive/radical and social constructivism. Whether knowledge is seen as socially situated or as individual construction has implications for the way we conceptualize learning.
Conceptual challenges are embedded in the teachers' attempt to understand the epistemological, philosophical, and psychological foundation of constructivism. One of the challenges of teachers embracing constructivist instructions in their classrooms is a lack of a clear understanding of what constructivism is. I must admit that the first time I heard of the term constructivist and its concepts were clearly elaborated to me, was when I was in my Educational Specialist degree program here at Georgia State having taught for 16 years before that. Windschitl (2002) argued that the most powerful determinant of the success or failure of constructivist instruction implementation in classrooms is the degree to which individual teachers understand the concepts of constructivism. Evidence of disconnect between constructivist theories and practices in classrooms have been observed in the literature. For example, Oakes et.al (2000) in a study of middle school educators participating in reforms, observed that " reform efforts to employ student-centered, constructivist pedagogy were routinely thwarted by the lack of opportunity for teachers to delve into theoretical underpinnings of the practices they were expected to enact" (p.xxii). Having been schooled in a traditional teacher-centered model of schooling, my perception of a constructivist learning environment was very limited. This observation also extends to my colleagues in an institution where the mathematics department consists of 14 teachers with a total of over 200 years of teaching experience. This observation poses a challenge to the reform communities and the teacher preparation institution to see to it that there are structures established to make a connection between theoretical perspectives and practices in our schools.
Windschitl (2002) further observes that unfortunately for teachers, the principles of instructions that derive from constructivist explanations for learning have not cohered into any comprehensible, widely applicable models. This leaves me and fellow mathematics teachers searching for constructivist ideas or models that have proven to be effective in the classrooms. To reemphasize the issue of disconnect between theory and practice, Cobb, Yackel, and Wood (1998) warns that "though constructivist theory is attractive when the issue of learning is considered, deep-rooted problems arise when attempts are made to apply it" (p.87). This warning comes against a backdrop of lack of visualization of what constructivist pedagogy looks like since most teachers were schooled in the traditional educational model. This lack of a visual perception of a constructivist pedagogy model, have led to a creation of naÃ¯ve conception of constructivism among teachers especially at the early stages of understanding the concept constructivism. In this naÃ¯ve constructivism, teachers find themselves placing significant amount of faith on the students' ability to structure their own learning, thus interfering with the development of a more sophisticated conception of constructivist teaching. The challenge then is for teachers to reconcile their current belief about learning which for the most part is traditional, with the epistemological conditions necessary to support a constructivist learning environment. Teachers' concerns in this light include whether the activities or tasks they design results in knowledge construction of the learners. In relation to reconciling their beliefs about teaching and learning, with constructivist ideas, teachers still question whether they should let their students adopt and internalize already existing knowledge or construct their own knowledge.
A second conceptual challenge that teachers are faced with, is an understanding of which constructivism is suitable for their teaching. As noted in the previous section, literature relevant to mathematics education has categorized constructivism into cognitive and social or cultural. Cognitive constructivism posits that meaningful learning is rooted in and indexed by personal experience and that learners maintain ideas that seem intuitively reasonable to them (Brown, Collins, & Duguid, 1989).
In contrast to cognitive constructivism is the social constructivism. Ernest (1994) argues that precise definition of social constructivism is a challenge and writes that there is "lack of consensus about what is meant by the term and what its underpinning theoretical bases are" (p.63). He argues that to a social constructivist theory both the individual cognitive sense making and social processes are essential. Social constructivism in this paper will focus on Vygotsky's socio-cultural theory of learning, a social constructivist theoretical perspective that originated in the work of Lev Vygotsky and was elaborated by members of his social-historical group that included Leontiev, and Luria. This theoretical perspective views knowledge as primarily a cultural product (Vygotsky, 1978). In this perspective, knowledge is shaped by both micro- and macro-cultural influences and evolves through increased participation within different communities of practice. Vygotsky's emphasis was on meaningful whole activities e.g. solving authentic mathematical problems, conducting scientific inquiry, creating and interpreting literary text, as opposed to decontextualized skill-building activities as the fundamental units of instruction in educational settings (Vygotsky, 1978). In trying to grasp the underpinnings of social and cognitive constructivism is the dilemma of designing tasks and instruction. The challenge is whether to design learning activities as a collection of individual work aimed at conceptual change as advocated by cognitive constructivism or as a community of learners whose development is measured by participating in a community of practice as advocated by the social constructivist.
