References:

Anderson (1993). Rules of the mind. Hillsdale, NJ: Erlbaum.
Ball, D. L. (1990). The mathematical understanding that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449-466.
Ball, D. L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. Phi Delta Kappa International, 77, 500-508.
Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83–104). Westport, CT: Ablex Publishing.
Ball, D. L., Hill, H. C., & Bass, H. (2005, Fall). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14- 46.
Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (pp. 443-456). New York: Macmillan.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407.
Baroody, A. J. (1990). How and when should place-value concepts and skills be taught. Journal for Research in Mathematics Education, 21, 281-286.
Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321-339.
Baroody, A. J., Lai, M., & Mix, K. S. (2006). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & O. N. Saracho (Eds.), Handbook of research on the education of young children (2nd ed., pp. 135-152). Mahwah, NJ: Lawrence Erlbaum Associates.
Bisanz, J., & LeFevre, J. A. (1992). Understanding elementary mathematics. In J. I. D. Campbell (Ed.), The nature and origins of mathematical skill (pp. 113-136). Amsterdam: Elsevier Science.
Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17, 25-64.
Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607-618.
Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27, 777-786.
Canobi, K. H., Reeves, R. A., & Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39(3), 521-534.
Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 5, 385-401.
Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179-202.
Cauley, K. M. (1988). Construction of logical knowledge: Study of borrowing in subtraction. Journal of Educational Psychology, 80, 202-205.
Cobb, P., Yackel, E., & Wood, T. (1992). Interaction and learning mathematics classroom situations. Educational Studies in Mathematics, 23, 99-122.
Comiti, C., & Ball, D. L. (1996). Preparing teachers to teach mathematics: A comparative perspective. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education, (Vol. 2 pp. 1123-1153). Dordrecht, The Netherlands: Kluwer.
Cowan, R., Dowker, A., Christakis, A., & Bailey, S. (1996). Even more precisely assessing children’s understanding of the order irrelevance principle. Journal of Experimental Child Psychology, 62
Cowan, R., & Renton, M. (1996). Do they know what they are doing? Children’s use of economical addition strategies and knowledge of commutativity. Educational Psychology, 16, 409-422.
Dixon, J. A., & Moore, C. F. (1996). The developmental role of intuitive principles in choosing mathematical strategies. Developmental Psychology, 32, 241-253.
Feigenson, L. (2005). A double-dissociation in infants representations of object arrays. Cognition, 13(2), 150-156.
Flores, A., Turner, E. E., & Bachman, R. C. (2005, October). Posing problems to develop conceptual understanding: Two teachers make sense of division of fractions. Teaching Children Mathematics, 117-121.
Frykholm, J. A. (1999). The impact of reform: Challenges for mathematics teacher preparation. Journal of Mathematics Teacher Education, 2, 79-105.
Fuson, K. C. (1986). Roles of representation and verbalization in the teaching of multi-digit addition and subtraction. European Journal of Psychology of Education, 1(2), 35-56.
Fuson, K. C. (1988). Teaching adapted to thinking. Journal for Research in Mathematics Education, 19, 263-267.
Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7, 343-403.
Fuson, K. C., & Briars, D. J. (1990). Base-ten blocks as a first- and second-grade learning/teaching approach for multidigit addition and subtraction and place-value concepts. Journal for Research in Mathematics Education, 21, 180-206.
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130-162.
Geary, D. C. (1994). Children’s mathematical development: Research and practical applications. Washington, DC: American Psychological Association.
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.
Gelman, R., & Meck, E. (1983). Preschoolers’ counting: Principles before skill. Cognition, 13, 343-359.
Gelman, R., & Williams, E. M. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesist. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology: Cognition, perception, and language (5th ed.), (Vol. 2, pp. 575-630). New York, NY: Willey.
Gould, P. (2005). Really broken numbers. Australian Primary Mathematics Classroom, 10(3), 4-10.
Graeber, A. O. (1999). Forms of knowing mathematics: What preservice teachers should learn. Educational Studies in Mathematics, 38, 189-208.
Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservice teachers’ misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 20, 95-102.
Halford, G. S. (1993). Children’s understanding: The development of mental models. Hillsdale, NJ: Erlbaum.
Harel, G., & Behr, M. (1995). Teachers’ solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31-51.
Hiebert, J., & LeFevre, J. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum.
Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-224). Hillsdale, NJ: Erlbaum.
Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98-122.
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251-284.
Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s Mathematics Professional Development Institutes. Journal of Research in Mathematics Education, 35, 330-351.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge” Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400.
Hill, H. C., Blunk, M. L., Charalambos, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4)¸430-511.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371-406.
Jackson, C. D., & Leffingwell, R. J. (1999). The role of instructors in creating math anxiety in student from kindergarten through college. The Mathematics Teacher, 92(7), 583-586.
Kahan, J. A., Cooper, D. A., & Bethea, K. A. (2003). The role of mathematics teachers’ content knowledge in their teaching: A framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6, 223-252.
