Preservice Teachers’ Conceptual and Procedural Knowledge of Fraction Operations: A Comparative Study of the United States and Taiwan

This study examined (a) the differences in preservice teachers’ procedural knowledge in four areas of fraction operations in Taiwan and the United States, (b) the differences in preservice teachers’ conceptual knowledge in four areas of fraction operations in Taiwan and the United States, and (c) correlation in preservice teachers’ conceptual knowledge and procedural knowledge of fractions in Taiwan and the United States. Participants were preservice teachers (N = 49) in a teacher education program in the United States and comparable Chinese preservice teachers (N = 47). Results indicated that Chinese preservice teachers performed better in procedural knowledge on fraction operations than American preservice teachers. No significant differences were found for conceptual knowledge on fraction division. Further, the correlation in this study showed that for Chinese and American preservice teachers, the relationship between conceptual and procedural knowledge of fraction operations was weak.     

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.     

Web‐Based Instruction on Preservice Teachers' Knowledge of Fraction Operations

This study determines whether web‐based instruction (WBI) represents an improved method for helping preservice teachers learn procedural and conceptual knowledge of fractions.. The purpose was to compare the effectiveness of web‐based instruction (WBI) with the traditional lecture in mathematics content and methods for the elementary school course. The results of this study suggest that the use of WBI in learning fractions is more effective. When compared with the traditional instruction, the WBI treatment results were significantly more effective for procedural and conceptual knowledge of fraction operations.     

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.     

Computational estimation skill of preservice teachers: operation type and teacher view

Despite the importance of computational estimation skill for the improvement of number sense, little research exists on preservice teachers’ estimation skills and their view on estimation in the US context. This study examined the computational estimation skill of 58 preservice elementary teachers (PSTs) and its relationship to their views of the meaning of estimation and the importance of teaching it. Three sets of instruments were used: an estimation task, a computational task, and a belief survey. Results indicated that PSTs performed differently depending on the types of operations on the estimation test. It was also found that different types of problems elicited different strategies. Furthermore, the intervention of the study, along with five other factors were found to significantly correlate with estimation skills. The five factors include PSTs’ mathematical knowledge, their reported confidence about estimation skills, their self-reported knowledge about calculator use in instruction, their views of estimation in teaching mathematics, and their definition of estimation. A negative correlation was documented for the knowledge of calculator use in instruction, and positive correlations were present for other factors. Implications are discussed in accordance with these findings.     

How Preservice Teachers Interpret and Respond to Student Geometric Errors

Recognizing and responding to students' thinking is essential in teaching mathematics, especially when students provide incorrect solutions. This study examined, through a teaching scenario task, elementary preservice teachers' interpretations of and responses to a student's work on a task involving reflective symmetry. Findings revealed that a majority of preservice teachers identified the student's errors from conceptual aspects of reflection rather than from procedural aspects. However, when they responded to the student's errors, preservice teachers tried to cope with them by invoking procedural knowledge. This study also revealed the three types of responses and two different forms of address by preservice teachers to student errors; these categories might provide insight into the difficulties arising in communication between students and teachers.     

Science Talk: Preservice Teachers Facilitating Science Learning in Diverse Afterschool Environments

The purpose of this study was to assess the impact a community‐based service learning program might have on preservice teachers' science instruction during student teaching. Designed to promote science inquiry, preservice teachers learned how to offer students more opportunities to develop their own ways of thinking through utilization of an afterschool science program that provided them extended opportunities to practice their science teaching skills. Three preservice teachers were followed to examine and evaluate the transfer of this experience to their student teaching classroom. Investigation methods included field observations and semi‐structured, individual interviews. Findings indicate that preservice teachers expanded their ideas of science inquiry instruction to include multiple modes of formative assessment, while also struggling with the desire to give students the correct answer. While the participants' experiences are few in number, the potential of afterschool teaching experience serving as an effective learning experience in preservice teacher preparation is significant. With the constraints of high‐stakes testing, community‐based service learning teaching opportunities for elementary and middle‐school preservice teachers can support both the development and refinement of inquiry instruction skills.     

Issues and Challenges for Middle School Mathematics Teachers in Inclusion Classrooms

The purpose of this qualitative study was to investigate typical middle school general education mathematics teachers' beliefs and knowledge of students with learning disabilities and inclusive instruction and to gain an understanding of the process of inclusion as it is implemented in middle school classrooms. In‐depth interviews, surveys, and classroom observations were conducted with seven teachers. The constant comparative method was used to analyze all interview and observation data. The findings reveal that even teachers who believe that inclusion is being successfully implemented are unclear about their responsibilities towards included students and the learning characteristics and specific mathematics teaching approaches that would be effective. The general educators feel that they were grossly under‐prepared during preservice and inservice for the realities of inclusion teaching. The study provides insights that can be used to enhance preservice and inservice programs for teachers and underscores the necessity for building teamwork and collaboration among general and special education middle school teachers.     

