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1.
This study examines new Turkish elementary school mathematics textbooks to provide perspectives on the quality of the tasks related to the proportion concept and the ways they are presented. Tasks were analysed for several dimensions with a particular focus on their level of cognitive demands (LCD). Tasks were distinguished in two groups in terms of LCD: lower-level demand and higher-level demand. The findings revealed that 75 % of the tasks were related to higher-level demand in that they requested a certain level of interpretation, required connecting knowledge and procedures related to each other, demanded responses with some explanation and reinforced students’ non-algorithmic thinking. Only 25 % of the tasks were related to a lower-level demand, and these tasks could be resolved by recalling and implementing rules, procedures and factual knowledge without reflecting upon the meaning behind them. Most of the tasks were presented in multiple representations and framed in non-mathematical contexts. All these task characteristics indicate that the new elementary school textbooks have the capacity to promote students’ proportional reasoning. The findings also inform the international community about crucial aspects of the curriculum reforms in Turkey and provide suggestions for teachers and textbook writers concerning the quality of the tasks and their selection and implementation in the classrooms.  相似文献   

2.
Explaining appears to dominate primary teachers’ understanding of mathematical reasoning when it is not confused with problem solving. Drawing on previous literature of mathematical reasoning, we generate a view of the critical aspects of reasoning that may assist primary teachers when designing and enacting tasks to elicit and develop mathematical reasoning. The task used in this study of children’s reasoning is a number commonality problem. We analysed written and verbal samples of reasoning gathered from children in grades 3 and 4 from three primary schools in Australia and one elementary school in Canada to map the variation in their reasoning. We found that comparing and contrasting was a critical aspect of forming conjectures when generalising in this context, an action not specified in frameworks for generalising in early algebra. The variance in children’s reasoning elicited through this task also illuminated the difference between explaining and justifying.  相似文献   

3.
Prospective elementary teachers must understand fraction division deeply in order to meaningfully teach this topic to their future students. This paper explores the nature of the subject content knowledge of fraction division possessed by a group of Taiwanese prospective elementary teachers at the beginning of their mathematics methods course. The findings provide preliminary evidence that many prospective Taiwanese elementary teachers have developed the knowledge package of fraction division as described by Ma (Knowing and teaching elementary mathematics: Teachers?? understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, 1999). The nature of various strategies used by these teachers provides further illustration of a secure common content knowledge that can serve as a benchmark for the development of mathematics courses for prospective teachers. However, the findings also show that the tasks of representing fraction division, through either word problems or pictorial diagrams, are challenging even for those highly proficient in elementary and middle school mathematics. The broader implications of this research for the international community are discussed, and recommendations for elementary teacher education programs are presented.  相似文献   

4.
Mathematics teacher education aims to improve teachers’ use of mathematical knowledge to support teaching and learning, an aspect of pedagogical content knowledge (PCK). In this study, we interviewed teachers to understand how they used mathematics to make sense of student solutions to proportional reasoning problems. The larger purpose was to find accurate ways of categorizing teachers’ ability to do this vital aspect of teaching and thereby to inform assessment, teacher education, and professional development. We conjectured that teachers’ PCK for proportional reasoning could be reliably described in terms of attention to quantitative meanings in story problem contexts and in terms of understanding naïve forms of proportional reasoning. Instead, our findings reveal that individual teachers used a variety of means to make sense of (1) cognitively similar student solutions to different tasks and (2) mathematically related steps of a student solution within a single task. These findings illustrate the complexity of PCK for the topic of proportional reasoning and suggest the limits of what can be inferred about teacher knowledge from teachers’ evaluations of student solutions. We discuss implications for teacher education and assessment.  相似文献   

5.
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6.
The purpose of this study was to investigate prospective mathematics teachers’ knowledge of common conceptions and misconceptions that sixth and seventh grade students had about multiplication of fractions. In addition, prospective teachers’ knowledge of the sources of these misconceptions and strategies they knew to overcome these misconceptions was examined. Data were collected from 17 Turkish prospective teachers at the end of the spring semester of the 2004–2005 academic year. A case study design was used in which data were collected through the Multiplication of Fractions Questionnaire and semi-structured interviews. The prospective teachers suggested many difficulties that elementary grade level students may have and stated that these difficulties stemmed from students’ lack of formal knowledge and rote memorization of the algorithms. In addition, the prospective teachers suggested many strategies that could be used to overcome these misconceptions or difficulties. These strategies could be grouped under three headings: strategies based on teaching methods, strategies based on formal knowledge of fractions, and strategies based on psychological constructs.  相似文献   

