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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

by

Sweet Rose P. Leonares

For additional information, please contact:

Author’s 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 seranoel2@gmail.com

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PREPARING PRE-SERVICE TEACHERS TO TEACH PROBABILITY

USING HEURISTICS

by

Sweet Rose P. Leonares1

ABSTRACT

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.

Introduction

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 latter’s 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 Vygotsky’s 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.

Newman’s 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 their identity and maintain confidentiality. This study utilized three instruments for data gathering. The first two were the teacher-made (or modified) Pretest and Posttest on Probability Problem Solving containing 7 questions each on Classical and Complementary Probability, Relative Frequency Probability, Addition Rules for mutually and not mutually exclusive events, Multiplication Rules for Independent and Dependent events, and Conditional Probability. All questions were of the typical variant. The pretest items were randomly arranged. A different set of items were given at the end of the semester as posttest with a different arrangement of the test items. The pretest and posttest items were constructed so that they were generally isomorphic: the given information, the key words/phrases, and the required probability. These tests were subjected to content and face validity. The third instrument contained the Newman’s Error Analysis (NEA) prompts and rubric. The prompts were the questions used in the pretest and posttest interviews to determine the level at which a student’s problem solving process had broken down. The rubric was used to determine the category of the error that the student committed (if any), in the course of solving a particular probability problem.

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Individual interviews were conducted after the pretest for the error analysis using NEA prompts prior to the beginning of instruction. This procedure assumed that a person wishing to obtain a correct solution to a word problem must proceed according to the following sequential steps: (1)Read the problem; (2) Comprehend what is read; (3) Carry out a mental transformation from the words of the question to the selection of an appropriate mathematical strategy; (4) Apply the process skills demanded by the selected strategy, and (5) Encode the answer in an acceptable written form, for which he or she will be assigned the NE (No error) category; otherwise it will be the error level corresponding to the task was not accomplished. A correct answer which, from the interview, was not associated with the correct concept would be assigned a corresponding Newman error category. On the other hand, a careless error, which was identified and corrected by the student during the interview, could not be attributed to any error level. A second interview using the same prompts was done after the posttest. Seventeen students were interviewed both times, but two of the posttest recordings were corrupted and could not be retrieved. The interviews were in line with phenomenographic research methodology wherein the think-aloud protocol revealed what was happening in a person’s head when he or she was performing the task. Additionally, from the test papers and the individual interviews the heuristics that a student adopted were identified and verified as to whether these were appropriate for the particular problem. Eventually, the classification was narrowed down to the four main categories of heuristics: representation (using a diagram, table, equation, formula, symbols), simplification (looking for patterns, restating the problem), generic (using guess-and-check, keywords, thinking of a similar problem) and none (that is, there was no attempt at all to solve the problem). Lectures were conducted on the different types of heuristics and incorporated the use of the most appropriate method as probability topics were discussed. Pedagogical methods focused on collaboration, class discussions, group activities and formative tests. Importance was also given to board work for assigned problems/homework. This was to make them aware of the value affirmation of one’s work from the teacher and peers, and at the same time, immediately correct any conceptual errors so as not to replicate these in future problem solving endeavors. The students were required to explain their own solutions; in the process, enhancing their communication skills. At the start of the semester, the students were not informed concerning the study. However, debriefing was conducted after the last interview was conducted. No negative reaction was raised upon full disclosure. Data analysis included the comparison of the error profiles of the students generated using the NEA rubric. These were considered to be a good measure of the improvement in student performance. Pretest and posttest interviews using the NEA prompts provided additional insight into the solving process of the student and the heuristics that were used, like the use of keywords, which were not captured from test papers. More importantly, the interviews provided a glimpse into the knowledge gaps that could have existed in the minds of the students. Results and Discussion Pretest results showed that the pre-service teachers had very little knowledge of probability at the beginning of the semester. Except for the most basic question (Item 1) wherein some students answered intuitively, the rest of the questions were identified to be difficult to understand. Posttest results, however, showed marked increase in the number of students getting correct answers in all items, except in Item 6 (Figure 1).

