Resources

Alexander, C. & Ambrose, R. (August 2010). Digesting student-authored story problems. Mathematics Teaching in the Middle School, 16, (1), 27-33.

Ambrose, R. (2004). Initiating Change in Prospective Elementary School Teachers’ Orientation to Mathematics Teaching by Building on Beliefs.  Journal of Mathematics Teacher Education 7(2), 91-119.

Ambrose, R., Baek, J., Carpenter, T. P., (2003). Children’s Invention of Multiplication and Division Algorithms, in Baroody, A. and Dowker (Eds.) The Development of Arithmetic Concepts and Skills: Recent Research and Theory. Mahwah, NJ: Erlbaum.

Ambrose, R. & Kenehan, G. (2009). Children's Evolving Understanding of Polyhedra in the Classroom, Mathematical Thinking and Learning, Vol 11 (3), 158 - 176.

Ambrose, R. & Molina, M. (in press). First-grade Latino English Language Learners´ performance on story problems in English versus Spanish. Canadian Journal of Science, Mathematics and Technology Education. 

Baek, J. M. (2008). Developing algebraic thinking through explorations in multiplication. In C. Greenes (Ed.), Algebra and algebraic thinking in school mathematics: NCTM 2008 Yearbook. Reston, VA: National Council of Teachers of Mathematics. 

Baek, J. M. (2005). Children’s mathematical understanding and invented strategies for multidigit multiplication. Teaching Children Mathematics, 12, 242-247.

Baek, J. (1998). Children’s invented algorithms for multidigit multiplication problems. In Morrow, L. (Ed). The teaching and learning of algorithms in school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Baek, J. M. & Flores, A. (2005). How does it feel? Teachers count on the alphabet, instead of numbers. Teaching Children Mathematics, 12, 54-59. 

Battey, D. & Chan, A. (2010). Building community and relationships that support critical conversations on race: The case of Cognitively Guided Instruction. In M. Q. Foote (Ed.), Mathematics Teaching & Learning in K-12: Equity and Professional Development. New York: Palgrave. 

Battey, D. & Franke, M. L. (2008). Transforming identities: Understanding teachers across professional development and classroom practice. Teacher Education Quarterly,35(3), 127-149.   

Battey, D. & Franke, M. (2015). Integrating professional development on mathematics and equity: Countering deficit views of students of color. Education and Urban Society, 47(4), 433-462, first published on August 29, 2013 doi:10.1177/0013124513497788.

Behrend, J. L. (2003). Learning-Disabled students make sense of mathematics. Teaching Children Mathematics, 9(1), 269-273. 

Behrend, J. L. (2001). Are rules interfering with children’s mathematical understanding? Teaching Children Mathematics, 8(1), 36-40. 

Behrend, J. L. & Mohs, L. C. (2005/2006). From simple questions to powerful connections: A two-year conversation about negative numbers. Teaching Children Mathematics, 12(5), 260-268. 

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2016). Leveraging structure: Logical necessity in the context of integer arithmetic. Mathematical Thinking and Learning, 18(3), 209–232.

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2016). Unlocking the structure of positive and negative numbers. Mathematics Teaching in the Middle School, 22(2), 84–91.

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning: An analysis of children’s thinking and the history of mathematics. Journal for Research in Mathematics Education, 45(1), 19–61.

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2014). Using order to reason about negative numbers: The case of Violet. Educational Studies in Mathematics, 86, 39–59. 

Bishop, J. P., Lamb, L. L., Philipp, R. A., Schappelle, B. P., & Whitacre, I. (2011). First graders outwit a famous mathematician. Teaching Children Mathematics, 17, 350–358.

Bray, W.S., & Blais, T. V. (2017). Stimulating base-ten reasoning with context. Teaching Children Mathematics, 24(2), 120-127.

Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving:  A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 24(5), 428-441.

Carpenter, T. P., Fennema, E., Franke, M. L. (1999). Cognitively Guided Instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3-20.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L, Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction, Second Edition Portsmouth, NH: Heinemann.

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.P., & Loef, M. (1989). Using knowledge of children's mathematics thinking in classroom teaching:  An experimental study.  American Educational Research Journal, 26 (4), 385-531.

