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Although improving racial equity in critical college courses such as introductory statistics is a laudable goal, making research-based progress toward that aim in a scalable manner remains a challenge. To translate psychological insights to benefit racially marginalized students, we implemented the “Better Book” approach, where instructors, researchers, and developers work together to improve an online textbook used in introductory statistics. The “Better Book” approach to equity assumes that racially marginalized students are a “canary in the coal mine”, alerting us to systemic issues that can affect a broader array of students. We started by finding places in the textbook where racially marginalized students reported higher perceptions of costs (the effort and time required to learn the content) than non-marginalized students. Then we drew upon suggestions from users to redesign the textbook where gaps in cost perceptions peaked. We then analyzed data from both the original and redesigned versions of the textbook to evaluate the impact on students who were subsequently enrolled in the course. Results showed that perceptions of cost were dramatically reduced in the experience of racially marginalized students but also the redesign resulted in an improved experience for all students.

We developed an interactive online textbook that interleaves R programming activities with text as a way to facilitate students' understanding of statistical ideas while minimizing the cognitive and emotional burden of learning programming. In this exploratory study, we characterize the attitudes and experiences of 672 undergraduate students as they used our online textbook as part of a 10-week introductory course in statistics. Students expressed negative attitudes and concerns related to R at the beginning of the course, but most developed more positive attitudes after engaging with course materials, regardless of demographic characteristics or prior programming experience. Analysis of a subgroup of students revealed that change in attitudes toward R may be linked to students' patterns of engagement over time and students' perceptions of the learning environment.

Research suggests that expert understanding is characterized by coherent mental representations featuring a high level of connectedness. This paper advances the idea that educators can facilitate this level of understanding in students through the practicing connections framework: a practical framework to guide instructional design for developing deep understanding and transferable knowledge in complex academic domains. We start by reviewing what we know from learning sciences about the nature and development of transferable knowledge, arguing that connectedness is key to the coherent mental schemas that underlie deep understanding and transferable skills. We then propose features of instruction that might uniquely facilitate deep understanding and suggest that the connections between a domain’s core concepts, key representations, and contexts and practices of the world must be made explicit and practiced, over time, in order for students to develop coherent understanding. We illustrate the practicing connections approach to instructional design in the context of a new online interactive introductory statistics textbook developed by the authors.

Producing content-related gestures has been found to impact students' learning, whether such gestures are spontaneously generated by the learner in the course of problem-solving, or participants are instructed to pose based on experimenter instructions during problem-solving and word learning. Few studies, however, have investigated the effect of (a) performing instructed gestures while learning concepts or (b) producing gestures without there being an implied connection between the gestures and the concepts being learned. The two studies reported here investigate the impact of instructed hand movements on students' subsequent understanding of a concept. Students were asked to watch an instructional video-focused on the concept of statistical model-three times. Two experimental groups were given a secondary task to perform while watching the video, which involved moving their hands to mimic the placement and orientation of red rectangular bars overlaid on the video. Students were told that the focus of the study was multitasking, and that the instructed hand movements were unrelated to the material being learned. In the content-match group the placement of the hands reinforced the concept being explained, and in the content-mismatch group it did not. A control group was not asked to perform a secondary task. In both studies, findings indicate that students in the content-match group performed better on the posttest, and showed less variation in performance, than did students in the content-mismatch group, with control students falling in between. Instructed hand movement-even when presented as an unrelated, secondary task-can affect students' learning of a complex concept.

BackgroundIn two studies we investigated whether removing opportunities to calculate could improve students' subsequent ability to solve similar word problems. Students were first asked to write explanations for three word-problems that they thought would help another student understand the problems. Half of the participants explained typical word problems (i.e., problems with enough information to make calculating an answer possible), while the other half explained the same problems with numbers removed, making calculating an answer impossible. We hypothesized that removing opportunities to calculate would induce students to think relationally about the word problems, which would result in higher levels of performance on subsequent transfer problems.ResultsIn both studies, participants who explained the non-calculable problems performed significantly better on the transfer test than participants who explained the typical (i.e., calculable) problems. This was so in spite of the manipulation not fully suppressing students' desire to calculate. Many students in the non-calculable group explicitly stated that they needed numbers in order to answer the question or made up numbers with which to calculate. There was a significant, positive relationship between the frequency with which students made up numbers and their self-reported mathematics anxiety.ConclusionsWe hypothesized that the mechanism at play was a reduction in instrumental thinking (and an increase in relational thinking). Interventions designed to help students remediate prior mathematical failure should perhaps focus less on the specific skills students are lacking, and more on the dispositions they bring to the task of "doing mathematics."

