Supporting Elementary School Teachers in Understanding, Assessing and Developing Children’s Mathematical Abilities in Taiwan

 

 

 

 

Hsin-Mei Edith Huang Department of Elementary Education, Taipei Municipal Teachers College, Taipei, Taiwan, R.O.C.

edith131@ms47.hinet.net

 

 

Abstract

Teachers have to understand the subject matter knowledge and are able to get it across to students seems to be the consensus that emerged by mathematics educators. Therefore, a teacher has to developed activities by paying more attention to children’s mathematics talk, problem-solving process, daily experience in and after school. Meanwhile, ways of supporting teachers, and encouraging them more involvement in research-practice connections programs must be paid attention. This paper focuses on the following two issues: first, providing an overview of current projects for supporting elementary school teachers in understanding , assessing and enhancing children’s mathematical ability in Taiwan; second, to recognize some of the tensions and teaching challenge potentially exiting in current cognitively based approach in mathematics instruction from classroom observations and interviews. Implications of research-practice connection for teachers will be discussed.

Introduction and background

Educational reform has been promoted in full flourish in Taiwan for the past few years. Professional education groups have been seriously engaging in the establishment of standards for the various academic disciplines. The Curriculum Guidelines for the Compulsory Education announced in 1999 are considering the desirability of a more integrated curriculum and the relationship between the curriculum of the school and the world outside（Ministry of Education, 2000）. The new curriculum, including mathematics, will be started for grade one students in 2001. The school mathematics curriculum that we are using now is 1993-curriculum, which was strongly influenced by the curriculum and evaluation standards of school mathematics from American (NCTM, 1989; 1991). Though there are differences between the new curriculum and the 1993-curriculum standards, both mathematics curricula take modern constructivist’s point of view that meaningful learning occurs only when children actively construct the information from new experience and connect to their own knowledge. In addition, children’s abilities to discuss and try out their ideas and challenge the ideas of others when they solve mathematical problems are emphasized in both curricula (Ministry of Education, 1993; 2000). To promote children’s development of powerful mathematical thinking and problem solving ability seems to be the consensus that emerged by mathematics educators.

The views that interpret how mathematics knowledge is learned by children is fully exploited in the 1993-curriculum of elementary school mathematics (Huang, 1996; Huang, 1999a). Constructivist conceptions of teaching and learning assign the essential to the way in which learners attempt to make sense of what they are learning rather than receive from teachers (Schoenfeld, 1994; Resnick, 1987; 1989). Researchers point to the fact that integrated and usable knowledge is possible when children develop multiple representations of ideas and, through their work in school and beyond, are engaged in activities that require them to apply this knowledge（Gardner, 1991; Kamii & Ewing, 1996） Hence, the term, "teacher" is taking a new meaning today. Teachers need to understand the subject matter knowledge they teach and are able to get it across to students（Carpenter, Fennema, Peterson, Chiang & Loef, 1989; Carpenter, Fennema & Franke, 1996; Haung & Lo, 1998; Huang, 1999a; 1999b; 1999c; 2000a）. Yet, they need to put that information and knowledge into a broader context, and nudge their students to construct knowledge (Marx, 1998; Campbell, Campbell & Dickinson, 1996). For this, when a teacher employs an activity within a classroom, he or she involves in listening to students and teaching them to listen to each other as they engage seriously with teacher, other students and new texts. A teacher has to develop activities by paying more attention to children’s mathematics talk, problem-solving process, daily experience in and after school as well as learning outcomes. In the past, whole class instruction, talk-and-chalk and rote-memorization were frequently used in mathematics teaching. Teachers using the direct teaching mode hold meaning in their heads, and their work was to transmit it to the heads of the students, and expected children to get good performance records in paper-and-pencil examinations. Under such situation as forementioned, discussion between teachers and children, and among peers did happen, but very rarely (Huang, 1996). However, mathematics teaching that is predominantly teacher showing and telling has been challenged because of the perceived ineffectiveness of its results over the past ten years ( Carpenter, Fennema & Franke, 1996; Simon, 1995; Simon, Tzur, Heinz, Schwan, & Kinzel, 1999).

