摘要
高中生数学问题提出能力发展进程是指在适当的教学下,高中生的数学问题提出能力如何随着时间的推移而日渐得到发展的进程,它描述学生在一段较长的时间内发展数学问题提出能力可能需要经历的途径。
研究以修订后的布卢姆教育目标分类学为理论基础,以设计研究为方法论指导,借鉴学习进程研究的研究范式,采用文献分析、问卷调查、教学模式研究等方法,经历能力发展进程假设的提出、检验和修正的研究过程,最后得到修正后的高中生数学问题提出能力发展进程。研究旨在弥补国内数学问题提出能力研究存在的概念界定不清晰、评价指标不明确等不足,以促进数学能力评估系统、教育研究与教学实践更有效地结合。
研究工作主要包括以下三个方面：
第一,提出高中生数学问题提出能力发展进程假设
高中生数学问题提出能力发展进程假设提出主要确定高中生数学问题提出能力发展目标、能力发展进程变量、发展水平、发展表现、教学建议及发展评价。研究首先通过对多国或州的课程标准、已有数学问题提出能力研究进行文献分析,确定数学问题提出能力发展进程变量、发展水平；其次通过对浙江三所不同学校的高一、高二学生的问卷调查,确定学生具体发展水平、发展表现及教学建议；能力发展目标则结合文献分析与问卷调查结果最终确定。发展评价主要指形成的能力测试卷。
第二,检验高中生数学问题提出能力发展进程假设
高中生数学问题提出能力发展进程检验主要通过与一位高二数学授课教师合作,在设计研究方法论指导下,由设计研究原型规划得出的问题变式教学模式原型,通过原型规划具身化、设计、实施针对提高学生数学问题提出能力的为期四周的教学。通过视频分析、前后测、嵌入式测试分析、对师生的访谈分析研究,比较学生在变式教学模式研究迭代过程前、中、后数学问题提出能力的变化情况。最后使用上述分析结果检验高中生数学问题提出能力发展假设。检验结果将同时用于修正针对学生数学问题提出能力发展的教学模式。
第三,修正高中生数学问题提出能力发展进程假设
高中生数学问题提出能力发展进程修正主要基于前面高中生数学问题提出能力发展进程检验结果,对问卷调查所得到的高中生数学问题提出能力发展进程假设进行修正。修正内容具体包括能力发展目标、能力发展表现及教学建议。
研究的关键性结论为修正后的高中生数学问题提出能力发展进程及修正后的针对学生数学问题提出能力发展的教学模式。研究发现随着年级的升高,学生只有在结构化情境中提出理解与联系性问题、反思与拓展性问题比例在随之提高,这种相关性达到显著水平,且在设计教学下,干预班学生也只有在结构情境中提出能力的优势有所提高。
研究借鉴学习进程研究范式探究高中生数学问题提出能力是一个新的尝试,也是未来数学能力研究的一个新起点。研究所得到的修正后的高中生数学问题提出能力发展进程还需更多实践的检验,从而能更加合理、准确、清晰地评价学生数学问题提出能力,帮助教师开展数学问题提出能力培养教学。
Developing progressions for mathematics problem posing are empirically grounded and testable hypotheses about how students'understanding of, and ability to pose mathematics problem and explanations and related problem posing activities grow and become more sophisticated over time, with appropriate instruction. These hypotheses describe the pathways students are likely to follow to the mastery of mathematics problem posing.
The research on developing progressions was based on revised Bloom's Taxonomy of Educational Objectives, put the design research as methodological guideline, learned from the research paradigm of learning progressions, and adopted some research methods including literature analysis, questionnaire investigation and teaching model. The research process of developing progressions included how to propose, validate and revise the hypotheses about mathematics problem posing, finally obtaining the revised developing progressions for mathematics problem posing. The main purpose of this dissertation was to make up the shortfall in research on domestic mathematics problem posing to improve the valid integration of mathematics competency evaluation system, educational research and teaching practice of mathematics competency, where the mathematics problem posing was vague in definition and unclear in evaluation index.
The research was mainly composed of three parts:
The first one was to pose the hypotheses of developing progressions for mathematics problem posing in grades10-11. The hypotheses include the target performances or developing goals, progress variables, levels of achievement and developing performances. First of all, the progress variables and levels of achievement were designed by literature analysis of the national curriculum standards and related research on mathematics problem posing. And then, the real levels of achievement and developing performances of students were decided by the questionnaire survey, which was performed among grades10-11students from3different schools. The target performances or developing goals were specified by the integration of literature analysis and questionnaire survey results. The hypotheses of developing progressions also embraced the corresponding instruction advice about mathematics problem posing.
