Learning Mathematics through Representations

A Curriculum Development and Research Project



Learning Mathematics through Representations (LMR) is a research-based curriculum unit for the teaching and learning of integers and fractions in the elementary grades, using the number line as the principal representational context. The curriculum builds on two core ideas: mathematical representations are fundamental to mathematical communication and learning, and curriculum units should be designed as well-orchestrated lesson sequences that support insight and understanding of representational forms.

The members of the LMR staff bring expertise in developmental and educational research, curriculum development, pre-service education and professional development, and elementary classroom teaching. The LMR team is led by Geoffrey B. Saxe and includes additional faculty and graduate students in the Graduate School of Education, University of California, Berkeley.

There are 3 units.

Resources for each unit include a Lesson Guide, Transparencies (can also be used with a document, designed to support full group discussions), Worksheets* (for in-class use), Homework, Classroom Poster**, Student Poster (with blanks to be filled in by the student), and a Completed Poster. Each of the above modules also contains the Guide Introduction for the complete curriculum.

*NOTE: The dimensions of the worksheets may change slightly when you print and copy on your printer and copier, and the result may be an inexact fit of Cuisenaire rods to worksheets. We suggest that you print and copy a test copy of each worksheet that uses rods (denoted by the "RODS" logo in the upper right) to verify that the correct rods fit between tickmarks (see answer keys for specific colored rods). You may find that you need to enlarge (or reduce) your copier settings slightly so that rods fit; for example, try slight adjustments like 105% or 106% enlargement.

**NOTE: The Class Poster file is letter-sized, but it should be printed as a poster large enough to be displayed in the classroom to the whole class. We suggest having the poster printed 2.5-3 feet wide, in black and white, and in draft quality (to minimize expense)


Institute of Education Sciences grant to Geoffrey B. Saxe (R305B070299 and R305B090026) funded the development of the Learning Mathematics through Representation (LMR) curriculum and associated research. The opinions expressed in the publications and curriculum materials are those of the authors and do not represent views of the Institute of Education Sciences or the U.S. Department of Education. We are grateful to the teachers and students who participated in the research, and to the many faculty and graduate students who contributed to the project.

Faculty and Team Leaders

  • Professor Geoffrey Saxe, Principal Investigator, Learning Mathematics through Representations, Graduate School of Education, UC Berkeley
  • Associate Adjunct Professor Maryl Gearhart, Graduate School of Education, UC Berkeley

Faculty Consultants

  • Professor Deborah Loewenberg Ball, School of Education, University of Michigan
  • Professor Hyman Bass, School of Education, University of Michigan

Contributing Teachers

  • Rick Kleine, Jefferson Elementary, Berkeley, CA
  • Jennifer Pfotenhauer, Malcolm X Elementary, Berkeley, CA

Former Graduate Student Researchers (UC Berkeley) who participated in the development and/or research on LMR.

Most of these former graduate students have now completed their degrees and launched their own careers:

  • Nicole Leveille Buchanan, Ph.D.
  • Jennifer Collett, Ph.D.
  • Kenton De Kirby, Ph.D.
  • Ronli Diakow, Ph.D.
  • Darrell Earnest, Ph.D.
  • Lina Chopra Haldar, Ph.D.
  • David Torres Irribarra, Ph.D.
  • Bona Kang, Ph.D., Ph.D.
  • Marie Le, Ph.D., Ph.D.
  • Katherine Lewis, Ph.D.
  • Anna McGee, Ph.D.
  • Alison Miller Singley, Ph.D.
  • Meghan Shaughnessy, Ph.D.
  • Yasmin Sitabkhan, Ph.D.
  • Joshua Sussman, Ph.D.
  • Lissa Tryer, M.A.
  • Ying Zheng, M.A.

Manuscripts, Journal Articles, and Book Chapters

  • Saxe, G. B., Sussman, J. (in preparation). Mathematics learning in language inclusive classrooms: Supporting the achievement of English learnings and their English proficient peers.

  • Collett, J., Gearhart, M. & Leveille Buchanan, N. (2018, January/February). Supporting students’ contributions to classroom discussions. Teaching Children Mathematics, 24(4), 236-243

  • Saxe, G. B., de Kirby, K., Kang, B., Le, M., Schneider, A. (2015). Studying cognition through time in a classroom community: The interplay between "everyday" and "scientific concepts." Human Development, 58, 5-44. DOI: 10.1159/000371560

  • Learning Mathematics through Representations. (2014, October). Alignment of the Learning Mathematics through Representations curriculum with the Common Core State Standards. Unpublished manuscript, Graduate School of Education, University of California, Berkeley.

  • Gearhart, M., Leveille Buchanan, N., Collett, J., Diakow, R., Kang, B., Saxe, G.B., McGee, A. (2014). Identifying opportunities to learn in mathematics discussions in heterogeneous eementary classrooms. Unpublished manuscript, University of California, Berkeley.

  • Leveille Buchanan, N., Collett, J., McGee, A., & Gearhart, M. (2014). Teachers’ orientations to a curriculum-supported pedagogical practice in mathematics. Unpublished manuscript, University of California, Berkeley.

  • Gearhart, M., Saxe, G. B., Earnest, D., Haldar, L. C., McGee, A., & Sitabkhan, Y. (2015). Embedding formative assessment in curriculum design: A research-based approach. Annual Perspectives in Mathematics Education (APME) 2015: Assessment to Enhance Teaching and Learning (pp. 219-231). Reston, VA: National Council of Teachers of Mathematics.

  • Saxe, G. B., de Kirby, K. Le, M., Sitabkhan, Y., & Kang, B. (2015). Understanding learning across lessons in classroom communities: A multi-leveled analytic approach. To appear in A. Bikner-Ahsbahs, G. Kaiser, N. Presmeg (Eds.) Doing (qualitative) research: Methodology and methods in mathematics education. ZDM research handbook series: Advances in Mathematics Education. pp. 253-318.New York, NY: Springer.

  • Gearhart, M., & Saxe, G. B. (2014, April). Differentiated instruction in shared mathematical contexts. Teaching Children Mathematics, 20(7), 426-435.

  • Saxe, G. B., Diakow, R., & Gearhart, M. (2013). Towards curricular coherence in integers and fractions: The efficacy of a lesson sequence that uses the number line as the principal representational context. Special Issue (Classroom-based interventions in mathematics education). ZDM: International Journal of Mathematics Education, (45)(3), 343-364. DOI 10.1007/s11858-012-0466-2.

  • Saxe, G. B., Shaughnessy, M., Gearhart, M., & Haldar, L. C. (2013). Coordinating numerical and linear units: Elementary students' strategies for locating whole numbers on the number line. Mathematical Thinking and Learning, 15, 235-258.

  • Pfotenhauer, J., Kleine, R., Sitabkhan, Y., & Earnest, D. (2013, May). Shifting understanding of mixed numbers. Teaching Children Mathematics, 19(9), 592.

  • Saxe, G. B., Gearhart, M., Shaughnessy, M. M., Earnest, D., Cremer, S., Sitabkhan, Y., Platas, L. M. & Young, A. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. Schwartz, T. Dreyfus and R. Hershkowitz (Eds.), Transformation of knowledge in classroom interaction (pp. 203-222). New York, NY: Routledge.

  • Saxe, G. B., Shaughnessy, M. M., Shannon, A., Garcia de Osuna, J., Chinn, R., & Gearhart, M. (2007). Learning about fractions as points on a number line. In W. G. Martin, M. E. Strutchens, & P. C. Elliott (Eds.), The learning of mathematics (pp. 221-238). Reston, VA: National Council of Teachers of Mathematics.