以作者查詢圖書館館藏 、以作者查詢臺灣博碩士 、以作者查詢全國書目 、勘誤回報 、線上人數：24 、訪客IP：54.225.39.142

姓名楊馥祤(Fu-Yu Yang) 查詢紙本館藏 畢業系所網路學習科技研究所 論文名稱解釋數學：透過科技支援創作與討論以增強小學生的數學溝通能力

(Explaining Mathematics: Technology Enhanced Elementary Students′ Mathematical Communication Abilities through Creation and Discussion)檔案[Endnote RIS 格式] [Bibtex 格式] 至系統瀏覽論文 ( 永不開放) 摘要(中)數學溝通是學習數學的重要方法和能力。本研究以兩個年級為研究對

象，檢驗學生如何使用平板電腦培養數學溝通能力。研究一維持一學期，

使用等組前後測實驗設計，組間情境變項為RPTMC 和傳統變項，而組內則

採前、後測時間變項。在RPTMC 實驗組中，鼓勵學生產生數學創作，包含

數學表徵、解法和解法解釋，作為自己的教材，將之和同學作交互同儕教

學以增加數學溝通的機會。採用數學小老師的系統支持學生的數學創作和

同儕交互教學活動。有51 位二年級學生參與此實驗，研究評估他們數學溝

通能力進步程度。控制組學生接受一對一自我數學學習和教師引導教學，

而實驗組則在同環境中以相同設備投入電腦支援的同儕交互教學。使用數

學溝通能力測驗和線上數學創作評估兩組學生的學習成效，結果指出實驗

組在測驗中表現明顯高於控制組。此外，本研究也分析學生的數學創作以

評量他們的形成性發展。結果顯示學生的數學創作變得更清楚且有效率。

換句話說，他們的數學表徵和解釋在學習活動後變得更為精確。

研究二也鼓勵學生產出數學創作，包括表徵、解釋、解釋型寫作、總

結性寫作和擬題作為溝通媒材。此研究設計一個新的數學解說員系統支持

學生的創作和討論活動。91 位學生參與其中並以數學溝通測驗、線上數學

創作和動機問卷評估其數學溝通能力，結果揭露創作和討論溝通活動可以

促進學生的數學溝通能力，且低成就學生展現明顯的進步，除了數學表徵

的表現以外。不同成就的學生在數學創作上，由於文字題特徵和難度，使

他們在面積單元的解釋型寫作和時間單元的解題展現明顯的差異。然而，

學生的總結性寫作表現未如預期，只有少數學生找到文字題的關鍵特徵。

此外，學生的動機問卷結果顯示他們對概念討論和擬題較有興趣。然而，學生的擬題和系統提供的問題相似，只有稍微改變。值得注意的是，他們的解釋型寫作、解題和對創作討論溝通活動的動機能夠預測數學溝通活動的表現。研究最後也討論了對教學設計、數學老師和數學教育研究者的影響。摘要(英)Mathematical communication is an important approach and ability when learning mathematics. This study examined how to foster students’ mathematical communication abilities by using tablet PCs. Two studies using students from different grades were conducted. In Study 1 , a pretest-posttest 2 x 2 factorial design with two levels of between-subject conditions (“RPTMC” and “traditional”), and two levels of within-subject time variables ("pre" and "post") were used.

In the experimental group, RPTMC, the students were encouraged to generate math creations, including mathematical representations, solutions, and solution explanations, as their instructional materials. They then reciprocally tutored classmates to increase opportunities for mathematical communication throughout the semester. A Math Teacher System (MTS) was designed for supporting students’ math creations and reciprocal peer-tutoring activities.

An experiment involved 51 second-graders to evaluate their improvement in mathematical communication abilities. While the control group received one-to-one self-learning of mathematical materials and teacher-led instruction, the experimental group was engaged in computer-supported reciprocal peer tutoring (RPTMC), in the same environment with the same materials. Both groups were evaluated using a mathematical communication assessment.

The results indicated that the experimental group outperformed the control group in the post-assessment. Additionally, students’ math creations were analyzed to assess their formative development. The results showed that students’ math creations became clearer and more efficient. In other words, their mathematical representational abilities and writing became more accurate after the learning activity.

In Study 2, students were also encouraged to generate math creations, including mathematical representations, solutions, explanatory writing, summarizing writing, and problem posing as their communication materials. A new mathematical communication system, Math Narrator System, was designed for supporting students’ math creations and discussion activities.

