Research

The focus of this article is the exploration of and an explanation of student researchers' affect and activity in an action research project. Using a hermeneutical theoretical framework we argue that the researcher group as a whole constructs a wave process and at the same time each individual researcher in the group creates a wave process that may be similar or different to the group. These processes shape each other, through phases of engagement and disengagement in the researcher cycle, and make the research experience richer. The article examines five separate researcher narratives, extracting excerpts, to show how these examples showcase this wave phenomenon. Ten figures are included. Two themes, activity and affect, are identified in the narrative excerpts provided, sub-categories such as roles on a team and context of research are explored in these themes. The importance of explicit discussion of researcher engagement and disengagement in wave cycles is discussed.

The goal of the Noticing Numeracy Now (N^3) research project is to determine the extent to which an innovative learning experience focused on the professional noticing of children's early numeracy thinking develops preservice teachers' capacities to attend, interpret, and respond appropriately to children's mathematical thinking. The N^3 project is being implemented at eight Kentucky public universities.

This study examines a state's large-scale professional development initiative focused on advancing teachers' mathematical knowledge for teaching early numeracy. Over three years, the teachers' demonstrated gains in their mathematical knowledge for teaching and reported significant positive changes in their mathematical beliefs and practices. Participating students demonstrated significant achievement gains on standardized assessments and many sustained mathematical proficiency through several grades. The purpose of the study was to identify the factors that contributed to the success of the professional development initiative. The findings of this study indicate the success of the initiative was based on the synergistic relationship of the conditions, culture, competencies and changes. This study contributes to the research literature of structuring effective development for the teaching and learning of mathematics.

The goal of this study is to develop the professional noticing abilities of preservice elementary teachers in the context of the Stages of Early Arithmetic Learning. In their mathematics methods course, the preservice elementary teachers participated in a researcher-developed multi-session module that progressively nests the three interrelated components of professional noticing – attending, interpreting, and deciding. A pre- and post-assessment was administered to measure their change in the three components of professional noticing. The preservice elementary teachers demonstrated significant growth in all three components.

This article describes the use of a case report, Multiplication As Original Sin (Corwin, The Journal of Mathematical Behavior, 1993), as an assignment in a mathematics course for preservice elementary teachers. In this case report, Corwin described her experience as a 6th grader when she revealed an invented algorithm. Preservice teachers were asked to write reflections and describe why Corwin's invented algorithm worked. The research purpose was: to learn about the preservice teachers' understanding of Corwin's invented multiplication algorithm (its validity); and, to identify thought-provoking issues raised by the preservice teachers. Rather than using mathematical properties to describe the validity of Corwin's invented algorithm, a majority of them relied on procedural and memorized explanations. About 31% of the preservice teachers demonstrated some degree of conceptual understanding of mathematical properties. Preservice teachers also made personal connections to the case report, described Corwin using superlative adjectives, and were critical of her teacher.

A group of mathematics educators set out to explore lesson study and fraction division. During the first implementation of the lesson, which we researched and created, we grappled with standard protocol for lesson study. What are the advantages and disadvantages of listening silently? When/why should teachers adhere strictly [to the lesson plan] or when/why should they allow for veering? If the lesson study group is silent and the teacher adheres to the plan, then opportunities for exploring other big mathematical ideas can potentially be lost in the implementation. The teacher's own inner voice [1] can be compromised. In what ways might lesson study coerce teachers to listen primarily to the voice of the lesson plan by assuming the voice of the discipline and the voices of the students were "built in" prior to the lesson implementation? In this article we describe, through narrative and transcripts, when/why we ultimately chose to deal with these questions.

The influential Common Core State Standards for Mathematics (CCSSM) expect students to start statistics learning during middle grades. Thus teacher education and professional development programs are advised to help preservice and in-service teachers increase their knowledge and confidence to teach statistics. Although existing self-efficacy instruments used in statistics education focus on students, the Self-Efficacy to Teach Statistics (SETS) instrument measures a teacher?s efficacy to teach key CCSSM statistical topics. Using the results from a sample of n = 309 participants enrolled in a mathematics education or introductory statistics course, SETS scores were validated for use with middle grades preservice teachers to differentiate levels of self-efficacy to teach statistics. Confirmatory factor analysis using the Multidimensional Random Coefficient Multinomial Logit Model supports the use of two dimensions, which exhibit adequate reliabilities and correspond to the first two levels of the Guidelines for Assessment and Instruction in Statistics Education adopted by the American Statistical Association. Item and rating scale analyses indicate that the items and the six-category scale perform as intended. These indicators suggest that the SETS instrument may be appropriate for measuring preservice teacher levels of self-efficacy to teach statistics.

