1 Investigating Filipino Mathematics Teachers’ Beliefs and Instructional Practices in
Developing Students’ Problem Solving Skills: A Case Study
1Maria Digi Anna Mance – Avila, 2Angela Fatima H. Guzon
1&2Ateneo de Manila University, Philippines
1Corresponding author: maria.mance@obf.ateneo.edu
Abstract
Purpose - In the Philippines, improving the problem solving competency of students remains a major concern. In order to understand the current situation of problem solving in the country, this study investigated how Filipino mathematics teachers incorporate problem solving into their lessons and how they develop problem solving among students. In particular, this study explored teachers’ beliefs about mathematics in relation to the following dimensions of their instructional practices: the types of tasks they use in classroom instruction and their practice of flexibility in problem solving.
Method - This study followed a case study design to explore teachers’ beliefs about mathematics in relation to various dimensions of their instructional practices. Three mathematics teachers from a public science high school were selected as participants of the study. In order to determine their beliefs and instructional practices, each teacher’s classroom instruction was observed, lesson plans were examined, a questionnaire on teachers’ beliefs was administered, and post- instruction interviews were conducted.
Findings - Analysis of the gathered data showed that all three teachers hold beliefs that were more associated with the constructivist view than with the transmissive view of mathematics. Two of the teachers used higher-level demands tasks the most frequently in their instruction, which reflect their constructivist beliefs. However, one teacher used lower- level demands tasks the most often, which is contrary to the constructivist perspective. Among the teachers, only one teacher showed evidence of flexibility in problem solving. Overall, it was observed that although the teachers predominantly hold constructivist beliefs, these beliefs were partially evident in their instructional practices.
Significance - The researcher believes that identifying factors that possibly interplay in the instructional practices of mathematics teachers from the preparation up to the actual implementation of the lesson can provide meaningful information about how these factors possibly interact and how the dynamics can be optimized and used to improve teaching and consequently, learning of mathematics, particularly in the area of problem solving.
Keywords: Problem Solving, Constructivist Beliefs, Transmissive Beliefs, Higher-Level Demands Tasks, Lower-Level Demands Tasks, Flexibility in Problem Solving
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IntroductionIn the Philippines, improving the problem solving competency of mathematics students remains a major concern. In fact, problem solving is one of the twin goals of mathematics at the basic education level, as illustrated in the latest K- 12 mathematics curriculum framework (Department of Education, 2013). While Filipino students perform well in knowledge acquisition, they demonstrate relative weakness in knowledge application through problem solving (Ogena, Laña, & Sasota, 2010). This is evident in the 2003 and 2008 Trends in International Mathematics and Science Study (TIMSS) test results. In 2003, the Philippines ranked 23rd out of 25 countries in fourth grade mathematics and 34th out of 38 countries in eighth grade mathematics. In 2008, although only the science high schools participated in the Advanced Mathematics category, the Philippines performed least among ten participating countries in mathematics overall and as well as in specific content areas and cognitive domains in terms of average scale score and percent correct responses. In 2018, the Philippines also joined the Programme for International Student Assessment (PISA) of the Organization for Economic Co-operation and Development (OECD) for the first time. The PISA results revealed that the Philippines scored 353 in Mathematics, which is significantly lower than the average of participating OECD countries and is classified as below Level 1 proficiency. Students who scored Level 1 proficiency can perform direct and straightforward mathematical tasks. In contrast, students who scored Level 3 proficiency can execute clearly described procedures, including those that require sequential decisions, with sound interpretations (Department of Education, 2019).
Teaching is an equally important constituent of learning. Hence, in order to understand why students find solving problems difficult, it may be useful to explore how teachers develop students’ problem solving skills. Teachers play a significant role as they are in charge of both the teaching and learning processes. Investigating several factors that are at play in the teachers’ classroom instruction may help shed light on students’ behaviour and achievement in problem solving. It is worthwhile therefore to investigate the instructional practices of Filipino mathematics teachers in developing students’ problem solving skills.
Several research studies claim that teachers’ practices in the classroom are linked to their beliefs about the nature, learning, and teaching of mathematics (Thompson, 1984, 1992; Pajares, 1992; McLeod & McLeod, 2002;
Schoenfeld, 1998; Wilson & Cooney, 2002; Phillip, 2007; Speer, 2008; Akinsola, 2009; Cai, Perry, Wong & Wang, 2009; Eichler & Erens, 2015; Brendefur & Carney, 2016). Raymond (1997) referred to beliefs as personal judgments about mathematics as a result of previous experiences in mathematics. Drawing on these research studies, it can be said that teachers’ beliefs play a major role in shaping the way they prepare for their classes, develop their lessons, and select which strategies to use to help the students achieve their learning goals. In order to fully understand mathematics teachers’ instructional practices, the current study aims to investigate the relationship that exists between teachers’
beliefs and practices during the planning and implementation stages of a lesson.
Other empirical studies on teachers' beliefs about mathematics also show that these beliefs influence how students learn mathematics and how they deal with mathematical tasks (Grigutsch, Raatz & Törner, 1998 as cited in Scheimmer, Krauss, Bruckmaier, Ufer & Blum, 2013). According to Lappan and Briars (1995), the tasks with which a teacher engages the students in studying mathematics are important because they have a substantial impact on the students' learning processes. If the teachers believe that the primary goal of teaching mathematics is to develop among the students the ability to think, reason, and solve problems, then it is expected that their mathematics instruction mainly involves high-level, cognitively challenging tasks (Stein & Lane, 1996). Drawing on these studies, the current study also aims to find out whether or not teachers’ beliefs about mathematics dictate the types of tasks they select and use for classroom instruction.
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Aside from the types of tasks, this study examines another specific dimension of a teacher’s instructional practice, which is flexibility in problem solving. Flexibility in this context is defined as the ability to select, create, or modify appropriate problem solving strategies. It is an important skill in problem solving because it enables an individual to approach a problem in multiple ways when the conditions warrant instead of adhering to one solution strategy. A research study by Stipek, Givvin, Salmon and MacGyvers (2001) reports that teachers' beliefs determine the degree to which they emphasize and encourage their students to explore mathematics problems and try alternative solution strategies. That is, teachers who believe that correctness is more important than understanding tend to ignore students' effort and the attempts to use multiple solutions. Drawing on this research study, the current study aims to investigate whether or not teachers foster a learning environment that elicits flexibility in problem solving among the students, and whether or not the teachers’ beliefs are aligned with their practice of flexibility in problem solving in the classroom. The researcher believes that identifying factors that possibly interplay in the instructional practices of mathematics teachers from the preparation up to the actual implementation of the lesson can provide meaningful information about how these factors interact and how the dynamics can be optimized and used to improve teaching and consequently, learning of mathematics. The results of this study can not only aid teachers in lesson preparation but can also help the teachers reflect on their current practices in a structured manner. Results of this study can also be used as a basis for establishing appropriate professional development programs.Theoretical and Conceptual Frameworks
The overarching theory used in the current study is Schoenfeld’s (1998) Theory of Teaching-in-Context, which represents teaching as a function of a teacher's beliefs, goals, and knowledge. However, the current study only focuses on beliefs, which Schoenfeld referred to as experiences and understandings of an individual coded into mental diagrams.