Ninety-one students were involved in this study. Mathematical communication abilitieswere measured with a mathematical communication assessment, online math creation, and questionnaires according to students’ achievement performance. The results revealed that the CD communication activity enhanced students’ mathematical communication abilities, and the low achievers showed significant progress in all abilities except for mathematical representation.

Regarding the math creations, students only showed significant differences in their writing in the area unit and solution in the time unit. This might have been due to the features and difficulty of the word problems. However, students’ summarizing writing did not meet the expectations, and few students identified the critical features of word problems. Further, the questionnaire showed that students′ motivation in mathematics communication were more spurred by the concept discussion and problem posing. Additionally, students’ problem posing exhibited similar problems as those found in the system provided with slight changes. Their mathematical explanatory writing, solutions, and motivation for the CD communication activity were found to be predictive of mathematical communication abilities. The conclusions of this study, along with related pedagogical implications for instructional designers, math teachers, and math educational researchers, are also discussed.關鍵字(中)★ 數學溝通能力

★ 同儕解釋

★ 數學寫作

★ 數學表徵

★ 解題關鍵字(英)★ Mathematical Communication Ability

★ Peer Explanation

★ Mathematical Writing

★ Mathematical Representation

★ Problem-Solving論文目次摘要 i

ABSTRACT iii

Acknowledgements v

Table of Content vi

List of Figures ix

List of Tables xi

Chapter 1 Introduction 1

1-1 Background and Motivation 1

1-2 Research Purposes and Questions 7

1-3 Definitions 9

1-4 Study Limitations 10

1-4-1 Time 10

1-4-2 Subjects 10

1-4-3 Content 10

1-4-4 Tools 10

Chapter 2 Literature Review 12

2-1 Mathematical Communication 12

2-2 Problem Solving 18

2-3 Mathematical Representation 20

2-4 Problem Posing 21

2-5 Mathematical Writing 24

2-6 Self Explanation and Peer Explanation 26

Chapter 3 System Design and Activity Flow 29

3-1 Study 1 29

3-1-1 System Design: Math Teacher System (MTS) 29

3-1-2 Activity Flow: RPTMC activity 35

3-2 Study 2 38

3-2-1 System Design: Math Narrator System (MNS) 38

3-2-2 Activity Flow: CD communication activity 52

Chapter 4 Methods 60

4-1 Study 1 60

4-1-1 Participants 60

4-1-2 Study Procedure 61

4-1-3 Evaluative Methods 62

4-2 Study 2 66

4-2-1 Participants 66

4-2-2 Study Procedure 67

4-2-3 Evaluative Methods 68

Chapter 5 Results 78

5-1 Study 1 78

5-1-1 Mathematical Communication Assessment 78

5-1-2 Math Creation 80

5-1-3 Findings and Influences 82

5-2 Study 2 87

5-2-1 Mathematical Communication Assessment 87

5-2-2 Math Creation 98

5-2-3 Questionnaire 117

5-2-4 Factors Facilitating Mathematical Communication Ability 120

Chapter 6 Discussion and Contribution 123

6-1 Empirical contribution 123

6-2 Theoretical contribution 127

6-3 Practical contribution 128

Chapter 7 Conclusion and Future Directions 130

Reference 132

Appendix 142

Test Instructions of Mathematical Communication Assessment 142

Pre-Test: Mathematical Communication Assessment 143

Evaluative Criteria of Pre-Test 150

Post-Test: Mathematical Communication Assessment 156

Evaluative Criteria of Post-Test 163

Questionnaire of the CD communication activity 169參考文獻Ainsworth, S., & Loizou, A. (2003). The effect of self-explaining when learning with text or diagrams. Cognitive Science, 27, 669–681.

Aleven, V. A., & Koedinger, K. R. (2002).An effective metacognitive strategy: Learning by doing and explaining with a computer-based Cognitive Tutor.Cognitive science, 26(2), 147-179.

Anderson, L. W., & Krathwohl, D. R. (Eds.). (2001). A taxonomy for learning, teaching and assessing: A revision of Bloom′s Taxonomy of educational objectives: Complete edition, New York : Longman.

Allen, B. (1991). Using writing to promote students metacognition in muthemutics. Mathematics Education Research Group of Australasia. Western Australia: Perth.

Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003).Transitioning from studying examples to solving problems: Effects of self-explanation prompts and fading worked-out steps.Journal of Educational Psychology, 95(4), 774–783.

Baker, S., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achievers. Elementary School Journal, 103(1), 51-73.

Bangert-Drowns, R. L., Hurley, M. M., & Wilkinson, B. (2004).The Effects of school-based writing-to-learn interventions on academic achievement: A Meta-analysis.Review of Educational Research, 74(1), 29–58.

Baroody, A. J., & Ginsburg, H. P. (1990).Children’s mathematical learning: A Cognitive view. In C. Maher & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 51–64). Reston, VA: National Council of Teachers of Mathematics.

Berghmans, I, Neckebroeck, F., Dochy, F., & Struyven, K. (2013).A Typology of approaches to peer tutoring. Unraveling peer tutors’ behavioural strategies. European Journal of Psychology of Education, 28(3), 703–723.

Bicer, A., Capraro, M. M., & Capraro, R. M. (2014).Integrating writing into mathematics classroom as one communication factor.The Online Journal of New Horizons in Education, 4(2), 58-67.

Bloom, Benjamin S. & David R. Krathwohl. (1956). Taxonomy of educational objectives: The classification of educational goals, by a committee of college and university examiners. Handbook 1: Cognitive domain. New York , Longmans.

Bonotto, C. (2013). ArtifaCD as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37-56.

Bossé, M. J., & Faulconer, J. (2008). Learning and assessing mathematics through reading and writing. School Science and Mathematics, 108, 8-19.

Brendefur, J., & Frykholm, J. (2000). Promoting mathematical communication in the classroom: two preservice teachers’ conceptions and practices. Journal of mathematics teacher education, 3(2), 125-153.

Burton, L., & Morgan, C. (2000). Mathematicians writing. Journu1 for Research in Mathematics Education, 31, 429-453.

Bruce, C. D., McPherson, R., Sabeti, F. M., & Flynn, T. (2011).Revealing significant learning moments with interactive whiteboards in mathematics.Journal of Educational Computing Research, 45(4), 433–454.

Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing.The Journal of Mathematical Behavior, 21(4), 401-421.

Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students′ learning. Educational Studies in Mathematics, 83(1), 57-69.

Chan, T. W., Roschelle, J., Hsi, S., Kinshuk, Sharples, M., Brown, T., Patton C., Cherniavsky, J., Pea R., Norris, C., Soloway, E., Balacheff, N., Scardamalia, M., Dillenbourg, P., Looi, C. K., Milrad, M., & Hoope, U. (2006). One-to-one technology enhanced learning: An Opportunity for global research collaboration. Research and Practice in Technology Enhanced Learning, 1(1), 3–29.

Chi, M. T. H. (2009). Active-Constructive-Interactive: A Conceptual Framework for Differentiating Learning Activities. Topics in Cognitive Science, 1, 73–105.

Chi, M. T. H., Bassok, M., Lewis, M., Reimann, P., & Glaser, R. (1989). How Students Study and Use Examples in Learning to Solve Problems. Cognitive Science, 13, pp. 145-182.

Chi, M. T. H., de Leeuw, N., Chiu, M., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439–477.

Chi, M. T. H., Roy, M., & Hausmann, R. G. M. (2008).Observing tutorial dialogues collaboratively: Insights about human tutoring effectiveness from vicarious learning.Cognitive Science, 32(1), 301–341.

Clarke, D. J., Waywood, A., & Stephens, M. (1993).Probing the structure of mathematical writing. Educational Studies in Mathematics, 25(3), 235-250.

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997).Reflective discourse and collective reflection.Journal for Research in Mathematics Education, 28(3), 258–277.

Cooke, B. D., & Buchholz, D. (2005). Mathematical communication in the classroom: A Teacher makes a difference. Early Childhood Education Journal, 32(6), 365–369.

Cox, R. (1999).Representation construction, externalised cognition and individual differences.Learning and Instruction, 9(4), 343–363.

Dacey, L., & Eston, R. (2002). Show and tell: Representing and communicating mathematical ideas in K–2 classrooms. Sausalito, CA: Math Solutions.

De Backer, L., Van Keer, H., & Valcke, M. (2012).Exploring the potential impact of reciprocal peer tutoring on higher education student’s metacognitive knowledge and regulation.Instructional Science, 40(3), 559–588.

