© 2010 Khon Kaen University, Thailand

International Journal of EducationVol.33, No.3, July.-September., 2010

pp. 175-184

Exploring Thai Students’ Kinds of Mathematical Connections in Open Approach

Wasukree Jaijan1 and Suladda Loipha2

1Doctoral Program in Mathematics Education,Faculty of Education,Khon Kaen University, Thailand

2Center for Research in Mathematics Education, Faculty of Education,Khon Kaen University, ThailandEmail: jaijan_nam@hotmail.com

Abstract This research aimed to explore Thai students’ kinds of mathematical connections in open approach. The research based on classroom that emphasis on problem solving as teaching approach. It is conducted by a qualitative research. Data were collected through teaching experiment. The target group was four five-grade students at Chumchon-banchonnabot School, under project of implementing Lesson Study and Open approach. The research instruments included 1) lesson plans, 2) field notes, 3) in-depth interview, 4) observing forms. The data were analyzed the conclusions were drawn which were presented in a descriptive form. The research findings found that Thai student’ five kinds of mathematical connections (Modeling, Structural, Representational, Procedure-Concept, and Between Strands of Mathematics) as they occurred in the classroom that emphases on problem-solving as teaching approach consist of: posing open-ended problem, students’ self-learning through problem solving, whole class discussion and comparison, and summarization through connecting students’ mathematical ideas emerged in the classroom. In addition, the structure of Japanese mathematics textbook could support in teaching approach as well as material designing which would affect the students in connections between strands of mathematics. The association between instructional material and problem situation could represent the students to access retrieve their own experience in interpreting the problem situation.

Introduction Nowadays, Thailand use mathematical curriculum in the educational core curriculum B.E. 2544, which is emphasizes on teaching approach in order to encourage learners to fully improve their capability by using child-centered approach. The main idea of the educational

Key words: Mathematical Connections, Open Approach, Problem Solving

176

reform is based on the National Educational Act 1999 focusing on educational reform in order to adapt the traditional learning culture of all tail people from transferring knowledge rather than learning through real life situations. This new core curriculum comprises six mathematical content areas: number and operations, algebra, geometry, measurement, data analysis and probability, and mathematical skills and processes. Mathematical learning process, which is necessary to be improved for learners are emphasized. To elaborate, these six mathematical learning processes are problem solving, reasoning, communication, representation connections, and creativity (The Ministry of Education, 2008). The difficulty of the teachers who teach mathematics in all over the country is responding to the core curriculum demand. They could not provide teaching approaches in order to emphasis on content areas and mathematical learning process even they have tried to use them when they teach mathematics (Secretary of Higher Educational Committee, 2009). Most of teaching approaches in school of Thailand are lecture. They use exercises in textbook based on only the right answers, which limit the students who have different ability to participate. The results from exploring Thai mathematics textbooks revealed that most textbooks consist of exercises for mathematical skills practice especially for calculating skill, which is used for revising mathematic principles that students have already studied (Inprasitha, 1997). One of the mathematical learning processes is mathematical connections since it is an important factor which is an important goal for schools (NCTM,1989; NCTM, 2000; The Ministry of Education, 2008). Making mathematical connections is a frequently stated goal for mathematics education (Evitts, 2004). Ideally, the use of real-world problems activity involves all students and promotes the establishment and use of connections (Hodgson, 1995). The desire to emphasize the connections in mathematics, to foster mathematical thinking in other disciplines, and to contextualize mathematics so that learners will see mathematics as a means to help make sense of their world is not a phenomenon unique to the late twentieth century (Coxford, 1995). The goal of emphasizing mathematical connections in the classroom is based on the premise that it is the students who will play the major role in making the connections (Ito-Hino, 1995). Thus, this research tried to explore Thai students’ kinds of mathematical connections in open approach. Romberg (1994) proposed that students’ mathematical learning in classroom was developed from problem situation (NCTM, 1989, p.11). For instructional organize in the classroom that emphases on problem-solving as teaching approach consist of: posing open-ended problem, students’ self-learning through problem solving, whole class discussion and comparison, and summarization through connecting students’ mathematical ideas emerged in the classroom (Inprasitha, 2010). The researcher used open approach as a teaching approach which the objective of teaching is supporting creative activity by students and mathematical thinking while they are solving the problems (Nohda, 2000).

