Bibliographies

Research Related to Early Algebra
  • Ball, D.L, & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W.G. Martin, & D. Schifter. A Research Companion to Principles and Standards for School Mathematics(pp. 27-44). Reston, VA: National Council of Teachers of Mathematics.
  • Blanton, M.L. & Kaput, J.J. (October 2003). Developing Elementary Teachers’ “Algebra Eyes and Ears”. Teaching Children Mathematics, (10)2, 70-77.
  • Blanton, M.L., Schifter, D., Inge, V., Lofgren, P., Willis, C., Davis, F. & Confrey, J. (2007). Early Algebra. In V.J. Katz (Ed.). Algebra: Gateway to a Technological Future (pp. 7-14). Washington, D.C.: The Mathematical Association of America.
  • Carpenter, T.P., Franke, M.L. & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School.Portsmouth, NH: Heinemann.
  • Greenes, C.E. & Rubenstein, R. (Eds.). (2008). Algebra and Algebraic Thinking in School Mathematics: 70th Yearbook.Reston, VA: National Council of Teachers of Mathematics. This book includes the following chapter:
    • Schifter, D., Bastable, V., Russell, S.J., Seyferth, L. & Riddle, M. Algebra in the Grades K-5 Classroom: Learning Opportunities for Students and Teachers. Pages 263-277.
  • Kaput, J.J., Carraher, D.W. & Blanton. M.L. (Eds.) (2008). Algebra in the Early Grades. New York: Lawrence Erlbaum Associates. This book includes the following chapters:
    • Bastable, V. & Schifter, D. Classroom Stories: Examples of Elementary Students Engaged in Early Algebra. Pages 165-184.
    • Carraher, D.W., Schliemann, A.D. & Schwartz, J.L. Early Algebra Is Not the Same as Algebra Early. Pages 235-272.
    • Schifter, D., Monk, S., Russell, S.J., & Bastable, V. Early Algebra: What Does Understanding the Laws of Arithmetic Mean in the Elementary Grades? Pages 413-447.
    • Tierney, C. & Monk, S. Children’s Reasoning about Change over Time. Pages 185-200.
  • Moses, R.P. & Cobb, C.E. (2002). Radical Equations: Civil Rights from Mississippi to the Algebra Project. Boston, MA: Beacon Press.
  • National Council of Teachers of Mathematics. (2000). The Algebra Strand, Pre-K-2 and 3-5. Principles and Standards in School Mathematics. Reston, VA: National Council of Teachers of Mathematics. Pages 37-40, 90-95, and 158-163.
  • Russell, Susan Jo; Schifter, Deborah; Kasman, Reva; Bastable, Virginia; & Higgins, Traci. (2017). But why does it work? Mathematical argument in the elementary classroom. Portsmouth, NH: Heinemann.
  • Russell, Susan Jo; Schifter, Deborah; Bastable, Virgina. (2011a). Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Porthsmouth, NH: Heinemann.
  • Russell, Susan Jo; Schifter, Deborah; & Bastable, Virginia (2011b). Developing algebraic thinking in the context of arithmetic. In Cai & E. Knuth (Eds.), The development of earlier algebraic thinking: Multiple perspectives.  New York: Springer.
  • Russell, S.J., Schifter, D. & Bastable, V. (January/February, 2006). Is It 2 More or Less? Algebra in the Elementary Classroom. Connect, Volume 19, Issue 3, 1-3.
  • Russell, S.J., & Vaisenstein, A. (September/October 2008). Computational fluency: Working with a struggling student. Connect, Volume 22, Issue 1, 8-12.
  • Schifter, Deborah; Russell, Susan Jo; & Bastable, Virginia (2009). Early algebra to reach the range of learners. Teaching Children Mathematics, 16:4, 230-237.
  • Schifter, Deborah; Monk, Steve; Russell, Susan Jo; & Bastable, Virginia. (2008). Early Algebra: What Does Understanding the Laws of Arithmetic Mean in the Elementary Grades? In J. Kaput, D. Carraher, and M. Blanton (Eds.) Algebra in the Early Grades. Mahwah, NJ: Lawrence Erlbaum Associates.  413-447.
