There are two theoretical frameworks that are pertinent in instructing elementary teachers on teaching mathematics: the developmental approach and the constructivist approach (Van de Walle et al., 2010). This discourse shall be delving deep into this viewpoint in the quest to establish their suitability and whether one is more preferable than the other, taking into account the present learning needs of mathematics students.
Generally, the motivations for the changes witnesses in the teaching of mathematics in schools are evidence of the fact that as the society changes, so do the need to develop technical aids in teaching the subject (Anghileri, 2006). These changes have also prompted psychologists to change their strategies whilst conducting research on ways in which children can be taught mathematics.
Throughout the previous century, the theorists of this approach have gone to the great length of bringing to the fore the discussion concerning the value of the children’s development base for teachers. More importantly, this period has seen to the modeling of learning concepts and development falling between the definitions of either behaviorist tradition or to the extreme the biological perspective, including entity ideas such as the claims that intelligence is fixed or rather the maturations’ perspective that children develop their intelligence on their own (Van de Walle et al., 2010). Anghileri (2006) defines this approach as a behaviorist. She notes that the approach relied on the conviction that the learning process of human beings and their behavior can only be explained as a response to a particular external stimulus. This approach presented the implication that knowledge was a transferable commodity, that it could be transferred from the teacher to the students in a class set up. Secondly, the approach also posited that techniques such as drill and practice were the most pertinent means through which students acquired such knowledge.
Anghileri (2006) further notes that the evident shortcomings of the behaviorist approach led to the subsequent questioning of its efficiency in imparting knowledge to the student. Undeniably, whereas the approach could aptly elaborate the intricacies involved in training pigeons to play ping-pong, it fell short of bringing to understanding the modalities involved in helping students to apply concepts in language, exploit their creative capacities and resolve all kinds of problems.
Constructivist Approach
The past decade, however, witnessed a turning point in perspective by theorists who instead endorsed the educational practices that are based on prevailing knowledge in regards to how children learn and develop (Freeman & Richards, 1996). These recent approaches that have been adopted in a bid to explain how children learn mathematics are termed as constructivist, and they are founded on the view that learning is not merely the transmission of knowledge from one person to another; it rather posits that students are actually active participants in the construction of their own knowledge, especially from their personal experiences (Anghileri, 2006).
Notably, the constructivist approach, as related to the teaching of mathematics, is broadly discussed through a variety of thematic concerns. Many scholars have employed various qualifiers in the advancement of the constructivist approach. This has led to such sub-branches like the individual and cognitive constructivism as propounded by Jean Piaget and social constructivism, as argued by Lev Vygotsky (Baker, McGaw & Peterson, 2007) and David Ausubel (Cakir, 2008). Other facets of the constructivist approach include socio-cultural constructivism (Branco & Valsiner, 2004); socio-transformative constructivism (Rodriguez, 1998), and contextual constructivism (Cobern, 1993). For the sake of this discourse, we are limited to delving into the dimensions of constructivism, as portrayed by Jean Piaget and Lev Vygotsky.
Understanding that human beings are born with reflexes that allow them to interact with their environment is fundamental in understanding Piaget’s approach to constructivism. The ability of human beings to adapt to their environment is, therefore, a product of the replacement of these varied reflexes through the construction of schemes and/or structures. There are two distinct channels through which this process of adaptation is facilitated. These are either accommodation or assimilation. These two channels are critical elements of constructivism. As a person interacts with their environment, they invariably create knowledge, and this, in turn, manipulates their cognitive structures; as a result of the continuous building of these knowledge, adaptation results. Whenever there comes up a difference between the mental structures that have been formulated and the environment that the learner operates in, the learner can alter their perception of their environment to match it with the order of the incoming information. This is achieved through the process of assimilation. Alternatively, the cognitive structures can change by themselves as a result of the interaction that occurs during accommodation. Whichever way, the individual interacts with his environment for the purpose of adaptation. (Piaget, 2001)
There are three types of knowledge that need to be present at the various stages of cognitive development, as proposed by Piaget (Driscoll, 2000). These include the physical, the logical-mathematical, and social knowledge. First of all, physical knowledge is accessed by learner through their encounter with the physical environment. It is therefore defined in terms of the experiences of the learner and how they perceive physical objects and the very nature of their concreteness. In essence, the physical knowledge is acquired through direct contact with environmental elements (Lutz & Huitt, 2004). On the other hand, the kind of knowledge referred to as the logical-mathematical knowledge consists of abstract reasoning. This knowledge is applied in concepts that are not necessarily dealing with the physical encounter with stimuli. It is different from physical knowledge in the sense that it can be discovered, it requires action before it can be acquired. Therefore, its acquisition comes through recurrent exposure and interaction with multiple objects in a number of places. This results into the creation and modification of the mental structures. In this case, generalizations and abstractions become the products of the manipulation of objects in varied patterns and contexts (Piaget, 1978). Thirdly, the social knowledge is specifically determined by the cultural context that an individual is exposed to. The process of acquiring this knowledge is contingent on the learner’s action as opposed to their perception of the physical objects. There is no hierarchy of importance in regards to the three types of knowledge are applicable at all the stages of cognitive stages of development (Piaget, 1978).
