The pilot study was designed to verify if any modifications to the open approach, which was adopted from a Japanese context, would have been required before any attempt was made at implementation in the context of primary schools in Jamaica. The main concern was how teachers would manage the class, facilitate discoveries and organise students’ solutions to increase their understanding of mathematical concepts.
4.2.1 Participating Teachers
Three teachers, two females and one male, volunteered to participate in the study. The female teacher who taught the co-educational class was in her late 40s. She has a Bachelor’s Degree in Primary Education and had been teaching for 25 years. She spent 17 consecutive years teaching grade two students, this was her second year teaching at grade four. She spoke in an even pleasant tone. She had excellent class control skills as students respected and obeyed her whenever she spoke. She volunteered for the study because she thought that in order for students to improve their academic performance, teachers must improve their teaching methods.
The other female teacher taught the all-girls’ class. She was in her late 30’s and had been teaching for 12 years. She had been teacher grade four for eight consecutive years prior to the implementation of the study. She has a Bachelor’s Degree in Primary Education and a Master’s Degree in Education. Her senior teacher duties kept her busy throughout the day.
She had an excellent command of the class as students obeyed her when she spoke. She commented that even though she knew of the open approach, she was afraid of trying it. She
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anticipated that she would benefit from participating in the research, and this motivated her to volunteer for the project.
The male teacher who taught the all-boys’ class was in his early 30’s. He has a Bachelor’s Degree in Primary Education and had been teaching for five years. He had a business-like attitude towards his work. His tone of voice changed depending on what he required from the students. A harsh tone was used if students became disruptive. He had excellent class control as all students responded appropriately whenever he spoke to them. His reason for volunteering was that he thought that the open approach could help students to take more responsibility for their learning.
The teachers claimed that they knew of open-ended problems from MOE workshops, but had not implemented them in their lessons. While they have used other forms of student-centred methods, they thought that open-ended problems posed unique challenges that were more difficult to manage. The common reason given for not teaching with open-ended problems was the fear that students may ask questions or produce solutions which they (the teachers) could not answer or explain. Teachers were also uncertain of how to manage multiple responses to a problem and how to support students learning in the open approach.
This attitude may be described as common among teachers in Jamaica and may be the reason the approach is not widely used.
4.2.2 Teacher Training
The teachers were trained to use the open approach method in the second week of September 2014. The main purpose was to acquaint them with using the method and to develop a common format for lessons. The three teachers participated in a four-day workshop. In days
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one and two, the teachers watched video recordings of lessons conducted in the open approach and discussed the main sections of the lessons, introduction, between desk instructions, organising students’ solutions, summing up and reflection. Day three was spent modifying textbook questions to create open-ended problems (Foong, 2002) and creating assessment rubrics for selected problems (Becker & Shimada, 1997). The focus was on changing the question to highlight a pre-determined concept and to ensure that they allowed for multiple solutions. Table 9 shows examples of questions that were modified from students’ regular textbooks. On day four, the teachers conducted mock lessons and continued discussions on using the open approach and open-ended problems. The teachers then participated in a lesson study practicum exercise, conducted as implementation of the pilot study.
Table 9. Modified Problems from Student’s Textbooks
4.2.3 Implementation
The 97 students from the three classes participated in the study. All parents gave their consent to students’ participation by signing a permission slip, and students were informed of their new roles in the classroom. Students were told not to rely solely on the teacher, but to try and solve the problems themselves, create their own solutions and give their own
Textbook closed-ended
questions Concept (s) Modified to open-ended problem Show that 5 + 3 = 10 - ____ Equivalency What pair of numbers can go in the
spaces to make the statement true?
______ + _____ = 15 Round 53.6 to the nearest
tens.
Rounding off.
Place value.
Decimals
Find eleven numbers that can be rounded off to 50.
What is the place value of 3 in
the number 238? Place value Using the digits 4, 5, 2 and 7, write different numbers with 5 in the tens
place.
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explanations. Social and socio-mathematical norms to govern each class were created.
Among the social rules generated were:
(a) Actively listen to each person’s contribution, (b) Do not talk while someone else is talking
(c) Only talk about things that are related to the topic of discussion (Lampert, 1990).
Some examples of sociomathematical norms included:
(a) Giving explanations for your answer, (b) Disagreeing politely,
(c) Commenting on individual’s ideas; not on the individual (Yackel & Cobb, 1996).
Implementation of the teaching intervention was done using a lesson study format where the researcher worked with each teacher individually, to plan, teach and evaluate each lesson.
In the lesson study practicum exercise, lessons were conducted with the open approach pedagogical method for three days each week. Modifications of lessons were done based on feedback from students. The researcher gradually reduced his participation in the process until the teacher was confident enough to implement the process alone. By the end of the second week, (fourth week in the month), all teachers were able to plan and implement their lessons without assistance from the researcher.
2.4.1 Data Collection and Analysis
The methods employed for data collection included participant observation, student journals, student’ feedback, students’ questionnaire, teacher interviews and student interviews.
Information collected was used to determine whether or not it would be feasible to conduct the main study.
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Analysis of information gathered from observations was carried out simultaneously with data collection and was used to set in motion corrective measures for revealed discrepancies.
Observations focused on teacher’s competence in using the open approach and on students’
aptitude for operating in the open approach. Data from students’ questionnaires, interviews and journals were analysed in terms of students’ academic ability, their willingness to participate in the open approach lesson, and on their opinion of learning with open-ended problems. The selection and placement of students was done in such a way that participants from the different classes were of similar academic ability and socio-economic status.