In the domain of mathematics, a teacher’s beliefs can be interpreted as constructs which have a strong shaping effect on how a teacher perceives and approaches mathematics as a discipline. Teachers may have beliefs about particular students and classes of students, the nature of the subject matter, the nature of the learning process, the nature of the teaching process, and the roles of various kinds of instruction. Schoenfeld identified these classes of beliefs as determinants of a teacher's moment-to-moment decisions and actions in a classroom. In the area of problem solving, these beliefs may influence how a teacher selects strategies to be used or avoided in solving a problem, how much time and effort to exert in solving that problem, and so on. The current study is rooted in this idea by Schoenfeld that beliefs establish the context in which a teacher’s instructional practices operate. Meanwhile, a teacher's knowledge includes a repertoire of useful information that he or she can bring and integrate into a teaching situation. As described by Schoenfeld (2011), what the teacher knows and does not know can be an affordance or a constraint with regard to what he or she can do in a classroom.
In order to assess the beliefs of mathematics teachers, the current study used the research of Voss, Kleickmann, Kunter, and Hachfeld (2013) as a conceptual framework. In this research, the transmissive and constructivist theories of learning were used in classifying the beliefs of mathematics teachers. Seeing mathematics as an established system of knowledge and procedures that must be covered is an indicator of a transmissive orientation (Swan, 2005). A teacher who follows this orientation regards learners as passive recipients of mathematical knowledge. This knowledge is passed on to the learners through teacher-directed lessons which focus on the demonstration and repetition of typical examples.
Hence, the usual classroom approach of a teacher who holds transmissive beliefs is to provide necessary demonstrations and to explain correct solutions in detail in order to facilitate the learning of the students. On the other hand, seeing mathematics as a dynamic body of interrelated ideas and reasoning processes is an indicator of a constructivist orientation (Swan, 2005). A teacher who follows this orientation believes that knowledge moves in more than one
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direction, and that the active engagement of students in learning mathematics is important (Prideaux, 2007). Therefore, a teacher who holds constructivist beliefs functions as a mediator whose task is to create a suitable learning environment which supports students’ active and independent construction of knowledge. According to Voss et al. (2013), although these two views of mathematics are distinct, they are not mutually exclusive and are not on opposite ends of a linear continuum. Hence, it is possible for a teacher to hold and endorse beliefs that are rooted in both the transmissive view and the constructivist view at the same time.The current study also used the mathematical task analysis guide by Stein, Smith, Henningsen (2000) in determining the type of tasks selected by the teachers for use in classroom instruction. Four types of tasks were identified according to levels of cognitive demand, namely: (1) memorisation tasks; (2) procedures without connections to concepts; (3) procedures with connections to concepts; and (4) doing mathematics tasks. Memorisation tasks involve either reproducing or committing previously encountered definitions, rules, and formulas to memory. These tasks do not involve procedures because a procedure does not exist or it is not necessary to complete the tasks. Rather, memorisation tasks only require exact reproduction of previously learned materials without any attempt to develop deep mathematical understanding among the students.
Procedures without connections to concepts are algorithmic in nature. These tasks require limited cognitive demand because the focus is to produce correct answers through standard solution procedures without paying attention to what these procedures mean and why they work. The use of procedures in this type of tasks is either explicitly stated or is evident from previous instruction or worked-out examples. In other words, tasks of this type may have been given right after a teacher’s demonstration or explanation on how to solve similar tasks, thereby suggesting a particular solution procedure to the students. On the other hand, procedures with connections to concepts emphasize the use of procedures to facilitate development of deeper levels of mathematical understanding. Instruction of these tasks suggests procedures that are broad and general instead of calling out specific procedures in order to require cognitive effort among the students. Multiple representations are used to enable the students to make connections, develop meaning, and understand conceptual ideas that underlie the procedures.
Lastly, doing mathematics tasks are tasks which require substantial cognitive effort because plain memorisation or standard solution procedures are no longer involved. Rather, students are expected to explore relevant knowledge and experiences, draw connections, and make appropriate use of these while working through the tasks.
Solving these tasks necessitates complex and non-algorithmic thinking and reasoning because the nature of the solution is not straightforward and may even involve varied pathways. Constraints which may limit possible solution strategies need to be actively checked while working through the tasks.
The aforementioned types of tasks are grouped under two broader terms, namely lower-level demands and higher-level demands. Lower-level demands tasks focus on recalling and reproducing basic facts, rules, formulas, and procedures without the need for deep understanding of concepts, whereas higher-level demands tasks emphasize making connections, reasoning, and applying. Solving lower-level demands tasks only requires minimum cognitive load among the students. On the other hand, working on higher-level demands tasks may result to a productive struggle and anxiety among the students because of the high cognitive load required.
According to Staub and Stern (2002), the development of thinking and reasoning processes is given more importance than the acquisition of specific knowledge in a constructivist-inspired classroom. Decker, Kunter and Voss (2015) also emphasized that a constructivist approach to teaching involves frequent use of engaging mathematics instruction and cognitively challenging tasks during class. Therefore, teachers who hold constructivist beliefs are expected to use tasks that elicit maximum cognitive effort among the students through thinking and reasoning in order to achieve their intended teaching and learning outcomes. Based on the mathematical task analysis guide by Stein et al.
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(2000), such tasks are called higher-level demands tasks, which include procedures with connections to concepts and doing mathematics tasks. In contrast, rote memorisation and drilling skills are emphasized in a transmissive-inspired classroom (Staub & Stern, 2002). Hence, teachers who hold transmissive beliefs tend to use tasks that only require memorisation and repetition of previously learned facts and procedures. These memorisation tasks and procedural tasks require minimum cognitive effort among the students, thereby restricting the development of the students’ mathematical understanding. According to Stein et al. (2000), such tasks are called lower-level demands tasks, which include memorisation tasks and procedures without connections to concepts. Drawing on these studies, the current study aims to find out whether or not teachers’ beliefs dictate the types of tasks that they use in their instruction.Another dimension of a teacher’s instructional practices investigated in this study is flexibility in problem solving. According to Underhill (1991), flexibility can be described from a constructivist view as the ability to explore relations between problems and strategies and to make connections between previously encountered problem solving situations and new ones. Developing flexibility among students can be achieved by teachers by modelling it in their mathematics instruction. Therefore, teachers who follow a constructivist view are expected to exemplify flexibility in problem solving in their instruction. In other words, they are expected to present a wide range of heuristics from which the students can draw when solving various problem solving situations. On the other hand, according to Voss et al.