  • Schifter, D. (1999). Reasoning about Operations: Early Algebraic Thinking, Grades K through 6. In L. Stiff and F. Curio (Eds.) Mathematical Reasoning, K-12: 1999 NCTM Yearbook.Reston, VA: National Council of Teachers of Mathematics. Pages 62-81.
  • Schifter, D. (2009). Representation-Based Proof in the Elementary Grades. In D.A. Stylianou, M. Blanton, & E. Knouth (Eds),Teaching and learning proof across the grades: A K-16 Perspective. Oxford: Routledge – Taylor Francis and National Council of Teachers of Mathematics.
  • Schweitzer, K. (2006). Teacher as Researcher: Research as a Partnership. In S.Z. Smith & M.E. Smith (eds.) Teachers Engaged in Research: Inquiry into Mathematics Classrooms, Grades Pre-K-2 (pp. 69-94). Greenwich, CT: Information Age Publishing.
Articles and Research that Informed the Development of the 2nd Edition
In developing the second edition of Investigations, the authors drew on current research and practice. The following list is a selection of some of the resources and references the authors found useful in their work and which also may be of interest to users of the curriculum.

  • Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire in trying to hear children. In T. Biddle, T. Good, & I. Goodson (Eds.), International handbook on teachers and teaching (pp. 769-817). Dordrecht, Netherlands: Kluwer Press.
  • Ball, D. L. & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.), Yearbook of the National Society for the Study of Education, Constructivism in Education (pp. 193-224). Chicago: University of Chicago Press.
  • Bass, H. (2003). Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective. Teaching Children Mathematics, 9(6), 322-327.
  • Boaler, J. (2003). When Learning No Longer Matters: Standardized Testing and the Creation of Inequality. Phi Delta Kappan, 84(7), 502-506.
  • Brownell, W. A. (2003). Meaning and Skill – Maintaining Balance. Teaching Children Mathematics, 9 (6), 310-316.
  • Cahnmann, M., & Remillard, J. T. (2002). What counts and how: Mathematics teaching in culturally, linguistically, and socioeconomically diverse urban settings. Urban Review, 34(3), 179-205.
  • Carpenter, T., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann.
  • Carpenter, T. & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, and J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference(Vol. 1, pp. 155-62). University of Melbourne, Australia.
  • Falkner, K. P., Levi, L., & Carpenter, T. (1999). Children’s Understanding of Equality: A Foundation for Early Algebra. Teaching Children Mathematics (Early Childhood Corner), 6(4), 232-236.
  • Flowers, J., Kline, K., & Rubenstein, R. N. (2003). Developing teachers’ computational fluency: examples in subtraction. Teaching Children Mathematics, 9(6), 330-334.
  • Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Westport, CT: Heinemann.
  • Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Westport, CT: Heinemann.
  • Hiebert, J., Carpenter, T. P., Fennema, E., Ruson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
  • Huinker, D., Freckman, J. L., & Steinmeyer, M. B. (2003). Subtraction Strategies from Children’s Thinking: Moving toward Fluency with Greater Numbers. Teaching Children Mathematics, 9(6), 347-53.
  • Izsak, A. & Fuson, K. (2000). Students’ understanding and use of multiple representations while learning two-digit multiplication. In M. Fernandez (Ed.), Proceedings of the Twenty-Second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education(Vol. 2, pp. 715-721). Tucson, AZ.
  • Jackson, K., & Remillard, J. T. (2005). Rethinking parent involvement: African American mothers construct their roles in the mathematics education of their children. School Community Journal,15(1), 51-74.
  • Kerekes, J. & Fosnot, C. T. (1998). Using Pictures with Constraints to Develop Multiplication Strategies. The Constructivist, 13(2), 15-20.
  • Keith, A. (2006). Mathematical argument in a second grade class: Generating and justifying generalized statements about odd and even numbers. In Smith, S. Z. & Smith, M. E. (Eds.), Teachers engaged in research: Inquiry into mathematics classrooms, grades pre-K-2(pp. 35-68). Greenwich, CT: Information Age Publishing.
