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13th National Convention on Statistics (NCS) EDSA Shangri-La Hotel, Mandaluyong City
October 3-4, 2016
PREPARING PRE-SERVICE TEACHERS TO TEACH PROBABILITY USING HEURISTICS
Sweet Rose P. Leonares
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Authors name Sweet Rose P. Leonares Designation Assistant Professor Affiliation University of St. La Salle Address #4 Bagong Lipunan St., Bacolod City 6100 Tel. no. (034)433-6835 E-mail firstname.lastname@example.org
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PREPARING PRE-SERVICE TEACHERS TO TEACH PROBABILITY
Sweet Rose P. Leonares1
Probability, together with statistics, comprise a strand of Mathematics in the DepEd K to 12 Curriculum. Studies have shown, however, that even in-service teachers do not have adequate understanding concerning probability that teaching it generally posed a challenge. This study aimed to prepare pre-service teachers to teach probability with emphasis on the use of heuristics. The research design was descriptive qualitative using the phenomenographic approach. The participants were fifteen third year BSEd Mathematics majors in an HEI in Bacolod City who enrolled in Probability Theory for the first semester, AY 2015-2016. A 7-item pretest and a parallel posttest consisting of typical probability problems were administered and the Newman Error Analysis was used both times to determine the category of error the student committed for each item. Instruction focused on the use of appropriate heuristics. Pretest results showed very minimal understanding of probability, with most students committing low-level Comprehension error. Only one student got 4 correct answers, the rest had at most 1. Heuristics used were mostly symbolic representation and restating the problem. Posttest results showed 7 students getting 3 or more correct answers, including 4 students who got 5. Heuristics used were more appropriate for a given problem. Higher-level errors were noted. There was a greater tendency to commit no error in some items compared to the pretest, indicating improved levels of understanding. This study recommends the use of heuristics as a pedagogical approach in teaching probability.
One of the branches of mathematics that is considered to be important for all students is probability. Individual and collective decisions made concerning everyday activities are influenced by it. Greer and Mukhopaday (2005) contended that since real life data are variable, there is the need to quantify how things vary and probability is a tool that helps to measure uncertainty. Hence, learning about probability is important.
The inclusion of Probability in the curriculum is a recognition, not only of its academic
importance, but also of its role in daily life. With the shift to the K to 12 curriculum in the Philippine Basic Education, probability concepts are introduced as early as Grade 1 with increasing degrees of difficulty up to Grade 12 (DepEd K to 12 Curriculum Guide for Mathematics).
The main concern that has been raised, though, is whether the in-service and pre-service
elementary and secondary teachers are prepared to teach this subject matter. Batanero, Godino & Roa (2004) pointed out that, in general, in-service teachers are not adequately trained to teach it. Batanero and Diaz (2012) identified specific issues regarding the training and preparation of teachers to teach probability. Their position is that correct and adequate preparation of the teachers, as well as the latters belief that probability is important for their students to learn, contribute to the effective teaching of probability. However, as Reston (2012) pointed out, many
1 Assistant Professor, University of St. La Salle, Bacolod City
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pre-service programs in teacher education do not provide adequate training for the teaching of probability.
Classroom evidence gathered by this researcher pointed to this inadequacy since most
students have pointed to coin-tossing, die-rolling, and card-drawing examples as the sum of their probability theory classroom experience in high school. These students were products of a high school curriculum which was supposed to cover probability in the fourth quarter of their fourth year mathematics class. Hence, there is a need to instill the importance of the subject matter in both the pre-service and in-service teachers and to identify more effective approaches in teaching probability.
This paper, therefore, aimed to address this gap by exploring how the use of heuristics
could help improve the pre-service teachers conceptual understanding of probability through problem solving in order to prepare them to teach the subject. Theoretical and Conceptual frameworks
This study is anchored on the enactivist theory of learning. Enactivism, a combination of constructivism and embodied cognition, theorizes that learning is drawn from the interaction between learner and environment that thinking and action are grounded in bodily actions, and that actions are not simply a display of understanding but they are themselves understandings (Sumara & Davis, 1997).
The enactivist view of mathematical knowledge is that it is located in the activity or in the
inter-activity of the learners wherein, as the events of the lesson unfold, it is more of the presence of interaction among the people inside the classroom, the learners and teacher, that contributes to greater increase in mathematical understanding, rather than simply an individual action (Davis, 1995).
Enactivism has a close connection to Vygotskys theory of the Zone of Proximal Development, the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers (Vygotsky, 1978). This may suggest that cognitive development is limited to a certain range, but with the help of social interaction such as assistance from a mentor (teacher or peer), students can understand concepts and schemes that they cannot know on their own. This requires and promotes active participation and collaboration among learners. This is especially true in solving problems in mathematics.
Mathematical problems, in contrast to exercises, represent realistic scenarios and are oftentimes more complex and require more mental work, since these are generally stated in ways wherein the form of the solution is not immediately identifiable.
One way of teaching probability is through word problems. Early examples deal with
games of chance using coins, dice, and cards. This, however, does not capture the true essence of the application of probability to real life. There is, therefore, the need to present the concepts of probability situated in everyday real-life experiences.
Probability problem solving can be quite difficult for students because, as suggested by
Garfield and Ahlgreen (1988) and Konold (1989), people have natural misconceptions about probabilistic concepts. In recognition of this perceived difficulty, articles have been written recommending how to teach concepts in probability (Corter & Zahner, 2007).
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In order for a student to successfully solve various types of problems, particularly the non-routine ones, he or she has to apply four types of mathematical abilities: namely, specific mathematics concepts, skills, processes, and metacognition (Yoong and Tiong, 2006). The use of heuristics falls under the process ability.
There are varied heuristic techniques, which include guessing and checking, looking for a
pattern, making an orderly list, drawing a picture, eliminating possibilities, solving a simpler problem, considering special cases, working backwards, solving an equation, restating the problem, thinking of a similar problem, using any of the following: symmetry, diagram, model, direct reasoning, a formula, tables or trees, symbolic representation, key words, and just being ingenious.
Newmans Error Analysis (NEA) is a diagnostic procedure that was developed in order to
determine if there is a change in conceptual understanding and the extent of the change, if any (Newman 1977, 1983). Newman maintained that when a person attempted to answer a standard, written mathematics word problem, that person had to be able to pass over a number of successive hurdles when the requested task during an interview has been successfully accomplished. This set of sequential procedures involves reading, comprehension, transformation, process skills, and encoding. The error analysis pinpoints the specific level at which the problem-solving process breaks down; and recommendations for intervention maybe designed to address such difficulty. Methodology The study used the descriptive qualitative method using the phenomenographic approach, which is a research tradition designed to answer questions about thinking and learning (Marton, 1986; in Ornek, 2008). The researcher was interested in probing how pre-service teachers experienced understanding and constructed new knowledge when they were taught the concepts of probability without using an algorithmic approach. Fifteen out of 17 third year students of the Bachelor of Secondary Education (Mathematics) program of the College of Education in a higher education institution in Bacolod City who were enrolled in Probability Theory for the first semester of academic year 2015 2016 were included in the study. All students were female. Names have been changed to protect