參考文獻
中文部分:
洪有情(2003)。青少年的數學概念學習研究-子計畫四:青少年代數運算概念發展
研究(3/3)。行政院國家科學委員會專題研究報告(報告編號:NSC
91-2522-S-003-016),未出版。
洪有情(2005)。青少年代數運算概念的「學習與教學」研究(1/3)。行政院國家科
學委員會專題研究報告(報告編號:NSC93-2521-S-003-007) ,未出版。
教育部(2005)。九年一貫課程數學領域綱要。台北:教育部。
莊松潔(2005)。不同年級學童在具體情境中未知數概念及解題歷程之研究:國立
中山大學教育研究所碩士論文,未出版,高雄市。
郭汾派、林光賢、林福來(1989)。國中生文字符號概念的發展。行政院國家科學委
員會專題研究報告(報告編號:NSC77-0111-S003-05A) ,未出版。
陳創義(2005)。青少年數學概念的學習與教學之研究-總計畫(3/3)。國科會專題
研究計畫成果報告。NSC 95-2521-S-003-011。
陳嘉皇(2007)。國小三年級學童代數推理教學與解題表現研究。高雄師大學報, 23,
125-150。
黃寶彰(2002)。六、七年級學童數學學習困難部分之研究:國立屏東師範學院數
理教育所碩士論文,未出版,屏東市。
楊凱琳、林福來(2006):探討高中數學溶入建模活動的支撐策略及促進參與教師反思的潛在機制。科學教育學刊,14(5),517-543。
戴文賓、邱守榕(1999)。國一學生由算術領域轉入代數領域呈現的學習現象與特徵。
科學教育,10,148-175。
戴政吉、侯美玲、詹勳國(2002)。算術到代數的學習研究。國教天地,150,8-15。
西文部份:
Abrams, J. P. (2001). Teaching mathematical modeling and the skills of representation. In
A.Cuoco (Ed.), The role of representation in school mathematics (pp. 269-282). Reston, VA.: NCTM.
Amit, M. & Klass-Tsirulnikov, B. (2005). Paving a way to algebraic word problems using
a nonalgebraic route. Mathematics Teaching in the Middle School, 10, 271-297.
American Association for the Advancement of Science. (1990). Science for all Americans: Project 2061. New York, NY: Oxford University Press.
Barnard, A. D. & Tall, D. O. (2001). A comparative study of cognitive units in mathematical thinking. Proceedings of the 25th conference of the international group for the psychology of mathematics education, 2, 89–96. Utrecht, the
Netherlands.
Bastable, V. & Schifter, D. (2008). Classroom stories: examples of elementary students
engaged in early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp.165-184). Mahwah, NJ: Lawrence Erlbaum Associates.
Behr, Erlwanger, & Nichols (1976). How children view equality sentence (PMDC
Technical Report No. 3). Tallahassee: Florida State University. (ERIC Document
Reproduction Service No. ED144802)
Booth, L. R. (1981). Strategies and errors in generalized arithmetic. In Equipe de Recherche Pe’dagogique (Eds.), Proceedings of the 6th conference of the international group for the psychology of mathematics education, 1, (pp.140-146). Israel: Weizmann, Institute of Science.
Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson.
Cai, J. (1998). Developing algebraic reasoning in the elementary grades. Teaching
Children Mathematics, 5, 225-228.
Cai, J. & Moyer, J. (2008). Develop algebraic thinking in earlier grades: some insights from international comparative students. In C. E. Greenes, & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp.169-182). Reston, VA: NCTM.
Carlson, M., Larsen, S., & Lesh, R. (2003). Intergrating a models and mdeling persective
with existing research and pratice. In R. Lesh & H. M. Doerr (Eds.), Beyond
constructivism: Models and modeling perspectives on mathematics teaching, learning, and problem solving (pp. 465-478). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Carpenter, T.P., Fennema, E.,&Franke, M. (1996). Cognitively guided instruction:A
knowledge base for reform in primary mathematics instruction. The Elementary
School Journal, 97, 3-30.
