Enhancing pre-service teachers’ fraction knowledge through open approach instruction

Journal of Mathematical Behavior 32 (2013) 309–330

Contents lists available at ScienceDirect

The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb

Enhancing pre-service teachers’ fraction knowledge through open approach instruction Cheng-Yao Lin a,∗ , Jerry Becker a , Yi-Yin Ko b , Mi-Ran Byun a a b

Southern Illinois University Carbondale, United States Indiana State University, United States

a r t i c l e

i n f o

Article history: Available online 12 June 2013 Keywords: Fraction Open approach instruction Pre-service elementary teachers

a b s t r a c t This study explores whether using the open approach instruction in teaching mathematics has a positive effect for enhancing pre-service teachers’ fraction knowledge. The test consisted of 32 items that were designed to examine pre-service teachers’ procedural and conceptual knowledge of fractions before and after receiving open approach instruction. The study was undertaken among students in four mathematics content and methods courses for the Elementary School Education program in a mid-western public university. The findings show that most of the teachers achieved improved learning outcomes through the open approach instruction. © 2013 Elsevier Inc. All rights reserved.

1. Introduction The importance of fractions in the school curriculum is recognized widely. However, fractions are a difficult topic for teachers to teach and for students to learn because they involve relations between quantities. Another reason is that one cannot count fractions because there are an infinite number of fractions between any two fractions or any two whole numbers. In order to teach fractions effectively, mathematics teachers should possess adequate understandings of mathematics. Research studies, however, have demonstrated that many pre-service teachers have difficulties with the concepts of fractions (Ball, 1990; Behr, Khoury, Harel, Post, & Lesh, 1997; Cramer, Post, & Del Mas, 2002; Davis & Thipkong, 1991; Li & Kulm, 2008; Li & Smith, 2007; Newton, 2008; Simon, 1993; Toluk-Uc¸ar, 2009). Ball (1990) found that pre-service teachers had significant difficulty with the meaning of division of fractions. They were unable to think flexibly and consciously about division operations as partitive or quotitive. Behr et al.’s (1997) study showed that pre-service teachers experienced difficulty understanding the operator construct of rational numbers. Research studies have also suggested that pre-service teachers are unable to explain fractions to students and why algorithms work (Chinnapan, 2000; Li & Kulm, 2008; Lubinski, Fox, & Thomason, 1998; Selden & Selden, 1997). According to the Principles and Standards for School Mathematics, “effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well” (National Council of Teachers of Mathematics [NCTM], 2000, p. 16). Ma (1999) suggested that effective teachers must have a profound understanding of mathematics because content knowledge is a foundation for them to use instructional approaches to teaching mathematics flexibly. That is, a teacher should be well equipped with content knowledge and should be prepared to represent mathematical ideas to students in ways that will connect their prior knowledge with the mathematics they are expected to learn.

∗ Corresponding author. Tel.: +1 618 453 4236. E-mail address: [email protected] (C.-Y. Lin). 0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmathb.2013.03.004

310

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

To achieve this goal, open approach teaching has been employed in Japan since 1970 in order to promote higher-order thinking in mathematics (Becker & Shimada, 1997; Hashimoto & Sawada, 1984; Nohda, 1986, 1995; Shimada, 1977). The philosophy of the open approach method in teaching mathematics is to encourage students to use their natural ways of mathematical thinking and to apply their previously learned knowledge to the process of solving open problems (Kwon, Park, & Park, 2006). Using open problems in classrooms has spread to other countries such as England, the Netherlands, Germany, the U.S., Australia, and Thailand (Inprasitha, 2006; Stacey, 1995) because it offers an alternative way of teaching, learning, and assessing mathematics learning. This teaching approach is often called an “open-ended” approach because the problems are selected to be solved in multiple ways (the process is open) or to have multiple correct solutions (the end products are open). In this approach, students analyze problems and problem-solving methods through a process of solving a problem first in one way and then discuss and evaluate a variety of solution methods that have been developed and presented by classmates (Hashimoto, 1987). There is also an emphasis on having students formulate their own new problems by changing one or more conditions of the problem (Becker, Silver, Kantowski, Travers, & Wilson, 1990; Hashimoto & Becker, 1999). When teaching mathematics using the open approach, students are engaged in solving open-ended problems, as well as in working on multiple intellectual processes such as exploring, testing, discussing, connecting, criticizing, communicating, investigating, generalizing, hypothesizing, and using reasoning to explain different ideas. As a result, the open approach can not only foster students’ thinking and but also help them develop a better understanding of the subject (Becker & Shimada, 1997; Hashimoto & Becker, 1999; Inprasitha, 2006; Kwon et al., 2006; Klavir & Hershkovitz, 2008; Miwa, 1991; Nohda, 2000; Sawada, 1997; Sullivan, 2009). Klavir and Hershkovitz (2008) showed that creative mathematical thinking could be encouraged by providing open-ended problems as an evaluation tool for both teachers and students. Kwon et al. (2006) found that divergent thinking in mathematics could be cultivated through an open-ended approach because openended problems allow various solutions and the ways students solve the problems based on their own scope and range of abilities. The open approach, which has multiple methods to solve the problem, offers students opportunities to communicate their ideas with each other and discuss their different problem-solving methods. The interactions among the pupils and the teacher in the open approach help students to improve their mathematical thinking and understanding (Miyakawa, 2006; Nohda, 2000; Yee, 2002). Through social interaction between students and teachers in mathematics classrooms, pupils’ feedback guided by the teacher was a relevant didactical contract (Miyakawa, 2006). Nohda (2000) emphasized the awareness of information processes, which consisted of the communications and interactions between the teacher’s explanations and students’ approaches to problem solving. Yee (2002) showed that primary teachers who used short open-ended problems enabled their students to demonstrate their thinking and understanding of mathematical concepts in a variety of ways. In fact, using the open approach to teaching mathematics can potentially change both students’ and teachers’ attitudes and beliefs about mathematics and develop their reasoning, communication skills, and making connections in mathematics (Hashimoto & Becker, 1999). Given that open problems are selected for all students, regardless of their mathematical abilities, skills, and interests, they are not intended only for students with high motivation and skills. Nohda (1986) indicated that even students with a lower motivation level could be involved in solving open problems. Appealing to students’ natural ways of thinking lies at the heart of the “open approach,” as students will produce different solutions and then will share their thinking with the rest of the class. Regarding the open approach, there are five different parts of the lesson plans: (1) introducing the problem or topic; (2) understanding the problem; (3) problem solving by the students; (4) comparison and discussion of students’ solutions; and (5) summary of the lesson by the teacher (Becker et al., 1990; Conway, 1996; Hashimoto, 1987; Hashimoto & Becker, 1999; Miwa, 1991; Miyakawa & Winsløw, 2009; Tougaw, 1994). The first two parts focus on students coming to understanding the problem, which plays a prominent role for students to think about mathematical problems that are related to but different from problems solved on a previous day (Hashimoto, 1987). And then students analyze problems and select problem-solving methods through processes of differing mathematical thoughts. For the next step, students have discussions for a variety of solution methods that have been developed and presented by classmates. The use of problems that permits a class of students to generate multiple solutions is a key feature of this approach to mathematics teaching, and it is associated with the development of students’ mathematical thinking abilities while they spend enough time finding their own solutions and discussing different methods and mistakes with others. Finally, the teacher summarizes and makes connections to reach the lesson’s objective by pulling all outcomes from discussions (Hashimoto & Becker, 1999). Despite the importance of using the open approach to teaching mathematics, assessment plays an important role in evaluating students’ progress to promote teachers’ teaching (NCTM, 2000). Assessment in the open approach is considered an integral part of classroom teaching (Nagasaki & Becker, 1993) and consists of the following four different features (Becker & Shimada, 1997; Hashimoto & Becker, 1999): Fluency, Flexibility, Originality, and Elegance. Fluency refers to the number of correct responses that the student produces. Flexibility refers to the number of different mathematical concepts and ideas that the student discovers. Originality refers to the extent that the student’s ideas are insightful, original or innovative. Elegance refers to the extent that the student expresses his or her thinking in mathematical notation. By using the open approach instruction, pre-service teachers can not only deepen their understandings of mathematics but also become capable of making instructional decisions that lead to meaningful activities and real world experiences for the students in their future classrooms. The aim of this study is to examine pre-service teachers’ knowledge of fractions through the use of open approach instruction.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

