Catalyzing Change in Early Childhood and Elementary Mathematics Task Force and Writing Team
in Catalyzing Change in Early Childhood and Elementary Mathematics Initiating Critical Conversations
Free access

DeAnn Huinker, Chair

University of Wisconsin-Milwaukee

Milwaukee, Wisconsin

 

Anne Marie Marshall

Lehman College, City University of New York

Bronx, New York

 

Nicole Rigelman

Portland State University

Portland, Oregon

 

Cathery Yeh

Chapman University

Orange City, California

 

David Barnes, Staff Liaison

National Council of Teachers of Mathematics

Reston, Virginia

CONTRIBUTING WRITERS

Robert Q. Berry III

University of Virginia

Charlottesville, Virginia

 

Trena Wilkerson

Baylor University

Waco, Texas

NCTM BOARD OF DIRECTORS

Robert Q. Berry III, President

University of Virginia

Charlottesville, Virginia

 

Trena L. Wilkerson, President-Elect

Baylor University

Waco, Texas

 

Ken Krehbiel, Executive Director

Reston, Virginia

 

Sarah Bush

University of Central Florida

Orlando, Florida

 

Linda Davenport

Boston Public Schools

Boston, Massachusetts

 

David Ebert

Oregon High School

Oregon, Wisconsin

 

Lorie Huff

Fayetteville Public School District

Fayetteville, Arkansas

 

DeAnn Huinker

University of Wisconsin–Milwaukee

Milwaukee, Wisconsin

 

Beth Kobett

Stevenson University

Stevenson, Maryland

 

Carol Matsumoto

Resource Teacher (Retired)

Winnipeg, Canada

 

Jennifer Outzs

Seminole Middle School

Seminole, Florida

 

Jeff Shih

University of Nevada

Las Vegas, Nevada

 

Jason Slowbe

Great Oak High School

Temecula, California

 

Daniel J. Teague

North Carolina School of Science and Mathematics

Durham, North Carolina

 

Denise Walston

Council of Greater City Schools

Washington, District of Columbia

  • Collapse
  • Expand

  • Adair, Jennifer Keys. 2015. The Impact of Discrimination on the Early Schooling Experiences of Children from Immigrant Families. Washington, DC: Migration Policy Institute.

    • Search Google Scholar
    • Export Citation

  • Adams, Melissa. 2018. “I Can Solve All the Problems”: Latinx Students (Re)Write Their Math Stories. In Re-humanizing Mathematics for Black, Indigenous, and Latinx Students, edited by Imani Goffney and Rochelle Gutiérrez, pp. 121–24. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Admad, Farah Z and Ulrich Boser. 2014. America’s Leaky Pipeline for Teachers of Color: Getting More Teachers of Color into the Classroom. Washington, DC: Center for American Progress.

    • Search Google Scholar
    • Export Citation

  • Aguirre, Julia, Beth Herbel-Eisenmann, Sylvia Celedon-Pattichis, Marta Civil, Trena Wilkerson, Michelle Stephan, Stephen Pape, and Douglas H. Clements. 2017. “Equity Within Mathematics Education Research as a Political Act: Moving from Choice to Intentional Collective Professional Responsibility.” Journal for Research in Mathematics Education 48, no. 2 (March): 124–47.

    • Search Google Scholar
    • Export Citation

  • Aguirre, Julia, Karen Mayfield-Ingram, and Danny Martin. 2013. The Impact of Identity in K–8 Mathematics: Rethinking Equity-based Practices. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Akay, Hayri, and Nihat Boz. 2009. “Prospective Teachers’ Views About Problem-Posing Activities.” Procedia-Social and Behavioral Sciences 1, no. 1 (December): 1192–98.

    • Search Google Scholar
    • Export Citation

  • Almeida, Rut, and Alicia Bruno. 2017. “Establishing Profiles on the Use of Number Sense.” Journal of Research in Mathematics Education 6, no. 1 (February): 56–84.

    • Search Google Scholar
    • Export Citation

  • Anthony, Glenda, Roberta Hunter, Jodie Hunter, and Shelley Duncan. 2015. “How Ambitious is Ambitious Mathematics Teaching?” SET: Research Information for Teachers, no. 2: 45–52. http://dx.doi.org/10.18296/set.0017.

    • Search Google Scholar
    • Export Citation

  • Arcavi, Abraham. 2003. “The Role of Visual Representations in the Learning of Mathematics.” Educational Studies in Mathematics 52, no. 3 (April): 215–41.

    • Search Google Scholar
    • Export Citation

  • Artiles, Alfredo, J., Janette K. Klinger, and William F. Tate. 2006. “Representation of Minority Students in Special Education: Complicating Traditional Explanations.” Educational Researcher 35, no. 6 (August): 3–5.

    • Search Google Scholar
    • Export Citation

  • Ashcraft, Mark H. 2002. “Math Anxiety: Personal, Educational and Cognitive Consequences.” Current Directions in Psychological Science 11, no. 5 (October): 181–85.

    • Search Google Scholar
    • Export Citation

  • Ashcraft, Mark H., and Alex M. Moore. 2009. “Mathematics Anxiety and the Affective Drop in Performance.” Journal of Psychoeducational Assessment 27, no. 3 (June): 197–205.

    • Search Google Scholar
    • Export Citation

  • Association of Mathematics Teacher Educators. 2017. Standards for Preparing Teachers of Mathematics. Raleigh, NC: AMTE. www.amte.net/standards.

    • Search Google Scholar
    • Export Citation

  • Au, Wayne, and Melissa Bollow Tempel. 2012. Pencils Down: Rethinking High-stakes Testing and Accountability in Public Schools. Milwaukee, WI: Rethinking Schools Publication.

    • Search Google Scholar
    • Export Citation

  • Baek, Jae M. 2006. “Children’s Mathematical Understanding and Invented Strategies for Multidigit Multiplication.” Teaching Children Mathematics 12, no. 5 (January): 242–47.

    • Search Google Scholar
    • Export Citation

  • Ball, Deborah Loewenberg, and Francesca M. Forzani. 2011. “Building a Common Core for Learning to Teach and Connecting Professional Learning to Practice.” American Educator 35, no. 2 (Summer): 17–21.

    • Search Google Scholar
    • Export Citation

  • Baker, Carrie, Hannah Chandler, Allyn Fisher, Pam Harris, and Kim Montague. 2019. Bridges in Mathematics, 2nd ed. Salem, OR: The Math Learning Center.

    • Search Google Scholar
    • Export Citation

  • Banilower, Eric R., P. Sean Smith, Kristen A. Malzahn, Courtney L. Plumley, Evelyn M. Gordon, and Meredith L. Hayes. 2018. Report of the 2018 NSSME+ (National Survey of Science and Mathematics Education). Chapel Hill, NC: Horizon Research.

    • Search Google Scholar
    • Export Citation

  • Barnett-Clarke, Carne, William Fisher, Rick Marks, and Sharon Ross. 2010. Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Baroody, Arthur J. 1987. Children’s Mathematical Thinking. New York: Teachers College Press.

  • Baroody, Arthur J. 2000. “Does Mathematics Instruction for Three- to Five-Year-Olds Really Make Sense?” Young Children 55, no. 4 (July): 61–67.

    • Search Google Scholar
    • Export Citation

  • Baroody, Arthur J. 2006. “Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them.” Teaching Children Mathematics 13 (August): 22–31.

    • Search Google Scholar
    • Export Citation

  • Baroody, Arthur J., Meng-Lung Lai, and Kelly S. Mix. 2006. “The Development of Young Children’s Early Number and Operation Sense and Its Implications for Early Childhood Education.” In Handbook of Research on the Education of Young Children, edited by Bernard Spodek and Olivia N. Saracho, pp. 187–221. Mahwah, NJ, US: Lawrence Erlbaum Associates Publishers.

    • Search Google Scholar
    • Export Citation

  • Baroody, Arthur J., Neet Priya Bajwa, and Michael Eiland. 2009. “Why Can’t Johnny Remember the Basic Facts?” Developmental Disabilities Research Reviews 15, no. 1, 69–79.

    • Search Google Scholar
    • Export Citation

  • Barta, James Jon, Marilyn Cuch and Virginia Exton. 2012. “When Numbers Dance for Mathematics Students: Culturally Responsive Mathematics Instruction for Native Youth.” In Voices of Native American Educators, edited by Sheila T. Gregory, pp. 145–66. Lanham, MD: Lexington Books

    • Search Google Scholar
    • Export Citation

  • Bartell, Tonya G. 2013. “Learning to Teach Mathematics for Social Justice: Negotiating Social Justice and Mathematical Goals.” Journal for Research in Mathematics Education, 44, no. 1 (January): 129–63.

