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An unsolved problem gets elementary and middle school students thinking and doing mathematics like mathematicians.

In today's world, data and statistical information permeate our lives, making it imperative that we educate students to be statistically literate. Statistical literacy is the ability to read and interpret statistical information to make informed decisions about events under conditions of uncertainty. Recently, the National Council of Teachers of Mathematics (NCTM) published a document, Catalyzing Change in High School Mathematics: Initiating Critical Conversations (2018), in which it proposed statistics as one of four essential content domains in secondary school mathematics and acknowledged quantitative literacy—the ability to reason both statistically and numerically—as a crucial life skill for all students. For a number of years, statistics has been an important content strand across grade levels in the school mathematics curricula of many countries. Thus, it is understandable that students and even teachers might perceive statistics simply as another topic in mathematics.

Researchers have hypothesized and, in some cases, shown that hypothetical learning trajectories (LT)s can be effective tools in teaching, but implementation, across individual teachers, varies. From the literature base, we have identified six ways researchers have hypothesized teachers might be able to use LTs in their work as teachers: formatively assess students over time, attend to student thinking, provide appropriate differentiation, design and modify mathematical tasks, choose appropriate learning goals, and relate lesson goals to broader curriculum goals. To gain greater insight from the teachers’ perspective, we followed one teacher, Mrs. Purl, and chronicled her efforts in using the LT to design and enact student interviews and lessons. During data collection, Mrs. Purl participated in in-the-moment noticing in which she noted events during interviews or instruction that she found significant. Following interviews and lessons, Mrs. Purl was asked to reflect on each of the events and describe what she noticed, what she could infer about student understanding, and instructional implications. Using literature on teacher noticing, we qualitatively analyzed each of the three components and then related them back to the ways in which researchers have hypothesized teachers might be able to use learning trajectories. Findings suggest that the LT supported Mrs. Purl in attending to students’ thinking, designing and modifying mathematical tasks, and choosing appropriate learning goals, but she did not refer to the LT to provide differentiation, formatively assess students over time, or relate lesson goals to broader curriculum goals.

For the last 4.5 years, I have been immersed in the work of editing the Journal for Research in Mathematics Education. I could talk for hours about reading manuscripts and reviews, writing decision letters, interacting with authors, editing manuscripts to prepare them for publication, my reflections on the research that has been published in the journal, and my reflections on the research that has not been published, but this talk is not about me. I want to focus on the journal itself, its past and its future, and what it means to us–the mathematics education research community. Also, I will be talking about unicorns, mastodons, and ants. So bear with me, this will not be a typical math ed. talk (and I might have gotten a little carried away with the mastodons). Let's begin this tale.

These evidence-based instructional strategies can lead to deeper mathematical conversations in upper elementary school classrooms.

Probability is no longer featured as an important domain of study in the elementary and lower secondary school mathematics curriculum in the United States. The majority of U.S. states have adopted a mathematics curriculum that has eliminated probability as a topic of study in the elementary grades and narrowed its focus in the lower secondary grades. I illustrate the decline of probability via an overview of the various standards documents that have guided mathematics curriculum and instruction for the past 25 years. This situation is not unique to the United States; many other countries around the world do not include probability in the mathematics curriculum for students under the age of 11. I discuss the implications of this curricular decision and raise questions for the consideration of the research community.

Our research addresses the role that context expertise plays when students compare data. We report findings from a study conducted in 3 countries: Australia, United States, and Thailand. In each country, six middle school students analyzed authentic data relating to selected students' areas of interest. We examined the data analysis processes and discussion among students as each country cohort worked in two groups of three, where only one group included a student with particular expertise with the data context. We found that students used context knowledge to (a) bring new insight or additional information to the task, (b) explain the data, (c) provide justification or qualification for claims, (d) identify useful data for the task at hand, and (e) state facts that may enhance the picture of the data but are irrelevant to the process of analyzing the data. Implications for practice are discussed.

As a prelude to the 2014 NCTM Research Conference, “Linking Research and Practice,” I take this opportunity to remind theJRMEcommunity about the responsibility that we, as researchers, bear in linking research and practice. I also make some suggestions about the role thatJRMEmight play in this work.

