Publication Type: Short Research Paper
Abstract: This study reports on two sixth grade classes’ development of fractional understanding as a result of teaching a hands-on unit of fractions. The activities in this study addressed common misconceptions related to fractions and sought to encourage students to adapt and refine their understanding of fractions. Both qualitative and quantitative data will be presented.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: The NCTM Standards (1989, 2000) highlight the need to recognize mathematics as more than a collection of skills and concepts to be mastered. Not only should mathematics instruction focus on developing concepts and skills, it should be done in such a way that students see mathematics as useable. The value of knowledge “lies in the extent to which it is useful in the course of some purposeful activity” and instruction should “persistently emphasize ‘doing’ rather than ‘knowing’ that” (NCTM, 1989, p. 7). These documents point to the importance of building on students’ previous experiences and the need to apply prior knowledge to increasingly more difficult situations. Students should be “responsible for what they have learned and for using that knowledge to understand and make sense of new ideas” (NCTM, 2000, p. 64). It is this distinction between learning about something and learning to use that knowledge in new settings that this paper addresses.

As the title suggests, this theme is studied in the context of fractions. There is a large body of literature, both empirical and theoretical, that focuses on what is involved in learning fractions when fractions are the focus of instruction. However, there is little research that explores how students learn to use what they have learned about fractions outside instruction on fractions. In response, this research explores how middle school students learn to use fraction knowledge, the fraction concepts and skills studied in formal curriculum units, in mathematical instructional settings where fractions are not the main focus of study, but rather support the development of other mathematical content. The research reported here is part of a larger study of sixth- and seventh-grade students. The purpose of this paper is to describe the practices that a class of sixth-grade middle school students engaged in when using knowledge learned about fractions in two contexts: (1) area and perimeter and (2) decimal operations.

Theoretical Framework
This work draws upon the situated nature of learning and the notion of practice where mathematical knowledge is a means rather than an ends. This notion of practice and “learning to use” is informed by work in the field of literacy. Barton (1994) and Scribner and Cole (1981) offer a “practice account” of literacy where literacy is best understood as a set of social practices that people use in certain situations. Rather than studying the separate skills that underlie reading and writing, it involves studying the social practices associated with a particular symbol system. In this study, this can be thought of as a shift from studying the separate skills that underlie fractions (i.e., adding or equivalent fractions) toward understanding how students make use of these skills.

Methodology
When studying literacy, literacy events and literacy practices are the basic units of analysis (Barton, 1994). In order to understand the intent of the research questions and data collection I begin by defining what I refer to as fraction literacy events and fraction literacy practices. Fraction literacy events are situations where students have to use their fraction knowledge and fraction literacy practices are stable identifiable patterns of behavior that students make use of during fraction literacy events. It is important to clarify that I am not trying to identify broad classroom practices or mathematical practices such as questioning others or justification that are found across mathematical content areas. The focus of data analysis was to identify specific practices students engaged in as part of learning to use fraction knowledge.

The participants are a class of 23 sixth-grade students and their teacher. This middle school, located in a small middle-class mid-western community, uses the Connected Mathematics Project (CMP) curriculum. The design of this curriculum provides an already existing setting in which to collect data. Fraction concepts and procedures are formally taught in two sixth-grade units. CMP does not reteach fractions in seventh or eighth grade. Rather, students revisit fraction concepts and procedures in other contexts across sixth, seventh, and eighth grade. One goal of the curriculum is to continually build and connect the ideas of one unit to others so students have to use previously developed concepts and procedures in new settings. A second goal, one regarding skill development, suggests that skill is more than symbol manipulation. “Skill means that students can use the mathematical tools, resources, procedures, knowledge and ways of thinking developed over time to make sense of new situations they encounter” (Lappan & Phillips, 1998, p. 83). This notion of skill is aligned with my research questions and the literacy framework this study draws upon.

Preliminary fieldwork included observing and taking notes of small-group and large-group discussions during two units where fractions were the focus of instruction. In addition, I interacted with students during small-group work so they could become comfortable with my presence and to identify four focus students for closer study. For formal data collection, four focus students, two boys and two girls, were identified. These students were representative of the class in terms of ability and worked together in small-group interaction. Data collection during nine fraction literacy events included field notes, video-recordings of small-group interaction of focus students and whole-class conversations, and collection of focus students’ written work. Two audio-recorded interviews were done with each focus student. Students were asked to reflect on conversations that took place during class. The interviews provided a setting in which to follow up on and test hunches that were emerging from the video data.