(2013), teachers who follow a transmissive view tend to restrict the demonstration of tasks to a single solution. Drawing on these studies, the current study aims to investigate whether or not there is an alignment between teachers’
constructivist beliefs or transmissive beliefs and their practice of flexibility in problem solving. In order to assess whether or not flexibility in problem solving is evident in the teachers’ instructional practices, two criteria for flexibility mapped out by Star and Seifert (2005) were used as a reference: (1) knowledge of multiple solutions; and (2) ability to create new solutions or re-invent existing solutions. The first criterion refers to an individual's knowledge of a rich repertoire of heuristics from which he or she can choose when solving problems. The second criterion refers to an individual's ability to innovate solutions because existing solutions which apply to one problem may not be appropriate for another problem. Hence, creating novel solutions may be necessary in order to solve the problem at hand.
Methodology
This study followed a case study design to investigate the instructional practices of three mathematics teachers from a specialised science high school in Laguna. The school is located at the centre of a city and is funded by the local government. It offers free secondary education to students who are gifted in the sciences and mathematics. Students who intend to enrol in the said school must belong to the upper 20% of the graduating class and must not have a grade lower than 85% in English, Science, and Mathematics during their last year in grade school. Only 250 students are accepted every year, and each class is composed of 30 to 40 students. The school puts special emphasis on subjects pertaining to science and technology with the end in view of preparing its students for careers in science and technology.
Since this study explores the instructional practices of teachers in mathematical problem solving, a public science high school is an ideal setting because of its specialized curriculum developed by the Department of Education which focuses on science and mathematics subjects.
With the assistance of the mathematics department chair, three mathematics teachers who are currently teaching public high school and senior high school students were selected as the participants of this study. Only three teachers were considered to allow the researcher to interview and observe each participant comprehensively within the given timeframe. Initially, the aim of the study was to focus on expert teachers only, but a minimum of ten years of teaching experience was specified to ensure that enough number of teachers would be qualified to participate.
Experienced teachers were selected because it is possible that novice teachers may not have fully developed their beliefs
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and may not have fully established their instructional practices. Additionally, in order to explore a variety of lessons in the study, different topics in mathematics across year levels were considered based on the schedule and suitability of each topic for the study. Some of the teachers were already at the last part of the lesson at the time of the observation, so the teachers in this study were chosen because they were just about to start their discussion of a new lesson. The topics that were selected should also include solving problems as indicated in the lesson plan of the teachers.The first teacher, referred to as Teacher A, currently teaches Algebra to Grade 7 students. She has been teaching for a total of 34 years – 18 years in a private school and 16 years in her current school. She graduated with a Bachelor’s degree in Accountancy and later on finished 18 units of professional education courses to pursue a career in the academe. The second teacher, Teacher B, currently teaches Statistics and Probability to Grade 10 students. She has been teaching for 15 years - four years in three different private schools and 11 years in her current school. She graduated with a Bachelor’s degree in Secondary Education Major in Mathematics and is presently finishing her Master’s degree in Mathematics Education. She has already completed 36 units and is about to take her comprehensive examination.
The last teacher, Teacher C, currently teaches Basic Calculus to Grade 11 students. She has been teaching for 16 years – three years in her current school and 13 years in five different private schools. She finished her Bachelor’s degree in Secondary Education Major in Mathematics and Master’s degree in Mathematics Education. She is currently pursuing her doctoral degree in Educational Management and has already completed 24 units.
The data used to investigate the beliefs and the instructional practices of the participating teachers were gathered through the following methods: non-participant classroom observation, lesson plan inspection, beliefs instrument, and post-instruction interview. An observation protocol served as a guide for the researcher during the classroom observation. The said protocol identified the following points for observation: materials used for instruction, structure of the lesson, number of tasks used by the teachers, and number of solutions presented by the teachers for each task. Meanwhile, lesson plans, textbooks, and other reference materials provided by the participating teachers were also inspected in order to check the types of tasks included in their classroom instruction. These tasks include tasks used for seatwork, board work, group work done in class, homework, quizzes, and worked-out examples presented by the teacher. The classification of tasks was based on the mathematical task analysis guide by Stein, Smith, Henningsen, and Silver (2000). The number of tasks for each type was divided by the total number of tasks used by each teacher in order to get the percentage distribution. The most frequently used type of tasks was noted as well as the number of higher- level demands tasks given by each teacher throughout the observation period.
The beliefs instrument used to measure the beliefs of the teachers in the current study was adopted from the study of Voss, Kleickmann, Kunter and Hachfeld (2013). All items in the instrument were designed to characterize whether the teacher respondent follows a transmissive view or a constructivist view of mathematics. The instrument consists of 44 items in total – 23 of which are associated with the transmissive view while the remaining 21 are associated with the constructivist view. Teacher respondents were instructed to rate their agreement with each item on a four-point response scale: strongly disagree, disagree, agree, and strongly agree. The items were categorized into seven subscales which apply into two content areas, namely: (1) beliefs on the nature of mathematics and (2) beliefs on learning and teaching of mathematics. To measure the beliefs on the nature of knowledge, two of the subscales were used, namely mathematics as toolbox and mathematics as process. The first subscale, which consists of five items, served to gather data on transmissive beliefs whereas the second subscale, which consists of four items, served to collect data on constructivist beliefs. Meanwhile, in order to measure the beliefs on the learning and teaching of mathematics, Voss et al. (2013) used five subscales. Two of these subscales, namely independent and insightful discursive learning (12 items) and confidence in the mathematical independence of students (five items), assessed beliefs that are associated with the constructivist view. On the other hand, the remaining three subscales, namely clarity of solution process (two
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items), receptive learning from examples and demonstrations (12 items), and automatization of technical procedures (four items), assessed beliefs that are associated with the transmissive view. To assess the overall beliefs of the teachers, the average of their scores in the four subscales under the transmissive view was calculated as well as the average of their scores in the three subscales under the constructivist view. An overall average score of 2 or lower means that the teacher does not agree with most of the statements associated with the corresponding view. On the other hand, an overall average score of 3 or higher means that the teacher agrees with most of the statements associated with the corresponding view. An overall average score between 2 and 3 indicates that the teacher probably holds beliefs that are associated with both views of mathematics.The interview was then conducted after the classroom observation, inspection of lesson plans, and administration of the beliefs instrument in order to give each teacher the opportunity to explain her answers in the questionnaire and clarify what had occurred in class as seen in the video clips, thereby ensuring that subsequent analysis was not solely based on the assumptions of the researcher. The interview encompassed items asking about their understanding of problem solving and how they incorporate problem solving into a lesson. The acquired audio and video recordings were then transcribed and translated to English when necessary. Results of the classroom observation, inspection of lesson plans, and beliefs instruments were analysed in light of the teachers’ answers in the interview for cross-checking.