  • Kline, K. & Flowers, J. (1999). The Impact of Developing Written Computation as a Representation of Mental Computation. In Hitt, F. & Santos, M. (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Groups for the Psychology of Mathematics Education (pp. 839-845). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Kline, K. & Grant, T. (2003). Developing the Building Blocks of Measurement with Young Children. In Clements, D (Ed.), The National Council of Teachers of Mathematics 2003 Yearbook, Learning and Teaching Measurement(pp. 46-56).
  • Kline, K. & Grant, T. (2002). Developing Elementary Teachers’ Knowledge of Content and Pedagogy Through Implementation of a Standards-Based Mathematics Curriculum. In Guyton, E. & Rainer, J. (Eds.), Teacher Education Yearbook X: Meeting and Using Standards in the Preparation of Teachers (pp. 67-80). Dubuque, Iowa: Kendall/Hunt Publishing Group.
  • Kline, K. & Grant, T. (2002). What Do Elementary Teachers Learn From Reform Mathematics Textbooks? In Thompson, D (Ed.), Proceedings of the Twenty Fourth Annual Meeting of the North American Chapter of the Psychology of Mathematics Education(pp. 1505-1513). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Kline, K. & Grant, T. (2001). What Impacts Teachers as They Implement a Reform Curriculum?: The Case of One Fifth Grade Teacher. In Speiser, R. & Maher, C. (Eds.), Proceedings of the Twenty Third Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (pp. 691-698). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Kline, K., Grant, T., & Archer, M. K. (2006.) Using Your Own Teaching as a Site for Research Into Practice. In S. Z. Smith and M. E. Smith (Eds.) Teachers Engaged in Research: Inquiry into mathematics classrooms, grades pre-K-2 (pp. 15-34). Greenwich, CT: Information Age Publishing.
  • Kline, K., Grant, T., Crumbaugh, C., Kim, O. K., & Cengiz, N. (2005). Exploring Elementary Teachers’ Use of a New Mathematics Curriculum. In Lloyd, G. M., Wilson, M., Wilkins, J. L. M., & Behm, S. L. (Eds.), Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education,[CD ROM] Eugene, OR: All Academic.
  • Kline, K., Grant, T., Crumbaugh, C., Kim, O. K., & Cengiz, N. (Forthcoming, 2008). Analyzing whole-group discussion to explore teachers’ emergent curriculum use. In J. Remillard, G. Lloyd & B. Herbel-Eisenman (Eds.), Teachers’ Use of Mathematics Curriculum Materials: Research Perspectives on Relationships Between Teachers and Curriculum. Routledge.
  • Ladson-Billings, G. (2002). I ain’t writin’ nuttin’: Permission to fail and demands to succeed in urban classrooms. In L. Delpit & J. K. Dowdy (Eds.), The skin that we speak: Thoughts on language and culture in the classroom (pp. 107-120). New York: The New Press.
  • Ladson-Billings, G. (1995). Making Mathematics meaningful in multicultural contexts. In W. Secada (Ed.), New Directions for Equity in Mathematics Education(pp. 126-145). Cambridge: Cambridge University Press.
  • Ladson-Billings, G. (1995). Toward a theory of culturally relevant pedagogy. American Educational Research Journal, 32, 465-491.
  • Lampert, M. (1992). Teaching and Learning Long Division for Understanding in School. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of Arithmetic for Mathematics Teaching(pp. 221-282). Mahwah, NJ: Lawrence Erlbaum Associates.
  • Lampert, M. (2001). Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press.
  • Lappan, G. & Bouck, M. (1998). Developing Algorithms for Adding and Subtracting Fractions. In The NCTM Yearbook: The Teaching and Learning of Algorithms in School Mathematics(pp. 183-197). Reston, VA: NCTM.
  • Ma, L. (1999). Knowing and Teaching Elementary School Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
  • Madell, R. (1985). Children’s Natural Processes. Arithmetic Teacher, 32(7), 20-22.
  • Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11-19.
  • Obidah, J. & Teel, K. M. (2001). Because of the kids. New York: Teachers College Press.
  • Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211-246.
  • Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for curricular change. Journal of Research in Mathematics Education, 35(5), 352-388.
