本研究根據國小四年級「數與量」和「代數」的能力指標課程，並利用與美國史丹佛大學國際教學科技中心所共同開發的Group Scribbles 2.0的電腦輔助學習軟體，將回饋機制的設計，融入到加減法未知數當中，內容結合共同解題的合作學習模式，以促進學生對於未知數的理解與喜愛。

本研究採準實驗研究，實驗架構以前測－後測－延宕後測的設計。研究對象為新竹某國小四年級三個班的學生（共83人）為樣本，進行一般傳統學習班級與個別操作學習班級的學習成就比較，及個別操作學習班級與合作學習班級的學習成就比較，以探討不同學習模式的差異。研究結果發現如下：

一、經由前後測發現，傳統學習班級的前後測成績是未達顯著，進步的程度幾乎是持平的情形，而個別操作學習班級與合作學習班級前後測成績皆有達顯著水準，尤其是合作學習班級進步幅度較大。

二、經由統計結果發現，個別操作學習班級與合作學習班級的加減法未知數測驗有達顯著差異，顯示在學習活動中加入合作學習策略，對學生學習加減法未知數有很大的幫助。

三、系統回饋提示部分，個別操作班級的平均使用次數高於合作班級，顯示在無合作的情況下，個別學習會較依賴回饋提示。接著從成就背景與成績進步情形分析，成就越低的學生會越依賴提示的使用，相對在成績進步也最多。

四、從學習態度量表與錄音訪談的紀錄得知，縱橫數謎系統可以激勵學生主動積極學習及促進小組團隊合作，培養學生對於學習的數學興趣、瞭解同儕互助的重要性、增加解題思考的能力。

五、從系統歷程的解題路徑分析，發現合作式的縱橫數謎題型，可以歸納出五種解題的模式，分別為各自計算型、交叉比對型、合作接力型、輔助驗證型、頭尾計算型五種型態。

六、從錄影觀察得知，合作學習班級的溝通互動模式有三種，分別為理想的溝通、私下交談的溝通及無反應的溝通。

七、比較三個班級的學習氣氛，其中合作學習班級較為熱絡，個別操作班級較為安靜，傳統班級較為低落。

The purpose of this study was to design a system of unknown numbers of addition and subtraction integrated with feedback mechanism according to the ability indicators of number/quantity and algebra, and by using Group Scribbles 2.0, the CSCL app, co-developed by SRI International. The content combined collaboration learning model for co-solving problems was to promote the pupils to understand and love unknown numbers.

The study adopted pretest-posttest experimental design with delayed test. The objects were eighty-three students from three classes studied in forth grade in an elementary school. Comparing the achievement of learning in traditional learning with in single-perform learning, and in single-perform learning with in collaboration learning.

The findings were as follows:

1. After analyzing the tests, there was no significant difference for pretest-posttest in the class of traditional learning, and there were significant differences in the classes of single-perform and collaboration learning, especially the collaboration one.

2. After analyzing the results of statistics, finding that there was significant difference for posttest between single-perform class and collaboration class. It showed that the collaboration strategy was helpful for pupils to learn the unknown numbers of addition and subtraction.

3. Concerning the feedbacks of system, the single-perform class used more number of times in average than collaboration class. It showed that single-learning students were more rely on using feedbacks when there were no collaboration. Within in, students who made the lower achievement, used the feedbacks more, and got the better grades.

4. According to the questionnaires of manner and sound records of interview, finding that the system of Cross Number Puzzle can inspire students to learn actively, promote team collaboration, cultivate interesting in math, understand the importance to help with each other, and increase the abilities of thinking and problem-solving.

5. Analyzing the solving paths from the process of system, finding that the question types of Cross Number Puzzle based on collaboration can be sorted out as five models: self-computing, overlap-comparing, co-relaying, assistant-check, and head-tail-computing.

6. Observing the videos, there were three interaction models in the collaboration class: Ideal, Tete-a-tete, and Unresponsive.

7. Comparing the atmosphere among the three classes: collaboration one was lively, single-perform one was calm, and the traditional one was low-spirited.