Results and Discussion Teachers’ Beliefs about Mathematics
Overall, the three teachers scored higher than 3 on items that are consistent with the constructivist view than on items that are consistent with the transmissive view. However, these teachers also agreed with particular items in the beliefs instrument that are associated with the transmissive view as indicated by their average scores. In particular, Teacher A scored 3.0 and above on the following subscales under the transmissive view: mathematics as a toolbox, receptive learning from examples and demonstrations, and automatization of technical procedures. Meanwhile, Teacher B scored 3.0 on only one subscale associated to the transmissive view, namely automatization of technical procedures, and scored lower than 3.0 on the remaining subscales. As for Teacher C, she scored 3.0 and above on all subscales under the transmissive view. Table 1 presents the descriptive analysis of the results for all subscales. Based on the results of the beliefs instrument, it can therefore be assumed that although these teachers predominantly follow the constructivist view of mathematics, they also possess characteristics that are rooted in the transmissive view of mathematics. Such condition is possible to happen according to the findings of Voss et al. (2013) since these two views are not mutually exclusive.
In other words, it is possible for a teacher to manifest characteristics that are linked to both views of mathematics at the same time.
Table 1
Descriptive Analysis of the Beliefs Instrument Responses
Number of items
Teacher A Teacher B Teacher C
μ SD μ SD μ SD
Constructivist beliefs Nature of Mathematics
Mathematics as a process 4 3.0 0 3.0 0.82 3.75 0.50
Learning and Teaching of Mathematics
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Independent and insightful discursive learning 12 3.75 0.45 3.75 0.45 3.58 0.51 Confidence in the mathematical independence ofstudents
5 3.4 0.55 3.4 0.55 3.2 0.45
Total 21 3.52 0.51 3.52 0.60 3.52 0.51
Transmissive beliefs Nature of Mathematics
Mathematics as a toolbox 5 3.0 0 2.8 0.84 3.4 0.55
Learning and Teaching of Mathematics
Clarity of solution procedure 2 2.0 0 1.0 0 3.0 0
Receptive learning from examples and demonstrations
12 3.33 0.49 2.68 0.49 3.5 0.52
Automatization of technical procedures 4 3.0 0 3.0 0 3.5 0.58
Total 23 3.09 0.51 2.61 0.72 3.44 0.51
Note: μ mean, SD standard deviation
Teacher A
Overall, Teacher A scored higher on items that are consistent with a constructivist view of mathematics (μ = 3.52) than on items that are consistent with a transmissive view of mathematics (μ = 3.09). With regard to her beliefs on the nature of mathematics, Teacher A equally scored on items that follow the transmissive view and the constructivist view. Hence, it can be assumed that Teacher A perceives mathematics both as a toolbox and a process. However, when asked in the interview which of the two is more likely for her to follow, she said that mathematics for her is dynamic rather than static. She believes that mathematics involves a continuous discovery of new rules, formulas, procedures, and concepts rather than it being a fixed system. Based on this answer, it can be said that Teacher A holds beliefs about the nature of mathematics that are more consistent with the constructivist view.
As for her beliefs on the learning and teaching of mathematics, the average of her responses to the two subscales under the constructivist view and the average of her responses to the three subscales under the transmissive view were determined and compared. Based on the results, Teacher A scored higher on items that are associated with the constructivist view (μ = 3.65) than on items that are associated with the transmissive view (μ = 3.11). In particular, Teacher A strongly agreed with most of the items that support independent and insightful discursive learning as indicated by her average score (μ = 3.75). For instance, she strongly agreed with statements such as “Students learn mathematics best when discovering solution procedures on their own in relatively simple tasks” and “Teachers should encourage students to look for their own solution of mathematical tasks, even if those are inefficient.” However, it is interesting to note that although Teacher A agreed with items that support independent learning, she also agreed with some items that support receptive learning (μ = 3.33). Based on her answers in the beliefs instrument, she strongly agreed that students learn best when solution processes are demonstrated, and that teachers should convey detailed procedures and numerous examples of how to solve tasks. In order to clarify which of the two approaches is more evident in her classroom instruction, further questions were asked in the interview by the researcher. She explained in the interview that her approach primarily depends on the students. That is, for fast learners, explaining the topic once would normally suffice, but for slow learners, her discussion needs to be detailed and guided. She further explained that she provides guide questions whenever necessary particularly if she sees that the students cannot understand and proceed with a task. This is clearly evident in her response in the interview shown below.
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Teacher A: My teaching approach depends on the students. For smart students, explaining the topic once would suffice, but for slow learners, the discussion really needs to be guided. If the students do not understand, guide questions are necessary. You cannot let them proceed if they do not know what to do. At times, they need step- by-step procedures.Based on the results of the beliefs instrument and the interview, it can be said that Teacher A employs a combination of both independent learning and receptive learning. Although her answers in the beliefs instrument indicate her strong inclination toward independent learning, her approach in class still largely depends on her students.
If she sees that none of her students can proceed with the tasks, she provides questions which serve as a scaffold to lead the students to the answer. In this context, Teacher A perceives receptive learning as more effective. Looking closely at her actual classroom practice, it was also observed that her usual routine was to begin with her own explanation of the concept and demonstration of how to solve tasks before allowing her students to solve tasks independently. From this point, it can be said that Teacher A’s instructional practice is primarily dictated by her beliefs about the needs of her students and not by her beliefs about the teaching and learning of mathematics.
Another interesting observation to note is the consistency of Teacher A’s responses on items that discuss clarity of solution process. Teacher A consistently disagreed that one should be restricted to the demonstration of a single solution when dealing with tasks with multiple solutions (μ = 2.0). Her answers in the beliefs instrument are similar to her answers in the interview, wherein she highlighted that it is expected for her students to have different solutions because some students have their own ways of solving. She therefore believes that the discussion should not be restricted to a single solution when dealing with tasks.
