本研究的目的是要探索不同種類的解答方式演練範例與問題提示策略對於國中三年級學生例行性問題解決與非例行性問題解決得分表現、類推表現與核証表現之影響，並分析學生在例行性問題解決與非例行性問題解決的類推策略。活動之教學設計以蛙跳遊戲為情境並提供相關電腦工具用來支援例行性與非例行性問題解決的類推與核證。本實驗共分兩個階段，在例行性問解解決任務階段，將學習者分為多解答方法演練範例組與單解答方法演練範例組，多解答方法演練範例組提供學生的範例中都有2個以上的解答方法，並要學生比較分析多種解答方法的優缺點;單解答方法演練範例組，所提供的範例則只提供一種較為常見的解答方法。兩組都有提供學生自我解釋解答步驟的學習單，以記錄學生的學習過程與思考策略。在非例行性問題解決任務階段，則將學習者分為精緻化反思問題提示組與專家解題程序問題提示組，精緻化反思提示組提供了三層的提示，由一般策略提示到特定策略提示，讓學習者有機會將思考過程更加精緻化並提供更多反思的機會; 專家解題程序提示組則是提供學習者專家的解題程序提示，希望學習者能模仿專家的解題行為與思考模式順利解決任務。本研究根據上述兩個階段的分組，採不等組前後測二因子之準實驗設計，選取台灣桃園縣某縣立國民中學國三學生四個班共120人為實驗研究對象，該校採常態男女合班的模式教學，隨機將四個班分別指定為多解答方法演練範例-精緻化反思問題提示組(M-R)，單解答方法演練範例-精緻化反思問題提示組(S-R)，多解答方法演練範例-專家解題程序問題提示組(M-P)與單解答方法演練範例-專家解題程序問題提示組(S-P)。學習單依照上述四組分別設計並於蛙跳問題教學網站提供相對應的網路學習教材。
經由統計與實徵資料分析所得主要結果如下: 1.先備知識與數學態度可以有效預測例行性問題解決之得分表現、類推表現與核證表現。2.精緻化反思問題提示組學生，在非例行性問題解決的得分表現、類推表現與核證表現，都顯著高於專家解題程序問題提示組。3.多解答方法演練範例，雖然在一開始的例行性問題解決階段，看不出任何優於單解答方法演練範例的效果，但在非例行性問題解決階段的類推與核證表現上，卻發現多解答方法演練範例教學的好處。4.類推與核證的關係是非常緊密的，發展較好的類推策略會影響到後續的核證品質，而發展較好的核證策略則會影響到後來的類推層次。5.多解答方法演練範例組的學生在科技輔助教學意見的看法上，比單解答範例演練組學生更為正向，特別是在情感方面與知覺易用性這兩個層面上。
最後根據研究結果與發現，提出若干建議以做為教師教學改進與未來研究之參考。

The purpose of this study was to explore the effects of multiple solution methods and elaborative reflection prompts on ninth graders’ generalizations and justifications in routine and non-routine problem solving. The Frog Leaping Computer Game was used as the context and web-based learning environment was provided for supporting generalizations and justifications in routine and non-routine mathematical problem solving.
A 2x2 (multiple solution methods: Multiple/Single; question prompts: elaborative Reflection/expert Problem-solving procedure) and factorial, quasi-experimental study was conducted to investigate generalizations and justifications of routine and non-routine problem solving performance. One hundred and twenty 9th graders from four classes in a public junior high school participated in the eight-week experimental instruction. These four classes were randomly assigned to the four groups (M-R, S-R, M-P, and S-P) to receive the one-hour weekly treatment. Worksheets and web-based learning materials were separately designed to record the four groups’ learning processes and thinking strategies.
Based on the data analysis of this study, the main results revealed that
1. Prior knowledge and mathematics attitude could significantly predict routine problem solving performance, generalization performance, and justification performance.
2. R group outperformed P group on non-routine problem-solving performance, generalization performance and justification performance.
