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1.
Multiplicative reasoning is critical for the development of more advanced mathematical concepts such as fractions and algebra, and therefore is a critical process for teachers and researchers to assess in an effective and efficient manner. Unfortunately, the vast majority of available assessments examine students’ skill with algorithms or memorized facts, with no necessary connection to students’ conceptual understanding of the content. This study reports on the continued development and validation of the Multiplicative Reasoning Assessment (MRA) for elementary grades. Data from 196 fourth and fifth grade students was examined using a Rasch modeling approach. Supplemental data from follow-up clinical interviews was also collected and analyzed. Results provide evidence for the reliability and validity of the MRA. Specifically, findings suggest the assessment adequately assesses students’ multiplicative reasoning, as an alternative to assessments of fact memorization or procedural fluency.  相似文献   

2.
Qualitative classroom data from video recordings and students' written work can play important roles in improving mathematics instruction. In order to take full advantage of these data sources, it is helpful to have a strong analytic lens to orient one's reflections on the data. One promising analytic lens is the National Research Council's five stands of mathematical proficiency framework. It prompts teachers to examine the extent to which their students have attained conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive dispositions. This article describes how a pair of prospective teachers used the five strands to analyze and reflect upon qualitative classroom data from a series of lessons they taught. The insights gained during the process can help others anticipate dynamics involved in using the five strands framework during practitioner research and reflection on classroom data.  相似文献   

3.
Although interim assessments are currently promoted as a mechanism for improving teaching and student learning, we know little about how teachers use this data to modify instruction. This article presents findings from a larger study on teachers’ use of interim assessment information in elementary mathematics. We address the following questions: (a) How do the Philadelphia teachers in our sample analyze benchmark assessment results, (b) how do they plan instruction based on these results, and (c) what are their reported instructional responses to such results? To answer these questions, we interviewed all 3rd- and 5th-grade teachers in five average- and above-average-performing elementary schools three times during the 2006–07 school year. We found that although the teachers in our study used interim assessment results to gain information about students’ learning in mathematics, teachers did not use interim assessments to make sense of students’ conceptual understanding. Furthermore, teachers’ tendency to interpret student errors as procedural missteps was paralleled by a trend toward procedural instructional responses.  相似文献   

4.
Based on Common Core Standards (2010), mathematics interventions should emphasize conceptual understanding of numbers and operations as well as fluency. For students at risk for failure, the concrete‐representational‐abstract (CRA) sequence and the Strategic Instruction Model (SIM) have been shown effective in teaching computation with an emphasis on conceptual understanding. No studies have investigated the effects of CRA and SIM for teaching multiplication with regrouping. Therefore, the purpose of this study was to replicate and extend the literature, teaching subtraction and multiplication with regrouping using CRA and SIM. Three students receiving tier three mathematics interventions participated. A multiple‐probe across behaviors design was used to show a functional relation. All of the students demonstrated increases in fluency across all regrouping tasks.  相似文献   

5.
In a study with ninth-grade mathematics students on learning the concept of variance, students experienced either direct instruction (DI) or productive failure (PF), wherein they were first asked to generate a quantitative index for variance without any guidance before receiving DI on the concept. Whereas DI students relied only on the canonical formulation of variance taught to them, PF students generated a diversity of formulations for variance but were unsuccessful in developing the canonical formulation. On the posttest however, PF students significantly outperformed DI students on conceptual understanding and transfer without compromising procedural fluency. These results challenge the claim that there is little efficacy in having learners solve problems targeting concepts that are novel to them, and that DI needs to happen before learners should solve problems on their own.  相似文献   

6.
The gap in achievement in mathematics between at-risk students and their more advantaged counterparts is a persistent problem of the U.S. education system. Although some research-based curricula and pedagogy have demonstrated promise in supporting students from diverse backgrounds to develop conceptual understanding and procedural fluency in mathematics, scaling up instructional change across a district organization is a significant challenge. The Middle School Mathematics and the Institutional Setting of Teaching (MIST) Project is a research–practice partnership seeking to understand how large urban school districts can support the development of rigorous and equitable middle-school mathematics instruction at scale. This article enumerates the goals and design of this multiyear, multidistrict partnership, and describes one illustrative example of how our partnership activities informed and supported one district's efforts to improve mathematics instruction over multiple years. General recommendations for district–researcher partnerships are discussed.  相似文献   

7.
This study utilized discourse-based instruction as an alternative method of instruction that emphasizes the teaching of mathematics by actively engaging students in mathematical discourse practices. A quasi-experimental study was employed to determine the effectiveness of mathematical discourse-based instruction in enhancing eleventh-grade students’ conceptual and procedural understanding of probability and statistics. A researcher-constructed test instrument was used for data collection from the experimental and control groups. The data analysis performed using the Kruskal-Wallis test showed that the experimental group outperformed the control groups in terms of conceptual and procedural knowledge. Furthermore, the results suggest that discourse-based instruction when appropriately designed and implemented can increase students’ understanding of mathematical topics.  相似文献   

