Theoretical backgrounds

1. Mathematical inquiry

Recently, both science and mathematics learning standards focus on their attention of inquiry for either promoting students to construct knowledge actively in process of problem solving, reasoning, and communication or encouraging students to explore patterns and relationships in data analysis, formulating conjectures, logic thinking and solving non-routine problems. (American Association for the Advancement of Science [AAAS], 1990, 1993; NCTM, 1989, 2000; National Research Council [NRC], 1989). “Doing mathematics” should be considered as an inquiry process in mathematics learning.

In order to develop mathematical thinking and the autonomy to solve challenging mathematical problems, students need to “do mathematics” (NCTM, 1989, 2000; NRC, 1989). Moreover, doing mathematics entails solving challenging problems, exploring patterns, formulating conjectures and examining them out, drawing conclusions and communication ideas, patterns, conjectures, conclusions and reasons (Baroody & Coslick, 1993). In brief, mathematical inquiry encourages students to construct mathematical knowledge actively and it must be underpinned by stances of constructivism. One of the hypotheses of constructivism is that knowledge is actively constructed by the cognizing subject, not passively received from the environment (Kilpatrick, 1987). Somehow, mathematical knowledge is not always constructed radically. Instead, sociocultural approaches contend that human thinking is inherently social in its origins (Kieran, Forman, & Sfard, 2001). Elbers (2003) replies the ideas with social perspective, he considers that when students are engaged in a community of inquiry, they could freely interact and collaborate with each other, and might have ample opportunities to make their own mathematical constructions and to discuss them in a social process of reflection. Empirical study result stands for these arguments as well. For example, Francisco (2013) holds a study with a group of six high school students working together on a challenging probability task as part of a larger, after-school, longitudinal study on students' development of mathematical ideas in problem-solving settings. The result shows that social settings, especially, collaborative activities can help promote students' mathematical understanding by providing opportunities for students to critically reexamine how they

sophisticated ways of reasoning.

2. Conjecturing and justifying

Conjecturing is an important part of an inquiry based approach (Cañadas, et al., 2007). Especially when individual confronts contexts of problems, he/she will actively propose conjectures, later, testing the conjectures, seeking counter examples for refuting it, and generalizing patterns of problems from systematical specializing strategy (Lakatos, 1976, 1978; Mason, 2002; Mason, Burton, & Stacey, 2010;

Polya, 1954). Lakatos (1976, 1978) advocates that mathematics is quasi-empirical as he thinks that mathematics is a dialogue when people negotiate with it. In addition, mathematics is not flawless, it always needs to be renegotiated or reconstructed when facing possible challenges or much more stringent criteria. Lakatos concludes that theoretical knowledge can be established in the process of conjecturing and refutation. Mason and Johnston-Wilder (2004, p.141) argue that “mathematicians rarely solve the initial problems they set themselves. Most often they specialize, they conjecture, they modify and remodify until they find a problem they can do”. Above all, it is reasonable to acknowledge that conjecturing is an ongoing process which is built on specializing and generalizing as an ascent and descent (Polya, 1954). As a result, Mason et al. (2010) propose an idea to describe the conjecturing process. They consent that the process of conjecturing hinges on being able to recognize a pattern, or depending on being able to make a generalization. In short, the conjecturing process could be described as a cyclic process of articulating a conjecture, checking the conjecture, refuting/accepting the conjecture, and recognizing the pattern (see figure 1).

Figure 1 Conjecturing process. Reprinted from Thinking mathematically (p. 59), by J. Mason, L.

Burton, & K. Stacey, 2010, Harlow, UK: Pearson Education Limited.

Moreover, generalizing and specializing are the two sides of a coin, in accordance with this view point, Mason (2002) points out two perceptions particularly, which are seeing the particular in the general and seeing the general through the particular. In addition, mathematics is perceived as the science of pattern and relationship. Consequently, exploring patterns, relations and functions is an essential focus of mathematics learning (AAAS, 1990; NTCM, 2000). Actually, generalizations are both objects for individual thinking and means for communication (Dörfler, 1991). For that reason, pattern-finding tasks in generalization can be considered as an important activity for getting students involved in a conjecturing atmosphere. Mason et al. (2010) synthesize these viewpoints, they think that specializing and generalizing are the backbone of the conjecturing process. Particularly, the problem solving phases, such as “entry”, “attack”, and “review” are owed much on specializing and generalizing (see figure 2). In addition, they see the “attack” phase is very much related to justifying and convincing, and it is also a crucial phase to seeing structural links.

Figure 2 Backbone of conjecturing. Reprinted from Thinking mathematically (p. 77 & 95), by J.

Mason, L. Burton, & K. Stacey, 2010, Harlow, UK: Pearson Education Limited.

Despite that the significance of conjecturing has been recognized by plenty of researchers (Davis, Hersh, & Marchisotto, 1995; Lakatos, 1976, 1978; Mason, et al., 2010), it could be recognized that evolving a conjecturing process in patterning approaches is one thing, justifying it for convincing others is quite another; even when students are able to generalize a pattern or a rule, few are able to explain why it occurs (Coe & Ruthven, 1994). Since that once you find the pattern, you need to state it carefully and clearly to convince yourself, convince a friend, and even to convince a skeptic (Mason et al., 2010).