The second one was to validate the hypotheses of developing progressions for mathematics problem posing in grades10-11. Under the methodological guideline of design research, the research process of validating the hypotheses went through design prototype of design research, specific the prototype, design and implement the teaching in grade11that focused on the mathematics problem posing during a four-week period. The results of analyzing the teaching videos, pretest, interim test, posttest and the interview to the teacher and students were used to validate the hypotheses of developing progressions for mathematics problem posing in grades10-11, and also revised the teaching model for improving students' mathematics problem posing.
The third one was to revise the hypotheses of developing progressions for mathematics problem posing in grades10-11. Based on the hypotheses and its validation of the developing progressions for mathematics problem posing in grades10-11, revising the hypotheses mainly involved the target performances or developing goals, developing performances and instruction advice.
The key findings of this research contributed to the revised developing progressions for mathematics problem posing in grades10-11and the revised teaching model for improving students' mathematics problem posing. The research also found that students increasingly posed the understanding problem and reflecting problem from the structured situation with the growth of students' grade.
The research drew lesson from learning progressions research, which was a new method of assessing mathematics competency, and it would be a start for the future competency study.
引文
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144 在首次测试卷编排中,将从自由情境中提出问题,从半结构化情境提出问题,从结构化情境中提出问题这三类题分别记为PA(Posing problem A类题),PB(Posing problem B类题),PC(Posing problem C类题),如PAl表示从自由情境中提出问题类第1题。
145 Tugrul Kara, Ercan Ozdemirb, Ali Sabri pekb, Mustafa Albayraka. The relation between the problem posing and problem solving skills of prospective elementary mathematics teachersfJ]. Procedia Social and Behavioral Sciences.2010.2:1577-1583
146 详见附录1
147 修正后的测试卷编排中,为区别于首次测试卷题目编排,将从自由情境中提出问题,从半结构化情境提出问题,从结构化情境中提出问题这三类题分别记为A(类题),B(类题),C(类题),如A1表示从自由情境中提出问题类第1题。
148 纪宏伟.基于SPSS的试卷分析与解读[J].职业教育研究.2011.8：169-170
149 Sig.是差异性显著的检验值,该值一般与0.05或0.01比较,本文中若sig小于0.05则表示差异显著,若者小于0.01则表示差异十分显著,下同。
150 代表0计分的问题,1-3分别依次代表记忆与操作类、理解与联系类、反思与拓展类问题。
151 由于前面是个百分比是四舍五入后的结果,所以它们之和可能不完全等同100%,下同
152 0代表0计分的问题,1-3分别依次代表记忆与操作类、理解与联系类、反思与拓展类问题。
153 0代表0计分的问题,1-3分别依次代表记忆与操作类、理解与联系类、反思与拓展类问题。
157 拉尔夫·泰勒,良方译.课程与教学的基本原理[M]：北京：人民教育出版社.1994：18
158 0代表0计分的问题,1-3分别依次代表记忆与操作类、理解与联系类、反思与拓展类问题。
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187 √表示四个环节都出现；×表示四个环节没有完全出现,括号内表示出现的几个环节(下同)
188 √表示四个环节中存在重复环节,括号内表示重复的环节；×表示没有出现重复的环节(下同)
189 √表示有环节之间顺序颠倒；×表示环节之间顺序没有出现颠倒(下同)
190 对应不同轮循环,括号内编码表示重复环节编码
191 非问题变式,辨析相关概念之间的异同,属于问题变式,虽然不是研究重点,但仍作为变式形式标出。
194 A表示自由情境类问题均值,B表示半结构化情境类问题均值,C表示结构化情境类问题均值,下同。
195 实验班总人数29,对照班总人数35
196 表中数字都是四舍五入的结果,因此在sum中的数值可能与前面三个数值相加不一致,下同。
197 这里的差值指相应表中实验班-对照班数值得到的差异,下同
211 Hayri Akay, Nihat Boz. The Effect of Problem Posing Oriented Analyses-Ⅱ Course on the Attitudes toward Mathematics and Mathematics Self-Efficacy of Elementary Prospective Mathematics Teachers[J]. Australian Journal of Teacher Education.2010.25(1):1-18
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