Carpenter, T.P., Franke, M. L., Jacobs, V. R., Fennema, E., and Empson, S. B. (1998). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3-20. 

Carpenter, T.P., Franke. M.L., Johnson, N.C., Turrou, A.C., Wager, A. A. (2017). Young children’s mathematics: Cognitively Guided Instruction in Early Childhood Education. Portsmouth, NH: Heinemann.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann.

Carpenter, T. P., Levi, L., Franke, M. L., Zeringue, J. K. (2005).  Algebra in the Elementary School: Developing Relational Thinking.  ZDM – The International Journal on Mathematics Education, 37(1).

Carpenter, T. P., and 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. 

Chan, A.G. (2010). Identity and practice: Preservice teacher learning within a practice-based mathematics methods course.  Unpublished doctoral dissertation, University of California, Los Angeles. 

Empson, S. B. (2003). Low-Performing Students and Teaching Fractions for Understanding: An Interactional Analysis. Journal for Research in Mathematics Education, 34: 4 305-343. 

Empson, S.B., (2001). Equal Sharing and the Roots of Fractions Equivalence. Teaching Children Mathematics, 7: 421-25. 

Empson, S.B., (1999) Equal Sharing and Shared Meaning: The Development of Fraction Concepts in a First-Grade Classroom. Cognition and Instruction, 17: 283-342.

Empson, S. B. (1995).  Using sharing situations to help children learn fractions. Teaching Children Mathematics, 2(2), 110-114.

Empson, S. B. & Jacobs, V. J. (2008). Learning to Listen to Children’s Mathematics. In T. Wood (Series Ed.) & P. Sullivan (Vol. Ed.), International handbook of mathematics teacher education, vol.1: Knowledge and beliefs in mathematics teaching and teaching development(pp. 257-281). Rotterdam, the Netherlands: Sense Publishers.

Empson, S. B., Junk, D., Dominguez, H., and Turner, E. (2006). Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking. Educational Studies in Mathematics. 

Empson, S. B. & Knudsen, J. (2003). Building on Children’s Thinking to Develop Proportional Reasoning. Texas Mathematics Teacher, L(2), 16-21. 

Empson, S. B., and Levi, L. (2011) Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann.

Empson, S. B., Levi, L., and Carpenter, T. P. (2010) The algebraic nature of fractions: developing relational thinking in elementary school in J Cai and E. Knuth (Eds) Early Algebraization: Cognitive, Curricular and Instructional Perspectives. New York: Springer 

Enyedy, N., Wischnia, S. & Franke, M.  (2008).  Classroom discourse : Contrastive and Consensus conversations.  Journal of Educational Research,  (2) 2./3.

Falkner, K. P., Levi, L., & Carpenter, T.P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232-236.

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction.  Journal for Research in Mathematics Education. 27, 4, 403-434.

Fennema, E., Carpenter, T. P., Jacobs, V. R., Franke, M. L., & Levi, L. (1998). A longitudinal study of gender differences in young children’s mathematical thinking. Educational Researcher, 27(5), 4 – 12. 

Fennema, E., Carpenter, T.P, Levi, L., Franke, M.L. & Empson, S. B. (2000). Children’s Mathematics: Cognitively Guided Instruction: A Guide for Workshop Leaders. Portsmouth, NH: Heinemann.

Franke, M.L. (2003). Fostering young children's mathematical understanding. In C. Howes (Ed.) Teaching 4- to 8-year olds: Literacy, math, multiculturalism, and classroom community. Baltimore, MD: Brookes.

Franke, M. L., Carpenter, T. P., & Battey, D. (2007). Content matters: The case of algebraic reasoning in teacher professional development. In J. Kaput, D. Carraher, & M. Blanton, (Eds.)Algebra in the Early Grades (pp. 333-359). Hillside, NJ: Lawrence Erlbaum. 

Franke, M. L., Carpenter, T., Fennema, E., Ansell, E., and Behrend, J. (1998).

Understanding Teachers' Self-Sustaining, Generative Change in the Context of Professional Development. Teaching and Teacher Education, 14(1), 67-80.

Franke, M.F., Carpenter, T.P., Levi, L., Fennema, E. (2001) “Capturing Teachers’ Generative Change: A Follow-up Study of Professional Development in Mathematics. American Educational Research Journal38 (3), 653-689.