As we begin a new calendar year and launch a new volume of JRME, it seems an appropriate time to reflect on the status of the journal and the changes that have been recently instituted in the submission and review process. As many of you are aware, JRME was not listed in the Thomson Reuters Journal Citation Reports for 2011. This oversight has been resolved, and we are pleased to report that the current Impact Factor for the journal is 1.552, which is slightly higher than in the past. The acceptance rate for JRME manuscripts remains fairly stable at about 8%. However, the number of submissions has been in decline since reaching an all-time high of 243 manuscripts in 2010; the total in 2012 was 193, and the projection for 2013 is less than that. This decline in submissions could be a result of the many options that are now available to authors for publishing mathematics education research. There may also be the perception that publishing in JRME is a lengthy process. Thus the editorial staff has been working diligently to make the review process as efficient as possible. Typically, the process—receipt of manuscript to decision—is completed within about 4 months. Our goal is that accepted manuscripts are published within a year of acceptance. JRME continues to maintain an international presence, and in the past year we received manuscripts from authors in 21 countries and sought reviews from colleagues in 25 countries, not including the United States and Canada. Moreover, JRME was ranked at the highest level for mathematics education research journals in a recent study conducted jointly by the Education Committee of the European Mathematical Society and the Executive Committee of the European Society for Research in Mathematics Education (Toerner & Arzarello, 2012).

“How many teeth have you lost?” Imagine second graders collecting data from their peers about how many teeth each child is missing and then creating their own data representations that mean something to them. Instead of showing her students how to create a bar graph of the data, this teacher asks them to display the information in some way that helps them make sense of their data. Students work in small groups, discuss their ideas with peers, and create their own data representations. Then they share their data with the entire class, discussing differences in their representations and interpretations.

Often referred to as “The Nation's Report Card,” the National Assessment of Educational Progress (NAEP) was designed in the 1960s as a tool for monitoring precollege student performance in various subject areas. The original design included assessment of nine-, thirteen-, and seventeen-year-old students. The first mathematics assessment was completed in 1973 with additional mathematics assessments following at two- to fouryear intervals. In contrast to assessments like the SAT, which are usually taken by college-bound students only, NAEP is given to a sampling of all students across the United States regardless of ability or aspiration. As such, it is the best available measure of mathematics achievement for the nation as a whole (Kenney and Kloosterman 2007).

Editors' Commentary In this chapter a group at Illinois State University describe how they used the scholarship of teaching and learning to investigate whether having preservice teachers participate in a mathematical research experience for undergraduates (REU) program influenced their beliefs about teaching and learning mathematics. Unable to find an appropriate survey instrument, they developed their own. They explain the organization and content of the survey, tell how they piloted and tested it for reliability, and describe how the results are being used to improve the REU program. Introduction The Scholarship of Teaching and Learning (SoTL) can serve as a critical tool for improving instruction at the post-secondary level. If we, mathematicians and mathematics educators, can systematically explore, share, and reflect on our understanding of quality collegiate instruction, improvement in overall course quality and student learning can follow. The multifaceted nature of student learning presents a challenge for any educational study. One such facet is student beliefs. For example, an effective instructional strategy for teaching an honors calculus course for mathematics majors may not translate to a calculus course designed for business majors because of the different attitudes and beliefs of these two groups. Differences in students' views of the nature of mathematics, the importance of building mathematical understanding, the beauty of mathematics, or the standards for reasoning and proof, all can influence the success of instruction. In this chapter, we describe a SoTL study designed to inform and improve a mathematics education program that engages future secondary teachers in authentic mathematics research experiences. One of the goals of the program, the REU Site: Mathematics Research Experience for Pre-service and In-service Teachers (REU), is to change teachers' beliefs about mathematics and about the teaching and learning of mathematics and thus to influence the way they teach. Impact of Beliefs on Secondary Mathematics Teachers Numerous researchers have cited the impact of beliefs on teaching and learning (Ernest, 1989; Philipp, 2007; Schoenfeld, 1985; Silver, 1985; Thompson, 1992).