3. In routine problem solving, M group did not outperform S group. However, M group outperformed S group on non-routine generalization performance and justification performance.
4. Generalization and justification are closed linked. Helping students develop their powerful generalizations would aid in their abilities to construct justifications. Furthermore, a focus on justification could help students develop the subsequent, more powerful generalizations.
5. M group had more positive perceptions toward the computer tools than R group, especially in the affective scale and perceived control scale.
Lastly, implications derived from these results were discussed and recommendations for both further instruction and future research were also provided.

Ahl, D. (1981). Computer games in mathematics education. Mathematics Teacher, 74(8), 653-656.
Aiken, L. R. (1976). Update of attitudes and other affective variables in learning mathematics. Review of Educational Research, 46, 293-311.
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183-198.
Amit, M., & Neria, D. (2007). Assessing a modeling process of a linear pattern task. Paper presented at the 13th conference of the International Community of Teachers of Mathematical Modeling and Applications, IN, USA.
Amit, M., & Neria, D. (2008). Rising to challenge: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111-129.
Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: Combining fading with prompting fosters learning. Journal of Educational Psychology, 95, 774-783.
Australian Education Council. (1994). Mathematics: A curriculum profile for Australian schools. Carlton. VIC: Curriculum Corporation.
Ayres, P., & Sweller, J. (2005). The split-attention principle. In R. E. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 135-146). Cambridge, UK: Cambridge University Press.
Bahr, C., & Rieth, H. (1989). The effects of instructional computer games and drill and practice software on learning disabled students’ mathematics achievement. Computers in the Schools, 6(3-4), 87-101.
Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18, 147–176.
Becker, J. R., & Rivera, F. (2004). An investigation of beginning algebra students’ ability to generalize linear patterns. In M. J. Hoins, & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 1, p. 286). Bergen, Norway: Bergen University College.
Bell, A. (1979). The learning of process aspects of mathematics. Educational Studies in Mathematics, 10, 361–387.
Betz, J. A. (1995). Computer games: increases learning in an interactive multidisciplinary environment. Journal of Educational Technology Systems, 24, 195-205.
Blanton, M., & Kaput, J. (2000). Generalizing and progressively formalizing in a third-grade mathematics classroom: Conversations about even and odd numbers. In M. L. Fernandez (Ed.), Proceedings of the 22nd Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 115-119). Columbus, OH: The Eric Clearinghouse for Science, Mathematics, and Environmental Education.
Blanton, M., & Kaput, J. (2002). Developing elementary teachers’ algebra “eyes and ears”: Understanding characteristics of professional development that promote generative and self-understanding change in teacher practice. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.
Bodemer, D., Plo¨tzner, R., Feuerlein, I., & Spada, H. (2004). The active integration of information during learning with dynamic and interactive visualisations. Learning and Instruction, 14, 325-341.
Bromme, R., & Stahl, E. (2002). Learning by producing hypertext from reader perspectives: Cognitive flexibility theory reconsidered. In R. Bromme, & E. Stahl (Eds.), Writing hypertext and learning: Conceptual and empirical approaches (pp. 39-61). London: Pergamon.
Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent. (Eds.), Proceedings of the 12th ICMI Study Conference: The future of the teaching and learning of algebra (pp. 155-162). Melbourne, Australia: The University of Melbourne.
Carpenter, T., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Research Report #002. Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science. Retrieved August 15, 2005 from http://www.wisc.wcer.edu/ncisla
Chang, K. E., Sung, Y. T., & Lin, S. F. (2006). Computer-assisted learning for mathematical problem solving. Computers & Education, 46, 140-151.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359-387.
Chen, C. H., & Bradshaw, A. C. (2007). The effect of web-based question prompts on Scaffolding knowledge integration and ill-structured problem solving. Journal of Research on Technology in Education, 39(4), 359-375.
Chi, M. T., & Glaser, R. (1985). Problem-solving ability. In R. J. Sternberg (Ed.), Human abilities: An information processing approach (pp. 227-250). New York: Freeman.
Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145-182.
Clark, R. C., & Mayer, R. E. (2002). e-Learning and the science of instruction: Proven guidelines for consumers and designers of multimedia learning. San Francisco, CA: John Wiley & Sons, Inc.
Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 2, 41–53.
Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive Apprenticeship. Teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning, and instruction (pp. 453-493). Hillsdale, NJ: Erlbaum.
Davis, R. B. (1986). Learning mathematics: The cognitive science approach to mathematics education (2nd ed.). Norwood, NJ: Ablex.
Davis, E. A. (2003). Prompting middle school science students for productive reflection: Generic and directed prompts. The Journal of Learning Sciences, 12(1), 91-142.
Davis, E. A., & Linn, M. (2000). Scaffolding students' knowledge integration: Prompts for reflection in KIE. International Journal of Science Education, 22(8), 819-837.
Davydov, G. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Soviet studies in mathematics, volume 2. Reston, VA: National Council of Teachers of Mathematics.
Department for Education and Skills. (2001). Frameworks for teaching mathematics: Years 7, 8, and 9. London: DfEs Publiscaitons.
De Jong, T., Ainsworth, S., Dobson, M., van der Hulst, A., Levonen, J., & Reimann, P. (1998). Acquiring knowledge in science and mathematics: The use of multiple representations in technology-based learning environments. In M. van Someren, P. Reimann, H. Boshuizen, & T. de Jong (Eds.), Learning with multiple representations (pp. 9-41). Oxford: Elsevier Sciences.
Dienes, Z. P. (1961). On abstraction and generalization. Havard Educaitonal Review, 3, 289-301.
Din, F., & Caleo, J. (2000). Playing computer games versus better learning. Paper presented at the Annual Conference of the Eastern Educational Research Association, Clearwater, FL, USA.February 16-19.
Dochy, F., Moerkerke, G., & Marten, R. (1996). Integrating assessment, learning, and instruction: Assessment of domain-specific and domain-transcending prior knowledge and program. Studies in Educational Evaluation, 22, 309-339.
Doerfler, W. (1991). Forms and means of generalization in mathematics. In A. Bishop (Ed.), Mathematical knowledge: Its growth through teaching (pp. 63-85). Mahwah, NJ: Erlbaum.
Egenfeldt-Nielsen, S. (2005). Beyond edutainment: Exploring the education potential of computer games. Ph. D. dissertation, IT University of Copenhagen.
Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM Mathematics Education. (Published Online: May. 19, 2009, doi: 10.1007/s11858-009-0184-6)
Ellis, A. B. (2007). Connections between generalizing and justifying: Students reasoning with linear relationships. Journal for Research in Mathematics Education, 38, 194–229.
Elshout, J. J., Veenman, M. V. J., & van Hell, J. G. (1993). Using the computer as a help tool during learning by doing. Computers in Education, 21(1-2), 115-122.
English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ:Lawrence Erlbaum.
English, L., &Warren, E. (1995). General reasoning processes and elementary algebraic understanding: Implications for instruction. Focus on Learning Problems in Mathematics, 17(4), 1-19.
English, L. D., & Warren, E. A. (1998). Introducing the variable through pattern exploration. The Mathematics Teacher, 91, 166-170.
Fennema, E., & Sherman, J. (1978). Sex-related differences in mathematics achievement: A further study. Journal for Research in Mathematics Education, 9, 189-203.
Fitzgerald, G. (1991). Using the computer with student with emotional and behavioral disorders. Reston, VA: Center for Special Education Technology.
Fogarty, R., Perkins, D. N., & Barell, J. (1992). How to teach for transfer. Palatine, IL: Skylight Publishing.
Garofalo, J., & Lester, F. (1985). Metacogintion, cognitive monitoring and mathematical performance. Journal for Research in Mathematics Education, 16, 163-175.
