2021年PISA数学——来自CCR的分析

数学是理解世界、公民身份和经济增长的基石。为了满足全社会对教育的需求,21世纪教育应该注重对知识理解的深度和多样性的培养。PISA关于数学能力的测试中,最重视学生运用数学推理来解决问题的能力。我们建议扩展数学过程的描述(表述、应用、解释、评估),并在PISA数学框架内确定这些处理过程为数学建模的主要组成部分,其中有七个最常用于寻找正确推理方法的推理工具:比较、比例推理、应用乘法量表、拆分、归并、由简入繁、概率推理和逻辑推理。PISA数学的素养领域涉及形状与空间、变化与关系、不确定性与数据、数量等,还要特别注意创造性思维能力、品格和元认知技能的培养。     

数学推理的本质和功能及其能力培养

数学推理包括演绎推理、合情推理及实践性推理等，但其本质在于演绎推理．思维功能和理解功能是数学推理的2个主要功能．在早期数学教育中，对推理能力尤其是演绎推理能力的培养至关重要．数学课程设置及教学实施要创造推理的环境和机会，使数学课堂形成良好的推理风气。     

Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles

This study is about prospective secondary mathematics teachers’ understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations—referent preserving versus referent transforming compositions—acting on these quantities. Although multiplicative structures can be modeled by additive structures, they have their own characteristics inherent in their nature. I situate my analysis within a framework of unit coordination with different levels of units supported by a theory of quantitative reasoning and theorems-in-action. Data consist of videotaped qualitative interviews during which prospective mathematics teachers were asked problems on multiplication and factorization of polynomial expressions in x and y. I generated a thematic analysis by undertaking a retrospective analysis, using constant comparison methodology. There was a pattern which showed itself in all my findings. Two student–teachers constantly relied on an additive interpretation of the context, whereas three others were able to distinguish between and when to rely on an additive or a multiplicative interpretation of the context. My results indicate that the identification and coordination of the representational quantities and their units at different categories (multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative reasoning and need to be emphasized in the teaching–learning process. Moreover, representational Cartesian products-in-action at two different levels, indicators of multiplicative thinking, were available to two research participants only.     

Multiplicative reasoning: teaching primary pupils in ways that focus on functional relations

This paper examines a problem described as widespread and long-standing in mathematics education: supporting pupils into multiplicative reasoning, a form of reasoning that has been noted as central to large tracts of secondary mathematics and beyond. Also noted, however, is a persistent perception of multiplicative situations only in terms of repeated addition – a perception held not only primary pupils, but also among primary teachers and curriculum developers. The focus of this paper is to synthesize literature on multiplicative reasoning as a conceptual field together with a sociocultural discussion of the role of mediating artifacts in the development of this conceptual field. Bringing MR into the primary classroom can then be achieved, I propose, through a pedagogy oriented toward model-eliciting and teacher appropriation of pupils’ models as pedagogic tools with the subsequent re-appropriation of refined models by pupils. This pedagogy is illustrated through the analysis of two vignettes from a teaching experiment which demonstrate the beginnings of MR as an emergent conceptual field in the classroom. The paper concludes that it is possible to move primary teaching and learning toward understanding the functional aspects of multiplicative reasoning, but that any such moves requires attention to teachers’ pedagogic and content knowledge.     

Proportional reasoning: The effect of two context variables,rate type,and problem setting

This study investigated the effects of two context variables, ratio type and problem setting, on the performance of seventh-grade students on a qualitative and numerical proportional reasoning test. Six forms of the qualitative and numerical tests were designed, each form using a single context (one of two settings for each of three ratio types). Different ratio types appear to have a stronger impact on the difficulty of the qualitative and numerical proportional reasoning problems than small differences in problem setting. However, the familiarity of problem setting did show an increasingly large effect on qualitative reasoning as the difficulty of ratio type increased. We also investigated the nature of the relationships between rational number skills, qualitative reasoning about ratios, and numerical proportional reasoning. Qualitative reasoning appears to be sufficient, but not necessary for numerical proportional reasoning. The evidence for the requisite nature of rational number skills for proportional reasoning was equivocal. The implications of these findings for science education are discussed.     

Teaching the partial products algorithm with the concrete representational abstract sequence and the strategic instruction model

Elementary standards include multiplication of single-digit numbers and students advance to solve complex problems and demonstrate procedural fluency in algorithms. The ability to illustrate procedural fluency in algorithms is dependent on the development of understanding and reasoning in multiplication. Development of multiplicative reasoning provides the foundation for advanced mathematics and algebraic reasoning. For students who struggle in mathematics, instruction in multiplication algorithms should ensure conceptual understanding so that students have a foundation for success in advanced mathematics. The concrete representational abstract (CRA) sequence addresses conceptual understanding and the strategic instruction model (SIM) supports procedural knowledge. The current pilot study combined these methods to teach elementary students the partial products algorithm. Twelve students in grades four and five participated in the study, receiving instruction from teachers in their school during an intervention period. Within a pre-experimental design, using pre- and postintervention data, students showed a significant change in performance. The article will describe and show how teachers implemented the CRA-SIM interventions and discuss implications for practice.     

