Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.     

Guiding Teachers in the Use of a Standards‐Based Mathematics Curriculum: Teacher Perceptions and Subsequent Instructional Practices After an Intensive Professional Development Program

Reforms in mathematics education call for K‐12 teachers to employ standards‐based pedagogies, which embody the National Council for Teachers of Mathematics' principles and standards. In order to effectively support teachers' implementation of standards‐based curricula, professional development must be provided that meets teachers' needs. The professional development program in this study focused on the implementation of a standards‐based mathematics curriculum entitled Investigations in Number, Data, and Space (Investigations). This study uses Guskey's framework as a guide to examining teachers' perceptions of the impact of the professional development that they received; their perceptions of mathematics teaching and learning; and how elements of the professional development translated into practice. Twenty‐two participants were randomly selected from the 53 professional development participants to be interviewed and observed during their mathematics teaching. Using a constant comparison method, the data sources in this study highlighted themes surrounding teachers' experiences with professional development and the implementation of the curricula. The analysis of the data sources in this study highlighted themes surrounding teachers' experiences with professional development: teachers as learners, teachers as self‐evaluators, shifting paradigms, enactment of professional development content into practice, and the influence of the state standardized mathematics test. The results of this study have several implications for future professional development and also highlight some of the more general issues that teachers face when attempting to enact new knowledge and skills into their practice.     

Enhancing Formative Assessment Practice and Encouraging Middle School Mathematics Engagement and Persistence

In the transition to middle school, and during the middle school years, students' motivation for mathematics tends to decline from what it was during elementary school. Formative assessment strategies in mathematics can help support motivation by building confidence for challenging tasks. In this study, the authors developed and piloted a professional development program, Learning to Use Formative Assessment in Mathematics with the Assessment Work Sample Method (AWSM) to build middle school math teachers' understanding of the characteristics of high‐quality formative assessment processes and increases their ability to use them in their classrooms. AWSM proved to be feasible to implement in the middle school setting. It improved teachers' practice of formative assessment, especially in their feedback practices, regardless of their pedagogical content knowledge at entry. Results from focus groups suggested that teachers were better able to implement ungraded practice and student self‐ and peer‐assessment after AWSM, and that students were more willing to engage in complex problem solving.     

Preservice teachers’ mathematics task modification for emergent bilinguals

Implementing mathematically challenging tasks is difficult for teachers when working with emergent bilinguals because cognitively demanding tasks in mathematics commonly have high language demand. Currently, inadequate teacher preparation for teaching emergent bilinguals is becoming a significant concern in the United States as this population of students is rapidly growing. This study investigated how two mathematics preservice teachers (PSTs) support middle school emergent bilinguals to understand cognitively demanding mathematical problems through task modification. Fieldwork with a concurrent intervention was designed for the PSTs to work with emergent bilinguals in a one‐on‐one setting. The PSTs modified cognitively demanding mathematics tasks and designed a lesson for the emergent bilinguals based on the modified tasks. The results revealed that the task modification made by the PSTs tended to shift from reducing cognitive demands in mathematics and language to maintaining the demands through learning strategies of contextual support.     

Modification and validation of the mixed-format Engineering Concept Assessment for middle school students using many-facet Rasch measurement

Integrating engineering into the K-12 science curriculum continues to be a focus in national reform efforts in science education. Although there is an increasing interest in research in and practice of integrating engineering in K-12 science education, to date only a few studies have focused on the development of an assessment tool to measure students’ understanding of engineering design. Most of the existing measures focus only on knowledge and understanding of engineering design concepts using multiple-choice items with the exception of the mixed-format Engineering Concept Assessment (ECA). Also, advanced measurement models are lacking application in the testing of such mixed-format assessments in science education. This study applied many-faceted Rasch measurement to the modified ECA for eighth-grade (ECA/M8) and a newly constructed rubric applied by five judges across 497 eighth-grade students’ responses after experiencing an integrated learning unit on the engineering design process. The results supported the fit of the items and rubric rating scales to the Rasch specifications. Recommendations are made for item wording, and further reliability and validity testing of the ECA/M8, and use of the ECA/M8 in science education and research.     

Components of Place Value Understanding: Targeting Mathematical Difficulties When Providing Interventions

Place value understanding requires the same activity that students use when developing fractional and algebraic reasoning, making this understanding foundational to mathematics learning. However, many students engage successfully in mathematics classrooms without having a conceptual understanding of place value, preventing them from accessing mathematics that is more sophisticated later. The purpose of this exploratory study is to investigate how upper elementary students' unit coordination related to difficulties they experience when engaging in place value tasks. Understanding place value requires that students coordinate units recursively to construct multi‐digit numbers from their single‐digit number understandings through forms of unit development and strategic counting. Findings suggest that students identified as low‐achieving were capable of only one or two levels of unit coordination. Furthermore, these students relied on inaccurate procedures to solve problems with millennial numbers. These findings indicate that upper elementary students identified as low‐achieving are not to yet able to (de)compose numbers effectively, regroup tens and hundreds when operating on numbers, and transition between millennial numbers. Implications of this study suggest that curricula designers and statewide standards should adopt nuances in unit coordination when developing tasks that promote or assess students' place value understanding.     

