Fundamental Fraction Knowledge of Preservice Elementary Teachers: A Cross‐National Study in the United States and Taiwan

The purpose of this paper is to show the similarities as well as the differences of fundamental fraction knowledge owned by preservice elementary teachers from the United States (N= 89) and Taiwan (N= 85). To this end, we examined and compared their performance on an instrument including 15 multiple‐choice test items. The items were categorized into four different types of fundamental fraction constructs, including part–whole relationship, quotient, equivalence, and meanings of operations. Each item was embedded in the area, linear, or set model except for the items constructed out of the meaning of operations. Several items were featured with a pictorial illustration. Quantitative analysis showed that U.S. preservice teachers were significantly outperformed by their Taiwanese counterparts overall. The difference between the two groups was statistically significant on 12 of 15 items. Findings suggest that preservice elementary teachers from both countries need to be better prepared in their understanding of the meaning of fraction multiplication or division operations. Findings also suggest that U.S. preservice elementary teachers need to be more knowledgeable in dealing with fraction problems embedded in a linear model. Further research is suggested to study the issues raised from the findings.     

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.     

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.     

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.     

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.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

A comparative analysis of British and Taiwanese students’ conceptual and procedural knowledge of fraction addition

This study examines students’ procedural and conceptual achievement in fraction addition in England and Taiwan. A total of 1209 participants (561 British students and 648 Taiwanese students) at ages 12 and 13 were recruited from England and Taiwan to take part in the study. A quantitative design by means of a self-designed written test is adopted as central to the methodological considerations. The test has two major parts: the concept part and the skill part. The former is concerned with students’ conceptual knowledge of fraction addition and the latter is interested in students’ procedural competence when adding fractions.

There were statistically significant differences both in concept and skill parts between the British and Taiwanese groups with the latter having a higher score. The analysis of the students’ responses to the skill section indicates that the superiority of Taiwanese students’ procedural achievements over those of their British peers is because most of the former are able to apply algorithms to adding fractions far more successfully than the latter. Earlier, Hart [1 Hart KM. Children's understanding of mathematics: 11–16. Oxford: Alden Press; 1981. [Google Scholar]] reported that around 30% of the British students in their study used an erroneous strategy (adding tops and bottoms, for example, 2/3 + 1/7 = 3/10) while adding fractions. This study also finds that nearly the same percentage of the British group remained using this erroneous strategy to add fractions as Hart found in 1981.

The study also provides evidence to show that students’ understanding of fractions is confused and incomplete, even those who are successfully able to perform operations. More research is needed to be done to help students make sense of the operations and eventually attain computational competence with meaningful grounding in the domain of fractions.     

Preservice Teachers' Understanding of Perimeter and Area

Fifty-four postgraduate (elementary school) preservice teachers were given four tasks, two to assess their understanding of perimeter and two to assess their understanding of area. The teachers were asked to prepare a question that would assess student understanding of perimeter. Then they were given three problems and asked to decide whether the problems had sufficient information for a solution. The type of question prepared for the first task and the number of preservice teachers who stated that the other three tasks had insufficient information indicate a procedural understanding of perimeter and area, rather than a conceptual and relational understanding     

Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study

In this study, we implemented one-on-one fractions instruction to eight preservice teachers. The intervention, which was based on the principle of Progressive Formalization (Freudenthal, 1983), was centered on problem solving and on progressively formalizing the participants’ intuitive knowledge of fractions. The objectives of the study were to examine the potential effects of the intervention and to uncover specific difficulties experienced by the preservice teachers during instruction. Results revealed improvement on one measure of conceptual knowledge, but not on a transfer task, which required the teachers to generate word problems for number sentences involving fractions. In addition, the qualitative analysis of the videotaped instructional sessions revealed a number of cognitive obstacles encountered by the participants as they attempted to construct meaningful solutions and represent those solutions symbolically. Based on the findings, specific suggestions for modifying the intervention are provided for mathematics teacher educators.     

Comparing the Development of Fractions in the Fifth‐ and Sixth‐Graders' Textbooks of Singapore,Taiwan, and the USA

This research examined the presentation of fractions in textbooks used by fifth and sixth graders in Singapore, Taiwan, and the United States. The specific textbooks examined were My Pals Are Here! Maths (MPHM) in Singapore; Kung Hsung (KH) in Taiwan; and Mathematics in Context (MiC) in the USA. Results show the problems posed in MiC put more emphasis on real‐life situations than KH textbooks in Taiwan and MPHM in Singapore. Designing materials that provide opportunities to connect mathematics content with applications in real life is consistent with recommendations from professional organizations. The activities in KH and MPHM tended to emphasize procedures, while the activities of MiC focused more on conceptual understanding and less on the development of procedures. An examination of the mathematics textbooks revealed that MPHM introduced and developed fractions the earliest among the three countries investigated and the content taught in MPHM was about one grade earlier than when the same content was experienced by students in KH and MiC.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

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.     

Chinese elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division

In this study, we investigated the extent of knowledge in mathematics and pedagogy that Chinese practicing elementary mathematics teachers have and what changes teaching experience may bring to their knowledge. With a sample of 18 mathematics teachers from two elementary schools, we focused on both practicing teachers’ beliefs and perceptions about their own knowledge in mathematics and pedagogy and the extent of their knowledge on the topic of fraction division. The results revealed a gap between these teachers’ limited knowledge about the curriculum they teach and their solid mathematics knowledge for teaching, as an example, fraction division. Moreover, senior teachers used more diverse strategies that are concrete in nature than junior teachers in providing procedural justifications. The results suggested that Chinese practicing teachers benefit from teaching and in-service professional development for the improvement of their mathematics knowledge for teaching but not their knowledge about mathematics curriculum.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

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.     

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.     

Evaluating students’ conceptual and procedural understanding of sampling distributions

Introductory statistics courses, which are important in preparing students for their daily lives, generally derive inferential statistics from informal knowledge. In this transition process, sampling distributions have an important place, yet research has shown that students often have difficulties with this concept. In order to increase their understanding of sampling distributions, students should have a strong conceptual foundation that is balanced with procedural knowledge. To address this issue, this study was designed to examine the relationship between college students’ procedural and conceptual knowledge of sampling distributions. With this aim in mind, an achievement test consisting of two sections – procedural and conceptual knowledge – was prepared. In answering the questions related to procedural knowledge, the participants were more successful in identifying the relationship between standard deviation of a population and sample means. However, they lacked theoretical knowledge about statements that they had heard or knew intuitively. Simulation activities provided in statistics courses may support students in developing their conceptual understanding in this regard.     

Primary teachers’ subject matter knowledge: decimals

The main objective of this study was to investigate primary teachers’ subject matter knowledge in the domain of decimals and more elaborately to investigate their performance and difficulties in reading scale, ordering numbers, finding the nearest decimal and doing operations, such as addition and subtraction. The difficulties in these particular areas are analysed and suggestions are made regarding their causes. Further, factors that influence this knowledge were explored. The sample of the study was 63 primary teachers. A decimal concepts test including 18 tasks was administered and the total scores for the 63 primary teachers ranged from 3 to 18 with a mean and median of 12. Fifty per cent of the teachers were above the mean score. The detailed investigation of the responses revealed that the primary teachers faced similar difficulties that students and pre-service teachers faced. Discrepancy on teachers’ knowledge revealed important differences based on educational level attained, but not the number of years of teaching experience and experience in teaching decimals. Some suggestions have been made regarding the implications for pre- and in-service teacher training.