Garris, R., Ahlers, R., & Driskell, J. E. (2002). Games, motivation, and learning: A research and practice model. Simulation & Gaming, 33(4), 441-467.
Ge, X., & Land, S. M. (2003). Scaffolding students’ problem-solving process in an ill-structured task using question prompts and peer interactions. Educational Technology Research and Development, 51(1), 21-38.
Ge, X., & Land, S. M. (2004). A conceptual framework for scaffolding ill-structured problem-solving processes using question prompts and peer interactions. Educational Technology Research and Development, 52(2), 5-22.
Ge, X., Chen, C. H., & Davis, K. (2005). Scaffolding novice instructional designers’ problem solving processes using question prompts in a web-based learning environment. Journal of Educational Computing Research, 33(2), 219-248.
Gee, J. P. (2003). What video games have to teach us about learning and literacy. NY: MacMillan.
Gredler, M. E. (1996). Games and simulations and their relationships to learning. In D. H. Jonassen (Ed.), Handbook of research for educational communications and technology (pp. 571-603). New York: Macmillan Library Reference.
Gredler, M. E. (2002). Educational games and simulations: A technology in search of research paradigm. In D. H. Jonassen (Ed.), Handbook of research for educational communications and technology (pp. 521-539). NY: MacMillan.
Große, C. S., & Renkl, A. (2005). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning and Instruction, 17(6), 612-634.
Große, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning. Learning and Instruction, 16, 122-138.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Harel, G., & Tall, D. (1991). The general, the abstract, and the generic. For the Learning of Mathematics, 11(1), 38–42.
Harskamp, E., & Suhre, C. (2007). Schoenfeld’s problem solving theory in a student controlled learning environment. Computers & Education, 49, 822-839.
Healy, L., & Hoyles, C. (2000). Visual and symbolic reasoning in mathematics: Making connections with computers? Mathematical Thinking and Learning, 1(1), 59-84.
Hwang, W. Y., Chen, N. S., & Hsu, R. L. (2006). Development and evaluation of multimedia whiteboard system for improving mathematical problem solving. Computers & Education, 46, 105-121.
Inkpen, K. (1994). We have never-forgetful flowers in our garden: Girls’ responses to electronic games. Journal of Computers in Mathematics and Science Teaching, 13(4), 383-403.
Ishida, J. (1997). The teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process. School Science and Mathematics, 97, 155–162.
Jenkins, H. (2002). Game theory. Technology Review, 29, 1-3.
Jonassen, D. H. (1997). Instructional design models for well-structured and ill-structured problem-solving learning outcomes. Educational Technology Research & Development, 45, 65-94.
Jonassen, D. H. (1999). Designing constructivist learning environments. In C. M. Reigeluth (Ed.), Instructional-design theories and models vol. 2: A new paradigm of instructional theory (pp. 215-140). NJ: Lawrence Erlbaum.
Kaput, J. (1999). Teaching and learning a new algebra with understanding. In E. Fennema, & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Erlbaum.
Ke, F. (2008). A case study of computer gaming for math: Engaged learning form gameplay? Computers & Education, 51(4), 1609-1620.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Macmillan Publishing Company.
Kiili, K. (2007). Foundation for problem-based gaming. British Journal of Educational Technology, 38(3), 394-404.
King, A. (1991). Effects of training in strategic questioning on children's problem-solving performance. Journal of Educational Psychology, 83(3), 307-317.
King, A. (1992), Facilitating elaborative learning through guided student-generated questioning. Educational Psychologist, 27(1), 111-1.
King, A., & Rosenshine, B. (1993). Effect of guided cooperative questioning on children’s knowledge construction. Journal of Experimental Education, 61(2), 127-148.
Knezek, G. (1997) Computers in education worldwide: Impact on students and teachers. (On line) Available: http://www.tcet.unt.edu/research/worldwd.htm.
Knuth, E., Slaughter, M., Choppin, J., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Eds.), Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1693-1700). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.