8.
In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: Everyday Mathematics (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that multiplication of fractions is developed in KM one semester earlier than in EM. However, the number of lessons devoted to the topic is similar in the two curricula. In contrast, division of fractions is developed at about the same time in both curricula, but due to different beliefs about the importance of the topic, KM contains five times as many lessons and about eight times as many problems about division of fractions as EM. Both curricula provide opportunities to develop conceptual understanding and procedural fluency. However, in EM, conceptual understanding is developed first followed by procedural fluency, whereas in KM, they are developed simultaneously. The majority of fraction multiplication and division problems in both curricula requires only procedural knowledge. However, multistep computational problems are more common in KM than in EM, and the response types are also more varied in KM.  相似文献   

9.
Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.  相似文献   

10.
The research interest underpinning this paper concerns the type of mathematical knowledge engineering students may acquire during their specialised education in terms of the conceptual and procedural dimensions of doing and using mathematics. This study draws on interviews with 25 qualified engineers from South Africa and Sweden regarding their views on the role of mathematics in engineering education, with special focus on the conceptual and procedural aspects of mathematical knowledge. A thematic analysis of the interview data led to the identification of two main themes. According to the conceptual view a predominantly conceptual approach is needed and valued more than procedural skills, while the balanced view emphasises a balance of conceptual understanding and procedural fluency as well as links between them. It is suggested that the mathematical education of engineers would need to be more conceptually oriented to prepare for the demands at the workplace.  相似文献   

11.
This research analyses preservice teachers’ knowledge of fractions. Fractions are notoriously difficult for students to learn and for teachers to teach. Previous studies suggest that student learning of fractions may be limited by teacher understanding of fractions. If so, teacher education has a key role in solving the problem. We first reviewed literature regarding students’ knowledge of fractions. We did so because assessments of required content knowledge for teaching require review of the students’ understanding to determine the mathematics difficulties encountered by students. The preservice teachers were tested on their conceptual and procedural knowledge of fractions, and on their ability in explaining the rationale for a procedure or the conceptual meaning. The results revealed that preservice teachers’ knowledge of fractions indeed is limited and that last-year preservice teachers did not perform better than first-year preservice teachers. This research is situated within the broader domain of mathematical knowledge for teaching and suggests ways to improve instruction and student learning.  相似文献   

12.
The present study aims to explore the use of assessment in mathematics content courses for future elementary school teachers. Analysis of self assessment data on mathematical understanding and peer assessment data on oral mathematical presentation showed that pre-service teachers had a balanced understanding of procedural knowledge and problem solving. Conceptual understanding was not in the structure of pre-service teachers’ mathematical knowledge. Understandings of conceptual knowledge, procedural knowledge, and problem solving had no meaningful effects on gains in mathematics performance. Aspects of oral mathematical presentation were associated with improved understanding of procedural knowledge and in particular conceptual knowledge. The result of the study calls for a conceptual approach to mathematical knowledge and sufficient mathematical problem solving in college-level mathematics content courses and in particular the infusion of assessment into college-level mathematics education for pre-service teachers.  相似文献   

13.
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   

14.
The Common Core Standards require demonstration of conceptual knowledge of numbers, operations, and relations between mathematical concepts. Supplemental instruction should explicitly guide students with specific learning disabilities (SLD) in these skills. In this article, we illustrate implementation of the concrete‐representational‐abstract (CRA) sequence and the Strategic Instruction Model (SIM) for teaching multiplication with regrouping to students with SLD. CRA combined with SIM has been shown to be effective in teaching computation for students with SLD, specifically for developing conceptual understanding. Four elementary students with SLD participated in this study. The researchers used a multiple‐probe design to show a functional relation. Students demonstrated increases in computational fluency; skills were maintained and generalized.  相似文献   

15.
Reform efforts in mathematics education arose, in part, in response to constructivist works on conceptual learning. However, little research has examined how students with learning disabilities (LD) respond to constructivist-oriented instruction in mathematics, particularly in moment-to-moment interactions. To understand the nature of constructivist-oriented mathematics instruction involving students with LD, the authors conducted a case study to analyze teacher–student interactions during constructivist-oriented small group instruction involving a student with LD. The student demonstrated, to a certain degree, the ability to reason mathematically when provided with appropriate opportunities and prompting. However, given the limited intervention time, his reasoning and problem solving did not seem to go beyond the semiconcrete level of operation, which may have inhibited his solving of complex word problems with large numbers. Findings indicate that more efforts are needed to support students, those with LD in particular, in their transitions from concrete or semiconcrete to abstract conceptual understanding and problem solving.  相似文献   