To further probe into Teacher A’s beliefs, questions on her definition of a problem and problem solving were asked in the interview as well. Teacher A described a problem as a task which involves practical application of mathematics principles learned in class. For her, it is extremely important to incorporate problem solving into every lesson because knowing where and how to apply the concepts in real-life situations is the very essence of teaching and learning mathematics. Without problem solving, students will not realize the importance of each lesson and as a result, they will lose their interest in learning mathematics. Furthermore, she emphasized that a problem should not just be answerable by yes or no and true or false. Rather, a problem should elicit “higher-order thinking” among the students when solving. When asked about her approach in developing the problem solving skills of her students, she highlighted the importance of practice. Excerpt from her transcribed interview is shown below:
Interviewer: How do you develop the problem solving skills of the students?
Teacher A: I bombard my students with problems – many problems. I don’t just give them homework. I also give them group work and lots of exercises to work on to develop their problem solving skills.
Teacher A pointed out in the interview that her approach in class is to bombard the students with several problems to allow them to develop their problem solving skills. However, it was observed that only 13% of the tasks that she presented in class were practical applications of the concepts. This observation indicates that when she said that she bombards her students with problems, these problems were not necessarily practical applications of the concepts taught in class. Therefore, it can be said that there is an inconsistency between her definition of a problem, as described in her interview, and the problems that she actually referred to and used in class.
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She also mentioned in another part of the interview that she promotes “learning by doing” in class because listening to the discussion is not enough in mathematics. Although her discussion of the lesson is important, she allots more time to the active engagement of the students by providing numerous problems for them to work on in order to develop their problem solving skills. This method reflects a constructivist view because it promotes active learning among the students. As stated by Decker, Kunter, and Voss (2015), the frequent use of engaging mathematics instruction and cognitively challenging tasks is a characteristic of a constructivist approach to teaching, as in the case of Teacher A based on her answers in the interview.In general, although Teacher A scored higher on items that follow the constructivist view than on items that follow the transmissive view, there are certain statements rooted in the transmissive view that she also agreed with, such as the belief that some students need to be shown detailed demonstrations of procedures particularly in cases when they do not know how to proceed with the given tasks. According to Voss et al. (2013), the transmissive view and the constructivist view are not mutually exclusive; therefore, it is possible for an individual to follow both views of mathematics at the same time. This may be the case of Teacher A.
Teacher B
Results of the beliefs instrument show that overall, Teacher B scored higher on items that are associated with the constructivist view (μ = 3.52) than on items that are associated with the transmissive view of mathematics (μ = 2.61). It is interesting to note that with regard to her beliefs about the nature of mathematics, her score on items that are rooted in the constructivist view (μ = 3.0) is close to her score on items that are rooted in the transmissive view (μ = 2.8).
Looking closely at her answers in the beliefs instrument, she strongly agreed that mathematical problems can be solved in many different ways correctly, which is an indicator of the constructivist view. At the same time, she also strongly agreed that almost all mathematical problems can be solved through direct application of familiar rules, formulas, and procedures, which is an indicator of the transmissive view. When asked to elaborate on her answers, she pointed out in the interview that although mathematics is dynamic, it is not always purely based on new ideas. Rather, mathematics is largely based on previously learned rules, formulas, and procedures. She believes that new ideas in mathematics are always anchored on existing knowledge, and that new methods of solving will not exist without tapping an individual’s prior knowledge and experience in solving. Her answers in the interview indicate that she is still more inclined towards the constructivist beliefs about the nature of mathematics than the transmissive beliefs because she regards mathematics as dynamic, and she acknowledged that it is possible to create new ideas and methods of solving in mathematics by building on previous knowledge and experience. The following is an excerpt from her transcribed interview.
Teacher B: A new concept or a new idea of solving will not exist if the students did not learn anything in the past. So new knowledge is based on the past. You will make connections because there is a reason why that is the solution that the students thought of.
As for her beliefs on the learning and teaching of mathematics, Teacher B scored higher on items that are consistent with the constructivist view (μ = 3.65) than on items that are consistent with the transmissive view (μ = 2.56).
It is worth noticing that in the beliefs instrument, Teacher B agreed with the items that support independent and insightful discursive learning, resulting in an average score of 3.75. In particular, she strongly agreed that learning goals in mathematics is best achieved if the students are allowed to discover their own solutions to tasks, and that teachers should encourage the students to think about their own ways of solving tasks. Her answers were also consistent when asked about her beliefs on receptive learning. Based on her responses in the beliefs instrument, she disagreed that students need to be shown detailed instruction on how to solve tasks. She explained in the interview that since her
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students are already in Grade 10, she expects that they are already capable of solving tasks without the need for step- by-step instruction. Based on the observations made by the researcher, Teacher B’s usual routine in class was to engage the students through group activities. She would normally instruct her students to divide into groups and answer tasks in the textbook collaboratively. It can therefore be said that Teacher B’s agreement towards independent and insightful learning is evident in her classroom practice. However, similar to Teacher A, if the students have queries, Teacher B emphasized that she normally provides guide questions to the students that will lead them to the answer.Another interesting detail is Teacher B’s strong disagreement towards restricting oneself to a single solution when dealing with tasks with multiple solutions as indicated by her answers in the beliefs instrument (μ = 1.0).
According to her, she always points out to her students that there can be more than one solution to a problem, and that they are free to think of their own methods of solving whichever they find easier to use. Her answers in the beliefs instrument and in the interview indicate that she promotes flexibility in problem solving among her students. Evidence of such in her actual classroom instruction is discussed in detail in the latter part of this chapter.
To probe deeper into her beliefs about problem solving, similar questions posed to Teacher A were also posed to Teacher B. When asked about her definition of a problem, Teacher B said that for her, there is a difference between a mathematical problem and a word problem. Excerpt from her transcribed interview is shown below.
Interviewer: How do you define a problem?
Teacher B: Everything we do in mathematics is considered a problem. However, a word problem is different from a mathematical problem per se. In a mathematical problem, numbers and variables are already given. Word problems are applications of the mathematical or numerical problems about the topic.
Based on her response, Teacher B perceives a mathematical problem different from a word problem in a sense that a mathematical problem does not have a context. In other words, Teacher B considers a naked number equation as a mathematical problem in which the numbers and variables are already given, whereas a word problem is the application of a mathematical problem embedded in a context. Her answers indicate that her definition of a problem is based on how a problem is presented, and that there is a distinction between a mathematical problem and a word problem in terms of context.