This study investigated how children organised and represented data and also examined relationships between their organisation and representation of data. Two protocols, one involving categorical data and the other involving numerical data, were used to interview 15 students, 3 from each of Grades 1 through 5. Although there were differences between Grade 1 students and the rest, the study suggested that numerical data was significantly harder for children to organise and represent than categorical data. Children beyond Grade 1 could make connections between organising and representing data for categorical data but their connections for numerical data were more tenuous. The process of reorganising numerical data into frequencies was not intuitive for the children in this study but they showed greater readiness in recognising and interpreting data that had already been reorganised as a frequency representation. Given this latter result, a pedagogical approach that asks students to make links between raw data and a frequency representation of it may prepare students to create and construct their own frequency representations.

Our goal in this chapter is to showcase the various theories and cognitive models that have been developed by mathematics and statistics educators regarding the development of probabilistic thinking from a “practitional” perspective. This perspective juxtaposes the work of mathematics and statistics educators with research on probability from the field of psychology. Next, we synthesize theories and models that have been developed for specific probability concepts and processes. These models suggest different levels or patterns of growth in probabilistic reasoning. Finally, we examine the role these models have had on instruction and curriculum recommendations.

This study designed and evaluated a teaching experiment in data exploration for second grade students. A research-based framework that incorporated a description of elementary students' statistical thinking on four constructs and across four levels of thinking informed the hypothetical learning trajectory for the teaching experiment. Qualitative evidence from four target students revealed that: (a) experiences with different kinds of data reduced children's idiosyncratic descriptions; (b) categorical data were more problematic than numerical data; (c) technology stimulated and influenced children's thinking in relation to organizing and representing data; (d) children displayed multifaceted conceptual knowledge of center and spread; and (e) children's contextual knowledge was a key factor in being able to analyze and interpret data. By the end of the intervention, the class data as a whole showed that at least 84% of the students was exhibiting Level 2 thinking or better on all constructs, i.e., they were no longer exhibiting idiosyncratic thinking.

Ellen, Jim, and Steve bought three helium-filled balloons and paid $2 for all three. They decided to go back to the store and buy enough balloons for everyone in the class.

Based on a review of research and a cognitive development model (Biggs & Collis, 1991), we formulated a framework for characterizing elementary children's statistical thinking and refined it through a validation process. The 4 constructs in this framework were describing, organizing, representing, and analyzing and interpreting data. For each construct, we hypothesized 4 thinking levels, which represent a continuum from idiosyncratic to analytic reasoning.We developed statistical thinking descriptors for each level and construct and used these to design an interview protocol. We refined and validated the framework using data from protocols of 20 target students in Grades 1 through 5. Results of the study confirm that children's statistical thinking can be described according to the 4 framework levels and that the framework provides a coherent picture of children's thinking, in that 80% of them exhibited thinking that was stable on at least 3 constructs. The framework contributes domain-specific theory for characterizing children's statistical thinking and for planning instruction in data handling.

This article reports some results of an informal study of very young children's reactions to some visual displays of data.

December is typically the time for year-in-review reflections, but given the publication schedule for JRME, a report of the status of the journal falls to the January issue. It has been a productive year for JRME. The number of manuscript submissions has increased by 17% when compared to this same time period last year.1 Although we are pleased to be receiving more submissions, the increase in manuscripts has not increased the acceptance rate. In fact, the acceptance rate for all manuscripts submitted to JRME has dropped from 8% to about 6%. However, the rate of acceptance for manuscripts that were fully reviewed (i.e., were appropriate for the journal but did not receive a desk reject decision or were not editorially reviewed2) was considerably greater, at about 22%. It is noteworthy that the majority of accepted manuscripts were resubmissions of papers that had previously received “revise and resubmit” decisions. This is a good indicator that the review process has been effective in providing authors with the kind of constructive criticism that can help them craft a publishable manuscript. Moreover, it points to the important role that reviewers play in the publication process.

In response to the need for studies that focus on learning and writing in mathematics, this study examined changes in students' probabilistic thinking and writing during an instructional program that emphasized transactional writing in a problem-solving context. Although correlations between probabilistic thinking and writing levels at the end of the study were not significant, students did make significant gains in both probability reasoning and writing. Analysis of target students' journals revealed that their writing incorporated both writing symbols and mathematical symbols. These symbols were more complementary for those students whose writing increased to the higher levels during instruction. Moreover, this growth appeared to be promoted by the teacher herself, who systematically sought verbal explanations of solutions and written interpretations of diagrams and numerical patterns.