English, L. (2003). Engaging students in problem posing in an inquiry-oriented mathematics classroom. In F. Lester, & R. Charles, Teaching mathematics through problem solving (pp. 187–198). Reston: National Council of Teachers of Mathematics.

Fantuzzo, J. W., King, J. A., & Heller, L. R. (1992).Effects of reciprocal peer tutoring on mathematics and school adjustment: A Component analysis.Journal of Educational Psychology, 84(3), 331–339.

Fiorella, L., & Mayer, R. E. (2014).Role of expectations and explanations in learning by teaching.Contemporary Educational Psychology, 39(2), 75–85.

Forman, E., & Ansell, E. (2002).Orchestrating the multiple voices and inscriptions of a mathematics classroom.Journal of the Learning Sciences, 1, 251–274.

Freitag, M. (1997). Reading and writing in the mathematics classroom. The Mathematics Educator, 8, 16-21.

Fuchs, L., Fuchs, D., Finelli, R., Courey, S., Hamlett, C., & Sones, E. (2006). Teaching third graders about real-life inathematical problem solving: A randomized controlled study. Elementary School, 106, 293-312.

Geist, L., Hatch, P., & Erickson, K. (2014).Promoting Academic Achievement for Early Communicators of All Ages.SIG 12 Perspectives on Augmentative and Alternative Communication, 23(4), 173-181.

Griffin, C. C., & Jitendra, A. K. (2008). Word problem solving instruction in inclusive third grade mathematics classrooms. Journal of Educational Research, 102(3), 187-202.

Hallgren, K. A. (2012). Computing inter-rater reliability for observational data: an overview and tutorial. Tutorials in quantitative methods for psychology, 8(1), 23.

Hausmann, R., Chi, M. T. H., & Roy, M. (2004). Learning from collaborative problem solving: An analysis of three hypothesized mechanisms. In K. Forbus, D. Gentner, & T. Regier (Ed.). (pp. 547–552). Mahwah, NJ: Erlbaum.

HegartyM., & KozhevnikovM. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91, 684–689.

Hwang, W. Y., Su, J. H., Huang, Y. M., & Dong, J. J. (2009).A Study of multi-representation of geometry problem solving with virtual manipulatives and whiteboard system.Journal of Educational Technology & Society, 12(3), 229–247.

Jitendra, A., Griffin, C., Haria, P., Leh, J., Adams, A., & Kaduvettoor, A. (2005). A comparison of single and multiple strategy instruction on third grade students′ mathematical problem solving. Journal of Educational Psychology, 99, 115-127.

Johnson, D. W., & Johnson, R. T. (1999).Making cooperative learning work.Theory into Practice, 38(2), 67–73.

Keller, J. (1987). Development and use of the ARCS model of motivational design. Journal of Instructional Development, 10(3), 2-10.

Keller, J. (2010). Motivational design for learning and performance: The ARCS model approach. New York: Springer.

Keller, J. M., & SuzukiK. (2004). Learner motivation and E-learning design: A multinationally validated process. Journal of Educational Media, 29(3), 229-239.

Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. Schoenfeld, Cognitive science and mathematics education (pp. 123–147). Hillsdale: Lawrence Erlbaum Associates.

King, A. (1994). Guiding knowledge construction in the classroom: EffeCD of teaching children how to question and how to explain. American Educational Research Journal, 31(2), 338–368.

King, A. (1998). Transactive peer tutoring: Distributing cognition and metacognition.Educational Psychology Review, 10(1), 57–74.

King, A., Staffieri, A., & Adelgais, A. (1998). Mutual peer tutoring: Effects of structuring tutorial interaction to scaffold peer learning. Journal of Educational Psychology, 90(1), 134–152.

Koedinger, K., Anderson, J. R., Hadley, W. H., & Mark, M. A. (1997). Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education, 8, 30–43.

Leung, S. (1994). On analyzing problem-posing processes: A study of prospective elementary teachers differing in mathematics knowledge. In J. da Ponte, & J. Matos, Proceedings of the 18th international conference of the International Group for the Psychology of Mathematics Education (pp. 168–175). Lisbon: University of Lisbon.

Leung, S. (2009). Research efforts on probing students’ conceptions in mathematics and reality: Structuring problems, solving problems, and justifying solutions. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay, Words and worlds: Modeling verbal descriptions of situations (pp. 213–225). New York: Sense.

