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Constructivism is seen as the way forward for mathematics education that has the potential to vastly improve the teaching and practise of mathematics in the classroom. Throughout this essay I will define constructivism and revert the concept of constructivism to the classroom, explore the various constructivism positions, take a look at constructivism in the mathematics classroom today and accompany the position of constructivism with its undoubted benefits and the teaching methodologies and characteristics that accompany this type of practise.
Constructivism is a philosophy of learning that solely focuses on learning from experiencing. "By reflecting on our experiences, we construct our own understanding of the world we live in" (Funderstanding, 2008). Constructivism emphasizes the importance of the knowledge, beliefs and skills that an individual develops through different experiences throughout their lives. It recognizes the creation of new understanding as a combination of prior learning, new information, and readiness to learn (Education Broadcasting Cooperation, 2004). In the case of constructivism individuals therefore make choices about what new ideas to accept into their understanding and how to incorporate these new ideas and perceptions into their pre-established outlook on the world. "Learning, therefore, is simply the process of adjusting our mental models to accommodate new experiences (Funderstanding, 2008).
Constructivism in the Classroom
"Constructivist philosophies focus on what students can do to integrate new knowledge with existing knowledge to create a deeper understanding of the mathematics" (Stiff, 2001). In the classroom, constructivism takes the idea of combining what they already know, with what they newly learn through experiential learning and applies it to a variety of teaching methodologies. Through this idea of teaching and learning students are encouraged through active learning techniques (e.g. group-work, real-world problem solving) to create more knowledge and then to reflect on and talk about what they are doing and how their understanding is changing (Education Broadcasting Cooperation, 2004). This type of learning is defined as a "constructive process in which students attempt to resolve problems that arise as they participate in the mathematical practices of the classroom" (Anderson, Reder & Simon, 2000). Therefore, constructivism in the mathematics classroom is learning mathematics by experiencing them through active/ engaging activities.
The Practise of Constructivism
Constructivism cannot be defined as one specific philosophy as there exists several forms thoroughly described throughout the literature. For example; Good, et al. (1993) describes 15 different forms of constructivism including; contextual, dialectical, empirical, humanistic, information-processing, methodological, moderate, Piagetian, post-epistemological, pragmatic, radical, rational, realist, social, and socio-historical (p. 74). To take a look at just two of these forms that shed light on the teaching of mathematics in schools I will focus on radical and social constructivism.
As discussed by Lee V. Stiff, from the national centre of teachers of mathematics (2001); Radical constructivismÂ is the philosophy that knowledge cannot be provided in some final form from parent to child or from teacher to student but must be actively assembled in the mind by each learner in his or her own way. What he is saying therefore is that students learn by constantly experiencing and developing their knowledge base through activity and reflection based on their own interpretation and how they process and perceive information rather than directly taking passed on information interpreted by another source as a final form of new knowledge without one's own elucidation. "The responsibility for expanding what one knows, or for constructing new knowledge, rests primarily on the learner and his or her efforts to achieve understanding" (Stiff, 2001).
Following from this, Stiff provides an outlook into Social constructivism. He explains that this form of constructivism states that "students can better build their knowledge when it is embedded in a social context" (2001). Thus, the interaction between teacher and students is enhanced when it involves a broader community of learners such as students working together. In this type of environment students are provided with the opportunity to engage with eachother and work together to achieve common goals. Through this active approach students can help one another create richer meanings for new mathematical content by offering up different ideas and views on different concepts. An example of where social constructivism can be incorporated into the mathematics classroom is through problem solving where interaction should take place between the students where they are encouraged to develop their own strategies for solving problem situations.
The Mathematics Teacher as a Constructivist
From the readings it is clear to say that the philosophy of constructivism identifies the students as an "active participant in the teaching and learning process" (Stiff, 2001). Taking this into account, the main role of the teacher is therefore, to facilitate the active engagement of all pupils through developing engaging learning activities where students interact with one another and "foster the integration and extension of knowledge among students" (Stiff, 2001). Teachers are advised to and should constantly vary their teaching styles and strategies in order to foster student engagement and interaction. "Good teachers use different strategies at different times for different purposes" (Stiff, 2001).
Constructivist teachers have many characteristics that set them apart from the 'traditional' teacher. As outlined by the Education Broadcasting Company (2004) what sets a constructivist teacher apart from a traditional teacher is that; constructivist teacher's have a dialogue with students in helping them to construct their own understanding and knowledge, their role is therefore interactive and rooted in negotiation, teachers set assessment that includes a variety of the students' work, observations, and points of view, as well as tests (emphasis on the process rather than product), and teachers use a variety of raw materials and interactive materials in an effort to facilitate discovery learning and prevent the students losing interest and becoming demotivated to learn mathematics. "The constructivist teacher sets up problems and monitors student exploration, guides the direction of student inquiry and promotes new patterns of thinking" (SEDL, 1995).