Franke, M. L. and Kazemi, E. (2001). Learning to Teach Mathematics: Focus on Student Thinking.  Theory Into Practice 40(2), 102-109.

Franke, M.L., & Kazemi, E. (2001).  Teaching as learning within a community of practice: Characterizing generative growth.  In T. Wood, B. Nelson, & J. Warfield (Eds.). Beyond classical pedagogy in elementary mathematics: The nature of facilitative teaching (pp. 47-74). Mahwah, NJ:  Erlbaum.

Franke, M. L., Kazemi, E., & Battey, D. (2007).  Understanding teaching and classroom practice in mathematics.  In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 225-256).  Greenwich, CT: Information Age Publishers.

Franke, M., Kazemi, E., Shih, J., Biagetti, S., & Battey, D. (2005). Changing teachers’ professional work in mathematics: One school’s journey. In T.A. Romberg,T.P. Carpenter, T. P., & F. Dremock (Eds.) Understanding mathematics and science matters (pp. 209-230).Mahwah, NJ: Erlbaum.

Franke, M. L., Kazemi, E., Turrou, A. C. (2018). Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom.Portland, ME: Stenhouse.

Franke, M., Webb, N., Chan, A., Battey D.,  Ing, M., Freund, D., De, T. (2009). Eliciting student thinking in elementary mathematics classrooms: Practices that support understanding.  Journal of Teacher Education. 


Franke, M. L., Webb, N. M., Chan, A., Ing, M., Freund, D., & Battey, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 364-379. 

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Oliver, A., and Human, P. (1997) A day in the life of one Cognitively Guided Instruction Classroom, in Hiebert, et. al., Making Sense: Teachers and learning mathematics with understanding. Portsmouth, NH: Heinemann. 

Jacobs, V. R. & Ambrose, R. C. (2008).  Making the most of story problems. Teaching Children Mathematics, 15, 260–266.

Jacobs, V. R., Ambrose, R. C., Clement, L., and Brown, D. (2006).  Using Teacher-Produced Videotapes of Student Interviews as Discussion Catalysts.  Teaching Children Mathematics 12(6), 276-295.

Jacobs, V. R., & Empson, S. B. (2016). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves. ZDM–The International Journal on Mathematics Education, 48(1–2), 185–197.

Jacobs, V. R., Empson, S. B., Pynes, S., Hewitt, A., Jessup, N., & Krause, G. (in press). Responsive teaching in elementary mathematics (RTEM) project.  In P. Sztajn & P. H. Wilson (Eds.), Designing professional development for mathematics learning trajectories.  New York: Teachers College Press.

Jacobs, V. R., Martin, H., Ambrose, R. C., Philipp, R. A. (2014).  Warning signs for taking over children’s thinking.  Teaching Children Mathematics, 21, 107–113.

Jacobs, V. R., Franke, M. L., Carpenter, T.P., Levi, L., & Battey, D. (2007) Professional Development Focused on Children’s Algebraic Reasoning in Elementary School.  Journal for Research in Mathematics Education, May 2007

Jacobs, V. R., & Kusiak, J. (2006).  Got tools? Exploring children's use of mathematics tools during problem solving.  Teaching Children Mathematics, 12, 470–477.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010).  Professional noticing of children’s mathematical thinking.  Journal for Research in Mathematics Education,41(2), 169–202. 

Jacobs, V. R., Lamb, L. L. C., Philipp, R. A., & Schappelle, B. P. (in press).  Deciding how to respond on the basis of children's understandings.  In M. G. Sherin, V. R., Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing:  Seeing through teachers' eyes.  New York: Routledge.

Jacobs, V. R., & Philipp, R. A. (2010). Supporting children's problem solving. Teaching Children Mathematics, 17(2), 98–105. 

Jacobs, V. R., & Philipp, R. A., (2004). Mathematical Thinking: Helping Prospective and Practicing Teachers Focus.  Teaching Children Mathematics, 11(4), 194-201.

Jaslow, L. & Jacobs, V. (2009, Spring).  Helping kindergartners make sense of numbers to 100.  The Journal of Mathematics and Science: Collaborative Explorations,11, 195–213.

