The Best of TCM, MTMS, and MT on Goals, Tasks, and Representations

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Teaching Children Mathematics (TCM), Mathematics Teaching in the Middle School (MTMS) and Mathematics Teacher (MT)
  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematics Success for All. Reston, VA: NCTM.

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  • Smith, Margaret Schwan, and Mary Kay Stein. 1998. “Selecting and Creating Mathematical Tasks: From Research to Practice.Mathematics Teaching in the Middle School 3, no. 5 (February): 34449.

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  • Daniel Brahier, brahier@bgsu.edu, is a professor of mathematics at Bowling Green State University in Ohio and has served as a mathematics classroom teacher, school administrator, consultant, and teacher educator.

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  • Baker, Scott, Nonie Lesaux, Madhavi Jayanthi, Joseph Dimino, C. Patrick Proctor, Joan Morris, Russell Gersten, Kelly Haymond, Micheal J. Kieffer, Sylvia Linan-Thompson, and Rebecca Newman-Gonchar. 2014. Teaching Academic Content and Literacy to English Learners in Elementary and Middle School (NCEE 2014-4012). Washington, DC: What Works Clearinghouse, National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, US Department of Education. https://ies.ed.gov/ncee/wwc/PracticeGuide/19

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  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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  • Randolph, Winifred, and Verne G. Jeffers. 1974. “A New Look for the Hundred Chart.Arithmetic Teacher 21, no. 3 (March): 2038. doi:http://www.jstor.org/stable/41188490

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  • Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2016. Elementary and Middle School Mathematics: Teaching Developmentally. 9th ed. New York: Allyn and Bacon.

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  • Ball, Deborah Loewenberg, and Hyman Bass.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.

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  • Ball, Deborah Loewenberg, and Heather Hill.Measuring Teacher Quality in Practice.” In Measurement Issues and Assessment for Teaching Quality, edited by Drew H. Gitomer, (pp. 8098). Thousand Oaks, CA: Sage Publications, 2008.

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  • Gross, Kenneth, Douglas Harris, and H. W. Meyers.Serious Mathematics for K–6 Teachers Leads to Higher Student Achievement.” Unpublished manuscript.

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  • Hill, Heather C., and Deborah Loewenberg Ball.Learning Mathematics for Teaching: Results from California's Mathematics Professional Development Institutes.Journal for Research in Mathematics Education 35, no. 5 (November 2004): 33051.

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  • Lampert, Magdalene M. Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press, 2001.

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  • Rowland, Tim, Fay Turner, Peter Huckstep, and Anne Thwaites. Developing Primary Mathematics Teaching: Reflecting on Practice with the Knowledge Quartet. London: Sage Publications, 2009.

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  • Kieran, Carolyn.Concepts Associated with the Equality Symbol.Educational Studies in Mathematics 12 (August 1981): 31726.

  • Matz, Marilyn.Towards a Process Model for School Algebra Errors.” In Intelligent Tutoring Systems, edited by Derick Sleeman and John Seeley Brown, 2550. New York: Academic Press, 1982.

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  • National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics: Discussion Draft. Reston, Va.: NCTM 1998.

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  • Black, Paul, and Dylan Wiliam. 1998. “Assessment and Classroom Learning.Assessment in Education: Principles, Policy & Practice 5 (1): 775.

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  • Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

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  • Council of Chief State School Officers (CCSSO). 2008. “Attributes of Effective Formative Assessment.Paper prepared for the Formative Assessment for Teachers and Students (FAST) State Collaborative on Assessment and Student Standards (SCASS) of the Council of Chief State School Officers (CCSSO) Washington, DC: Council of Chief State School Officers. http://www.ccsso.org/Resources/Publications/Attributes_of_Effective_Formative_Assessment.html

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  • Engel, Susan. 2015. The Hungry Mind: The Origins of Curiosity in Childhood. Cambridge: Harvard University Press.

  • Gopnik, Alison, Andrew N. Meltzoff, and Patrician K. Kuhl. 1999. The Scientist in the Crib: What Early Learning Tells Us about the Mind. 1st ed. New York: William Morrow.

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  • Heritage, Margaret. 2010. Formative Assessment and Next-Generation Assessment Systems: Are We Losing an Opportunity? Washington, DC: Council of Chief State School Officers.

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  • Kahneman, Daniel. 2011. Thinking, Fast and Slow. 1st ed. New York: Farrar, Straus and Giroux.