When asked about how she defines problem solving, Teacher B mentioned that when solving a problem, she expects her students to understand and relate the context of the problem to the lesson proper. She also emphasized that critical thinking should be involved in the process because the students need to strategize a plan for attacking the problem. Similar to Teacher A, Teacher B also expressed firm belief in the importance of incorporating problem solving into the lesson because according to her, it is through problem solving that students realize the practical applications of the concepts learned in class. Otherwise, teaching the concepts is useless. In another portion of the interview, she highlighted that being able to connect the concepts to real-life contexts serves as a motivation for her students to learn mathematics. Her approach, therefore, is to vary the structure of her instruction. There are cases in which her discussion of the concept comes before problem solving. There are also cases that her students begin with solving problems, and then she integrates the concept into the problems. She mentioned that in most cases, she uses the problems as a springboard for discussion to capture the interest of her students. According to her, her students find the latter more interesting than the former because they understand and remember the lesson better through the problem. Based on Teacher B’s description, her classroom practice is not always in the traditional tell-then-practice format. She also employs the problem-solve-instruct strategy (Loibl, Roll, & Rummel, 2016) or the explore-instruct strategy (Loehr,
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Fyfe, & Rittle-Johnson, 2014), which are both labelled under the broader term exploratory learning (DeCaro, DeCaro,& Rittle-Johnson, 2015). An exploratory learning approach combines both a constructivist method and a direct instruction method, wherein exploration activities such as problem solving come before the discussion of the teacher on pertinent procedures and concepts, as in the case of Teacher B. Relating this to her beliefs with regard to her confidence in the mathematical independence of students (μ = 3.4), it can be said that her practice as she described in the interview and her answers in the beliefs instrument are consistent because based on the beliefs instrument, she strongly agreed that students have the capability to find solutions for many mathematical tasks on their own. Meanwhile, her classroom practice also allows the students to solve problems independently before receiving instruction from her on relevant procedures and concepts.
As for her strategy in developing the problem solving competency of her students, she emphasized in the interview that mathematics for her is a skill and a mental exercise which is why it is important to provide numerous tasks for the students to practice on. She further explained that practice is essential in increasing the confidence of the students in problem solving. In the beliefs instrument, she also agreed that frequent practicing of algorithmic tasks is essential to acquire numerical factual knowledge. Based on Teacher B’s answers in both the interview and beliefs instrument, it can be said that she perceives frequent practicing of mathematical tasks as an effective strategy in developing the problem solving skills and the numerical factual knowledge of the students.
Altogether, Teacher B’s beliefs about mathematics are more consistent with the constructivist view than with the transmissive view based on the overall results of the beliefs instrument. This is supported by her answers in the interview which showed her strong agreement towards independent learning and demonstration of multiple solutions when solving tasks – which are attributes consistent with the constructivist view of mathematics.
Teacher C
Similar to the first two teachers, Teacher C had a higher overall score on items that are linked to the constructivist view (μ = 3.52) than on items that are linked to the transmissive view (μ = 3.44). However, the difference in her overall scores is small, which indicates that Teacher C probably holds beliefs that are associated with both constructivist and transmissive views of mathematics at the same time. To confirm whether or not Teacher C holds beliefs that are rooted in both views, her answers in each subscale of the beliefs instrument and in the interview were assessed.
When asked about her beliefs on the nature of mathematics, Teacher C scored higher on items that follow the constructivist view (μ = 3.75) than on items that follow the transmissive view (μ = 3.4). In particular, she strongly agreed that mathematics consists of learning, remembering, and applying a collection of processes and rules, which is consistent with the transmissive view. At the same time, she also strongly agreed that mathematics subsists on new ideas, which is consistent with the constructivist view. When asked to explain her answers in the beliefs instrument, Teacher C pointed out in the interview that she regards mathematics both as a toolbox and a process because in order to come up with new solutions, one needs to have a deep knowledge first of fundamental rules, formulas, and procedures. That is, solving a problem requires recalling previous knowledge and building on this knowledge to create new solutions.
Similar to Teacher B, Teacher C expects her students to tap into their prior knowledge and experience as they attempt to solve a problem. Excerpt from her transcribed interview is shown below:
Teacher C: When you solve, how will you know that your solution is new? It’s a combination of both because you cannot find something new without recalling your previous knowledge. Otherwise, what will be your basis if you don’t have prior knowledge at all?
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Based on her response, Teacher C recognized the fact that it is possible to find something new in mathematics by tapping into one’s prior knowledge. In other words, students do not only stop at memorising rules, formulas, and procedures. Rather, they need to understand and apply these rules, formulas, and procedures in order to come up with new ways of solving. Her response indicates that she still regards mathematics more as a process than as a toolbox because there is a continuous integration of new knowledge into existing knowledge involved. This means that mathematics for her is dynamic, which is an attribute rooted in the constructivist view of mathematics.With regard to her beliefs on the learning and teaching of mathematics, Teacher C also scored almost similarly on items that follow the constructivist view (μ = 3.47) and the transmissive view (μ = 3.44). While she strongly agreed that it is important for students to discover how to solve tasks independently, she also strongly agreed that students learn best when solutions are demonstrated by the teacher. In the interview, she explained that she strongly agreed with both statements because her approach is a combination of independent learning and receptive learning. Normally, she allots half of the class time to her own discussion of the concept and the remaining half to problem solving. This structure is confirmed by the observations of the researcher. Teacher C’s usual approach in class was to begin with her own discussion of the concept and demonstration of how to solve tasks before allowing her students to solve tasks independently. However, Teacher C mentioned that it is a case-by-case approach depending on her perceived needs of the students. Although she believes that it is important for students to complete the problems on their own, there are cases that her students are requesting her to show examples first particularly if the problems are too difficult. In the case of her students who belong to the top section, she mentioned that she usually lets them explore independently because she observed that most of the time, these students can answer a given problem even without examples and guide questions. In other words, she allows her students to explore on their own if they can, but she also gives step-by-step demonstrations to those who cannot proceed with the problems. She emphasized, however, that she still provides a follow-up discussion to ensure that all pertinent details about the topic has been covered. Her answers in the beliefs instrument and in the interview indicate that her instructional practice is primarily dictated by her beliefs about the needs of the students, which is also the case of Teacher A.
As for her beliefs on the clarity of solution procedures, she consistently agreed that it is safer and better to restrict oneself to the demonstration of one single solution when dealing with tasks with multiple solutions, as indicated by her answers in the beliefs instrument (μ = 3.0). When asked to explain in the interview why she agreed with the statements that support the demonstration of a single solution, she pointed out that although there are many solutions to a problem, it is usually better to stick to one solution when solving due to time constraints. According to her, given only one hour per session, it would be impossible to show all solutions to a problem especially during exams, so she reminds her students to use only the solution that they are most comfortable with. Based on Teacher C’s explanation, it can be said that her practice of flexibility in problem solving is mainly influenced by time considerations in class.
Similar to Teacher A and Teacher B, Teacher C was also asked to define a problem and problem solving. Based on her answers in the interview, Teacher C differentiates an exercise and a problem in a sense that a problem requires the students to go through several steps in order to arrive at an answer. She emphasized that a problem, such as a word problem, cannot be solved right away. One needs to analyse and go through a process first before he or she can find the answer. In another part of the interview, she also described problem solving as a process. This is clearly evident in her response shown below.
