## Linking Behaviorism, Cognitivism and Constructivism Approaches to the Teaching of Mathematics

Gladys Nakalema, Imelda Kemeza, Mary Kobusingye, Dennis Zami Atibuni and Eunice Ndyareeba Contact : Imelda Kemeza email:ikemeza@must.ac.ug, imekeza@gmail.com World over teachers are considered to be nation builders because of their responsibility to transmit knowledge and shape experiences for individuals from one generation to the next (Sarsan, 2006). Currently in Uganda, there is an increasing demand on teachers to improve the quality of teaching and learning of mathematics in schools. Hindering the response to this desire are the challenges faced by the teachers as they execute their duties. Some of these challenges include having big classes, inadequate instructional materials, inadequate parental support, negative attitudes towards mathematics and monotony of the teaching methods (Millennium Villages Project, 2009). These challenges have been aggravated by inception of the Universal Secondary Education, which has drastically increased student enrolment, yet the school resources have not been increased and in the process affecting the teaching methodology (Fontana, 1995). Most notably the methodologies used in the teaching and learning process are bound to have a larger impact on the learning process. This comes right from the grassroots of teaching mathematics because if a child does not understand the meaning of numbers and of numerical relationships then his or her later achievement in mathematics is greatly compromised (Sun, 2009). It is therefore important for teachers to explore the challenges they face in the teaching and learning process for possible solutions (Millennium Villages Project, 2009). This article focuses on instruction methods used in the classroom while teaching mathematics. It is debatable whether behaviorist theories or cognitive theories are effective in the teaching and learning process. The behaviorist theory at its most extreme places emphasis on studying observable behaviour; that is upon responses made towards a stimulus provided by the environment. Ignoring internal factors like thinking and interpretation among others. The behaviourists therefore place more emphasis on the connections made between the stimulus, the response and the conditions under which they occur (Fontana, 1995). Skinner and others behaviourists viewed the teacher’s job as modifying the behavior of students by setting up situations to reinforce students when they exhibit desired responses. This can be done through step -wise presentation of stimuli and reinforcing the occurrence of a desired response. The responses are then linked together involving lower-level skills forming a learning “chain” to teach higher-level skills hence the birth of direct instruction approach. This Approach is also referred to as Programmed Learning (Farrant, 1980) or explicit teaching (Cornway 1997). This implies that while teaching mathematics a teacher should breakdown material into simple manageable units which follow a particular sequence of known to the unknown. The learners should be taken through the different concepts of statistics progressing towards more complex activities like plotting a cumulative frequency curve, and finding the inter-quartile range. For example if one is teaching statistics, starting with simple ungrouped data would be favorable for the learners. When the learner is able to find the mean, he or she is reinforced, then he or she should attempt to find the median; if he or she succeeds then reinforcement follow and so on. Farrant (1980) emphasized the need for the structure of the subject matter to be sequenced in such a way that learners are able to proceed without the normal help of a teacher. This lays great emphasis on self-based learning which is fundamental in mathematics learning. This can be encouraged through giving exercises which are a step ahead of what has been taught in class. This prompts a learner to solve a problem with no or little help from the teacher. Programmed learning puts the teacher in an authoritative position, since he or she is in control of the learning program by designing learning through a series of assignments. The learner is directed towards a desired goal and is given a couple of tasks to accomplish. These tasks are then evaluated by the teacher to check whether the learner can move onto the next step (Farrant, 1980). The behaviourist theory also gave birth to rote learning, the drill and practice method which put emphasis on the explain-practice-memorise teaching model. These methods do not focus on understanding of concepts but the production of a desired response (Sun, 2009). Despite the flaws attached to these methods, they are effective in studying some concepts like learning colors, multiplication tables and squares of numbers. The theory puts emphasis on overt behavior which makes it very popular in education systems that dwell much on the achievement tests or examinations. As a result learners do not concentrate on understanding concepts but more on giving the desired responses in the examinations. This kind of norm leads to mathematics anxiety as the learners try to drill and practice in order to avoid failure. Like most education practitioners, it is believed that personal and process oriented teaching which places emphasis on comprehension of concepts and process is preferred to the direct instruction approach of teaching (Karimi & Venkatesan, 2009). Cornway (1997) advances that cognitive theorists including Lev Vygotsky, Jean Piaget and Jerome Bruner propose that children actively construct knowledge and this construction of knowledge happens in a social context. Cognitive psychologists therefore attempt to explain the learning process by understanding the thought process of an individual inferred from the behavior the individual exhibits (Wade & Travis, 2003). According to the cognitive theorists, teaching is a process through which a teacher facilitates the development of insight or understanding in the learner. Learners organize concepts and ideas through accommodation or assimilation (Chauhan, 1995). A teacher is therefore expected to model the child’s experiences by providing moderately challenging tasks. Such tasks give the teacher an ‘intellectual scaffolding’ position which helps the child progress form one level to another (Cornway, 1997). Jerome Bruner, a renowned cognitive psychologist, was influential in defining discovery learning. The discovery method of learning is the kind that encourages the learner to explore, manipulate and interpret the environment (Conway, 1997). Here the teacher abandons his role as a teacher and sets the pupils free to pursue their own learning as their interest and knowledge guide them. The assumption here is that the learner will easily remember what they discovered on