Rote and Traditional Teaching Methods in Mathematics Have Often Contributed to a Lack Of

Rote and traditional teaching methods in mathematics have often contributed to a lack of mathematical understanding for learners. The focus of traditional mathematical instruction appears to be on developing mathematical fluency rather than understanding (Fuson, 2009). Learners experiencing difficulties in mathematics require supports for developing meaning and connecting prior knowledge with new concepts.

When observing learners attempting to solve mathematical problems, the methods employed by students are often very different from the culturally valued and efficient strategies enforced in traditional instruction. With guidance from instructors or more capable peers, learners can move beyond more primitive methods toward more desirable methods, exploring a richer mathematical understanding. For this transition to be realized, it is important that classrooms provide environments where math concepts are supported by meaningful situations and visual supports that aid learners in developing meaning (Fuson, 2009).

Vygotsky (1978) proposed that learning occurs within a zone of proximal development (ZPD) for learners. This zone is the distance between the actual development level determined by individual problem solving ability and the potential development level as determined by problem solving under guidance from instructors or more capable peers (Vygotsky, 1978). Teachers are able to best influence learning and help learners reach full potential by providing authentic tasks within a student’s ZPD. “(T)eaching has to start from the spontaneous concepts of the children and at the same time run ahead of actual development: learning is only good if it runs ahead of the actual development, if it is a pacemaker of development, and directed onto the zone of proximal development.” (Steinbring, 2007, p. 96). A major difficulty when working with students in the classroom is that teachers are required to plan activities for entire groups or classes who will likely not share a common ZPD.

Learners differ in cognitive abilities, background knowledge and learning preferences (Huebner, 2010). It has become increasingly important for teachers to implement differentiation in instruction to address the diversity of student needs in the classroom. Hall (2002) described differentiated instruction as “a process to teaching and learning for students of differing abilities in the same class. The intent of differentiating instruction is to maximize each student's growth and individual success by meeting each student where he or she is and assisting in the learning process.” Identifying student ability level is important to determining an individual ZPD to which instruction can be tailored for maximum development. When teachers are aware of student prior knowledge and abilities, they are better able to make decisions about the ways students learn. Students need time and variety in problem solving contexts to develop understanding. Instruction is not effective when learners are asked to work with problems that they are capable of solving independently or with problems that they could not find solutions for even with guidance. Maintaining position within the ZPD is essential to effective learning.

Differentiated instruction is developed in response to student differences and is accomplished by integrating meaningful assessment with instruction while continually reflecting upon and adjusting content, process and product to meet student needs (Huebner, 2010). Challenges for instructors include understanding the mathematical needs and readiness of individual learners while in the collective classroom environment. Meaningful tasks should be designed so that learners are able to show mathematical thinking and not just the ability to follow rote learned procedures or routines (Literacy and Numeracy Secretariat, 2008). Instruction in mathematics should focus on both mathematical fluency and understanding. Introducing elements of choice in activities including how tasks will be carried out and assessed are important to ensure that all students have the opportunity to make contributions to problem solving solutions. The sharing of knowledge and strategies among learners builds a collective knowledge. Setting tasks within students’ ZPDs with elements of choice and opportunities for differing perspectives allow students at various levels of mathematical development to benefit and grow. When students are encouraged to demonstrate their individual knowledge, skills and abilities, all learners can benefit and grow mathematically from the collective knowledge. It is important to consider that it is not always essential for all learners to respond in identical or prescribed manners.

Learning within the ZPD requires scaffolding of concepts for learners. Scaffolding should be adapted to the level of the learner and decrease as skill levels increase to provide a continuously moving ZPD (Hung, 2002). Maintaining learner position within the ZPD is a deliberate strategy that engages learners, connecting prior knowledge with new content as instructors strive to assist students in achieving predetermined outcomes (Marsh & Ketterer, 2005).

An effective strategy for mediation of a task includes having learners mediate tasks for classmates. The methods and strategies used by learners will differ from the methods employed by teachers (Iju & Kellog, 2007). Working with other students forces learners to share personal knowledge and theories. Reciprocal teaching occurs as a result where students find themselves asking questions, summarizing, clarifying and refining understanding (Hung, 2002).

Iju & Kellog (2007) discuss ideas of cognition described by Vygotsky who observed that psychological processes develop through the mediation of tools and people. The internalization of concepts and activities are first realized through interactions with others. The interactions between teachers and students often take place in different modes than those between students and peers, using different language. While teachers will often illustrate problem solving in a specific manner, discourse between learners allows for a greater co-construction of knowledge. While there is often no planned scaffolding in discussion between peers, as knowledge and understanding is shared, solutions are developed with deeper mathematical understanding.

Learners construct knowledge through interaction with others and internalize meanings of language, symbols and objects. Learners engage internal speech in thinking, problem solving and reasoning (Marsh & Ketterer, 2005). The use of tools is engrained within social content and contributes to internal knowledge construction in conjunction with social interaction. Tools allow learners to engage with the external environment.