  • Remillard, J. T., & Cahnmann, M. (2005). Researching mathematics teaching in bilingual-bicultural classrooms. In T. McCarty (Ed.), Language, learning, power, and schooling. Hillsdale, NJ: Erlbaum.
  • Remillard, J. T., & Geist, P. (2002). Supporting teachers professional learning through navigating openings in the curriculum. Journal of Mathematics Teacher Education, 5(1), 7-34.
  • Remillard, J. T., & Jackson, K. (2006). Old math, new math: Parents’ experiences with standards-based reform. Mathematical Thinking and Learning, 8(3), 231-259.
  • Richardson, K. (1997). Too Easy for Kindergarten, Just Right for First Grade. Teaching Children Mathematics (Early Childhood Corner), 3(8), 432-37.
  • Russell, S. J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7(3), 154-158.
  • Russell, S. J. (1999). Mathematical reasoning in the elementary grades. In Lee V. Stiff & Frances R. Curcio (Eds.), Developing mathematical reasoning in grades K-12, 1999 Yearbook(pp. 1-12). Reston, VA: National Council of Teachers of Mathematics.
  • Russell, S. J. (2006). What does it mean that 5 has a lot?: From the world to data and back. In G. Burrill (Ed.), Thinking and reasoning with data and chance, 2006 Yearbook. Reston, VA: National Council of Teachers of Mathematics.
  • Russell, S. J., Eston, R., Rook, J., Scott, M., & Sweeney, L. (2003). How to focus the mathematics curriculum on solving problems. In F. K. Lester & R. I. Charles (Eds.), Teaching mathematics through problem solving, prekindergarten-grade 6(pp. 85-100). Reston, VA: National Council of Teachers of Mathematics.
  • Russell, S. J., & Vaisenstein, A. (September/October 2008). Computational fluency: Working with a struggling student. Connect, Volume 12 Issue 1, 8-12.
  • Schifter, D. (2001). Learning to see the invisible: What skills and knowledge are needed to engage with students’ mathematical ideas? In T. Wood, B.S. Nelson, J. Warfield (Eds.). Beyond Classical Pedagogy: Teaching Elementary School Mathematics(pp. 109-134). Mahwah, NJ: Lawrence Erlbaum Associates.
  • Schifter, D. & Szymaszek, J. (2003). Structuring a Rectangle: Teachers Write to Learn about Their Students’ Thinking. In D. Clements and G. Bright (Eds.) Learning and Teaching Measurement: 2003 NCTM Yearbook (pp. 143-156). Reston, VA: National Council of Teachers of Mathematics.
  • Whitenack, J. & Yackel, E. (2002). Making Mathematical Arguments in the Primary Grades: The Importance of Explaining and Justifying Ideas. (Principles and Standards). Teaching Children Mathematics, 8(9), 524-527.

 

Articles and Research Related to the 1st Edition
This list includes research that contributed to or directly informed the development of the 1st edition of the Investigations curriculum. It also includes articles that examine the thinking and learning of students engaged with 1st edition activities, materials, or software. 

  • Battista, M. T. (1999).Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. Journal for Research in Mathematics Education, 30(4), 417-448.
  • Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998).Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503-532.
  • Battista, M. & Clements, D. H. (1998). Finding the number of cubes in rectangular cube buildings. (Research into Practice)Teaching Children Mathematics, 4(5), 258-264.
  • Battista, M. T. & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes.Journal for Research in Mathematics Education, 27, 258-292.
  • Clements, D. H., & Battista, M. T. (2001). Logo and geometry.Journal for Research in Mathematics Education Monograph Series.
  • Clements, D. H., Battista, M. T., Sarama, J., Swaminathan, S. & McMillen, S. (1997).Students’ development of length concepts in a Logo-based unit on geometric paths. Journal for Research in Mathematics Education, 28, 70-95.
  • Clements, D. H., Battista, M. T., Samara, J., & Swaminathan, S. (1997). Development of students’ spatial thinking in a unit on geometric motions and area.The Elementary School Journal, 98(2). 171-186.
  • Clements, D. H., & Sarama, J. (1995). Design of a Logo environment for elementary geometry.Journal of Mathematical Behavior, 14, 381-398.