There is a growing body of research indicating that flipped classrooms are associated with increased levels of student engagement, as compared to engagement in "traditional" settings. Much of this research, however, occurs in post-secondary classrooms and is based upon self-reported engagement data. This study seeks to extend existing flipped classroom research by assessing behavioral engagement in flipped and non-flipped settings using observational instruments in three pre-college settings. Contrary to widely-reported results, this study found an increase in engagement in only one of the three observed settings. Analyses of the classroom contexts and teachers' actions in the three settings suggests that student engagement is not solely a function of instructional strategy (flipped versus non-flipped), but is also affected by student characteristics and teachers' skill and expectations.

APOS Theory is applied to study student understanding of directional derivatives of functions of two variables. A conjecture of mental constructions that students may do in order to come to understand the idea of a directional derivative is proposed and is tested by conducting semi-structured interviews with 26 students. The conjectured mental construction of directional derivative is largely based on the notion of slope. The interviews explored the specific conjectured constructions that student were able to do, the ones they had difficulty doing, as well as unexpected mental constructions that students seemed to do. The results of the empirical study suggest specific mental constructions that play a key role in the development of student understanding, common student difficulties typically overlooked in instruction, and ways to improve student understanding of this multivariable calculus topic. A refined version of the genetic decomposition for this concept is presented.

Tracing the path from a numerical Riemann sum approximating the area under a curve to a definite integral representing the precise area in various texts and online presentations, we found 3 semiotic registers that are used: the geometric register, the numerical register, and the symbolic register. The symbolic register had 3 representations: an expanded sum, a sum in sigma notation, and the definite integral. Reviewing the same texts, we found that in the presentation of double and triple integrals, not a single textbook continues to present the numerical register and the expanded sum representation of the symbolic register. They are implied and the expectation appears to be that students no longer need them. The omission of these representations is quite ubiquitous and correspondingly affects millions of students. Materials that present the missing numerical register representation and the expanded sum representation of the symbolic register throughout topics associated with double and triple integrals have been created. This paper presents the results of a clinical study on the improvement of student comprehension of multivariable integral topics when these representations are included.

Tracing the path from a numerical Riemann sum approximating the area under a curve to a definite integral representing the precise area in various texts and online presentations, we found 3 semiotic registers that are used: the geometric register, the numerical register, and the symbolic register. The symbolic register had 3 representations: an expanded sum, a sum in sigma notation, and the definite integral. Reviewing the same texts, we found that in the presentation of double and triple integrals, not a single textbook continues to present the numerical register and the expanded sum representation of the symbolic register. They are implied and the expectation appears to be that students no longer need them. The omission of these representations is quite ubiquitous and correspondingly affects millions of students. Materials that present the missing numerical register representation and the expanded sum representation of the symbolic register throughout topics associated with double and triple integrals have been created. This paper presents the results of a clinical study on the improvement of student comprehension of multivariable integral topics when these representations are included.

A test project at the University of Puerto Rico in Mayagüez used GeoGebra applets to promote the concept of multirepresentational fluency among high school mathematics preservice teachers. For this study, this fluency was defined as simultaneous awareness of all representations associated with a mathematical concept, as measured by the ability to pass seamlessly among verbal, geometric, symbolic, and numerical representations of the same mathematical object. The preservice teachers in this study attended a seminar where they were introduced to the underlying concepts and the pedagogical advantages of multirepresentational fluency. For select topics, this idea was reinforced with interactive GeoGebra applets that allowed preservice teachers to alter a parameter and simultaneously view how it changes all four associated representations simultaneously. A qualitative study found that this approach appeared to (a) promote the use of multirepresentational fluency in problem solving approaches used among preservice teachers, (b) change preservice teachers' perceptions of what it means for a student to understand a concept, and (c) change the nature of evaluations that preservice teachers felt were appropriate for high school students.

Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit discussions of these representations are not necessary. For example, in the presentation of partial derivatives, textbook presentations assume that students will understand the slope in a specified direction in without it ever being explicitly presented. Our preliminary interviews indicated that such assumptions may be erroneous, so we created and tested materials that explicitly use representations commonly omitted in textbook presentations of the differential and integral calculus of functions of two and three variables. This paper discusses the classroom activities that were created to include these missing representations, as well as the results when they were implemented in classroom instruction.