Leung, S. (2013). Teachers implementing mathematical problem posing in the classroom: challenges and strategies. Educational Studies in Mathematics, 83(1), 103-116.

Leung, S., & Silver, E. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9(1), 5–24.

Lin, C. S., Shann, W. C., & Lin, S. C. (2008). Reflections on mathematical communication from Taiwan math curriculum guideline and PISA 2003. Retrieved from http://www.criced.tsukuba.ac.jp/math/apec/apec2008/papers/PDF/16.Lin_Su_Chun_Taiwan.pdf

Lin, Y. H., & Lee, Q. Y. (2004). A study of ability on mathematical communication for students of elementary schools. Journal of Educational Measurement and Statistics, 12, 233-268.(In Chinese)

Marks, G., & Mousley, J. (1990). Mathematics, education and genre: Dare we make the process writing mistake again? Language and Education, 4, 117-135.

Mason, J. (2010). Effective questioning and responding in the mathematics classroom. Retrieved from http://xtec.cat/centres/a8005072/articles/effective_questioning.pdf

McIntosh, M. E. (1991). No time for writing in your class?.The Mathematics Teacher, 84(6), 423-433.

McNamara, D. (2004). SERT: Self-explanation reading training. Discourse Processes, 38, 1-30.

Montague, M. (1997). Cognitive strategy instruction in mathematics for students with learning disabilities. Journal of learning disabilities, 30(2), 164-177.

Mooney, C., Hansen, A., Ferrie, L., Fox, S., & Wrathmell, R. (2012).Primary mathematics: Knowledge and understanding. Exeter,UK: Learning Matters.

Moschkovich, J. (2012). Mathematics, the common core, and language: Recommendations for mathematics instruction for ELs aligned with the common core. In Understanding Language: Language, Literacy, and Learning in the Content Areas (pp. 17-31) Stanford, CA: Understanding Language.

Mosston, M., & Ashworth, S. (2002). Teaching physical education (5th ed.). San Francisco, CA: Benjamin Cummings.

Nathan, M. J., & Koedinger, K. R. (2000).An investigation of teachers′ beliefs of students′ algebra development.Cognition and Instruction, 18(2), 209-237.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.Reston, VA: National Council of Teachers of Mathematics.

National Governors Association.(2010). Common core state standards.retrieved on August 20, 2016 at http://www.corestandards.org/Math/Practice/

Neria, D., & Amit, M. (2004). Students preference of non-algebraic representations in mathematical communication′. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 409-416).

Newell, S. (2008). Dirty Whites:" Ruffian-Writing" in Colonial West Africa. Research in African Literatures, 39(4), 1-13.

OECD (2016), PISA 2015 Assessment and Analytical Framework: Science, Reading, Mathematic and Financial Literacy, PISA, OECD Publishing, Paris.

Ottmar, E. R., Rimm-Kaufman, S. E., Larsen, R. A., & Berry, R. Q. (2015). Mathematical knowledge for teaching, standards-based mathematics teaching practices, and student achievement in the context of the responsive classroom approach. American Educational Research Journal, 52(4), 787-821.

Pape, S., & Tchoshanov, M. (2001). The Role of Representation(s) in Developing Mathematical Understanding. Theory into Practice, 40(2), 118-127.

Patton, M. (2002). Qualitative research and evaluation methods (3 ed.). Thousand Oaks, CA: Sage.Paiva, F., Glenn, J., Mazidi, K., Talbot, R., Wylie, R., Chi, M. T., ... & Nielsen, R. D. (2014, June). Comprehension seeding: Comprehension through self explanation, enhanced discussion, and inquiry generation. In International Conference on Intelligent Tutoring Systems (pp. 283-293). Springer International Publishing.

Paiva, R. O. A., Santa Pinto, I. I. B., da Silva, A. P., Isotani, S., & Jaques, P. (2014, June). A systematic approach for providing personalized pedagogical recommendations based on educational data mining. In International Conference on Intelligent Tutoring Systems (pp. 362-367). Springer International Publishing.

Pigott, H. E., Fantuzzo, J. W., & Clement, P. W. (1986). The Effect of reciprocal peer tutoring and group contingencies on the academic performance of elementary school children.Journal of Applied Behavior Analysis, 19(1), 93–98.

Poch, A. L., van Garderen, D., & Scheuermann, A. M. (2015). Students’ Understanding of Diagrams for Solving Word Problems A Framework for Assessing Diagram Proficiency. TEACHING Exceptional Children, 47(3), 153-162.

