Browsing by Author "Tillema, Erik"
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Item Combinatorics Problems: a Constructive Resource for Finding Volumes of Fractional Dimension?(2019) Tillema, Erik; Liu, Jinqing; Bharaj, Pavneet Kaur; School of EducationFractions and volume are two challenging domains, which initially come together in the Common Core State Standards in Mathematics (CCSS-M) in the 7th grade where students learn about volumes of rectangular prisms with fractional dimensions. However, relatively little research has been conducted on how students’ reason about these volumes. To address this dearth of research, we designed a teaching experiment based on a central conjecture that combinatorics problems could be a constructive resource in the development of volumes with fractional dimension. In this paper, we demonstrate the central conjecture by providing two cases of how pre-service secondary teachers (PSSTs) reasoned with volumes of fractional dimension. A contribution of this study is that it offers an expansion and novel combination of a number of empirically grounded theoretical constructs.Item Generalization across Domains: The Relating-Forming-Extending Generalization Framework(2017-10) Ellis, Amy; Tillema, Erik; Lockwood, Elise; Moore, KevinGeneralization is a critical aspect of doing mathematics, with policy makers recommending that it be a central component of mathematics instruction at all levels. This recommendation poses serious challenges, however, given researchers consistently identifying students' difficulties in creating and expressing normative mathematical generalizations. We address these challenges by introducing a comprehensive framework characterizing students' generalizing, the Relating-Forming-Extending framework. Based on individual interviews with 90 students, we identify three major forms of generalizing and address relationships between forms of abstraction and forms of generalization. This paper presents the generalization framework and discusses the ways in which different forms of generalizing can play out in activity. [For complete proceedings, see ED581294.]Item Helping Students Explore the Cartesian Coordinate System(2017-07-01) Tillema, Erik; Gatza, AndrewThis paper explores a problem-based approach to developing the Cartesian coordinate system as a set of whole number, integer, and rational number ordered pairs. We share our approach, discuss student work, and outline a sequence of problems and key conversations for classroom discussion that we have used with this approach.Item Not Just Mathematics, "Just' Mathematics: Investigating Mathematical Learning and Critical Race Consciousness(2021-07) Gatza, Andrew Martin; Tillema, Erik; Morton, Crystal; Willey, Craig; Cross Francis, DionneThis study is situated at the confluence of three calls for research within mathematics education: 1) work using novel approaches for studying students’ understanding of nonlinear meanings of multiplication; 2) work using discrete mathematics to explore social issues related to equity; and 3) work at the intersection of mathematical learning and critical race consciousness—specifically, social justice mathematics initiatives that explicitly address racism and the learners’ perspectives. The design research methodology of the study with 8th grade students provides practical curricular and pedagogical steps for doing work at the intersection of mathematical learning and race and racism; offers domain-specific learning insights; and merges theory and practice in conceptualizing the multiple complexities of learning and development in situ to create new possibilities for a more just mathematics education. Findings from this study offer insights at the intersection of the evolution of students’ establishment of nonlinear meanings of multiplication and critical race consciousness development. Specifically, this study identifies two schemes that students use to establish a nonlinear meaning of multiplication (SARC Scheme and RA Scheme), illustrates students’ growing racism awareness, and highlights how these initiatives can be mutually supportive in helping to normalize conversations about race and racism.Item Opportunities for Generalizing Within Pre-Service Secondary Teachers’ Symbolization of Combinatorial Tasks(2019) Burch, Lori; Pinheiro, Weverton Ataide; Tillema, Erik; School of EducationItem The Processes and Products of Students' Generalizing Activity(2017-10) Tillema, Erik; Gatza, AndrewGeneralization has been a major focus of curriculum standards and research efforts in mathematics education. While researchers have documented many productive contexts for generalizing and the generalizations students make, less attention has been given to the processes of generalizing. Moreover, there has been less work done with high school students in advanced mathematical contexts. To address these issues we use a model of learning that enables us to make explicit the processes of generalizing. We exemplify this model of learning in the context of an interview study with high school students working on cubic relationships.Item Three Facets of Equity in Steffe's Research Programs(2017-10) Tillema, Erik; Hackenberg, AmyThe [National Council of Teachers of Mathematics] 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 facets of equity. We also suggest opportunities for researchers working within this framework to deepen their commitments to issues of equity. [For complete proceedings, see ED581294.]Item Units Coordination, Combinatorial Reasoning, and the Multiplication Principle: The Case of Ashley, an Advanced Stage 2 College Student(Taylor & Francis, 2024) Tillema, Erik; Antonides, Joseph; School of EducationThe multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research literature, however, has not captured, generally, how students establish these unit relationships. We provide a second order model of an advanced stage 2 college student, Ashley, who had no prior combinatorics instruction, as she engaged in solving combinatorics problems that we considered to involve the MP. Our findings suggest that Ashley began by interpreting combinatorics problems using her whole number iterative units coordination scheme. Through engagement with the teacher-researcher, Ashley constructed combinatorial composites using a pairing operation, units coordination, and units simplification. We also found that Ashley was able to create a three-level-of-unit structure in activity, and to use notation that she produced to re-instantiate the reasoning that produced this unit structure. Doing so provides novel insights into how advanced stage 2 students, especially those at the college level, can use notation to manage complex unit relationships.Item Urban Sixth Graders Reason about Combinatorics Problems(Office of the Vice Chancellor for Research, 2011-04-08) Tillema, Erik; Tan, Paul; Mockler, Samantha15 sixth grade students at three different developmental levels solved combinatorics problems as a basis for reasoning about multi-digit multiplication. Each student was interviewed three times. The first interview was an un-recorded selection interview, which was used to identify the student’s developmental level. The second and third interviews were video recorded, and involved students in solving combinatorics problems. The results from the study include: (1) models of how students at different developmental levels solved the combinatorics problems; and (2) a framework for integrating research on mathematical cognition and urban education.