In two dimensions (2D), representations associated with slopes are seen in numerous forms before representations associated with derivatives are presented. These include the slope between two points and the constant slope of a linear function of a single variable. In almost all multivariable calculus textbooks, however, the first discussion of slopes in three dimensions (3D) is seen with the introduction of partial derivatives. The nature of the discussions indicates that authors seem to assume that students are able to naturally extend the concept of a 2D slope to 3D and correspondingly it is not necessary to explicitly present slopes in 3D. This article presents results comparing students that do not explicitly discuss slopes in 3D with students that explicitly discuss slopes in 3D as a precursor to discussing derivatives in 3D. The results indicate that students may, in fact, have significant difficulty extending the concept of a 2D slope to a 3D slope. And that the explicit presentation of slopes in 3D as a precursor to the presentation of derivatives in 3D may significantly improve student comprehension of topics of differentiation in multivariable calculus.

The University of Puerto Rico in Mayaguez (UPRM) has found that there are disadvantages to a semester long remedial mathematics course that is administered during the freshmen year to students with mathematics deficiencies in STEM (Science, Technology, Engineering and Math) programs. Correspondingly, the UPRM designed and implemented an Accelerated, Online, Adaptive Approach to Developmental Mathematics that was administered the summer before students in STEM fields entered the UPRM. This article discusses the theoretical factors in the design of the online course and presents a study of the effectiveness of these two approaches based on (i) the success rate of students being able to enter Precalculus and (ii) the success of students from the two programs upon entering Precalculus.

This article reviews the history — through different political periods — and current status of mathematics education in Puerto Rico. It chronicles the struggles that have informed educational policies in the forging of a national identity in the field — policies that will hopefully foster the development of a society that values mathematical ideas.

The goal of this study is to better understand preservice elementary teachers' changes in attitudes towards mathematics in connection with their participation in a module aimed at developing professional noticing capacities. This module, typically implemented in the mathematics methods course, involves practice with the three interrelated components of professional noticing – attending, interpreting, and deciding. Pre- and post-assessments of participants' mathematical attitudes and professional noticing capacities were administered to measure change in these areas. Participants demonstrated significant growth in their professional noticing capabilities and mostly positive attitudinal change; however, there was no significant correlation between the changes on the respective measures.

The goal of this study is to develop the professional noticing abilities of pre-service elementary teachers in the context of the Stages of Early Arithmetic Learning. In their mathematics methods course, ninety-four pre-service elementary teachers from three institutions participated in a researcher-developed five-session module that progressively nests the three interrelated components of professional noticing – attending, interpreting, and deciding. The module embeds video excerpts of diagnostic interviews of children doing mathematics (representations of practice) to prepare the pre-service teachers for similar work. The module culminates with pre-service teachers implementing similar diagnostic interviews (approximations of practice) to gain experience in the three component skills of professional noticing. A pre- and post-assessment was administered to measure pre-service teachers' change in the three components. A Wilcoxon Signed Ranks test was conducted and found the pre-service elementary teachers demonstrated significant growth in all three components. Selected pre-service elementary teacher responses on the pre- and post-assessment are provided to illustrate sample growth in the pre-service teachers' abilities to professionally notice. These results, the first in an ongoing study, indicate the potential that pre-service teachers can develop professional noticing skills through this module. Continued data collection and analysis from the ongoing study by these authors and future, longer-term emphasis on professional noticing for pre-service teachers should be studied.

The chapters in this volume of Studies in Graduate and Professional Student Development represent current research on mathematics teaching assistants (MTAs) in the United States. The papers collected in this volume represent a variety of lenses focused on this area of inquiry, including the professional lives and needs of MTAs as well as challenges they face as they develop into teachers of college mathematics.

Not long ago nearly all materials and programs to help mathematics graduate students learn to teach were the product of the collective wisdom of experienced teachers of college matheamtics. Although these products were often used to help graduate students learn about teaching-related issues, little research existed to inform the design of goals and approaches used in the products. This situation is beginning to change as the mathematics education research community works to amass information about graduate students' characteristics, experiences, and needs for professional development. This growing area of research borrows heavily from the more established base of research on K-12 teachers and their preparation. This chapter in the edited volume provides a tour of how this research area developed and the discipline-specific context in which that development occurred.

Teacher noticing and the related construct of professional noticing of children's mathematical thinking have proven to be fertile ground for education researchers. Professional noticing is a framework for a teaching practice consisting of three component parts: attending, interpreting, and deciding. The current study investigates the conceptions and enactment of professional noticing of 24 elementary and middle grades teachers participating in professional learning programs that incorporated professional noticing. These teachers demonstrated a wide range of interpretations of professional noticing which varied in consistency with respect to the literature in this area. This diversity of conceptions is seen as a consequence of teachers having different definitions and scopes of application for professional noticing. This study adds to current discussions about the meaning and role of professional noticing by considering the perspective of practitioners, a group whose input is often secondary to education researchers but whose conceptions and enactment of such noticing is critical for student success.