Kinds of mathematical connections The National Council of Teachers of Mathematics described the process standard for “connections” in terms of its goals for students: Instructional programs from prekindergarten trough grade 12 should enable all students to recognize and use connections among mathematical ideas, [to] understand how mathematical ideas interconnect and build on one

177

another to produce a coherent whole, [and to] recognize and apply mathematics in contexts outside of mathematics. (NCTM, 2000: p. 64)Coxford (1995) elaborated upon the possible ways that students can experience mathematical connections. These experiences (Coxford, 1995: pp.3-4; NCTM, 1989): linking procedural and conceptual knowledge, using mathematics in other curriculum areas, using mathematics in daily life, seeing mathematics as an integrated whole, applying mathematical thinking and modeling to solve real-life problem, using the connections among mathematical topics, and recognizing equivalent representations of the same concept. In particular, the NCTM perspective on connections for grades 9-12 called attention to the importance of two kinds of connections – modeling connection, which link problem situations and their mathematical representation, and mathematical connections among equivalent representations (NCTM, 1989: p.146). This research of what constitute important mathematical connections leads to my identification of five fundamental kinds of connections. These are modeling connections, structural connections, representational connections, procedure-concept connections, and connections between strands of mathematics (Evitts, 2004).

Conceptual framework Five kinds of mathematical connections (Evitts, 2004) as follows: 1. Modeling Connections constitute links between the world of mathematics and the real world (or the daily lives) of students. 2. Structural Connections recognizing the sameness of two mathematical ideas or constructs is an important goal for school mathematics and emphasizes mathematical structure. 3. Representational Connections mathematical relationships can be represented in graphical, numerical, symbolic, pictorial, and verbal forms. 4. Connections between conceptual and procedural knowledge constitute another dimension of mathematical connections. When concepts are linked to procedures, the prevailing, rule-oriented perception of mathematics could be diminished. 5. The connections among various stands of mathematics contribute to the conception of mathematics as an integrated whole.

Research objective To explore Thai students’ kinds of mathematical connections in open approach.

Methodology Research design In this research, qualitative research was conducted by focusing on the usage of natural setting. The researcher as an instruments paid attention in the process and description of meaning by using ethnographic study. the researcher’s role determination by learning as well as participation in school using innovation of Lesson Study and Open Approach during 2006-2009. The researcher to study context and participation in construct the lesson plan with teachers, school coordinator, research team, and teacher observer. Every Monday of the week, by using Japanese mathematics textbook, observing instruction on every Monday and Wednesday, reflecting by the whole school every Wednesday, teaching experiment since

178

the researcher obtained direct experience in students’ mathematical learning and reasoning, case study in school under project of mathematics teaching professional development by innovation of Lesson Study and Open Approach. Participants The target group included 27 five-grade students with the age 10-11 years old, including 12 male and 15 female students. In the classroom, group activity was performed. There were 4-5 students each group. The learning achievement was not considered. Therefore, The target group was four five-grade students with prominent characteristic as; when he obtained the open-ended problem situation, he started thinking immediately with geometric style of thought that could connect to other content areas. He dared to think, express, was not afraid of answering wrong. According to Piaget’s Theory of Intellectual Development, children in this age could use their brain in reasonable thinking. However, most of thinking and reasoning processes were based on concrete things. The strong point of children in this age, was the starting in reason, reversible thinking, viewing incidences with various aspects, were able to set rules and apply in classifying things in groups, at Chumchon-banchonnabot School, under project of implementing Lesson Study and Open approach. The teacher teaching this classroom, has participated in and used the innovation of Lesson Study and Open Approach for 4 years (2006-2009) in five-grade, mathematical content areas , regular participating in the phase of collaboratively plan, collaboratively do, collaboratively reflection. Being in classroom, the students’ thinking process could be studied with details in thinking approach and reasoning.