Kazemi, E., & Franke, M. L. (2004). Teacher Learning in Mathematics:  Using Student Work to Promote Collective Inquiry. Journal of Mathematics Teachers Education, 7(3), 203-235. 

Lamb, L. L., Bishop, J. P., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2018). A cross-sectional investigation of students’ reasoning about integer addition and subtraction: Ways of reasoning, problem types, and flexibility. Journal for Research in Mathematics Education, 49(5), 575 – 613. 

Lamb, L. L., Bishop, J. P., Philipp, R., Whitacre, I., Schappelle, B., & Lewis, M. L. (2012). Developing symbol sense for the minus sign. Mathematics Teaching in the Middle School, 18(1), 5–9.

Lamb, L. C., Philipp, R. A., Jacobs, V. R., & Schappelle, B. P. (2009). Developing teachers’ stances of inquiry: Studying teachers’ evolving perspectives.  In D. Slavit, T. Holmlund Nelson, & A. Kennedy (Eds.),Perspectives on supported collaborative teacher inquiry, (pp. 16–45).  New York: Taylor & Francis.

Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., Franke, M. (2010) Using designed instructional activities to enable novices to manage ambitious mathematics teaching (pp. 129-141).  In  M.K. Stein & L. Kucan (Eds.) Instructional explanations in the discipline.  New York: Springer.

Levi, L. & Ambrose, R. (2018). Cognitively Guided Instruction and Formative Assessment. In E. Silver and V. Mill (Eds.) A Fresh Look at Formative Assessment in Mathematics Teaching: Leveraging Connections to Tasks, Discourse, Equity, and More.Reston, VA: National Council for Teachers of Mathematics.

Levi, L. (2004).  Are students in a reform mathematics class ill-equipped for traditional mathematics instruction?  Texas Mathematics Teacher, Spring 2004, 24 – 27.

Levi, L. (2000) Gender Equity in Mathematics Education.   Teaching Children Mathematics, October 2000.

Maldonado, L., Turner, E. E., Dominguez, H. & Empson, S. B. (2009) English Language Learners Learning from and Contributing to Discussions. Mathematics for All: Instructional Strategies for Diverse Classrooms. National Council of Teachers of Mathematics.

Molina, M. and Ambrose, R. (2007) Fostering Relational Thinking While Negotiating the Meaning of the Equal Sign. Teaching Children Mathematics, September 2006, p. 111 – 118. 

Moscardini, L & Sadler, S. (2018). Scottish Attainment Challenge Cognitively Guided Instruction Project 2016-2018 Final Report. Glasgow, Scotland: University of Strathclyde. DOI: 10.13140/RG.2.2.21616.30728

Moscardini, L., (2015). Primary special school teachers' knowledge and beliefs about supporting learning in numeracy. Journal of Research in Special Educational Needs.15, (1), 37–47.  

Moscardini, L. (2014) Developing equitable elementary mathematics classrooms through teachers learning about children’s mathematical thinking: Cognitively Guided Instruction as an inclusive pedagogy. Teaching and Teacher Education, 43, 69–79.

Moscardini, L. (2010). ‘I like it instead of maths’: how pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems.  British Journal of Special Education, Vol 37 Number 3.

Moscardini, L. (2009). Tools or crutches? Apparatus as a sense-making aid in mathematics teaching with children with moderate learning difficulties. British Journal of Support for Learning, 24,(1), 35-41. 

Philipp, R. A. (2008). Motivating prospective elementary school teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17(2) 7-26.

Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257-315). Reston, VA: National Council of Teachers of Mathematics.

Philipp. R. A., Ambrose, R. Lamb, L. L. C., Sowder, J., T., Schappelle, B. P., Sowder, L., Thanheiser, E., and Chauvot, J. (2007). Effects of early field experiences on the mathematical content knowledge of prospective elementary school teachers:  An experimental study.  Journal for Research in Mathematics Education. (28)5, 438 – 476.

Philipp, R. A., & Thanheiser, E. (2010). Showing your students you care:  Seeing the individual trees in the classroom forest. New England Mathematics Journal, 42 (May, 2010), 8-17.

Pierson, J, Lamb, L, Philipp, R., Schappelle, B., & Whitacre, I. (in press). Children’s Reasoning about “Numbers Under Zero.” Teaching Children Mathematics.

