Amy Hackenberg Research Collection

Permanent link for this collectionhttps://hdl.handle.net/2022/26748

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Teaching practices for differentiating mathematics instruction for middle school students
(Mathematical Thinking and Learning, 2020-02-27) Hackenberg, Amy; Creager, Mark; Eker, Ayfer
Three iterative, 18-episode design experiments were conducted after school with groups of 6–9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI emerged: proactively tailoring instruction to students’ mathematical thinking while developing a cohesive classroom community. Analysis of 10 episodes across experiments yielded five teaching practices that facilitated DI: using research-based knowledge of students’ mathematical thinking, providing purposeful choices and different pathways, inquiring responsively during group work, attending to small group functioning, and conducting whole class discussions across different thinkers. The latter three practices, at times, impeded DI. This study is a case of using second-order models of students’ mathematical thinking to differentiate instruction, and it reveals that inquiring into research-based knowledge and inquiring responsively into students’ thinking are at the heart of differentiating mathematics instruction.
Students' distributive reasoning with fractions and unknowns
(Educational Studies in Mathematics, 2016) Hackenberg, Amy; Lee, Mi Yeon
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.
Mathematical caring relations: A challenging case
(Mathematics Education Research Journal, 2010) Hackenberg, Amy
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.
The fractional knowledge and algebraic reasoning of students with the first multiplicative concept
(The Journal of Mathematical Behavior, 2013) Hackenberg, Amy
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.
Students' Reasoning With Reversible Multiplicative Relationships
(Cognition and Instruction, 2010) Hackenberg, Amy
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 thought—they 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.
"Approximate" multiplicative relationships between quantitative unknowns
(The Journal of Mathematical Behavior, 2017) Hackenberg, Amy; Jones, Robin; Eker, Ayfer; Creager, Mark
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.
Tiering Instruction for Seventh-Grade Students
(The Mathematics Teacher, 2020-02) Hackenberg, Amy; Jones, Robin; Borowski, Rebecca
Relationships Between Students' Fractional Knowledge and Equation Writing
(Journal for Research in Mathematics Education, 2015-03) Hackenberg, Amy; Lee, Mi Yeon
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.
Mathematical Caring Relations in Action
(Journal for Research in Mathematics Education, 2010-05) Hackenberg, Amy
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.
Making Quilts without Sewing: Investigating Planar Symmetries in Southern Quilts
(The Mathematics Teacher, 2005-11) Anthony, Holly; Hackenberg, Amy
In this article, we first give a brief introduction to symmetries of the plane. Then we describe our analyses of the different combinations of planar symmetries – the wallpaper patterns – displayed by seventeen Tennessee quilts made by Holly Anthony's grandmothers. The different possible combinations of symmetries that coexist in a planar pattern are called wallpaper patterns, where the word wallpaper is used to indicate the nature of a planar pattern continuing indefinitely, not to refer literally to wallpaper. Finally, we outline an activity for "making quilts without sewing" that enables high school students to develop their understanding of planar symmetries and wallpaper patterns. This activity allows students and their teachers to incorporate the culture and traditions of quilting into their study of geometry. Thus in our work with quilts, "appropriate consideration of symmetry provides insights into mathematics and into art and aesthetics”.
Units coordination and the construction of improper fractions: A revision of the splitting hypothesis
(The Journal of Mathematical Behavior, 2007) Hackenberg, Amy
This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth-grade students’ multiplicative structures and their construction of improper fractions. Students’ multiplicative structures are the units coordinations that they can take as given prior to activity—i.e., the units coordinations that they have interiorized. This research indicates that the construction of improper fractions requires having interiorized three levels of units. Students who have interiorized only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. These findings call for a revision in Steffe's hypothesis (Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20, 267–307) that upon the construction of the splitting operation, students’ fractional schemes can be regarded as essentially including improper fractions. While the splitting operation seems crucial in the construction of improper fractions, it is not necessarily accompanied by the interiorization of three levels of units.
TIERING INSTRUCTION ON SPEED FOR MIDDLE SCHOOL STUDENTS
(Proceedings of the Forty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2019-11) Aydeniz, Fetiye; Hackenberg, Amy; Matyska, Robert