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Figure 1. Number of students with correct answers in the Pretest and Posttest

There was minimal use of heuristics in the pretest items, except for some forms of representation (symbolic), simplification (restating the problem) and generic methods (guess-and-check) (Figure 2). None was used whatsoever in some of the items. There was the predominant use of representation heuristics in the posttest in all items, except in Item 6; this could be attributed to the training the students have had as Mathematics majors, where the conscious use of symbolic representation is almost second nature. In classroom discussions, representation was emphasized as an important component of probability problem solving (e.g., defining an event). In addition, other strategies such as the use of diagram or table where applicable, which generally allowed the solver a fuller understanding of the problem that could have helped him or her to tackle it better (Tiong et al., 2005). Generic heuristics may help solve the problem directly, or may help in finding suitable representations. An intuitive guess could be the answer to the problem or the use of keywords could lead the problem solver to the appropriate formula to be used. Since keywords are generally inherent in probability problems, this explains the use of the heuristic in all items, especially in the posttest. Pretest use of the heuristic, however, was limited mostly to guessing the answer. The use of simplification methods, specifically, looking for patterns, was relatively higher in Items 4a and 4b since this was the most appropriate heuristic for the type of problem that was posed. Posttest results also showed generally greater utilization of simplification heuristics than in the pretest (Figure 2).

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Figure 2. Comparison of the frequency of heuristics use by category between the

Pretest and Posttest, by Item

The following examples show how the students made use of heuristics. Josie’s solution (Figure 3) consisted of drawing a bar graph in an attempt to present the different proportions. However, the graphs did not accurately capture the information provided in the problem. Her posttest solution, however, was based on the accurate interpretation of the keywords in the form of proportions.

Figure 3. Josie’s solution to Pretest and Posttest, Item 4

During the pretest interview for item 2, Heidi acknowledged that her answer was probably wrong as she had no basis for coming up with such a solution (Figure 4). For her posttest solution, she identified the key phrase “not influenced” but interpreted it to mean that the two events are mutually exclusive. This is reflected in the Venn diagram that she constructed. While she correctly interpreted the required probability to involve the operation of intersection, her answer, which was based on the Venn diagram, was zero.

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Figure 4. Heidi’s solution to Pretest and Posttest, Item 2 Mila’s pretest answer consisted simply of adding the numbers (Figure 5). In the interview, she admitted that she had no idea how to solve the problem, so she only guessed the answer by adding the two values. Many had the same answer. In her posttest solution, she identified the relationship between the two events as independent. She did not define L and M, but correctly identified the probability values associated with each of these letters, which are the beginning letters of the words ‘laptop’ and ‘MP4’, respectively. The correct formula was used, and generated the correct answer.

Figure 5. Mila’s solution to Pretest and Posttest, Item 2 Heidi’s answer to pretest Item 3 was incorrect (Figure 6). But her posttest answer presents an example of a student who recognized the error on her own while explaining her solution during the interview. She was allowed to write what, to her, were her corrections, which turned out to be correct. During the interview she identified the relationship as not mutually exclusive.

Figure 6. Heidi’s solution to Pretest and Posttest, Item 3

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Lanie’s pretest solution to Item 5 consisted of simplifying the problem but her solution, although intuitive, did not lead to the correct answer (Figure 7). Her posttest solution showed the use of symbolic representation, a table, a formula and, from the interview, the use of keywords. The events were clearly defined such that, even if the formula used the letters only, it was easy to relate them to the definitions. Also, the table headings and stubs were completely labelled. She affirmed that answering the problem was easier with a table.

Figure 7. Lanie’s solution to Pretest and Posttest, Item 5 Heidi’s solutions for both pretest and posttest (Item 6) are shown in Figure 8. The fractions in the pretest have denominators which correspond to the number of good tetrapaks on the first and on the second selections. This is indicative of an intuitive conception of an experiment without replacement although the overall solution is incorrect. The posttest solution was correct, except for the symbolic representation of the probability that the second question will be answered by a male student if the first question was also answered by a different male student. Using her notation, it should have been written as P(M2/M1), instead of P(M2).

Figure 8. Heidi’s solution to Item 6. Top: Pretest; bottom: Posttest

Figure 9 compares the levels of error committed by the students based on the NEA per item per type of test (pretest vs. posttest). The items were arranged according the increasing levels of difficulty. No student committed a reading error in both pretest and posttest, which was expected. However, it is not necessarily true that if a student is able to read all the words, then he or she automatically comprehends what he or she has read. The interviews were replete with examples of this dilemma.

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Common errors in the pretest were Comprehension and Transformation in the first four items, with the number of students committing the first type increasing as the level of difficulty of increased. While Transformation errors seem to have decreased with increasing difficulty, it should be remembered that the error categories in the NEA are hierarchical. Therefore, the fact that the Comprehension errors had increased actually implies that there were only a few who hurdled that level and could possibly have committed Transformation error. Hence, the greater the number of students belonging to lower levels of error, the less the number belonging to the higher levels. Improvements in terms of the common errors committed from the pretest to the posttest were generally observed for all items. The error level, except for Item 6; with 26.7% of the students still having that difficulty.