An overview of current projects for supporting elementary school teachers in understanding , assessing and enhancing children’s mathematical ability

It is a big challenge for a teacher to change his or her teaching method from direct teaching mode to cognitively based approach, and his or her role shifts from a demonstrator and a problem solver to a problem poser, from a main presenter to a monitor of children’s discussion. How to integrate teaching materials as student-centered learning, assessing and how to develop children’s mathematical ability are important issues in teacher education. At the same time, methods of supporting teachers and encouraging them to have more involvement in research-practice connection programs must be paid attention to. Therefore, combined efforts from researchers and innovators were made to inform elementary school teachers and the public by inservice education programs, workshops, open seminars and so on in these later years (Leung & Wu, 2000). For the purpose of supporting elementary school teachers’ professional development and understanding children’s mathematical thinking, programs promoted in the past few years in Taiwan include the following:

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2. Small cooperative group discovery project and topic-based work. Children need activities which they can express their own meaning. Meaning cannot really be fully told because the process of meaning involves doing and communication with others (Feinberg, 1998). This doing entails the specialized doings that we call talking, listening, reading, manipulating, writing, gathering and analyzing information as well as reasoning（Keys, 1995）. One of the most powerful ways to encourage children to cooperatively accomplish real things is project approach and topic-based work. Such learning capitalizes on the familiar experiences of children, offers multiple ways of active interaction with people, objects and their physical environment and must extend over a period of time (Krajcik, Blumenfeld, Marx & Soloway, 1994; Otto, 1996). Meanwhile parents can also become involved in the project based learning by helping to provide materials, as well as serving as judges for special events. Here is one mathematical project-" Travel around Taiwan Island One Week By Train" that was designed by Teacher Tsai Shu-ying for the fifth grade students. Miss Tsai is an experienced teacher from Experimental Elementary School of Taipei Municipal Teachers College, her teaching experience is over 20 years. Two tasks of this project were showed as Figure 1 and Figure 2. It took Teacher Tsai four years (from 1996 to 1999) to complete this project and developed it into multiple medias CAI. Four mathematical topics integrated in this project are as follows: time addition and subtraction, conversions within time system, rate and proportionality（Tsai & Lee, 2000）. Assisting by the CAI, children have learned to recognize various tracks around the island, to check train time table, to make a travel plan, as well as to estimate expense on tickets. Then children had to complete a written report and an oral presentation. Children got scores from self-evaluation, peer-evaluation and teacher evaluation. In addition to solving the problems in this project, children learned to integrate language, social study, art for introducing scenery, and local cultural features. Students appeared genuinely interested in using the project approach. Even though no attempt was made to quantify this interest, but it was easy to observe during the presentation and from children’s final thoughts in written repots（Tsai & Lee, 2000）. Children’s writing gave the teacher a better understanding of the quality of children’s mathematical ability as reflected in their works and assignments.

Figure 1: Various tracks around Taiwan island.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2. The number of kilometer between every station of various tracks around Taiwan.

 

 

2. On integration as a relationship between the mathematics curriculum and the classroom outside. It is common to find the dilemma in old mathematics curriculum and direct instruction mode in Taiwan that students is the irrelevance of their mathematics course work in their lives out of school. Furthermore, most teachers’ mathematics instruction is based on a textbook used in isolation from its application. Consequently students’ perceptions of the applications of mathematics in daily life remain within the limited boundaries of balancing checkbooks or simple counting and measurement context（Manouchehri, 1997）. For the purpose of increasing children’s awareness of real life applications of mathematics, some explorations of school play ground, local park, supermarket and so on are useful in extending students’ notion of mathematical applicability（Huang, 1999b）. Such explorations include questions that potentially make the investigation of many topics from arithmetic, money, measurement, geometry, and therefore are more accessible and meaningful to students. Here I will share two examples of school playground mathematics developed by Wan-Fu public elementary school in Taipei and Teacher Wang Hung-ying respectively. The first one shows as Figure 3. It was a task- " Where are the mysterious shapes?" for lower grades students which was derived from " Booklet of Mathematics Walkway Around Wan-Fu Public Elementary School" (translation from Chinese)（The 7^th Parent Association of Wan-Fu public elementary school, 1997, p. 26）. In this task, children have to observe, recognize and to count the number of various shapes that are embedded in the multiple composition play equipment. The lower grades children apply their knowledge about shape and counting skills to complete this task from observing the equipment that they are familiar.

 

 

Figure 3. A task of school playground mathematics-"Where are the mysterious shapes?" .

Figure 4. Objects found within school playground.