Examining preservice teachers' responses to area conservation tasks

The purpose of this work was to explore how elementary preservice teachers responded to area conservation tasks. We administered written pre-assessments, followed by semi-structured interviews with 23 preservice teachers, asking them to respond to and reason with area conservation tasks. Findings highlighted several interesting preservice teachers' struggles when assessing area conservation tasks. In many cases, preservice teachers exhibited struggles similar to students, especially with regards to the justification of their area conservation claims. We provide recommendations to assist preservice teachers in their development of mathematical content knowledge in their teacher education programs, so that in the future they may better plan area lessons that promote procedural fluency from conceptual understanding in area measurement.     

Towards a framework for developing students' fraction proficiency

The importance of the knowledge of fractions in mathematical learning, coupled with the difficulties students have with them, has prompted researchers to focus on this particular area of mathematics. The term ‘fraction proficiency' used in this article refers to a person's conceptual comprehension, procedural skills and the ability to approach daily situations involving fractions. In the area of fractions, there has been a call for more research to study how, and where, efforts should be focused in order to integrate the various aspects of fraction knowledge for students, and even for teachers, to help them develop proficiency in fractions. Thus, the article presents a theoretical synthesis of the specialized literature in the learning and teaching of fractions, with the aim of proposing a framework for developing students' fraction proficiency. The frameworks presented in the article may shed light upon the implications for the design of fraction instruction, which should focus on developing a multi-faceted knowledge of fractions, rather than simply isolating one facet from the others.     

Investigating Changes in Preservice Teachers’ Conceptions of STEM Education Following Video Analysis and Reflection

Nationally, there is a steadily increasing emphasis on the improvement of STEM education. This includes the integration of STEM subjects that have been traditionally taught separately, making it critical that prospective STEM educators are equipped to teach using integrated STEM approaches. Connected, an important challenge is providing preservice STEM teachers with experiences in which they can develop an understanding of how to optimize learning through integrated STEM instruction. A potentially effective way to foster this conceptualization is through video analysis of integrated STEM practices. To investigate this possibility, here we present a semester‐long study focused on engaging preservice STEM teachers with observing, analyzing, and reflecting about instructional STEM practices through a video‐based intervention. Findings suggest that viewing and reflecting on integrated STEM practices may enhance preservice STEM teachers' conceptions of integrated STEM approaches, representing a practical method of preservice STEM teacher professional development.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

Diagram Literacy in Preservice Math Teachers,Computer Science Majors,and Typical Undergraduates: The Case of Matrices,Networks, and Hierarchies

According to the National Council of Teachers of Mathematics (2000), children need to learn how to create and use mathematical diagrams to represent and reason about phenomena in the world. The author proposes a model of diagram literacy that includes six types of knowledge required for diagrammatic competence - implicit, construction, similarity, structural, metacognitive, and translational. A study is reported that examined college students' diagram literacy for three interrelated mathematical diagrams - matrices, networks, and hierarchies. Three groups of students participated: preservice, secondary-level, math teachers; computer science majors; and typical undergraduates. The results of the study are reassuring in some ways concerning the level of diagram literacy possessed by students at the culmination of current K through 12 instruction and by teachers of future secondary students. However, the results also point to areas in which preservice math teachers should be better prepared if the goals for students' diagram literacy are to be met.     

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.     

Diagram Literacy in Preservice Math Teachers, Computer Science Majors, and Typical Undergraduates: The Case of Matrices, Networks, and Hierarchies

According to the National Council of Teachers of Mathematics (2000), children need to learn how to create and use mathematical diagrams to represent and reason about phenomena in the world. The author proposes a model of diagram literacy that includes six types of knowledge required for diagrammatic competence - implicit, construction, similarity, structural, metacognitive, and translational. A study is reported that examined college students' diagram literacy for three interrelated mathematical diagrams - matrices, networks, and hierarchies. Three groups of students participated: preservice, secondary-level, math teachers; computer science majors; and typical undergraduates. The results of the study are reassuring in some ways concerning the level of diagram literacy possessed by students at the culmination of current K through 12 instruction and by teachers of future secondary students. However, the results also point to areas in which preservice math teachers should be better prepared if the goals for students' diagram literacy are to be met.     