311

2. Theoretical framework The construction and basis of open approach instruction is closely related to conceptual and procedural knowledge and profound understanding of fundamental mathematics (PUFM) that Ma (1999) depicted in her study. Conceptual and procedural knowledge are regarded as two complementary aspects concerning mathematics content knowledge (Groth & Bergner, 2006). On the other hand, PUFM has been considered as a comprehensive and significant ability for elementary teachers in teaching mathematics. Open approach teaching can be used to establish students’ profound mathematical thinking and to balance with conceptual and procedural knowledge. Taken together, these three features of PUFM all play a significant role in pre-service teachers’ fraction knowledge. 2.1. Listen 2.1.1. Read phonetically Conceptual knowledge of mathematics can be described as a network that constructs cognitive connections out of relevant information and sorts out the unrelated information (Groth & Bergner, 2006, p. 39). Hiebert and Lefevre (1986) explained, “a unit of conceptual knowledge cannot be an isolated piece of information; by definition it is a part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information” (p. 4). Conceptual knowledge typically needs conscious thinking (Kadijevich & Haapasalo, 2001) because it involves “knowing why” in the discipline of mathematics education. Take pre-service teachers’ conceptual knowledge of division of fractions, for instance. Although pre-service teachers could calculate 1(3/4)÷(1/2) accurately, very few of them could come up with a mathematically accurate representation to explain why the procedure or rule works (Ball, 1990; Ma, 1999). With respect to procedural knowledge of mathematics, it often requires unconscious steps (Kadijevich & Haapasalo, 2001, p. 157). Procedural knowledge is composed of two different types: one consists of “the formal language, or symbol representation system, of mathematics” (Hiebert & Lefevre, 1986, p. 6), while the other is made up of “the algorithms, or rules, for completing mathematical tasks” (Hiebert & Lefevre, 1986, p. 6). Both categories of procedural knowledge could be applied to the case of fractions. Regarding the division problem 1:1(3/4)÷(1/2), distinguishing the difference between symbols in 1(3/4) and (13/4) falls on the former type of procedural knowledge. Recognizing an algorithm – invert and multiply – for solving the problem 1(3/4)÷(1/2) falls on the latter type of procedural knowledge. Ma (1999) developed PUFM as a thorough and comprehensive understanding of elementary mathematics. PUFM also emphasizes the teachers’ conceptual as well as procedural understanding of mathematics. Not only is PUFM a conceptual understanding of elementary mathematics, but Ma also described that PUFM includes the following characteristics: “[PUFM] is the awareness of the conceptual structure and basic attitudes of mathematics inherent in elementary mathematics and the ability to provide a foundation for that conceptual structure and instill those basic attitudes in students” (p. 124). Overall, teachers with PUFM tend to focus on both mathematical concepts and procedures, cherish multiple viewpoints as well as various approaches when solving a mathematical problem, and possess great capacities for incorporating previous concepts students have already learned with brand-new concepts along with those to be studied later. These characteristics of teachers with PUFM are consistent with the crucial skills that teachers need to proceed in open approach teaching. In a similar vein, Ball, Thames, and Phelps (2008) stated that preparing teachers to be equipped with foundational to the subject is really helpful for them to teach students effectively. As the quality of mathematics teaching depends on teachers’ mathematical content knowledge (Ball, Hill, & Bass, 2005), this study shows changes of pre-service elementary mathematics teachers’ PUFM as consequence of using the open approach instruction. 3. Research questions Considering the existing research about per-service teachers’ knowledge of fractions and the open approach to teaching mathematics, the aim of this study is to learn about the effects of an Open-ended Approach Instruction (OAI) on teachers’ understandings and computational abilities with respect to fractions. Specific research questions are as follows. (1) Is there any improvement for pre-service teachers in different courses of their fractional procedural knowledge (PK)/conceptual knowledge (CK) after having been taught using an open approach course? (2) Is there any PK/CK improvement for the eight fractional dimensions – fraction concepts (CON), equivalent fractions (EQU), fraction ordering (ORD), fraction addition (ADD), fraction subtraction (SUB), fraction multiplication (MUL), fraction division (DIV), and fraction transferring (TRA) – after pre-service teachers have been taught using an open approach? (3) Did pre-service teachers perform differently in PK and CK across each fractional dimension? This research parallels the work of Chapman (2007) in pre-service teachers’ content knowledge. Chapman (2007) used arithmetic word problems as the basis of an approach to facilitate pre-service elementary teachers’ development of mathematical knowledge for teaching arithmetic operations. The results indicated that the tasks can provide an effective and meaningful basis to help pre-service teachers to develop deeper understandings of the arithmetic operations and their relationships and pedagogical-mathematical knowledge for teaching the arithmetic operations conceptually.