壹、中文文獻

[1] 丁惠琪（2000）。合作學習應用在國小數學教學之探究。國立台北師範學院課程與教學研究所碩士論文，臺北。
[2] 方建良（2003）。「合」樂融融的數學課：以四年級「四則運算」之補救教學為例。國教世紀，208，85-100。
[3] 方振邦（2003）。績效管理。北京：中國人民大學。
[4] 呂玉琴（1989）。在國小實施代數教學的可能性研究。臺北師院學報，2，263-286。
[5] 呂玉琴（1991）。分數概念文獻探討。台北師院學報，4，573-606。
[6] 吳秉恩（1986），組織行為學。台北：華泰書局。
[7] 李淑芬（1997）。超本文網路合作學習環境中互動之研究。國立交通大學傳播研究所碩士論文，新竹。
[8] 林生傳（1988）。新教學理論與策略。台北：五南圖書。
[9] 周立勳 (1995)。小組獎勵對國小兒童分組學習表現的影響。嘉義師院學報，9，175-222。
[10] 林佩璇、黃政傑（1996）。合作學習。台北：五南圖書。
[11] 黃政傑、吳俊憲（2006）。合作學習－發展與實踐。台北：五南圖書。
[12] 林建仲、鄭宗文（2001）。資訊與教育雜誌，85，57。
[13] 孟瑛如、吳東光（1999）。數學學習障礙與多媒體教材之發展應用。特殊教育季刊，72，13-18。
[14] 林達森（2000）。合作建構教學與認知風格對國中學生生物能量概念學習之效應。國立臺灣師範大學科學教育研究所博士論文，臺北。
[15] 林達森（2002）。合作學習在九年一貫課程的應用。教育研究資訊，10(2)，87-103。
[16] 林碧珍（1988）。兒童水平概念發展之研究。國教世紀，23(5)，2-20。
[17] 林鴻源（1998）。線上測驗回饋型態對國小學生學習影響之分析研究。國立臺南師範學院國民教育研究所碩士論文，臺南。
[18] 洪榮昭（1986）。電腦輔助教學之設計原理與應用。臺北：松崗電腦圖書。
[19] 高文民（1996）。國小數學科新課程的趨勢。進修學訊年刊，2，32-38。
[20] 高台茜（2001）。未來教室學習－以無線網路應用為基礎的認知學徒制學習環境，台大教與學，9。
[21] 教育部（2000）。國民教育九年一貫課程綱要。台北：教育部。
[22] 教育部（2003）。國民中小學九年一貫課程綱要數學學習領域。台北：教育部。
[23] 張育綾（2008）。潛在類別分析國小五年級學生四則運算規則之縱貫研究。國立臺中教育大學數學教育系碩士班。台中。
[24] 張春興（1996）。教育心理學－三化取向的理論與實踐。台北：東華書局。
[25] 張新仁、許桂英（2004）。國小數學領域採合作學習之教學成效。教育學刊，23，111-136。
[26] 張簡明旺、黃志賢、高慧蓮（2002）。小組合作學習與學生科學本質。屏師科學教育，15，3-11。
[27] 湯清二（1996）。建構教學與回饋。教育實習輔導，2(1)，62-65。
[28] 趙文敏（1985）。數學史。台北：協進圖書。
[29] 蔡文淵（1996）。企業員工溝通滿足量表發展之研究。國立成功大學企業管理研究所碩士論文。台南。
[30] 蔣治邦、鍾思嘉（1991）。低年級學生加減概念的發展。教育心理與研究，14，35-68。
[31] 劉秋木（1996）。國小數學科教學研究。台北：五南。
[32] 鄭雅文（1996）。國民小學校長領導型態與教師溝通滿足感關係之研究。國立台南師範學院國民教育研究所碩士論文。台南。
[33] 謝堅（2000）。實驗課程中四則運算教材的設計。臺灣省國民教育研習會（主編），國民小學高年級數學科新課程概說（pp.78-97）。臺灣省國民教育研習會，台北。
[34] 鐘樹椽（1996）。國小學生電腦合作學習之研究－小組友誼、獎勵結構和學生因素之探討。台北：師大書苑。

