1 Background

Researchers have emphasized more and more on the issues of enhancing students’ mathematical competency recently in mathematics education (Lesh and Zawojewski, 2007; Niss, 2003). Developing students’ modelling ability is one effective teaching strategy (Lesh and Doerr, 2003; Niss, 2003).

Recently, the issues of the model and modelling perspective gradually receive more and more attention in Taiwan. Some empirical research has focused on the investigations of modelling contests. Some talked about modelling teaching that related to specific mathematical contents such as linear function, parabolic equations. Their results showed students’ positive learning motivations or learning effects increased after modelling teaching. Other empirical research of teachers’ education has focused on the problem-solving strategies of pre-service teachers and the results show the modelling activities and teaching need to correspond to students’ experience. Also, technology is worth considering. Other research has investigated the latent mechanism underlying the case teacher’s reflection in the modelling context whilst collaborating with the researcher and the results showed the effects of teachers’ professional development in practice. These empirical studies reveal the approval of modelling teaching in Taiwan.

On the other hand, most mathematics teachers have taught in lecture style and transference of mathematical knowledge to students in the school context. Students just need to listen to what teachers say and there is a lack of thinking by themselves. Over a long period of time, students have become used to memorising the formula and solving routine mathematical problems. Under these circumstances, students hardly develop multiple mathe­matical competencies in their school mathematics classes.

Recently, we tried to phase in the MEAs to our mathematics classrooms in order to amend the situation and promote students’ thinking, explaining and interpreting opportunities. The crucial reason we used modelling activities was that in such activities, students have to describe, manipulate, predict, and verify (Lesh and Doerr, 2003). We hoped that students could enhance the descriptions and interpretations of what they saw and observed and the ability of problem solving through MEAs. Although empirical studies related to modelling teaching showed positive results of enhancing students’ mathematical learning, this still was an uncommon teaching strategy in the Taiwanese context. So, the first problem we needed to face and overcome was to let these teachers learn how to implement MEAs in their mathematics classes. We designed a course to foster these teachers to become involved in MEAs and modelling teaching. The purpose of the study was to know what the Taiwan mathematics teachers thought of MEAs and modelling teaching.

2 Theoretical Frame

To answer the question, the theoretical approach focused on the discussion of MEA and the six principles of designing MEAs. Lesh and Doerr (2003) refer to “Case Studies for Kids” as many cases of MEAs. Each case consists of four main parts: newspaper articles, readiness questions, problem statements, and the process of sharing solutions. The purpose of the newspaper articles and readiness questions was to introduce the students to the context of the problem. Students can become more familiar with the situations of the case via reading the article and readiness questions just like a warm-up period. The problem statements should be the central part of the teaching and teachers present these to the students according to the grade level and previous experiences they have. Whether the students could identify the client they were working for and the product they should create must be verified. Then comes the process of sharing solutions and it is the stage of presentations of solutions when the teacher tries to encourage students to not only listen to the other groups’ presentations but also to try to understand the other groups’ solutions and consider how well these solutions meet the needs of the client.

On the other hand, Lesh and Doerr (2003) mention six principles to evaluate the quality of a modelling activity and these were also crucial points that we consi­dered. The construction principle ensured that the solutions to the activity required the construction of an explicit description, explanation, procedures, or justified prediction for a given mathematically significant situation. The reality principle, also called the meaningfulness principle, required the activity to be designed so that students can interpret it meaningfully from their different levels of mathematical ability and general knowledge, and also pose a problem that could happen in real life. The self-assessment principle ensures that the activity contains criteria that students can identify and use to test and revise their solutions and also include information that students can assess the usefulness of their alternative solutions. The documentation principle ensures the activity requires students to create some form of documentation that can reveal explicitly how they are thinking about the situation. Share-ability and re-usability principles require students to produce more generalized solutions that others can also use or solutions that can be reused in other similar situations. Effective prototype principle ensures the solution of the activity is as simple as possible yet mathematical and significant and provides useful prototypes for interpreting other similar situations.