312

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Table 1 Summary of fractional items used in the study. Question number

Fractional content

PK

CK

1, 2, 3, 4

Concepts

5, 6, 7, 8

Equivalence

9, 10, 11, 12

Order

13, 14, 15, 16

Operation: addition

17, 18, 19, 20

Operation: subtraction

21, 22, 23, 24

Operation: multiplication

25, 26, 27, 28

Operation: division

29, 30, 31, 32

Transfer

Student knows how to do the problem with parts of the unit Student knows how to do the equivalent fractions Student knows how to do the fraction ordering Student knows how to do fraction addition problem Student knows how to do fraction subtraction Student knows how to do fraction multiplication Student knows how to do fraction division Student knows how to do transfer fraction in number line

Student knows how to explain the part and whole relationship Student knows how to explain equivalent fractions are equal Student knows how to explain fractions are in order Student knows how to explain fraction addition Student knows how to explain fraction subtraction Student knows how to explain fraction multiplication Student knows how to explain fraction division Student knows how to explain fractions are in correct number line

4. Methods 4.1. Participants Participants were 125 undergraduate students (106 female, 19 male) aged 18–23 who attended four mathematics content methods courses for elementary, special education, and/or early childhood teacher education majors at a Midwestern university in the United States. They were selected by convenience sampling; that is, the participants were contacted by the first author and were recruited on the basis of their willingness to participate in the study. Four courses, CI/Math 120, CI/Math 220, CI/Math 321, and CI/Math 322, are the mathematics courses required by the university for the Elementary School. CI/Math 120 and CI/Math 220 are required courses for all undergraduate students in the elementary, special education, and early childhood teacher education programs, whereas CI/Math 321 is a required course for all undergraduate students in only the elementary and special education teacher education programs. Forty-six participants, who were freshmen and were aged 18–19, came from CI/Math 120; 16 participants, who were sophomores and were aged 19–20, came from CI/Math 220; 48 participants, who were juniors and were aged 20–21, came from CI/Math 321; and 15 participants, who were juniors and were aged 21–22, came from CI/Math 322. All four courses were divided into 16 weeks and met 4 h per week. The four courses (CI/Math 120, 220, 321 and 322) run in a hierarchical system. For example, students who enroll in CI/Math 120 and 220 and do not receive at least a grade of C are not eligible to enroll in CI/Math 321. Moreover, students in CI/Math 120 gain introductory mathematical knowledge, including fraction-based knowledge, whole numbers, integers, rational numbers, and the essential arithmetic operations. At the CI/Math 220 level, students concentrate on learning rational and irrational numbers. At the CI/Math 321 level, students are instructed in three-dimensional geometry and are introduced to the fundamental ideas of statistics and probability, which are based on the mastery of fractional knowledge obtained in CI/Math 120 and 220. Students in CI/Math 322 are expected to obtain the knowledge of algebra, and recognize relations and functions, as well as their applications to real-life problems. All four courses, focusing on multiple representations of mathematical concepts, provide not only procedural and conceptual knowledge in mathematics but also pedagogical content knowledge regarding teaching mathematics for the elementary school. With regard to this study, students who enrolled in the four courses were taught about knowledge of fractions for the first six weeks. Especially, many students in CI/Math 321 and 322 transferred from another college with different content curriculum, so this review process associated with rational numbers was necessary. Since this study does not target participants based on age, gender, or other characteristics, such characteristics were not considered in the analysis of the data. 4.2. Instrument The Fraction Knowledge Test (see Appendix A), comprised of 32 items that were adapted from Cramer et al. (2002) and Ma (1999), was designed to access pre-service elementary mathematics teachers’ procedural and conceptual knowledge of fractions. The fractional items specifically designed to measure knowledge of fractions in areas related to eight dimensions: (1) concepts (CON), (2) equivalence (EQU), (3) order (ORD), (4) operation: addition (ADD), (5) operation: subtraction (SUB), (6) operation: multiplication (MUL), (7) operation: division (DIV), and (8) transfer (TRA). These dimensions were selected in order to assess pre-service teachers’ fraction knowledge from Long and DeTemple (1996). Four items were designed for each fractional dimension (CON, EQU, ORD, ADD, SUB, MUL, DIV, and TRA). Within each fractional dimension, the four items were divided into two sets of questions—one was procedural knowledge and the other one was conceptual knowledge. The summary of each item used in the Fraction Knowledge Test is presented in Table 1.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

1. Calculate

313

1 1 1 1 + . Use pictures to explain + ."What are the different ways to get the 2 4 2 4

solution? 2. Calculate

1 1 1 1 - . Use pictures to explain - . What are the different ways to get the 2 4 2 4

solution? 3. Calculate

1 1 1 1 x" . Use pictures to explain x" ."What are the different ways to get the 2 4 2 4

solution? 4. Calculate solution? "

1 1 1 1 ÷ . Use pictures to explain ÷ . What are the different ways to get the 2 4 2 4

Fig. 1. An example of the worksheet.

The fractional instrument was piloted with forty pre-service teachers who were enrolled in different mathematics methods courses. A panel of specialists, including three mathematics educators, ascertained content validity of the test. The instrument items were modified following their suggestions, which were used both in the pre-test and post-test to measure pre-service teachers’ understandings of fraction concepts. The internal reliability (Cronbach’s alpha) of the test was found to be .86.

4.3. Description of the instruction In order to help pre-service teachers carry out current reforms and approaches regarding teaching fractions (NCTM, 2000), it is essential to improve their mathematical knowledge and computational skills. The present study examined the use of open approach instruction to determine if it would enhance pre-service teachers’ content knowledge and pedagogical knowledge of fractions. There were 24 instructional sessions with each of the four groups of pre-service teachers, and each session for this study lasted approximately 50 min. This study progressed in four steps: In the first step, the instructors administered a pre-test that was designed to diagnose pre-service teachers’ understandings, as well as overall proficiency and misconceptions of fractions. Next, the instructors used an “open approach” (Becker & Selter, 1996; Becker & Shimada, 1997; Hashimoto & Becker, 1999). In the “open approach,” instructors designed plans based on an open approach teaching pedagogy, and developed worksheets and other materials (see Fig. 1), which provided understanding and practice for the pre-service teachers. Instruction was focused on a diversity of approaches to reach solutions. In this approach, the main activities are under the teacher-led discussions between students. An example of lesson development for open approach teaching is described in the following dialog.

(T represents the teacher and S the student): 1. Introducing the problem: T: Imagine that you are teaching

÷ . What are the different ways to get the solution?

2. Understanding the problem: T: Do you understand the problem? Can you use the different ways to get the solution of ÷

?

314

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

S4:

÷

315

have same denominators which means we are working with same portions.

Now we take numerator 2 ÷ 4. Next, we complete the division of denominator. Take 5 ÷ 5 equals 1. Thus, our answer is

.

4. Comparison and discussion of students’ solutions: T: To compare S1 and S2 method, S1 only shows procedural knowledge and does not exaplain why we have to multiply by its reciprocal. On the contrary, S2 method does explain why we use that method to solve the problem. 5. Summary of the lesson by the teacher: T: In summary, there are many ways to get the solution of

÷ . S1 shows the

traditional way to solve division of fractions. S2 explains how to use the standard algorithm for dividing fractions. S3 and S4 use different ways to get the answer to

÷ .