The structure of prospective kindergarten teachers’ proportional reasoning

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.     

Equivalence and relational thinking: preservice elementary teachers’ awareness of opportunities and misconceptions

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.     

Development and validation of a path analytic model of students' performance in chemistry

This article reports the development and validation of an integrated model of performance on a chemical concept - volumetric analysis. From the chemical literature a path-analytic model of performance on volumetric analysis calculation was postulated based on studies utilizing the proportional reasoning schema of Piaget and the Cumulative learning theory of Gagne. This integrated model hypothesized some relationships among the variables: direct proportional reasoning, inverse proportional reasoning, prerequisite concepts (content) and performance on volumetric analysis calculations. This model was postulated for the two groups of students involved in the study - that is those who use algorithms with understanding and those who use algorithms without understanding. Two hundred and sixty-five grade twelve chemistry students in eight schools (14 classes) in the lower mainland of British Columbia, Canada participated fully in the study. With the exception of the test on volumetric analysis calculations all the other tests were administered prior to the teaching of the unit on volumetric analysis. The results of the study indicate that for subjects using algorithms without understanding, their performance on VA problems is not influenced by proportional reasoning strategies while for those who use algorithms with understanding, their performance is influenced by proportional reasoning strategies.     

Educating Teachers to Teach Multiplicative Structures in the Middle Grades

The middle-grades mathematics related to multiplicative structures has undergone careful scrutiny over the past decade. Researchers have identified the types of reasonings involved; the difficulties students have with the concepts and why these difficulties might occur; and the interconnections within this content area. On the basis of this research we make four recommendations for the preparation and professional development of teachers. The recommendations deal with different but related forms of reasoning: quantitative reasoning, multiplicative reasoning, proportional reasoning, and reasoning with rational numbers. Problematic issues that follow from the recommendations are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Situated Simulation-Based Learning Environment to Improve Proportional Reasoning in Nursing Students

Proportional reasoning is the basis for most medication calculation processes and is fundamental for high-quality care and patient safety. We designed a simulated Medication Mathematics (siMMath) environment to support proportional reasoning in transitioning via concreteness fading between two mediators. The first mediator is simulated nursing tools of medication preparation. The second is a ratio-table setup which is used as a goal representation, which enables one to spatially hold in place different quantities in their relative proportion. We conducted a two-part study with nursing students. Part 1 was a quasi-experimental pretest–intervention–posttest design assessing the effectiveness of learning, by evaluating four categories of medical calculation questionnaire items (solid medications, unit conversion, concentrations, infusion rates). We used the Noelting proportional reasoning test to evaluate the generalizability and abstraction of proportional reasoning. Part 1 included an experimental group (n = 96) learning with siMMath, and a comparison group (n = 73) learning with an equation-based lecture approach. Part 2 employed a case study design to characterize the learning process. The experimental group’s learning gains were significantly higher than the comparison group’s for the two most challenging categories of the medication calculation problems questionnaire, namely concentrations and infusion rates. Furthermore, the experimental group’s learning gains were significantly higher than the comparison group’s for formal operational reasoning on the Noelting test. Students who used a ratio-table setup scored significantly higher on the Noelting posttest questionnaire. Nursing students who learned with the siMMath environment overcame difficulties in proportional reasoning to the highest levels and extended this understanding to other contexts.     

类比法在经典电磁理论研究中的两大成功范例

类比是人类认识客观世界的一种基本的思维方法,类比法在科学研究中具有重大意义。本文就经典电磁理论建立过程中,运用类比方法取得成功的两个经典范例即库仑定律和麦克斯韦的电磁场理论,从一般方法论的角度,分析了类比法在科学研究中的作用。     

Supervision in Secondary Education

It is widely accepted that including writing activities in the learning process positively impacts student achievement and leads to greater depth of student understanding. This writing is often missing in the math classroom though, when the focus is misplaced on rote procedures. In these classrooms students learn mathematical processes but have little depth of understanding into the mathematical foundations, nor have an ability to clearly express their mathematical reasoning. This article promotes the use of Internet-based chat, forums, and blogs as the environment in which necessary mathematical writing can occur. Zemelman, Daniels, and Hyde provide a best practice framework through which the benefits of chat, forum, and blog writing are obvious. Student engagement with material increases in a cooperative environment, where a real audience and purpose for writing is clear, and student ownership in personal learning grows. In addition, students mature in traditional reading and writing literacy and further develop critical thinking skills.     