Procedure In school context, Chumchon-banchonnabot School is a primary school in Northeastern Region, under project of mathematics teaching professional development by innovation of Lesson Study and Open Approach, for 3 years (2006-2009). For the process of Lesson Study, consisted of 3 major phases including the collaboratively plan, collaboratively do, collaboratively reflection on the lesson. In 2007, the collaboratively plan was constructed during the first session in writing lesson plan, it was not clearly determined. In 2008, it was revised into writing the plan together as the class level. The participants were the teacher, teacher observers, school coordinator, research team, internship students in school by scheduling as after class every Monday (3:30 p.m.- 6:30 p.m.). In classroom observation in every Wednesday, the whole school would reflect together. The teachers in each level, teacher observer, internship students in school, school coordinator, and research team reflected the students’ thinking process as well as the found problem for modifying the collaboratively plan further. The researcher started from classroom observation in the second semester of 2006-2009 school year for studying the classroom nature, thinking approach, and characteristic of each student for selecting the target group. In developing the collaboratively plan with the teacher, teacher observers, internship students in school, school coordinator collaborated in observing the classroom and reflection.

179

Instrument 1. Field note for recording behaviors, ways of thinking find answers from students while they were solving mathematic problems and explaining their solution. 2. Lesson plans of the area of a parallelogram 3. In-depth Interview 4. Observing forms from Mathematic problem activity which used open-ended problems from the Japanese primary school mathematic textbook for creating lesson plans, observing teaching and reporting the result.

Data collection and analysis Data collected for analysis in this study the researcher implemented it during the first semester of 2008 school year, with the target group. The researcher recorded behavior participating in the students’ thinking approach, implemented by the teacher in classroom using participant observation from looking at and describing the existed natural phenom-enon without interrupting while the target groups were thinking for solving the problem. After finishing activity, the researcher immediately interviewed target group, the interview was implemented as a conversation for assuring the students’ thinking approach. Later on, the teacher was interviewed. The issues for interviewing included the teacher’s role in presenting the open-ended problem, collection of students’ problem solving approach, discussion in problem solving, organization and connection of students’ problem solving approach, duration for collecting data for 2 hours, collecting work piece from participating in activities by taking a picture of work piece. After finishing instructional approach, the researcher immediately interviewed target group for understanding the thinking approach, and investigating the validity of data.

Findings and discussionProblem situation “I just now big”:

Measure length in each side of square a, b, c2. Compare area of square a, b, c3. Explain how to calculate area of square a, b, c as much as possible.

Each section contains descriptions and supporting examples from the data. To avoid adding unnecessary length and, in many case, repetitive versions of other students’ work, target group is represented for every type of connection. 1. Modeling connections A modeling connection occurred when student identified some aspect of her or his mathematical knowledge that could be used to portray some real-world component from the problem situation.

180

Figure 1: students’ idea of picture a

Figure 2: students’ idea of picture b

Figure 3: students’ idea of picture c Modeling connections were activated by questions students asked aloud as they considered the real-world characteristics they identified as necessary to express mathematically.

2. Structural connections A structural connection occurred when student used and analyzed similarities between a real or mathematical component and another real or mathematical compo-nent.

Figure 4: students’ idea of picture a

Figure 5: students’ idea of picture b

Figure 6: students’ idea of picture c

181

Structural connections arose because of finding similarity between an aspect of the problem situation and, in most cases, a recalled problem situation that seemed more familiar.

3. Representational connections Representational connections were evident when a student used two (or more) representational forms to talk about a mathematical idea.

Figure 7: students’ idea of picture a

Figure 8: students’ idea of picture b

Figure 9: students’ idea of picture c All of student used representations to convey the overall features of the problem situation and to denote particular features of a problem situation.