  • Moschkovich, Judit. 2012. “Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for ELs Aligned with the Common Core.Understanding Language Conference, Stanford University, January 1314. http://ell.stanford.edu/sites/default/files/pdf/academic-papers/02-Jmoschkovich%20Math%20FINAL_bound%20with%20appendix.pdf

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  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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  • Wiliam, Dylan. 2011. Embedded Formative Assessment. Bloomington, IN: Solution Tree Press.

  • Clarke, Doug, Barbara Clarke, and Anne Roche. 2011. “Building Teachers’ Expertise in Understanding, Assessing and Developing Children's Mathematical Thinking: The Power of Task-Based, One-to-One Interviews.ZDM Mathematics Education 43 (6): 90113. doi:http://dx.doi.org/10.1007/s11858-011-0345-2

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  • Clarke, Doug, Anne Roche, and Annie Mitchell. 2007. “Year Six Fraction Understanding: A Part of the Whole Story.” In Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia, edited by Jane Watson and Kim Beswick, pp. 20716, vol. 2, Mathematics: Essential Research, Essential Practice. Hobart: MERGA.

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  • Clarke, Doug, Anne Roche, Karina Wilkie, Vince Wright, Jill Brown, Ann Downton, Marj Horne, Rose Knight, Andrea McDonough, Matthew Sexton, and Chris Worrall. 2013. “Demonstration Lessons in Mathematics Education: Teachers’ Observation Foci and Intended Changes in Practice.Mathematics Education Research Journal 25 (2): 20730. doi:http://dx.doi.org/10.1007/s13394-012-0058-z

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  • Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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  • Hiebert, James, and Diana Wearne, 1997. “Instructional Tasks, Classroom Discourse and Student Learning in Second Grade Arithmetic.American Educational Research Journal 30 (2): 393425. doi:http://dx.doi.org/10.3102/00028312030002393

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  • Hilton, Geoff, Annette Hilton, Shelley L. Dole, Merrilyn Goos, and Mia O’Brien. 2012/2013. “Reasoning Impaired: A Hands-On Activity Helps Promote Sight-Impaired Students’ Understanding of Relative and Comparative Sizes.Mathematics Teaching in the Middle School 18 (5): 28791.

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  • Lamon, Susan J. 2006. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers. 2nd ed. Mahwah, NJ: Lawrence Erlbaum.

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  • Lamon, Susan J. 2007. “Rational Numbers and Proportional Reasoning: Towards a Theoretical Framework for Research.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester, Jr., pp. 62967, vol. 1. Reston, VA: National Council of Teachers of Mathematics.

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  • Lesh, Richard, Thomas Post, and Merlyn Behr. 1988. “Proportional Reasoning.” In Number Concepts and Operations in the Middle Grades, edited by James Hiebert and Merlyn Behr, pp. 93118, vol. 2, Research Agenda for Mathematics Education. Reston, VA: National Council of Teachers of Mathematics and Mahwah, NJ: Lawrence Erlbaum.

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  • Stein, Mary Kay, and Suzanne Lane. 1996. “Instructional Tasks and the Development of Student Capacity to Think and Reason: An Analysis of the Relationship between Teaching and Learning in a Reform Mathematics Project.Educational Research and Evaluation 2 (1): 5080. doi:http://dx.doi.org/10.1080/1380361960020103

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  • Ainsworth, Shaaron. 1999. “The Functions of Multiple Representations.Computers and Education 33 (2–3): 13152.

  • Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

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  • Heinze, Aiso, Jon Star, and Lieven Verschaffel. 2009. “Flexible and Adaptive Use of Strategies and Representations in Mathematics Education.ZDM 41 (September): 53540.

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  • Kaplan, Jerome D., and Mark Moskowitz. 2000. Mathematics Problem Solving. New York: Triumph Learning.

  • Kuper, Emily G., and Patrick Kimani. 2013. “Responding to Students’ Work on a Rich Task.Mathematics Teaching in the Middle School 18 (March): 43138.

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  • Lappan, Glenda, and Diane J. Briars. 1995. “How Should Mathematics Be Taught?” In 75 Years of Progress: Prospects for School Mathematics, edited by Iris Carl, pp. 13156. Reston, VA: National Council of Teachers of Mathematics.

