The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

Formative Use of Intuitive Analysis of Variance

Students’ informal inferential reasoning (IIR) is often inconsistent with the normative logic underlying formal statistical methods such as Analysis of Variance (ANOVA), even after instruction. In two experiments reported here, student's IIR was assessed using an intuitive ANOVA task at the beginning and end of a statistics course. In both experiments, students were provided feedback regarding the normative logic underlying ANOVA and how their reasoning compared with it. Additionally, students in Experiment 2 were given an assignment in which they analyzed and interpreted other students’ performance on the intuitive ANOVA task. Results indicate that the feedback combined with the assignment (which required active explanation of both normative and non-normative reasoning applied to the task) led to more normative inferential reasoning at the end of the course, whereas providing feedback alone did not. Implications are discussed for using the intuitive ANOVA task as a formative classroom tool to help students improve their conceptual understanding of ANOVA.     

Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities

Contributing to a growing body of research addressing secondary students’ quantitative and covariational reasoning, the multiple case study reported in this article investigated secondary students’ quantification of ratio and rate. This article reports results from a study investigating students’ quantification of rate and ratio as relationships between quantities and presents the Change in Covarying Quantities Framework, which builds from Carlson, Jacobs, Coe, Larsen, and Hsu’s (2002) Covariation Framework. Each of the students in this study was consistent in terms of the quantitative operation he or she used (comparison or coordination) when quantifying both ratio and rate. Illustrating how students can engage in different quantitative operations when quantifying rate, the Change in Covarying Quantities Framework helps to explain why students classified as operating at a particular level of covariational reasoning appear to be using different mental actions. Implications of this research include recommendations for designing instructional tasks to foster students’ quantitative and covariational reasoning.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers

This paper reports the findings of an investigation of 11 preservice secondary school teachers' interpretations of the development of proportional reasoning strategies used by middle school students. The preservice teachers examined samples of solution strategies generated by middle school students in proportional reasoning situations and prepared written responses of their views concerning the developmental levels indicated in the students' work. Each preservice teacher also participated in an hour‐long interview, in which the researchers asked for elaboration and clarification of the written responses and, in some cases, challenged the preservice teachers to consider alternative interpretations for the middle school students' work. The interviews were audiotaped for later analysis by the investigators, and key aspects of both the written and audiotaped responses were entered into a spreadsheet and later tabulated into categories indicating trends in the preservice teachers' interpretations. Some implications for the preparation of preservice middle school science and mathematics teachers are included.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

The introduction of real numbers in secondary education: an institutional analysis of textbooks

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallard's Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks. It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block.     

Declining participation in post-compulsory secondary school mathematics: students’ views of and solutions to the problem

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.     

Urban Third Grade Teachers' Practices and Perceptions in Science Instruction with English Language Learners

The study examined relationships among key domains of science instruction with English language learning (ELL) students based on teachers' perceptions of their classroom practices (i.e., what they think they do) and actual classroom practices (i.e., what they are observed doing). The four domains under investigation included: (1) teachers' knowledge of science content; (2) teaching practices to support scientific understanding; (3) teaching practices to support scientific inquiry; and (4) teaching practices to support English language development during science instruction. The study involved 38 third‐grade teachers participating in the first‐year implementation of a professional development intervention aimed at improving science and literacy achievement of ELL students in urban elementary schools. Based on teachers' self‐reports, practices for understanding were related to practices for inquiry and practices for English language development. Based on classroom observations in the fall and spring, practices for understanding were related to practices for inquiry, practices for English language development, and teacher knowledge of science content. However, we found a weak to non‐existent relationship between teachers' self‐reports and observations of their practices.     

AHP多路径排序度量转换中的非线性问题

由于不同测量条件下的测量结果不是线性可加,AHP用矩阵乘法实现多路径序转换值得商榷.自隶属度从只取"1或0"两个值扩展到可取[0,1]区间上一切实数,可表征界于"是"与"不是"之间所有可能"部分是"模糊状态时起,对二值逻辑的研究已拓展到研究近似推理的模糊逻辑.这是逻辑的一个新的研究方向,目的是在隶属度转换过程中,通过对人类近似推理本领进行规范,使得到的目标值是"真值"在当前条件下的最优近似.模糊逻辑的量化方法是数值计算;推理依据是区分权滤波的冗余理论;实质性计算是由冗余理论导出的、实现隶属度转换的非线性去冗算法;所建的隶属度转换模型也是不同测量条件下高维状态空间上测量结果的非线性可加模型.将一维测量数据映射到高维状态空间上表为隶属度向量,可借助隶属度转换模型解决AHP多路径序转换的非线性计算.     