Results
Across all nine events students engaged in the practice of determining appropriateness. This practice includes determining an appropriate quantity, an appropriate way to operate, or an appropriate form of representation. One case (others will be included in talk and paper) involved finding the width of a rectangular storm shelter with a floor area of 24 square meters and a length of 5 1/3 meters. A lengthy conversation took place where various decimals were offered to represent 5 1/3 when operating to find the missing width.
C: I did 5.3 ¥ 4.5 and got 24.3.
B: I did 5.3 ¥ 4.5 and got an even 24.
R: Well 5 1/3 is not equal to 5.3. It is 5.3 with a line over it. So let’s say that times 4.5 which equals 23.99999. It is pretty close to 24.
Later in the conversation a student shows that if you multiply the fractions 5 1/3 ¥ 4 1/2 the result is exactly 24.
Teacher: So that works. 5 1/3 rows of 4 1/2. But how did you get that?
C: Maybe like I did. [Earlier he offered that he did 24 ÷ 5.333]
Teacher: Okay, but you got 4.528. What would explain that?
C: I rounded off.
T: That’s the problem with sometimes switching to a decimal. If I don’t I can get the exact answer.
In the end, five approaches were rejected because using approximations for 5 1/3 did not lead to an exact area of 24. Nonetheless there are times, for example, when estimating, where using 5.3 to represent 5 1/3 is appropriate. There are many situations where using decimal form rather than fraction form is appropriate. However, in this situation switching to decimal form and operating is not appropriate when you want an exact measure. In this episode and others, students’ conversations revolved around questions of the following nature: Can I do that here? How would I use that idea here? Are these two forms equivalent here?

Conclusions
Across the events, the conversations students and teacher engaged in involved issues that are different from conversations that take place when learning about fractions. As students try to make connections between fractions and the contexts where they are used, they need support understanding when to use them and what form is appropriate. Results support the notion that a single concept does not develop in isolation but in relation to other concepts across various contexts, wordings and symbolisms (Vergnaud, 1988). It points to the need for teachers to see student reasoning in such situations as attempts to engage in sense-making.

Relationship to PME-NA Goals
This research grows out of the study of classroom practice. It seeks to answer questions regarding how students move toward proficient use through study of practices students engage in when learning to use fractions.

Key Words
fractions, middle school student practices

References
Barton, D. (1994). Literacy: An introduction to the ecology of written language. Malden,
MA: Blackwell Publishers Inc.

Lappan, G. & Phillips, E. (1998).Teaching and learning in the connected mathematics
project. In L. Leutzinger (Eds.) Mathematics in the middle (pp. 83-92). Reston,
VA: National Council of Teachers of Mathematics.

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

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

Scribner, S. & Cole, M. (1981). The psychology of literacy. Cambridge, MA: Harvard
University Press.

Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.) Number
concepts and operations in the middle grades. Reston, VA: National Council of
Teachers of Mathematics.