3 Methodological Approach

Methodologically, the research is qualitatively oriented and the applied empirical methods concerning choice of samples, data collection, data analysis, and data interpretation are based on the theoretical attempts of Grounded Theory (Strauss and Corbin, 1998).

3.1 Samples

A total of 16 secondary mathematics teachers participated in the study. The information on these teachers’ background is given in Table 16.1.

Table 16.1 Background information of samples
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3.2 Research Process

The process of this study is linked to a master’s degree program in education for in-service teachers with 2 h per week for 9 weeks and includes two stages. First, these teachers were divided into four groups with three to five teachers in a group as the role of students engaged in three MEAs, such as “Who saved the oriental cherry trees?”, “Parking Lot” and “Volleyball problems.” They cooperatively discussed solving one MEA every 2 weeks (see Fig. 16.1). They also wrote reflection journals to compare these MEAs and show their understanding of modelling pedagogy. Secondly, each group was asked to design one MEA and used the Six Principles of designing a MEA (Lesh and Doerr, 2003) to evaluate these MEAs by themselves. The evaluative process also showed their perception of mathematics and understanding of MEAs.

Fig. 16.1
figure 1_16

Teachers engaged in the “Parking Lot” problem

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3.3 Data Collection

The sources of data collections included the learning sheets that showed teachers’ strategies for the three MEAs and the result of the MEA they designed, researchers’ observation journals, reflection journals, open-ended questionnaires, interview reports, and video tapes and audio tapes of the classes.

Learning sheets. These teachers solved three MEAs in groups and wrote down their solving strategies, and letters to the client of these MEAs. These sheets showed teachers’ thinking during modelling activities.

MEA. Every group of teachers designed one MEA, according to their experience of solving MEA and their understanding of MEA. They also introduced their designed MEA to other teachers and evaluated these MEAs with the six principles. These MEAs showed these teachers’ perceptions of MEA.

Observation journals. Researchers kept observation journals every week. We wrote our reflections about every class in our program and kept notes about these teachers’ questions and changes.

Reflection journals. Teachers’ reflection journals were collected every week. The participants were encouraged to reflect on these MEAs and the experiences of solving MEAs. They were also asked to compare the similarities and differences between these three MEAs.

Open-ended questionnaire. The open-ended questionnaire was administrated on the first day and the last day of the program, in order to give an opportunity for teachers to reflect on their beliefs about mathematics, teaching, and learning.

Interview reports. Four teachers, especially those who were interested in modelling teaching and activities, joined the interview and shared their ideas about ­modelling teaching and MEAs with us. Much interesting data were provided for the research through these informal talks.

Video tapes and audio tapes. Every class for 9 weeks was videotaped. When these teachers discussed in groups, we also audiotaped the process of their discussion to keep the details of their group discussion.

3.4 Data Analysis

These qualitative data such as teachers’ reflection journals, interview reports, and video tapes of the classes were read, coded, and categorised repeatedly by the two authors. In doing analysis data, we started to make sense of the data by “making interpretations.” This process continued using “open coding” to discover categories. In this process, we used the “make comparisons” procedure to conceptualize our data by taking apart each observation, every oral or written comment, and we gave each emerging category a “name”.

The teachers’ thinking about modelling teaching and MEAs were the main focus of the study. The richest part of the data came from the reflection journals of these teachers which were written during the program. First, we quoted the teachers’ writings that mentioned MEAs and modelling teaching and checked the frequencies of different quotations. Then, we categorized these quotations into the same categories. These categories conceptualized these teachers’ thoughts about MEAs and modelling teaching. Second, we also compared the results of the MEA they designed with Six Principles of designing a MEA (Lesh and Doerr, 2003) and attempted to reveal these teachers’ thinking about MEAs and modelling teaching. The following was the result of the analysis of data regarding the focal point. The quotations and paraphrases included in the following paragraphs are representatives of the range of the teachers’ thinking.