Mason (2002) further states that once a conjecture is made, it needs to be challenged, justified, and possibly reconstructed. Blanton and Kaput (2002) propose that justification induces a habit of mind

justifying is using statements accepted by the classroom community and employs forms of reasoning that are valid and known to classroom community, as well as it is communicated with forms of expression that are appropriate and known to the classroom community. Consequently, conjecturing accommodates fruitful opportunities for reasoning in behalf of justifying conjectures. As a result, Zack and Graves (2001) adopt a sociocultural perspective to investigate discourse and its role in how children and teachers make meaning of mathematics in a fifth grade inquiry-based classroom for exploring the relationship between discourses and knowing Therefore, “participating in a conjecturing atmosphere in which everyone is encouraged to construct extreme and paradigmatic examples, and to try to find counter-examples involves learners in thinking and constructing actively” (Mason & Johnson-Wilder, 2004, p.142). To sum up, when students are engaged in the context of inquiry-based conjecturing activity, they need to articulate the conjecture and specialize the conjecture systematically to validate it.

Furthermore, they also have to propose an effective pattern based on the process of generalization to persuade themselves and others clearly and carefully. Even more, if the situation becomes complicated, they need to build on specializing and generalizing as an ascent and descent, in an ongoing process of conjecturing.

III. Methodology

1. Research setting

This study is conducted for developing eleventh students’ conjecturing and justifying power on finding problems. Since the study is implemented in an inquiry-based classroom, the pattern-finding activity is orchestrated in steps of presentation of a problem, whole-class discussion about the methods for solving the problem, and summing up by the teacher (Shimizu, 1999). Further, qualitative case study approach (Yin, 2009) is adopted in this study.

The pattern-finding activity is designed to support students’ development of conjecturing and justifying power. There are six tables of numbers (Figure 3) from 1~63 that are constructed by transforming of number systems from decimal to binary in students’ worksheet. For instance, decimal number 10 could be represented as the binary number 1010, that is, number 10 is equal to (1)*23+(0)*22+(1)*21+(0)*20. Among the six tables, table 1 contains the binary numbers with the first digit is 1. The rest may be deduced by analogy, such as table 5 contains the binary numbers with the fifth digit is 1. Hence number 10 can be found in tables 2 and 4 only. Some of the research subjects have

learned the binary system in the "information and technology" lessons, but not with this case. Hence, randomly placing the numbers in each table but not sequencing them in order might increase the degree of difficulty of generalizing the hidden pattern. In addition, there are three questions designed in this game: (1) What properties can you find from these six tables? (2) Can you induce any generality of the tables from the properties you found? (3) Can you propose any conjecture of how the game works and justify your conjecture?

Figure 3 Number tables for pattern-finding activity

2. Participants

Fourteen eleventh graders participate in a four-week extra-curricular program aimed to develop the power of conjecturing and justifying. Students are purposefully selected from seven different senior high schools with varied achievements. In addition, five of the students have chosen to be research subjects due to their fruitful performances. These schools are all located in the suburb of central Taiwan. The teacher of this study is one of the authors, who holds the viewpoint that students should construct mathematical knowledge actively and students’ self-efficacy of math-learning might be fostered in the community of inquiry.

3. Number pattern generalizing activity

The activity is held throughout two phases, the first phase is started with a game asking a student to select a number from 1~63 and bear it in mind firstly, and then showing the student these six number tables sequentially for she/he to examine whether the selected number is in the table or not. In the end, the teacher notifies the student the accurate number she/he selected. After playing the game for several

At the end of personal session, we interview the students who have proposed conjectures in their own ideas; four of research subjects are purposefully select with maximum variation sampling strategy. Each critical case carries highly potential ideas of solving the hidden pattern of activity and can be seen a capable peer for leading collaborative learning. As regards the second phase, students are reorganized into four heterogeneous groups for following four selected representative students respectively (Louise, Isaac, Wendy and Bob) to find out the hidden patterns. Secondly, teacher walks around groups and poses productive questions for students. Finally, students’ findings are integrated and validated through classroom community.

4. Data collection and analysis

Across a 4-week period, the whole activity procedures are videotaped and all the critical, extreme or unique, and revelatory cases are interviewed simultaneously while proceeding of group discussion at the second phase. Most of questions focus on the progress of students’ development of specializing and generalizing power. All audio recordings are later transcribed verbatim. Further, field notes of class observation are taken during classroom video recordings and later expanded to help the researcher understand and orchestrate the story line of the study outcomes. Students’ worksheets and reflections are also collected for interpretation of students’ conjecturing behavior. Multiple evidence sources (Patton, 1987) are collected for constructing validity and triangulating evidence. Furthermore, all data are connected to a series of evidenced chain for enhancing reliability (Yin, 2009).

As regards data analysis strategy, first of all we generalize results systematically from the phenomenon observation (Strauss & Corbin, 1990). Secondly, students’ mathematical conjecturing behavior are analyzed with theory driven. Competitive expositions are formulated from the results of critical cases. Finally, we take cross-case analysis and base on the level of findings of four groups respectively to construct a holistic view of students’ conjecturing and justifying power on the pattern-finding activity.