Therefore, students’ responses or answers were used by the teacher to provide experience to students in learning something new. Students combined what is newly learned with their previous knowledge, skills and mathematical ways of thinking (Becker & Shimada, 1997; Hashimoto & Becker, 1999; Shimada, 1977). 4.4. Data collection and analysis The primary source of data was pre-service elementary teachers’ responses to the Fraction Knowledge Test. The participants were asked to complete a computational test in the first day of each course (pre-test), and the same items were given to the pre-service teachers 6 weeks later after using an open approach instruction (post-test). The participants were given approximately 120 min to answer 32 items used in the Fraction Knowledge Test, and Fig. 2 shows mastering computational skills model. Open approach instruction on computational skills was begun after the pre-test, and further open approach instruction on computational skills was given after the post-test.

Fig. 2. Mastering computational skills model.

316

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Table 2 Within-subjects effects on PK and CK in various course levels. Level

Test

N

M

SD

14.85 25.87 15.56 25.88 19.69 29.77 20.333 28.467

8.377 4.500 8.294 5.584 7.209 2.587 6.673 3.563

5.35 14.20 4.69 14.19 8.250 23.146 6.933 16.133

3.837 5.540 3.945 6.504 4.849 4.771 4.096 5.792

df



F

p

PK Math 120 Math 220 Math 321 Math 322

PK PK PK PK PK PK PK PK

1 2 1 2 1 2 1 2

CK CK CK CK CK CK CK CK

1 2 1 2 1 2 1 2

46 16 48 15

45

114.876* *

.658

<.001

15

29.084

.672

<.001

47

102.780*

.562

<.001

14

16.427*

.540

.001

45

213.248*

.658

<.001

15

45.320*

.629

<.001

CK Math 120 Math 220 Math 321 Math 322 *

46 16 48 15

*

47

460.746

.466

<.001

14

26.915*

.529

<.001

p < .01.

Data analysis began by examining the pre-service teachers’ responses to the pre-test and post-test. Two points were counted for each item, so the total score of both the pre-test and post-test was 64. More specifically, the total score of procedural knowledge’s (PK) and conceptual knowledge’s (CK) items was 32, respectively. For grading of pre-service teachers’ answers, each item was scored 2 points for a fully correct answer, 1 point for a partially correct answer, and 0 for an incorrect answer. Basically, pre-service teachers received full credit (2 points) when they showed a clear representation of fractional operations with a step-by-step process. Pre-service teachers received partial credit (1 point) if they used the correct algorithm but made an arithmetic mistake, or if they skipped one of the important steps of representations. Also, if pre-service teachers represented the main ideas of fractions but failed to find the final answer, they received partial credit (1 point). We did not give the pre-service teachers any credit for showing just a wrong answer with a random diagram such as a circle, square, and bar. For scoring reliability, the test was scored by three graduate research assistants independently. When items caused disagreement among the graders, the research team (three faculty members and the three graders) had discussions to decide reasonable apportionment of the grades. After grading pre-service teachers’ responses to the pre- and post-test, the scores were translated into SPSS files for further analysis. The analysis of the first research question, “Is there any improvement for pre-service teachers in different courses of their fractional procedural knowledge (PK)/conceptual knowledge (CK) after having been taught using an open approach course?”, was based on the repeated measure tests in Fraction Knowledge Test scores. The pre-test and post-test scores were used to analyze various courses according to PK or CK separately. To answer the second research question, “Is there any PK/CK improvement in the eight fractional dimensions – fraction concepts (CON), equivalent fractions (EQU), fraction ordering (ORD), fraction addition (ADD), fraction subtraction (SUB), fraction multiplication (MUL), fraction division (DIV), and fraction transferring (TRA) – after pre-service teachers have been taught using an open approach course?”, similar repeated measures in pre-test and post-test scores were used to analyze the improvements in each fractional dimension according to PK or CK separately. Finally, to answer the third research question, “Did pre-service teachers perform differently in PK and CK across each fractional dimension?”, the post-test scores of PK and CK were compared in each fractional dimension (CON, EQU, ORD, ADD, SUB, MUL, DIV, and TRA). 5. Results In this section, we first focus on the descriptive statistics of pre-service teachers’ responses to the pre- and post-test, and then we examine the correctness of their answers. 5.1. Descriptive statistics of pre-service teachers’ responses Regarding the examination of PK, Table 2 shows that all four levels of pre-service teachers (levels of CI/Math 120, 220, 321 and 322) performed differently at a .01 significant alpha level. We can also see that effect sizes were medium ( = .65, .67, .56 and .54, respectively), which indicated that the highest proportion of total variability in the four level samples could be attributed to the difference of pre-test group scores, respectively. Similar results were revealed in the comparisons of CK. Pre-service teachers in levels of CI/Math 120, 220, 321 and 322 performed differently at a .01 significant alpha level with the effect sizes medium ( = .66, .63, .47 and .53, respectively). As can be seen in Table 3, pre-service teachers performed differently between pre-test and post-test scores (p < .001) in each fractional dimension (CON, EQU, ORD, ADD, SUB, MUL, DIV, and TRA), no matter what their procedural knowledge or

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

317

Table 3 Differences between pre-test and post-test PK/CK scores for eight fractional dimensions (N = 125). Dimension

Test

M

SD

Wilks’ 

SSa

Partial ␩2

F

PK CON EQU ORD ADD SUB MUL DIV TRA

PK PK PK PK PK PK PK PK PK PK PK PK PK PK PK PK

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2.25 3.09 2.57 3.38 2.17 3.22 2.37 3.62 2.48 3.53 2.17 3.23 2.12 3.36 1.05 3.03

1.505 1.390 1.233 1.132 1.232 1.176 1.807 .989 1.708 1.070 1.743 1.336 1.919 1.287 1.284 1.365

CK CK CK CK CK CK CK CK CK CK CK CK CK CK CK CK

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2.02 3.03 2.18 3.15 1.15 2.65 .20 2.44 .14 1.92 .02 .83 .00 .43 .83 2.63

1.564 1.387 1.192 1.177 1.500 1.503 .712 1.616 .592 1.518 .173 1.234 .000 .995 1.228 1.480

47.158

30.538*

.188

.750

43.041

*

43.886

.250

.729

73.684

49.045*

.271

.704

104.846

55.435*

.296

*

.812

.753

72.635

43.215

.247

.787

74.741

35.647*

.213

.754

102.350

43.005*

.246

.429

262.015

175.577*

.571

.743

67.504

45.579*

.257

.680

62.560

62.118*

.320

*

CK CON EQU ORD ADD SUB MUL DIV TRA a *

.580

150.376

95.604

.420

.378

336.094

217.041*

.622

.455

209.383

158.282*

.545

*

.687

43.850

60.199

.313

.842

12.214

24.696*

.158

.474

216.541

146.238*

.526

Type III sum of squares by Greenhouse–Geisser method. p < .001.