In the process of analysis, many themes emerged from the data. The following themes represent the nature of the findings of the research. Finally, we analyzed and interpreted these data into three themes: positive thinking about MEAs and modelling teaching, negative thinking about modelling teaching, and weaknesses of designing MEAs.

3.4.1 Positive Thinking about MEAs and Modelling Teaching

According to these teachers’ reflection journals, interview reports, and video tapes of the classes, we grouped the advantages into four aspects, which are shown as follows:

3.4.1.1 Close-in Real Life Situation

A total of 8 of 14 teachers expressed this point of view. They regarded MEAs and modelling teaching relates to a real life situation intently. For example, T22 said “when she experienced MEA, she felt that mathematics can be constructed from the learning activity of real life experience”. T3 mentioned that “It was full of math in the modelling process and we used mathematical language to deal with the problem which was in connection with real life.” T7 and T24 both voiced the idea that MEAs are closer to real life situations than the textbooks.

3.4.1.2 Enhancement of Mathematical Competencies

As, MEAs are all open-ended problems and are accompanied by a lot of information, so teachers approved for enhancing students’ competencies relating to learning mathematics. T10 said that “developing the modelling ability can promote students’ problem solving ability.” T7 referred to “creative thinking ability, conjectural ability, induction and categorization, built the model.” T21 thought that “we ask students to think an integral problem with the concepts which they learned before, and they needed to use mathematical competencies of logical thinking, data gathering and data analyzing…. MEAs are divergent problems and the abilities that students developed were comprehensive. It made students to learn, search for information and analyzed data actively.” T30 pointed out that “the focus of mathematical ­modelling was different than traditional problem solving and changed into, transformed, and explained the situation, recognized potential problems, built the model, re-interpreted the premise, hypothesis and biases of mathematical solution.”

3.4.1.3 Advantages of Modelling Teaching

On the other hand, they also spoke of the advantages of implementing MEAs in school mathematics classes. T12 noted that “In the process, students needed to talk to each other and utilised peers’ thought to inspire themselves to think the problem.” T3 emphasized that “students can learn how to communicate with others, establish good relationship with peers and understand the importance of respect.” T28 pointed out that “the mathematical content was not too difficult for students and it didn’t make students feel scary.” T30 liked the way of group discussion and he thought it was helpful to students to think through problems and also convey their opinions.

3.4.1.4 MEAs as Supplementary Materials

As for the possibilities for implementing MEAs in school mathematics classes, teachers in the interview thought that MEAs could work well as supplementary ­materials, and modelling activities and teaching could be regarded as the corporation or training for supplementary curriculum. We found that the teachers who taught the mathematical corporation accepted MEAs and they agreed with MEAs as supplementary materials more easily.

3.4.2 Negative Thinking about Modelling Teaching

These teachers mentioned many obstacles to implementing MEAs in school mathematics classes. This included the weak connection to the current school curriculum, the influence of the entrance examinations, and mathematical content being too easy.

3.4.2.1 Out of School Curriculum

In Taiwan, the main curriculum standards come from the government. Although there are different versions of textbooks, most content in textbooks is in traditional mode, such as examples for teachers and exercises for students. Also, teachers in Taiwan are used to teaching with textbooks and relying on the content of textbooks. The most common obstacle these teachers mentioned was the weak connection to the current school curriculum. T5 mentioned that “so far, I doubted that whether MEAs can [be] put into the current mathematics classes and maybe this will be one of my goals in the future.” T21 said that “MEAs [had] almost no direct connections with current mathematical textbooks of junior high school.” T24 pointed out that “It seemed not helpful for students to learn school mathematics.” T3 wondered “how to transform the materials in school math into appropriate MEAs?” T10 thought that “not all units in school math were suitable for transforming into MEAs and it was not necessary to use modelling teaching.” So, these teachers were really concerned with the connection between MEAs and the current school mathematics curriculum, and they would accept modelling teaching into their classes only when they were sure that the connection was close.