Students' gendered perceptions of mathematics in middle grades single‐sex and coeducational classrooms

With the introduction of single‐sex classroom settings in coeducational public schools, there is an ongoing debate as to whether single‐sex education may reduce or reinforce traditional stereotypes and gender roles. In this article we present findings from a study that investigated the extent to which mathematics is perceived as a gendered domain among adolescent students enrolled in single‐sex classes and coeducational classes. Further we analyzed the relationships between student characteristics, class‐type, and teacher variables on students' perceptions of gender in mathematics. Findings from this study challenge the traditional view of mathematics as a male domain. Female participants more frequently considered mathematics to be a female domain than the male participants. Male participants, on the other hand, typically did not stereotype the mathematics as a gendered domain. Results from this study do not indicate, for girls at least, that participation in single‐sex classes results in a greater propensity to stereotype mathematics as a gendered domain than would be the case in coeducational classes. This study contributes to the evolving discourse and understanding of adolescents' gendered attitudes and beliefs towards mathematics—especially in light of stereotyped assertions that have a bearing on efforts to promote the learning of mathematics and science.     

Students' use of STEM content in design justifications during engineering design‐based STEM integration

Engineering design‐based STEM integration is one potential model to help students integrate content and practices from all of the STEM disciplines. In this study, we explored the intersection of two aspects of pre‐college STEM education: the integration of the STEM disciplines, and the NGSS practice of engaging in argument from evidence within engineering. Specifically, our research question was: While generating and justifying solutions to engineering design problems in engineering design‐based STEM integration units, what STEM content do elementary and middle school students discuss? We used naturalistic inquiry to analyze student team audio recordings from seven curricular units in order to identify the variety of STEM content present as students justified their design ideas and decisions (i.e., used evidence‐based reasoning). Within the four disciplines, fifteen STEM content categories emerged. Particularly interesting were the science and mathematics categories. All seven student teams used unit‐based science, and five used unit‐based mathematics, to support their design ideas. Five teams also applied science and/or mathematics content that was outside the scope of the units' learning objectives. Our results demonstrate that students integrated content from all four STEM disciplines when justifying engineering design ideas and solutions, thus supporting engineering design‐based STEM integration as a curricular model.     

The Effectiveness of Process‐Oriented Guided Inquiry Learning to Reduce Alternative Conceptions in Secondary Chemistry

A nonequivalent, control group design was used to investigate student achievement in secondary chemistry. This study investigated the effect of process‐oriented guided inquiry learning (POGIL) in high school chemistry to reduce alternate conceptions related to the particulate nature of matter versus traditional lecture pedagogy. Data were collected from chemistry students in four large high schools and were analyzed using analysis of covariance. The results show that POGIL pedagogy, as opposed to traditional lecture pedagogy, resulted in fewer alternate conceptions related to the particulate nature of matter. Male and female students in the POGIL group posted better posttest scores than their traditional group peers. African‐American and Hispanic students in the POGIL group exhibited achievement gains consistent with Caucasian and Asian students. Further studies are needed to determine the value of POGIL to address achievement gap concerns in chemistry.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Body‐based activities in secondary geometry: An analysis of learning and viewpoint

Body‐based activities have the potential to support mathematics learning, but we know little about the impact they have in the classroom. This study compares high school geometry students learning through either body‐based or analogous non‐body‐based activities over the course of a two‐week unit on similarity. Pre‐ and post‐tests revealed that while students in both conditions showed gains in content area comprehension over the course of the study, the body‐based condition showed significantly greater gains. Further, there were differences in the language students used to describe the learning activities at the end of the unit that may have contributed to the differences in learning gains. The students in the body‐based condition included more mathematical and nonmathematical details in their recollections. Additionally, students in the body‐based condition were more likely to recall their experiences from a first‐person perspective, while students in the control condition were more likely to use a third‐person perspective.     

Preservice Teachers of High School Mathematics: Success,Failure, and Persistence in the Face of Mathematical Challenges

The goal of this study was to identify variables related to success and resilience in an undergraduate, high school mathematics teacher education program. Over a five‐year period, we tracked the academic performance and achievement motivation goals of multiple cohorts of students. Students who successfully completed their degrees had higher grade point average (GPAs) upon entering the program, earned higher grades in their first college mathematics course, and failed fewer courses than students who left the program or university. Learning and performance motivational goals did not predict success in the program. Performance goals decreased over time. Nearly half the successful students repeated one or more mathematics courses. Ten students completed their degrees, obtained a teaching license, and are teaching despite the need for multiple repetitions of the same mathematics courses. These persistent students did not differ from their peers in motivational goals. Our results suggest that although students with higher GPAs and initial mathematics grades were more likely to complete the program, students who experienced challenges in mathematics courses were able to succeed. We discuss the implications of these results for recruiting, advising, and retention of students in mathematics education programs.     

