Multiple Representations


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Multiple representations are ways to symbolize, describe and refer to the same mathematical entity. They are used to understand and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds.

Higher-order thinking


Use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills REF(2, 3, 13) (KateGillin http://ktgee24.wordpress.com/2010/07/18/multiple-representations/) (Peterhorn33 http://peterhorn33.wordpress.com/2010/07/13/response-to-article-on-multiple-representations/) (GailARice http://gailly123.wordpress.com/2010/07/19/multiple-representations-in-math/). The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are example of such mathematically sophisticated activities. REFS Estimation, another complex task, can strongly benefit from multiple representations REF(8) (aheleniak http://continuities.wordpress.com/2010/04/25/sharing-multiple-representations-of-systems/#comment-911).

Curricula that support starting from conceptual understanding, then developing procedural fluency, for example, AIMS Foundation Activities REF (9) (aheleniak http://teachy79.wordpress.com/2010/07/18/multiple-representations-on-youtube/), frequently use multiple representations.

Supporting student use of multiple representations may lead to more open-ended problems, or at least accepting multiple methods of solutions and forms of answers. Project-based learning units, such as webquests, typically call for several representations. REF (prm_arcadia http://prmarcadia.wordpress.com/2010/07/19/week-1-multiple-representation-in-math-education/).

Motivation


Some representations, such as pictures, videos and manipulatives, can be motivating because of their richness, possibilities of play, technologies involved, or connections with interesting areas of life REF(13) (rsheffler http://rsheffler.wordpress.com/2010/07/18/multiple-representations/). Tasks that involve multiple representations can sustain intrinsic motivation in mathematics through supporting higher-order thinking and problem solving.

Assessment


Tasks that involve construction, use and interpretation of multiple representations can lend themselves to rubric assessment REF (12) (semhouston1 http://semhouston1.wordpress.com/2010/07/18/multiple-representations-outcome-article/) and to other assessment types suitable for open-ended activities.

Special education and differentiated instruction


Students with special needs may be weaker in their use of some of the representations. For these students, it may be especially important to use multiple representations, for two purposes. First, including representations that currently work well for the student ensures the understanding of current mathematical topic. Second, connections among multiple representations within the same topic strengthens overall skills in using all representations, even those currently problematic. REF(2) (KateGillin http://ktgee24.wordpress.com/2010/07/18/multiple-representations/)

Using multiple representations can help differentiate instruction by addressing different learning styles REF (13, 15) (GailARice http://gailly123.wordpress.com/2010/07/19/multiple-representations-in-math/) (Snyder.Doug http://snyderdoug.blogspot.com/2010/07/following-powerpoint-presentation.html) (rpstock http://bobbo1.wordpress.com/2010/07/13/multiple-representations-in-math/)

Qualitative and quantitative reasoning


Visual representations, manipulatives, gestures, and to some degree grids can support qualitative reasoning about mathematics. Instead of only emphasizing computational skills, multiple representations can help students make the conceptual shift to the meaning and use of mathematical entities, and to develop algebraic thinking. By focusing more on the conceptual representations of algebraic problems students will become more capable problem solvers. REF(3) (Peterhorn33 http://peterhorn33.wordpress.com/2010/07/13/response-to-article-on-multiple-representations/)

NCTM Representations standard


National Council of Teachers of Mathematics has a standard dealing with multiple representations. In part, it reads REF (10) (prm_arcadia http://prmarcadia.wordpress.com/2010/07/19/week-1-multiple-representation-in-math-education/)
"Instructional programs should enable all students to do the following:
  • Create and use representations to organize, record, and communicate mathematical ideas
  • Select, apply, and translate among mathematical representations to solve problems
  • Use representations to model and interpret physical, social, and mathematical phenomena"

Four most frequent school math representations


While there are many representations used in mathematics, the secondary curricula heavily favor numbers (often in tables), formulas, graphs and words. REF (11) (prm_arcadia http://ctlonline.org/blog/?p=357&cpage=1#comment-393)

Systems of manipulatives


Several curricula use extensively developed systems of manipulatives and the corresponding representations. For example, Cuisinaire rods REF(14) ((Snyder.Doug http://mathematics24x7.ning.com/forum/topics/multiple-representations?id=2957074%3ATopic%3A4701&page=2#comments), Montessori beads REF, and Algebra Tiles REF.

