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Publications


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Publications


Journal Articles

Ball, D. L., Ben-Peretz, M., & Cohen, R. B. (2014). Records of practice and the development of collective professional knowledge. British Journal of Educational Studies, 62(3), 317–335.

Phelps, G., Corey, D., DeMonte, J., Harrison, D., & Ball, D. L. (2012). How much English language arts and mathematics instruction do students receive? Investigating variation in instructional time. Educational Policy, 26(5), 631–662.

Corey, D., Phelps, G., Ball, D. L., DeMonte, J., & Harrison, D. (2012). Explaining variation in instructional time: An application of quantile regression. Educational Evaluation and Policy Analysis, 34(2), 146–163.

Charalambos, C. Y., Hill, H. C., & Ball, D. L. (2011). Prospective teachers’ learning to provide instructional explanations: How does it look and what might it take? Journal of Mathematics Teacher Education, 14(6), 441–463.

Learning Mathematics for Teaching Project. (2011). Measuring the mathematical quality of instructionJournal of Mathematics Teacher Education, 14(1), 25-47.

Ball, D. L., & Forzani, F. M. (2011, Summer). Building a common core for learning to teach, and connecting professional learning to practiceAmerican Educator, 35(2), 17–21, 38–39.

Ball, D. L., & Forzani, F. M. (2010). Teaching skillful teachingEducational Leadership, 68(4), 40–45.

Ball, D. L., & Forzani, F. M. (2010). What does it take to make a teacher? Phi Delta Kappan, 92(2), 8–12.

Thames, M. H. & Ball, D. L. (2010). What mathematical knowledge does teaching require? Knowing mathematics in and for teachingTeaching Children Mathematics, 17(4), 220–225.

Ball, D. L., Forzani, F. M. (2009). The work of teaching and the challenge for teacher educationJournal of Teacher Education, 60(5), 497–511.

Ball, D. L., Hill, H. (2009). The curiousand crucialcase of mathematical knowledge for teachingPhi Delta Kappan, 91(2), 68–71.

Ball, D. L., Sleep, L., Boerst, T., & Bass, H. (2009). Combining the development of practice and the practice of development in teacher educationElementary School Journal, 109, 458–476.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Hill, H. C., Blunk, M., Charalambous, C., Lewis, J., Phelps, G., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory studyCognition and Instruction, 26(4), 430–511.

Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of provingJournal of Mathematics Teacher Education, 11, 307–332.

Ball, D. L., Lewis, J. & Thames, M. H. (2008). Chapter 1: Making mathematics work in schoolJournal for Research in Mathematics Education, Monograph 14, 13–201.

Delaney, S., Ball, D. L, Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: Adapting U.S. measures for use in IrelandJournal of Mathematics Teacher Education, 11(3), 171–197.

Hill, H., Ball, D. L., & Schilling, S. G. (2008). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of studentsJournal for Research in Mathematics Education, 39(4), 372–400.

Ball, D. L., & Forzani, F. M. (2007). What makes education research "educational"? Educational Researcher36(9), 529–540.

Hill, H. C., Ball, D. L., Blunk, M., Goffney, I. M., & Rowan, B. (2007). Validating the ecological assumption: The relationship of measure scores to classroom teaching and student learning. Measurement: Interdisciplinary Research & Perspective, 5(2), 107–118.

Ball, D. L., Goffney, I. M., & Bass, H. (2005).  The role of mathematics instruction in building a socially just and diverse democracyThe Mathematics Educator, 15(1), 2–6.

Ball, D. L., Ferrini-Mundy, J., Kilpatrick, R. J., Milgram, J., Schmid, W., & Schaar, R. (2005). Reaching for common ground in K-12 mathematics educationNotices of the American Mathematical Society, 52(9), 1055–1058.

Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 30(3), p. 14–17, 20–22, 43–46.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievementAmerican Educational Research Journal, 42(2), 371–406.

Ball, D. L., & Rowan, B.  (2004). Introduction: Measuring instructionThe Elementary School Journal, 105(1), 3–10.