4. Procedure-concept connections A procedure-concept connection is a cognitive link through which a person employs procedural knowledge that he or she identified with a particular mathemati-cal concept.

Figure 10: students’ idea of picture a

Figure 11: students’ idea of picture b

182

Figure 12: students’ idea of picture c There were multiple instances of students trying to recall a procedure for a particular aspect of their work

5. Connections between Strands of mathematics A connection between strands, or branches, of mathematics exists when skills and concepts from one strand of mathematics are used to analyze a problem that seems to reside in another strand.

Figure 13: students’ idea of picture a

Figure 14: students’ idea of picture b

Figure 15: students’ idea of picture c

Strand associations were often stated when particular features of the problem situation were identified as characteristic of a certain strand. The significance of connections among strands of mathematics in an era of integrated curricula may be debatable.

Conclusion The students made and utilized different mathematical connections as she or he worked in open approach. Every student demonstrated the capacity to make and use connection. Modeling connections were generated by questions that students asked about the real-world situation. Students sought to mathematics their questions with some aspect

183

of mathematics that they knew. Structural connections were of several kinds. Students used whole class discussion and comparison to seek similarity between the posing problem and another situation. Representational Connections extended beyond figure, students used a strategic representation to draw conclusions about a formula or to create a framework to reason about the problem situation. Procedure-Concept Connections were identified in which the formula or procedure preoccupied the student’s attention. Students who could find or sought a conceptual basis for the formulas were more successful in reconstructing them. Connections between strands were as evident in the problem situation, most likely because of the problems that were used. Students did make associations with several strands, referred to number and geometry are closely connected in developing the function concept (Geddes & Fortunato, 1993).

Acknowledgements This research was financially supported by Center for Research in Mathematics Education (CRME) and Centre of Excellence in Mathematics (CEM).

ReferencesCoxford, F.A.(1995). The Case for Connections. In P. A. House (Ed.), Connecting

mathematics across the curriculum, 1995 Yearbook. (pp. 3-21). Reston, VA: National Council of Teachers of Mathematics.

Evitts, T. (2004). Investigating the mathematical connections that pre-service teachers use and develop while solving problems from reform curricula. Doctoral Dissertation, The Pennsylvania State University. (Unpubished).

Geddes, D., & Fortunato, I. (1993). Geometry: Research and classroom activities. In D.T. Owens, Ed., Research ideas for the classroom: Middle grades mathematics (pp. 199-222). New York: Macmillan Publishing Company.

Hodgson, T.R. (1995). Connections as problem-solving tool. In P. A. House & A. F. Coxford (Eds.), Connecting mathematics across the curriculum, 1995 Yearbook. Reston, VA: National Council of Teachers of Mathematics.

Inprasitha, M. (1997). Problem Solving: A basis to reform mathematics instruction, Reprinted from the Journal of the National Research Council of Thailand, 29(2).

Inprasitha, M. (2010). One Feature of Adaptive lesson study in Thailand designing learning unit. (pp.193-206). Proceeding of the 45th Korean national meeting of mathematics education dongkook university, Gyeongju, Korea. October 8-10, 2010.

Ito-Hino, K.(1995). Students’ Reasoning and Mathematical Connections in the Japanese Classroom. In P. A. House (Ed.), Connecting mathematics across the curriculum,1995 Yearbook. (pp. 233-245). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston,VA: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston,VA: NCTM.

Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics

184

classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of. 24th Conf. of the Int. Group for the Psychology of Mathematics Education: (Vol.1, pp.39-54). Hiroshima, Japan: PME.

Romberg, T. A. (1994). Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates.

Secretary of Higher Education Commission. (2009). Proceeding of workshop in mathematics teaching by educational innovation of lesson study and open approach: Center for research in mathematics education, Khon Kaen Thailand. [n.p.]

The Ministry of Education. (2008). The basic education curriculum B.E. 2008 (A.D. 2008). Bangkok: Agricultural Cooperative Association of Thailand.