7.
In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed a new assessment instrument. These video-based prediction assessments engage prospective teachers in a video analysis of a child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and then use that model to predict how the child will respond to a subsequent task. In this paper, we share data concerning the evolution and effectiveness of the instrument. Results from implementation indicate moderate to high degrees of inter-rater reliability in using the rubric to assess prospective teachers’ models and predictions. They also indicate strong correlation between participation in the experimental course and prospective teachers’ performances on the video-based prediction assessments. Such findings suggest that prediction assessments effectively evaluate the pedagogical content knowledge that we are seeking to foster among the prospective teachers.  相似文献   

8.
The aim of this paper is to propose a theoretical model to analyze prospective teachers?? reasoning and knowledge of real numbers, and to provide an empirical verification of it. The model is based on Sierpinska??s theory of theoretical thinking. Data were collected from 59 prospective teachers through a written test and interviews. The data indicated that mathematical tasks on real numbers, based on Sierpinska??s theory, could be categorized according to whether they require reflective, systemic or analytic thinking. Analysis of the data identified three different groups of prospective teachers reflecting different types of theoretical thinking about real numbers. The interviews confirmed the empirical data from the written test, and provided a better insight into the thinking and characteristic features of the prospective teachers in each group. The analysis also indicated that the participants were more successful in tasks requiring systemic and analytic thinking, and only when this was achieved were they able to solve problems which required reflective thinking. Implications for teaching related to the findings of the study are discussed.  相似文献   

9.
Recent work in case-based reasoning (CBR) reinforces the importance of situated learning, expert cases, and authentic tasks and activities for novice learners. As novices engage CBR environments, they apprentice in the experts’ practices while developing the understanding, knowledge and skill of a given community. This study examined how prospective teachers, as novices in a semester-long course, engaged and developed expert-like practices using the case knowledge of experienced teachers who teach with technology. By engaging experienced teachers’ knowledge and skill via Web-enhanced cases, prospective teachers refined their understanding of teaching culture and teaching with technology as they transitioned to the teaching community.  相似文献   

10.
Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005) claimed that the development of proportional reasoning relies on various kinds of understanding and thinking processes. The critical components suggested were individuals’ understanding of the rational number subconstructs, unitizing, quantities and covariance, relative thinking, measurement and “reasoning up and down”. In this study, we empirically tested a theoretical model based on the one suggested by Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005), as well as an extended model which included an additional component of solving missing value proportional problems. Data were collected from 238 prospective kindergarten teachers. To a great extent, the data provided support for the extended model. These findings allow us to make some first speculations regarding the knowledge that prospective kindergarten teachers possess in regard to proportional reasoning and the types of processes that might be emphasized during their education.  相似文献   

11.
We report on interview results from a classroom teaching experiment in a Number and Operations course for prospective elementary teachers. Improving the number sense of this population is an important goal for mathematics teacher education, and researchers have found this goal to be difficult to accomplish. In earlier work, we devised a local instruction theory for the development of number sense, which focused on whole-number mental computation. In this study, the local instruction theory was applied to the rational-number domain, with the help of a framework for reasoning about fraction magnitude, and it guided instruction in the content course. We interviewed seven participants pre- and post-instruction, and we found that their reasoning on fraction comparison tasks improved. The participants made more correct comparisons, reasoned more flexibly, and came to favor less conventional and more sophisticated strategies. These improvements in number sense parallel those that we found previously in mental computation. In addition to the overall results, we highlight two cases of improvement that illustrate ways in which prospective elementary teachers’ reasoning about fraction magnitude can change.  相似文献   

12.
This documentary account situates teacher educator, prospective teacher, and elementary students’ mathematical thinking in relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning skill—creating and analyzing representations. The author examines responses of prospective teachers to a visual representation task and, in turn, their examination of school children’s responses to mathematical tasks. The analysis revealed the initial tendency of prospective teachers to create pictorial representations and highlights the importance of looking beyond the pictures created to how prospective teachers use mathematical models. In addition, the challenges prospective teachers face in moving beyond a ruled-based conception of mathematics and a right/wrong framework for assessing student work are documented. Findings suggest that analyzing representations helps prospective teachers (and teacher educators) rethink their teaching practices by engaging with a culture of teaching focused on reading for multiple meanings and posing questions about student thinking and curriculum materials.  相似文献   