Use of technology


Use of computer tools to create and to share mathematical representations opens several possibilities. It allows to link multiple representations dynamically. For example, changing a formula can instantly change the graph, the table of values, and the text read-out for the function represented in all these ways. Technology use can increase accuracy and speed of data collection and allow real-time visualization. It also supports collaboration REF(5) (GailARice http://gailly123.wordpress.com/2010/07/19/multiple-representations-in-math/)

Computer tools may be intrinsically interesting and motivating to students, and provide a familiar and comforting context students already use in their daily life. (GASS2, http://gk1as.wordpress.com/2010/07/16/math-concepts/)

Spreadsheet software (insert examples - Excel, Open Office, Google Docs...) is widely used in many industries, and helping students see the applications can make math alive for them (GASS2, http://gk1as.wordpress.com/2010/07/16/math-concepts/) Most spreadsheet programs provide dynamic links among formulas, grids and several types of graphs.

Carnegie Learning curriculum is an example of emphasis on multiple representations and use of computer tools. REF (4) (Peterhorn33 http://peterhorn33.wordpress.com/2010/07/13/response-to-article-on-multiple-representations/)

GeoGebra is a free software dynamically linking geometric constructions, graphs, formulas and grids REF. It can be used in a browser and is light enough for older or low-end computers. REF (6) (esivel http://esivel.wordpress.com/ed526/#comment-2).

Project Interactivate REF (7) has many activities linking visual, verbal and numeric representations. There are currently 159 different acitivites available, in many areas of math, including numbers and operations, probability, geometry, algebra, statistics and modeling. (esivel http://esivel.wordpress.com/ed526/#comment-2).

Concerns


There are concerns that technology for working with multiple representations can become a distraction from mathematical content, and an end in itself. (GASS2, http://gk1as.wordpress.com/2010/07/16/math-concepts/)

Care should be taken that informal representations do not prevent students from progressing toward formal, symbolic mathematics. (aheleniak, Wednesday discussion).

References


  1. http://demirus.com/2010/06/10/math-textbooks-in-the-future (GASS2)
  2. S. Ainsworth, P. Bibby, and D. Wood, “Information technology and multiple representations: New opportunities – new problems,” Journal of Information Techology for Teacher Education 6, no. 1 (1997): 93. (KateGillin http://ktgee24.wordpress.com/2010/07/18/multiple-representations/)
  3. B. Moseley and M. Brenner, Using Multiple Representations for Conceptual Change in Pre-algebra: A Comparison of Variable Usage with Graphic and Text Based Problems., 1997, http://eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED413184. (Peterhorn33 http://peterhorn33.wordpress.com/2010/07/13/response-to-article-on-multiple-representations/)
  4. http://www.carnegielearning.com/secondary-curricula/
  5. http://www.ioste2008.org/109/how-can-instructional-technology-make-teaching-and-learning-more-effective-in-the-schools
  6. M. Hohenwarter and J. Preiner, “Dynamic mathematics with GeoGebra,” Journal of Online Mathematics and its Applications 7 (2007).
  7. http://www.shodor.org/interactivate/activities/
  8. http://continuities.wordpress.com/2010/04/25/sharing-multiple-representations-of-systems
  9. http://www.youtube.com/watch#!v=4c5SunC5Lbs
  10. http://standards.nctm.org/document/chapter6/rep.htm
  11. http://ctlonline.org/blog/?p=357
  12. http://www.losmedanos.edu/deved/documents/m25_student_work.multiple_representations_000.pdf
  13. www.pictorialmath.com/updates_downloads/PresenterGuide.ppt
  14. http://www.etacuisenaire.com/
  15. J. Schultz and M. Waters, “Why Representations?” Mathematics Teacher 93, no. 6 (2000): 448–53.