Publication Type: Short Research Paper
Abstract: We examine elementary school teachers’ prior understandings of concept of fractions within different contexts. We argue that teacher' understanding on fractions significantly improves when they are offered opportunities to investigate fractions in new context, specifically, in the situations when they can not rely on procedural approaches. This leads to cognitive conflict when approaching problems involving continuous quantities, and wholes consisting of several objects. These situations arise when problems related to Egyptian fraction are introduced. Pre-service teachers' investigations of these problems have lead to deeper conceptual understanding of fair share concept (in the new context), comparing fractions, relations between different wholes, fractions part-part-whole representations (one bigger fraction represented as sum of two smaller fractions), connection of different fraction representations with numerical measure (number of cuts), and on the different level it lead to understanding why Egyptians used specific unit fraction representation. Pre-service teachers (treatment group, 15, and control group, 60) participated in the study. Pre- and post-tests were administered before and after the learning cycle. Students were involved in mathematical project investigations in the context of word problems related to Egyptian fractions. At the end of the cycle pre-service teachers designed and implemented the lesson related to Egyptian fractions in elementary classrooms. The lessons captured through field notes were analyzed and compared. Pre-service teachers' reflections, learning journals, and pre- post- surveys were collected and analyzed. The treatment group was compared to similar groups of pre-service teachers from several previous semesters. The results indicated a statistically significant improvement in posttest scores (in the treatment group). Student interviews and surveys indicated that bringing new contexts in teaching and learning fractions (1) helped students to see the practical applications of fractions in the new light, (2) illustrated that addition of fractions can be illustrated without common denominator procedure, and (3) made study of mathematics more interesting and meaningful for pre-service teachers.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: Many students experience great difficulty when studying topics related to fractions, especially division of fractions. One explanation for this may be that learning how to divide fractions is often taught devoid of meaning. The lack of sense making in carrying out algorithms without making connections to concrete or other types of representations contributes to the belief that mathematics is more an arbitrary set of rules and procedures than a subject that can and should be understood. In this paper we explore the flexibility and durability of knowledge that students acquire when they study this topic in a way that encourages understanding.

INTRODUCTION AND THEORETICAL FRAMEWORK
In previous research (Bulgar, 2002; Bulgar, 2003a; Bulgar, 2003b; Bulgar, Schorr & Maher, 2002), we discussed the conceptual development of ideas relating to division of fractions amongst fourth grade students involved in a teaching experiment. Therein, it was also reported that when this teaching experiment was replicated as part of the regular teaching practice of the first author, similar outcomes were achieved. In this paper, we continue to examine the development of ideas relating to fractions amongst the latter group of students. More specifically, we report on the mathematical activity of the students during the following year as they explore ideas relating to division of fractions. This is done with a specific focus on flexibility in mathematical thinking.

Many students have experienced difficulty in solving problems involving fractions (cf. Tzur, 1999; Davis, Hunting, and, Pearn, 1993; Davis, Alston, and Maher, 1991; Steffe, von Glasersfeld, Richards and Cobb, 1983; Steffe, Cobb and von Glasersfeld, 1988). Other researchers have also documented how teachers can help in facilitating the development of ideas relating to fractions (c.f. Steencken, 2001; Steencken and Maher, (2002); Ma, 1999; Cobb, Boufi, McClain and Whitenack, 1997). In the original teaching experiment mentioned above, a fundamental premise was that in order to help students build a conceptual basis for considering these ideas, teachers must relinquish instructional approaches which emphasize rote memorization and the execution of rules and procedures, moving toward instructional practices which are more student centered and provide the opportunity to build concepts and ideas as students are engaged in mathematical activities that promote understanding (Davis & Maher, 1997; Maher, 1998; Cobb, Wood, Yackel & McNeal,1993; NCTM, 2000; Klein and Tirosh, 2000; Schorr, 2000; Schorr and Lesh, 2003). Kamii and Dominick (1997) compared students who were taught algorithmically with a group that was not, finding that the non-algorithmic group was more successful and that even when they made errors, the errors resulted in more reasonable responses.

Discussions of flexible thought exist both in cognitive psychology and in mathematics education. Mathematics education studies generally consider flexibility to be the capacity to exhibit various invented strategies or a large repertoire of problem-solving strategies (cf. Heirdsfield, 2002). In elaborated definitions, other researchers, such as Spiro & Jehng (1990) characterize cognitive flexibility as the ability to spontaneously restructure one’s knowledge, in adaptive response to radically changing situational demands. Krutetskii (1969), a mathematics educator, characterizes flexible thinking as reversibility of thought. Gray and Tall (1994) characterize flexible thinking in terms of an ability to move between interpreting notation as an instruction to do something (procedural use of notation) and as an object to think with and about (conceptual use of notation). Warner, Alcock, Coppolo & Davis (2003) developed a list of characteristic behaviors that indicate mathematical flexible thought. They identify students who think flexibly as those who exhibit some or all of the following behaviors:
· interpret their own or someone else’s idea (e.g. through questioning it and thus showing it to be valid or invalid; through using, reorganizing or building on it);
· use the same idea in different contexts;
· sensibly raise hypothetical problem situations based on an existing problem: creating “What if…?” scenarios;
· use multiple representations for the same idea; and,
· link representations for the same idea.