Polya, G. (1973). How to solve it: A New aspect of mathematical method. Princeton, NJ: Princeton University Press.

Porter, M., & Masingila, J. (2000). Examining the effect of writing on conceptual and procedural knowledge in calculus. Educutional Studies in Mathematics, 42, 165-177.

Presmeg, N. (1986a). Visualization and mathematical giftedness. Educational Studies in Mathematics, 17, 297–311.

Presmeg, N. (1986b). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.

Pugalee, D. K. (2004).A comparison of verbal and written descriptions of students′ problem solving processes.Educational Studies in Mathematics, 55(1-3), 27-47.

Ririe, J., & Redford, K. (2008). Communication and Discourse. Retrieved from Strategies for Teaching Elementary Mathematics: http://mathteachingstrategies.wordpress.com/2008/11/24/communication-discourse/

Roscoe, R. D., & Chi, M. T. H. (2007). Understanding tutor learning: Knowledge building and knowledge telling in peer tutors’ explanations and questions. Review of Educational Research, 77(4), 534–574.

Roscoe, R. D., & Chi, M. T. H. (2008).Tutor learning: The role of explaining and responding to questions.Instructional Science, 36(4), 321–350.

Rosenthal, D. J., & Resnick, L. B. (1974). Children′s solution processes in arithmetic word problems. Journal of Educational Psychology, 66(6), 817.

Roy, M., & Chi, M. (2005). The self-explanation principle in multimedia learning. In R. Mayer, The Cambridge handbook of multimedia learning (pp. 271-286). Cambridge: Cambridge University.

Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475-522.

Sedig, K. (2008). From play to thoughtful learning: A design strategy to engage children with mathematical representations. Journal of Computers in Mathematics and Science Teaching, 27(1), 65-101.

Sfard, A. (2000). On reform movement and the limits of mathematical discourse. Mathematical Thinking and Learning, 2, 157-189.

Shepard, R. G. (1993). Writing for conceptual development in mathematics. The Journal of Mathematical Behavior,12(3), 287-293.

Shimizu, Y., & Lambdin, D. V. (1997). Assessing students’ performance on an extended problem-solving task: A Story from a Japanese classroom.The Mathematics Teacher, 90(8), 658–664.

Siegler, R. (2005). Children′s learning. American Psychologist, 60, 769-778.

Sierpinska, A. (1998). Three epistemologies, three views of communication: Constructivism, social cultural approaches, interactionism. In H. Steinbring, M. G. Bartonlini Bussi, & A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 30–62). Reston, VA: National Council of Teachers of Mathematics.

Silver, A. E. (1994). On Mathematical Problem Posing. For the learning of mathematics, 14(1), 19-28.

Silver, E. A., & Smith, M. S. (1996). Building discourse communities in mathematics classrooms: A worthwhile but challenging journey. In P. C. Elliott, Communication in mathematics, K-12 and beyond: 1996 yearbook (pp. 20-28). Reston, VA: National Council of Teachers of Mathematics.

Silver, E., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521-539.

Silver, E., & Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135.

Siswono, T. (2010). Leveling students’ creative thinking in solving and posing mathematical problem. Indonesian Mathematical Society Journal on Mathematics Education, 1(1), pp. 20–41.

Smith, G. G., Ferguson, D., & Caris, M. (2003). The web versus the classroom: Instructor experiences in discussion-based and mathematics-based disciplines. Journal ofEducational Computing Research, 29(1), 29–59.

Sorsana, C. (2005). Beliefs and conversational skills among children: Reflections on the dialogic management disagreements within reasoning. Psychology of Interaction, 19/20, 39-97.

Sriraman, B. (2009). The characteristics of mathematical creativity. Zentralblatt für Didaktik der Mathematik, 41(1-2), pp. 13–27.

Stahl, G. (2006). Supporting group cognition in an online math community: A cognitive tool for small-group referencing in text chat. Journal of Educational Computing Research, 35(2), 1–16.

Stahl, G. (2009). Social practices of group cognition in virtual math teams. In S. Ludvigsen, A. Lund, & R. Säljö (Eds.), Learning in social practices. ICT and new artifacts: Transformation of social and cultural practices (pp.190-205). London, UK: Pergamon.

