Three approaches to the Doughnut task highlight how representing functions in multiple ways can support student understanding in interpreting key features of functions within a context.
Brenner, Mary E., Richard E.Mayer, BryanMoseley, TheresaBrar, RichardDurán, Barbara SmithReed, and DavidWebb.1997. “Learning by Understanding: The Role of Multiple Representations in Learning Algebra.” American Educational Research Journal34, no. 4 (Winter): 663–89.
Brenner, Mary E., Richard E.Mayer, BryanMoseley, TheresaBrar, RichardDurán, Barbara SmithReed, and DavidWebb.1997. “Learning by Understanding: The Role of Multiple Representations in Learning Algebra.” American Educational Research Journal34, no. 4 (Winter): 663–89.)| false
Garofalo, Joe, and Christine P.Trinter.2012. “Tasks That Make Connections through Representations.” Mathematics Teacher106, no. 4 (November): 302–7.
Garofalo, Joe, and Christine P.Trinter.2012. “Tasks That Make Connections through Representations.” Mathematics Teacher106, no. 4 (November): 302–7.)| false
Just, Angela, and Jennifer D.Cribbs.2020. “Focusing on Visual Representations in Mathematics.” Mathematics Teacher: Learning and Teaching PK–12113, no. 12 (December): e100–e106.
Just, Angela, and Jennifer D.Cribbs.2020. “Focusing on Visual Representations in Mathematics.” Mathematics Teacher: Learning and Teaching PK–12113, no. 12 (December): e100–e106.)| false
Lloyd, Gwendolyn, SybillaBeckmann, Rose MaryZbiek, and ThomasCooney.2010. Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9–12. Reston, VA: National Council of Teachers of Mathematics.
Lloyd, Gwendolyn, SybillaBeckmann, Rose MaryZbiek, and ThomasCooney.2010. Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9–12. Reston, VA: National Council of Teachers of Mathematics.)| false
Lobato, Joanne, AmyEllis, and Rose MaryZbiek.2010. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades6–8. Reston, VA: National Council of Teachers of Mathematics.
Lobato, Joanne, AmyEllis, and Rose MaryZbiek.2010. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades6–8. Reston, VA: National Council of Teachers of Mathematics.)| false
National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO).2010. Common Core State Standards for Mathematics. Washington, DC: NGA Center and CCSSO. http://www.corestandards.org.
National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO).2010. Common Core State Standards for Mathematics. Washington, DC: NGA Center and CCSSO. http://www.corestandards.org.)| false
Piez, Cynthia M., and Mary H.Voxman.1997. “Multiple Representations—Using Different Perspectives to Form a Clearer Picture.” Mathematics Teacher90, no. 2 (February): 164–66.
Piez, Cynthia M., and Mary H.Voxman.1997. “Multiple Representations—Using Different Perspectives to Form a Clearer Picture.” Mathematics Teacher90, no. 2 (February): 164–66.)| false
Ronau, Robert N., DanMeyer, and TerryCrites.2014. Putting Essential Understanding of Functions into Practice in Grades 9–12. Reston, VA: National Council of Teachers of Mathematics.
Ronau, Robert N., DanMeyer, and TerryCrites.2014. Putting Essential Understanding of Functions into Practice in Grades 9–12. Reston, VA: National Council of Teachers of Mathematics.)| false
Smith, Margaret S., Amy F.Hillen, and Christy L.Catania.2007. “Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms.” Mathematics Teaching in the Middle School13, no. 1 (August): 38–44.
Smith, Margaret S., Amy F.Hillen, and Christy L.Catania.2007. “Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms.” Mathematics Teaching in the Middle School13, no. 1 (August): 38–44.)| false
Van den Kieboom, Leigh A., and Marta T.Magiera.2012. “Cultivating Algebraic Representations.” Mathematics Teaching in the Middle School17, no. 6 (February): 352–57.
Van den Kieboom, Leigh A., and Marta T.Magiera.2012. “Cultivating Algebraic Representations.” Mathematics Teaching in the Middle School17, no. 6 (February): 352–57.)| false
Wilkie, Karina J.2016. “Students’ Use of Variables and Multiple Representations in Generalizing Functional Relationships prior to Secondary School.” Educational Studies in Mathematics93, no. 3 (November): 333–61.
Wilkie, Karina J.2016. “Students’ Use of Variables and Multiple Representations in Generalizing Functional Relationships prior to Secondary School.” Educational Studies in Mathematics93, no. 3 (November): 333–61.)| false