Royea, Diana A
Effects of Lesson Sequencing on Preservice Teachers’ Mathematical Knowledge of Place-Value.
Masters thesis, Concordia University.
The Effects of Lesson Sequencing on Preservice Teachers' Place-Value Knowledge
Elementary students' mathematical achievement is a focal point of mathematics education research. Place-value is a foundational topic in the elementary mathematics curriculum. In order to teach place-value in a manner that is in line with mathematics reform practices, teachers must possess strong conceptual, procedural, and specialized content knowledge (SCK) of place-value. At the same time, preservice teachers tend possess mathematical knowledge that is conceptually and procedurally weak. This study used a pretest-posttest design to investigate the effects of lesson sequencing on preservice teachers' conceptual, procedural, SCK, and transfer knowledge of place-value. Preservice teachers were assigned to one of three conditions: Concepts-first, Procedures-First, or Iterative. All of the participants were exposed to the same eight lessons, four conceptual and four procedural. The differences between the conditions was the order the lessons were received in. The results were analyzed quantitatively and where there were significant effects, those results were further analyzed from a qualitative perspective. Quantitative results indicated that there was a significant time × group interaction for conceptual knowledge. The Iterative condition significantly outperformed the Concepts-first and the Procedures-first conditions. While there was no main effect of condition on procedural knowledge, SCK, and transfer, there was a main effect of time for all three of these knowledge types. Furthermore, qualitative analyses revealed that the pathway of conceptual knowledge acquisition was affected by lesson sequencing. Finally, limitations, future research, and practical implications of this study are discussed.
|Divisions:||Concordia University > Faculty of Arts and Science > Education|
|Item Type:||Thesis (Masters)|
|Authors:||Royea, Diana A|
|Date:||5 September 2012|
|Thesis Supervisor(s):||Osana, Helena P|
|Keywords:||teacher education; place-value; teacher education; lesson sequencing; concepts-first; iterative; procedures-first|
|Deposited On:||25 Oct 2012 16:33|
|Last Modified:||05 Nov 2016 02:20|
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
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