Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

Integrating history and philosophy in mathematics education at university level through problem-oriented project work

Through the last three decades several hundred problem-oriented student-directed projects concerning meta-aspects of mathematics and science have been performed in the 2-year interdisciplinary introductory science programme at Roskilde University. Three selected reports from this cohort of project reports are used to investigate and present empirical evidence for learning potentials of integrating history and philosophy in mathematics education. The three projects are: (1) a history project about the use of mathematics in biology that exhibits different epistemic cultures in mathematics and biology. (2) An educational project about the difficulties of learning mathematics that connects to the philosophy of mathematics. (3) A history of mathematics project that connects to the sociology of multiple discoveries. It is analyzed and discussed in what sense students gain first hand experiences with and learn about meta-aspects of mathematics and their mathematical foundation through the problem-oriented student-directed project work.     

Professional Development Strategically Connecting Mathematics and Science: The Impact on Teachers' Confidence and Practice

The press to integrate mathematics and science comes from researchers, business leaders, and educators, yet research that examines ways to support teachers in relating these disciplines is scant. Using research on science and mathematics professional development, we designed a professional development project to help elementary teachers improve their teaching of mathematics and science by strategically connecting these disciplines. The purposes of this study are: (a) to identify changes in teachers' confidence and practice after participating in the professional development and (b) to identify different ways to connect mathematics and science during the professional development. We use a Likert‐scale survey to assess changes in teachers' confidence related to teaching mathematics and science. In addition, we report on a thematic analysis of teachers' written responses to open‐ended questions that probed teachers' perceived changes in practice. We analyze field notes from observations of project workshops to document different types of opportunities for connecting mathematics and science. We conclude with implications for future professional development that connects mathematics and science in meaningful ways, as well as suggestions for future research.     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

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.     

Incorporating Mathematics into the Science Program of Students Labeled “At‐Risk”

Twenty three at‐risk high school female students who had failed mathematics and science in a traditional school setting were the subjects of this study that integrated mathematics and science lessons over a period of four weeks. Using a combination of direct instruction, calculations, graphing, hands‐on projects, and discussion, the topic of mechanical advantage was studied, to find out how well students understood the topic. The study found that these students who initially knew very little about mechanical advantage, and who did not see any need to use mathematics in the study of science, indicated an increased understanding of mechanical advantage, and also seemed to realize that integrating mathematics and science enhanced learning.     

Measurement Instruction in the Context of Scientific Investigations With Diverse Student Populations

The purpose of this study was to investigate the impact of a hands‐on science curriculum, which integrates mathematics and supports English language development, on third‐grade students' mathematics achievement—specifically the measurement subscale of the statewide assessment. The data drew from a larger five‐year research and development project consisting of reform‐based science curriculum units and teacher workshops designed to promote effective science instruction while integrating mathematics and supporting English language development. The third‐grade curriculum places a strong emphasis on measurement skills in the context of science inquiry. The third‐grade students' performance on the measurement subscale of the statewide mathematics assessment in the treatment schools was compared with that of comparison schools using a hierarchical linear model. Students at the treatment schools performed significantly higher than students at the comparison schools. The results provide evidence that an integrated approach to mathematics and science instruction can benefit diverse student groups.     

'Model your genes the mathematical way'--a mathematical biology workshop for secondary school teachers

This article describes a mathematical biology workshop givento secondary school teachers of the Danville area in Virginia,USA. The goal of the workshop was to enable teams of teacherswith biology and mathematics expertise to incorporate lessonplans in mathematical modelling into the curriculum. The biologicalfocus of the activities is the lactose operon in Escherichiacoli, one of the first known intracellular regulatory networks.The modelling approach utilizes Boolean networks and tools fromdiscrete mathematics for model simulation and analysis. Theworkshop structure simulated the team science approach commonin today's practice in computational molecular biology and thusrepresents a social case study in collaborative research. Theworkshop provided all the necessary background in molecularbiology and discrete mathematics required to complete the project.The activities developed in the workshop show students the valueof mathematical modelling in understanding biochemical networkmechanisms and dynamics. The use of Boolean networks, ratherthan the more common systems of differential equations, makesthe material accessible to students with a minimal mathematicalbackground. High school students can be exposed to the excitement of mathematicalbiology from both the biological and mathematical point of view.Through the development of instructional modules, high schoolbiology and mathematics courses can be joined without havingto restructure the curriculum for either subject. The relevanceof an early introduction to mathematical biology allows studentsnot only to learn curriculum material in a innovative setting,but also creates an awareness of new educational and careeropportunities that are arising from the interconnections betweenbiological and mathematical sciences. The materials used in this workshop are available at a websitecreated by the directors: http://polymath.vbi.vt.edu/mathbio2006/.     