Students learn many concepts in the introductory statistics course, but even our most successful students end up with rigid, ritualized knowledge that does not transfer easily to new situations. In this article we describe our attempt to apply theories and findings from learning science to the design of a statistics course that aims to help students build a coherent and interconnected representation of the domain. The resulting "practicing connections" approach provides students with repeated opportunities to practice connections between core concepts (especially the concepts of statistical model, distribution, and randomness), key representations (R programming language and computational techniques such as simulation and bootstrapping), and real-world situations statisticians face as they explore variation, model variation, and evaluate and compare statistical models. We provide a guided tour through our curriculum implemented in an interactive online textbook (CourseKata.org) and then provide some evidence that students who complete the course are able to transfer what they have learned to the learning of new statistical techniques.

In this article we propose the use of Bayesian networks as a potentially promising way to model usable knowledge. Using the Classroom Video Analysis (CVA and CVA-M) assessments as a lab model for studying teachers' usable knowledge, we first explored whether we can identify the knowledge (pieces) underlying teachers' written responses. In the CVA approach we ask teachers to respond to short video clips of authentic classroom instruction based on different prompts that are designed to simulate common teaching tasks. We then explored the affordances of Bayesian networks to functionally model usable knowledge as an interconnected dynamic knowledge system consisting of different knowledge pieces and connected pathways weighted by situation-specific relevance and applicability. We explore the implications of these models for studying the development and growth of usable knowledge and propose the use of Bayesian networks as a novel and potentially promising way to model usable knowledge and for understanding how knowledge is used in teaching.

In this study, we explored the potential for machine scoring of short written responses to the Classroom-Video-Analysis (CVA) assessment, which is designed to measure teachers' usable mathematics teaching knowledge. We created naïve Bayes classifiers for CVA scales assessing three different topic areas and compared computer-generated scores to those assigned by trained raters. Using cross-validation techniques, average correlations between rater- and computer-generated total scores exceeded 0.85 for each assessment, providing some evidence for convergent validity of machine scores. These correlations remained moderate to large when we controlled for length of response. Machine scores exhibited internal consistency, which we view as a measure of reliability. Finally, correlations between machine scores and another measure of teacher knowledge were close in size to those observed for human scores, providing further evidence for the validity of machine scores. Findings from this study suggest that machine learning techniques hold promise for automating scoring of the CVA.

Developing understanding of fractions involves connections between nonsymbolic visual representations and symbolic representations. Initially, teachers introduce fraction concepts with visual representations before moving to symbolic representations. Once the focus is shifted to symbolic representations, the connections between visual representations and symbolic notation are considered to be less useful, and students are rarely asked to connect symbolic notation back to visual representations. In 2 experiments, we ask whether visual representations affect understanding of symbolic notation for adults who understand symbolic notation. In a conceptual fraction comparison task (e.g., Which is larger, 5 / a or 8 / a?), participants were given comparisons paired with accurate, helpful visual representations, misleading visual representations, or no visual representations. The results show that even college students perform significantly better when accurate visuals are provided over misleading or no visuals. Further, eye-tracking data suggest that these visual representations may affect performance even when only briefly looked at. Implications for theories of fraction understanding and education are discussed.

If teacher value-added estimates (VAEs) are to be used as indicators of individual teacher performance in teacher evaluation and accountability systems, it is important to understand how much VAEs are affected by the data and model specifications used to estimate them. In this study we explored the effects of three conditions on the stability of VAEs and evaluated their relative impact. We varied how we accounted for differences among students in their prior learning; whether we estimated VAEs from single or multiple cohorts of students; and the number of students contributing to the VAE for each teacher. Using data from one of the largest school districts in the nation, we created a single, complete data set and used it to estimate several sets of VAEs for each of the 3,651 5th grade mathematics teachers. We found that approximately two thirds of teachers' were stable in that they remained in the same performance group across all conditions. We also found that differences in number of students used for VAEs accounted for up to one third of teacher reclassifications into different performance groups; single versus multiple cohort models accounted for about one fifth; and different methods for controlling for student prior learning accounted for about one sixth (16%) of teacher reclassifications. We relate our findings to characteristics of our data and discuss implications for educational policy. (Contains 7 tables and 5 figures.)

