Amy Hackenberg

Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures. These students were challenged to represent multiplicative relationships between unknowns: They changed the given relationship, did not think of the relationship as multiplicative until after concerted work, and used numerical values in lieu of unknowns. Our account for these challenges is that students needed to simplify the involved units coordinations. Ultimately students abstracted the relationship as multiplicative, but the exact relationship was not certain or had to be constituted in activity. Implications for teaching are explored.

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The study included six students with each of three different multiplicative concepts, which are based on how students create and coordinate composite units (units of units). Students participated in two 45-min semi-structured interviews and completed a written fraction assessment. This paper reports on how 12 students operating with the second and third multiplicative concepts demonstrated distributive reasoning in equal sharing problems and in taking fractions of unknowns. Students operating with the second multiplicative concept who demonstrated distributive reasoning appeared to lack awareness of the results of their reasoning, while students operating with the third multiplicative concept demonstrated this awareness and the construction of more advanced distributive reasoning when they worked with unknowns. Implications for relationships between students’ fractional knowledge and algebraic reasoning are explored.

To understand relationships between students' fractional knowledge and algebraic reasoning in the domain of equation writing, an interview study was conducted with 12 secondary school students, 6 students operating with each of 2 different multiplicative concepts. These concepts are based on how students coordinate composite units. Students participated in two 45-minute interviews and completed a written fractions assessment. Students operating with the second multiplicative concept had not constructed fractional numbers, but students operating with the third multiplicative concept had; students operating with the second multiplicative concept represented multiplicatively related unknowns in qualitatively different ways than students operating with the third multiplicative concept. A facilitative link is proposed between the construction of fractional numbers and how students represent multiplicatively related unknowns.

This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. > Problem . The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. > Method . The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. > Results . The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. > Implications . This paper has the implication that radical constructivism has the potential to impact practice.

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.

To investigate relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, the first author conducted a case study of a seventh grade student, Willa. Willa’s fractional knowledge—specifically her reversible iterative fraction scheme and use of fractions as multipliers—influenced how she wrote equations to represent multiplicative relationships between two unknown quantities. The finding indicates that implicit use of powerful fractional knowledge can lead to more explicit use of structures and relationships in algebraic situations. Curricular and instructional implications are explored.

To understand relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of 3 different multiplicative concepts participated. This paper reports on the 6 students with the most basic multiplicative concept, who were also pre-fractional in that they had yet to construct the first genuine fraction scheme. These students' emerging iterating operations facilitated their algebraic activity, but the lack of a disembedding operation was a significant constraint in developing algebraic equations and expressions.

To understand relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The study targeted a balanced mix of students with 3 different multiplicative concepts, which are based on how students coordinate composite units (units of units). Students participated in two 45-minute semi-structured interviews and completed a written fractions assessment. This paper reports on how students with the second and third multiplicative concepts demonstrated the use of a distributive operation in fraction and algebraic problem solving. Fractional knowledge is regarded as important for algebraic reasoning (Kilpatrick & Izsak, 2008; National Mathematics Advisory Panel [NMAP], 2008; Wu, 2001), in part because such knowledge is a basis for typical algebra topics such as ratios and slopes of lines. More generally, fractional knowledge is important for learning algebra because it helps students advance their multiplicative reasoning (cf. Thompson & Saldanha, 2003). For example, generating strategies for multiplying fractions can help students develop a distributive operation (Hackenberg & Tillema, 2009): to determine 1/5 of ¾ of a yard a student can take 1/5 of each of the three one-fourths of the yard. So, 1/5 of ¾ is 1/5 of (¼ + ¼ + ¼). Thinking of 1/5 of ¾ in this way might be considered algebraic because it highlights how the distributive property emerges from reasoning and is powerful for solving problems. However, little research has focused on how students' ways of operating with fractions may influence their algebraic reasoning (Lamon, 2007). The purpose of this paper is to examine the distributive reasoning with fractions and unknowns of 18 middle and high school students who participated in a clinical interview study. The study was designed to investigate relationships between students' quantitative reasoning with fractions and their algebraic reasoning in the area of equation solving. In this paper we assess whether students with different multiplicative concepts demonstrated the use of a distributive operation in their quantitative reasoning with fractions and their algebraic reasoning. The research questions addressed in this paper are: 1) Do middle and high school students show evidence of a distributive operation in solving problems that involve sharing multiple units fairly? If so, how do they reason distributively? 2) Do middle and high school students show evidence of a distributive operation in solving problems involving unknowns and fractions? If so, how do they reason distributively? 3) What differences are there in the distributive reasoning of students with different whole number multiplicative concepts? A Quantitative and Operational Approach Quantitative Reasoning Following Thompson and colleagues (1993; Smith & Thompson, 2008), we conceive of quantity and quantitative reasoning as a basis for helping students build fractional knowledge and algebraic reasoning. A quantity is a property of one's concept of an object, and to conceive of a quantity requires a person to conceive of a measurement unit, of the property as subdivided into some number of these measurement units, and of a way to enumerate the number of these units to find a value of the quantity (cf. Thompson, 1993).