Lev Vykotsky, on his part, identifies society and culture as great determinants of cognitive development. He employs social interaction as to lay out the concepts needed for learning and development. He argues that the development of an individual’s mind involves the interweaving of the biological development of their body, on one hand, and the appropriation of their cultural and material heritage that exist in the present to coordinate people with each other and their physical world (Lutz & Huitt, 2004). According to Wink and Putney (2002), there are three major principles that underlie the Vykotsky approach to constructivism. For one, social interaction plays a crucial role in the development of cognitive abilities of a leaner. Nicholl (2012) views this principle as meaning that without the learning that occurs as a result of social interaction or without self awareness or the application of symbols that allows people to think in ways that are more complex, we would invariably be subjected to slavery to situations, thereby, responding directly to the environments that we are subjected to.
Vykotsky’s second principle simple imply that the potential for the development of a person’s cognitive abilities are limited to a particular time span (Kearsley, 2001); whereas, the third principle assert that the only means of understanding how individuals acquire knowledge is by studying learning in an environment in which the process of learning is studied (Lutz & Huitt, 2004).
It can therefore be concluded that both the society and culture are crucial to the social development theory as proposed by Vykotsky. This is evident from his argument that the higher mental functions need to be processed through an external stage in the context of social occurrences (Lutz & Huitt, 2004). This allows their merging into the integral part of the thinking of learner. This dialectical discovery eventually becomes complex as much as it is a continuous process (Wink &Putney, 2002). According to the Vykotsky’s, it is suggested that each and every person has their unique potential range for learn concepts. This range of learning is what Vykotsky defined as the zone of proximal development.
In this zone, at any given point, there are three levels of ability that are possible at any point of development. This includes whatever an individual can learn without guidance or help, that a person cannot learn even if they are helped and that which they can learn with help (Lutz & Huitt, 2004).
The central element, according to this approach, is the potential of a learner to develop as opposed to the snapshot that can be provided through asking the learner to independently accomplish some particular mathematical tasks (Lutz & Huitt, 2004). Consequently, the cognitive capacities of a learner can be measured by simply evaluating the tasks provided to the learner. This zone of proximal development is represented by the difference between actual levels of development, which ate measured by the ability to solve problems and, the levels of a learner’s potential that are characterized by the learner’s ability to resolve problems when they are under guidance of either an adult or a fellow competent peer (Lutz & Huitt, 2004).
Many scholars have considered the approaches propounded by Piaget, Ausubel and Vygotsky as offering quite divergent of the development of learning or rather cognition. For one, Vygotsky and Ausubel are considered as providing teaching recommendations that are more explicit than those that are provided by Piaget and there are glaring similarities that can be pinpointed between the processes of cognition that are proposed by the three theorists (Cakir, 2008).
Piaget proposes that both children and adults employ mental patterns or schemes in their attempt to appreciate and control behavior and cognition (Cakir, 2008). In essence, they interpret new material or experiences by relating them to existing schemes. This demands that for the new material to become fully absorbed, it should fit into a present scheme. In close relation to Piaget proposal, Ausubel argues that significant information is often saved in the networks of closely related informational facts connected facts that are reckoned to be called schemata. This means that for one to clearly understand and be able to use in practice the new information, it is necessary for it to fit in the existing schemata for it to be successfully assimilated. The two theorists therefore concur that new concepts are more readily learned and assimilated than the new information that relates to less established schemes or patterns (Cakir, 2008).
On the other hand, Vygotsky approach has been lauded for its focus on culture as the basis for cognition and also for proposing that each learner has got their zone, which depends on two crucial factors: the developmental threshold that is necessary for learning and upper limit of the current ability of the learner to absorb the material that is under consideration (Cakir, 2008).
In that regard, mathematical knowledge is not acquired by student just by sitting in their classroom and listening to their teachers; it is rather something that students take initiative in constructing by themselves through establishing meaning and mental connection of concepts. Therefore, the learning outcome of students vary depending upon the framework for understanding that a particular student has developed and this limits the teacher’s role to that of providing an enabling environment, which can stimulate active participation (Anghileri, 2006).
Duckworth (2006) also notes that research has demonstrated that the constructivist view, by the teachers, that the students’ mindset are somewhat consistent with that of their child-centered practices. Other studies demonstrate that such practices encourage qualities such as the motivation to learn and solve problem, which are highly valued in children. For instance, studies demonstrate that preschool and kindergarten teachers who advocate for approaches that are child centered are prone to employing a variety of engaging and authentic activities while teaching mathematics in their classrooms (Stipek & Byler, 1997). Additionally, other studies have established that there is a link between the uses of activity based approaches that are characteristically instructional and the constructivist view of the mind (Duckworth, 2006).