    • Search Google Scholar
    • Export Citation

  • Battey, Dan, and Luis Levya. 2016. “A Framework for Understanding Whiteness in Mathematics Education.” Journal of Urban Mathematics Education 9, no. 2 (December): 49–80.

    • Search Google Scholar
    • Export Citation

  • Beilock, Sian. 2011. Choke: What the Secrets of the Brain Reveal about Getting It Right When You Have To. New York: Simon and Schuster.

  • Berry, John, and Susan H. Picker. 2000. “Your Pupils’ Images of Mathematicians and Mathematics.” Mathematics in School 29, no. 2 (March): 24–26.

    • Search Google Scholar
    • Export Citation

  • Berry, Robert G., Mark Ellis, and Sherick Hughes. 2013. “Examining the History of Failed Reforms and Recent Stories of Success: Mathematics Education and Black Learners of Mathematics in the United States.” Race, Ethnicity, and Education 17, no. 4 (March): 540–68.

    • Search Google Scholar
    • Export Citation

  • Berry, Robert Q., III. 2008. “Access to Upper-Level Mathematics: The Stories of Successful African American Middle School Boys.” Journal for Research in Mathematics Education 39, no. 5 (November): 464–88.

    • Search Google Scholar
    • Export Citation

  • ________. 2016. “Informing Teachers About Identities and Agency: Using the Stories of Black Middle School Boys Who Are Successful with School Mathematics.” In More Lessons Learned From Research: Helping All Students Understand Important Mathematics, vol. 2, edited by Edward Silver and Patricia Anne Kenney, pp. 25–37. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • ________. 2019. “Examining Equitable Teaching Using the Mathematics Teaching Framework.” President’s Messages (blog), National Council of Teachers of Mathematics. www.nctm.org

    • Search Google Scholar
    • Export Citation

  • Blanton, Maria L., and James J. Kaput. 2003. “Developing Elementary Teachers’ Algebra Eyes and Ears.” Teaching Children Mathematics 10, no. 2 (October): 70–78.

    • Search Google Scholar
    • Export Citation

  • Blazar, David. 2015. “Effective Teaching in Elementary Mathematics: Identifying Classroom Practices that Support Student Achievement.” Economics of Education Review 48 (October): 16–29.

    • Search Google Scholar
    • Export Citation

  • Boaler, Jo, and Megan E. Staples. 2008. “Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School.” Teachers College Record 110, no. 3 (March): 608–45.

    • Search Google Scholar
    • Export Citation

  • Boaler, Jo. 2011. “Changing Students’ Lives through the De-Tracking of Urban Mathematics Classrooms.” Journal of Urban Mathematics Education 4, no. 1 (July): 7–14.

    • Search Google Scholar
    • Export Citation

  • Boaler, Jo. 2014. “Research Suggests that Timed Tests Cause Math Anxiety.” Teaching Children Mathematics 20, no. 8 (April): 469–74.

    • Search Google Scholar
    • Export Citation

  • ________. 2015. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching. New York: Wiley.

    • Search Google Scholar
    • Export Citation

  • Bonotto and Dal Santo. 2015. “On the Relationship Between Problem Posing, Problem Solving, and Creativity in the Primary School.” In Mathematical Problem Posing: From Research to Effective Practice, edited by Florence Mihaela Singer, Nerida F. Ellerton, and Jinfa Cai, pp. 103–23. New York: Springer.

    • Search Google Scholar
    • Export Citation

  • Booth, Julie L., and Kristie J. Newton. 2012. “Fractions: Could they Really be the Gatekeeper’s Doorman?” Contemporary Educational Psychology 37, no. 4 (October): 247–53.

    • Search Google Scholar
    • Export Citation

  • Bray, Wendy S., and Tanya Vik Blais. 2017. “Stimulating Base-ten Reasoning with Context.” Teaching Children Mathematics 24, no. 2 (October): 120–27.

    • Search Google Scholar
    • Export Citation

  • Breyfogle, M. Lynn, and Courtney M. Lynch. 2010. “van Hiele Revisited.” Mathematics Teaching in the Middle School 16, no. 4 (November): 232–38.

    • Search Google Scholar
    • Export Citation

  • Burden, Pamela. 2018. The Mathematics of Opportunity: Rethinking the Role of Math in Educational Equity. Berkeley, CA: Just Equations.

  • Bushart, Brian. 2017. “Doing Math with #ElemMathChat.” Brian Bushart’s Blog: Teaching to the Beat of a Different Drummer. https://bstockus.wordpress.com.

    • Search Google Scholar
    • Export Citation

  • Bushart, Brian. 2019. “The Annotated Numberless Word Problem.” Brian Bushart’s Blog: Teaching to the Beat of a Different Drummer. https://bstockus.wordpress.com.

    • Search Google Scholar
    • Export Citation

  • Carpenter, Thomas P., Elizabeth Fennema, Megan L. Franke, Linda Levi, and Susan B. Empson. 2015. Children’s Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • Carpenter, Thomas, Megan L. Franke, and Linda Levi. 2003. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • CAST. 2017. Universal Design for Learning Guidelines Version 2.0. Wakefield, MA: Author.

  • Chao, Theodore, Eileen Murray, and Rochelle Gutiérrez. 2014. NCTM Equity Pedagogy Research Brief: What Are Classroom Practices That Support Equity-Based Mathematics Teaching? Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Chapin, Suzanne, Alice Koziol, Jennifer MacPherson, and Carol Rezba. 2002. Navigating through Data Analysis and Probability in Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Civil, Marta. 2007. “Building on Community Knowledge: An Avenue to Equity in Mathematics Education.” In Improving Access to Mathematics: Diversity and Equity in the Classroom, edited by Suad Nasir and Paul Cobb, pp. 105–117. New York: Teachers College Press.

    • Search Google Scholar
    • Export Citation

  • Claessens, Amy, and Mimi Engel. 2013. “How Important is Where You Start? Early Mathematics Knowledge and Later School Success.” Teachers College Record, 115, no. 6 (June): 1–29.

    • Search Google Scholar
    • Export Citation

  • Claessens, Amy, Greg Duncan, and Mimi Engel. 2009. “Kindergarten Skills and Fifth-grade Achievement: Evidence from ECLS-K.” Economics of Education Review 28, no. 4: 415–27.

    • Search Google Scholar
    • Export Citation

  • Clark, Faye B., and Constance Kamii. 1996. Identification of Multiplicative Thinking in Children in Grades 1–5. Journal for Research in Mathematics Education 27, no.1 (January): 41–51.

    • Search Google Scholar
    • Export Citation

  • Clarke, Barbara, Jill Cheeseman, and Doug Clarke. 2006. “The Mathematical Knowledge and Understanding Young Children Bring to School.” Mathematics Education Research Journal 18, no. 1 (May): 78–102.

    • Search Google Scholar
    • Export Citation

  • Clements, Douglas H. 1999. “Subitizing: What Is It? Why Teach It?” Teaching Children Mathematics 5, no. 7 (March): 400–05.

  • Clements, Douglas H., and Julie Sarama. 2000. “Young Children’s Ideas About Geometric Shapes.” Teaching Children Mathematics 6, no. 8 (April): 482–88.

    • Search Google Scholar
    • Export Citation

  • ________. 2007. “Early Childhood Mathematics Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 461–555. Charlotte, NC: Information Age.

    • Search Google Scholar
    • Export Citation

  • ________. 2014. Learning and Teaching Early Math: The Learning Trajectory Approach, 2nd ed. New York: Routledge.

  • Clements, Douglas H., Julie Sarama, and Beth L. MacDonald. 2019. “Subitizing: The Neglected Quantifier.” In Constructing Number, edited by Norton Anderson and Martha W. Alibali, pp. 13–45. New York: Springer.

    • Search Google Scholar
    • Export Citation

  • Coburn, Cynthia E., Kelly McMahon, Graciela Borsato, Abigail Stein, Natalie Jou, Seenae Chong, Rebekah LeMahieu, Megan Franke, Sonia Ibarra, and Deborah Stipek. 2018. Fostering Pre–K to Elementary Alignment and Continuity in Mathematics in Urban School Districts: Challenges and Possibilities. Stanford, CA: Stanford University Policy Analysis for California Education.