the content knowledge was. This might be because attending open approach instruction could improve pre-service teachers’ mathematical knowledge. However, Fig. 3 shows that the effect sizes varied across the eight dimensions for PK and CK. Effects shown on PK were higher than CK on almost all dimensions, except in the DIV dimension. This is due to the fact that the pre-service teachers who were taught by an open approach developed a deeper understanding of the logic behind the procedure. Considering (1/2) divided by (1/4), the pre-service teachers learned the meaning of how many (1/4) goes into (1/2). In addition, they were able to explain their reasoning of what concepts underlined the supporting solutions to (1/2) divided by (1/4) rather than just apply the rule of invert and multiply. This is especially apparent in the ADD and SUB dimensions, in which the discrepancies of effects were biggest. Although no effective difference was shown in PK and CK, another high effect after an open approach instruction was shown in the dimension of TRA. Pre-service teachers showed higher levels of understanding on the TRA’s PK and CK. On the other hand, however, the lowest CK effect shown in the DIV dimension indicated pre-service teachers had more difficulty understanding DIV’s CK than PK. This is the only dimension in which pre-service teachers did

Fig. 3. Effect sizes comparisons between PK and CK in each dimension. Note. Eight fractional dimensions: fraction concepts (CON), equivalent fractions(EQU), fraction ordering(ORD), fraction addition (ADD), fraction subtraction (SUB), fraction multiplication (MUL), fraction division (DIV), and fraction transferring (TRA).

318

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Table 4 Within-subjects effects and adjusted means by covariates on the differences of PK and CK (N = 125). Dimension CON EQU ORD ADD SUB MUL DIV TRA a b *

KW PK CK PK CK PK CK PK CK PK CK PK CK PK CK PK CK

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

M

Ma

SD

3.09 3.03 3.38 3.15 3.22 2.65 3.62 2.44 3.53 1.92 3.23 .83 3.36 .43 3.03 2.63

3.303 3.254 3.637 3.407 3.380 2.803 3.819 2.740 3.705 2.102 3.412 .859 3.500 .390 3.163 2.854

1.390 1.387 1.132 1.177 1.176 1.503 .989 1.616 1.070 1.518 1.336 1.234 1.287 .995 1.365 1.480

Wilks’ 

r

SSb

F

p

Partial 2

.973*

.986

.088

1.772

.185

.014

*

.867

.983

.983

2.266

.135

.017

.754*

.899

6.511

14.501*

.000

.101

.399*

.834

28.106

25.661*

.000

.166

.414*

.817

29.544

28.810*

.000

.183

*

*

.236

.862

24.307

20.610

.000

.138

.180*

.789

33.305

34.485*

.000

.211

.827*

.856

7.196

21.786*

.000

.144

Covariates appearing in the model are evaluated at the following values: PK 1 = 17.17, CK 1 = 6.53. Type III sum of squares by Greenhouse–Geisser method. p < .01.

not show as much improvement on their content knowledge as on their procedural knowledge after an open approach instruction. By using two pre-test as covariates, six fractional dimensions (ORD, ADD, SUB, MUL, DIV, and TRA) revealed differences between the PK and CK (all p’s < .001. See Table 4). Through open approach instruction, pre-service teachers performed higher PK scores than CK scores among these fractional dimensions, but the effects of differences were not large. The highest effect size was shown in the DIV dimension (Partial 2 = .211). Even though they performed higher understandings of PK (adjusted mean = 3.50), pre-service teachers still had less understanding in the DIV’s CK (adjusted mean = .39). On the other hand, only the CON and EQU dimensions showed no differences between the PK and CK (p = .185, and .135, respectively). Pre-service teachers received similar understanding of PK and CK after attending open approach instruction. This might be due to their highly correlated understandings in PK and CK for these two dimensions (Pearson r = .973 and .867, respectively). But this correlation needs to be investigated more deeply because of a high correlation still displayed in the TRA dimension (Pearson r = .827). Adjusted post-test mean scores of PK and CK scores in each fractional dimension are also shown in Fig. 4. 5.2. Correctness of pre-service teachers’ answers Tables 5 and 6 show the percent of correct, partially correct, and incorrect answers to questions of PK and CK provided by pre-service teachers, respectively. Regarding the questions of PK in the pre-test, Table 5 shows that less than 50% of the participants accurately answered Questions 1, 7, 11, 23, and 31, whereas the majority of pre-service teachers provided correct responses to the remaining questions. Considering the problems of PK in the post-test, more than 71% of the participants were able to answer each question accurately. With respect to the items of CK in the pre-test, Table 6 indicates that only over 60% of the pre-service teachers responded to Questions 4 and 6 accurately. For the CK’s questions in the post-test, while less than 20% of the participants correctly answered Questions 22, 24, 26, and 28, the majority of the pre-service teachers solved the remaining questions correctly. Overall, the majority of the teachers gave correct responses to questions of PK and CK after receiving an open approach instruction. As evidenced in Tables 5 and 6, working on questions of PK seemed to be easier

Fig. 4. Mean differences between PK and CK post-test scores for fractional dimensions under two pre-test covariates scores controlled (PK = 17.17, CK = 6.53). Note. Eight fractional dimensions: fraction concepts (CON), equivalent fractions(EQU), fraction ordering(ORD), fraction addition (ADD), fraction subtraction (SUB), fraction multiplication (MUL), fraction division (DIV), and fraction transferring (TRA).

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

319

Table 5 Percent of pre-service teachers’ responses of PK. Question number

Test Pre-test

Q1 Q3 Q5 Q7 Q9 Q11 Q13 Q15 Q17 Q19 Q21 Q23 Q25 Q27 Q31

Post-test

Correct

Partially correct

Incorrect

Correct

44% 69% 87% 39% 86% 23% 62% 58% 71% 57% 62% 47% 57% 50% 9%

0% 2% 9% 1% 0% 0% 0% 1% 0% 0% 0% 1% 0% 0% 0%

56% 29 4% 60% 14% 77% 38% 41% 29% 43% 38% 52% 43% 50% 91%

73% 88% 96% 79% 96% 72% 99% 91% 97% 87% 88% 80% 93% 82% 71%

Partially correct 0% 1% 2% 1% 0% 0% 0% 0% 1% 0% 0% 2% 0% 1% 0%

Incorrect 27% 11% 2% 20% 4% 28% 1% 9% 2% 13% 12% 18% 7% 13% 29%

for the participants than the items of CK before they were taught by open approach instruction. More specifically, more than half of the participants provided accurate responses to 10 questions involved in PK and two items of CK, respectively. After receiving an open approach instruction, over 50% of the participants were able to give correct answers to all PK’s questions and to 10 CK’s items. While there were six questions of CK in the post-test in which only a few pre-service teachers responded accurately, using open approach instruction seem to correspond to the pre-service teachers correctly providing answers to PK’s and CK’s items. Given that the pre-service teachers seemed to experience difficulty working on problems of CK before and after receiving an open approach instruction, the following examples illustrate (a) a correct response—giving a correct answer with accurate representations, (b) a partially correct response—using correct representations to show an improper fraction, and (c) an incorrect response—giving correct representations with an incorrect fraction (Problem 16). Problem 16.