Kulik, J. (1994). Meta-analytic studies of findings on computer-based instruction. In E. Baker, & H. O’Neil (Eds.), Technology assessment in education and training (pp. 9-33). New York: Lawrence Erlbaum.
Lakatos, I. (1976). Proofs and refutations. Cambridge, England: Cambridge University Press.
Lakatos, I. (1986). A renaissance of empiricism in the recent philosophy of mathematics. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 29–48). Boston: Birkhauser.
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7, 231–258.
Lee, C. Y. (in press). Pattern generalization and justification in game-based mathematical E-learning environments: The differences between the good problem-solver group and the poor problem-solver group.
Lee, C. Y., & Chen, M. P. (2007). Using computer multimedia to dissolve cognitive conflicts of mathematical proof. The Electronic Journal of Mathematics and Technology, 1(2), 95-106.
Lee, C. Y., & Chen, M. P. (2008). Taiwanese junior high school students' mathematics attitudes and perceptions toward virtual manipulatives. British Journal of Educational Technology. (Published Online: Aug. 22, 2008, doi:10.1111/j.1467-8535.2008.00877.x)
Lee, C. Y., & Chen, M. P. (2009). A computer game as a context for non-routine mathematical problem solving: The effects of type of question prompt and level of prior knowledge. Computers & Education, 52(3), 530-542.
Lepper, M., & Malone, T. (1987). Intrinsic motivation and instructional effectiveness in computer-based education. In R. E. En Snow, & M. J. Farr (Eds.), Aptitudes, learning and instruction, II: Cognitive and affective process analysis (pp. 255-286). Hillsdale, NJ: Lawrence Erlbaum.
Lester, F. K. (1994). Musings about mathematical problem-solving research: 1970-1994. Journal for Research in Mathematics Education, 25, 660-675.
Lin, X. (2001). Designing metacognitive activities. Educational Technology Research and Development, 49, 23-40.
Lin, X., Hmelo, C., Kinzer, C. K. & Secules, T. J. (1999). Designing technology to support reflection. Educational Technology Research and Development, 47, 43-62.
Lin, X., & Lehman, J. D. (1999). Supporting learning of variable control in a computer-based biology environment: Effects of prompting college students to reflect on their own thinking. Journal of Research in Science Teaching, 3(7), 837-858.
Lou, Y., Abrami, P., & d’Apollonia, S. (2001). Small group and individual learning with technology: A meta-analysis. Review of Educational Research, 71(3), 449-521.
Malone, T. W. (1981). Toward a theory of intrinsically motivating instruction. Cognitive Science, 5, 333-369.
Mandinach, E. (1987). Clarifying the ‘‘A’’ en CAI for learners of different abilities. Journal of Educational Computing Research, 3(1), 113-128.
Mason, J. (1996). Expressing generality and roots of algebra. In L. Lee (Ed.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer Academic.
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.
Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51.
McFarlane, A., Sparrowhawk, A., & Heald, Y. (2002). Report on the educational use of games: An exploration by TEEM of the contribution which games can make to the education process. (On line) Available: http://reservoir.cent.uji.es/canals/octeto/es/440.
Meijer, J., & Riemersma, F. (2002). Teaching and testing mathematical problem-solving by offering optional assistance. Instructional Science, 30, 187-220.
Mitchelmore, M. C. (2002). The role of abstraction and generalization in the development of mathematical knowledge. ERIC document reproduction service no. ED 466 962. Retrieved from ERIC Web site: http://www.eric.ed.gov. Accepted 22 June 2006.
Moreno, R. (2002). Who learns best with multiple representations? Cognitive theory implications for individual differences in multimedia learning. Paper presented at World Conference on Educational Multimedia, Hypermedia, & Telecommunications. Denver, CO.
Moseley, B. (2005). Students’ early mathematical representation knowledge: The effects of emphasizing single or multiple perspectives of the rational number domain in problem solving. Educational Studies in Mathematics, 60, 37-69.