    • Search Google Scholar
    • Export Citation

  • Cohen-Vogel, Lora. 2011. “‘Staffing to the Test’: Are Today’s School Personnel Practices Evidence Based?” Educational Evaluation and Policy Analysis 33, no. 4 (December): 483–505.

    • Search Google Scholar
    • Export Citation

  • Confrey, Jere. 2010. “‘Both And’—Equity and Mathematics: A Response to Martin, Gholson, and Leonard.” Journal of Urban Mathematics Education 3, no. 2 (December): 25–33.

    • Search Google Scholar
    • Export Citation

  • Cooper, Linda L. and Ming C. Tomayko. 2011. “Understanding Place Value.” Teaching Children Mathematics 17, no. 9 (May): 558–67.

  • Curcio, Frances R. 1987. “Comprehension of Mathematical Relationships Expressed in Graphs.” Journal for Research in Mathematics Education 18, no. 5 (November): 382–93.

    • Search Google Scholar
    • Export Citation

  • Cvencek, Dario, Manu Kapur, and Andrew N. Meltzoff. 2015. “Math Achievement, Stereotypes, and Math Self-concepts Among Elementary-School Students in Singapore.” Learning and Instruction 39, no. 1 (May): 1–10.

    • Search Google Scholar
    • Export Citation

  • Davenport, Linda Ruiz, and Connie S. Henry. 2019. “Strategies for Building Authentic Fact Fluency.” In No More Fact Frenzy, edited by Linda Ruiz Davenport, Connie S. Henry, Douglas H. Clements, and Julie Sarama, pp. 43–88: Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • Delpit, Lisa D. 2012. Multiplication is for White People: Raising Expectations for Other People’s Children. New York: The New Press.

  • Devlin, Keith. 2012. “Patterns? What Patterns?” Devlin’s Angle (blog), Mathematical Association of America. www.devlinsangle.blogspot.com.

    • Search Google Scholar
    • Export Citation

  • Dougherty, Barbara J., Alfinio Flores, Everett Louis, and Catherine Sophian. 2010. Developing Essential Understanding of Number and Numeration: Pre-K–Grade2. Reston: VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Drake, Corey, James P. Spillane, and Kimberly Hufferd-Ackles. 2001. “Storied Identities: Teacher Learning and Subject-Matter Context.” Journal of Curriculum Studies 33, no. 1 (November): 1–23.

    • Search Google Scholar
    • Export Citation

  • Drefs, Michelle, and Lissa D’Amour. 2014. “The Application of Ambiguous Figures to Mathematics: In Search of the Spatial Components of Number.” Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education. Vancouver, BC: PME.

    • Search Google Scholar
    • Export Citation

  • Duncan, Greg J., Chantelle J. Dowsett, Amy Claessens, Katherine Magnuson, Aletha C. Huston, Pamela Klebanov, Linda S. Pagani, Leon Feinstein, Mimi Engel, Jeanne Brooks-Gunn, Holly Sexton, Kathryn Duckworth, and Crista Japel. 2007. “School Readiness and Later Achievement.” Developmental Psychology 43, no. 6 (November): 1428–46.

    • Search Google Scholar
    • Export Citation

  • Dweck, Carol S. 2006. Mindset: The New Psychology of Success. New York: Random House.

  • Dyson, Nancy I., Nancy C. Jordan, Jessica Rodrigues, Christina Barbieri, and Luke Rinne. 2018. “A Fraction Sense Intervention for Sixth Graders With or At Risk for Mathematics Difficulties.” Remedial and Special Education (November): 1–11.

    • Search Google Scholar
    • Export Citation

  • Education Trust. 2015. Checking In: Do Classroom Assignments Reflect Today’s Higher Standards? Washington, DC: The Education Trust.

  • Eisenhart, Margaret, Hilda Borko, Robert Underhill, Catherine Brown, Doug Jones, and Patricia Agard. 1993. “Conceptual Knowledge Falls through the Cracks: Complexities of Learning to Teach Mathematics for ­Understanding.” Journal for Research in Mathematics Education 24, no. 1 (­January): 8–40.

    • Search Google Scholar
    • Export Citation

  • Ellis, Mark W. 2008. “Leaving No Child Behind Yet Allowing None Too Far Ahead: Ensuring (In)Equity in Mathematics Education Through the Science of Measurement and Instruction.” Teachers College Record 110, no. 6 (February): 1330–56.

    • Search Google Scholar
    • Export Citation

  • Espinosa, Linda M. 2013. Early Education for Dual Language Learners: Promoting School Readiness and Early School Success. Washington, DC: Migration Policy Institute. www.migrationpolicy.org.

    • Search Google Scholar
    • Export Citation

  • Falkner, Karen P., Linda Levi, and Thomas P. Carpenter. 1999. Children’s Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics 6, no. 4 (December): 232–36.

    • Search Google Scholar
    • Export Citation

  • Farmer, Gabrielle, Brian N. Verdine, Kelsey Lucca, ToriAnne Davies, Raissa Dempsey, Kathryn Hirsh-Pasek, and Roberta M. Golinkoff. 2013. “Putting the Pieces Together: Spatial Skills at Age 3 Predict to Spatial and Math Performance at Age 5.” Presented at the Society for Research in Child Development, Seattle, WA.

    • Search Google Scholar
    • Export Citation

  • Featherstone, Heather, Sandra Crespo, Lisa Jilk, Joy Oslund, Amy Parks and Marcy Woods. 2011. Smarter Together! Collaboration and Equity in the Elementary Math Classroom. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Flores, Alfinio. 2007. “Examining Disparities in Mathematics Education: Achievement Gap or Opportunity Gap?” The High School Journal 91, no. 1 (October): 29–42.

    • Search Google Scholar
    • Export Citation

  • Foreman, Linda, and Albert Bennett. 1995. Visual Mathematics, Course 1. Salem, OR: Math Learning Center.

  • Fosnot, Catherine Twomey, and Maarten Dolk. 2001. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • Frankenstein, Marilyn. 1983. “Critical Mathematics Education: An Application of Paulo Freire’s Epistemology.” Journal of Education 165, no. 4 (Fall): 315–39.

    • Search Google Scholar
    • Export Citation

  • Franklin, Christine A., and Denise S. Mewborn. 2008. “Statistics in the Elementary Grades: Exploring Distributions of Data.” Teaching Children Mathematics 15, no. 1 (August): 10–16.

    • Search Google Scholar
    • Export Citation

  • Bargagliotti, Anne, Christine Franklin, Pip Arnold, Rob Gould, Leticia Perez, and Denise Spangler. 2020. Pre-K-12 Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report II. Alexandria, VA: American Statistical Association and Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Freeman, Daniel W. and Theresa A. Jorgensen. 2015. “Moving Beyond Brownies and Pizza.” Teaching Children Mathematics 21, no. 7 (March): 413–20.

    • Search Google Scholar
    • Export Citation

  • Freire, Paulo. 2000/1970. Pedagogy of the Oppressed. 30th Anniversary Edition. Translated from Spanish by Myra Bergman Ramos. New York: Continuum.

    • Search Google Scholar
    • Export Citation

  • Frye, Douglas F., Arthur J. Baroody, Margaret Burchinal, Sharon M. Carver, Nancy C. Jordan, and Judy McDowell. 2013. Teaching Math to Young Children: A Practice Guide (NCEE 2014–4005). Washington, DC: Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/PracticeGuides

    • Search Google Scholar
    • Export Citation

  • Fuchs, Lynn S. Robin F. Schumacher, Jessica Long, Jessica Namkung, Carol L. Hamlett, Paul T. Cirino, Nancy C. Jordan, Robert Siegler, Russell Gersten, and Paul Changas. 2013. “Improving At-risk Learners’ Understanding of Fractions.” Journal of Educational Psychology 105, no. 3 (August): 683–700.

    • Search Google Scholar
    • Export Citation

  • Fuller, Bruce and Cynthia Garcia Coll. 2010. “Learning from Latinos: Contexts, Families, and Child Development in Motion.” Developmental Psychology 46, no. 3 (May): 559–65.

    • Search Google Scholar
    • Export Citation

  • Fuson, Karen C., Mindy Kalchman, and John D. Bransford. 2005. “Mathematical Understanding: An Introduction.” In How Students Learn: History, Mathematics, and Science in the Classroom, edited by M. Suzanne Donovan and John D. Bransford, 217–526. Washington, DC: National Academies Press.

    • Search Google Scholar
    • Export Citation

  • González, Norma, Luis C. Moll, and Cathy Amanti. 2005. Funds of Knowledge: Theorizing Practices in Households, Communities, and Classrooms. London, England: Routledge.