Teacher A: In mathematics, a problem requires a process. Although exercises also have processes, in problem solving, several steps are necessary before you can answer the question. For example, you cannot solve a word problem right away. There is a process – you have to analyse and think of the solution.
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When asked to describe how she incorporates problem solving into her discussion, she highlighted that she normally separates the discussion of the concepts from problem solving. According to her, given only one hour per session, it is difficult to incorporate problem solving right away. Hence, her approach is to separate the two so that the students can focus on problem solving itself. Based on her answer, it can be said that Teacher C regards the discussion of a concept and problem solving as two separate entities. Shown below is an excerpt of her transcribed interview.Teacher C: In my case, I separate the two. I have to teach them the concept first. After that, they have to apply the concept. I separate problem solving so they can focus on it. Considering that we only have one hour, you cannot incorporate problem solving right away. You have to separate so that there will be focus.
In general, although Teacher C scored higher on items that are associated with the constructivist view than on items that are associated with the transmissive view, the difference in her overall scores is small (nearly 0.03). In the interview, she defined mathematics both as a process and as a toolbox, but based on the researcher’s analysis, Teacher C still recognized the fact that there is a continuous integration of new knowledge into existing knowledge in mathematics, which is a belief consistent with the constructivist view. On another note, she described her approach in class as a combination of independent learning and receptive learning, which are attributes rooted in the constructivist beliefs and the transmissive beliefs of mathematics, respectively. Looking closely at Teacher C’s responses in the beliefs instrument and in the interview, it can be said that she holds beliefs that are associated with both views of mathematics.
Types of Tasks
This section presents the types of tasks used by the teachers in their classroom instruction in order to answer the second research question. The tasks gathered from the lesson plans, textbooks, and classroom observations were categorized according to the types of tasks identified by Stein et al. (2000), namely: memorisation tasks, procedures without connections to concepts, procedures with connections to concepts, and doing mathematics tasks. Table 2 summarizes the types of tasks used by the three teachers in their instruction throughout the data gathering period. Based on the analysis of the tasks, Teacher A used procedures without connections to concepts the most often, while Teacher B and Teacher C used procedures with connections to concepts the most frequently in their instruction.
Table 2.
Summary of Tasks Used by the Teachers in Classroom Instruction Teacher Number of
Sessions
Total Number of Tasks
Lower-Level Demands Higher-Level Demands Memorisation
tasks
Procedures without connections
Procedures with connections
Doing mathematics
tasks
A 5 45 0 34 (75.6%) 8 (17.8%) 3 (6.7%)
B 8 75 0 11 (14.7%) 43 (57.3%) 21 (28.0%)
C 5 17 0 2 (11.8%) 12 (70.6%) 3 (17.7%)
Total 18 137 0 47 (34.3%) 63 (46.0%) 26 (19.7%)
Teacher A
Based on the analysis of the tasks, Teacher A used lower-level demands tasks (75.6%) more frequently than higher- level demands tasks (24.5%). In particular, she used procedures without connections the most often (75.6%), followed by procedures with connections (17.8%) and doing mathematics tasks (6.7%). It should be worth noticing that Teacher
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A did not use any memorisation task in her classroom instruction. Rather, she focused on tasks that emphasize the use of familiar rules and procedures in solving.When asked to differentiate an easy task from a difficult one, Teacher A explained that a task is easy for her if the solution is simple and straightforward such as mere plugging in of values, whereas a task is considered difficult if critical thinking and decision-making are already involved while working through the task. When asked in the interview how Teacher A selects the tasks she uses in her instruction, she explained that she usually begins with easy tasks before she bombards the students with difficult tasks in order to help the students gain interest and confidence in solving. Based on her observation, most of the students lose their interest and confidence when they are given difficult tasks at the start of a lesson because only the top students are able to complete the tasks. Furthermore, most of the students get paralyzed as they attempt to solve difficult tasks, and so when they are given easy tasks later on, they are no longer motivated to finish the remaining tasks. Thus, her approach is to begin with easy tasks then gradually shift to difficult tasks in order to scaffold the learning of the students. This approach helps the students in a sense that once they are done answering the easy tasks, they become confident and interested because they are able to finish the tasks. Therefore, once they are given difficult tasks, they are no longer demotivated to proceed with the tasks.
Teacher A pointed out that when choosing tasks for use in her classroom instruction, she normally copies the tasks from multiple sources. Also, she mentioned that she does not prefer using tasks that are procedural and structured because it is expected for her students to have different solutions. According to her, as long as the answer is correct and the students are able to justify the answers, she allows them to use different solution strategies. This is clearly evident in her response shown below.
Teacher A: I use different types of tasks because a procedural task is not advisable. I don’t require procedures because sometimes the students have their shortcuts. As long as the answer is correct and they are able to justify, not using procedures would be fine. Smart students, in particular, have many techniques, so you cannot force them. Some would even specify what is given, what is asked – I don’t require that. I prefer not using a structure.
Contrary to her answers in the interview, majority of the tasks that she presented in class during the observation are classified as procedures without connections to concepts, which emphasize the use of standard procedures. On the other hand, only three doing mathematics tasks were used by Teacher A throughout her classroom instruction. When the researcher inspected the lesson plan and the textbook that the teacher provided, it was observed that Teacher A copied all tasks directly from one textbook. This inconsistency is probably due to the lack of preparation time because in another portion of the interview, Teacher A explained that although she uses other reference materials, most of the time, she only copies the tasks from the textbook specified in the curriculum which are mostly procedural in nature in order to save time. In addition, she mentioned that she rarely modifies tasks from the book because it is difficult and time consuming. Therefore, although Teacher A does not prefer using procedural tasks, she ends up using this type of tasks in her actual classroom instruction due to insufficient preparation time.
Teacher B
According to the results of the tasks analysis, more higher level demands tasks (85.3%) were given throughout Teacher B’s classroom instruction than lower-level demands tasks (14.7%). In particular, she used procedures with connections to concepts the most often (57.3%), followed by doing mathematics tasks (28.0%) and procedures without connections to concepts (14.7%). Similar to Teacher A, Teacher B did not use any memorisation task in her classroom instruction.