Supporting disciplinary and interdisciplinary knowledge development and design thinking in an informal,pre‐engineering program: A Workplace Simulation Project

In this paper, we examined students’ engagement in an implementation of a Workplace Simulation Project (WSP). The WSP was designed to actively engage students in learning disciplinary content by inviting engineers from industry to have a physical presence within the school building to collaborate with teachers and students to complete projects which simulate the tasks authentic to their work. We focus on the first year implementation of the program that partnered a high school in the rural Midwest with an engineering unit of a government organization. Using a multiple methods study design, we analyzed disciplinary and interdisciplinary pre and posts test along with students’ interviews to determine learning gains as well as students’ interpretations of creative and critical thinking as experienced in the project and their knowledge of the engineering design process. Effect sizes showed that students in the WSP group had notable gains over the control group participants. Additionally, students’ knowledge of core elements of the design process were identified in inductive analyses of the interviews. Findings from this study will provide usable knowledge about effective ways to support systems and design thinking and ways to support expert‐novice collaboration to ensure success.     

Preparing preservice elementary teachers to teach engineering: Impact on self‐efficacy and outcome expectancy

Research indicates there is a need for teachers to experience multiple mastery experiences with engineering teaching in order to improve teaching engineering self‐efficacy. To prepare future K–5 teachers to teach the engineering design process, one science methods course integrated 2‐day engineering mini‐units into the class meeting and school‐based field experience. The preservice teachers participated as students in an exemplar mini‐unit and then designed their own mini‐unit, which they later taught to K–5 students. Pre‐ and post‐testing of the preservice teachers indicated significant improvement in engineering pedagogical content knowledge self‐efficacy, engagement self‐efficacy, and disciplinary self‐efficacy. Significant improvement was not observed in engineering outcome expectancy.     

Applying weak equivalence of categories between partial map and pointed set against changing the condition of 2‐arms bandit problem

The main interest of the organizational learning is that the agent balances between exploring and exploiting. When the agent decides the answer in finite time, it gives rise to the problem that is trade‐off between exploring and exploiting. This problem never arises in the given enough time. The agent must decide answer from his imperfect information, when the time is given finite for the task. 2‐arms bandit problem is often taken into consideration in this problem. In previous article, we proposed the model for weak identity that is dynamically changing the hierarchy of his knowledge. Recently, the method of the heuristic model is attracted to approach this problem. In this article, we proposed different approaches for 2‐arms bandit problem, and the model we proposed can adapt to the environment when the condition of the problem is changed and shows another approach to the organizational learning. © 2010 Wiley Periodicals, Inc. Complexity 16: 10‐21, 2011     

The ice‐fishing problem: the fundamental sloshing frequency versus geometry of holes

We study an eignevalue problem with a spectral parameter in a boundary condition. This problem for the Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a half‐space covered by a rigid dock with some apertures (an ice sheet with fishing holes). The dependence of the fundamental eigenvalue on holes' geometry is investigated. We give conditions on a plane region guaranteeing that the fundamental eigenvalue corresponding to this region is larger than the fundamental eigenvalue corresponding to a single circular hole. Examples of regions satisfying these conditions and having the same area as the unit disk are given. New results are also obtained for the problem with a single circular hole. On the other hand, we construct regions for which the fundamental eigenfrequency is larger than the similar frequency for the circular hole of the same area and even as large as one wishes. In the latter examples, the hole regions are either not connected or bounded by a rather complicated curves. Copyright © 2004 John Wiley & Sons, Ltd.     

Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

Expanding the Role of K‐5 Science Instruction in Educational Reform: Implications of an Interdisciplinary Model for Integrating Science and Reading

Addressed is the current practice in educational reform of reducing time for science instruction in favor of traditional reading/language arts instruction. In contrast, presented is an evidence‐based rationale for increasing instructional time for K‐5 science instruction as an educational reform initiative. Overviewed are consensus interdisciplinary research and complementary multi‐year findings of the Science IDEAS model demonstrating the effectiveness of integrating conceptually‐relevant reading within science instruction in improving student achievement in both science and reading comprehension. Based on research summarized, increasing time for integrated K‐5 science is advocated as a meaningful reform‐based approach to science learning and reading comprehension proficiency that, in turn, better prepares students for subsequent success in science and content‐area reading comprehension across upper elementary and middle school grades (3–8).