2000 NCTM Principles and Standards
The theoretical approaches discussed above are essential in teaching children mathematical concepts. However, as discussed by Van de Walle et al. (2010), the ‘2000 NCTM Principles and Standards’ is a crucial document that teachers need to acquaint themselves with, if at all they are going to achieve maximum effectiveness with teaching children mathematical concepts. These principles and standards as reflected in this document lay out the important components that should be inculcated in the high quality mathematics programs in schools. The program gives much prominence the need for well supported and well prepared teachers and, also recognizes the significance of a system that is carefully organized to assess the learning of students and the effectiveness of the program. Thirdly, the program underscores the necessity for all stakeholders including the students, teachers, parents, administrators and the community at large to play a part in the quest to build a high quality program for all students. (NCTM, 2000)
The principles proposed in this document reflect the basic precepts that are considered as fundamental to the provision of high quality teaching of mathematics. There are six principles in total that are proposed in the document. They include equity, learning, curriculum, teaching, assessment and technology (NCTM, 2000). According to the principle of equity, excellence in teaching mathematics is dependent on the strong support that the teacher offers to the child. The principle makes the assumption that all children can be taught and learn mathematical concepts of high-quality instruction, their physical challenges, social-economic background and personal characteristics notwithstanding. Rather than demanding that every student be accorded identical instruction, the principle of equity prescribes that the teacher makes reasonable and appropriate accommodation and that they also provide challenging content in order to boost access and attainment for all students (Van de Walle et al. 2010).
Secondly, curriculum as principle represents coherent activities that are focused on learning important concepts in mathematics. In this case, mathematical concepts are built on one another to help the understanding of the child and deepen their knowledge of the concept, thus multiplying their ability to apply mathematics. This implies that an effective mathematics curriculum will lay much emphasis on concepts that will prepare the child to further study the subject and also solve problems in any environment they find themselves in whether it be at home, school or work place in the future. When a curriculum is well articulated, it challenges the student to deepen their knowledge on more sophisticated mathematical ideas as they further their studies (Van de Walle et al. 2010).
Teaching as a principle posits that effective impartation of mathematical concepts requires the understanding of what the students already know and what they need to learn; it therefore also involves the teacher challenging and supporting the student to acquire these new concepts that they need to learn. Teaching is imperative to achieving the understanding of mathematical concepts by students, their ability to use them to solve problems and generally their confidence in doing mathematics (Van de Walle et al. 2010).
Students must also learn mathematics with understanding as they build new knowledge from the experience and previously acquired knowledge. The conceptual understanding of mathematical concepts cannot be gainsaid. The students become effective learners only after they can draw parallels on the factual knowledge and procedural proficiency on one side with conceptual knowledge on the other side (Daniels & Shumow, 2003). In this state, students are able to identify the significance of reflecting on their thinking and even draw lessons from the errors that they have made. They also achieve competence and confidence in their capability to handle difficult problems, thereby becoming more willing to persevere whenever the task presents challenges (Van de Walle et al. 2010).
Additionally, the principle of assessment supports the learning of important mathematical concepts and provides invaluable information to both student and teacher. Assessment should be placed at the center of the student learning since it contributes to the understanding of mathematical concepts (Daniels & Shumow, 2003). Teachers should formulate assessment in the manner that it informs and guides them to arrive at instructional decisions. They should also formulate the assessment in the manner that the task that they select for the students conveys a message to the students about the most pertinent mathematical knowledge and performance. More importantly, the feedback that is derived from assessment enables the students in assuming responsibility for their own learning by setting learning goals and achieve independence in the whole learning process (Van de Walle et al. 2010).
Finally, technology is essential in the teaching and learning of mathematical concepts. This is because it does influence the teaching process itself by enhancing the learning of students. Through the use of technology, students can attain a deeper understanding of mathematical concepts. This could be through conduction investigations in mathematical concepts and allowing students to emphasize decision-making, reflection, reasoning and the solving of problems (Daniels & Shumow, 2003).
On the other hand, the standards proposed by the ‘2000 NCTM Principles and Standards’ document include the descriptions that mathematics instructions are given, which enable the students to familiarize with what they need to know or do in order to solve a particular mathematical problem. The standards are further categorized as content standards and process standards. The content standards describe the five basic contents that students need to be acquainted with including: numbers and operation, algebra, geometry, measurement, data analysis and probability. The process standards, on the other hand, demonstrate the ways through which content knowledge should be acquired and applied. The process standards include problem solving, reasoning and proof, communication, connection and representation.
Conclusion
In this discourse, we have looked at both the theoretical and practical underpinning of teaching children mathematical concepts. In the theoretical framework, we have looked at the developmental aspect and ruled it out as an effective concept to be adopted by teachers to teach mathematics to children. This is informed by the knowledge that the approach refuses to acknowledge human beings as active participants in the learning process. We have therefore endorsed the constructivist approach, taking into account its strengths as highlighted by scholars such as Vykotsky and Piaget. This is because the approach appreciates the interaction between the learner and his environment and how this crucial matrix helps in the comprehension of patterns, thus, concepts.
Finally, we have studied the 2000 NCTM Principles and Standards document, which are lauded by Van de Walle et al (2010) as the best blue print for teachers who are teaching mathematical concepts to children. The document proposes principles and standards that offer a robust guideline to teachers in their quest to impart knowledge that is direly required to solve mathematical problems.
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