Charbonneau, L. (1996). From Euclid to Descartes: Algebra and its relation to
Geometry. In N. Bednarz, C. Kieran, and L. Lee (Eds.), Approaches to algebra,
perspective for research and teaching (pp.15-37). Dordrecht, the Netherlands:
Kluwer Academic Publishers.
Carraher, D. W., Schliemann, A. D., & Brizuela, B. (2000). Early algebra, early
arithmetic: Treating operations as functions. Paper presented at the 22nd Meeting of
the PME. North American Chapter, Tucson, AZ(Available on CD-ROM).
Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high
school algebra classroom. New York: Teachers College Press.
Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, G. W. Martin, & D. Schifter (Eds.), A research companion to the principles and standards for school mathematics (pp. 123-135). Reston, VA: National Council of Teachers of Mathematics.
Chen, K. J., Chin, E. T., Tuan, H. L. (2007). An investigation of modeling activity workshop on elementary mathematics teachers’ teaching. Paper presented at the 4rd East Asia Regional Conference on Mathematics Education. Penang, Malaysia.
Chen, K. J. & Chin, E. T. (2008). Implementing modeling activity to enhance
student’s conceptual understanding and active thinking. Paper presented at the 32th
Annual Meeting of the International Group for the Psychology of mathematics
Education, PME32, Mecico.
Chin, E-T. (2008). Mathematical proof as formal precept in advanced mathematical
thinking. Mediterranean Journal for Research in Mathematics Education, 7 (2),
121-133.
Confrey, J. (1991). Steering a course between Vygotsky and Piaget. Educational
Researcher, 20 (8), 28-32.
Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural
perspectives in the context of developmental research. Educational Psycologist, 31,
175-190.
Cramer, K. (2003). Using a translation model for curriculum development and classroom
instruction. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and
modeling perspectives on mathematics teaching, learning, and problem solving (pp.
465-478). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Dark, M. (2003). A model and modeling perspective on skills for the high performance
workplace . In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and
modeling perspectives on mathematics teaching, learning, and problem solving (pp.
465-478). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
De Bock, D., Verschaffel, L., Janssens, D. & Claes, K. (2000). Involving pupils in an
authentic context: Does it help them to overcome the ’illusion of linearity’? In
T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th annual conference of the
international group for the psychology of mathematics education, 2, (pp.233-240).
Japan: Hiroshima University
Denzin, N. K. & Lincoln, Y. S. (1994). Handbook of qualitative research. Thousand Oaks, CA: Sage.
Diefes-Dux, H. A., Moore, T., Zawojewski, J. Imbrie, P. K., & Follman, D. (2004). A
framework for posing open-ended engineering problem:Model-eliciting activities.
from the World Wide Web http://rlab.cs.utep.edu/~freudent/stem-ed/papers/1719.pdf
Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson.
Doerr, H. M. & English, L. (2003). A modeling perspective on students'
mathematical reasoning about data. Journal for Research in Mathematics
Education, 34 (2), 110-136.
Dreyfus, Hershkowitz and Schwarz, (2001). The construction of abstract knowing in
interaction. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th
conference of the International Group for the Psychology of Mathematics of
Education, 2, (pp.377-384). Utrecht, the Netherlands: Freudenthal Institute.
Driscoll, M. (1999). Fostering algebra thinking: A guide for teacher grades. 6-10.
Portsmouth, N.H.: Heinemann.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D.
Tall (Ed.), Advanced mathematical thinking (pp. 95-126). Dordrecht: Kluwer.
Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In
E. Dubinsky & G. Harel (Eds.), The concepts of function: Aspects of epistemology
and pedagogy (pp. 85-106). Washington, DC: Mathematical Association of
America.
English, L. & Watters, J. (2004). Mathematical modeling in the early school years
Mathematics Education Research Journal, 16 (3), 59-80 .
English, L. D. (2009). Promoting interdisciplinarity through mathematics modeling.
ZDM Mathematics Education, 41, 161-181.
Ernest, P. (1991). The philosophy of mathematics education. London: The Falmer
Press.
Ferrucci, B. J., Kaur, B., Carter, J. A., & Yeap, B. (2008). Using a model approach
to enhance algebraic thinking in the elementary school mathematics classroom.