Unlike a child's observable, physical interactions with mathematical tools (e.g. physically touching blocks in order to count them), the subtle manifestations of imagery construction can be considerably more challenging to identify and act upon. Although there have been substantive examinations of mental imagery in a variety of mathematical contexts (i.e., spatial patterns, geometric rotation, etc.) there is a paucity of study regarding the nature of mathematical imagery with respect to initial counting acts. Towards that end, we conducted clinical interviews and longitudinal teaching experiments to ascertain the salient features of early quantitative mental imagery. Our findings indicate that children construct imagined units that are variably connected to the mathematical tool of the moment. Moreover, while this variability appears congruent with existing mathematical progressions, attending to nuances in children's mental imagery provides a platform for more refined instructional design. Indeed, identification of and attention to the child's quantitative imagery in whatever form it may take is essential to maximizing mathematical experiences.

This article aims to use the Stages of Early Arithmetic Learning to help readers develop an appreciation for the complex nature of counting and adding skills. Additionally, classroom activities are provided that are suitable to teaching students at varying stages of this early numeracy progression.

In this article, we describe a lesson study process aimed at examining and refining division of fractions experiences with pre-service teachers. Through a deliberate, research-based design process, the authors constructed a lesson focused on explicating the nature and mechanics of the traditional fraction division algorithm; however, implementation revealed unexpected yet powerful mathematical experiences that existed aside from the primary lesson goals. Specifically, pre-service teachers experienced significant disequilibrium regarding the shifting nature of "the whole" when working on a particular sub-set of lesson tasks. The authors describe their design process, implementation, and present several conclusions gleaned from this experience.

This chapter reports on an NSF-funded collaboration between faculty from mathematics, meteorology, and psychology to write and test a set of classroom voting questions for introductory statistics. Each of the four authors presents an example lesson plan with voting questions. Their plans include the following topics: "Box and Whiskers Plots," "Hypothesis Testing," "Expected Values," and "Methods for Reporting Statistical Results."

There is a nearly universal awareness among educators regarding the importance of assessment. Testing data drives educational policy, revenue streams, and curriculum development. Ultimately, though, assessment must focus on the determination of student learning needs. The purpose of this document is to clearly describe three distinct types of assessment that one finds in mathematics education, and emphasize the importance of dynamic features (dynamism) in the diagnostic assessment process.

This editorial describes the different interpretations of mathematical fluency and argues in favor of meaning-driven approaches to mathematics teaching and learning.

Thoughtful implementation of the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) presents an opportunity for increased emphasis on the development of mathematical understanding among students. Granted, ascertaining the mathematical understanding of an individual student is highly complex work and often exceedingly difficult. Although textbooks may provide practitioners with considerable overarching instructional guidance, to complete the picture, mathematics teachers must often focus on individual children. In some instances, they might consider verbal explanations or work samples to gain insight into one's thinking; however, quite often these avenues do not provide a complete or accurate portrayal of a student's understanding. Indeed, a student may unwittingly offer an explanation that differs from her actual strategy (CCSSI 2010). Similarly, work samples may feature deceptive or insufficient details to truly gauge the student's thinking. In these instances, a systematic approach to appraise the mathematical moment is required to fully appreciate the student's true understanding and then respond with effective instructional tactics. This article begins by describing the three interrelated phases of the "Professional Noticing of Children's Mathematical Thinking" (Jacobs, Lamb, and Philipp 2010) framework for teachers to better understand and act on their students' mathematical conceptions and practices. The second half of the article addresses how to put the framework into practice through the progression of the three phases (Attending, Interpreting, and Deciding).

Moving children's mathematical thinking beyond reliance on physical materials is important and challenging work. The authors examine the different manner in which students interact with physical materials to negotiate arithmetic tasks, and provide descriptions of diagnostic and instructional tools that may be used to help children develop quantitative mental imagery.

To maximize learning of whole-number and arithmetic operations, it is important to give instructional attention to several different aspects of number. This article uses transcripts drawn from videotaped records of one-on-one, intervention sessions with two low-attaining first-graders, to contrast the strengths and weaknesses of the students with respect to different aspects of number. The article includes a discussion of the instructional implications of taking account of the different aspects.

Relational thinking is a necessary and vital component for true conceptual understanding of mathematical thinking and application. Teachers and administrators who realize and nurture this pedagogical component through the study of vertical knowledge, collaboration, and ongoing professional development are solidifying a strong foundational mathematical journey for their students.

Use of clinical interview is becoming a significant aspect of many numeracy projects. It is important for teachers to identify children's understanding and misconceptions at all stages in the learning cycle, and the clinical interview appears to be an appropriate technique for gathering information on children's thinking. This paper explores the development of a conceptual framework used as a basis for an investigation into cognitive aspects associated with mental computation. Examples of tasks from clinical interviews which were based on this conceptual framework are described.