数学能力的脑与认知科学研究及其对早期数学教育的启示

数学能力是基础性的认知能力,包括数量、空间和逻辑推理等认知能力。早期数学教育有助于在儿童发育和发展的关键期为儿童奠定认知和神经基础,从而培养儿童抽象而精确的数学思维能力与问题解决能力。脑与认知科学研究表明,儿童生来具有数的概念,体现在两个独立的数的核心表征系统,一是大数系统,模糊估计、粗略表征物体的数量幅度;二是小数系统,精确计数、清晰表征每一个物体。早期数学教育可以借鉴当前丰富的脑与认知科学研究成果,将科学理论和教学实践相结合,利用儿童先天具备的数学潜质,逐渐深入而广泛地培养儿童的数学技能。培养儿童的早期数学能力需要家庭、学校和社会的共同努力。     

Relative and absolute thinking in visual estimation processes

This study has two main goals: (1) to investigate the processes involved in visual estimation (part I of the study), and (2) to investigate the processes of judgment in visual estimation situations, which mostly involved proportional reasoning (part II). The study was conducted with 9-year old children in the third grade. Four strategies were expressed by the children in visual estimation situations. Exposure to a unit in the Agam project, designed to enhance visual estimation capabilities resulted in changes in the children's strategies. These changes reflected the processes by which children overcame their limited ability to process visual information. The development of proportional reasoning was investigated through a series of judgment situations. Although, as was expected, most of the children showed an additive behavior, these situations stimulated some children towards qualitative proportional reasoning, where easy/difficult considerations played an important role.     

初中数学发散性思维能力培养策略

《初中数学课程标准(2011版)》指出,数学课程能使学生掌握必备的基础知识和基本技能,培养学生的抽象思维和推理能力,培养学生的创新意识和实践能力数学的发散性思维能力是"问题解决"的基础,是培养数学推理能力和创新意识前提要求。数学发散性思维作为用学科自身的品质陶冶人、启迪人、充实人。"问题解决"是人的高级数学思维。高级思维的学习,可以使学生充分享受思维的快乐,可以让思维自由飞翔。本文就初中数学发散思维的培养谈几点体会,通过创设问题情景、设置开放性试题、发挥学科优势等教学策略,着力培养初中学生的数学发散性思维能力,实现有效教学。     

Development and validation of a scientific (formal) reasoning test for college students

We present a multiple-choice test, the Montana State University Formal Reasoning Test (FORT), to assess college students' scientific reasoning ability. The test defines scientific reasoning to be equivalent to formal operational reasoning. It contains 20 questions divided evenly among five types of problems: control of variables, hypothesis testing, correlational reasoning, proportional reasoning, and probability. The test development process included the drafting and psychometric analysis of 23 instruments related to formal operational reasoning. These instruments were administered to almost 10,000 students enrolled in introductory science courses at American universities. Questions with high discrimination were identified and assembled into an instrument that was intended to measure the reasoning ability of students across the entire spectrum of abilities in college science courses. We present four types of validity evidence for the FORT. (a) The test has a one-dimensional psychometric structure consistent with its design. (b) Test scores in an introductory biology course had an empirical reliability of 0.82. (c) Student interviews confirmed responses to the FORT were accurate indications of student thinking. (d) A regression analysis of student learning in an introductory biology course showed that scores on the FORT predicted how well students learned one of the most challenging concepts in biology, natural selection.     

Messy but meaningful: exploring the transition to reform-based pedagogy with teachers of mathematics and coordinators in Ontario,Canada

The RE4MUL8 Project involved the creation of an online/mobile resource for Intermediate Division (Grade 7 and 8) teachers of mathematics. This resource showcases video documentaries of seven key mathematics topic lessons (fractions, integers, proportional reasoning, composite shapes and solids, solving equations, and, patterning and algebraic thinking), as delivered by seven teachers in Ontario, Canada who were nominated by their respective District School Boards as being, or becoming, highly effective practitioners in the area of reform-based mathematics education. As part of a qualitative case study research design, these teachers, often along with their math coordinators, were then interviewed following the lesson, and shared reflections on the lesson itself and, more generally, on their ongoing journey towards reform-based mathematics teaching. This paper reports on three major themes that emerged from these discussions, namely, problem-based learning, the reality and necessity of ‘messy time’ transition to reform-based pedagogy, and, balancing instructional planning and practices.     

"离散数学"教学探析

“离散数学”课程是计算机专业的重要课程,有较强的理论性和专业实用性。教学内容多、抽象性强是这门课的特点;提高学生的逻辑思维能力、推理能力和创新能力是这门课程的教学要求。这就要求教师要培养学生的学习兴趣,提高他们分析和解决问题的能力。