In G. E. Greens, & R. Rubenstein (Eds.), Algebra and algebraic thinking in school
mathematics (pp.195-210). Reston, VA: National Council of Teachers of
Mathematics.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, Netherlands:
Reidel.
Goldin, G. (2008). Perspectives on presentation in mathematical learning and problem
solving. In L. D. English (Ed.), Handbook of international research in mathematics
education (pp. 176-201). Taylor and Francis.
Gravemeijer, K & Doorman, M.(1999). Context problems in realistic mathematics
education: A Calculus course as an example. Educational Studies in Mathematics,
39, 111-129.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing,
modeling, and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.),
Communicating and symbolizing in mathematics: Perspectives on discourse, tools,
and instructional design (pp.225-274). Mahwah, NJ: Lawrence Erlbaum Associates.
Gravemeijer, K. (2007). Emergent modelling as a precursor to mathematical modelling.
In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.), Modelling and application
in mathematics education (pp.485-490) NY: Springer Science+business Media, LLC.
Gray, E. & Tall, D. (1994). Duality, ambiguity and flexibility: A "proceptual" view of
simple arithmetic. Journal for Research in Mathematics Education, 25(2), 16-40.
Gray, E., Pitta, D. & Tall, D. (1997). The nature of the object as an integral component of numerical processes, Proceedings of the 21th conference of the international group for the psychology of mathematics education, 1, 115– 130.Lahti, Finnland.
Gray, E., Pitta, D., Pinto, M. & Tall, D. (1999). Knowledge Construction and diverging
thinking in elementary and advanced mathematics. Educational Studies in
Mathematics, 38 (1-3), 111-133.
Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain.
Journal for Research in Mathematics Education, 22, 170-218.
Hall, R. (2000). Videorecording as theory. In E. Kelly, & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 647-664). Mahwah, NJ: Lawrence Erlbaum Associates.
Harper, E. W. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18,
75-90.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and
algebra. Educational Studies in Mathematics, 27, 59-78.
Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of
mathematical power by “algebrafying” the K-12 curriculum. In National Council
of Teacher Mathematics and Mathematical Science Education Board (Eds.), The
nature and role of algebra in the K-14 curriculum: Proceedings of a national
symposium (pp. 25-26). Washington, DC: National Research Council, National
Academic Press.
Kaput, J. & Blanton, M. (2001). Algebrafying the elementary mathematics
experience. Part Ι: Transforming tasks structures. In H. Chick, K. Stacey, & J.Vincent
(Eds.), The future of the teaching and learning of algebra (pp. 344-351). Melboume,
Australia: The University of Melboume.
Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. W.
Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp.5-15). Mahwah,
NJ: Lawrence Erlbaum Associates.
Kaput, J., Noss, R., & Hoyles, C. (2008). Developing new notations for a learnable
mathematics in the computational era. In L. English (Ed.), Handbook of
international research in mathematics education(second edition)(pp. 697-719).
Taylor and Francis.
Kho, Tek-Hong (1987). Mathematical models for solving arithmetic problem.
Proceedings of fourth Southeast Asian conference on mathematical
Education (pp.345-351). Singapore: Institute of Education.
Kieran, C. (1979). Children’s operational thinking within the context of bracketing
and order of operations. In D. Tall (Ed.), Proceedings of the 3th conference of the
international group for the psychology of mathematics education (pp. 128-133). UK:
Warwick University.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies
in Mathematics, 12, 317-326.
Kieran, C. (1984). A comparison between novice and more-expert algebra students on
tasks dealing with the equivalence of equations. In J. M. Moser (Ed.), Proceedings of
the 6th conference of the international group for the psychology of mathematics
education (pp. 83-91). Madison, University of Wisconsin.
Kieran, C. (1989). The early learning of algebra: A structural perspective. Research
agenda for mathematics education. Reston, VA: Author.
Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P.
Nesher & J. Lilpatrick (Eds), Mathematics and cognition: A research synthesis by
the international group for the psychology of mathematics education (pp. 96-112).
New York: Cambridge University Press.
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.
Kieran C., Boileau A., & Garaçnon M. (1996). Introducing algebra: A functional
approach in a computer enviroment. In N. Bednarz, C. Kieran and L. Lee (Eds.),
Approaches to algebra: perspectives for research and teaching (pp. 257-293).