We examine the distinction between teaching and teachers as it relates to instructional improvement. Drawing from work outside of education on improvement systems and from analyzing the Japanese system of lesson study, we contend that a focus on teaching can shape a coordinated system for improvement whereas a focus on teachers, common in the United States, leads to elements that are uncoordinated and often work against the continuous, steady improvement of classroom teaching. We propose that the concept of systems for improvement and its instantiation in Japanese K-8 education offer opportunities to reexamine U.S. efforts to improve teaching and shift these efforts toward a more promising direction.

Originating in Asia, lesson study is gradually spreading around the globe. As evident from the papers in this issue, we have much to learn as it is implemented in a variety of cultural contexts. In this article we reflect on the goals of lesson study, the organizational supports required to sustain the practice in various contexts, and the benefits that may be derived from making more explicit the connections between lesson study and the wider field of improvement science. Both research and practice can benefit from learning from, and about, the process of importing cultural routines.

People of lower social status tend to have greater emotional responses to stress. The present study assessed whether lower social status was related to greater emotional responses in anticipation of a naturalistic stressor: academic exams among college students. College students in an introductory statistics class (N = 252; 75.81% female; 18.41% Latino, 25.10% White, 43.93% Asian, 12.56% different racial backgrounds) completed two course exams as part of this naturalistic prepost-experimental design. They provided four reports of positive, depressive, and anxious emotion – one the day before and one immediately after each exam. As hypothesized, multilevel models (ratings nested within participants) predicting emotion indicated that students with lower mother's education had less positive emotion, more depressive emotion, and more anxious emotion the day prior to academic exams than students with higher mother's education (proportional reductions in variance [PRV] =.013–.020). Specifically, lower mother's education was associated with poorer well-being before but not after the exam. Exploratory models revealed that differences in emotion by mother's education were strongest for students with lower exam scores (PRV =.030–.040). Socioeconomic status may influence college students' anticipatory distress prior to academic exams, which may impact health and academic performance. [ABSTRACT FROM AUTHOR]

Many students graduate from K-12 mathematics programs without flexible, conceptual mathematics knowledge. This article reviews psychological and educational research to propose that refining K-12 classroom instruction such that students draw connections through relational comparisons may enhance their long-term ability to transfer and engage with mathematics as a meaningful system. We begin by examining the mathematical knowledge of students in one community college, reviewing results that show even after completing a K-12 required mathematics sequence, these students were unlikely to flexibly reason about mathematics. Rather than drawing relationships between presented problems or inferences about the representations, students preferred to attempt previously memorized (often incorrect) procedures (Givvin, Stigler, & Thompson, 2011; Stigler, Givvin, & Thompson, 2010). We next describe the relations between the cognition of flexible, comparative reasoning and experimentally derived strategies for supporting students' ability to make these connections. A cross-cultural study found that U.S. teachers currently use these strategies much less frequently than their international counterparts (Hiebert et al., 2003; Richland, Zur, & Holyoak, 2007), suggesting that these practices may be correlated with high student performance. Finally, we articulate a research agenda for improving and studying pedagogical practices for fostering students' relational thinking about mathematics. (Contains 4 figures.)

In a prior issue of "MathAMATYC Educator," we reported on our efforts to find out what community college developmental mathematics students understand about mathematics (Stigler, Givvin, & Thompson, 2010). Our work painted a distressing picture of students' mathematical knowledge. No matter what kind of mathematical question we asked, students tended to respond with computational procedures, which they often applied inappropriately and incorrectly. Their knowledge of mathematical concepts appeared to be fragile, and weakly connected to their knowledge of procedures. But we also found some reason for hope. First, we found that when students were able to provide conceptual explanations for procedures, they often produced correct answers. Second, though students rarely used reasoning on their own to solve problems, they could reason under the right conditions. (Contains 2 figures.) [For Part 1, see EJ890224.]

This study investigates, through an experimental design, the effectiveness of a professional development program on teacher knowledge and practices and on student learning. The program consisted of a series of video-based modules designed to respond to needs of U.S. teachers, as highlighted by findings from the 1999 Third International Mathematics and Science Video Study. Sixty-four 6th-grade teachers from five low-performing inner-city schools participated in the study and were randomly assigned to treatment and control groups. Measures included fidelity of implementation, teacher knowledge and practice, and student mathematics learning. The program did not impact significantly teacher knowledge or practices as measured in the study. An effect was found on mathematics learning for students whose teachers reached a certain level of mathematics content knowledge. Discussion of findings includes lessons learned about conducting and studying professional development, particularly in low-performing schools. (Contains 1 figure, 4 footnotes, and 4 tables.)