    • Search Google Scholar
    • Export Citation

  • Gresalfi, Melissa, and Paul Cobb. 2011. “Negotiating Identities for Mathematics Teaching in the Context of Professional Development.” Journal for Research in Mathematics Education 42, no. 3 (May): 270–304.

    • Search Google Scholar
    • Export Citation

  • Grissom, Jason A., Demetra Kalogrides and Suanna Loeb. 2017. “Strategic Staffing: How Performance Pressures Affect the Distribution of Teachers Within Schools and Resulting Student Achievement.” American Educational Research Journal 54, no. 6 (July): 1079–1116.

    • Search Google Scholar
    • Export Citation

  • Gutiérrez, Rochelle and Ezekiel Dixon-Román. 2011. “Beyond Gap Gazing: How Can Thinking About Education Comprehensively Help Us (Re)Envision Mathematics Education?” In Mapping Equity and Quality in Mathematics Education (Volume 1), edited by Bill Atweh, Mellony Graven, Walter Secada and Paola Valero, pp. 21–34. Springer: Netherlands.

    • Search Google Scholar
    • Export Citation

  • Gutiérrez, Rochelle, and Sonya Irving. 2012. “Student Centered Learning: Latino/a, Blacks, and Mathematics.” A white paper commissioned by the Nellie Mae Foundation Boston: Nellie Mae. https://files.eric.ed.gov/fulltext/ED561299.pdf.

    • Search Google Scholar
    • Export Citation

  • Gutiérrez, Rochelle. 2018. “The Need to Rehumanize Mathematics.” In Re-Humanizing Mathematics for Black, Indigenous, and Latinx Students, edited by Imani M. Goffney and Rochelle Gutiérrez, pp. 1–10. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Gutstein, Eric. 2006. Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice. New York: Routledge.

  • Gutstein, Eric. 2012. “Connecting Community, Critical, and Classical Knowledge in Teaching Mathematics for Social Justice.” In Alternative Forms of Knowing (in) Mathematics, edited by Swapna Mukhopadhytay and Wolf-Michael Roth, pp. 299–311. Leiden, Netherlands: Brill.

    • Search Google Scholar
    • Export Citation

  • Henry, Valerie J., and Richard S. Brown. 2008. “First-Grade Basic Facts: An Investigation into Teaching and Learning of an Accelerated, High-Demand Memorization Standard.” Journal for Research in Mathematics Education 39, no. 2 (March): 153–83.

    • Search Google Scholar
    • Export Citation

  • Hiebert, James, and Diana Wearne. 1993. “Instructional Tasks, Classroom Discourse, and Students’ Learning in Second-Grade Arithmetic.” American Educational Research Journal 30, no. 2 (June): 393–425.

    • Search Google Scholar
    • Export Citation

  • Howden, Hilde. 1989. “Teaching Number Sense.” Arithmetic Teacher 36, No. 6 (February): 6–11

  • Hufferd-Ackles, Kimberly, Karen C. Fuson, and Miriam Gamoran Sherin. 2015. “Describing Levels and Components of a Math-Talk Learning Community.” In More Lessons Learned From Research, Volume 1: Useful and Usable Research Related to Core Mathematical Practices, pp. 125–34. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Huinker, DeAnn, and Melissa Hedges. (in development). Early Math: Teaching with Learning Trajectories. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Huinker, DeAnn, and Victoria Bill. 2017. Taking Action: Implementing Effective Mathematics Teaching Practices, Grades K–5. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Huinker, DeAnn. 1998. “Letting Fraction Algorithms Emerge Through Problem Solving.” In Teaching and Learning of Algorithms in School Mathematics edited by Lorna Morrow, pp. 170–82. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • ________. 2013. “Dimensions of Fraction Operation Sense.” In Defining Mathematics Education, edited by Francis Fennell and William Speer, pp. 373–80. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • ________. 2015. “Representational Competence: A Renewed Focus for Classroom Practice in Mathematics.” Wisconsin Teacher of Mathematics 67, no. 2 (Spring): 4–8.

    • Search Google Scholar
    • Export Citation

  • Huinker, DeAnn, and Beth Schefelker. 2019. “Taking Action: Implementing Effective Mathematics Teaching in the Elementary Grades.” Presentation at the Annual Meeting of the National Council of Teachers of Mathematics, San Diego, CA, April 4, 2019.

    • Search Google Scholar
    • Export Citation

  • Humphreys, Cathy, and Ruth Parker. 2015. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4–10. New York: Stenhouse Publishers.

    • Search Google Scholar
    • Export Citation

  • Immordino-Yang, M. H. 2015. Emotions, Learning, and the Brain: Exploring the Educational Implications of Affective Neuroscience. New York: Norton.

    • Search Google Scholar
    • Export Citation

  • Immordino-Yang, Mary Helen, Linda Darling-Hammond, and Christina Krone. 2018. The Brain Basis for Integrated Social, Emotional, and Academic Development. Washington, DC: The Aspen Institute.

    • Search Google Scholar
    • Export Citation

  • Jacob, Simon N., and Andreas Nieder. 2009. “Notation-Independent Representation of Fractions in the Human Parietal Cortex.” Journal of Neuroscience 29, no. 14 (April): 4652–57.

    • Search Google Scholar
    • Export Citation

  • Jones, Dustin L., Megan Brown, Alisha Dunkle, Lindsay Hixon, Nicole Yoder, and Zach Silbernick. 2015. “The Statistical Content of Elementary School Mathematics Textbooks.” Journal of Statistics Education 23, no. 3: 1–22.

    • Search Google Scholar
    • Export Citation

  • Kagan, Sharon L, Jude Carroll, James P. Comer and Catherine Scott-Little. 2006. “Alignment: A Missing Link in Early Childhood Transitions.” Young Children 61, no. 5 (September), 26–30.

    • Search Google Scholar
    • Export Citation

  • Kamii, Constance. 1996. “Place Value: An Explanation of its Difficulty and Educational Implications for the Primary Grades.” Journal of Research in Childhood Education 1, no. 2 (February): 75–86.

    • Search Google Scholar
    • Export Citation

  • Kanold, Timothy D. and Mathew R. Larson. 2015. Beyond the Common Core: A Handbook for Mathematics in a PLC at Work. Bloomington, IN: Solution Tree Press.

    • Search Google Scholar
    • Export Citation

  • Kaput, James J. 2008. “What is Algebra? What is Algebraic Reasoning?” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 5–18. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Karp, Karen S., and Barbara Dougherty. 2019. Putting Essential Understanding of Number and Numeration into Practice: Pre-K–Grade 2. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Karp, Karen S., Sarah B. Bush, and Barbara J. Dougherty. 2019. “Avoiding the Ineffective Keyword Strategy.” Teaching Children Mathematics 25, no. 7 (May): 428–35.

    • Search Google Scholar
    • Export Citation

  • Kisker, Ellen Eliason, Jerry Lipka, Barbara L. Adams, Anthony Rickard, Dora Andrew-Ihrke, Eva Evelyn Yanez, and Ann Millard. 2012. “The Potential of a Culturally-Based Supplemental Mathematics Curriculum to Improve the Mathematics Performance of Alaska Native and Other Students.” Journal for Research in Mathematics Education 43, no. 1 (January): 75–113.

    • Search Google Scholar
    • Export Citation

  • Kling, Gina, and Jennifer M. Bay-Williams. 2015. “Three Steps to Mastering Multiplication Facts.” Teaching Children Mathematics 21, no. 9 (May): 549–59.

    • Search Google Scholar
    • Export Citation

  • Knudson, Joel. 2019. Pursuing Equity and Excellence in Mathematics: Course Sequencing and Placement in San Francisco. San Mateo, CA: California Collaborative on District Reform.

    • Search Google Scholar
    • Export Citation

  • Knuth, Eric J., Ana C. Stephens, Nicole M. McNeil, and Martha W. Alibali. 2006. “Does Understanding the Equal Sign Matter? Evidence from Solving Equations.” Journal for Research in Mathematics Education 37, no. 4 (July): 297–312.

    • Search Google Scholar
    • Export Citation

  • Ladson-Billings, Gloria. 2001. Crossing Over to Canaan: The Journey of New Teachers in Diverse Classrooms. San Francisco: Jossey-Bass.

  • Lambert, Rachel. 2015. “Constructing and Resisting Disability in Mathematics Classrooms: A Case Study Exploring the Impact of Different Pedagogies.” Educational Studies in Mathematics 89, no. 1 (January): 1–18.