Developed from Noddings’s (2002) care theory, von Glasersfeld’s (1995) constructivism, and Ryan and Frederick’s (1997) notion of subjective vitality, a mathematical caring relation (MCR) is a quality of interaction between a student and a mathematics teacher that conjoins affective and cognitive realms in the process of aiming for mathematical learning. In this paper I examine the challenge of establishing an MCR with one mathematically talented 11-year-old student, Deborah, during an 8-month constructivist teaching experiment with two pairs of 11-year-old students, in which I (the author) was the teacher. Two characteristics of Deborah contributed to this challenge: her strong mathematical reasoning and her self-concept as a top mathematical knower. Two of my characteristics also contributed to the challenge: my request that Deborah engage in activity that was foreign to her, such as developing imagery for quantitative situations, and my assumption that Deborah’s strong reasoning would allow her to operate in the situations I posed to her. The lack of trust she felt at times toward me and the lack of openness I felt at times toward her impeded our establishment of an MCR. Findings include a way to understand this dynamic and dissolve it to make way for more productive interaction.

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008; Steffe, 2002). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thoughtthey had to carry out some activity, review its results, and then carry out more activity in order to solve the problems. The other two could foresee results of their schemes prior to implementing them; their schemes were anticipatory. One of these two also constructed reciprocal relationships, an advanced form of reversibility. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity, a particular whole number multiplicative concept. The study also reveals that even students with this multiplicative concept will be challenged to construct reciprocal relationships. Suggestions for further inquiry on student learning in this area, as well as implications for classroom practice and teacher preparation, are considered.

Directly or indirectly, The Fractions Project has launched several research programs in the area of students’ operational development. Research has not been restricted to fractions, but has branched out to proportional reasoning (e.g., Nabors 2003), multiplicative reasoning in general (e.g., Thompson and Saldanha 2003), and the development of early algebra concepts (e.g., Hackenberg accepted). This chapter summarizes current findings and future directions from the growing nexus of related articles and projects, which can be roughly divided into four categories. First, there is an abundance of research on students’ part-whole fraction schemes, much of which preceded The Fractions Project. The reorganization hypothesis contributes to such research by demonstrating how part-whole fraction schemes are based in part on students’ whole number concepts and operations. Second, several researchers have noted the limitations of part-whole conceptions and have advocated for greater curricular and instructional focus on more advanced conceptions of fractions (Mack 2001; Olive and Vomvoridi 2006; Saenz-Ludlow 1994; Streefland 1991). The Fractions Project has elucidated these limitations while articulating how advancement can be realized through the construction of key schemes and operations that transcend part-whole conceptions. In particular – and deserving of its own (third) category – research on fraction schemes has highlighted the necessity and power of the splitting operation in students’ development of the more advanced fraction schemes, such as the reversible partitive fraction scheme and the iterative fraction scheme.

In a small-scale, 8-month teaching experiment, the author aimed to establish and maintain mathematical caring relations (MCRs) (Hackenberg, 2005c) with 4 6th-grade students. From a teacher's perspective, establishing MCRs involves holding the work of orchestrating mathematical learning for students together with an orientation to monitor and respond to energetic fluctuations that may accompany student—teacher interactions. From a student's perspective, participating in an MCR involves some openness to the teacher's interventions in the student's mathematical activity and some willingness to pursue questions of interest. In this article, the author elucidates the nature of establishing MCRs with 2 of the 4 students in the study and examines what is mathematical about these caring relations. Analysis revealed that student—teacher interaction can be viewed as a linked chain of perturbations; in student—teacher interaction aimed toward the establishment of MCRs, the linked chain tends toward perturbations that are bearable (Tzur, 1995) for both students and teachers.

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.