The nation is facing a crisis in its community colleges: more and more students are attending community colleges, but most of them are not prepared for college-level work. The problem may be most dire in mathematics. By most accounts, the majority of students entering community colleges are placed (based on placement test performance) into "developmental" (or remedial) mathematics courses (e.g., Adelman, 1985; Bailey et al., 2005). The organization of developmental mathematics differs from school to school, but most colleges have a sequence of developmental mathematics courses that starts with basic arithmetic, then goes on to pre-algebra, elementary algebra, and finally intermediate algebra, all of which must be passed before a student can enroll in a transfer-level college mathematics course.

This study explores the relationship between teacher knowledge and student learning in the area of mathematics by developing and evaluating an innovative approach to assessing teacher knowledge. This approach is based on teachers' analyses of classroom video clips. Teachers watched 13 video clips of classroom instruction and then provided written comments on the interactions of the teacher, students, and content. The quality of teachers' analyses, coded using an objective rubric, are shown to be reliable and valid, relating both to another widely used measure of teacher knowledge and to teachers' own students' learning (from pre- to posttest). (Contains 1 table and 3 notes.)

Many have proposed the creation of a practitioner knowledge base as a foundation for the continuous improvement of teaching. In this article, we discuss the promise, and the challenges, involved in the creation and use of such a knowledge base. We begin by reflecting on the difficulties of changing teaching, a cultural and contextual activity. We then discuss 3 phases of the knowledge-building process--creation, accumulation, and utilization--describing how each phase is influenced by the cultural nature of teaching, and the role that practitioners might play in each phase. Finally, we discuss implications for teacher education and professional development.

A video-based program on lesson analysis for pre-service mathematics teachers was implemented for two consecutive years as part of a teacher education program at the University of Lazio, Italy. Two questions were addressed: What can preservice teachers learn from the analysis of videotaped lessons? How can preservice teachers' analysis ability, and its improvement, be measured? Two groups of preservice teachers (approximately 140 in total) participated in the program. A three-step lesson analysis framework was applied to three lesson videos: (1) goal(s) and parts of the lesson; (2) student learning; and, (3) teaching alternatives. Preservice teachers' ability to analyze lessons was measured through an open-ended pre- and post-assessment. In the assessment, preservice teachers were asked to mark and comment on events (in a lesson not included in the program) that they found interesting for: teachers' actions/decisions; students' behavior/learning; and, mathematical content. A coding system was developed based on five criteria: elaboration, mathematics content, student learning, critical approach, and alternative strategies. In both studies, the ability to analyze instruction improved significantly on all five criteria. These data suggest promising directions for the development of both an instrument to measure lesson analysis abilities and a model for teacher learning.

International comparisons are made to improve mathematics teaching in United States. In 1995, the Trends in International Mathematics and Science Study (TIMSS) examined national samples of 8th grade mathematics lessons from Germany, Japan and United States.

An international study of mathematics and science education tells us more about what we need to know to improve math and science professional development. Students need regular opportunities to explore mathematical relationships to develop high levels of understanding in addition to developing skills. Teachers in high-achieving countries offer such opportunities significantly more than do U.S. teachers.

This study examines the influence of a dramatic art-based history program for fifth-grade students on both their learning and enjoyment of history. The program, called "Performing History," reflects theories of effective use of drama in the classroom as well as successful ways to teach history. The program presents historical information as part of a coherent story embedded in a musical drama that students rehearse and perform. Students who participated in the program in fifth grade were assessed on a test of history knowledge when they were in sixth grade and compared to a control group that had not had access to the program. The sixth graders also were asked to rate how much they enjoy history. Results indicate that students who participated in the program showed higher levels of history knowledge and enjoyment of history than those in the control group, controlling for prior scores on standardized achievement tests. The effect on knowledge was partially mediated by level of enjoyment of history, which also predicted later scholastic achievement in history. The study suggests that dramatic art-based instruction improves historical knowledge, and that success may be attributed in part to an increase in student enjoyment. Items from the History Test are appended. (Contains 3 tables and 4 figures.)

Investigates 8th grade mathematics teaching in Italy as videotaped and analyzed following the same procedures in the Third International Mathematics and Science Study (TIMSS). Compares results with German, Japanese, and U.S. lessons videotaped in TIMSS. Indicates that Italian lessons are relatively homogenous among themselves yet distinct from those in the other countries. (Contains 16 references.) (Author/ASK)