  • Henry, J. J., & Clements, D. H. (1999).Challenges for teachers attempting to integrate a mathematics innovation. Journal of Research on Technology in Education, 31(3), 240-260.
  • Mokros, J. & Russell, S. J. (1995).Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), 20-39.
  • Sarama, J., Clements, D. H., Swaminathan, S., McMillen, S., & González Gómez, R. M. (2003). Development of mathematical concepts of two-dimensional space in grid environments: An exploratory study.Cognition and Instruction, 21, 285-324.
  • Sarama, J., Clements, D. H., & Henry, J. J. (1998).Network of influences in an implementation of a mathematics curriculum innovation. International Journal of Computers for Mathematical Learning, 3, 113-148.
  • Sarama, J., Clements, D. H. (1995).Redesigning Logo: the turtle metaphor in mathematics education. Unpublished doctoral dissertation, State University of New York, Buffalo, NY.
The Impact of the 1st and 2nd editions
  • Flowers, J. M. (1998). A study of proportional reasoning as it relates to the development of multiplication concepts. Unpublished doctoral dissertation, University of Michigan, Ann Arbor, MI.
  • Gatti, Guido G. and Giordano, K. (2008). Investigations in Number, Data, and Space Efficacy Study. Pittsburgh, PA: Gatti Evaluation Inc.
  • Goodrow, A. M. (1998). Children’s construction of number sense in traditional, constructivist, and mixed classrooms. Unpublished doctoral dissertation, Tufts University, Medford, MA.
  • Kehle, Paul; Essex, Kathy; Lambdin, Diana and McCormick, Kelly. (2007.) What Did They Learn? A Longitudinal, Comparative, and Focused Study of a Prepublication Version of Investigations in Number, Data, and Space.
  • Mokros, J., Berle-Carmen, M., Rubin, A., & Wright, T. (1994). Full-year pilot grades 3 and 4: Investigations in Number, Data, and Space. Cambridge, MA: TERC.
  • Mokros, J. (2000). The Investigations curriculum and children’s understanding of whole number operations. Cambridge, MA: TERC.
  • Mokros, J., Berle-Carmen, M., Rubin, A., & O’Neill, K. (1996). Learning operations: Invented strategies that work. Paper presented at the Annual Meeting of the American Educational Research Association. New York, NY.
  • Sconiers, Sheila. The ARC Center Tri-State Student Achievement Study. COMAP, 2003.
  • Smith, S. Z., & Smith, M. E. (2006, March). Assessing elementary understanding of multiplication concepts. School Science and Mathematics 106(3), 140-149.
Policy and Practice
Publications related to educational research, effective practice, curriculum materials, and some of the other complex issues involved when making decisions about curricula and instruction.

  • Burkhardt, H., & Schoenfeld, A. H. (2003). Improving educational research: Toward a more useful, more influential, and better-funded enterprise. Educational Researcher, 32(9), 3-14.
  • Donovan, et al. (Eds.). (2000). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.
  • Hiebert, J. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education, 30(1), 3-19.
  • Kilpatrick, J., Martin, W. G., & Schifter, D. (2003). A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • Leinwand, S. & Burrill, G. (Eds.). (2001). Improving Mathematics Education: Resources for Decision Making. Washington, D.C.: National Academy Press.
  • National Council of Teachers of Mathematics. (2000). Principles and Standards in School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • National Council of Teachers of Mathematics. The Curriculum and Evaluation Standards for School Mathematics(1989), The Professional Standards for Teaching Mathematics (1991), and The Assessment Standards for School Mathematics (1995). Reston, VA: NCTM.
  • National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
  • National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, D.C.: National Academy Press.
  • National Research Council. (2002). Helping Children Learn Mathematics. Mathematics Learning Study Committee, J. Kilpatrick and J. Swafford, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Pelle Grino, J.W. et al. (Eds.). (2001). Knowing what students know: The science and design of educational assessment. Washington, D.C.: National Academy Press.
  • Schoenfeld, A. H. (2002). Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity. Educational Researcher, 31(1), 13-15.
  • Stigler, James W. and Hiebert, James. (1999). The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. NY, NY: Free Press.