The importance of mathematics has been widely recognized, the pursuit of the effectiveness of mathematics teaching has also become the consensus of the international mathematics education. This study was carried out in the first theme of the Third Wave project (The Third Wave Project: The Values Approach to Optimising Mathematics Education), its purpose was to explore the similarities and differences of perceptions of effective mathematics teaching and learning and the effectiveness of classroom activities, based on the perspectives of the primary mathematics teachers and their students with different levels of mathematical ability and to discuss the effective classroom activities. On the basis of the literature review, I have sorted out the rubrics of the effective mathematics teaching and classroom activities for effective mathematics teaching and learning as the theoretical framework of this study. A fifth grade mathematics teacher from Changchun City, Jilin Province and her six students in different mathematical ability level were selected. This study drew on the methods in the first theme of the Third Wave project, including interviews, lesson observations, teaching videos and teacher questionnaires to collect data. In terms of effective mathematics teaching, we found that (1) Both the teacher and the students emphasized the ‘knowledge and skills’, ‘mathematical thinking’, ‘problem-solving’ and ‘emotions and attitudes’. However, students mentioned the ‘efficiency’ but the teachers did not; the teacher believed that effective mathematics teaching is conducive to the professional development of teachers, while students did not mention this. (2) The three ability groups of students mentioned the ‘knowledge and skills’, ‘emotions and attitudes’ as well as the ‘efficiency’ both. But there were also differences, which are shown through the following aspects: the higher and average ability level group students can perceive problem-solving; mathematical thinking was only highlighted by the students at the average ability level. In terms of classroom activities for effective mathematics teaching and learing, we found that (1) in the first level classification, both the teacher and the students mentioned most frequently were ‘students’ behaviors’, followed by ‘teachers’ behaviors’, the least was the‘teachers and students’ common behaviors’, where they had similar trend in the first level classification. However, the number of classroom activities for effective mathematics teaching and learning mentioned by the teacher was more than that the students perceived. In addition, three groups of students v mentioned the ‘teachers and students’ common behaviors’, but the teacher did not mention. In the second level classification, ‘students’ movement behaviors’ such as ‘hands-on’, ‘student’s presetation’ were most frequently mentioned, but they had different trend in the second level classification. (2) The three ability groups of students had similar trend in the first level classification, that was ‘students’ behaviors’, ‘teachers’ behaviors’ and the ‘teachers and students’ common behaviors’ from the most to the least. However, the lower ability level group mentioned ‘teachers’ behaviors’ and ‘students’ behaviors’ far more than the other two groups. In the second level classification, they mentioned most frequently were ‘students’ movement behaviors’, but they had different trend in the second level classification. In addition, the lower ability level group mentioned ‘teacher’ verbal behavior’ and ‘students’ movement behavior’ far more than the other two groups. Based on the similarities and differences between the perceptions of teachers and students, from the following three aspects, including the perception of effective mathematics teaching and learning, communication between teachers and students, and the selection of classroom activities, this study suggested improving the effectiveness of mathematics teaching, to provide reference for pre-service teacher education and in-service training and teachers' teaching and training.

數學的重要性被世界所公認，追求數學課堂教學的有效性也成為國際數學教 育界的共識。本研究是在第三波研究（The Third Wave Project: The Values Approach to Optimising Mathematics Education）主題一的課題下開展的，旨在探 索同一班級中的教師與學生和不同數學能力水準的學生對有效數學教學以及有 效數學教學的課堂活動感知的異同，以期結合師生兩者的感知來探討有效的課堂 教學活動。 筆者在文獻探討的基礎上，整理出有效數學教學以及有效數學教學課堂活動 的類目表，作為本研究的分析框架，選取中國吉林省長春市某小學五年級的一名 數學教師和她所教班級的 6 名不同數學能力水準的學生作為個案研究的對象，借 鑒主題一的研究方法，通過訪談、課堂觀察、教學錄像和教師問卷的方式收集資 料。 研究結果發現：在有效數學教學的認識方面：（1）教師和學生相比，都提 到了「知識與技能」、「數學思考」、「解決問題」和「情感與態度」方面。但 是，學生提到了「有效率」，教師沒有提到；教師認為有效的數學教學有利於教 師專業發展，學生沒有提到這一點。（2）三個能力組別的學生相比，都提到了 「知識與技能」、「情感與態度」和「有效率」這三個方面。但也存在差異，具 體體現在「解決問題」被較高和中等能力水準組的學生感知；「數學思考」只被 能力水準中等組別的學生所強調。 iii 在對有效數學教學課堂活動的認識方面：（1）教師與學生相比，在一級類 目上，教師和學生提到次數最多的都是「學生行為」，其次是「教師行為」，最 少的是「師生共同行為」，在一級類目的感知趨勢上相同。不過，教師提到的有 效數學教學的課堂活動的次數要遠比學生感知到的多。另外，三組學生都提到了 「師生共同行為」，但教師並沒有提到。在二級類目的感知上，教師和學生提到 次數最多的都是「學生動作行為」，如「動手操作」、「學生展示」等，但二級 類目的感知趨勢不同。（2）三組學生相比，在一級類目上的感知趨勢相同，從 多到少依次為「學生行為」、「教師行為」和「師生共同行為」。但能力水準較 低組的學生提到的「教師行為」和「學生行為」的次數明顯多於其他兩組。在二 級類目上，三組學生提到次數最多的都是「學生動作行為」，但二級類目的感知 趨勢不同；另外，能力水準較低組的學生對「教師語言行為」和「學生動作行為」 提及的次數要遠多於其他兩組。 本研究依據師生之間感知情況的一致性與差異性，從有效數學教學的認識、 師生之間的溝通、課堂活動的選擇方面提出提高數學教學有效性的建議，以期對 職前和在職教師的教育與培訓提供借鑒。