their own (Farrant, 1980). When teaching mathematics therefore, a teacher should give the learner an opportunity to infer relationships between concepts with self- guidance. For example when teaching sets If A= [3,6,2,9,8,0] B=[8,6,3,9,2,0] When learners are told to compare set A with Set B, they will be able to discover that sets A and B have similar elements. The teacher then brings in the concept of ‘equal sets’ And if asked to compare sets C and D; Where C=[3,1,5,8,7,0] D=[8,6, 3,9, 2 ,0], Learners will be able to discover that sets C and D are not equal but have some common elements. The teacher then brings in the concept of ‘intersection between two sets’. As indicated by Farrant (1980) the discovery method deviated from the normal routine of the teacher acting as an authority to being a guide. The pupil is enabled to make use of the information around them. As a result this information is better assimilated and remembered than the information that is memorized. Furthermore the Gestalt psychologists proposed that during the learning process learners look at the patterns rather than isolated events. Gestalts’ views of learning have been incorporated into the cognitive theories which emphasize that the teacher has to help the learner find a realistic and worthwhile goal. The teacher must know what is familiar to students so as to introduce elements of novelty (Chauhan, 1996). The major difference between gestaltists and behaviorists is the locus of control over the learning activity: the individual learner is more key to gestalts than the environment that the behaviorists emphasize. Two key assumptions underlie this cognitive approach: (1) that the memory system is an active organized processor of information and (2) that prior knowledge plays an important role in learning. Therefore, when teaching mathematics the teachers should look beyond behavior to explain brain-based learning. Cognitivists consider how human memory works to promote learning. For example, the physiological processes of sorting and encoding information and events into short term memory and long term memory are important to educators working under the cognitive theory. Adams (2007) argues that as children who find mathematics difficult are slow and inaccurate counters, hence they have a lower working (short-term) memory span, a distinct functional deficit when performing mental arithmetics which requires the retention of operands and a counting sequence. As a result frequent calculation errors can occur due to information being lost through decay, thus inducing the formation of weak associations between operands and their sums in long-term memory. This in turn decreases the probability of such children developing an efficient math-fact retrieval strategy (Wikipedia, The Free Encyclopedia 2008). One of the paradigms of learning that has been given little attention is the constructivism approach which views learning as a process that enables the learner to actively construct or build new ideas or concepts based upon current and past knowledge or experience. In other words, “learning involves constructing one’s own knowledge from one’s own experiences.” Constructivist learning, therefore, is a very personal endeavor, whereby internalized concepts, rules, and general principles may consequently be applied in a practical real-world context (Wikipedia, The Free Encyclopeadia, 2008). In this case constructivism interfaces for behaviorism and cognitivism. The approach is very feasible in mathematics due to its dynamic nature as a subject. The learner can be given an opportunity to explore their surroundings and elicited to relate the surroundings with the already existing body knowledge. The teacher acts as a facilitator who encourages students to discover principles for themselves and to construct knowledge by working to solve realistic problems. This kind of practice empowers learners to solve problems in real life by utilizing what is learnt in school. In summary, when teaching mathematics there are three main approaches of teaching: behaviorism, cognitivism, and constructivism. Behaviorism focuses on the objectively observable aspects of learning. Cognitivism looks beyond learners’ behaviors to explain brain-based learning. Constructivism is a process in which the learner actively constructs or builds new concepts and ideas based on prior knowledge and experience. A mathematics teacher should always strive at hitting a balance in those three aspects. References Adams, J. W. (2007). Individual differences in mathematical ability: genetic, cognitive and behavioural factors. Journal of research in special needs education, 7(2), 97-103. Chauhan, S. S., (1996). Advanced Educational Psychology 6th ed. New Delhi. Conway, J. (1997). Educational Technology’s Effect on Models of Instruction. Retrieved from http://udel.edu/~jconway/EDST666.htm#dirinstr on 15/03/2010 at 12.32pm Sun, H. V.(2009) Investigating feelings towards mathematics among Chinese kindergarten children. In: Thirty-second annual conference of the Mathematics Education Research Group of Australasia, 5-9 July 2009, Wellington, NZ. Farrant, J. S. (1980) Principles and Practice of Education . London: Longman Group. Fontana, D. (1995). Psychology for Teachers. 3rd ed. New York: Palgrave Karimi, A. & Venkatesan, S. (2009). Cognitive Behavior Group Therapy in Mathematics Anxiety. Journal of the Indian Academy of Applied Psychology, 35, 299-303. Millennium Villages Project, 2009 Wade, C., & Tavris, C., (2003) Psychology. New Jersey: Pearson Education, Inc. Wikipedia: The Free Encyclopeadia (2008). Learning Theories (Education). Retrieved from http://en.wikipedia.org/wiki/Learning_theory_(education) on March 22, 2010 at 12.45pm

hi,thanks for your good article.it was great ,could you explain more about cognitivism and constructivism.

thanks

@Gladys Nakalema Hi Gladys this is agood project keep it up

@nyombi hacsam. (ucu student)

Hello Nyombi, if you want to start your own blog try wordpress.com blogger.com or weebly,com. They are free. If you want to contribute here, submit your article and it will be published after it had been approved.

Hello Nyombi, if you want to start your own blog try wordpress.com blogger.com or weebly,com. They are free. If you want to contribute here, submit your article and it will be published after it has been approved.

dear presenter, thank you for the good task of which you have done excellently.

please may take me as part of research student because i am much influenced by your paper and please what are the basic procedures to creating a blog.

thank you.

an interesting journal and possibility to publish:

http://www.spreadcorp.org/ajote/index.htm

Publishing it in a journal is a good Idea.

Dear Imelda this is a very nice article but a bit too long to post on the blog. Would you like to publish it in a journal?