    • Search Google Scholar
    • Export Citation

  • Larson, Matt. 2016. A Renewed Focus on Access, Equity, and Empowerment. President’s Message. https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/A-Renewed-Focus-on-Access,-Equity,-and-Empowerment.

    • Search Google Scholar
    • Export Citation

  • Lawler, Brian R. 2012. “The Fabrication of Knowledge in Mathematical Education: A Postmodern Ethic Toward Social Justice.” In Towards an Education for Social Justice: Ethics Applied to Education, edited by Tony Cotton, pp. 93–108. Bern, Switzerland: Peter Lang.

    • Search Google Scholar
    • Export Citation

  • Lee, Jaekyung. 2012. “College for All: Gaps Between Desirable and Actual P–12 Math Achievement Trajectories for College Readiness.” Educational Researcher 41, no. 2 (March): 43–55.

    • Search Google Scholar
    • Export Citation

  • Lee, Joon Sun, and Ginsburg, Herbert P. 2009. “Early Childhood Teachers’ Misconceptions about Mathematics Education for Young Children in the United States.” Australasian Journal of Early Childhood 34, no. 4 (September): 37–46.

    • Search Google Scholar
    • Export Citation

  • LeMahieu, Paul. 2011. “What We Need in Education is More Integrity (and Less Fidelity of Implementation.” Carnegie Common Blogs. October 11, 2011. https://www.carnegiefoundation.org/blog.

    • Search Google Scholar
    • Export Citation

  • Lesh, Richard, Tom R. Post, and Merlyn Behr. 1987. “Representations and Translations Among Representations in Mathematics Learning and Problem Solving.” In Problems of Representation in the Teaching and Learning of Mathematics, edited by Claude Janvier, pp. 33–40. Hillsdale, NJ: Lawrence Erlbaum.

    • Search Google Scholar
    • Export Citation

  • Li, Xiaobao, Meixia Ding, Mary Margaret Capraro, and Robert M. Capraro. 2008. “Sources of Differences in Children’s Understandings of Mathematical Equality: Comparative Analysis of Teacher Guides and Student Texts in China and the United States.” Cognition and Instruction 26, no. 2: 195–217.

    • Search Google Scholar
    • Export Citation

  • Lortie-Forgues, Hugues, Jing Tian, and Robert S. Siegler. 2015. “Why is Learning Fraction and Decimal Arithmetic so Difficult?” Developmental Review 38 (December): 201–21.

    • Search Google Scholar
    • Export Citation

  • Louie, Nicole. 2017. “The Culture of Exclusion in Mathematics Education and its Persistence in Equity-oriented Teaching.” Journal for Research in Mathematics Education 48, no. 5 (November): 488–519.

    • Search Google Scholar
    • Export Citation

  • Louie, Nicole L. 2018. “Culture and Ideology in Mathematics Teacher Noticing.” Educational Studies in Mathematics 97, no. 1 (June): 55–69.

    • Search Google Scholar
    • Export Citation

  • Loveless, Tom. 2013. The 2013 Brown Center Report on American Education: How Well Are American Students Learning? Washington, DC: Brookings Institution.

    • Search Google Scholar
    • Export Citation

  • Lubienski, Sarah Theule. 2007. “What Can We Do about Achievement Disparities?” Educational Leadership 65, no. 3 (November): 54–59.

    • Search Google Scholar
    • Export Citation

  • Markovits, Zvia, and Judith Sowder. 1994. “Developing Number Sense: An Intervention Study in Grade 7.” Journal for Research in Mathematics Education 25, no. 1 (January): 4–29.

    • Search Google Scholar
    • Export Citation

  • Markworth, Kim, Jenni McCool, and Jennifer Kosiak. 2016. Problem Solving in All Seasons: Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Marshall, Anne Marie, Alison Castro Superfine, and Reality Canty. 2010. “Star Students Make Connections.” Teaching Children Mathematics 17, no: 1 (August): 38–47.

    • Search Google Scholar
    • Export Citation

  • Marshall, Anne Marie. 2008. “Understanding Opportunities to Practice What We Preach: Mathematical Experiences of Mathematics Education Doctoral Students.” PhD diss., University of Maryland. ProQuest Dissertations and Theses Global (3339450).

    • Search Google Scholar
    • Export Citation

  • Martínez, José Manuel, and Laura Ramírez. 2018. “Angling for Students’ Mathematical Agency.” Teaching Children Mathematics 24, no. 7 (May): 424–31.

    • Search Google Scholar
    • Export Citation

  • Matthews, Percival, Bethany Rittle-Johnson, Katherine McEldoon, and Roger Taylor. 2012. “Measure for Measure: What Combining Diverse Measures Reveals about Children’s Understanding of the Equal Sign as an Indicator of Mathematical Equality.” Journal for Research in Mathematics Education 43, no. 3 (May): 316–50.

    • Search Google Scholar
    • Export Citation

  • Messick, Samuel. 1995. “Validity of Psychological Assessment: Validation of Inferences from Persons’ Responses and Performances as Scientific Inquiry into Score Meaning.” American Psychologist 50, no. 9 (September): 741–49.

    • Search Google Scholar
    • Export Citation

  • Mix, Kelly S. and Yi-Ling Cheng. 2012. “The Relation Between Space and Math: Developmental and Educational Implications.” In Advances in Child Development and Behavior, vol. 42, edited by Janette B. Benson, pp. 197–243. Burlington, MA: Elsevier Academic Press.

    • Search Google Scholar
    • Export Citation

  • Mohamed, Mohini, and Jacinta Johnny. 2010. “Investigating Number Sense Among Students.” Procedia Social and Behavioral Science 8 (January), 317–24.

    • Search Google Scholar
    • Export Citation

  • National Academies of Sciences, Engineering, and Medicine (NAS). 2018. How People Learn II: Learners, Contexts, and Cultures. Washington, DC: National Academies Press.

    • Search Google Scholar
    • Export Citation

  • National Association for the Education of Young Children (NAEYC). 2019. Advancing Equity in Early Childhood Education. Washington, DC: NAEYC.

    • Search Google Scholar
    • Export Citation

  • National Association for the Education of Young Children (NAEYC) and National Council of Teachers of Mathematics (NCTM). 2010. Early Childhood Mathematics: Promoting Good Beginnings. Joint Position Statement of NAEYC and NCTM [Updated]. Washington, DC: NAEYC.

    • Search Google Scholar
    • Export Citation

  • National Assessment Governing Board. 2020. Mathematics Framework for the 2025 National Assessment of Educational Progress. Washington, DC: National Assessment Governing Board, U.S. Department of Education. https://www.nagb.gov/

    • Search Google Scholar
    • Export Citation

  • National Center for Education Statistics (NCES). 2017. National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment. Washington, DC: National Center for Education Statistics, U.S. Department of Education. https://nces.ed.gov/nationsreportcard/nqt/

    • Search Google Scholar
    • Export Citation

  • ________. 2019. The Nation’s Report Card. Washington, DC: National Center for Education Statistics, U.S. Department of Education. https://www.nationsreportcard.gov.

    • Search Google Scholar
    • Export Citation

  • National Council of Supervisors of Mathematics (NCSM) and TODOS: Mathematics for ALL. 2016. Mathematics Education Through the Lens of Social Justice: Acknowledgement, Actions, and Accountability. Aurora, CO: NCSM and TODOS.

    • Search Google Scholar
    • Export Citation

  • National Council of Teachers of Mathematics (NCTM). 1980. An Agenda for Action: Recommendations for School Mathematics in the 1980s. Reston, VA: NCTM.

    • Search Google Scholar
    • Export Citation

  • ________. 1989. Curriculum and Evaluation Standards for School Mathematics. ­Reston, VA: NCTM.

  • ________. 2000. Principles and Standards in School Mathematics. Reston, VA: NCTM.

  • ________. 2006. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: NCTM.

  • ________. 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

  • ________. 2016. Large-Scale Mathematics Assessments and High-Stakes Decisions. Reston, VA: NCTM. https://www.nctm.org/positions.

  • ________. 2018. Catalyzing Change in High School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.

  • National Education Association (NEA). 2015. Research Spotlight on Academic Ability Grouping. http://www.nea.org/tools/16899.htm.

  • National Governors Association Center for Best Practice and Council for Chief State School Officers (NGA Center and CCSSO). 2010. Common Core State Standards in Mathematics. Washington, DC: NGA Center and CCSSO. http://www.corestandards.org.

    • Search Google Scholar
    • Export Citation

  • National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: National Academies Press.