Using a Functional Model to Develop a Mathematical Formula

The unifying theme of models was incorporated into a required Science Capstone course for pre‐service elementary teachers based on national standards in science and mathematics. A model of a teeter‐totter was selectedfor use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing the mathematical relationship for a Class 1 lever, M1D1=M1D1. In this study, 20 student groups (n=72) collected data using the model in an inquiry‐based activity. All groups developed the qualitative relationship, 13 groups developed a correct mathematical formula, 6 groups developed one‐half of the relationship (X = mass × distance), and 1 group attempted to develop a procedural relationship. The pre‐service elementary teachers used a variety of model types in the activity including visual/pictorial, functional/physical and mathematical‐both graphs and formulas. The use of the teeter‐totter model as a visual and functional model of a mathematical formula was a factor in developing the mathematical relationship.     

A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance

The problem of students entering university lacking basic mathematical skills is a critical issue in the Australian higher-education sector and relevant globally. The Maths Skills programme at La Trobe University has been developed to address under preparation in the first-year science cohort in the absence of an institutional mathematics support centre. The programme was delivered through first-year science and statistics subjects with large enrolments and focused on basic mathematical skills relevant to each science discipline. The programme offered a new approach to the traditional mathematical support centre or class. It was designed through close collaboration between science subject coordinators and the project leader, a mathematician, and includes resources relevant to science and mathematics questions written in context. Evaluation of the programme showed it improved the confidence of the participating students who found it helpful and relevant. The programme was delivered through three learning modes to allow students to select activities most suitable for them, which was appreciated by students. Mathematics skills appeared to increase following completion of the programme and student participation in the programme correlated positively and highly with academic grades in their relevant science subjects. This programme offers an alternative model for mathematics support tailored to science disciplines.     

Assessing science students’ attitudes to mathematics: a case study on a modelling project with mathematical software

This article reports the attitudes of students towards mathematics after they had participated in an applied mathematical modelling project that was part of an Applied Mathematics course. The students were majoring in Earth Science at the National Taiwan Normal University. Twenty-six students took part in the project. It was the first time a mathematical modelling project had been incorporated into the Applied Mathematics course for such students at this University. This was also the first time the students experienced applied mathematical modelling and used the mathematical software. The main aim of this modelling project was to assess whether the students’ attitudes toward mathematics changed after participating in the project. We used two questionnaires and interviews to assess the students. The results were encouraging especially the attitude of enjoyment. Hence the approach of the modelling project seems to be an effective method for Earth Science students.     

Supporting Middle School Teachers’ Implementation of STEM Design Challenges

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers’ content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers’ beliefs about engaging underperforming students in challenging problems. A unique aspect involved low‐achieving students and their teachers learning alongside each other as they co‐participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students’ use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

“Everything is Math in the Whole World”: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students

This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions related to ensuring rich mathematics in the contexts of projects that were personally and socially meaningful to the students. Our analysis offers insights into critical mathematics education with elementary aged students, and has the potential to counter dominant deficit perspectives of Latina/o youth. Additionally, the findings of this study inform critical approaches to teaching mathematics in schools attended by marginalized students in order to reverse prevalent trends of our educational system failing these students.     

