The Advantages of an STS Approach Over a Typical Textbook Dominated Approach in Middle School Science

Two sections of middle school science were taught by two longtime teachers where one used an STS approach and the other followed the more typical textbook approach closely. Pre‐ and post assessments were administered to one section of students for each teacher. The testing focused on student concept mastery, general science achievement, concept applications, use of concepts in new situations, and attitudes toward science. Videotapes of classroom actions were recorded and analyzed to determine the level of the use of STS teaching strategies in the two sections. Information was also be collected that gave evidence of and noted changes in student creativity and the continuation of student learning and the use of it beyond the classroom. Major findings indicate that students experiencing the STS format where constructivist teaching practices were used to (a) learn basic concepts as well as students who studied them directly from the textbook, (b) achieve as much in terms of general concept mastery as students who studied almost exclusively by using a textbook closely, (c) apply science concepts in new situations better than students who studied science in a more traditional way, (d) develop more positive attitudes about science, (e) exhibit creativity skills more often and more uniquely, and (f) learn and use science at home and in the community more than did students in the textbook dominated classroom.     

Metaphor Clusters: Characterizing Instructor Metaphorical Reasoning on Limit Concepts in College Calculus

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student metaphorical reasoning, we examined 11 college instructors' metaphorical reasoning on limit concepts. This paper focused on previous research of metaphor clusters observed among students to answer the following: (a) Do college instructors use metaphorical reasoning to conceptualize the meaning of a limit? (b) Can we characterize instructor metaphorical reasoning similar to those observed among students? (c) Will an instructor's self‐identification of metaphor clusters be consistent with our metaphor coding? We found that college instructors' perspectives vary, either graphical or algebraic, in their explanations of limit items. All the instructors used metaphors, and instructor metaphorical reasoning aligned with student metaphor clusters. Instructors tended to change their metaphors with respect to the limit item. Instructors were not aware of their use of metaphors, nor were they aware of their inconsistency in their choice of metaphor. We believe that instructor awareness of their own distinct perspectives and metaphors would assist students' understanding of limit concepts.     

The Advantages of an Inquiry Approach for Science Instruction in Middle Grades

The purpose of the study was to examine the effectiveness of the Iowa Chautauqua Professional Development Program in terms of changes in concept mastery, use of process skills, application of science concept and skills, student attitudes toward science, student creativity, and student perceptions regarding their science classrooms. Participants were 12 teachers who agreed to participate in an experimental study where an inquiry approach was utilized with one section and traditional strategies in another section. A total of 24 sections of students were enrolled in inquiry sections (365 students) and traditional sections (359 students). The data collected were analyzed using quantitative methods. The results are tabulated and contrasted for students enrolled in the two sections for each teacher. The results indicate that student use and understanding of science skills and concepts in the inquiry sections increased significantly more than they did for students enrolled in typical sections in terms of process skills, creativity skills, ability to apply science concepts, and the development of more positive attitudes.     

Characteristics of Second Graders' Mathematical Writing

This study compared the characteristics of second graders' mathematical writing between an intervention and comparison group. Two six‐week Project M2 units were implemented with students in the intervention group. The units position students to communicate in ways similar to mathematicians, including engaging in verbal discourse where they themselves make sense of the mathematics through discussion and debate, writing about their reasoning on an ongoing basis, and utilizing mathematical vocabulary while communicating in any medium. Students in the comparison group learned from the regular school curriculum. Students in both the intervention and comparison groups conveyed high and low levels of content knowledge as indicated in archived data from an open‐response end‐of‐the‐year assessment. A multivariate analysis of variance indicated several differences favoring the intervention group. Both the high‐ and low‐level intervention subgroups outperformed the comparison group in their ability to (a) provide reasoning, (b) attempt to use formal mathematical vocabulary, and (c) correctly use formal mathematical vocabulary in their writing. The low‐level intervention subgroup also outperformed the respective comparison subgroup in their use of (a) complete sentences and (b) linking words. There were no differences between groups in their attempt at writing and attempts at and usage of informal mathematical vocabulary.     

Proof validation and modification in secondary school geometry

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos’ notion of local counterexample that rejects a specific step in a proof. By using Toulmin’s framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students’ learning of proof and proving.     

The Border Crossings of a Multicultural Science Education Enthusiast

Jill was a preservice science education student who wanted to make science more accessible to all students. This study is an examination of the “borders” she encountered as she completed her student teaching in a cultural setting that was different from her own. Her student teaching experience was documented through interviews, participant observations, field notes, lesson plans, and a journal. An inductive analysis of the documents and a context chart of the coded data revealed that Jill encountered the (a) cultural border of her students, (b) cultural border of science instruction, and (c) cultural border of the school. While some borders were crossed, others were not. This study suggests that during field experiences, preservice teachers may encounter multiple cultural borders, some consistent and some inconsistent with their instructional philosophy. As student teachers work with diverse populations, supervisors and cooperating teachers need to recognize the borders student teachers will encounter and encourage student teachers to examine their beliefs about practice as a means to acknowledge and understand the encountered borders.     

An analysis of the amount and characteristics of writing prompts in Grade 3 mathematics student books

Curriculum guidelines and professional organizations’ recommendations lack details about how often and how much students should write in mathematics and what characteristics should define their writing. This study presents an analytic framework that addresses how often students are prompted in student mathematics books to write, how much they may be encouraged to write, and the characteristics of the writing prompts. Consequently, 2,095 writing prompts in student books across 10 comprehensive Grade 3 resources were analyzed. Findings indicate a marked variation in how often and how much students are positioned to write. Most prompts have students explain what they did to solve a problem and why about number concepts, with most pressing for procedures. The greatest percentage of prompts had students write about their own solutions and do not urge them to include specific writing features.     