  • ________. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press.

  • ________. 2006. Learning to Think Spatially: GIS as a Support System in the K–12 Curriculum. Washington, DC: National Academies Press.

    • Search Google Scholar
    • Export Citation

  • ________. 2009. Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, DC: National Academies Press.

  • ________. 2015. Transforming the Workforce for Children Birth Through Age 8: A Unifying Foundation. Washington, DC: National Academies Press.

    • Search Google Scholar
    • Export Citation

  • Neill, Monty. 2010. “A Better Way to Assess Students and Evaluate Schools.” Education Week. (June). http://www.edweek.org/ew/articles/2010/06/18/36neill.h29.html.

    • Search Google Scholar
    • Export Citation

  • Nguyen, Tutrang, Tyler W. Watts, Greg J. Duncan, Douglas H. Clements, Julie S. Sarama, Christopher Wolfe, and Mary Elaine Spitler. 2016. “Which Preschool Mathematics Competencies are Most Predictive of Fifth Grade Achievement?” Early Childhood Research Quarterly 36: 550–60.

    • Search Google Scholar
    • Export Citation

  • Nichols, Sharon L., Gene V. Glass, and David C. Berliner. 2012. “High-stakes Testing and Student Achievement: Updated Analyses with NAEP Data.” Education Policy Analysis Archives 20, no. 20 (July): 1–31.

    • Search Google Scholar
    • Export Citation

  • Oakes, Jeannie, and Marisa Saunders. 2008. Beyond Tracking: Multiple Pathways to College, Career, and Civic Participation. Cambridge, MA: Harvard Education Press.

    • Search Google Scholar
    • Export Citation

  • Ontario Ministry of Education. 2012. Paying Attention to Spatial Reasoning K–12. Toronto, ON: Queen’s Printer for Ontario. http://www.edu.gov.on.ca/eng/literacynumeracy/lnspayingattention.pdf.

    • Search Google Scholar
    • Export Citation

  • Ontario Ministry of Education. 2014. Paying Attention to Fractions: K–12. Ontario, Canada: Author. http://www.edu.gov.on.ca/eng/literacynumeracy/LNSAttentionFractions.pdf.

    • Search Google Scholar
    • Export Citation

  • Oppland-Cordell, Sarah, and Danny Bernard Martin. 2015. “Identity, Power, and Shifting Participation in a Mathematics Workshop: Latin@ Students’ Negotiation of Self and Success.” Mathematics Education Research Journal 27, no. 1 (March): 21–49.

    • Search Google Scholar
    • Export Citation

  • Parke, Carol S., and Suzanne Lane. 1997. “Learning from Performance Assessments in Math.” Educational Leadership 54, no. 4 (January): 26–29.

    • Search Google Scholar
    • Export Citation

  • Parks, Amy Noelle. 2015. Exploring Mathematics Through Play in the Early Childhood Classroom. New York: Teachers College Press and Reston, VA: NCTM.

    • Search Google Scholar
    • Export Citation

  • Parrish, Sherry D. 2011. “Number Talks Build Numerical Reasoning.” Teaching Children Mathematics 18, no. 3 (October): 198–206.

  • Perry, Rebecca R., and Catherine C. Lewis. 2017. “How Can Students Learn Fraction (De)Composition?” Teaching Children Mathematics 24, no. 1 (September): 31–39.

    • Search Google Scholar
    • Export Citation

  • Philipp, Randolph A., and Casey Hawthorne. 2015. “Unpacking Referent Units in Fraction Operations.” Teaching Children Mathematics 22, no. 4 (November): 240–47.

    • Search Google Scholar
    • Export Citation

  • Porter, Andrew, Jennifer McMaken, Jun Hwang, and Rui Yang. 2011. “Common Core Standards: The New U.S. Intended Curriculum.” Educational Researcher 40, no. 3 (April): 103–16.

    • Search Google Scholar
    • Export Citation

  • Reardon, Rick. 2011. “The Widening of the Achievement Gap Between the Rich and the Poor: New Evidence and Possible Explanations.” In Whither ­Opportunity? Rising Inequality, Schools, and Children’s Life Chances, edited by Greg J. Duncan and Richard J. Murnane, pp. 91–115. New York: Russell Sage Foundation.

    • Search Google Scholar
    • Export Citation

  • Reigosa-Crespo, Vivian, Eduardo González-Alemañy, Teresa León, Rosario Torres, Raysil Mosquera, and Mitchell Valdés-Sosa. 2013. “Numerical Capacities as Domain-Specific Predictors Beyond Early Mathematics Learning: A Longitudinal Study.” PloS One 8, no. 11 (e79711): 1–11.

    • Search Google Scholar
    • Export Citation

  • Reys, Robert, Barbara Reys, Göran Emanuelsson, Bengt Johansson, Alistair McIntosh, and Der Ching Yang. 1999. “Assessing number sense of students in Australia, Sweden, Taiwan, and the United States.” School Science and Mathematics 99, no. 2 (February): 61–70.

    • Search Google Scholar
    • Export Citation

  • Richardson, Kathy. 2012. How Children Learn Number Concepts: A Guide to the Critical Learning Phases. Bellingham, WA: Math Perspectives.

    • Search Google Scholar
    • Export Citation

  • Richland, Lindsey E., James W. Stigler, and Keith J. Holyoak. 2012. “Teaching the Conceptual Structure of Mathematics.” Educational Psychologist 47, no. 3 (July): 189–203.

    • Search Google Scholar
    • Export Citation

  • Rigelman, Nicole, and Kellie Petrick. 2014. “Student Mathematicians Developed Through Formative Assessment Cycles.” In Annual Perspectives in Mathematics Education: Using Research to Improve Instruction, edited by Karen Karp and Amy Roth-McDuffie, pp. 215–27. Reston, VA: NCTM.

    • Search Google Scholar
    • Export Citation

  • Rigelman, Nicole R. 2009. “Eliciting High-Level Student Mathematical Discourse: Relationships Between the Intended and Enacted Curriculum.” In The Role of Mathematics Discourse in Producing Leaders of Discourse: The Montana Mathematics Enthusiast Journal, edited by Libby Knott, 153–72. Charlotte, NC: Information Age.

    • Search Google Scholar
    • Export Citation

  • Rodrigues, Jessica, Nancy I. Dyson, Nicole Hansen, and Nancy C. Jordan. 2016. “Preparing for Algebra by Building Fraction Sense.” Teaching Exceptional Children 49, no. 2 (November): 134–41.

    • Search Google Scholar
    • Export Citation

  • Ross, Sharon H. 1989. “Parts, Wholes, and Place Value: A Developmental View.” The Arithmetic Teacher 36, no. 6 (February): 47–51.

  • Ross, Sharon H. 1990. “Children’s Acquisition of Place-Value Numeration Concepts: The Roles of Cognitive Development and Instruction.” Focus on Learning Problems in Mathematics 12, no. 1 (Winter): 1–17.

    • Search Google Scholar
    • Export Citation

  • Ross, Sharon R. 2002. “Place Value: Problem Solving and Written Assessment.” Teaching Children Mathematics 8, no. 7 (March): 419–24.

    • Search Google Scholar
    • Export Citation

  • Russell, Susan Jo, and Karen Economopoulos. 2008. Investigations in Number, Data, and Space, 2nd ed. Parsippany, NJ: Pearson Learning Group.

    • Search Google Scholar
    • Export Citation

  • Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • Schifter, Deborah, Reva Kasman, Virginia Bastable, and Traci Higgins. 2017. But Why Does it Work? Mathematical Argument in the Elementary Classroom. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation

  • Sarama, Julie, and Douglas H. Clements. 2009. Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. New York and London: Routledge.

    • Search Google Scholar
    • Export Citation

  • Schifter, Deborah, Stephen Monk, Susan Jo Russell, and Virginia Bastable. 2008. “Early Algebra: What Does Understanding the Laws of Arithmetic Mean in the Elementary Grades?” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 413–47. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Schoenfeld, Alan H. 2014. “What Makes for Powerful Classrooms, and How Can We Support Teachers in Creating Them? A Story of Research and Practice, Productively Intertwined.” Educational Researcher 43, no. 8 (November): 404–12.

    • Search Google Scholar
    • Export Citation

  • Schwartz, David M. 2004. How Much Is a Million? New York: Scholastic.

  • Seeley, Cathy L. 2009. Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century. Sausalito, CA: Math Solutions.