16.
Nine prospective secondary mathematics teachers were interviewed about their teaching internship experience. The results of these interviews revealed that 7 of the 9 participants professed to value reform-oriented teaching and conceptual understanding in mathematics, yet all were paired with cooperating teachers who seemed to value traditional instruction and procedural understanding in mathematics. We explored the reasons that some of these student interns had positive experiences with their cooperating teachers and university supervisors while others had negative experiences. We found that the participants valued (a) critical feedback that was constructive and contained concrete recommendations for improvement, (b) freedom to use their own teaching methods, and (c) a friendly and supportive relationship with their mentors. The differing teaching philosophies of student teachers and their cooperating teachers contributed to negative experiences only when student teachers were not allowed freedom in their teaching methods.  相似文献   

17.
The authors examined the relationships among teacher classroom practices, student motivation, and mathematics achievement in high school. The data for this study was drawn from the base-year data of High School Longitudinal Study of 2009. Structural equation modeling method was used to estimate the relationships among variables. The results indicate that conceptual teaching positively affected student mathematics achievement, whereas procedural emphasis in mathematics instruction had a negative effect. Teacher support influenced student mathematics achievement indirectly through students' mathematics self-efficacy, and also influenced students' interest in mathematics courses. Finally, students with higher levels of family socioeconomic status and prior achievement were more likely to have teachers who use conceptual teaching strategies. Students with higher prior achievement were also more likely to perceive higher levels of teacher support. The findings have theoretical and practical implications.  相似文献   

18.
The purpose of the study reported in this article is to examine the impact of curriculum on instruction. Over a three-year period, we observed 579 algebra-related lessons in grades 6-8. Approximately half the lessons were taught in schools that had adopted a Standards-based mathematics curriculum called the Connected Mathematics Program (CMP), and the remainder of the lessons were taught in schools that used more traditional curricula (non-CMP). We found many significant differences between the CMP and non-CMP lessons. The CMP lessons, emphasized the conceptual aspects of instruction to a greater extent than the non-CMP lessons and the non-CMP lessons emphasized the procedural aspects of instruction to a greater extent than the CMP lessons. About twice as many CMP lessons as non-CMP lessons were structured to use group work as a method of instruction. During lessons, non-CMP students worked individually on homework about three times as often as CMP students. When it came to text usage, CMP teachers were more likely than non-CMP teachers to work problems from the text and to follow lessons as laid out in the text. However, non-CMP students and teachers were more likely than CMP students and teachers to review examples or find formulas in the text. Surprisingly, only small proportions of the CMP lessons utilized calculators (16%) or manipulatives (11%).  相似文献   

19.
The present study compared the relative effects of hands-on and teacher demonstration laboratory methods on declarative knowledge (factual and conceptual) and procedural knowledge (problem-solving) achievement. Of particular interest were (a) whether these relationships vary as a function of reasoning ability and (b) whether prior knowledge and reasoning ability predict student achievement. Ninth-grade physical science students were randomly assigned to classes taught by either a hands-on or a teacher demonstration laboratory method. Students' reasoning ability and prior knowledge of science were assessed prior to the instruction. The two instructional methods resulted in equal declarative knowledge achievement. However, students in the hands-on laboratory class performed significantly better on the procedural knowledge test than did students in the teacher demonstration class. These results were unrelated to reasoning ability. Prior knowledge significantly predicted performance on the declarative knowledge test. Both reasoning ability and prior knowledge significantly predicted performance on the procedural knowledge test, with reasoning ability being the stronger predictor.  相似文献   

20.
Although skilled mathematics teachers and teacher educators often “know” when interruptions in the flow of a lesson provide an opportunity to modify instruction to improve students’ mathematical understanding, others, particularly novice teachers, often fail to recognize or act on such moments. These pivotal teaching moments (PTMs), however, are key to instruction that builds on student thinking about mathematics. Video of beginning secondary school mathematics teachers’ instruction was analyzed to identify and characterize PTMs in mathematics lessons and to examine the relationships among the PTMs, the teachers’ decisions in response to them, and the likely impacts on student learning. These data were used to develop a preliminary framework for helping teachers learn to identify and respond to PTMs that occur during their instruction. The results of this exploratory study highlight the importance of teacher education preparing teachers to (a) understand the mathematical terrain their students are traversing, (b) notice high-leverage student mathematical thinking, and (c) productively act on that thinking. This preparation would improve beginning teachers’ abilities to act in ways that would increase their students’ mathematical understanding.  相似文献   

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