Explain how you determined your answer for problem 15 (Calculate 1(2/3) + (3/4)).

(a) Correct Megan’s example is representative of 2% of the pre-service teachers who generated a correct answer of 1(2/3) + (3/4) on the pre-test. This pre-service teacher wrote her explanations, “I would use a rectangular model. I would draw the fraction out to represent each number 1(2/3) plus (3/4). Next, I would cut each box into equal portions as 1(12/8) plus (9/12). Then

Table 6 Percent of pre-service teachers’ responses of CK. Question number

Test Pre-test

Q2 Q4 Q6 Q8 Q10 Q12 Q14 Q16 Q18 Q20 Q22 Q24 Q26 Q28 Q30 Q32

Post-test

Correct

Partially correct

Incorrect

Correct

36% 62% 71% 27% 36% 19% 2% 2% 2% 1% 1% 0% 0% 0% 32% 7%

2% 6% 20% 5% 3% 3% 4% 4% 4% 2% 0% 0% 0% 0% 2% 2%

62% 32% 9% 68% 62% 78% 94% 94% 94% 98% 99% 100% 100% 100% 66% 90%

70% 87% 94% 67% 64% 67% 62% 50% 42% 33% 18% 9% 10% 10% 78% 56%

Partially correct 2% 1% 2% 6% 5% 9% 18% 14% 31% 22% 19% 14% 9% 9% 2% 6%

Incorrect 29% 12% 4% 27% 32% 24% 20% 36% 27% 45% 63% 78% 82% 82% 20% 38%

320

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

I would add the squares inside the boxes to get the answer 2 and (5/12).” Because this teacher’s answer with explanations were correct, her work was coded as correct. (b) Partially correct Camille’s example is representative of the work of 4% of the pre-service teachers who provided a partially correct answer of 1(2/3) + (3/4)on the pre-test. This pre-service teacher wrote, “I would draw 1 and 2/3 plus 3/4 in a rectangular model. Three times four equal to twelve, so twelve is the common denominator. I add the shaded squares, so the answer is 1 and 17/12.” However, this teacher forgot to change 17/12to 1 and 5/12. Thus, her written work was coded as partially correct. (c) Incorrect Jillian’s example given above is representative of the responses from 94% of the pre-service teachers who gave an incorrect answer of 1(2/3) + (3/4) on the pre-test. This teacher wrote, “I would use pictures of circles showing the parts and a whole. Visual representations show the common denominators and the addition to the wholes and parts. I would draw 1 and (2/3) plus 3/4 in a circle and take 1 plus 2 plus 3 over 3 plus 4, so my answer is 5/7.” While this teacher used pictures to help her add two fractions, she seemed to possess inadequate understanding of fractional properties. Thus, her work was coded as incorrect. As seen previously in Table 6, pre-service teachers’ conceptual knowledge of fractions grew stronger in all addition and subtraction. However, multiplication and division of fractions showed little growth. Fig. 6 presents a sample of correct and incorrect responses of fraction multiplication. To compare with Figs. 5 and 6 responses, we can easily tell that solving items of fraction addition is much easier for pre-service teachers than the items of fraction multiplication. As shown previously in Tables 5 and 6, providing a correct answer to the item of PK in the pre- and post-test seemed is easier for pre-service teachers than the items of CK. Fig. 7 present the representative responses from one pre-service teacher who used different ways to solve Question 15 before and after receiving the open approach. In the first example, the pre-service teacher wrote 1(2/3) as 1 and (2/3) and took fraction add fraction and whole number add whole number in the pre-test. In the second example, the pre-service teacher made 1(2/3) as improper fraction and found a common denominator and added the fraction in the post-test. These two representative examples show that the pre-service teachers were able to employ different strategies for solving a fractional problem after attending the open-approach teaching class. While the majority of pre-service teachers had difficulty solving fractional problems of conceptual knowledge (see Table 6), Fig. 8 presents two examples from one pre-service teacher’s responses to Question 14 of conceptual knowledge before and after receiving the open approach instruction. The first example shows that the pre-service teacher used rectangular model to present (1/4) + (2/3) in the pre-test. This teacher used area model to explain why (1/4) + (2/3) in the post-test. This figure shows that the teacher was able to explain her solution after receiving an open approach. 6. Discussions and implications The purpose of this study is to discover whether there is any improvement of pre-service teachers’ fractional procedural knowledge (PK)/conceptual knowledge (CK) after receiving open approach at different levels of mathematics courses. This section discusses the results in relation to pre-service teachers’ overall understandings of fractions, their overall performance of PK and CK, and their different performance between PK and CK after receiving an open approach instruction. Implications of this work for teaching and learning mathematics in teacher education and future research are also addressed. 6.1. Pre-service teachers’ overall understandings of fractions In general, using open approach instruction significantly improved pre-service teachers’ achievement in procedural knowledge and conceptual knowledge of fractions. It is interesting to note that the pre-service teachers in CI/Math 120 showed a significant difference between two tests. This might be because the pre-service teachers in CI/Math 120 class received this open approach instruction for the first time and the participants in other classes had experienced this teaching approach in their previous semester. The multiplicity of possible answers, as well as approaches for finding answers, create an ideal environment to enhance students’ creativity and high order thinking (Nohda, 1995). Hence, it is not surprising that pre-service teachers receiving this treatment showed significant improvements in their performance of procedural knowledge and conceptual knowledge of fractions. The results indicate that the open approach improved the participants’ overall understandings of fractions in all the dimensions. In particular, there is evidence that the participants achieved a high level of conceptual understanding of basic fractional operations, including equivalent fractions, addition, and subtraction (see Figs. 2 and 3). The open approach provided rich opportunities for pre-service teachers to experience fractional operations through the use of multiple ways of solving problems. As this study was just six weeks of instruction instead of the whole academic year, future studies should add more instructional time to confirm the results of the effects of open approach instruction on pre-service teachers’ understandings of fractions. Given that this research did not have any comparison group and used the identical pre, the different scores shown in the teachers’ pre and post-test might be due to other factors, such as practice effect. Future studies in this area

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

321

Fig. 5. Sample of correct, partially correct, and incorrect responses of fraction addition.

should include a comparison group and avoid using the same pre-test and post-test in order to examine the impact of open-ended approach instruction on pre-service teachers’ fraction knowledge. 6.2. Pre-service teachers’ overall performance of PK and CK Overall, the participants’ performance improved evenly on all strands of procedural knowledge from basic fractional ideas to multiplication and division. The participants made progress toward their procedural fluency with fractions, which shows that the open approach instruction helps learners develop their computational skills. While uncovering fundamental mathematical concepts, the use of open-ended instructional tasks lead pre-service teachers to versatile problem solving strategies and the use of multiple representations. Although this study shows that using open approach instruction enhances pre-service teachers’ understandings of a fractional problem and its mathematical connections, it is still unclear how this teaching method contributes specifically to participants’ knowledge transfer. Further research is needed to explore how open approach instruction helps learners connect their existing knowledge of fractions and the mathematical concepts they are learning.