The Impact of Professional Noticing on Teachers' Adaptations of Challenging Tasks

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.     

Recursive and explicit rules: How can we build student algebraic understanding?

Differing perspectives have been offered about student use of recursive and explicit rules. These include: (a) promoting the use of explicit rules over the use of recursive rules, and (b) encouraging student use of both recursive and explicit rules. This study sought to explore students’ use of recursive and explicit rules by examining the reasoning of 25 sixth-grade students, including a focus on four target students, as they approached tasks in which they were required to develop generalizations while using computer spreadsheets as an instructional tool. The results demonstrate the difficulty that students had moving from the successful use of recursive rules toward explicit rules. In particular, two students abandoned general reasoning, instead focusing on particular values in an attempt to construct explicit rules. It is recommended that students be encouraged to connect recursive and explicit rules as a potential means for constructing successful generalizations.     

Technological Tools in the Introductory Statistics Classroom: Effects on Student Understanding of Inferential Statistics

While technology has become an integral part of introductory statistics courses, the programs typically employed are professional packages designed primarily for data analysis rather than for learning. Findings from several studies suggest that use of such software in the introductory statistics classroom may not be very effective in helping students to build intuitions about the fundamental statistical ideas of sampling distribution and inferential statistics. The paper describes an instructional experiment which explored the capabilities of Fathom, one of several recently-developed packages explicitly designed to enhance learning. Findings from the study indicate that use of Fathom led students to the construction of a fairly coherent mental model of sampling distributions and other key concepts related to statistical inference. The insights gained point to a number of critical ingredients that statistics educators should consider when choosing statistical software. They also provide suggestions about how to approach the particularly challenging topic of statistical inference. This revised version was published online in July 2006 with corrections to the Cover Date.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

Using Similarities and Differences: A Meta‐Analysis of Its Effects and Emergent Patterns

The purpose of the study was to update previous meta‐analytic findings on the effectiveness of using similarities and differences as an instructional strategy. The strategy includes facilitating student comparison, classification, use of analogies, and use of metaphors. Previously, Marzano, Pickering, and Pollock reported a mean effect size of 1.61. For the present meta‐analysis, literature was searched to locate experimental studies meeting the following inclusion criteria: published between 1998 and 2008; examined effects of facilitating student comparison, classification, use of analogies, and use of metaphors; measured student academic achievement as an outcome; involved students in kindergarten through grade 12; and provided data necessary to compute effect sizes. Based on the eligible research, the overall mean effect size (Hedges' g) was .65, confirming earlier results indicating that using similarities and differences positively influences student achievement. Type of control group, however, moderated the effect. Larger effect sizes were associated with control groups receiving textbook‐guided instruction versus those receiving interactive instruction. Emergent patterns were observed for the positive influence of long‐term instruction, systematic instruction, supportive cuing, and opportunity for reflection and discussion. Results support recommendations to guide students through analogical reasoning about, and classification of, important concepts and relationships in content‐area instruction.     

Changes in Student Attitudes Regarding Science When Taught by Teachers Without Experiences With a Model Professional Development Program

This study focuses on two main issues concerning changes in student attitudes toward science study and their perceptions of its usefulness in their lives. Information has been gathered concerning how student attitudes toward science have changed for teachers and schools not involved with any funded professional development project. Pretesting and posttesting were administered with such “control” groups at the same intervals corresponding with the data collected from students with teachers enrolled in five funded Professional Development projects over the 1981–2008 interim. The grade levels used by the National Assessment of Education Progress in their 1977 assessment of science were used; it focused on students in grades 3, 7, and 11. The results indicate a steady decline in student positive attitudes concerning their science study as grade levels increase. Conversely, the student perceptions of the usefulness of their science study as related to daily living, further science study, and for potential careers remained much the same over the 30‐year interim is a second focus. Generally, results indicate that traditional teaching and major use of textbooks cause increasingly negative student attitudes about science while not producing major changes in their perceptions of its usefulness in their lives.     

Do students really understand what an ordinary differential equation is?

Differential equations (DEs) are important in mathematics as well as in science and the social sciences. Thus, the study of DEs has been included in various courses in different departments in higher education. The importance of DEs has attracted the attention of many researchers who have generally focussed on the content and instruction of DEs. However, DEs are complex issues that students may find difficulty to understand. The limited research in this literature points to the need for more studies on students’ conceptions, and understanding of DEs and their basic concepts. The objective of this study is to fill this need by revealing the understanding, difficulties and weaknesses of the students who are successful in algebraic solutions, in relation to the concepts of DEs and their solutions. For this purpose, 77 students were asked 13 DE questions (6 of them about algebraic solution, and the rest about interpreting DEs and their solutions). From an analysis of the students’ answers, it was concluded that the students who were quite successful in algebraic solutions, indeed did not fully understand the related concepts, and they had serious difficulties in relation to these concepts.     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

Understanding How Students Develop Mathematical Models

in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mismatches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technology-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in problem situations that elicit the development of significant mathematical models.     

Understanding How Students Develop Mathematical Models

in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mismatches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technology-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in problem situations that elicit the development of significant mathematical models.