Pedagogical challenges of implementing a constructivist learning environment emerge from the more complex approaches to designing curriculum and learning experiences that constructivism calls for. Fundamental to this challenge is managing to balance between systems prescribed learning objectives (like in the Georgia Performance Standard GPS) with opportunities for students' to construct their own knowledge. Teachers' dilemma here is on whether to base their teaching on students' existing ideas rather than the adopted GPS learning objectives. Teachers find themselves in a dilemma of whether to honor students' attempts to think for themselves while at the same time remaining realistic to the accepted disciplinary ideas some of which trace their origin to the absolutist view of teaching mathematics. Unfortunately, the teaching style emphasized in absolutist philosophical spaces is one that tends to rely on practices that promote rote learning, memorization of formula, one-way to solve problems, and adherence to procedures and drill. In a learning environment where some students are performing below grade level in reading, mathematics, and language arts (as assessed by their CRCT performance) implementing constructivist instruction can prove to be a challenge. One of my greatest challenges in implementing lesson instructions advocated for by constructivists is the demand for designing conceptual questions that promote critical thinking. This is a balancing act between designing conceptual questions that take into consideration the learning difficulties of some of my students while at the same time factoring in the cognitive demand required. As Sfard (2000) describes it "whatever is done out of a sole concern about mathematics is likely to hurt the child, and whatever is done for the sake of the child invariably compromises some mathematical contents and skills" (p.354). In addition the need for teachers to become facilitators of learning calls for professional development on skills and strategies necessary for mastering the art of facilitation.
In this constructivist learning environment teachers must not only be familiar with the principles underlying a topic of study but also be prepared for a variety of ways in which these principles can be explored by the learners. Classroom management concerns, where new kinds of discourse like open-ended learning tasks and collaborative work is another pedagogical challenge mathematics teachers have to deal with. Included also in this pedagogical challenge is the time related dilemma. Teachers still find themselves challenged as to the most effective and equitable way of allocating time in the classroom. Time constraints are also evident in training and implementation of collaborative learning/teaching communities, designing curriculum and innovative activities for students.
Discussion and Suggestions
In implementing teaching practices advocated for the by constructivist reform communities, this paper propagates the stand that teachers should be made aware of the complexity and challenges involved. To transform practice that can sustain progressive educational change, researchers, reformers, and practitioner must jointly fashion a vision for constructivism that involves more than theories of learning or instruction (Windschitl, 2002). Education reformers and stakeholders, who gravitate towards a constructivist approach to teaching, should be challenged to fashion professional development programs that are cognizant of the challenges that come with implementing a constructivist learning environment. For a successful constructivist reform effort, a host of teacher skills will be required. Conceptual understanding challenge calls for structures and models that will assist teachers in making sense of constructivism to an extent they are able to flexibly apply its principles in various instructional contexts. These structures will also facilitate teachers' recognition of the limitations and constraints in a constructivist environment.
Understanding pedagogical challenges will call for structures and models that aim at promoting an understanding that teachers will be challenged to become critically cognizant of the dynamics in their own classroom culture and attend to patterns of the classroom discourse as well as the thinking that goes with them. Equivalently, teachers will be challenged to learn to capitalize on rather than suppress differences in students' existing understanding due to different backgrounds. Finally this paper informs us of the need to form communities of learners within the mathematics education community in our schools that can start conversations around the subject of implementing constructivism within learning environments