Kluwer Academic Publishers.
Kieran, C. (2004). Algebraic thinking in the early grades: What is it? Mathematics
Educator, 8, 51-139.
Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutierrez
& P. Boero, (Eds.), Handbook of research on the psychology of mathematics
education past, present and future (pp.11-50). Sense Publishers.
Kilpatrick, J. (1985). A retrospective account of the past twenty- five years of research
on teaching mathematical problem solving. In E. A. Silver (Ed.), Teaching and
learning mathematical problem solving: Multiple research perspectives (pp. 1-16).
Hillsdale, NJ: Erlbaum.
Knuth, E. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in mathematics Education, 31, 500-508.
Knuth, E.J., Stephens, A. S., McNeil, N. M., Weinberg, A., & Alibali, M.W.
(2006). Does understanding the equal sign matter? Evidence from solving
equation. Journal for Research in Mathematics Education, 5(4), 297-312.
Krutetskii, V. A. (1976). The psychology of mathematics abilities in school children.
Chicago. University of Chicago Press.
Küchemann, D. (1978). Children's understanding of numerical variables. Mathematics
in School, 7 (4), 23-26.
Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's understanding of
Mathematics (pp. 102-119). London: John Murray.
Lakatos, I.M. (1976). Proofs and refutations. Cambridge: Cambridge University
Press.
Lakoff, G. & Numez, R. (2000). Where mathematics comes from. New York: Basic
Books.
LeFevre, D. M. (2004). Designing for teacher learning: Video-based curriculum design.
In J. Brophy (Ed.), Advanced in research on teaching : Vol. 10. Using video in
teacher education (pp. 235-258). Oxford, UK: Elsevier.
Lehrer, R., & Schauble, L. (2000). Model-based reasoning in mathematics and
science. In R. Glaser (Ed.), Advances in instructional psychology: Educational
design and cognitive science (Vol 5, pp. 101-159). Mahwah NJ: Lawrence Erlbaum
Associates.
Lesh, R., & Doerr, H. M. (1998). Symbolizing, communicating, and mathematizing:
Key components of models and modeling. In P. Cobb & E. Yackel (Eds.),
Symbolizing, communicating, and mathematizing (pp.361-384). Mahwah, NJ:
Lawrence Erlbaum Associates, Inc.
Lesh, R., & Kelly, A. (2000). Multitiered teaching experiments. In E. Kelly, & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 197-230). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Lehrer, R. (2000). Iterative refinement cycle for videotape analyses of conceptual change. In E. Kelly, & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 665-708). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Doerr, H. M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Zawojewski, J .S. (2007). Problem solving and modeling. In F. K. Jr. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763-804). Greenwich, CT: Information Age Publishing.
Lester, F. K., & Kehle, P. (2003). From problem solving to modeling: The evolution of
thinking about research on complex mathematics activity. In R. Lesh & H. M.
Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on
mathematics teaching, learning, and problem solving (pp. 465-478). Mahwah, NJ:
Lawrence Erlbaum Associates, Inc.
Leung, S. S. & Chuang, S. (2005). Implications on algebra learning: A comparison
on three case studies of unknown concepts and equation solving in Taiwan. Paper
presented at the 3rd East Asia Regional Conference on Mathematics Education.
Shanghai, Nanjing, and Hangzhou, People's Republic of China.
Linchevski, L. & Livneh, D. (1999). Structure sense: The relationship between
algebraic and numerical contexts. Educational Studies in Mathematics, 40,
173-196.
Lima, R. N. & Tall, D. (2006). The concept of equation: What have students met
before? (Eds.)Proceedings of the 30th Conference of the International Group for the
Psychology of Mathematics Education. vol. 4, 233–241.Czech Republic: Prague
university,
Lincoln, Y. S. & Guba, E. G. (1985). Naturalistic inquiry. Berverly Hills, CA: Sage.
Loving, C. C. (1991). The scientific theory profile: A philosophy of science model for
science teachers’. Journal of Research in Science Teaching, 28, 823-838.
