Research and Publications

Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity, and it varies by the situation at hand. This book offers educators the inspiration to develop a deeper understanding of procedural fluency, along with a plethora of pragmatic tools for shifting classrooms toward a fluency approach.

Fluency in mathematics is more than deptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity. The Figuring Out Fluency Series offers educators a deeper understanding of procedural fluency, along with pragmatic tools for shifting a classroom toward a fluency approach. This book in the series addresses instruction and practice focused on addition and subtraction of whole numbers.

Fluency in mathematics is more than deptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity. The Figuring Out Fluency Series offers educators a deeper understanding of procedural fluency, along with pragmatic tools for shifting a classroom toward a fluency approach. This book in the series addresses instruction and practice focused on multiplication and division of whole numbers.

Fluency in mathematics is more than deptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity. The Figuring Out Fluency Series offers educators a deeper understanding of procedural fluency, along with pragmatic tools for shifting a classroom toward a fluency approach. This book in the series addresses instruction and practice focused on addition and subtraction of fractions and decimals.

Fluency in mathematics is more than deptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity. The Figuring Out Fluency Series offers educators a deeper understanding of procedural fluency, along with pragmatic tools for shifting a classroom toward a fluency approach. This book in the series addresses instruction and practice focused on multiplication and division of fractions and decimals.

Mastering the basic math facts for addition, subtraction, multiplication and division is a cornerstone of the Kentucky Academic Standards. This book offers everything a teacher needs to teach, assess, and communicate with parents about basic math fact instruction.

This book provides K-12 math coaches and teacher leaders a condensed explanation of critical mathematics coaching and teaching actions along with the tools to needed to do the work of navigating a coaching conversation and planning, facilitating and evaluating professional development.

The stories in this book represent the efforts along the continuum of change including work that is just starting, to initiatives in progress, to examples of advanced implementation. Each story shares an approach addressing one or more of the four key recommendations from Catalyzing Change:

• Broaden the purposes of learning mathematics
• Create equitable structures in mathematics
• Implement equitable instruction
• Develop deep mathematical understanding

These stories share efforts at the district and state levels as well as within schools and highlight the challenges and successes of implementing equitable teaching practices in classrooms everywhere.

This book identifies and addresses critical challenges in high school mathematics to ensure that each and every student has the mathematical experiences necessary for his or her future personal and professional success. These challenges include:

• explicitly broadening the purposes for teaching high school mathematics beyond a focus on college and career readiness;
• dismantling structural obstacles that stand in the way of mathematics working for each and every student;
• implementing equitable instructional practices;
• identifying Essential Concepts that all high school students should learn and understand at a deep level; and
• organizing the high school curriculum around these Essential Concepts in order to support students' future personal and professional goals.

The stories in this book represent the efforts along the continuum of change including work that is just starting, to initiatives in progress, to examples of advanced implementation. Each story shares an approach addressing one or more of the four key recommendations from Catalyzing Change:

• Broaden the purposes of learning mathematics
• Create equitable structures in mathematics
• Implement equitable instruction
• Develop deep mathematical understanding

These stories share efforts at the district and state levels as well as within schools and highlight the challenges and successes of implementing equitable teaching practices in classrooms everywhere.

This article presents a large-scale longitudinal study of hundreds of students across the state of Kentucky that participated in a dual-focus mathematics intervention initiative when they were in the third grade. Rather than an exclusive focus on intervention, this initiative focused on both (i) high quality pull-out intervention and (ii) coherence between pull-out intervention and classroom instruction. The study found that over half of the third grade intervention students that participated in this initiative were classified as “novice” (the lowest possible performance category) on state standardized mathematics assessments at the end of the third grade. However, over the course of the following four years, the novice reduction rate of these students was significantly (p < .01) greater than other novices in Kentucky that did not participate in the initiative. These findings indicate that when implementing intervention initiatives to help students that are struggling with mathematics, it may be important to establish coherence between pull-out intervention and classroom instruction. The long term impact of this approach among traditionally underrepresented minorities suggest that this publication may provide insight into important equity issues where long-term analyses may sometimes be needed to capture the full impact of intervention initiatives.

While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three elementary schools in a school district implements mandatory automaticity drills for 10 minutes each day while the remaining two elementary schools, with the same curriculum and very similar demographics, do not. This study looks at (a) the impact that schoolwide implementation of automaticity drills has on schoolwide computational math skills as measured by the ITBS and (b) the relationship between automaticity and conceptual understanding as measured by statewide standardized testing. The results suggest that while there may be an association between automaticity and higher performance on standardized tests, caution should be taken before assuming there are benefits to promoting automaticity drills. These results are consistent with those that support a process-driven approach to automaticity based on familiarity with properties and strategies associated with the base-ten number system; they are not consistent with those that support an answer-driven approach to automaticity based on memorization of answers.

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
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Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
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Middle and high school students are part of Generation Z (birth years 1997–2012), a group that is emerging as very different from their parents and teachers. This article considers ways that generational research about Generation Z learners and NCTM's Effective Mathematics Teaching Practices can be used to inform and innovate practice in the mathematics classroom. Research suggests that while Generation Z learners are digitally engaged, they often display a lack of tech savvy. Generation Z students thrive on personalization and are often uncomfortable with collaborative learning. And the social movements of the early 2000s have shaped their world view. The authors provide resources, such as real-world tasks rooted in social problems, and instructional suggestions for teachers. Teachers who consider the characteristics and preferences of Generation Z in their planning can enact mathematics lessons that better connect to Generation Z learners. With the Effective Mathematics Teaching Practices as a guide, teachers can design and deliver innovative lessons that support Generation Z learners to promote deeper understanding of mathematical content.
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Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
Read the article

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
Read the article

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
Read the article

This book expands upon the guiding principles at the heart of Math Recovery® instruction, exploring their connections with learning theory, practical application in the classroom and their wider links to agreed concepts of high-quality mathematics teaching.

Teacher noticing and related variants have ascended in prominence among the mathematics education research community. While the component processes of such noticing (e.g., attending, interpreting and deciding) have been cast as interrelated, capturing the relationships amongst the components has been more elusive. We focused on the component processes of teacher noticing with particular attention given to interrelatedness. Specifcally, we were interested in how and the extent to which the component processes of professional noticing (attending, interpreting, deciding) are thematically connected when preservice elementary teachers are engaged in an assessment approximating professional noticing. We refer to this thematic linkage in this paper as coherence. Our fndings suggest a complex interplay between the creation and continuation of themes when enacting professional noticing, and the quality of such noticing.
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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.
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Differing research worldviews have typically resulted in interpretations at odds with one another. Yet, leveraging distinct perspectives can lead to novel interpretations and theoretical construction. Via an empirically grounded research commentary, we describe the value of such activity through the lens of previously reported findings. This synthesis of research from dissimilar scholarly traditions is one example of how paradigms in related but sometimes disconnected fields were used to provide a more comprehensive model of foundational numeracy development. While critique and skepticism may be valuable scholarly tools, we argue that such practices should be balanced with openness and belief towards ideas from worldviews different than our own. This balance can provide new and creative interpretations and extend our collective research power.
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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.

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 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.

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.

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.

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.

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 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.

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.

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.

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).

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

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.

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.