3.4.2.2 Out of Entrance Examinations

In Taiwan, students need to pass the entrance examinations to enter senior high schools and universities. These teachers always emphasise students’ grades in these examinations as the purpose of their teaching. So, the entrance examinations of senior high schools and colleges are also the main factor why teachers resist modelling teaching coming into their classes. T22 said “my school is a typical private senior high school that emphasized the rate of entering colleges, so students’ grades were the most important thing.” T23 mentioned that “how to connect [the prevailing] education system (exam system) will be the first barrier in reality!” T30 referred to “the first consideration of students and teachers was to get higher scores in the entrance exams to colleges.”

3.4.2.3 Other Obstacles of Modelling Teaching

Other obstacles such as “I cannot convey the mathematical concepts which students wanted most. (T3)” “We spent too much time letting students solve and discuss the MEA, so that we cannot achieve the rate of progress of school math. (T24)” “Students and I were not familiar with modelling teaching and MEAs, so we may have the attitude of rejecting this teaching mode…. The group discussion makes chaos in the classroom, and students can’t keep their concentration. (T30)” “I thought that was a challenge for me to end the open MEA. I don’t know what to do and it seemed not very interesting. (T23)”. Therefore, it will be a tough challenge to tune these teachers’ minds to accept modelling teaching in this kind of background and trend. After arranging these data and themes, we found that teachers in senior high schools or vocational schools displayed more obstacles to modelling teaching than did teachers in junior high school according to the teachers’ reflection journals.

3.4.3 Weaknesses of Designing MEAs

In the second part of this course, these teachers designed one MEA in every group. Four groups produced four MEAs, and it deserved to mention that we just introduced to these teachers the six principles of designing MEAs. Also, they did not read the literature about the model and modelling perspective. The understanding of MEAs they showed was simply according to the experience which they engaged in during the three MEAs.

Here, we describe four MEAs that they designed; they are shown in Table 16.2.

Table 16.2 MEAs designed by each group
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In general, these teachers feel that MEAs need to be realistic situations and rela­ting to students’ life experiences. Also, they keep the problem as open as possible in order to get a model of the solution. Furthermore, the authors tried to use the six principles (Lesh and Doerr, 2003) of designing MEAs to check the MEAs produced by these teachers. Table 16.3 showed the authors’ interpretation based on the corresponding six principles. They also revealed obstacles to designing MEAs. The principles which were achieved easily are the Reality Principle and the Model. Construction Principle, but the other four were hardly present. It meant that the ways of promoting teachers’ ­ability for designing MEAs will still be an issue to be addressed in the future.

Table 16.3 MEAs and the corresponding six principles
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4 Conclusions and Implications

After the 9-week course, these teachers revealed many thoughts about MEAs and modelling teaching. We summarized the teachers’ thinking into three aspects shown in Table 16.4. We found that these teachers agreed that MEAs were useful to enhance students’ problem-solving ability and they had positive attitudes toward MEAs and modelling teaching. But they thought that there were still many obstacles to implementing MEAs in their mathematics classes. On the basis of the MEAs that they designed in the end, it was shown that they were still lacking the ability to design MEAs to fit with the six principles.

Table 16.4 Teachers’ thinking about MEAs and modelling teaching
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In terms of this study, we noticed that a literature review of MEAs and modelling may be important for these teachers to understand how to implement MEAs in their classrooms and to design MEAs. Because of the lack of theoretical background, they just pay attention to the surface characteristic of MEAs. Besides, we found that strengthening the connections between MEAs and the school mathematics curriculum and improving the modelling teaching so as to relate to teaching practice closely are two important factors which influence these teachers’ ideas regarding MEAs and modelling.