MacGregor, M., & Stacey, K. (1993). Cognitive models underlying students’
formulation of simple linear equations. Journal for Research in Mathematics
Education, 24(3), 217-232.
MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notion:
11-15. Educational Studies in Mathematics, 33(1), 1-19.
Makar, K., & Confrey, J. (2007). Moving the context of modeling to the forefront:
Preservice teachers’ investigations of equity in resting. In W. Blum, P. L. Galbraith, H.
W. Henn, & M. Niss (Eds.), Modelling and application in mathematics education
(pp.485-490). NY: Springer Science business Media, LLC.
Maki, D. P., & Thompson, M. (1973). Mathematics models and applications: with
emphasis on the social, life, and management sciences. New Jersey: Pretice-Hall, Inc.
Mary, S. R., James, G. G., & Joan I. H. (1983). Development of children’s
problem-solving ability in arithmetic. In H. P. Ginsburg (Ed.), The Development of
mathematical thinking (pp.153-200). London: Academic Press.
Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.), Problem of
representation in the teaching and learning of mathematics (pp. 73-81). Hillsdale,
NJ: Lawrence Erlbaum.
Mason, J. (1989). Mathematical abstraction as the result of delicate shift of attention.
For the Learning of Mathematics, 9 (2), 2-8.
Mason. J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C.
Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and
teaching (pp. 65-86). Dordrecht, Netherlands: Kluwer Academic Publishers.
Mason, J. (2008). Making use of children’s power to produce algebraic thinking. In
J. J. Kaput, D. W. Carraher, & M. L. Blanton, (Eds.), Algebra in the early grades
(pp.57-94). Mahwah, NJ: Lawrence Erlbaum Associates.
Merseth, K. K. (1996). Cases and case methods in teacher education. In J. Sikula
(Ed.), Handbook of research on teacher education (pp.722-744). New York:
Macmillan.
Nathan, M. J., & Koellner, K. (2007). A framework for understanding and
cultivating the transition from arithmetic to algebraic reasoning. Mathematical
thinking and learning, 9 (3), 179-192.
National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and
standards for school mathematics. Reston, VA: Author.
Niss, M., Blum, W., & Galbraith, P. (2007). Modelling and application in
mathematics education (pp.3-32). NY: Springer Science business Media, LLC.
Nussbaum, J. (1989). Classroom conceptual change: Philosophical perspectives.
International of Science Education, 11, 530-540.
Piaget, J. (1970). Genetic epistemology. NY: Columbia University Press.
Piaget, J. (1985). The equilibration of cognitive structures. Cambridge MA: Harvard.
Peterson, P. L., Carpenter, T.P., & Fennema, E. (1989). Teachers’ knowledge of
student’s knowledge in mathematics problem solving: Correlation and analysis.
Journal of Educational Psychology, 81, 558-569.
Pimm, D. (1995). Symbols and meanings in school mathematics. Routledge,
London-New York.
Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.
Radford L. G..(2001). The historical origins of algebraic thinking. In R. Sutherland, A.
Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp.1-24). Kluwer
Academic Publishers.
Riley, M, S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s
problem-solving ability in arithmetic. In H. Ginsberg (Ed.), The development of
mathematical thinking (pp.153-196). New York: Academic Press.
Rittle-Johnson, B., &Alibali, M. W. (1999). Conceptual and procedural knowledge of
mathematics: Does one lead to the other? Journal of Educational Psychology, 91,
175-189.
Rodwell, M. K. (1998). Social work constructivist research. New York: Garland Pub.
Romberg, T. A. (1992). Perspectives on scholarship and research methods. In D. A.
Grouws (Eds.), Handsbook of research on mathematics teaching and learning
(pp. 49-64). New York: Macmillan.Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2006). Bringing out the
algebraic character of arithmetic: From children`s ideas to classroom practice.
Mahwah, NJ: Erlbaum.
Schoenfeld, A. H. (1982). Some thoughts on problem-solving research and
mathematics education. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem
solving: Issues in research (pp. 27-37). Philadelphia: Franklin Institute Press.
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 and learning (pp. 334-370). New York:
Macmillan.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on
processes and objects as different sides of the same coin. Educational Studies in
Mathematics, 22, 1-36.
Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification - the case of
algebra, Educational Studies in Mathematics, 26, 191-228.
Shulman, J. (1992). Case methods in teach education. New York: Teachers
College Press.
Smith, M. S., Silver, E. A., Stein, M. K., Henningsen, M. A., Boston, M., & Hughes, E. K. (2005). Improving instruction in algebra: Using cases to transform mathematics
teaching and learning, Vol. 2. New York: Teachers College Press.
Stacey, K. (1989). Finding and using pattern in linear generalizing problems.
Educational Studies in Mathematics, 20, 147-232.
Steele, D. (2005). Using writing to access students’ schema knowledge for algebraic
thinking. School Science and Mathematics, 105, 54-142.
Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical
meanings and strategies. New York: Springer-Verlag.
Stwez, Frank, & Hartzler, J. S. (1991). Mathematical modeling in the secondary school
curriculum: A resource guide of classroom exercise. Reston, VA.: National Council
of Teachers of Mathematics.
Swafford, J.,& Langgrall, C. (2000). Grade 6 students’preinstructional use of equations to describe and represent problem situation. Journal for Research in Mathematics Education, 31, 89-112.
Tabach, M., & Friedlander, A.(2008). The role of context in learning beginning
algebra. In G. E. Greens, & R. Rubenstein (Eds.), Algebra and algebraic thinking
in school mathematics (pp.195-210). Reston, VA: National Council of Teachers of
Mathematics.
Tall, D. (2004). Thinking through three worlds of mathematics. In Proceedings of the 28th
Conference of the international Group for the Psychology. Vol. 2, (pp. 423-430).
Norway: Bergen University College.
Tall, D. (2007a). Embodiment, symbolism and formalism in undergraduate mathematics
Education. Paper presented at 10th Conference of the Special Interest Group of the
Mathematical Association of America on Research in Undergraduate Mathematics
Education, San Diego, California, USA.
Tall, D. (2007b). Teachers as Mentors to encourage both power and simplicity in active mathematical learning. Paper presented at The Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, Abu Dhabi.
Tall, D. (2007c). Mathematical Growth. Retrieved November 20, 2007, from the World Wide Web: http://www.davidtall.com/
Tall, D., Gray, E., Ali, N. B., Crowley, L., DeMarois, P., McGowen, M. (2001). Symbols and the Bifurcation between procedural and conceptual thinking, Canadian Journal of Science, Mathematics and Technology Education 1, 81–104.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.
Terry, W. (2006). Learning and memory: Basic principles, process and procedures
(3rded.). Boston: Allyn & Bacon.
Threlfall, J. (1999). Repeating patterns in the primary years, In A. Orton (Ed.),
Pattern in the Teaching and Learning of Mathematics (pp. 18-30). Cassell: London.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the
research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching
and learning (pp. 127-146). New York: Macmillan.
Van Amerrom, B. A. (2003). Focusing on informal strategies when linking arithmetic
to early algebra. Educational Studies in Mathematics, 54, 63-75.
Von Glasersfeld, E. (1995). Radical constructivism: A way to knowing and learning.
Washington, DC: Falmer Press.
Warren, E. & Cooper, T. J.(2008). Pattern that support early algebraic thinking in the
elementary school. In C. E. Greenes, & R. Rubenstein (Eds.), Algebra and
algebraic thinking in school mathematics (pp.113-126). Reston: NCTM.
Watson, A., Spirou, P., & Tall, D. O. (2003). The relationship between physical
Embodiment and mathematical symbolism: The concept of vector. The
Mediterranean Journal of Mathematics Education, 12, 73-97.
Williams, G. (2000). Collaborative problem solving and discovered complexity. In J.
Bana (Ed.), Proceedings of 23th annual conference of the mathematics education
Research group of australasia. 2, (pp.656-663).
Fremantle:MERGA.
Zawojewski, J., Chamberlin, M., Hjalmarson, M., & Lewis, C. (2006). The role of
design experiments in teacher professional development. In A. E. Kelly, R. A. Lesh
& J. Baek (Eds.), Handbook of design research in mathematics, science and
technology education. Mahwah, NJ: Erlbaum.