Figure 9. Comparison of Errors Committed between the Pretest and Posttest, by item Process errors tended to dominate from Items 4a to 3, but as the level of difficulty increased, so did the number of students committing the lower level which is Transformation error. This could imply that the students knew what they were supposed to do (hence, hurdling the Comprehension error), but they could not identify the procedure or process that they were supposed to apply in order to successfully solve the problem. The results presented in Figure 9 agree with the study of Ellerton and Clements (1996) that different questions produced quite different error patterns and would appear to have important implications for curriculum and test developers and for classroom teachers. Teachers need to be reminded that many “correct responses may be given by students who do not really understand the concepts being tested” and that different kinds of errors are likely to be committed for different kinds of tasks. The authors further indicated that the high percentage of Comprehension and Transformation errors in studies using the Newman procedure in widely differing contexts has provided “unambiguous evidence of the importance of language in the development of mathematical concepts,” which raises the difficult issue of what educators can do to improve a learner’s comprehension of mathematical text or ability to identify an appropriate sequence of operations that will solve a given problem.

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This study highlighted the relatively limited ability of the students in comprehending what the problems required. One possible reason is that the structure of probability word problems is different from those in algebra or calculus, which they have been used to. However, since probability problems are replete with key words that identify or imply the type of relationship that may exist between events and the operation that is required in order to successfully complete the solution, emphasis was given to being able to identify these keywords and understand what they mean. The researcher believes that making the students more aware of the importance of keywords contributed to the movement towards higher-level (and no) error categories. A relative measure of improvement in terms of the levels of error hurdled can be seen from Table 1. The row labels reflect the changes in the levels. The highest frequency occurred with an improvement of 1 level with 34.3%. A two-step increment happened 17.1% of all the possible instances, a three-step improvement was observed in 1 out of every 5 instances. The highest possible improvement with a hurdle of four categories, which occurred when a student leveled up from committing a Comprehension error to a No Error solution, occurred 15.2% of the time. Only in 8 out of the 105 instances were there no changes between the pretest and posttest error levels (other than a NE). Did the pre-service teachers learn anything at all? Figure 1 presents the number of students successfully completing the solution for each problem in both pretest and posttest. It is clear that the number of students getting correct answers have increased in each item, except for Item 6. Figure 10 shows how each student scored in the pretest and posttest. The posttest interview results indicated that the pre-service students believe that classroom discussions and activities have, to a certain extent, prepared them for their eventual teaching responsibilities, especially for probability. They specifically mentioned the use of heuristics as an important pedagogical tool. The results corroborate in part the results of the study conducted by Dollard (2011). He emphasized that mathematics teacher educators, specifically those teaching probability, cannot assume that the pre-service mathematics teachers enter their classrooms with adequate knowledge of the subject matter. This is true in this study where the knowledge level of the students concerning some concepts of as measured by the pretest was almost nil or at most intuitive. Table 1. Distribution of the Improvement in the Categories of Error Committed from Pretest to Posttest Level

Improvement in the level of Error

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C P (2) 1 4 4 2 3 1 1 16

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Total 15 15 15 15 15 15 15 105

Note: NE – No error E – Encoding error P – Process error T – Transformation error C – Comprehension error X – Careless error (#) – number of levels hurdled

Figure 10. Scores of the pre-service teachers in the pretest and posttest

Conclusion The use of heuristics in the pretest was limited to writing the given information in symbolic form, restating the problem in the dialect, guesswork, or none at all. On the other hand, there was substantial and intentional use of the strategies in the posttest, and as observed from the written output as well as from the interviews, the methods used in solving the problems were appropriate. There was the ubiquitous use of symbolic representation, formulas and keywords. Posttest results showed marked increase in the number students who got correct answers for each item as well as the number of correctly-solved items per student. Seven students got 3 or more correct answers, including 4 students who got 5. They have improved in terms of the category of errors that were committed (from lower- to higher-level). In general, they have a fairly good understanding of Classical, Relative frequency, Complementary probability and the addition rule for mutually exclusive events. They had a moderate grasp of the concepts of the Additive Rule for not mutually exclusive events, the Multiplicative Rule for independent events, and

