# 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 T
*he 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,* - 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,* - Battista, M. & Clements, D. H. (1998). Finding the number of cubes in rectangular cube buildings. (Research into Practice)
*Teaching Children Mathematics,* - Battista, M. T. & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes.
*Journal for Research in Mathematics Education,* - 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,* - 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,* - Clements, D. H., & Sarama, J. (1995). Design of a Logo environment for elementary geometry.
*Journal of Mathematical Behavior,* - Henry, J. J., & Clements, D. H. (1999).Challenges for teachers attempting to integrate a mathematics innovation.
*Journal of Research on Technology in Education,* - Mokros, J. & Russell, S. J. (1995).Children’s concepts of average and representativeness.
*Journal for Research in Mathematics Education,* - 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,* - 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,* - 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 Mathematic*s (1991), and*The Assessment Standards for School Mathematics* - 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.