貳、 英文文獻

[1] Alexander, J. M., & Shih, S. S. (2000). Interacting effects of goal setting and self- or other-referenced feedback on children’s development of self-efficacy and cognitive skill within the Taiwanese classroom. Journal of Educational Psychology, 92(3), 536-543.
[2] Anderson, J. R., & Kosslyn, S. (1984). Tutorials in learning and memory. San Franciso: W.H.Freeman.
[3] Applefield, J. M., Huber, R., & Mahnaz, M. (2000). Constructivism in theory and practice: toward a better understanding. High School Journal, 84(2), 35–59.
[4] Bangert-Drowns, R.L., Kulick, C. C., Kulik, J.A., & Morgan, M. T. (1991). The instructional effect of feedback in test-like events. Review of Educational Research, 61, 213-238.
[5] Barnard, C. I. (1968). The functions of the executive (30 Ed.). Cambridge, Massachusetts: Harvard University Press.
[6] Baroody, A. J. (1998). Fostering Children’s Mathematics Power: An investigative approach to K-8 Mathematics instruction.
[7] Barros, B., & Verdejo, M. F. (2000). Analysing student interaction processes in order to improve collaboration. The DEGREE approach. International Journal of Artificial
Intelligence in Education, 11, 221-241.
[8] Brown, A. L., & Campione, J. C. (1994). Guided discovery in a community of learners. In Classroom Lessons: Integrating Cognitive Theory and Classroom Practice (pp.
229-272). MIT Press/Bradford Books, Cambridge, MA.
[9] Carey, D., Carey, R., Willis, D. & Willis, J. (1992). Technology and Teacher Education Annual. Charlottesville: Association for the Advancement of Computing in Education.
[10] Carnine, D. (1997). Instructional design in mathematics for students with learning disabilities. Journal of learning disabilities, 30(2), 134-141.
[11] Cohen, V. B. (1985). A reexamination of feedback in computer-based instruction: Implications for instructional design. Educational Technology, 25(1), 33-37.
[12] Collins, M., Carnine, D., & Gersten, R. (1987). Elaborated corrective feedback and the acquisition of reasoning skills: A study of computer-assisted instruction. Exceptional Children, 154(3), 254-262.
[13] Copley, J. (1992). The integration of teacher education and technology: a constructivist model. In D. Carey, D. Carey, D. Willis, and J. Willis (Eds.).Technology and Teacher Education, Charlottesville, VA: AACE, 681.
[14] English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Hillsdale: Erlbaum.
[15] English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83-106.
[16] Fuson, K. C. (1992a). Research on Whole Number Addition and Subtraction. In D. A. Grouws (Ed.). Handbook of research on Mathematics Teaching and Learning (pp.243-275). New York: Macmillan Pub.
[17] Fuson, K. C. (1992b). Research on learning and teachimg addition and suntraction of whole numbers. In G. Leinhardt, R. Putnam & A. Hattrup (Eds.). Analysis of arithmetic for mathematics teaching (pp.53-187). Hillsdale, New Jersey: Lawrence Erlbaum Associates.
[18] Gibb, E. G. (1956). Children’s thinking in the process of subtraction. Journal of experimental education, 25, 71-80.
[19] Goswami, U. (1992). Analogical reasoning in children, Hillsdale, NJ: Lawrence Erlbaum Associates.
[20] Greenbaum, H. H. (1982). The Audit of Organizational Communication ,Contemporary Perspectives in Organizational Behavior. Boston: Allyn & Bacon.
[21] Greeno, J. G., & Heller, J. I. (1978). Development of children’s problem-soving ability in arithmetic. In H.P. Ginsburg (Ed.). The development of mathematical thinking
(pp.153-196). New York: Academic Press.
[22] Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics,18,75-90.
[23] Johnson, D. W., & Ahlgren, A. (1976). Relationship between student attitudes about cooperation and competition and attitudes toward schooling. Journal of Educational
Psychology, 68(1), 92-102.
[24] Johnson, D. W., & Johnson, R. T. (1987). Learning together and alone: Cooperative, competitive, and individualistic. Englewood Cliffs, NJ: Prentice Hall.
[25] Johnson, D. W., & Johnson, R. T. (1989). Leading the cooperative school. Edina, MN: Interaction.
[26] Johnson, D. W., & Johnson, R. T. (1990). Social skills for successful group work. Educational Leadership, 47(4), 29-33.