Thames, M. H., & Ball, D. L. (2004). [Review of the book Learning Discourse: Discursive Approaches to Research in Mathematics Education, by C. Kieran, E. A. Forman, & A. Sfard]. Mathematical Thinking and Learning, 6(4), 421–433.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers' mathematical knowledge for teaching. Elementary School Journal, 105(1), 11–30.

Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s mathematics professional development institutes. Journal for Research in Mathematics Education, 35(5), 330–351.

Cohen, D., Raudenbush, S., & Ball, D. L. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 1–24.

Cohen, D. K., & Ball, D. L. (2001). Making change: Instruction and its improvement. Kappan83(1), 73–77.

Ball, D. L. (2000). Bridging practices: Intertwining content and pedagogy in teaching and learning to teach. Journal of Teacher Education, 51(3), 241–247.

Ball, D. L., & Bass, H. (2000). [Review of the book The Number Devil, by H. M. Enzensberger]. Notices of the American Mathematical Society, 47(1), 51–56.

Chazan, D. & Ball, D. L. (1999). Beyond being told not to tellFor the Learning of Mathematics, 19(2), 2–10.

Price, J., & Ball, D. L. (1998). Challenges of liberatory pedagogy in mathematics and teacher education. Theory Into Practice, 37(4), 256–264.

Ball, D. L. (1997). From the general to the particular: Knowing our students as learners of mathematics. Mathematics Teacher, 90(9), 732–737.

Price, J., & Ball, D. L. (1997) “There's always another agenda”: Marshalling resources for mathematics reform. Journal of Curriculum Studies, 29(6), 637–666. 

Ball, D. L. & Wilson, S. W. (1996). Integrity in teaching: Recognizing the fusion of the moral and the intellectual. American Educational Research Journal, 33(1), 155–192.

Ball, D. L. (1996). Teacher learning and the mathematics reforms: What do we think we know and what do we need to learn? Phi Delta Kappan, 77(7), 500–508.

Ball, D. L. & Cohen, D. K. (1996). Reform by the book: What is–or might be–the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–8, 14.

Wilson, S. W., & Ball, D. L. (1996). Helping teachers meet the standards: New challenges for teacher educators. Elementary School Journal, 97(2), 121–138.

Wilson, S. M., Peterson, P. L., Ball, D. L., & Cohen, D. K. (1996). Learning by all. Phi Delta Kappan, 77(7), 468–476.

Ball, D. L. (1995). Blurring the boundaries of research and practice. Remedial and Special Education, 16(6), 354–363.

Ball, D. L. (1995). Transforming pedagogy: Classrooms as mathematical communities: A response to Timothy Lensmire and John PryorHarvard Educational Review, 65(4), 670–677.

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373–397.

Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education. American Educator, 16(2), 14–18, 46–47.

Mosenthal, J., & Ball, D. L. (1992). Constructing new forms of subject matter instruction: Subject matter knowledge in inservice teacher education. Journal of Teacher Education, 43(5), 361–371.

Ball, D. L., & Schroeder, T. L. (1992). Implementing the Professional Standards for Teaching Mathematics: Improving teaching, not standardizing it. The Mathematics Teacher, 85(1), 67–72.

Ball, D. L. (1991). What's all this talk about "discourse"? Arithmetic Teacher, 39 (3), 44–48.

Ball, D. L. (1990). Breaking with experience in learning to teach mathematics: The role of a preservice methods course. For the Learning of Mathematics, 10(2), 10–16.

Ball, D. L. (1990). Reflections and deflections of policy: The case of Carol Turner. Educational Evaluation and Policy Analysis, 12(3), 263–275.

Ball, D. L. (1990). Prospective elementary and secondary teachers' understandings of division. Journal for Research in Mathematics Education, 21(2), 132–144.

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal, 90(4), 449–466.

Cohen, D. K., & Ball, D. L. (1990). Policy and practice: An overview. Educational Evaluation and Policy Analysis, 12(3), 347–353.

Cohen, D. K., & Ball, D. L. (1990). Relations between policy and practice: A commentaryEducational Evaluation and Policy Analysis, 12(3), 249–256.

Friel, S. N., Ball, D. L., Cooney, T. J., & Lappan, G. (1990). Envisioning change in the practice of mathematics teaching: The NCTM's Professional Standards for Teaching Mathematics. School Science and Mathematics, 90(6), 510–515.

Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics8(1), 40–48.

Ball, D. L., & Feiman-Nemser, S. (1988). Using textbooks and teachers' guides: A dilemma for beginning teachers and teacher educators. Curriculum Inquiry, 18(4), 401–423.

Ball, D. L., & McDiarmid, G. W. (1988). Research on teacher learning: Studying how teachers’ knowledge changes. Action in Teacher Education, 10(2), 17–24.

Book Chapters

Ball, D. L. & Hoover, M. (2014). Transforming research to transform mathematics instruction. In Y. Li, E. A. Silver, & S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 549–557). New York: Springer.

Thames, M. H. & Ball, D. L. (2013). Making progress in mathematics education: Lessons learned—past, present, and future. In K. Leatham (Ed.), Vital directions for mathematics education research (pp. 15–44). New York: Springer.

Lewis, J., & Ball, D. L. (2009). Defining the problem of equity in teaching elementary school mathematics. In D. Featherman, M. Hall, & M. Krislov (Eds.), The next 25 years: Affirmative action and higher education in the United States and South Africa (pp. 244–258). Ann Arbor, MI: University of Michigan Press.

Ball, D. L., & Hill, H. C. (2008). Measuring teacher quality in practice. In D. H. Gitomer (Ed.), Measurement issues and assessment for teaching quality (pp. 80–98). Thousand Oaks, CA: SAGE Publications.

Ball, D. L., & Bass, H. (2008). The role of mathematics in education for democracy. In G. Fenstermacher (Series Ed.) & D. Coulter, & J. Wiens (Vol. Eds.), Yearbook of the National Society for the Study of Education: Vol. 107 (1). Why do we educate in a democratic society? (pp. 171–184). Malden, MA: Blackwell Publishing.

Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers' mathematical knowledge: What knowledge matters and what evidence counts? In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111–155). Charlotte, NC: Information Age Publishing.

Cohen, D. K. & Ball, D. L. (2007). Innovation and the problem of scale. In B. Schneider & S. McDonald (Eds.), Scale-up In education: Ideas in principle (Volume I) (pp. 19–36). Lanham, MD: Rowman & Littlefield. An earlier draft of this paper was presented at a 2003 meeting (“Conceptualizing Scale-Up”), sponsored by the Data Research and Development Center of The University of Chicago, and supported by the National Science Foundation, in connection with the Interagency Research Initiative (IERI).

Boaler, J., Ball, D. L., & Even, R. (2003). Preparing mathematics education researchers for disciplined inquiry: Learning from, in, and for practice. In A. Bishop & J. Kilpatrick (Eds.), International handbook of mathematics education (pp. 491–521). Dordrecht, Netherlands: Kluwer.

Ball, D.L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, (pp. 3–14). Edmonton, AB: CMESG/GCEDM.

Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, and D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics, (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.

Ball, D. L. (2002). What do we believe about teacher learning, and how can we learn with and from our beliefs? In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vols. 1–4) (pp. 3–19). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In Proceedings for the International Congress of Mathematicians (pp. 907–920). Beijing, China: Higher Education Press.

Cohen, D. K., Raudenbush, S. & Ball, D. L. (2002). Resources, instruction, and research. In F. Mosteller & R. Boruch (Eds.), Evidence matters: Randomized trials in education research, (pp.80–119). Washington, DC: Brookings Institution Press.

Ball, D. L. (2001). Teaching with respect to mathematics and students. In T. Wood & B. Scott Nelson (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 11–22). Hillsdale, NJ: Erlbaum.

Ball, D. L., Lubienski, S., and Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.) (pp. 433–456). New York: Macmillan.

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 the teaching and learning of mathematics (pp. 83–104). Westport, CT: Ablex.

Ball, D.L. (2000). Foreword. In M. K. Stein, M. S. Smith, M. A. Henningsen, & E. Silver, Implementing standards-based mathematics instruction: A casebook for professional development (pp. ix–xiv). New York: Teachers College Press. 

Ball, D. L. (2000). Working on the inside: Using one's own practice as a site for studying mathematics teaching and learning. In Kelly, A. & Lesh, R. (Eds.). Handbook of research design in mathematics and science education, (pp. 365–402). Dordrecht, Netherlands: Kluwer.