Pierson, J., Lamb, L., Philipp, R., Schappelle, B., & Whitacre, I. (2010).A Developing Framework for Children’s Reasoning About Integers. In P. Brosnan, D. Erchick, & L. Flevares (Eds) Proceedings of the 32ndAnnual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 695-702). Columbus, OH: The Ohio State University.

Schwerdtferger, Julie K., Chan, A. (2007). Counting Collections. Teaching Children Mathematics, March 2007, p. 356 – 361. 

Steinberg, R. M., Empson, S. B., Carpenter, T. P. (2004). Inquiry into Children’s Mathematical Thinking as a Means to Teacher Change.  Journal of Mathematics Teacher Education, 7(3), 237-267.

Turner, E., Celedon-Pattichis, S., Marshall, M. & Tennison, A. (2009). "Fíjense amorcitos, les voy a contar una historia":  The Power of Storyto Support Solving and Discussing Mathematical Problems among Latino/a Kindergarten Students(pp. 23-43).  In D. White & J. Spitzer (Eds).Mathematics for Every Student: Responding to Diversity, Grades PreK-5 (pp. 19-42).Reston, VA: NCTM.

Turner, E. E., Junk, D., & Empson, S. B. (2007). The power of paper-folding tasks: Supporting multiplicative thinking and rich mathematical discussion. Teaching Children Mathematics,13(6), 322-329.

Warfield, J., & Yttri, M. J. (1999). Cognitively Guided Instruction in one kindergarten classroom. In J. V. Copley (Ed.). Mathematics in the early years. Reston, VA: NCTM. 

Webb, N. , Franke, M., De, T, Chan A., Freund, D., Shein, P., Melkonian, D. (2009).  “Explain to your Partner”: Teachers’ instructional practices and students’ dialogue in small groups.   Cambridge Journal of Education. 

Webb, N. M., Franke, M. L., Ing, M., Chan, A., De, T., Freund, D., & Battey, D. (2008). The Role of teacher instructional practices in student collaboration. Contemporary Educational Psychology, 33(3), 360-381.

Whitacre, I., Bishop, J. P., Philipp, R., Lamb, L. L., & Schappelle, B. (2014). Dollars and sense: Students’ integer perspectives. Mathematics Teaching in the Middle School,20(2), 84–89.

Whitacre, I., Bishop, J. P., Lamb, L. L., Philipp, R. A., Schappelle, B. P., & Lewis, M. L. (2012). Happy and sad thoughts: An exploration of children’s integer reasoning. Journal of Mathematical Behavior, 31, 356–365.

Whitacre, I., Bouhjar, K., Bishop, J. P., Philipp, R. A., Schappelle, B. P., & Lamb, L. L. (2016). Regular numbers and mathematical worlds. For the Learning of Mathematics,36(2), 20–25.

Whitacre, I., Lamb, L., Azuz, B., Bishop, J. P., Philipp, R. A., & Schappelle, B. P. (2017). Integer comparisons across the grades: Students’ justifications and ways of reasoning. Journal of Mathematical Behavior, 45, 47–62.

Whitacre, I., Schoen, R. C., Champagne, Z., & Goddard, A. (2017). Relational thinking: What’s the difference? Teaching Children Mathematics, 23(5), 303–308.

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CGI trainings were initially developed for classroom teachers, but we now know that training is most effective when administrators, special educators, and instructional aides participate as well.

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Research indicates that it takes a significant amount of experience with CGI before teachers are effectively able to lead trainings themselves, but it is our goal to help districts build capacity over time. This usually involves a scaffolded approach with future trainers shadowing experienced trainers or serving as apprentices.

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CGI is a research base and professional development program in which children’s thinking, not a textbook, drives instructional decisions. However, a CGI teacher will be able to use and adapt any textbook skillfully and thoughtfully to meet the needs of their students.

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CGI research informed the Common Core State Standards and there is a great deal of alignment between the two in both the content standards and the Standards for Mathematical Practice. 

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Standardized tests aligned to the Common Core demand students go beyond assessing computational fluency and require students demonstrate conceptual knowledge and problem solving skills. CGI prepares students for both computational fluency and conceptual understanding.

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