Based on a synthesis of the literature and observations of young children over two years, a framework for assessing probabilistic thinking was formulated, refined and validated. The major constructs incorporated in this framework were sample space, probability of an event, probability comparisons, and conditional probability. For each of these constructs, four levels of thinking, which reflected a continuum from subjective to numerical reasoning, were established. At each level, and across all four constructs, learning descriptors were developed and used to generate probability tasks. The framework was validated through data obtained from eight grade three children who served as case studies. The thinking of these children was assessed at three points over a school year and analyzed using the problem tasks in interview settings. The results suggest that although the framework produced a coherent picture of children‘s thinking in probability, there was ‘static’ in the system which generated inconsistencies within levels of thinking. These inconsistencies were more pronounced following instruction. The levels of thinking in the framework appear to be in agreement with levels of cognitive functioning postulated by Neo-Piagetian theorists and provide a theoretical foundation for designers of curriculum and assessment programs in elementary school probability. Further studies are needed to investigate whether the framework is appropriate for children from other cultural and linguistic backgrounds.

Dans cette etude, les AA. ont cherche a savoir de quelle facon l'approche de l'enseignement des mathematiques chez de futurs enseignants de cette discipline peut etre modifiee par une reflexion sur leur conceptions du metier et une collaboration entre collegues

Karen and Leon had been constructing their booth for the school fair. They had made a “pull the strings” game in which half the strings had one end painted red and the other half of the strings had one end painted blue. The strings are randomly placed over a board with the colored end hidden. After the first string is pulled, it is replaced to a random location so that ten strings are available for the second pull. A prize would be given whenever two strings of the same color were pulled. Leon was concerned that the game was not fair.

Once again, the wheels of change are turning at JRME. This issue marks the beginning of Volume 47, which will be the last volume under my editorship. To facilitate the transition to the new editor, there is a 1-year period during which the work of the journal is shared by the editor and the editor-designate. I am pleased to announce that Jinfa Cai, Professor of Mathematical Sciences at the University of Delaware, is the JRME editor-designate. His term as JRME editor will span Volumes 48–51, from 2017–2020. Although volumes of the journal correspond to calendar years, the tasks of the editor follow a different timeline, which is governed by production schedules. Thus, the work of the editor-designate begins long before his or her name appears as editor on the journal cover. For example, Jinfa and the University of Delaware editorial team began handling all new manuscript submissions on September 1, 2015. The Illinois State team continued to process manuscripts that were already in the system and conducted reviews for resubmissions of manuscripts that had previously received revise and resubmit decisions. As of January 1, Jinfa and his team will have assumed responsibility for all aspects of the submission and review process. During this final phase of the transition, my role, with the assistance of Amanda Fain, will be to work with authors to complete the final editing of articles that will appear in Volume 47. Another aspect of the transition involves the special section editors, who are appointed by the JRME editor. As you may have noticed from previous issues, some new appointments have already occurred: Steve Williams is now serving as the Research Commentary editor, and Sarah Lubienski has taken on the role of Book Review editor. Randy Groth will continue as the Monograph editor. The gradual shifting of responsibilities between the JRME editor and the editor-designate in this transition period allows for a seamless change in the stewardship of the journal. While the work at one editorial office is winding down, the work at the other is ramping up. I would like to thank the NCTM Board of Directors for supporting both editorial offices during this period of transition. Journal for Research in Mathematics Education 2016, Vol. 47, No. 1, 2

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) speaks of the necessity of providing effective mathematics education for all students. Noting that “the social injustices of past schooling practices can no longer be tolerated” (p. 4), the standards document calls for a mathematics content that is “what we believe all students will need if they are to be productive citizens in the twenty-first century. If all students do not have the opportunity to learn this mathematics, we face the danger of creating an intellectual elite and polarized society” (p. 9), Similarly, the National Research Council's Mathematical Sciences Education Board noted that two themes underlie current analysis of American education: “equity in opportunity and… excellence in results” (1989. 28–29). Although the NCTM's standards and other reform documents have been critiqued as addressing the issue of equity in terms of “enlightened self-interest” as opposed to seeking justice (Secada 1989), these documents have called attention to educational disparity. The issue today is how to make the goal of equity a reality in classrooms. To do otherwise would be to assign “mathematics for all” to the status of a slogan, a catchy phrase but having no meaning in practice.