Khoury, H. A., Zazkis, R. (1994). On fractions and non-standard representations: Pre-service teachers’ concepts. Educational Studies in Mathematics, 27, 191-204.
Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. O. (1988). Results from the fourth NAEP assessment of mathematics: Number, operations, and word problems. Arithmetic Teacher, 35(8), 14-19.
Kribs-Zaleta, C. (2006). Invented strategies for division of fractions. In S. Alatorre, J. L. Cortina, M. Saiz,, & A. Mendez (Eds.), Proceedings of the 28thAnnual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education: Vol. 2. Rational and whole numbers (pp. 371-376). Mérida, Mexico: Universidad Pedagógica Nacional.
Labinowicz, E. (1985). Learning from children: New beginnings for teaching numerical thinking. Menlo Park, CA: Addison-Wesley Publishing Company
Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Erlbaum.
Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16-32.
Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26, 422-441.
McClain, K. (2003). Supporting preservice teachers’ understanding of place value and multidigit arithmetic, Mathematica Thinking and Learning, 5, 281-306.
Mestre, J. P. (2002). Probing adults’ conceptual understanding and transfer of learning via problem posing. Applied Developmental Psychology, 23, 9-50
Ministère de L’Éducation, Loisirs et Sport (2001). Quebec education program: Preschool and
elementary education. Quebec: Ministère de L’Éducation, Loisirs et Sport.
Miura, I. T., Okamoto, Y., Kim, C. C., Steere, M., & Fayol, M. (1993). First graders’ cognitive representation of number and understanding of place value: Cross-national comparisons – France, Japan, Korea, Sweden, and the United States. Journal of Educational Psychology, 85, 24-30.
Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teachers learn? Journal for Research in Mathematics Education, 40(5), 491-529.
National Council of Teachers of Mathematics. (2009). Principles and standards for school mathematics. Reston, VA: NCTM.
Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45, 1080-1110. Olivier, Murray, & Human (1990).
Olivier, A., Murray, H., & Human, P. (1990). Building on young children’s informal arithmetical knowledge. In G. Booker, P. Cobb, & T. N. Mendicuti (Eds.), Proceedings of the 14th Psychology of Mathematics Education Conference, 3 (pp. 297-304). Oaxtepec, Mexico.
Osana, H. P., Lacroix, G. L., Tucker, B. J., & Desrosiers, C. (2006.). The role of content knowledge and problem features on preservice teachers’ appraisal of elementary mathematics tasks. Journal of Mathematics Teacher Education, 9, 347-380.
Osana, H. P., & Pitsolantis, N. (in press). Addressing the struggle to link form and understanding in fractions instruction. British Journal of Educational Psychology.
Osana, H. P., & Royea, D. A. (2011). Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study. Journal of Mathematical Behavior, 30, 333-352.
Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3). Hillsdale, NJ: Erlbaum.
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189.
Rittle-Johnson, B., & Koedinger, K. R. (2002). Comparing instructional strategies for integrating conceptual and procedural knowledge. In D. S. Mewborn, P. Sztajin, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (eds.), Proceedings of the 24th Annual Meeting for the International Group for the Psychology of Mathematics Education (pp.969-978). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Rittle-Johnson, B., & Koedinger, K. (2009). Iterating between lessons on concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 483-500.
Rittle-Johnson, B., & Siegler, R. S. (1998).The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). London: Psychology Press.
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362.
Rizvi, N. F. & Lawson, M. J. (2007). Prospective teachers’ knowledge: Concept of division. International Educations Journal, 8, 377-392.
Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281.
Sarama, J., & Clements, D. H. (2009). "Concrete" computer manipulatives in mathematics education. Child Development Perspectives, 3, 145-150.
Saxe, G. B., Gearhart, M., & Nasir, N. S. (2001). Enhancing students’ understanding of mathematics: A study of three contrasting approached to professional support. Journal of Mathematics Teacher Education, 4, 55-79.
Saxton, M., & Towse, J. N. (1998). Linguistic relativity: The case of place value in multi-digit numbers. Journal of Experimental Child Psychology, 69, 66-79.
Shrager, J., & Siegler, R. S. (1998). A model of children’s strategy choices and strategy discoveries. Psychological Science, 9, 405-410.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
Siegler, R. S. (1991). In young children’s counting, procedures precede principles. Educational Psychology Review, 3(2), 127-135.
Siegler, R. S., & Crowley, K. (1994). Constraints on learning in nonprivileged domains. Cognitive Psychology, 27, 194-226.
Silverman, J., & Thompson, J. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499-511.
Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233-254.
Simon, M. & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25, 472-494.
Sohn, M. H., & Carlson, R. A. (1998). Procedural frameworks for simple arithmetic skills. Journal of Experiemental Psychology: Learning, Memory, and Cognition, 24(4), 1052-1067.
Sophian, C. (1997). Beyond Competence: The significance of performance for conceptual development. Cognitive Development, 12, 281-303.
Sowder, J., Armstrong