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Conditional probability. However, there still seems to be a struggle to understand the concepts of dependent and sequential events and the multiplicative rule that applies to these relationships. Overall, the pre-service teachers had indicated that they are, to a certain extent, prepared to teach probability through their exposure to the subject matter, especially with the use of heuristics and other pedagogical approaches that they had been introduced to. Implications The results of the study look promising where pre-service training is concerned. Considering their status at the beginning of the study and their accomplishments at the end, it could be said with cautious optimism that while not all of the prescribed topics were discussed in class, the preparation that they were exposed to, both cognitively and pedagogically and in the context of an enactivist learning environment, would have started to prepare them for the task of teaching Probability in the K to 12 curriculum. Recommendations Currently, Probability is taught as a three-unit subject. The time allotted would not be sufficient to cover the topics necessary for a more comprehensive understanding of the concepts. If it is possible, another 3-unit subject (Probability Theory II) could be offered as a continuation of the first. The effectiveness of the use of heuristics in solving probability problems should lead to the intentional use of the strategies in teaching Probability Theory. The topic on Heuristics could, therefore, be included in the syllabus for the course. Due to time limitations and the scope of the study, certain pedagogical methods were not tested or applied. Future researches could dwell on the effects of manipulatives and simulations on the students’ level of understanding. This study covered only seven concepts and the problems that were posed were only of the typical variant. Researches could be done to determine the effect of using atypical or complex variants of the probability problem.

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References Batanero, C. & Diaz, C. (2012). Training school teachers to teach probability: Reflections and

challenges. Chilean Journal of Statistics, 3(1), 3-13. Batanero, C. Godino, J. & Roa, R. (2004). Training teachers to teach probability. Journal of

Statistics Education, 12(1). Corter, J. & Zahner, D. (2007). Use of external visual representations in probability problem

solving. Statistics Education Research Journal, 6(1), 22-50. Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For

the Learning of Mathematics. 15(2). 2-9. Dollard, C. (2011). Preservice elementary teachers and the fundamentals of probability.

Statistics Education Research Journal, 10(2), 27-47. Ellerton, N. F., & Clements, M. A. (1996). Newman error analysis. A comparative study involving

Year 7 students in Malaysia and Australia. In P. C. Clarkson (Ed.), Technology and mathematics education (pp. 186-193). Melbourne: Mathematics education Research Group of Australasia.

Garfield, J. & Ahlgreen, A. (1988). Difficulties in learning basic concepts in probability and

statistics: Implications for research. Journal for Research in Mathematics Education, 19(1), 44-63.

Greer, G. & Mukhopadhyay, S. (2005). Teaching and learning the mathematization of

uncertainty: Historical, cultural, social, and political contexts. In G. A. Jones (Ed.) Exploring probability in school: Challenges for teaching and learning (pp. 297 – 324). New York: Springer.

Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59-98. Newman, M. A. (1977). An analysis of sixth-grade pupils’ errors on written mathematical tasks.

Victorian Institute for Educational Research Bulletin, 39, 31-43. Newman, M. A. (1983). Strategies for diagnosis and remediation. Sydney: Harcourt, Brace

Jovanovich. Ornek, F. (2008). An overview of a theoretical framework of phenomenography in qualitative

education research: An example from physics education research. Asia-Pacific Forum on Science Learning and Teaching, 9(2).

Reston, E. (2012). Exploring inservice elementary mathematics teachers’ conceptions of

probability through inductive teaching and learning methods. 12th International Congress on Mathematics Education. COEX, Seoul, South Korea.

Sumara, D. & Davis, B. (1997). Enactivist theory and community learning toward a complexified

understanding of action research. Education Action Research, 5(3), 403 – 422.

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Tiong, J. Y. S., Hedberg, J., & Lioe, L. T. (2005). A metacognitive approach to support heuristic solution of mathematical problems. Singapore: National Institute of Education.

Vygotsky, L. S. (1978). Mind in society: the development of higher psychological processes.

Cambridge: Harvard University Press. Walpole, R. (1982). Introduction to statistics. (3rd Ed.) New York: Macmillan. Yoong, W.K. & Tiong, J. (2006). Developing the repertoire of heuristics for mathematical

problem solving: Student problem exercises and attitude. Technical report for project CRP38/03 TSK. Singapore: National Institute of Education.

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APPENDICES

Appendix 1 Pretest and Posttest Questions

PRETEST POSTTEST

1. In a class about Asian Civilizations there are 35 students, of which 10 are males. The teacher of this class has prepared class cards, one for every student. Every meeting the teacher conducts a graded oral recitation concerning the topic assigned for the day. In order to determine who is going to recite, he shuffles the class cards and randomly selects one. What is the probability that for a particular question, a female student will have to answer?