    • Search Google Scholar
    • Export Citation

  • Shah, Prachi E., Heidi M. Weeks, Blair Richards, and Niko Kaciroti. 2018. “Early Childhood Curiosity and Kindergarten Reading and Math Academic Achievement.” Pediatric Research 84, no. 3 (September): 380–86.

    • Search Google Scholar
    • Export Citation

  • Sherin, Miriam Gamoran, Victoria R. Jacobs, and Randolph A. Philipp. 2011. “Situating the Study of Teacher Noticing.” In Mathematics Teacher Noticing: Seeing Through Teachers’ Eyes, edited by Miriam Gamoran Sherin, Victoria R. Jacobs, and Randolph A. Philipp, pp. 3–13. New York: Routledge.

    • Search Google Scholar
    • Export Citation

  • Shetterly, Margot Lee, and Winifred Conkling. 2018. Hidden Figures: The True Story of Four Black Women and the Space Race. New York: Harper Collins.

    • Search Google Scholar
    • Export Citation

  • Siebert, Daniel, and Nicole Gaskin. 2006. “Creating, Naming, and Justifying Fractions.” Teaching Children Mathematics 12, no. 8 (April): 394–400.

    • Search Google Scholar
    • Export Citation

  • Siegler, Robert S., Lisa K. Fazio, Drew H. Bailey, and Xinlin Zhou. 2012. “Fractions: The New Frontier for Theories of Numerical Development.” Trends in Cognitive Sciences 17, no. 1 (January): 13–19.

    • Search Google Scholar
    • Export Citation

  • Siegler, Robert, Thomas Carpenter, Francis (Skip) Fennell, David Geary, James Lewis, Yukari Okamoto, Laurie Thompson, and Jonathan Wray. 2010. Developing Effective Fractions Instruction for Kindergarten through 8th Grade: A Practice Guide. Washington, DC: Institute of Education Sciences. https://ies.ed.gov/ncee/wwc/PracticeGuides.

    • Search Google Scholar
    • Export Citation

  • Siemon, Dianne, Margarita Breed, and Jo Virgona. 2005. “From Additive to Multiplicative Thinking: The Big Challenge of the Middle Years.” In Mathematics: Celebrating Achievement, edited by Judy Mousley, Leicha Bragg, and Coral Campbell. Melbourne: Mathematical Association of Victoria.

    • Search Google Scholar
    • Export Citation

  • Silver, Edward A. 1994. “On Mathematical Problem Posing.” For the Learning of Mathematics 14, no. 1 (February): 19–28.

  • Silver, Edward A. 1997. “Fostering Creativity Through Instruction Rich in Mathematical Problem Solving and Problem Posing.” ZDM 29, no. 3 (June): 75–80.

    • Search Google Scholar
    • Export Citation

  • Skinner, Debra, Donna Bryant, Jennifer Coffman, and Frances Campbell. 1998. “Creating Risk and Promise: Children’s and Teachers’ Co-constructions in the Cultural World of Kindergarten. The Elementary School Journal 98, no. 4 (March): 297–310.

    • Search Google Scholar
    • Export Citation

  • Sleeter, Christine E., La Vonne I. Neal and Kevin K. Kumashiro. 2015. Diversifying the Teacher Workforce: Preparing and Retaining Highly Effective Teachers. New York: Routledge.

    • Search Google Scholar
    • Export Citation

  • Small, Marian. 2018. Fun and Fundamental Math for Young Children: Building a Strong Foundation in PreK–Grade 2. New York: Teachers College Press.

    • Search Google Scholar
    • Export Citation

  • Small, Marian. 2017. Good Questions: Great Ways to Differentiate Mathematics Instruction in the Standards-Based Classroom, 3rd ed. New York: Teachers College Press, Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Sowder, Judith T. 1992. “Making Sense of Numbers in School Mathematics.” In Analysis of Arithmetic for Mathematics Teaching, edited by Gaea Leinhardt, Ralph Putnam, and Rosemary A. Hattrup, pp. 1–51. Hillsdale, NJ: Lawrence Erlbaum.

    • Search Google Scholar
    • Export Citation

  • Souto-Manning, Marina, and Christina Hanson Mitchell. 2010. “The Role of Research in Fostering Culturally-Responsive Practices in Preschool ­Classrooms.” Early Childhood Education Journal 37, no. 4 (January): 269–77.

    • Search Google Scholar
    • Export Citation

  • Starkey, Gillian S., and Bruce D. McCandliss. 2014. “The Emergence of ‘Groupitizing’ in Children’s Numerical Cognition.” Journal of Experimental Child Psychology 126 (October): 120–37.

    • Search Google Scholar
    • Export Citation

  • Stefanakis, Evangeline H. 2010. Differentiated Assessments: How to Assess the Learning Potential of Every Student. San Francisco, CA: Jossey-Bass.

    • Search Google Scholar
    • Export Citation

  • Steig, William. 2005. Which Would Your Rather Be? New York: Harper Collins.

  • 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, no. 1: 50–80.

    • Search Google Scholar
    • Export Citation

  • Stein, Mary Kay, Julia Heath Kaufman, Milan Sherman, and Amy F. Hillen. 2011. “A Challenge at the Crossroads of Policy and Practice.” Review of Educational Research 81, no. 4 (December): 453–92.

    • Search Google Scholar
    • Export Citation

  • Stigler, James W., and James Hiebert. 1999. The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. New York: Simon and Schuster.

    • Search Google Scholar
    • Export Citation

  • Stinson, David, Christopher C. Jett, and Brian A. Williams. 2013. “Counterstories from Mathematically Successful African American Male Students: Implications for Mathematics Teachers and Students.” In The Brilliance of Black Children in Mathematics, edited by Jacqueline Leonard and Danny B. Martin, pp. 221–46. Charlotte, NC: Information Age Publishing.

    • Search Google Scholar
    • Export Citation

  • Stipek, Deborah. 2017. “Playful Math Instruction in the Context of Standards and Accountability.” Young Children 72, no. 3 (July): 8–13.

    • Search Google Scholar
    • Export Citation

  • Stipek, Deborah. 2019. “Should Children Be Ready for Kindergarten—Or Should Kindergarten Be Ready for Children?” Education Week. (June). https://www.edweek.org/ew/articles/2019/03/26/should-children-be-ready-for-kindergarten–or-should.html.

    • Search Google Scholar
    • Export Citation

  • Strauss, Mark S., and Lynne E. Curtis. 1981. “Infant Perception of Numerosity.” Child Development 52, no. 4 (December): 1146–52.

  • Stylianou, Despina A., and Edward A. Silver. 2004. “The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences.” Mathematical Thinking and Learning 6, no. 4 (September): 353–87.

    • Search Google Scholar
    • Export Citation

  • Su, Francis Edward. 2017. “Mathematics for Human Flourishing.” American Mathematical Monthly 124, no. 6 (June/July): 483–93.

  • Su, Francis Edward. 2019. “The Importance of Play in Mathematics Education.” Presentation at the Annual Meeting of the National Council of Supervisors of Mathematics, San Diego, CA, April 2, 2019.

    • Search Google Scholar
    • Export Citation

  • Swinburne, Stephen R. 2010. Whose Shoes? A Shoe for Every Job. Honesdale, PA: Boyds Mills Press.

  • Szydlik, Jennifer E., Eric Kuennen, and Carol E. Seaman. 2009. “Development of an Instrument to Measure Mathematical Sophistication.” In Proceedings for the Twelfth Conference of Research in Undergraduate Mathematics Education. Raleigh, NC: Mathematical Association of American. http://sigmaa.maa.org/rume/crume2009.

    • Search Google Scholar
    • Export Citation

  • Takanishi, Ruby. 2016. First Things First! Creating the New American Primary School. New York: Teachers College Press.

  • Tan, Paulo, Rachel Lambert, Alexis Padilla, and Rob Wieman. 2018. “A Disability Studies in Mathematics Education Review of Intellectual Disabilities: Directions for Future Inquiry and Practice.” Journal of Mathematical Behavior 54 (November): 1–13.

    • Search Google Scholar
    • Export Citation

  • Terman, Lewis. M., Virgil Everett Dickson, Arthur Howard Sutherland, Raymond Franzen, Charles R. Tupper, and Grace Maxwell Fernald. 1922. Intelligence Tests and School Reorganization. Yonkers-on-Hudson, NY: World Book Company.