322

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Fig. 6. Sample of correct and incorrect responses of fraction multiplication.

Because of the participants’ remarkable progress in their procedural knowledge in all dimensions and in their conceptual understanding of basic fractional operations, such as addition and subtraction, their conceptual knowledge of fraction multiplication and division was expected to be similar. Despite their fluency in procedural skills involving multiplication and division, the participants showed a general lack of conceptual knowledge in these two dimensions, which require a rich base of models and informal scenarios for conceptual sense-making. In order to develop pre-service teachers’ understandings of

Fig. 7. Examples of one pre-service teacher’s procedure knowledge response before and after receiving the open approach instruction.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

323

Fig. 8. Examples of one pre-service teacher’s conceptual knowledge response before and after receiving the open approach instruction.

fractions, mathematics instructors may consider paying more attention to individuals’ misunderstandings than problemsolving skills. While the majority of the participants showed the improvement of their PK and CK after receiving an open approach instruction, this study only included eight dimensions of fractions (CON, EQU, ORD, ADD, SUB, MUL, DIV and TRA). Given that being able to solve fractional problems and to explain concepts accurately are both important for pre-service teachers for their further teaching, further research is needed to include more aspects of fractions to examine pre-service teachers’ procedural and conceptual knowledge of fractions.

6.3. Pre-service teachers’ different performance of PK and CK The results of this study show that pre-service teachers’ scores of PK were higher than CK in almost all dimensions, excluding the fraction division (DIV) dimension. In particular, we found that pre-service teachers’ CK of fractional addition and fractional subtraction showed more improvement than PK after an open approach instruction. The results of the study also show that pre-service teachers had higher PK scores than CK scores among these fractional dimensions through an open approach instruction although the effect of the differences was not large. These findings might be because using open approach instruction helps pre-service teachers improve not only their fraction knowledge achievement in both PK and CK but also their mathematical thinking. As an example of addition of fractions, looking at the results of the pre-test, pre-service teachers tended to rely only on one way without understanding the concept of the common denominator, or some teachers could not even solve simple fraction additions due to their misconceptions. After receving open approach instruction, as seen previously in Figs. 4 and 5, the results show that pre-service teachers appeared to conceptualize procedural rules in their various answers, as well as constructed their own models to express the meaning of a common denominator.

324

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

While the pre-service teachers expressed their own strategies for solving problems, they could develop their fraction knowledge by improving in both procedural knowledge and conceptual knowledge, which is consistent with Miyakawa’s (2006) and Yee’s (2002) findings. Miyakawa (2006) pointed out that a teacher needs to ask students’ reasons for a given answer, clarify their statements, and bring them to a common solution by respecting their ideas instead of merely giving students direct feedback. This relevant didactical interaction between the teacher and the pupils helps students to improve their mathematical thinking. Likewise, Yee (2002) found that the open approach enables a teacher to see students’ thinking from their work, as students are encouraged to show their different ideas with step by step. In this study, the instructor helped the pre-service teachers to improve their mathematical thinking in a role as a guide among the their social interactions when negotiating mathematical ideas. Another feature of the results is that using open approach instruction helps pre-service teachers to develop creative thinking skills. In this instruction, pre-service teachers had many chances to discuss various ways to solve problems with their peers. Such learning environment helped pre-service teachers promote their creative thinking while they were gathering their fraction knowledge packages, which supports Imai’s (2000) and Kwon et al.’s (2006) results. Imai (2000) reported that students who overcame the reliance on one method through the open-ended situations in mathematics could achieve significantly higher scores in divergent thinking. Similarly, Kwon et al. (2006) found that the treatment group students who had open-ended problem experiences performed better than the comparison students overall on each component of divergent thinking skills, which includes fluency, flexibility, and originality. Hashimoto and Becker (1999) further indicated that the open approach to teaching mathematics effectively is consistent with current reform suggestions (NCTM, 2000), which has emphasized teaching and learning mathematics through the use of open-ended curricular tasks. As the open approach has been shown to have a significant effect in fostering students’ mathematical thinking, problem solving ability, and motivation toward mathematics (Inprasitha, 2006; Nohda, 1995), much more research is needed to investigate how this instructional approach promotes pre- and in-service teachers’ mathematical reasoning.

7. Conclusion The results of this research indicate that open approach instruction helps pre-service teachers improve their procedural knowledge and conceptual knowledge of fractions. Using open approach instruction provides opportunities for pre-service teachers to learn mathematics meaningfully by discussing mathematical ideas, developing different strategies for solving problems, and generating multiple solutions. We recommend that mathematics teacher education programs review and take into consideration students’ needs to prepare pre-service teachers’ future teaching by using open approach instruction in the mathematics classroom effectively. The results of this study also suggest that mathematics education faculty encourage pre-service teachers to use different ways of solution strategies in the classroom in order to support the development of their content and pedagogical knowledge of fractions. Investigation of open approach instruction on pre-service teachers’ knowledge of fractions is only just beginning. Further research can involve exploring the relationship between in-service teachers’ perceptions of teaching fractions with open approach instruction and students’ achievement, as well as the pre- and in-service teachers’ perspectives on using open approach to teaching fractions. These findings might be helpful for designing mathematics teacher education programs and professional development programs to better promote pre- and in-service teachers’ understandings and instructional approaches regarding teaching fractions.

Acknowledgement The authors wish to thank Ruba Miqdadi, Tsai-Wei Huang, and anonymous reviewers for their helpful comments on earlier versions of this article.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Appendix A.