A design experiment with 18 students in a regular seventh grade math class was conducted to investigate how to differentiate instruction for student’s diverse ways of thinking during a 26-day unit on proportional reasoning. The class included students operating with three different multiplicative concepts that have been found to influence rational number knowledge and algebraic reasoning. The researchers and classroom teacher tiered instruction during a 5-day segment of the unit in which students worked on problems involving speed. Students were grouped relatively homogenously by multiplicative concept and experienced different number choices. Students operating at each multiplicative concept demonstrated evidence of learning, but all did not learn the same thing. We view this study as a step in supporting equitable approaches to students’ diverse ways of thinking, an aspect of classroom diversity.
EXPLORING DIFFERENTIATION WITH MIDDLE SCHOOL TEACHERS
(Proceedings of the Thirty-ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2017-10) Aydeniz, Fetiye; Hackenberg, Amy; Jones, Robin
STUDENTS’ MEANINGS FOR EXTENSIVE QUANTITATIVE UNKNOWNS
(Proceedings of the Thirty-ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2017) Aydeniz, Fetiye; Borowski, Rebecca; Hackenberg, Amy; Jones, Robin
A series of three design experiments was conducted with middle school students to investigate relationships between students’ rational number knowledge and algebraic reasoning. After the first experiment a change was made in the investigation of students’ construction of extensive quantitative unknowns. Students were asked to represent in pictures and equations the values for an unknown height measured in two different, multiplicatively-related measurement units. The work of 13 students operating at two levels of multiplicative reasoning was analyzed to identify differences and similarities. Students operating at the lower level of reasoning required substantial support to construct unknowns with implicit quantitative relationships, while students operating at the higher level of reasoning constructed unknowns with explicitly embedded units.
THREE FACETS OF EQUITY IN STEFFE’S RESEARCH PROGRAMS
(Proceedings of the Thirty-ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2017-10) Hackenberg, Amy; Tillema, Erik
The NCTM research committee made a recent, urgent call for mathematics education researchers to “examine and deeply reflect on our research practices through an equity lens.” With this in mind, we use this paper to reflect on the ways in which Steffe’s work has contributed to three factors of equity. We also suggest opportunities for researchers working within this framework to deepen their commitments to issues of equity.
Musings on Three Epistemic Algebraic Students
(University of Wyoming Press, 2014) Hackenberg, Amy
An epistemic subject is “that which is common to all subjects at the same level of development, whose cognitive structures derive from the most general mechanisms of the co-ordination of actions” (Beth & Piaget, 1966, p. 308). Epistemic algebraic students are dynamic models of students’ ways and means of operating that are taken to characterize a stage in the development of students’ algebraic activity and that can be used to communicate purposefully with actual algebraic students. The purpose of this paper is to present the author’s working, and sometimes second-order, models of three epistemic algebraic students, and to generate comments and questions about learning trajectories for these students. These three epistemic students are distinct in that they are operating with three qualitatively different multiplicative concepts, which are based on how students create and coordinate units of units. The paper focuses on relationships between students’ fractional knowledge and algebraic reasoning in order to contribute to understanding how students’ arithmetical and quantitative reasoning can be a basis for algebraic thinking.
PRE-FRACTIONAL MIDDLE SCHOOL STUDENTS’ ALGEBRAIC REASONING
(Proceedings of the Thirty-fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2012-11) Hackenberg, Amy; Lee, Mi Yeon
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.
STUDENTS’ DISTRIBUTIVE REASONING WITH FRACTIONS AND UNKNOWNS
(Proceedings of the Thirty-third Annual Meeting of PME-NA, 2011-10) Hackenberg, Amy; Lee, Mi Yeon
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.
PARTICIPANT RESEARCH ESSAY FOR DIME RESEARCH TEAM
(University of Wyoming Press, 2011) Hackenberg, Amy
SIXTH GRADERS’ CONSTRUCTION OF QUANTITATIVE REASONING AS A FOUNDATION FOR ALGEBRAIC REASONING
(Proceedings of the Twenty-eighth Annual Meeting of PME-NA, 2006-11) Hackenberg, Amy
In a year-long constructivist teaching experiment with four 6th grade students, their quantitative reasoning with fractions was found to form an important basis for their construction of algebraic reasoning. Two of the four students constructed anticipatory schemes for solving problems that could be solved by an equation such as ax = b. In doing so, these students operated on the structure of their schemes. In solving similar problems, the other two students could not foresee the results of their schemes in thought—they had to carry out activity and then check afterwards to determine whether their activity had solved the problems. Operating on the structure of one’s schemes is argued to be fundamentally algebraic, and a hypothesis is proposed that algebraic reasoning can be constructed as a reorganization of quantitative operations students use to construct fractional schemes.

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