In 1996, Paul Ernest outlined some 'pedagogical implications" of constructivism in the classroom with the teacher as a facilitator. He claimed that the teacher must; provide sensitivity towards attentiveness to the learners previous constructions, use cognitive conflict techniques to help students challenge their own thinking, pay attention to the way that students learn and adapt their thinking, use multiple representations of information, be aware of the importance of goals for the learner and be aware of the importance of social contexts (street knowledge v school knowledge). Ernest sets out clear guidelines for the role of the constructivist teacher but characteristics can change with a changing classroom environment especially as pupils in a constructivist environment have autonomy over their own learning, therefore teachers must be ready to adapt and be prepared for changes in environment.
Overall, as outlined by Ishii (2003) constructivist teachers should; encourage and accept student autonomy and initiative, use raw data and primary sources, use cognitive terminology, allow student responses to drive lessons, inquire about students' understandings, encourage students to engage in dialogue, encourage student inquiry and discussion by asking quality, open-ended questions and nurture students' natural curiosity. By doing this, students will be provided with the opportunity and necessary environment to promote critical thinking and problem solving of mathematical concepts which will enhance their understanding and promote a possible liking and interest in what is a negatively perceived subject.
The Benefits of Constructivism for Mathematics
Constructivism is the way forward for mathematics teaching. It is seen right throughout the new project maths curriculum in secondary schools and throughout mathematics pedagogical curricula in colleges. It provides countless benefits towards the implication and practise of mathematics in the classroom. It has the potential to improve practice in the classroom for the enhancement of pupil engagement and interpretation of the mathematical curriculum content. Some noteworthy benefits include the following;
Children learn more and enjoy learning more when they are actively involved rather than passive learners.
Education works best when it concentrates on thinking and understanding, rather than on rote memorization
Constructivist learning is transferable throughout other subject areas.
Constructivism gives students ownership of what they learn.
Constructivist assessment engages the students' initiatives and personal interpretations.
By grounding learning activities in an authentic, real-world context, constructivism stimulates and engages students.
Constructivism promotes social and communication skills by creating a classroom environment that emphasizes collaboration and exchange of ideas.
Students must learn how to articulate their ideas clearly as well as to collaborate and negotiate on tasks effectively by sharing in group projects.
(Education Broadcasting Cooperation, 2004)
With all these benefits it is clear that pupils are been taken away from the traditional alternative that offers merely; rote learning, passive attention and limited student engagement.
Constructivism in the Mathematics Classroom Today
Whilst the benefits far outweigh the negatives, the implementation of the constructivist style of learning faces numerous challenges today, starting with its acceptance among 'traditional style' educators. As with everything, it is always going to be difficult to change the already indented beliefs and values of a teacher. Traditional style teachers believe constructivism to de-value the role of the teacher and 'expert knowledge'. But "contrary to criticisms by some (conservative/traditional) educators, constructivism does not dismiss the active role of the teacher or the value of expert knowledge" (Education Broadcasting Cooperation, 2004). Instead constructivism alters the role of the teacher by emphasizing the student as more in charge of constructing their own knowledge and the teacher taking a step back to guide the students in the interpretation and building of this new knowledge, therefore students are not directly targeted with piles of factual information that they cannot comprehend as their own. "Constructivism transforms the student from a passive recipient of information to an active participant in the learning process" (Education Broadcasting Cooperation, 2004).
Another limitation that constructivism faces is through the students. The students already will have long-formed habits through the traditional style of teaching that will be hard to change and some students may be pro-foundly against the new style of engaging and critical thinking their way through maths whilst others will be more interested and value mathematics more as a subject relevant for life. Other issues relating to students is; as students come from a variety of backgrounds and ways of thinking which include pre-determined myths, beliefs on mathematics, cultural influences, peer and family influences etc., "when presented with information in the classroom that contradicts existing ideas, a student may try to accommodate both interpretations, rather than change deeply held beliefs" (SEDL, 1995). The best way to counteract such a problem is by easing the students in slowly to the new style of learning as "content is embedded in culture and it is difficult to separate the two" (SEDL, 1995). Classrooms therefore need to promote the exchange of personal views from all students so that students can learn from eachother and build upon interpretation of their own knowledge against others' knowledge and predetermined knowledge. Constructivist classrooms can provide this type of learning environment through interactive and engaging activities.
So as we can see, throughout this essay, constructivism is a learning theory that most certainly has the potential to improve practice in mathematics classrooms. "Constructivism has done a service to science and mathematics education: by alerting teachers to the function of prior learning and extant concepts in the process of learning new material, by stressing the importance of understanding as a goal of science instruction, by fostering pupil engagement in lessons, and other such progressive matters" (Matthews, 2000). Therefore, constructivism has made educators aware that there needs to be a move from the 'traditional' style of mathematics in order for there to be a deeper change in the teaching and learning of mathematics so that the beliefs and values of all can be altered for the better of mathematics education for future generations.