Engineering design and the development of knowledge for teaching among preservice science teachers

The goal of this article is to inform professional understanding regarding preservice science teachers’ knowledge of engineering and the engineering design process. Originating as a conceptual study of the appropriateness of “knowledge as design” as a framework for conducting science teacher education to support learning related to engineering design, the findings are informed by an ongoing research project. Perkins’s theory encapsulates knowledge as design within four complementary components of the nature of design. When using the structure of Perkins’s theory as a framework for analysis of data gathered from preservice teachers conducting engineering activities within an instructional methods course for secondary science, a concurrence between teacher knowledge development and the theory emerged. Initially, the individuals, who were participants in the research, were unfamiliar with engineering as a component of science teaching and expressed a lack of knowledge of engineering. The emergence of connections between Perkins’s theory of knowledge as design and knowledge development for teaching were found when examining preservice teachers’ development of creative and systematic thinking skills within the context of engineering design activities as well as examination of their knowledge of the application of science to problem‐solving situations.     

Non‐Traditional Preservice Teachers and Their Mathematics Efficacy Beliefs

In Florida, recent legislative changes have granted community colleges the ability to offer baccalaureate degrees in education, frequently to non‐traditional students. Based on information obtained from the literature covering preservice teachers' math knowledge, teachers' efficacy beliefs about math, and high‐stakes mathematics testing, a study examined a population of preservice teachers in a new Florida teacher preparation program. The research investigated relationships surrounding non‐traditional preservice teachers' characteristics such as: ages, high‐stakes math failures, lower division mathematics history, and math methods course performance, in relation to their efficacy beliefs about mathematics. Results revealed that preservice teachers' ages, lower division mathematics history, and math methods course performance, did have a significant relationship with their math efficacy beliefs, as measured by the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI); the variable of high‐stakes math failures did not. Additionally, a multiple regression model including the aforementioned variables did predict preservice teachers' MTEBI scores, but did not generalize to the greater population. The findings from this study can assist new teacher preparation programs in isolating variables that identify preservice teachers who are at risk for poor mathematical attitudes; can posit avenues for fostering positive math beliefs in preservice teachers; and can recommend further research in this area.     

Examining prospective mathematics teachers' pedagogical content knowledge on fractions in terms of students' mistakes

The aim of the study is to examine prospective mathematics teachers’ pedagogical content knowledge in terms of knowledge of understanding students and knowledge of instructional strategies which are the subcomponents of pedagogical content knowledge. The participants of this research consist of 98 prospective teachers who are studying in two universities in Turkey. The participants were selected with the purposive sampling method which is one of the non-random sampling methods. Case study method, which is based on the qualitative research approach, was used. The answers given by secondary school students to fraction-related open-ended questions in the study of Soylu and Soylu were used as the data collection tool. The obtained data were analyzed via the content analysis technique. The analyses showed that the prospective mathematics teachers’ pedagogical content knowledge on fractions was not at an adequate level in identifying and correcting students’ errors. However, it was observed that the prospective teachers experienced more difficulty in the knowledge of instructional strategies compared to the knowledge of understanding students.     

Transferring specialized content knowledge to elementary classrooms: preservice teachers’ learning to teach the associative property

This study explores how preservice teachers (PSTs) transfer the intended specialized content knowledge (SCK) to elementary classrooms. Focusing on the case of the associative property of multiplication, we compared three PSTs’ SCK during enacted lessons in fourth grade classrooms with their own learning in professional development (PD) settings. Findings revealed the PSTs’ successes and challenges in unpacking an example task, especially in areas of making connections between concrete and abstract representations and asking deep questions that target quantitative interactions. Factors that may have supported or hindered PSTs’ SCK transfer include the complex nature of teacher knowledge, the PD effort and the outside factors such as the support from textbooks and cooperating teachers. Implications for teacher education and directions for future research are discussed.     

Communication in Mathematics: Preparing Preservice Teachers to Include Writing in Mathematics Teaching and Learning

“Math was strictly math, from what I remember.” This is a comment about using writing in mathematics from a preservice elementary teacher enrolled in a methods course. Comments such as these concern teacher educators who wish to prepare elementary teachers to include writing in mathematics instruction. A teacher development experiment was completed to discover how to improve preservice teachers’ abilities and attitudes toward using writing in mathematics. The preservice teachers made use of a graphic organizer to facilitate writing in the college math methods class, then practiced teaching writing with the same graphic organizer and in the math classes in an elementary classroom. Reflections of the preservice teachers illustrated this was a positive practice. The preservice teachers also concluded that writing in mathematics is valuable to instruction and would include it in their teaching.