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4. Mathematical task-based assessment. For the purpose of getting more detailed information of individual child’s mathematical thinking and understanding, teachers design several task-based assessments concerning what children have learned from mathematics lessons during a period of time, and then teachers interview with every child one-to-one in the middle or at the end of each semester. Some parents who are interested in are invited to engage in this assessment at times. Students’ performances are recorded by teachers and parents. Here are brief descriptions about two mathematical tasked-based assessments that were held in two schools. First, "A Super buyer" assessment for the first grade students in Wu&emdash;Kung public elementary school in Taipei city (recorded by video tape and field noted by the author in March, 1997). Teachers and several parents decorated and set up the shopping stores before assessment. Children used their pocket money for shopping, and then wrote down what they bought (in Chinese characters) and calculated how much money they spent as well as how much money they remained on shopping list. After finishing the shopping activity, children’s shopping lists were collected and reviewed by teachers. Second, "Shapes assemble assessment’ for the second grade students that was held in Nuan Nuan public elementary school in Keelung city. Figure 5 shows one of the task-based assessment of shapes combination. Teachers provided various shapes for each child to assemble and to create various shapes and then pasted them on a piece of work sheet . After finishing tasks, each child counted the shapes and told the teacher (or the parent who hold this assessment) the numbers of triangle, square, and rectangles respectively on their work sheet（Field notes and Data collected by the author in June, 2000）.

    

    

   Figure 5. Shapes assemble assessment for the second grade students.

    

   While Integrating the results of interviewing teachers, parents, and children from both forementioned schools , the teachers expressed that they gained more information about children’s thinking, and the ability to use other nonthreatening questions to understand what children learned, the parents who took part in the activity expressed that the assessments gave them more access to children’s thinking and views in problem solving from interviews. Besides, children expressed that the assessment was like a game rather than an examination. As a result, children didn’t feel nervous about mathematical assessment. Parents and students alike derived a great deal of interest and enjoyment from this assessment.

   Authentic assessment tasks bridge the gap between school and real mathematics, and teachers gain much more information about children’s thinking, and application of mathematics to real world problem solving from such assessments（Stenmark, 1991）.

    

    

    

    

5. Problem posing and solving activities. Problem solving and problem posing are natural curriculum partners, and problem posing enrich children’s understanding and abilities in problem solving（English, 1998）. In direct teaching mode, a teacher poses a problem and solves it, and then explains the way of solving the problem in front of children. Children learn by imitating the teacher’s solution. Under current mathematics curriculum reform, a teacher pose a problem and then asks children to come up with various ways of solving the problem instead of solving it by himself or herself (Leung & Wu, 2000). Meanwhile, students also pose mathematical problems that they are going to solve, for instance, children were invited to write down their mathematics ideas in diaries. Their diaries may consist of various issues relating to mathematics learning, such as incomplete work during the day, new ideas they try after school or thoughts about the activities they experienced in mathematics classes（Leung & Wu, 2000; Sternmark, 1991）. In addition, mathematical problem posing and story design from children story books are also in children’s favor. Here are examples of mathematical diary and a mathematical story book respectively.

Figure 6. An example of mathematical diary.

Figure 7. A symmetric figure problem posing of mathematical story adapted from Toy Story.

 

 

 

 

 

Figure 8. Solutions for a symmetric figure problem of mathematical story adapted from Toy Story.

 

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6. Teacher as an educational action researcher. In Dewey’s perspectives, teachers would engage in reflecting about and testing possible solutions to practical problems (Dewey, 1990 ). The teaching reflection is also the primary source of professional development in mathematical teaching for teachers. Teachers plan and enact projects, reflects, reflect on their classroom teaching experiences, and then return to the group to share experiences and problem solving strategies with other teachers. Reflecting on action included using personal journals as instruments to describe the daily or weekly experience of change; besides, teachers may write case reports of their teaching for others to read (Krajcik, et al., 1994; Campbell, Campbell & Dickinson, 1996). For enhancing teachers’ professional development, university personnel and experts in fields help teachers through cycles of collaboration, enactment, and reflection. Thus, teachers and university personnel get more connection between theory and practices and new visions of instruction for each other.