Muis, K. R. (2004). Personal epistemology and mathematics: A critical review and synthesis of research. Review of Educational Research, 74(3), 317-377.
National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
O’Neil, H., Wainess, R., & Backer, E. (2005). Classification of learning outcomes evidence from the computer games literature. Curriculum Journal, 16(4), 455-474.
Orton, A., & Orton, J. (1994). Students’ perception and use of pattern and generalization. In J. P. daPonto & J. F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (Vol. III, pp. 407-414). Lisbon, Portugal: PME Program Committee.
Paas, F., Renkl, A., & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments. Educational Psychologist, 38, 1-4.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.
Pegg, J., & Redden, E. (1990). Procedures for, and experiences in, introducing algebra in New South Wales. Mathematics Teacher, 83, 386–391.
Polya, G. (1957). How to solve it. New York: Anchor Books.
Provenzo, E. (1992). The video generation. American School Board, 179(3), 29-32.
Quinn, C. N. (1994). Designing educational computer games. In K. Beattie, C. McNaught and S. Wills (Eds.), Interactive multimedia in university education: Designing for change in teaching and Learning (pp. 45-57). Elsevier Science BV, Amsterdam.
Radford, L. (1996). Some reflections on teaching algebra through generalization. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 107–111). Dordrecht, The Netherlands: Kluwer.
Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Sa´iz, & A. Me´ndez (Eds.). Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 2–21). Me´rida, Me´xico: Universidad Pedago´gica Nacional.
Reeves, L. M., & Weisberg, R. W. (1994). The role of content and abstract information in analogical transfer. Psychological Bulletin, 115, 381-400.
Reid, D. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33, 5-29.
Renkl, A. (1997). Learning from worked-out examples: A study on individual differences. Cognitive Science, 21, 1-29.
Renkl, A. (2005). The worked-out-example principle in multimedia learning. In R. E. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 229-246). Cambridge, UK: Cambridge University Press.
Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23, 90-108.
Rieber, L. P. (1996). Seriously considering play: Designing interactive learning environments based on the blending of microworlds, simulations and games. Educational Technology Research and Development, 44(2), 43-58.
Rivera, F. (2007). Visualizing as a mathematical way of knowing: Understanding figural generalization. Mathematics Teacher, 101(1), 69–75.
Rivera, F. D., & Becker, J. R. (2005). Figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School, 11(4), 198–203.
Robertson, M. E., & Taplin, M. L. (1995). Patterns and relationships: Conceptual development in mathematical (small scale) and real (large scale) space. Paper presented at the 1995 Australian Association for Research in Education conference, Hobart.
Rosenshine, B., Meister, C., & Chapman, S. (1996). Teaching students to generate questions: A review of the intervention studies. Review of Educational Research, 66(2), 181-221.
Sanford, R., Ulicsak, M., Facer, K., & Rudd, T. (2006). Teaching with games: Using commercial off-the-shelf games in formal education. Retrieved from http://www.futurelab.org.uk/download/pdfs/reserach/TWG_report.pdf (assessed 30.4.07).
Scardamalia, M., Bereiter, C., & Steinbach, R. (1984). Teach ability of reflective processes in written composition. Cognitive Science, 8, 173-190.
Scardamalia, M., Bereiter, C., McLean, R. S., Swallow, J., & Woodruff, E. (1989). Computer supported intentional learning environment. Journal of Educational Computing Research, 5, 51-68.
Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (pp. 145-152). Utrecht, The Netherlands: Kluwer.
Schoenfeld, A. H. (1983). The wild, wild, wild, wild, wild world of problem solving: A review of sorts. For the Learning of Mathematics, 3, 40–47.
Schoenfeld, A. H. (1985). Mathematical problem-solving. Orlando: Academic Press.
Schoenfeld, A. H. (1987). Cognitive Science and Mathematics Education. Hillsadale, NJ: Lawrence Erlbaum.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching (pp. 334-370). New York: MacMillan Publishing.
