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Behaviorism Its Implication To Mathematics Education

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BEHAVIOURISM: ITS IMPLICATION TO MATHEMATICS EDUCATION

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BEHAVIOURISM: ITS IMPLICATION TO MATHEMATICS EDUCATION

1 Junn Ree B Montilla 1 junnree@g.msuiit.edu

The theory of behaviourism holds that the act of learning is based around a series of stimulus-response mechanisms; as such, education can be considered as the process of training a learner to respond in particular ways to a set of recognized prompts. Viewing learning in this way allows psychologists to pose and test questions about learning in an empirical way, and does not require the researcher to posit the existence of any hypothetical mental state. The behaviourist psychologist Edward Thorndike describes a mechanistic approach to teaching and learning mathematics in his books The Psychology of Arithmetic on 1992 and The Psychology of Algebra on 1923. He puts forward a law of exercise in these books which says that connections between the stimulus and response are strengthened as they are used, and a law of effect, which says that responses that lead to positive outcomes and feedback are also strengthened. This view of mathematical learning has been influential in the widespread use of rote and practice methods. Some learners have also been seen to develop an automated response to more advanced mathematical topics: for example, upon seeing a trigonometry question, many learners have been trained to label the sides of the triangle, then to select the appropriate ratio, and follow a standard learned set of procedures to arrive with the answer. Transfer in behaviourism, as a phenomenon that can occur when situations possess a sufficient degree of commonality in their stimuli; if we set a learner two problems that look sufficiently similar, then they will recognise the similarities and respond appropriately. While there is some disagreement about the interpretation of experimental evidence regarding transfer, studies do seem to support this idea in a broad sense. D. Detterman suggests in his research that transfer is both uncommon and difficult, but when it does happen it occurs between situations which are very similar. One immediate implication of cross-curricular practice is that teachers should strive for consistency in the way different subjects present similar tasks and actively direct pupils’ attention to instances where transfer is being called for. The difficulty of even near transfer can be perceived in mathematics lessons, when pupils are able to answer questions on a topic if they are worded in a certain way, but are unable to tackle questions that are differently phrased or structured. Catrambone and Holyoak (1990) investigated pupils’ ability to solve probability problems and found little evidence of transfer when pupils had previously been trained on questions centred on a specific subgoal. They suggested that teachers should therefore give pupils practice on questions phrased in a variety of ways. Reed et al. (1985) also found limited evidence of transfer when investigating how pupils fared solving algebra word problems. Surprisingly, when pupils had a solution to a problem in front of them their ability to solve equivalent problems improved, but their ability to solve problems that were only similar did not improve. Bassok and Holyoak (1993) found evidence of different degrees of transfer between different combinations of subjects, and also an asymmetry in the direction of transfer. In one experiment they investigated pupils’ performances in problems in algebra and physics that were structurally isomorphic; for example, a question on arithmetic progressions could be mapped to a question about the motion of a vehicle travelling with constant acceleration in a straight line. They found that it was more common for pupils to transfer skills from algebra to physics than vice versa. In another experiment they constructed similar problems in algebra and finance, and observed a greater degree of transfer than had been the case between algebra and physics. Although the results discussed above can be understood in terms of situations with different levels of similarity in their stimuli, another possibility is that the pupils had built up their understandings in a way that made transfer to algebra a qualitatively different task for learners who had studied finance

There is a need to consider the amount of time learners are actively taught or supervised. In a direct instruction classroom, the teacher presents information and develops concepts through lecture and demonstration. As learners question, respond to teacher queries, react to assignments, and do practice exercises, elaborations are given that are designed to clarify and strengthen understanding. Mastery Learning. Mastery learning permits the teacher to go on as long as a learners’ learning pace is not hurried just to keep up with the rest of the class. This is based on the realization that we all learn at different rates. Mastery learning implies that each learner will master a subordinate skill before proceeding to the next skill level. When using mastery learning with direct instruction, the teacher begins a unit of study by using techniques involving lecture, demonstrations, and review, along with drill and practice. At an appropriate interval, more often about 3 – 5 days, the teacher deviates from the normal routine to administer some form of evaluation to assess learner understanding. If only we could dissect each learners’ brain and somehow tell what has been learned, the processes that are most likely to succeed, and the propensities for learning. If that could be done, perhaps a better connection between theory and practice could be established. Then, learning theories could be more solid and identifiable. We cannot dissect a learners’ brain. We are reduced to studying tendencies and creating

theories. We heard many times, “ Is there a difference between theory and practice?” Many

theories will “ sound good ” or “ make sense ” on paper. Often, theories are created out of

observation, knowledge of learner learning, thought, and discussion. The background is often built in a specific, sometimes controlled, setting or with a limited number of cases. The real world of the classroom often does not resemble the theory environment. Putting theory into practice is not that simple. Implementation of a learning theory also assumes the teacher is well versed in the related ideas. That assumption is often unrealistic from the standpoint of a teacher who was exposed to the theory or strategy in a class or in-service session but is swamped with all the details related to classroom production. We think back to our own learning environments, searching for clues about how we

learned different skills and concepts. The assumption is that if we are “normal” and if we can

figure out how we learned to perform a given task, we might gain some insight into what can be done to help students learn as they advance through the world of mathematics. Considering time and memory, the likelihood of our remembering minute details from our learning is not great. Besides, most of us who are involved in the teaching and learning of mathematics are

not “typical” representatives of students found in secondary mathematics classes. Students and

how they perceive or learn mathematics change from year to year.

References

Bassok, M., & Holyoak, K. (1993). Pragmatic knowledge and conceptual structure: determinants of transfer between quantitative domains. In: D. K. Detterman & R. J. Sternberg (Eds) Transfer on Trial intelligence, cognition, and instruction (pp. 68–98). (Norwood: Ablex).

Brumbaugh, D. K. and Rock, D. (2013). Teaching Secondary Mathematics (Fourth Edition). Routledge 711 Third Avenue, New York, NY 10017: Taylor & Francis Group

Catrambone, Richard & Holyoak, Keith. (1990). Learning subgoals and methods for solving probability problems. Memory & cognition. 18. 593-603. 10/BF03197102.

Detterman, D. L. (1993). The case for the prosecution: Transfer as epiphenomenon. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction. Norwood, NJ: Ablex.

Driscoll J. (1994). Reflective practice for practise. Senior Nurse. Vol Jan/Feb. 47 -

Esler, W. K. , & Sciortino, P. (1991). Methods for teaching: An overview of current practices. Raleigh, NC: Contemporary Publishing Co.

Hamilton, E. (1924). The Psychology of Arithmetic. By E. L. Thorndike. Pp. xvi 314. 9s. net. 1922. (The Macmillan Co.) - The Psychology of Algebra. By E. L. Thorndike and Others. Pp. xi 483. 10s. 6d. net. 1923. (The Macmillan Co.). The Mathematical Gazette, 12 (171), 174-176. doi:10/

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