13.
This study investigates algebraic reasoning and self-regulation skills among elementary school teachers who participated in a professional development program either with IMPROVE metacognitive questioning (PD + Meta) or with no metacognitive guidance (PD). Sixty-four Israeli teachers participated in a 3-year program designed to enhance mathematical knowledge. Results indicated that the PD + Meta teachers outperformed the PD teachers on various algebraic procedural and real-life tasks regarding conceptual mathematical explanations. In addition, the PD + Meta group outperformed the PD group in using self-monitoring and evaluation strategies in algebraic problem solving. We discuss educational and practical implications.  相似文献   

14.
As engineering learning experiences increasingly begin in elementary school, elementary teacher preparation programs are an important site for the study of teacher development in engineering education. In this article, we argue that the stances that novice teachers adopt toward engineering learning and knowledge are consequential for the opportunities they create for students. We present a comparative case study examining the epistemological framing dynamics of two novice urban teachers, Ana and Ben, as they learned and taught engineering design during a four-week institute for new elementary teachers. Although the two teachers had very similar teacher preparation backgrounds, they interpreted the purposes of engineering design learning and teaching in meaningfully different ways. During her own engineering sessions, Ana took up the goal not only of meeting the needs of the client but also of making scientific sense of artifacts that might meet those needs. When facilitating students' engineering, she prioritized their building knowledge collaboratively about how things work. By contrast, when Ben worked on his own engineering, he took up the goal of delivering a product. When teaching engineering to students, he offered them constrained prototyping tasks to serve as hands-on contexts for reviewing scientific explanations. These findings call for teacher educators to support teachers' framing of engineering design as a knowledge building enterprise through explicit conversations about epistemology, apprenticeship in sense-making strategies, and tasks intentionally designed to encourage reasoning about design artifacts.  相似文献   

15.
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16.
The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.  相似文献   

17.
In this paper, we focus on Finnish pre-service elementary teachers’ (N?=?269) and upper secondary students’ (N?=?1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6?=?83. How could you conclude from this relationship (without using long-division algorithm) what 491:6?=?? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency.  相似文献   

18.
Researchers have argued that integrating early algebra into elementary grades will better prepare students for algebra. However, currently little research exists to guide teacher preparation programs on how to prepare prospective elementary teachers to teach early algebra. This study examines the insights and challenges that prospective teachers experience when exploring early algebraic reasoning. Results from this study showed that developing informal representations for variables and unknowns and learning about the two interpretations of the equal sign were meaningful new insights for the prospective teachers. However, the prospective teachers found it a conceptual challenge to identify the relationships contained in algebraic expressions, to distinguish between unknowns and variables, to bracket their knowledge of formal algebra and to represent subtraction from unknowns or variables. These findings suggest that exploring early algebra is non-trivial for elementary prospective teachers and likely necessary to adequately prepare them to teach early algebra.  相似文献   

19.
This study examined awareness of equivalence and relational thinking exhibited by 30 preservice elementary teachers in order to assess their initial preparedness to engage students in these two important aspects of early algebraic reasoning. Findings indicated that preservice teachers collectively demonstrated an awareness of relational thinking both in identifying opportunities offered by tasks to engage students in this thinking and in identifying this thinking in samples of student work. However, in proposing difficulties students might have with selected tasks, few participants demonstrated the understanding that many elementary school students hold misconceptions about the meaning of the equal sign. Implications of these findings for preservice and inservice teacher education are discussed.  相似文献   

20.
Recognizing meaning in students’ mathematical ideas is challenging, especially when such ideas are different from standard mathematics. This study examined, through a teaching-scenario task, the reasoning and responses of prospective elementary and secondary teachers to a student’s non-traditional strategy for dividing fractions. Six categories of reasoning were constructed, making a distinction between deep and surface layers. The connections between the participants’ reasoning, their teaching response, and their beliefs about mathematics teaching were investigated. We found that there were not only differences but also similarities between the prospective elementary and secondary teachers’ reasoning and responses. We also found that those who unpacked the mathematical underpinning of the student’s non-traditional strategy tended to use what we call “teacher-focused” responses, whereas those doing less analysis work tended to construct “student-focused” responses. These results and their implications are discussed in relation to the influential factors the participants themselves identified to explain their approach to the given teaching-scenario task.
Sandra CrespoEmail:
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