Schroder, T. L., & Lester, F. K. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New Direcetions for Elementary School Mathematics, 1989 Year book of the National Council of Mathematics (pp. 31-42. Reston, VA: NCTM.
Simon, M. A., & Blume, G.W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31.
Skemp, R. R. (1986). The psychology of learning mathematics. NY: Penguin Books.
Sowder, L., & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher, 91, 670–675.
Spiro, R. J., & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matters. In D. Nix, & R. J. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology (pp. 163-205). Hillsdale, NJ: Erlbaum.
Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1991). Cognitive flexibility, constructivism, and hypertext: Random access instruction for advanced knowledge acquisition in ill-structured domains. In T. Duffy, & D. Jonassen (Eds.), Constructivism and the technology of instruction (pp. 57-76). Hillsdale, NJ: Erlbaum.
Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. Journal of Secondary Gifted Education, 14, 151–165.
Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20, 147–164.
Stacey, K., & MacGregor, M. (1997). Building foundations for algebra. Mathematics Teaching in the Middle School, 2, 253–260.
Steffe, L., & Izsak, A. (2002). Pre-service middle-school teachers’ construction of linear equation concepts through quantitative reasoning. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Eds.), Proceedings of the 24th Anuual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1163-1172). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Educaiton.
Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building students’ capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.
Stigler, J. W., Gallimore, R., & Hiebert, J. (2000). Using video surveys to compare classrooms and teaching across cultures: Examples and lessons from the TIMSS video studies. Educational Psychologist, 35, 87-100.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12, 257-285.
Sweller, J., Van Merrie¨nboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251-296.
Szombathely, A., & Szarvas, T. (1998). Ideas for developing students’ reasoning: A Hungarian perspective. Mathematics Teacher, 91, 677–681.
Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1994). Toward a theoretical account of strategy use and sense-making in mathematics problem solving. In Proceedings of the 16th annual conference of the cognitive science society (pp. 836-841). Hillsdale, NJ: Erlbaum.
Tabachnick, B. G., & Fidell, L. S. (2007). Using Multivariate Statistics (5th ed.). Boston: Pearson Education.
Tarmizi, R. A., & Sweller, J. (1988). Guidance during mathematical problem solving. Journal of Educational Psychology, 80, 424-436.
Thompson, P. W. (1988). Quantitative concepts as a foundation for algebra. In M. Behr (Ed.), Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 163–170). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Van Dormolen, J. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8, 27–34.
Van Someren, M. W., Boshuizen, H. P. A., De Jong, T., & Reimann, P. (1998). Introduction. In M. van Someren, P. Reimann, H. Boshuizen, & T. De Jong (Eds.), Learning with multiple representations (pp. 1-5). Oxford: Elsevier Sciences.
Van Someren, M. W., Boshuizen, H. P. A., De Jong, T., & Reimann, P. (1998). Introduction. In M. van Someren, P. Reimann, H. Boshuizen, & T. De Jong (Eds.), Learning with multiple representations (pp. 1-5). Oxford: Elsevier Sciences.
Van Streum, A. (2000). Representations in applying functions. International Journal of Mathematics in Science and Technology, 31(5), 703-725.
Voss, J. F., & Post, T. A. (1988). On the solving of ill-structured problems. In M. H. Chi, R. Glaser & M. J. Fan (Eds.), The nature of expertise (pp. 261-285), Hillsdale, NJ: Lawrence Erlbaum.
Wang, M. C., & Peverly, S. T. (1986). The self-instructive process in classroom learning contexts. Contemporary Educational Psychology, 11, 370-404.
Wong, R. M. F., Lawson, M. J., & Keeves, J. (2002). The effects of self-explanation training on student’s problem solving in high-school mathematics. Learning and Instruction, 12, 233-262.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379–402.
Zimmerman, B. (1990). Self-regulated learning and academic achievement: An overview. Educational Psychologist, 25(1), 3-17.