Applying Mathematical Concepts with Hands‐On,Food‐Based Science Curriculum

This article addresses the current state of the mathematics education system in the United States and provides a possible solution to the contributing issues. As a result of lower performance in primary mathematics, American students are not acquiring the necessary quantitative literacy skills to become successful adults. This study analyzed the impact of the Food, Math, and Science Teaching Enhancement Resource (FoodMASTER) Intermediate curriculum on fourth‐grade students' mathematics knowledge. The curriculum is a part of the FoodMASTER Initiative, which is a compilation of programs utilizing food, a familiar and necessary part of everyday life, as a tool to teach mathematics and science. Students exposed to the curriculum completed a 20‐item researcher‐developed mathematics knowledge exam (intervention n = 288; control n = 194). Overall, the results showed a significant increase in mathematics knowledge from pretest to posttest. These findings suggest that the food‐based science activities provided the students with the context in which to apply mathematical concepts to an everyday experience. Therefore, the FoodMASTER approach was successful at improving students' mathematics knowledge while building a foundation for becoming quantitatively literate adults.     

A Study of STEM Assessments in Engineering,Science, and Mathematics for Elementary and Middle School Students

The purpose of this study was to develop, scale, and validate assessments in engineering, science, and mathematics with grade appropriate items that were sensitive to the curriculum developed by teachers. The use of item response theory to assess item functioning was a focus of the study. The work is part of a larger project focused on increasing student learning in science, technology, engineering, and mathematics (STEM)‐related areas in grades 4–8 through an engineering design‐based, integrated approach to STEM instruction and assessment. The fact that the assessments are available to school districts at no cost, and represent psychometrically sound instruments that are sensitive to STEM‐oriented curriculum, offers schools an important tool for gauging students' understanding of engineering, science, and mathematics concepts.     

Implications of Informal Education Experiences for Mathematics Teachers' Ability to Make Connections Beyond Formal Classroom

The Common Core Standard for Mathematical Practice 4: Model with Mathematics specifies that mathematically proficient students are able to make connections between school mathematics and its applications to solving real‐world problems. Hence, mathematics teachers are expected to incorporate connections between mathematical concepts they teach and their applications to solving problems arising in real‐world situation. Clearly, it is assumed that the teachers themselves are able to make such connections. On the other hand, research shows that mathematics teachers find it difficult to make those connections. In this paper, we present the results of the study that investigated the ways in which exploring mathematics in informal sites, and in particular science museum, assist teachers with making connections between school mathematics and its applications in real world.     

Mathematical aspects of educating architecture designers: a college study

This paper considers a second-year Mathematical Aspects in Architectural Design course, which relies on a first-year mathematics course and offers mathematical learning as part of hands-on practice in architecture design studio. The 16-hour course consisted of seminar presentations of mathematics concepts, their application to covering the plane by regular shapes (tessellations), and an architecture design project. The course follow-up examined the features of mathematical learning in the studio environment using qualitative methods. It showed students’ curiosity and motivation to deepen in mathematical subjects and use them in their tessellation design projects. The majority of the students refreshed and practically applied their background mathematical knowledge, especially in calculus, on a need-to-know basis.     

The impact of integrated STEM modeling on elementary preservice teachers’ self‐efficacy for integrated STEM instruction: A co‐teaching approach

The purpose of the current study was to evaluate the impact of co‐taught integrated STEM methods instruction on preservice elementary teachers’ self‐efficacy for teaching science and mathematics within an integrated STEM framework. Two instructional methods courses (Elementary Mathematics Methods and Elementary Science Methods) were redesigned to include STEM integration components, including STEM model lessons co‐taught by a mathematics and science educator, as well as a special education colleague. Quantitative data were gathered at three time points in the semester (beginning, middle, and end) from 55 preservice teachers examining teacher self‐efficacy for integrated STEM teaching. Qualitative data were gathered from a purposeful sample of seven preservice teachers to further understand preservice teachers’ perceptions on delivering integrated STEM instruction in an elementary setting. Quantitative results showed a significant increase in teacher self‐efficacy across all three time points. Item‐level analysis revealed that self‐efficacy for tasks involving engineering and assessment (both formative and summative) were low across time points, while self‐efficacy for tasks involving technology and flexibility were consistently high. Qualitative results revealed that the preservice teachers did not feel adequately prepared by university‐level science and mathematics courses, in terms of content knowledge and integration of science and mathematics for elementary students.     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.