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When asked about how Teacher B selects the tasks that she uses in her classroom instruction, she explained that she normally copies the tasks from the reference material specified in the curriculum. However, if she thinks that the students need more tasks so they can master the lesson, she resorts to other textbooks. Similar to Teacher A, Teacher B finds it difficult to modify or create new tasks, so her usual practice is directly copy the tasks from the reference materials. When asked to define an easy task and a difficult task, Teacher B considers a task easy if it is direct to the point and if it only requires the students to use the current lesson. On the other hand, she considers a task difficult if it requires the students to go through a process and to apply previously learned lessons. Teacher B mentioned that she normally gives more easy tasks than difficult tasks because of the table of specifications (TOS) of the school, which requires mathematics teachers to give a higher percentage of easy tasks than difficult tasks. However, since it was observed that more higher-level demands tasks were given by Teacher B than lower-level demands tasks, it can be assumed that Teacher B considers some higher-level demands tasks as easy because previously learned lessons are not involved. For instance, the previous task on permutation lock is categorized as a higher-level demands tasks, specifically a doing mathematics task, but Teacher B may consider it easy because it only focuses on the current lesson, and it does not require the students to use previously learned lessons.Teacher C
Based on the results of the tasks analysis, Teacher C used higher-level demands tasks (88.3%) more frequently than lower-level demands tasks (11.8%) throughout her classroom instruction. In particular, she used procedures with connections to concepts the most often (70.6%), followed by doing mathematics tasks (17.7%) and procedures without connections to concepts (11.8%). It is interesting to note that similar to the first two teachers, Teacher C did not use any memorisation task in her classroom instruction. Another interesting observation is that both Teacher B and Teacher C emphasized the use of procedures with connections to concepts in their instruction.
Similar to the first two teachers, Teacher C also minimally used doing mathematics tasks in her classroom instruction. When asked about how she selects the tasks that she uses in the classroom, Teacher C mentioned that she normally uses different reference materials such as textbooks and internet sources. This was confirmed by the researcher when Teacher C presented all the reference materials that she used in preparing the current lesson. She also mentioned that she usually allows her students to solve easy tasks first before proceeding to difficult tasks to help them manage their time in answering. Otherwise, the students may get stuck in one task and may not be able to proceed to the remaining tasks. When she presented the tasks on tangent lines and rates of change in class, it was observed that she started the discussion with tasks classified as procedures without connections to concepts, followed by tasks classified as procedures with connections to concepts. She presented the doing mathematics tasks to her students in the latter part of her discussion on rates of change. She gradually shifted from lower-level demands tasks to higher-level demands tasks. Based on these observations, it can be said that Teacher C’s answer in the interview with regard to how she presents the tasks to her students is consistent with her actual practice.
Flexibility in Problem Solving
The following section discusses evidence of flexibility in problem solving in the classroom instruction of the teachers in order to answer the third research question. To determine whether flexibility in problem solving is evident in the instructional practices of the mathematics teachers, two criteria identified by Star and Seifert (2005) were used in the current study, namely: (1) knowledge of multiple solutions; and (2) ability to create new solutions or re-invent existing solution.
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Teacher ATo determine whether Teacher A practices flexibility in problem solving in her classroom instruction, her discussion on operations on polynomials was observed. Throughout the five-day classroom observation, there were two instances in which evidence of flexibility was seen in her instruction. These instances occurred during an individual board work wherein students were asked to solve tasks that involve division of a polynomial by a monomial. Some of the students were randomly selected by the teacher to show their solutions on the board. In the first instance, the given expression was the following:
2(12x − 10)
4 +10(3x + 2) 5
The strategy of one student was to distribute 2 and 10 to the terms inside the parentheses before dividing the resulting terms by the denominators 4 and 5. The solution of the student is shown below.
2(12x − 10)
4 +10(3x + 2) 5
= 24x − 20
4 +30x + 20 5
= 6x − 5 + 6x + 4
= 12x − 1
The final answer of the student was 12𝑥 − 1, which Teacher A said it was correct. She then asked the rest of the students if they could think of another way of solving the given. Excerpt of her discussion is shown below:
Teacher A: Is there another way of solving this task instead of distributing? There is a simpler way of solving this. If you divide 2 by 4 and 10 by 5, do you think you will have the same answer? Please try if you will get the same answer. Instead of distributing, you can simply divide 2 by 4 and 10 by 5.
Teacher A called another student to solve the same task on the board using the said steps in order to check whether the final answer would be the same. She wanted to show the class that although a different solution was used, the second student would still get the same final answer as the first student. Teacher A then clarified that the students can use either of the two solutions, but she pointed out that the second solution would require less time and less effort.
The solution of the second student is shown in the next page.
2(12x − 10)
4 +10(3x + 2) 5
= 12x − 10
2 + 2(3x + 2)
= 6x − 5 + 6x + 4
= 12x − 1
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In the next task, one of the students asked if her solution could be accepted although it was different from what was shown on the board. The given expression was:2a(a + 4)
a +2a(3a − 4) 2a
Similar to the previous task, the solution that was shown on the board used distribution first. That is, the term 2a was multiplied to each of the terms inside the parentheses before proceeding to division.
2a2+ 8a
a +6a2− 8𝑎 2a
What the student did, on the other hand, was to divide 2a by a in the first expression and 2a by 2a in the second expression, resulting in a simpler expression:
2(a + 4) + 3a − 4
Teacher A noted that the solution of the student was also correct and was, in fact, easier and faster to use. In the third task, special products of binomials were included in the given expression. This time, however, Teacher A did not allow the students to use the long method in getting the square of binomials. The given expression is shown below:
(x − 4)2+ (x + 4)2 2x
Teacher A randomly selected a student to show the solution to class. The student initially used the FOIL method (First-Outer-Inner-Last) to get the square of the expressions (𝑥 − 4) and (𝑥 + 4). But halfway through the student’s solution, Teacher A asked the student to use special products of binomials instead. In response to the student's solution, Teacher A remarked, “I don’t want you to use the long method in special products.” In another part of the discussion, Teacher A pointed out that although the long method is acceptable, she prefers using shortcuts especially if special products are involved. Throughout the five-day classroom observation, Teacher A showed evidence of flexibility in problem solving in only two instances. In particular, she presented alternative solutions to only two tasks out of 45 tasks discussed in class. However, it is important to note that while in some instances, Teacher A encouraged her students to think of simpler ways to solve tasks, there were also instances that she restricted the students from using longer methods of solving.
To probe deeper into Teacher A’s practice of flexibility in her classroom instruction, further questions were asked in the interview. Based on her answers, Teacher A agreed that there can be other solutions to a problem, and that students should not be restricted to a single solution when solving. According to her, as long as the answer is correct and the students are able to justify the solution, she allows the use of multiple solutions. Moreover, it is difficult for her to restrict her students from using other solutions because based on her observations, some students will always come up with shorter ways to solve a problem. Excerpt from the transcribed interview is shown below.
Teacher A: I do not restrict myself to one solution. If my students have other solutions, that is okay as long as the answer is correct. It is difficult to restrict because some students have shortcuts in solving.
Interviewer: If a student presents a different solution other than what you discussed in class, do you entertain?