4.2.5 Pilot Test
The pilot test was conducted in September 2014 to determine how well the created open-ended problems might measure Jamaican fourth grade students’ understanding of mathematics. The pilot test also served to confirm each measure’s dimensionality, explore item parameters, and investigate measure reliability.
Test items were created from topics outlined in the grade four curriculum as stated in chapter two of this study. The twelve open-ended items on the test covered topics in types of numbers, number operations, place value, equivalent fractions, adding fractions and formulating problems. There were two items for each topic on the test. These items were modified from students’ textbook. The test was given to 42 grade four students in a neighbouring school. The researcher selected these students because their results on the grade three test were similar to those of the prospective participants in the study.
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Solutions were assessed using the criteria of fluency, flexibility and originality as previously described. The reliability of each item was judged based on the number of students who were able to produce correct solutions. Two items were removed as more than 80% of the participant’s scored maximum marks on these items. One item was removed as more than 80% of the participants omitted it or attempted it with very little success. It was at times difficult for the researcher to ascertain whether an error on the part of a student was due to misunderstanding or miscalculation. Closed items for testing students’ computational ability were added to clarify this difficulty. For verification, the final test instrument consisted of 17 items; 8 closed and 9 open-ended problems. Four of the open-ended problems were paired with closed questions and the other five remained “stand alone” problems (see Appendix A for test items).
4.2.6 Results of the Pilot Study
Before implementing the pilot study, teachers gave suggestions regarding expected challenges for both teachers and students. The major challenge pertaining to the conceptual framework was on establishing and maintaining a supportive environment. That is, teachers were worried about moving around the room to observe and give feedback to individual students. Previously, teachers used the student-centred method once or twice a month, but the research required them to do this for three days each week. It was thought that this action would be tiring and difficult. It was also difficult to walk within the small spaces between the desks that the layout of the classroom allowed. From the students’ standpoint, it was expected that some students would have difficulty adjusting to their new role in the open approach classroom. Students were accustomed to listening to the teacher and applying given strategies. The research however, required that they contributed more to class
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discourse and created their own strategies for solving a given problem. The teachers thought that students would not readily adjust to such unfamiliar activities. Corrective measures were planned to overcome these and other challenges, however, some challenges faced were different from the expected challenges previously stated. Please see Table 10.
Table 10. Foreseen Challenges and Actual Events
4.2.7 Summary of Pilot Study
Results from the pilot study showed the potential for implementing the open approach in Jamaican classroom. The teachers were capable of facilitating students learning in the classroom. Students were eager to learn with this approach. They explored math at their own pace and showed signs of increased understanding of mathematical concepts. Both boys and
Foreseen Challenges Corrective Measures Actual Events (Results)
Teachers
Co-ordinating students’
responses so that they learn something from the
“chaos”.
Emphasise the underlying concept behind each strategy and the
connection between two strategies. Teach the
“why”.
Corrective measure worked well.
Each lesson focused on one or two main concept (s). Teachers guided students to compare two strategies at a time and to defend their choice.
Being mobile for the duration of the class to support students.
Rest while standing and observing students at work.
Teachers occasionally sat at their desks during the lesson.
Class control
(daydreaming, noise, distractions, lack of pencil or book).
Create stimulating activities that nine-year-olds could relate to. Have extra pencils and books available.
Students did not show many signs of distractions. Actions reflecting a lack of interest gradually reduced during the intervention.
Students
Willingness to share ideas and participate in classroom discourse
Offer incentives for participation.
Students were willing to show their solutions and often created
extensions to the problem.
Working collaboratively. Structure the lesson to encourage group work. Students’ willingness to work in groups increased over time.
Writing in Journal.
Provide reflection time in the lesson. Teachers to ask students to verbalised their thoughts and then to list them on the board.
Slower students copied all the sentences from the board instead of only the ones relevant to what they had learnt.
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girls seemed to welcome the idea of using their own strategies to solve the problem. It was however, necessary to make three changes to the structure of the lessons based on the results of the pilot study.
Change 1. Relate a concept to students’ experience
Teachers created problems relating to daily life to which the students could relate. It was observed that students tended to create more diverse and unique solutions when such problems were stated, as opposed to when problems were from abstract or unfamiliar contexts. Due to this observation, most lessons or problems were given names that reflected daily life activities, for example “Sharing Sweets,” or “Shopping at the Market.”
Change 2: The Open Environment
Exploring extensions of the problem in the same class period was allowed. Nohda (1991) spoke of the possibility of extending a problem in class; however, most lessons observed in Japan consisted of one open-ended problem. During the pilot study, the teachers realised that some students often tried to create their own extension to the problem. For example, in asking students to use five digits to create different numbers, some students also tried to find the largest number and smallest number the four digits could produce. Extension of the problem was assigned as additional activities. One rationale for this was that, apart from further exploration with the problem, students were familiar with having more than one activity in a lesson. The extra or extended activity kept students interested and on task.
Extensions to the problem often allowed students to solve a wide range of related examples and non-examples (Davis, 2006; Hoosain, 2001). As such, extensions to the problem were
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welcomed and encouraged and were included in the lesson. The teachers also began to include expected extensions to the problem in their lesson plans.
Change 3: Brief Small Group Discussion
Brief small group discussion was used when students had difficulty producing a response to a comment or question made by a peer or by the teacher. In these cases, along with allowing students to think about the comment or question before responding, students would be asked to share their opinions with others in the class. Students were at this point, allowed to move about the class to listen to each other or share their opinions as they saw fit. Students were then called back to their seats and responses were solicited from different members of the class. These short group discussions lasted from about 30 seconds to about 2 minutes and were carried out during general class discussions.