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 Effects on Students' Cognitive Achievement When Using the Cooperative Learning Method in Earth Science Classrooms

This study investigated the effects of cooperative learning instruction versus traditional teaching methods on students' earth science achievement in secondary schools. A total of 770 ninth-grade students enrolled in 20 sections of a required earth science course participated in this nonequivalent control group quasi-experiment. The control groups (n= 10) received a traditional approach, while the experimental groups (n= 10) used cooperative strategies. Study results include (a) no significant differences were found between the experimental groups and the control groups when overall achievement (F= 0.13, p > .05), knowledge-level (F= 0.12, p > .05), and comprehension-level (F= 0.34, p > .05) test items were considered; and (b) students who worked cooperatively performed significantly better than students who worked alone on the application-level test items (F= 4.63, p < .05). These findings suggest that cooperative-learning strategies favor students' earth science performance at higher but not lower levels of cognitive domains in the secondary schools.     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Representing scientific activity: Affordances and constraints of central design and enactment features of a model‐based inquiry unit

This research explores how explaining an anchoring phenomena and engaging students in investigations, as central designs of a model‐based inquiry (MBI) unit, afforded or constrained the representation of scientific activity in the science classroom. This research is considered timely as recent standards documents and scholars in the field have highlighted the significance of identifying what features of scientific activity are important and how these can be represented for students in classrooms. Through taking advantage of qualitative research methods to closely examine the enactment of an MBI unit, both affordances and constraints were identified for each design. More specifically, explaining an anchoring phenomenon provided a context for more authentically framing the work of students, while investigations afforded students insight into the role these play in the refinement of models. Further, the teacher's attempts to support student reasoning and, at times, reasoning for students when they were found struggling were the most salient constraints identified connected to explaining an anchoring phenomenon and engaging students in investigations.     

Using the Tower of Hanoi puzzle to infuse your mathematics classroom with computer science concepts

This article suggests that logic puzzles, such as the well-known Tower of Hanoi puzzle, can be used to introduce computer science concepts to mathematics students of all ages. Mathematics teachers introduce their students to computer science concepts that are enacted spontaneously and subconsciously throughout the solution to the Tower of Hanoi puzzle. These concepts include, but are not limited to, conditionals, iteration, and recursion. Lessons, such as the one proposed in this article, are easily implementable in mathematics classrooms and extracurricular programmes as they are good candidates for ‘drop in’ lessons that do not need to fit into any particular place in the typical curriculum sequence. As an example for readers, the author describes how she used the puzzle in her own Number Sense and Logic course during the federally funded Upward Bound Math/Science summer programme for college-intending low-income high school students. The article explains each computer science term with real-life and mathematical examples, applies each term to the Tower of Hanoi puzzle solution, and describes how students connected the terms to their own solutions of the puzzle. It is timely and important to expose mathematics students to computer science concepts. Given the rate at which technology is currently advancing, and our increased dependence on technology in our daily lives, it has become more important than ever for children to be exposed to computer science. Yet, despite the importance of exposing today's children to computer science, many children are not given adequate opportunity to learn computer science in schools. In the United States, for example, most students finish high school without ever taking a computing course. Mathematics lessons, such as the one described in this article, can help to make computer science more accessible to students who may have otherwise had little opportunity to be introduced to these increasingly important concepts.     

Elements of Design‐Based Science Activities That Affect Students' Motivation

The primary purpose of this study was to examine the ways in which a 12‐week afterschool science and engineering program affected middle school students' motivation to engage in science and engineering activities. We used current motivation research and theory as a conceptual framework to assess 14 students' motivation through questionnaires, structured interviews, and observations. Students reported that during the activities they perceived that they were empowered to make choices in how to complete things, the activities were useful to them, they could succeed in the activities, they enjoyed and were interested in the hands‐on activities and some presentations, they felt cared for by the facilitators and received help when they were stuck or confused, and they put forth effort. Based on our examination of data across our three data sources, we identified motivating opportunities that were provided to students during the activities. These motivating opportunities can serve as examples to help both formal and informal science educators better connect motivation theory to practice so that they can create motivating opportunities for students. Furthermore, this study provides a methodological example of how students' motivation can be examined during the context of authentic science and engineering instruction.