325

326

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

6. Explain how you determined your answer for problem 5.

7. Determine whether the two fractions are equivalent. and

8. Explain how you determined your answer for problem 7.

9. Circle the larger fraction: and

10. Explain how you determined your answer for problem 9.

11. Find a fraction between the two given fractions. and

12. Explain how you determined your answer for problem 11.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

13. Calculate

14. Explain how you determined your answer for problem 13.

15. Calculate

=

16. Explain how you determined your answer for problem 15.

17. Calculate

18. Explain how you determined your answer for problem 17.

19. Calculate

20. Explain how you determined your answer for problem 19.

327

328

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

21. Calculate

=

22. Explain how you determined your answer for problem 21.

23. Calculate

=

24. Explain how you determined your answer for problem 23.

25. Calculate

=

26. Explain how you determined your answer for problem 25.

27. Calculate

=

28. Explain how you determined your answer for problem 27.

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

329

References Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132–144. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14–17, 20–22, 43–46. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. Becker, J. P., Silver, E. A., Kantowski, M. G., Travers, K. J., & Wilson, J. W. (1990). Some observations of mathematics teaching in Japanese elementary and junior high schools. Arithmetic Teacher, 38(2), 12–21. Becker, J. P., & Shimada, S. (1997). The open-ended approach: A new proposal for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics. Becker, J. P., & Selter, C. (1996). Elementary school practices. In A. Bishop, A. Bishop, et al. (Eds.), International Handbook of Mathematics Education (pp. 511–564). Dordrecht: Kluwer, Academic Publishers. Behr, M. J., Khoury, H. A., Harel, G., Post, T., & Lesh, R. (1997). Conceptual units analysis of preservice elementary school teachers’ strategies on a rational number as operator task. Journal for Research in Mathematics Education, 28(1), 48–69. Chapman, O. (2007). Facilitating preservice teachers’ development of mathematics knowledge for teaching arithmetic operations. Journal of Mathematics Teacher Education, 10(4–6), 341–349. Chinnapan, M. (2000). Preservice teachers’ understanding and representation of fractions in a JavaBars environment. Mathematics Education Research Journal, 12(3), 234–253. Conway, K. D. (1996). The effects of the “Open Approach” to teaching mathematics on elementary preservice teachers’ problem solving performance, attitudes toward mathematics, and beliefs about mathematics. Carbondale, IL: Southern Illinois University (Unpublished Doctoral dissertation). Cramer, K., Post, T. R., & Del Mas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the Rational Number Project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144. Davis, E. J., & Thipkong, S. (1991). Preservice elementary teachers’ misconceptions in interpreting and applying decimals. School Science and Mathematics, 91(3), 93–99. Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers’ conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37–63. Hashimoto, Y. (1987). Classroom practice of problem solving in Japanese elementary schools. In J. Becker, & T. Miwa (Eds.), Proceedings of the U.S.–Japan seminar on mathematical problem solving (pp. 94–119). Columbus, OH: ERIC/SMEAC Clearinghouse. Hashimoto, Y., & Becker, J. (1999). The open approach to teaching mathematics – Creating a culture of mathematics in the classroom: Japan. In L. J. Sheffield (Ed.), Developing mathematically promising students (pp. 101–119). Reston, VA: National Council of Teachers of Mathematics, Inc. Hashimoto, Y., & Sawada, T. (1984). Research on the mathematics teaching by developmental treatment of mathematical problems. In T. Kawaguchi (Ed.), Proceedings of the ICMI-JSME regional conference on mathematical education (pp. 309–313). Japan: Japan Society of Mathematical Education. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–28). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Imai, T. (2000). The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students. International Journal of Mathematical Education in Science and Technology, 31, 187–193. Inprasitha, M. (2006). Open-ended approach and teacher education. In In Proceedings of the APEC International conference on Innovative Teaching Mathematics through Lesson Study Tsukuba, Japan. Kadijevich, Dj., & Haapasalo, L. (2001). Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17, 156–165. Klavir, R., & Hershkovitz, S. (2008). Teaching and evaluating ‘open-ended’ problems. International Journal for Mathematics Teaching and Learning, 20(5), 23. Kwon, O., Park, J., & Park, J. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61. Li, Y., & Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: The case of fraction division. Zentralblatt für Didaktik der Mathematik, 40, 833–843. Li, Y., & Smith, D. (2007). Prospective middle school teachers’ knowledge in mathematics and pedagogy for teaching – The case of fraction division. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (pp. 185–192). Seoul, the Republic of Korea: Psychology of Mathematics Education. Long, C. T., & DeTemple, D. W. (1996). Mathematical reasoning for elementary teachers. Reading, MA: Addison-Wesley. Lubinski, Ch., Fox, T., & Thomason, R. (1998). Learning to make sense of division of fractions: One K-8 preservice teacher’s perspective. School Science and Mathematics, 98(5), 47–53. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahway, NJ: Lawrence, Erlbaum. Miwa, T. (1991). A comparative study on classroom practices of mathematical problem solving between Japan and the U.S. Tsukuba Journal of Educational Study in Mathematics, 10, 81–84. Miyakawa, T. (2006). A study of “good” mathematics teaching in Japan. In Proceedings of the APEC international symposium on innovation and good practice for teaching and learning mathematics through lesson study Khon Kaen, Thailand. Miyakawa, T., & Winsløw, C. (2009). Didactical designs for students’ proportional reasoning: An “open approach” lesson and a “fundamental situation”. Educational Studies in Mathematics, 72(2), 199–218. http://dx.doi.org/10.1007/s10649-009-9198-y Nagasaki, E., & Becker, J. P. (1993). Classroom assessment in Japanese mathematics education. In N. Webb (Ed.), Assessment in the mathematics classroom (pp. 40–53). Reston, VA: NCTM. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110. Nohda, N. (1986). A study of “open-approach” method in school mathematics. Tsukuba Journal of Educational Study in Mathematics, 5, 119–131. Nohda, N. (1995). Teaching and evaluating using “open-ended problems” in the classroom. Zentralblatt für Didaktik der Mathematik, 27(2), 57–61. Nohda, N. (2000). A study of “open-approach” method in school mathematics teaching. In Proceedings of the ICME-9 Conference Makuhari, Japan. Sawada, D. (1997). NCTM’s standards in Japanese elementary schools. Teaching Children Mathematics, 4(1), 20–23. Selden, A., & Selden, J. (1997). Preservice teachers’ conceptions of mathematics and how to teach I. The Mathematical Association of America. Retrieved from. http://www.maa.org/t and l/sampler/rs 3.html Shimada, S. (Ed.). (1977). On lessons using open-ended problems in mathematics teaching. Tokyo: Mizuumishobo. Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233–254. Stacey, K. (1995). The challenges of keeping open problem-solving open in school mathematics. Zentralblatt für Didaktik der Mathematik, 27(2), 62–77. Sullivan, P. (2009). Constraints and opportunities when using content-specific open-ended tasks. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (vol. 1). Palmerston North, NZ: MERGA. Toluk-Uc¸ar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175.

330

C.-Y. Lin et al. / Journal of Mathematical Behavior 32 (2013) 309–330

Tougaw, P. (1994). Study of the effect of using an “open approach” to teaching mathematics upon the mathematical problem solving behaviors of secondary school students. Yee, F. (2002). Using short open-ended mathematics questions to promote thinking and understanding. In P. Spagnolo, & A. Rogerson (Eds.), Proceedings of the 4th international conference on the humanistic renaissance in mathematics education Palermo, Italy, (Unpublished doctoral dissertation, Southern Illinois University, Carbondale, IL).