    • Search Google Scholar
    • Export Citation

  • Thompson, Patrick W., Randolph Philipp, and Barbara A Boyd. 1994. “Calculational and Conceptual Orientations in Teaching Mathematics.” In Professional Development for Teachers of Mathematics, edited by Douglas B. Aichele and Arthur F. Coxford, pp. 79–92. Reston, VA: National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Thornton, Carol A. 1978. “Emphasizing Thinking Strategies in Basic Fact ­Instruction.” Journal for Research in Mathematics Education 9, no. 3 (May): 214–27.

    • Search Google Scholar
    • Export Citation

  • Tierney, Cornelia, and Mary Berle-Carman. 1995. Fair Shares: Fractions. Cambridge, MA: Dale Seymour.

  • TNTP. 2018. The Opportunity Myth. New York: TNTP.

  • Torbeyns, Joke, Michael Schneider, Ziqiang Xin, and Robert S. Siegler. 2015. “Bridging the Gap: Fraction Understanding is Central to Mathematics Achievement in Students from Three Different Continents.” Learning and Instruction 37 (June): 5–13.

    • Search Google Scholar
    • Export Citation

  • Tournaki, Nelly. 2003. “The Differential Effects of Teaching Addition through Strategy Instruction Versus Drill and Practice to Students With and Without Learning Disabilities.” Journal of Learning Disabilities 36, no. 5 (September): 449–58.

    • Search Google Scholar
    • Export Citation

  • Tripathi, Preety N. 2008. “Developing Mathematical Understanding through Multiple Representations.” Mathematics Teaching in the Middle School 13, no. 8 (April): 438–45.

    • Search Google Scholar
    • Export Citation

  • Turner, Erin, Higinio Dominguez, Luz Maldonado, and Susan Empson. 2013. “English Learners’ Participation in Mathematical Discourse: Shifting Positioning and Dynamic Identities” Journal for Research in Mathematics Education 44, no. 1 (January): 199–234.

    • Search Google Scholar
    • Export Citation

  • U.S. Department of Housing and Urban Development (HUD). 2018. The 2018 Annual Homeless Assessment Report to Congress. Washington, DC: HUD.

    • Search Google Scholar
    • Export Citation

  • Vamvakoussi, Xenia, Wim Van Dooren, and Lieven Verschaffel. 2013. “Brief Report. Educated Adults are Still Affected by Intuitions about the Effect of Arithmetical Operations: Evidence from a Reaction-Time Study.” Educational Studies in Mathematics 82, no. 2 (February): 323–30.

    • Search Google Scholar
    • Export Citation

  • van Hiele, Pierre M. 1999. “Developing Geometric Thinking through Activities That Begin with Play.” Teaching Children Mathematics 5, no. 6 (February): 310–16.

    • Search Google Scholar
    • Export Citation

  • Verdine, Brian N., Roberta Michnick Golinkoff, Kathryn Hirsh-Pasek, and Nora S. Newcombe. 2014. “Finding the Missing Piece: Blocks, Puzzles, and Shapes Fuel School Readiness.” Trends in Neuroscience and Education 3, no. 1: 7–13.

    • Search Google Scholar
    • Export Citation

  • Vergnaud, Gérard. 1988. “Multiplicative Structures.” In Number Concepts and Operations in the Middle Grades, edited by James Hiebert and Merlyn Behr, pp. 141–161. Hillsdale, NJ: Lawrence Erlbaum.

    • Search Google Scholar
    • Export Citation

  • Wai, Jonathan, David Lubinski, and Camilla P. Benbow. 2009. Spatial Ability for STEM Domains: Aligning Over 50 Years of Cumulative Psychological Knowledge Solidifies its Importance. Journal of Educational Psychology 101, no. 4: 817–35.

    • Search Google Scholar
    • Export Citation

  • Walkington, Jackie. 2005. “Becoming A Teacher: Encouraging Development of Teacher Identity Through Reflective Practice.” Asia-Pacific Journal of Teacher Education 33, no. 1 (March): 53–64.

    • Search Google Scholar
    • Export Citation

  • Watanabe, Tad. 2006. “The Teaching and Learning of Fractions: A Japanese Perspective.” Teaching Children Mathematics 12, no. 7 (March): 368–74.

    • Search Google Scholar
    • Export Citation

  • Watts, Tyler W., Greg J. Duncan, Robert S. Siegler, and Pamela E. Davis-Kean. 2014. “What’s Past is Prologue: Relations Between Early Mathematics Knowledge and High School Achievement.” Educational Researcher 43, no. 7 (October): 352–60.

    • Search Google Scholar
    • Export Citation

  • Weisberg, Deena Skolnick, Kathy Hirsh-Pasek, Roberta Michnick Golinkoff, Audrey K. Kittredge, and David Klahr. 2016. “Guided Play: Principles and Practices.” Current Directions in Psychological Science 25, no. 3 (June): 177–82.

    • Search Google Scholar
    • Export Citation

  • Whitin, Phyllis, and David Whitin. 2004. New Visions for Linking Literature and Mathematics. Urbana, IL: National Council of Teachers of English.

    • Search Google Scholar
    • Export Citation

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

  • Wills, John, and Judith Haymore Sandholtz. 2009. “Constrained Professionalism: Dilemmas of Teaching in the Face of Test-Based Accountability.” Teachers College Record 111, no. 4 (February): 1065–114.

    • Search Google Scholar
    • Export Citation

  • Woleck, Kristine Reed. 2003. “Tricky Triangles: A Tale of One, Two, Three Researchers.” Teaching Children Mathematics 10, no. 1 (September): 40–45.

    • Search Google Scholar
    • Export Citation

  • Wright, Bob, Joanne Mulligan, Rita Stewart, and Janette Bobis. 1996. “Research on Early Number Learning.” In Research in Mathematics Education in Australasia 1992–1995, edited by Bill Atweh, Kay Owens, and Peter Sullivan, pp. 281–310. Australia: Mathematics Education Research Group of Australasia.

    • Search Google Scholar
    • Export Citation

  • Wright, Brian L., and Shelly L. Counsell. 2018. The Brilliance of Black Boys: Cultivating School Success in the Early Grades. New York: Teachers College Press.

    • Search Google Scholar
    • Export Citation

  • Wyborney, Steven. 2017a. “Splat!” Steve Wyborney’s Blog: I’m on a Learning ­Mission. https://www.stevewyborney.com/?p=893.

  • Wyborney, Steven. 2017b. “The Fraction Splat! Series!” Steve Wyborney’s Blog: I’m on a Learning Mission. https://www.stevewyborney.com/?p=1028.

    • Search Google Scholar
    • Export Citation

  • Yang, Der-Ching, Mao-Neng Li, and Chih-I Lin. 2008. “A Study of the Performance of 5th Graders in Number Sense and Its Relationship to Achievement in Mathematics.” International Journal of Science and Mathematics Education 6 (July): 789–807.

    • Search Google Scholar
    • Export Citation

  • Yang, Der-Ching. 2005. “Number Sense Strategies Used by Sixth Grade Students in Taiwan.” Educational Studies 31, no. 3 (September): 317–33.

    • Search Google Scholar
    • Export Citation

  • Yeh, Cathery, Mark W. Ellis, and Carolee Koehn Hurtado. 2017. Reimagining the Mathematics Classroom: Creating and Sustaining Productive Learning Environments. Reston, VA: NCTM.

    • Search Google Scholar
    • Export Citation

  • Yeh, Cathery. 2018. “Democratic Accountability in the Neoliberal Era: The Politics of Teaching and Teacher Education.” Policy Futures in Education 16, no. 6 (June) 764–80.

    • Search Google Scholar
    • Export Citation

  • Yeh, Cathery. 2019. “Countering Deficit Myths of Students with Dis/Abilities and Conceptualizing Possibilities: A Culturally Responsive and Relational Approach to Mathematics” TODOS Live! (webinar). https://vimeo.com/353856573.

    • Search Google Scholar
    • Export Citation

  • Young, C. B., Sarah Wu, and Vinod Menon. 2012. “The Neurodevelopmental Basis of Math Anxiety.” Psychological Science 23, no. 5 (May): 492–501.

    • Search Google Scholar
    • Export Citation

  • Zhang, Xiaofen, MA (Ken) Clements, and Nerida F. Ellerton. 2015. “Engaging Students With Multiple Models of Fractions.” Teaching Children Mathematics 22, no. 3 (October): 138–47.

    • Search Google Scholar
    • Export Citation

  • Zimba, Jason. 2011. Examples of Structure in the Common Core State Standards’ Standards for Mathematical Content. Tucson, AZ: Institute for Mathematics and Education, University of Arizona.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 538 116 7
PDF Downloads 614 129 8