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.

Ball, D. L. (1999). Crossing boundaries to examine the mathematics entailed in elementary teaching. In T. Lam (Ed.), Contemporary mathematics (pp. 15–36). Providence: American Mathematical Society.

Ball, D. L. & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In G. Sykes and L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). San Francisco: Jossey Bass.

Ball, D. L. & Lampert, M. (1999). Multiples of evidence, time, and perspective: Revising the study of teaching and learning. In E. Lagemann & L. S. Shulman (Eds.), Issues in education research: Problems and possibilities (pp. 371–398). San Francisco: Jossey Bass.

Lampert, M. & Ball, D. L. (1999). Aligning teacher education with contemporary K-12 reform visions. In G. Sykes and L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 33–53). San Francisco: Jossey Bass.

Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire in trying to hear children. In B. Biddle, T. Good, & I. Goodson (Eds.), International handbook on teachers and teaching (Vol. II), (pp. 679–718). Dordrecht, Netherlands: Kluwer Press.

Ball, D. L. (1997). Developing mathematics reform: What don't we know about teacher learningbut would make good working hypotheses. In S. N. Friel & G. W. Bright (Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (pp. 77–111). Lanham, NY: University Press of America.

Comiti, C., & Ball, D. L. (1996). Preparing teachers to teach mathematics: A comparative perspective. In A. Bishop & C. Kreidt-Keitel (Eds.), International handbook on mathematics education (pp. 1123–1153). Dordrecht, Netherlands: Kluwer Press.

Ball, D. L. (1995). Connecting to mathematics as part of learning to teach. In D. Schifter (Ed.), What's happening in math class? Vol. 2: Reconstructing professional identities (pp. 36–45). New York: Teachers College Press.

Ball, D. L. (1993). Moral and intellectual, personal and professional: Restitching practice. In M. Buchmann & R. E. Floden, Detachment and concern: Conversations in the philosophy of teaching and teacher education (pp. 193–204). New York: Teachers College Press.

Ball, D. L. (1993). Preface. In D. Schifter and C. Fosnot, Reconstructing mathematics education: Stories of teachers meeting the challenge of reform (pp. xi–xii). New York: Teachers College Press.

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. Carpenter, E. Fennema, & T. Romberg, (Eds.), Rational numbers: An integration of research (pp. 157–196). Hillsdale, NJ: Erlbaum.

Ball, D. L. & Rundquist, S. (1992). Collaboration as a context for joining teacher learning with learning about teaching. In D. K. Cohen, M. W. McLaughlin, & J. E. Talbert, (Eds.), Teaching for understanding: Challenges for policy and practice (pp. 13–42). San Francisco: Jossey Bass.

Ball, D. L. (1992). The permutations project: Mathematics as a context for learning about teaching. In S. Feiman-Nemser and H. Featherstone (Eds.), Exploring teaching: Reinventing an introductory course (pp. 18–33). New York: Teachers College Press.

Ball, D. L. (1992). Teaching mathematics for understanding: What do teachers need to know about the subject matter? In M. Kennedy (Ed.), Teaching academic subjects to diverse learners (pp. 63–83). New York: Teachers College Press.

Ball, D. L. (1991). Research on teaching mathematics: Making subject matter part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Teachers’ subject matter knowledge and classroom instruction (Vol. 2) (pp. 1–48). Greenwich, CT: JAI Press.

Ball, D. L., & McDiarmid, G. W. (1990). The subject matter preparation of teachers. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 437–449). New York: Macmillan.

McDiarmid, G. W., Ball, D. L., & Anderson, C. W. (1989). Why staying ahead one chapter doesn't really work: Subject-specific pedagogy. In M. Reynolds (Ed.), The knowledge base for beginning teachers (pp. 193–205). New York: Pergamon and the American Association of Colleges of Teacher Education.

Papers and Technical Reports

Ball, D. L. & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures. Paper prepared based on keynote address at the 43rd Jahrestagung für Didaktik der Mathematik held in Oldenburg, Germany, March 1–4, 2009.

Stylianides, A. J. & Ball, D. L. (2004). Studying the mathematical knowledge needed for teaching: The case of teachers' knowledge of reasoning and proof. Paper prepared for the 2004 Annual Meeting of the American Educational Research Association, San Diego, CA, April 14, 2004.