Given that the subjects of this study constructed powerful ideas related to division of fractions, we conjectured that this knowledge would be flexible and therefore extend to other problem-tasks – and in subsequent school years. We further conjectured that the flexible knowledge would be robust, being able to be recalled after time had elapsed, so that previously built ideas could be retrieved, modified, and extended to formulate new knowledge and understanding.

METHODS AND PROCEDURES
Background, setting and subjects
This study took place during the 2001-2002 school year, when the subjects, fourteen girls, were in 6th grade. Thirteen of these students had been taught mathematics by the same teacher, the first author of this paper, during 5th grade. This small parochial school in New Jersey attracts children from several surrounding communities. Conditions, established during the 5th grade, were set up to create a classroom community in which student inquiry and discovery were of paramount importance. In this academically heterogeneous class, discourse was of supreme importance. The classroom community was one in which students’ ideas were always respected. Students were questioned and encouraged to explain their solutions, developing their own sense of accuracy. Alternate strategies were encouraged, shared and discussed, as students were invited to discuss their thinking and to submit ideas in writing. Students were not taught algorithms. When they recognized patterns and could justify that these patterns were valid, they created generalizations, which they could apply to future problems. Questions were used to elicit explanations, to guide students towards persuasive justifications of their solutions and to redirect them when they were engaged in faulty reasoning. Justification of solutions became a part of the classroom culture.

Data
The data examined here consist of artifacts of actual student work, which were collected over the course of approximately six weeks. Written notes from the teacher were attached to some of the work, usually in the form of questions and answers to these questions also appear in the students’ writing.

Tasks
The primary task studied, “Tuna Sandwiches”, was created by the first author to be isomorphic with the problem done the previous year called “Holiday Bows” , which introduces division of a natural number by a common fraction. The problem follows.

Mr. Tastee’s restaurant serves 4 different kinds of sandwiches. A junior sandwich contains ¼ lb of tuna; a regular sandwich contains 1/3 lb of tuna; a large sandwich contains ½ lb of tuna and a hero sandwich contains 2/3 lb of tuna. Tuna comes in cans that are 1lb, 2 lb, 3lb and 5 lb.

How many of each type of sandwich can you make from each size can? Find a clear way to record your information. You will need to write a letter to the restaurant owner, Mr. Tastee, and give him your findings.

One of the goals in creating the “Tuna Sandwiches” problem was for it to lend itself to be represented by an area model rather than a linear model, as was the case with “Holiday Bows”. Another was to see if the ideas established during the previous year could be applied to this new context.

When students indicated that they were proficient in using symbolic notation associated with this problem, they were given the problem 2 ÷ 3/4 and ultimately 5/8 ÷ 2 1/2, which they had to solve and provide justification for the solution.

RESULTS AND DISCUSSION
In the full paper, we will share the work of several students, showing that most recognized that the tuna problem was similar in structure (although the two embodied different representations) to the Holiday Bow problem that they had worked on the previous year. Further, they spontaneously began using symbolic notation to represent their ideas. When posed with the numerical problems, several students, whose work we will indicate, justified their solutions using representations, such as drawings of Cuisenaire ® Rods and story problems to facilitate their explanations.

CONCLUSIONS
Students were able to retrieve ideas they had built during the previous school year, and the ideas were used and extended appropriately. In the full paper, we will illustrate how students demonstrated flexible thought in the way they demonstrated their grasp of division of fractions and extended their understanding to more complex division of fractions problems. We will link what is observed in their work to the characteristics of flexible thought that were identified above.

Publication Type: Short Research Paper
Abstract: This study explored elementary teachers’ transformations of textbooks into practice in teaching fractions in the context of recent efforts to reform mathematics education. This study focused on the first chain of teachers’ transformations for classroom instructions—that of selecting activities with their textbooks. Factors that support and constrain teachers’ textbook transformation approaches are also explored. This study revealed three activity selection patterns in terms of cognitive demand. Three influential factors, textbook cognitive demands, curriculum policy, teachers’ view on textbooks were identified.