Summary This article describes the learning experiences of young children meeting stem-and-leaf plots for the first time.

There are many venues for engaging in scholarly discourse about mathematics education, including conferences, webinars, social networking sites, blogs, university classes, and colloquia. Given the dynamic nature of these forums, one might lose sight of the role of academic journals in contributing to scholarly discourse in the field. The integrity of JRME and the high standards to which its publications are held support the notion of the journal as an archive, a means for recording and disseminating knowledge. However, JRME also influences the direction of mathematics education research and contributes to the ongoing discourse of the mathematics education community.

This issue of the Journal for Research in Mathematics Education (JRME) is a special one indeed. Not only is this the Equity Special Issue, but it is the first issue of my editorship. I am delighted to begin my term as editor with the print publication of these important papers. The articles in this issue address equity in terms of identity and power, and it occurred to me that this editorial should do the same. Undoubtedly, the editor of JRME is in a position of power. The responsibility of that power is something I take very seriously. I am mindful of it as I read every manuscript that is submitted to the journal and as I write letters to authors explaining why their manuscript has not been accepted for publication or asking for revisions to work I know they have labored over. Thankfully, it is a shared responsibility, and I am fortunate to work with a dedicated editorial team that includes Associate Editors Nerida Ellerton and Tami Martin; Assistant Editor Josh Hertel; and Staff Associate Amanda Fain. Cognizant of that responsibility, we see ourselves as stewards of JRME, serving the mathematics education community. Our goal is to publish articles that address the pressing problems of the field, stimulate discussion, and present different viewpoints. To that end, we view each manuscript submitted with regard to its potential contribution to mathematics education practices, policies, and research. We aim for a broad and balanced range of articles that reflect multiple research perspectives, examine various facets of mathematics education, and maintain the high standards of disciplined inquiry that are hallmarks of JRME. Great care is taken in selecting reviewers who can provide authors with expert commentary on their work and offer us—the editorial team—insights from multiple vantage points so that we can make an informed decision about whether or not a paper should be published. Regardless of the decision, ours is an educative role. That is, we provide detailed feedback to authors to help improve the manuscript or give direction to future research endeavors, helping authors polish their writing to convey their ideas effectively. Stewardship also entails other tasks to maintain the health of the journal such as inviting young scholars to participate in the review process, updating and expanding the pool of existing reviewers, promoting JRME’s international visibility (especially through interactions with members of the International Advisory Board), representing the journal at professional meetings, and staying abreast of leading research and topical issues in mathematics education. Editorial Copyright © 2013 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Recent research in mathematics instruction requires educators to rethink long-established beliefs about teaching, learning, and assessment. In particular, this research underscores the need for problem solving and higher level thinking in mathematics. Consistent with these recommendations, this article presents and illustrates four promising themes for mathematics instruction that have emerged from research involving students with learning disabilities. These themes—(a) providing a broad and balanced mathematics curriculum; (b) engaging students in rich, meaningful problem tasks; (c) accommodating the diverse ways in which children learn; and (d) encouraging students to discuss and justify their problem-solving strategies and solutions—suggest ways for rethinking the teaching and learning of mathematics in relation to students with learning disabilities.

The purpose of this study was to investigate 6th-grade students' use of equations to describe and represent problem situations prior to formal instruction in algebra. Ten students were presented with a series of similar tasks in 6 different problem contexts representing linear and nonlinear situations. The students in this study showed a remarkable ability to generalize the problem situations and to write equations using variables, often in nonstandard form. Although students were often able to write equations, they rarely used their equations to solve related problems. We describe students' preinstructional uses of equations to generalize problem situations and raise questions about the most appropriate curriculum for building on students' intuitive knowledge of algebra.

In this study we evaluated the thinking of 3rd-grade students in relation to an instructional program in probability. The instructional program was informed by a researchbased framework that included a description of students' probabilistic thinking. Both an earlyand a delayed-instruction group participated in the program. Qualitative evidence from 4 target students revealed that overcoming a misconception in sample space, applying both part-part and part-whole reasoning, and using invented language to describe probabilities were key patterns in producing growth in probabilistic thinking. Moreover, 51% of the students exhibited the latter 2 learning patterns by the end of instruction, and both groups displayed significant growth in probabilistic thinking following the intervention.