[27] Johnson, D. W., & Johnson, R. T. (1991). Learning together and alone: Cooperative, competitive, and individualistic. Third Edition. Englewood Cliffs, NJ: Prentice Hall.
[28] Johnson, D. W., & Johnson, R. T. (1993). Cooperative learning and feedback in technology-based instruction. In Dempsey, J., & Sales, G. C. (Ed.). Interactive instruction and feedback. Englewood Cliffs, NJ: Educational Technology Publications.
[29] Keh, N. C. (1992). Students’ use of teacher feedback during badminton instruction. Unpublished doctoral dissertation, Louisiana State University, LA: Baton Rouge.
[30] Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.).Handbook of research on mathematics teaching and learning. (pp.390-419). New York: Macmillan Pub.
[31] Knowles, M. S. (1975). Self-Directed Learning: A guild for leaders and teachers. New York:Cambridge Association Press.
[32] Kulhavy, R. W., & Stock, W. A. (1989). Feedback in written instruction: The place of response certitude. Educational Psycholohy Review, 1, 279-308.
[33] Marshall, S. P., Pribe, C. A., & Smith, J. D. (1987). Schema knowledge structure for representing and understanding arithemetic story problems. Tech. Rep. Contract No. N00014-85-K-0661.
[34] Michaels, J. W. (1977). Classroom reward structures and academic performance. Review of Educational Research, 47(1), 87-98.
[35] Milson, Fred (1973 ). A Introduction to Group Work Skill. London: Routledge and Kegan Paul.
[36] Roschelle, J., DiGiano, C., Koutlis, M., Repenning, A., Phillips, J., Jackiw, N., & Suthers, D. (1999). Developing educational software components. Computer, 32(9), 50-58. Piscataway, NJ: IEEE Computer Society.
[37] NCTM (2000). Principle and standards for school mathematics. Reston, VA:NCTM.
[38] Nesher, P., & Hershkovitz, S. (1994). The role of schemes in two-step Problems : Analysis and research findings. Educational Studies in Mathematics, 26, 1-23.
[39] Piaget, J. (1977). The development of thought: Equilibration of cognitive structures. (A. Rosin, Trans.). New York: Viking Penguin Inc.
[40] Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66, 211-277.
[41] Riel, M. M., & Levin, J. A. (1990). Building electronic communities: Success and failure in computer networking. Instructional Science, 19(2), 145-169.
[42] Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problem-soving ability in arithmetic. In H.P. Ginsburg (Ed.). The development of
mathematical thinking (pp.153-196). New York: Academic Press.
[43] Rink, J. E. (1985). Teaching physical education for learning. St. Louis: Times Mirror/Mosby.
[44] Roper, A. L. (1977). Feedback in computer-assisted instruction. Programmed Learning and Educational Technology, 14(1), 43-49.
[45] Sales, G. C. (1998). Designing feedback for CBI: Matching Feedback to learning and outcomes, Computers in the Schools, (pp225-239).
[46] Schmidt, R. A. (1991). Motor learning & performance from principles to practice, Champaign, IL: Human Kinetics Publications.
[47] Schmidt, R. A., & Wrisberg, C. A. (2000). Motor learning and performance (2nd Ed.). Champaign, IL: Human Kinetics.
[48] Siedentop, D. (1991). Developing teaching skills in physical education. (3rd Ed.). Mountain View,CA:Mayfield.
[49] Slavin, R. E. (1983). Effects of Cooperative Learning on Mainstreamed Academically Handicapped Children. Final Report. Johns Hopkins Univ., Baltimore, MD. Center for
Social Organization of Schools. ED249719.
[50] Slavin, R. E. (1989). Cooperative learning and student achievement. In R. E. Slavin(Ed.). School and classroom organization. Hillsdale,NJ:Lawrence Erlbaum Associates.
[51] Slavin, R. E. (1990). Cooperative learning: Theory, research, and practice. New Jersey: Prentice Hall.
[52] Slavin, R. E. (1995). Cooperative learning: Theory, research, and practice (2nd Ed.).Massachusetts: Allyn & Bacon.
[53] Swigger, K., & Brazile, R. (1997). The Virtual Collaborative University. Computers & Education, 29(2/3), 55-61.
[54] Waldrop, P. B., Justin, J. E., & Adams, T. M. (1986). A comparison of three types of feedback in a computer-assisted instruction task. Educational Technology, 26(2), 43-45.
[55] Webb, N. M. (1982). Group composition,group interaction, and achievement in cooperative small groups. Journal ofEducational Psychology, 74(4), 475-484.