The tension and obstacles which arose for the teachers

Central in the cognitively based approach eliciting various intellectual operations or the confrontation of the students with problem situations in from of problem solving techniques, questioning, discussions, which create a feeling of bafflement resulting in the use of intellectual operation(s) with an organized body of knowledge in order to solve the problem. The cognitively based approach that contemporary projects represent is based on considerably more sophisticated knowledge of instruction and children’s learning（Keys, 1995; Krajcik, et al., 1994; Otto, 1996）. From the results of interviews, some teachers who tried and implemented the cognitively based approach into teaching practice, were so impressed with the results that they were convinced that it was superior to the direct teaching. However, teachers face challenges in orchestrating the features of collaboration, investigation, and addressing a driving question as pointed by previous research (e.g., Jaworski, 1997; Krajcik, et al., 1994; Richardson, 1990). Teachers have to make efforts for sustaining students’ cognitive engagements and motivating them to work productively together. The tensions and obstacles emerging from practical teaching situations to impede teachers to put the research-practice suggestions into practice in Taiwan are as follows:

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2. With most population heavily concentrated in the urban areas around, it’s common to find classes with more than thirty children in elementary schools, while some schools in remote area have only a few children each class. Too many teachers are fully occupied in the classroom setting under the big class situation. To increase the number of teachers and decrease the number of students for every class in elementary schools is one of the most important issue for education bureaus and government.

    

3. It takes teachers’ much vigor and time to make efforts for transforming classroom into active learning environment. Other than twenty-five sessions per week for teaching loads, teachers need more time for peer discussion to design activities and projects（Huang, 2000 c）. Meanwhile, it is also a challenge for schools to change the current situation - arbitrary division for the subject matter and time blocks taught by teachers, thus creating learning experiences that periodically demonstrate the relationship of the disciplines.

    

4. Teachers’ pedagogical content beliefs, knowledge and experience influence what teachers understand, what they adopt and how they implement changes（Carpenter, et al., 1989; Carpenter, Fennema & Franke, 1996; Krajcik, et al., 1994; Huang, 1999b）. Quite a few teachers see the curriculum as a fixed set of ideas to be transmitted, as a result they are less likely to allow children to explore their own thinking in searching for solutions to self-generated questions. Furthermore, there is a general agreement among teachers’ views that mathematical knowledge is too abstract for children to understand especially when they involved too many abstract ideas or operational procedures. Even though teachers have awareness of children’s learning difficulty, many teachers have a preference for direct instruction mode than cognitively based instruction（Huang & Lo, 1998; Huang, 1999b; 2000a）. It is essential to provide more workshops and opportunities that allow teachers to interact and have conversations about curriculum standards, research-practice connection projects and classroom activities.

    

5. Quite a few teachers do not have adequate knowledge about multiple assessments the evaluation of children’s progress in learning. Basically teachers are aware of the advantage of authentic assessments, while they are unable to implement portfolio assessment and provide more description about children’s cognition development in learning（Huang, 2000b）. Many teachers just collected students’ works without looking them through in depth. Teachers need more learning about authentic assessment. Furthermore, teachers and parents are quite supportive for traditional examination in Taiwan. In addition to the current reform of multiple assessments for entrance high school and university, providing more workshops about multiple assessments for teachers and parents is also a vital investment.

Conclusion

Getting teachers to change is difficult (Duffy & Roehler, 1986; Richardson, 1990). The sentiment concerning the resistance of teachers and their unwillingness to apply research is shared by many researchers. As a researcher, being able to participate in the teaching practice of the various classrooms, I am in a privileged position to stand back and reflect on the complexities of the theory-practice interface. In order to help children to achieve more meaningful learning, teacher-effectiveness scholars have tried various approach in supporting and changing teachers’ teaching practice as well as understanding children’s mathematical thinking. Working with teachers is quite a meaningful approach for both researchers’ and teachers’ professional development. As far as we can see currently, the cognitively approach should be considered a success. The reason for being optimistic is that most inservice teachers and preservice teachers included have learned the constructivist conceptions of teaching and learning. Furthermore both the children’s discussion ability and problem solving performances of mathematics experiments （Huang, 1996; Leung & Wu, 2000）and curriculum reform are strong enough to persuade teachers to change. There is still a long way to go for this mathematical curriculum reform in Taiwan. However, in addition to providing supports for teachers, waiting time for teachers’ professional development is one way of implementing research-practice theory. Waiting time may then affect teachers’ practice as the concept is filtered through their beliefs and understandings of context（Richardson, 1990）.

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