Steele, D. (2005). Using writing to access students′ schemata knowledge for algebraic thinking. School Science and Mathematics, 105(3), 142–154.

Steele, D. F., & Arth, A. A. (1998). Math instruction and assessment: Preventing anxiety, promoting confidence. Schools in the Middle, 7(3), 44-48.

Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students′ problem posing in school mathematics. In P. C. Clarkson, Technology in mathematics education (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia.

Sür, B., & Delice, A. (2016, January). The examination of teacher student communication process in the classroom: mathematical communication process model. In SHS Web of Conferences (Vol. 26).EDP Sciences.

Taiwan, M. O. E. (2003). General Guidelines of Grade 1-9 Curriculum of Elementary and Junior High School Education—mathematics learning areas. Taipei, Taiwan: Author.

Tajika, H., Nakatsu, N., Nozaki, H., Neumann, E., & Maruno, S. (2007). Effects of self‐explanation as a metacognitive strategy for solving mathematical word problems1. Japanese Psychological Research, 49(3), 222-233.

Teuscher, D., Kulinna, P. H., &Crooker, C. (2016). Writing to Learn Mathematics: An Update. The Mathematics Educator, 24(2), 56–78

Thompson, D. R., & Chappell, M. F. (2007). Communication and Representation as Elements in Mathematical Literacy. Reading & Writing Quarterly: Overcoming Learning Difficulties, 23(2), 179-196.

Topping, K. J. (2005).Trends in peer learning.Educational Psychology, 25(6), 631–645.

Tsuei, M. (2012).Using synchronous peer tutoring system to promote elementary students’ mathematics learning.Computers & Education, 58(4), 1171–1182.

Uesaka, Y., & Manalo, E. (2011). The effeCD of peer communication with Diagrams on students’ math word problem solving processes and outcomes. Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 312–317). Austin, TX: Cognitive Science Society.

Van de Walle, J. (2007). Elementary and middle school mathematics: Teaching developmentally (6 ed.). Boston: Allyn and Bacon: Texas.

van Garderen, D., & Montague, M. (2003). Visual-Spatial Representation, Mathematical Problem Solving, and Students of Varying Abilities. Learning Disabilities Research & Practice, 18(4), 246–254.

van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36(3), 145-160.

VanLehn, K., Jones, R. M., & Chi, M. T. (1992).A model of the self-explanation effect.The journal of the learning sciences, 2(1), 1-59.

Verschaffel, L., Van Dooren, W., Chen, L., & Stessens, K. (2009). The relationship between posing and solving division-with-remainder problems among flemish upper elementary school children. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay, Words and worlds: Modelling verbal descriptions of situations (pp. 143–160). Rotterdam: Sense.

Walker, E., Rummel, N., & Koedinger, K. R. (2011). Designing automated adaptive support to improve student helping behaviors in a peer tutoring activity. International Journal of Computer-Supported Collaborative Learning, 6(2), 279–306.

Webb, N. M., & Mastergeorge, A. M. (2003). The development of students’ helping behavior and learning in peer-directed small group. Cognition and Instruction, 21(4), 361–428.

Whitin, D., & Whitin, P. (2000a). Exploring mathematics through talking and writing. In M. Burke, & F. Curcio, Learning mathematics for a new century: 2000 yearbook (pp. 213–222). Reston, VA: National Council of Teachers of Mathematics.

Whitin, P., & Whitin, D. J. (2000b).Math is language too: Talking and writing in the mathematics classroom. Reston, VA: National Council of Teachers of Mathematics.

Wilcox, B., & Monroe, E. E. (2011). Integrating Writing and Mathematics. The Reading Teacher, 64(7), 521-529.

Yuan, X., & Sriraman, B. (2010). An exploratory study of relationships between students’ creativity and mathematical problem posing abilities. In B. Sririman, & K. Lee, The elements of creativity and giftedness in mathematics (pp. 5-28). Rotterdam: Sense.

Zhe, L. (2012). Survey of Primary Students′. Mathematical Representation Status and Study on the Teaching Model of. Mathematical Representation. Journal of Mathematics Education, 5(1), 63-76.

Zinsser, W. (1998). Writing to learn. New York: Harper and Rowe.指導教授陳德懷 審核日期2016-8-31 推文facebook plurk twitter funp google live udn HD myshare reddit netvibes friend youpush delicious baidu 網路書籤Google bookmarks del.icio.us hemidemi myshare