Instructional tools and strategies can be classified into three categories within the ZPD: societal-cultural, communities of practice and school-classroom (Hung, 2002). Learners acquire beliefs, values and ways of thinking from members of society. Within communities of practice, learners develop specific skills and application of knowledge while from the school-classroom perspective, learners acquire knowledge that is relevant to the community in which they belong (Hung, 2002). Within each category, there is a potential level learners can attain with assistance.

Tools can be used to support knowledge construction in the classroom and are particularly effective when they allow for bridging between mathematical concepts and real-world applications (Hung, 2002). When tools allow students to monitor and evaluate the cognitive processes being demonstrated, students are able to reflect on learning and understanding. Observing others also allows learners to construct knowledge from multiple perspectives. Hung (2002) suggests that learning in the Vygotsky classroom should include cooperative learning, tutoring among students and opportunities for students to self monitor behavior and cognition.

Narratives are effective instructional tools for the communication of knowledge (Hung, 2002). Engaging in mathematical discourse is a social practice. Learners acquire mathematical language in a social environment, therefore communicating mathematically becomes socially meaningful. Using video in mathematics instruction can offer benefits to learning in the presentation of problems in relevant, real life contexts. Bringing the real world to the classroom fosters a community of practice in which learners are able to apply principles in context. The narratives constructed from interaction with video makes thinking visible and open to comment and reflection (Hung, 2002). Discussion generated from interaction with video creates a situation for knowledge building and transforms the act of writing into the social act of communication. Video narratives are constructed within individual learner contexts reflecting different perspectives and prior knowledge. Viewing multiple perspectives of a single account may facilitate further learning in individuals who may be at different stages in the ZPD. Vygotsky’s work on cognitive development suggests that “the most significant moment in the course of intellectual development…occurs when speech and practical activity, two previously completely independent lines of development, converge.” (Vygotsky, 1978, p.24)

Vygosky’s work suggests that knowledge is culturally specific. Language, symbols and tools are used within cultures as vehicles for learning and thinking and learners make meaning through connections between these vehicles (Fuson, 2009). Mathematical concepts can be very abstract in nature and as such, it is of even greater importance to learners that they are able to make connections between specific language, symbols and visual representations to engage in effective problem solving strategies and work toward developing fluency and understanding.

Marsh & Ketterer (2005) suggest that effective learning is situated in activity, context and culture within a community of practice. Knowledge construction occurs as a result of participation in authentic tasks through social collaboration. Bruner (1966) proposed that learners would better retain concepts if allowed to discover them independently as opposed to passively through traditional rote learning methods. This opportunity would be fostered when asking students to construct video narratives.

Social interaction and communication are essential elements of Vygotsky’s work on cognition. As learners engage in discourse, they internalize meanings and develop reasoning strategies (Steele & Reynolds, 1999). Standards documents from the National Council of Teachers of Mathematics (NCTM) (1989, 1991, 1995, 2000) emphasize the important roles of communication and social interaction in the development of mathematical understanding. “Communication plays and important role in helping children construct links between their informal, intuitive notions and the abstract language and symbolism of mathematics; it also plays a key role in helping children make important connections among physical, pictorial, graphic, symbolic, verbal and mental reprensentations of mathematical ideas.” (NCTM, 1989, 26)

Experience enables learners to develop language. Mathematical language can be difficult to discern meaning from when presented in absence from experience. Introducing mathematical language in conjunction with physical experiences, objects or tools in meaningful, relevant contexts with the ZPD increases the likelihood that learners will be effective when communicating mathematically about strategies and thinking in problem solving situations (Steele & Reynolds, 1999).

References:

Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.

Differentiating Mathematics Instruction. September, 2008. The Literacy and Numeracy Secretariat. Retrieved March 6, 2010 from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/different_math.pdf

Fuson, K. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching?. Cognitive Development, 24(4), 343-361.

Hall, T. (2002). Differentiated instruction [Online]. Wakefield, MA: CAST. Retrieved on March 6, 2010 from

Huebner, T. (2010). Differentiated Instruction. Educational Leadership, 67(5), 79-81.

Iju, G., & Kellogg, D. (2007). The ZPD and whole class teaching: Teacher-led and student-led interactional mediation of tasks. Language Teaching Research, 11(3), 281-299.

Marsh, G., & Ketterer, J. (2005). Situating the Zone of Proximal Development. Online Journal of Distance Learning Administration, 8(2). Retrieved on March 6, 2010 from http://www.westga.edu/~distance/ojdla/summer82/marsh82.htm

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and Evaluation Standards for School Mathematics. Retrieved March 6, 2010 from

Steele, D., & Reynolds, A. (1999). Learning mathematical language in the zone of proximal development. Teaching Children Mathematics, 6(1), 38.

Steinbring, H. (2007). Epistemology of mathematical knowledge and teacher--learner interaction. ZDM, 39(1/2), 95-106.

Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA.: Harvard University Press.

Wei Loong Hung, D. (2002). Learning through video-based narratives within the cultural zone of proximal development. International Journal of Instructional Media, 29(1), 125-137.