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  • National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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  • Schroeder, Thomas L., and Frank K. Lester. 1989. “Developing Understanding in Mathematics via Problem Solving.” In New Directions for Elementary School Mathematics, edited by Paul R. Trafton and Albert P. Shulte, pp. 3142. Reston, VA: National Council of Teachers of Mathematics.

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  • Smith, Margaret S., Elizabeth K. Hughes, Randi A. Engle, and Mary Kay Stein. 2009. “Orchestrating Discussions.Mathematics Teaching in the Middle School 14 (May): 54856.

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  • Stein, Mary Kay, and Margaret S. Smith. 1998. “Mathematical Tasks as a Framework for Reflection: From Research to Practice.Mathematics Teaching in the Middle School 3 (December): 26875.

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  • Boaler, Jo. 2013. “The Stereotypes That Distort How Americans Teach and Learn Math.The Atlantic. http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-about-math-that-hold-americans-back/281303/

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  • Burkhardt, Hugh. 2012. “Engineering Good Math Tests.Education Week 32 (6): 23.

  • Buxton, Cory A. 2006. “Creating Contextually Authentic Science in a ‘Low-Performing’ Urban Elementary School.Journal of Research in Science Teaching 43 (7): 695721. doi:ttp://dx.doi.org/10.1002/tea.20105

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  • Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

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  • Friesen, Sharon. 2013. “Discipline-Based Inquiry Rubric.Calgary, AB: University of Calgary. http://www.galileo.org/research/publications/rubric.pdf

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  • Gladwell, Malcolm. 2008. Outliers: The Story of Success. New York: Little Brown and Co.

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  • Steen, Lynn Arthur. 2001. Mathematics and Democracy: The Case for Quantitative Literacy. Washington, DC: National Council on Education and the Disciplines and the Woodrow Wilson National Fellowship Foundation.

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  • Billings, Esther M. H., Tarah L. Tiedt, and Lindsey H. Slater.Algebraic Thinking and Pictorial Growth Patterns.Teaching Children Mathematics 14 (December 2007/January 2008): 3028.

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  • Friel, Susan N., and Kimberly A. Markworth.A Framework for Analyzing Geometric Pattern Tasks.Mathematics Teaching in the Middle School 15 (August 2009): 2433.

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  • Fulton, Brad S., and Bill Lombard. The Pattern and Function Connection. Emeryville, CA: Key Curriculum Press, 2001.

  • Lee, Lesley, and Viktor Freiman.Developing Algebraic Thinking through Pattern Exploration.Mathematics Teaching in the Middle School 11 (May 2006): 42833.

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  • Peitgen, Heinz-Otto, Hartmut Jürgens, Dietmar Saupe, Evan Maletsky, Terry Perciante, and Lee Yunker. Fractals for the Classroom: Strategic Activities Volume One. New York: NCTM and Springer-Verlag, 1991.

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  • Phillips, Elizabeth. Patterns and Functions. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5–8. Reston, VA: National Council of Teachers of Mathematics, 1991.

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  • Rivera, Ferdinand.Visualizing as a Mathematical Way of Knowing: Understanding Figural Generalization.Mathematics Teacher 101 (August 2007): 6975.

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  • Rivera, Ferdinand D., and Joanne Rossi Becker.Figural and Numerical Modes of Generalizing in Algebra.Mathematics Teaching in the Middle School 11 (November 2005): 198203.

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  • Schaaf, Oscar, Richard Brannan, Mary-Ann Debrick, Judith Johnson, Glenda Kimerling, Scott McFadden, Jill Mc Kenney, and Mary Ann Todd. Problem Solving in Mathematics: Grade 9 Algebra. Palo Alto, CA: Dale Seymour Publications, 1983.

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  • Smith, Margaret S., Amy F. Hillen, and Christy L. Catania.Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms.Mathematics Teaching in the Middle School 13 (August 2007): 3844.

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  • National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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  • Rasmussen Chris, Michelle Zandieh, Karen King, and Anne Teppo. 2005. “Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking.” Mathematical Thinking and Learning 7 (1): 5173.

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  • Leitze, Annette Ricks, and Nancy A. Kitt.Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts.Mathematics Teacher 93 (2000): 46266, 520.

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  • Shulman, Lee S.Those Who Understand: Knowledge Growth in Teaching.Educational Researcher 15, no. 2 (1986): 414.

  • Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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  • National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

  • Schroeer, Dietrich. 1984. Science, Technology, and the Nuclear Arms Race. New York: Wiley.

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  • National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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