Ball, D. L. (2001). Developing a mathematically proficient public: What are the problems, what do we know about them, and what would it take to solve them? Paper prepared for the Aspen Institute congressional conference on "Promoting Excellence in the New Economy: The Challenges to National Policy," St. Petersburg, Florida, February 16–19, 2001.

Cohen, D. K. & Ball, D. L. (1999). Instruction, capacity, and improvement. (CPRE Research Report No. RR-043). Philadelphia, PA: University of Pennsylvania, Consortium for Policy Research in Education (CPRE).

Kennedy, M. M., Ball, D. L., & McDiarmid, G. W. (1993). A study package for examining and tracking changes in teachers' knowledge (Technical Series 93-1). East Lansing, Michigan: The National Center for Research on Teacher Education.

Committee Reports

(2011-2013). Chair, Michigan Council for Educator Effectiveness.

Michigan Council for Educator Effectiveness. (2013). Building an improvement-focused system of educator evaluation in Michigan: Final recommendations. (Video of Ball's September 11, 2013, testimony before the Michigan House and Senate Education committees is available here)

(20022008). Chair, Teachers and Teaching Task Group, The National Mathematics Advisory Panel (Panel Chair, Larry R. Faulkner).

National Mathematics Advisory Panel. (2008). Foundations for success. Washington, DC: U.S. Department of Education.

National Mathematics Advisory Panel. (2008). Chapter 5: Report of the Task Group on Teachers and Teacher Education. Washington, DC: U.S. Department of Education.

(20022006). Chair, (with Ruhama Even). The Professional Education and Development of Teachers of Mathematics: The 15th ICMI Study.

Ruhama, E. & Ball D.L. (Eds.). (2009). The professional education and development of teachers of mathematics: The 15th ICMI study. New ICMI Study Series, 11.

(1999–2003). Chair, RAND Mathematics Study Panel: Improving the Quality of Educational Research and Development.

Ball, D.L. (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education. RAND Science & Technology Policy Institute.

(1999–2001). Member, Mathematics Learning Study Committee (Chair, Jeremy Kilpatrick).

National Research Council. (2001). Adding it up: Helping children learning mathematics. J. Kilpartrick, 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.

(1999–2000). Member, Commission on Improving U.S. Mathematics Education for the 21st Century (Chair, Senator John Glenn).

National Commission of Science and Mathematics Teaching for the 21st Century. (2000). Before it's too late: A report to the nation from The National Commission on Mathematics and Science Teaching for the 21st Century. Washington, DC: U.S. Department of Education.

(19891991). Chair, Mathematics Teaching Working Group, The NCTM Commission of Teaching Standards for School Mathematics (Commission Chair, Glenda Lappan).

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

Books

Even, R. & Ball D. L. (Eds.). (2009). The professional education and development of teachers of mathematics: The 15th ICMI study. New York: Springer.

Lampert, M. & Ball, D. L. (1998). Teaching, multimedia, and mathematics: Investigations of real practice. New York: Teachers College Press.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, East Lansing.

Videos of Teaching

Mathematics Teaching and Learning to Teach, University of Michigan. (2016). Naming One-Third on the Number Line.

TeachingWorks. (2015). Elementary Mathematics Laboratory (EML) Videos. (Available only to 2015 EML participants by subscription through the University of Michigan Teaching and Learning Exploratory.)

TeachingWorks. (2014). Elementary Mathematics Laboratory (EML) Videos. (Available only to 2014 EML participants by subscription through the University of Michigan Teaching and Learning Exploratory.)

Mathematics Teaching and Learning to Teach, University of Michigan. (2012). Grand Rapids Elementary Mathematics Laboratory (GREML) Videos. (Available by subscription through the University of Michigan Teaching and Learning Exploratory.)

Mathematics Teaching and Learning to Teach, University of Michigan. (2010). Mamadou-Half-Rectangle.

Mathematics Teaching and Learning to Teach, University of Michigan. (2010). BetsyProof-Start.

Mathematics Teaching and Learning to Teach, University of Michigan. (2010). SeanNumbers-Ofala.