A school surveyed the favorite snacks of 1000 of its students. Of the 625 male students who were surveyed, 200 liked ice cream, 125 preferred pizza, while 300 chose hamburger. Among those surveyed, 300 liked ice cream, and 450 chose hamburgers. A student is selected at random from the group. Find the probability that the student is not male.

2. The probability that a person vacationing in Bacolod will visit Mambukal is 0.33 and that he will visit the Ruins is 0.92. His decision to visit one site is not affected by his decision to visit the other. What is the probability that a person vacationing here will visit both sites?

The probability that a student attending a certain college in Cebu will purchase a laptop is 13/20. The probability that he will buy an MP4 player is 3/10. If buying an MP4 is not influenced by whether he bought a laptop or not, what is the probability that he will buy the laptop and the MP4?

3. Twenty percent of the students in a chemistry class are physics majors, sixty percent are chemistry majors, and twelve percent of them are majoring in both chemistry and physics. What is the probability that a randomly chosen student will be a chemistry major or physics major (or both)?

A geology professor has two graduate assistants helping with her research, one younger (A) than the other. The probability that the older (B) of the two will be absent on any given day is 0.08; the probability that the younger of the two will be absent on any given day is 0.06; and the probability that both will be absent on any given day is 0.02. Find the probability that either of the two assistants is absent.

4. Darlene has four suitors: Andoy, Bugoy, Caloy, and Dodoy. According to her, Bugoy has twice the chances of Andoy and Caloy, while Dodoy has three times the chances of Bugoy of becoming her boyfriend. Only one of these suitors will eventually become her boyfriend.

Miko anticipates that she will be exempted from taking the final exams in some subjects. She thinks that she has three times as many chances of being exempted in Linear Algebra as Probability Theory, five times as many chances of being exempted in Filipino as Probability theory, and twice as

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What is the probability that Darlene’s boyfriend will a. be Bugoy? b. either be Dodoy or Caloy?

many chances of being exempted in Probability Theory as Assessment. Find the probability that Miko will be exempted from taking the final exam in a. Filipino. b. any Math subject.

5. A class in advanced physics is composed of 15 juniors, 30 seniors, and 5 graduate students. The final grades showed that 7 of the juniors, 12 of the seniors, and 3 of the graduate students received a 1.5 for the course. If a student is selected at random, find the probability that the student is a senior, if it is known that he/she got a grade of 1.5 (Walpole, 1982).

A school surveyed the favorite snacks of 1000 of its students. Of the 625 male students who were surveyed, 200 liked ice cream, 125 preferred pizza, while 300 chose hamburger. Among those surveyed, 300 liked ice cream, and 450 chose hamburgers. A student is selected at random from the group. If it is known that the student is female, find the probability that she likes pizza.

6. Find the probability of randomly selecting two good tetrapaks of milk in succession from a cooler containing 20 tetrapaks of which 5 have spoiled.

1. In a class about Asian Civilizations there are 35 students, of which 10 are males. The teacher of this class has prepared class cards, one for every student. Every meeting the teacher conducts a graded oral recitation concerning the topic assigned for the day. In order to determine who is going to recite, he shuffles the class cards and randomly selects one. What is the probability that for two successive questions, two male students are called to answer one question each?

Appendix 2 Newman Error Analysis Prompts

Code Type of Error Prompt Explanation

R Reading

“Read the question to me. If you don’t know a word, tell me.”

Could not read a key word or symbol in the written problem to the extent that this prevented him/her from proceeding further along an appropriate problem-solving path

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C Comprehension “Tell me what the question asked you to do.”

Able to read all the words in the question, but has not grasped the overall meaning of the words and, therefore, was unable to proceed further along an appropriate problem-solving path

T Transformation

“Now tell me what method you used to find the answer.”

Had understood what the questions wanted him/her to find out but was unable to identify the operation, or sequence of operations, needed to solve the problem.

P Process

“Show me how you get your answer, and “talk aloud” as you do it, so that I can understand how you are thinking.”

Identified an appropriate operation, or sequence of operations, but did not know the procedures necessary to carry out these operations accurately.

E Encoding

“Tell me, what is the answer to the question? Point to your answer.”

Correctly worked out the solution to a problem, but could not express this solution in an acceptable written form.

X Careless

If, when reworking a question using the Newman Error Analysis the student is able to correctly answer the question, the original error is classified as a careless error, or in a second attempt he/she gets the correct answer and, after the teacher has listened to the answers to the Newman requests, the teacher is convinced that the child originally made a careless slip

NE No Error The student successfully

completes the solution and gets the correct answer.