Multiple representations (mathematics education)

Last updated

In mathematics education, a representation is a way of encoding an idea or a relationship, and can be both internal (e.g., mental construct) and external (e.g., graph). Thus multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, 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. [1] Representations are thinking tools for doing mathematics.

Contents

Higher-order thinking

The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] 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 examples of such mathematically sophisticated activities. [ citation needed ] Estimation, another complex task, can strongly benefit from multiple representations [5]

Curricula that support starting from conceptual understanding, then developing procedural fluency, for example, AIMS Foundation Activities, [6] 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. [ citation needed ]

Motivation

Some representations, such as pictures, videos and manipulatives, can motivate because of their richness, possibilities of play, use of technologies, or connections with interesting areas of life. [4] Tasks that involve multiple representations can sustain intrinsic motivation in mathematics, by supporting higher-order thinking and problem solving.

Multiple representations may also remove some of the gender biases that exist in math classrooms. For example, explaining probability solely through baseball statistics may potentially alienate students who have no interest in sports. When showing a tie to real-life applications, teachers should choose representations that are varied and of interest to all genders and cultures.[ neutrality is disputed ]

Assessment

Tasks that involve construction, use, and interpretation of multiple representations can lend themselves to rubric assessment, [7] and other assessment types suitable for open-ended activities. For example, tapping into visualization for math problem solving manifests multiple representations. [1] These multiple representations arise when each student uses their knowledge base and experience—to create a visualization of the problem domain on the way toward a solution. Since visualization can be categorized into two main areas, schematic or pictorial, [8] most students will make use of one method, or sometimes both methods to represent the problem domain.

Comparison of the different visualization tools created by each student is an excellent example of multiple representations. Further, the instructor may glean from these examples elements which they incorporate into their grading rubric. In this way, it is the students that provide the examples and standards against which scoring is done. This crucial factor places each student on equal footing and links them directly with their performance in class.[ citation needed ]

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 the current mathematical topic. Second, connections among multiple representations within the same topic strengthens overall skills in using all representations, even those currently problematic. [2]

It is also helpful to ESL/ELL (English as a Second Language/English Language Learners) to use multiple representations. The more one can bring a concept to "life" in a visual way, the more likely the students will grasp what the teacher is talking about. This is also important with younger students, who may have not had a lot of experience or prior knowledge on the topics that are being taught.

Using multiple representations can help differentiate instruction by addressing different learning styles,. [4] [9]

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, and to develop algebraic thinking. By focusing more on the conceptual representations of algebraic problems, students have a better chance of improving their problem solving skills. [3]

NCTM representations standard

National Council of Teachers of Mathematics has a standard dealing with multiple representations. In part, it reads [10] "Instructional programs should enable all students to do the following:

Four most frequent school mathematics representations

While there are many representations used in mathematics, the secondary curricula heavily favor numbers (often in tables), formulas, graphs and words. [11]

Systems of manipulatives

Several curricula use extensively developed systems of manipulatives and the corresponding representations. For example, Cuisinaire rods, [12] Montessori beads [ citation needed ], Algebra Tiles, [13] Base-10 blocks, counters.

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 and experimentation. [14] It also supports collaboration. [15]

Computer tools may be intrinsically interesting and motivating to students, and provide a familiar and comforting context students already use in their daily life.

Spreadsheet software such as Excel, LibreOffice Calc, Google Sheets, is widely used in many industries, and showing students the use of applications can make math more realistic. 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. [16] More specifically, Carnegie learning focuses the student not only on solving the real life scenarios presented in the text, but also promotes literacy through sentence writing and explanations of student thinking. In conjunction with the scenario based text Carnegie Learning provides a web based tutoring program called the "Cognitive Tutor" which uses data collected from each question a student answers to direct the student to areas where they need more help.

GeoGebra is free software dynamically linking geometric constructions, graphs, formulas, and grids. [17] It can be used in a browser and is light enough for older or low-end computers. [18]

Project Interactivate [19] has many activities linking visual, verbal and numeric representations. There are currently 159 different activities available, in many areas of math, including numbers and operations, probability, geometry, algebra, statistics and modeling.

Another helpful tool for mathematicians, scientists, engineers is LaTeX. It is a typesetting program that allows one to create tables, figures, graphs etc., and to provide one with a precise visual of the problem being worked on.

Concerns

There are concerns that technology for working with multiple representations can become an end in itself, thereby distracting the students from the actual mathematical content.[ citation needed ]

Additionally, it is also objected that care should be taken, so that informal representations do not prevent students from progressing toward formal, symbolic mathematics.[ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Word problem (mathematics education)</span> Mathematical exercise presented in ordinary language

In science education, a word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation. As most word problems involve a narrative of some sort, they are sometimes referred to as story problems and may vary in the amount of technical language used.

<span class="mw-page-title-main">Mathematics education</span> Teaching, learning, and scholarly research in mathematics

In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge.

<span class="mw-page-title-main">Graphing calculator</span> Electronic calculator capable of plotting graphs

A graphing calculator is a handheld computer that is capable of plotting graphs, solving simultaneous equations, and performing other tasks with variables. Most popular graphing calculators are programmable calculators, allowing the user to create customized programs, typically for scientific, engineering or education applications. They have large screens that display several lines of text and calculations.

A cognitive tutor is a particular kind of intelligent tutoring system that utilizes a cognitive model to provide feedback to students as they are working through problems. This feedback will immediately inform students of the correctness, or incorrectness, of their actions in the tutor interface; however, cognitive tutors also have the ability to provide context-sensitive hints and instruction to guide students towards reasonable next steps.

Principles and Standards for School Mathematics (PSSM) are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada. It is the primary model for standards-based mathematics.

Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds annual national and regional conferences for teachers and publishes five journals.

<span class="mw-page-title-main">Manipulative (mathematics education)</span> Educational aid

In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience.

<span class="mw-page-title-main">Core-Plus Mathematics Project</span> High school mathematics program

Core-Plus Mathematics is a high school mathematics program consisting of a four-year series of print and digital student textbooks and supporting materials for teachers, developed by the Core-Plus Mathematics Project (CPMP) at Western Michigan University, with funding from the National Science Foundation. Development of the program started in 1992. The first edition, entitled Contemporary Mathematics in Context: A Unified Approach, was completed in 1995. The third edition, entitled Core-Plus Mathematics: Contemporary Mathematics in Context, was published by McGraw-Hill Education in 2015.

Traditional mathematics was the predominant method of mathematics education in the United States in the early-to-mid 20th century. This contrasts with non-traditional approaches to math education. Traditional mathematics education has been challenged by several reform movements over the last several decades, notably new math, a now largely abandoned and discredited set of alternative methods, and most recently reform or standards-based mathematics based on NCTM standards, which is federally supported and has been widely adopted, but subject to ongoing criticism.

Investigations in Numbers, Data, and Space is a K–5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts, United States. The curriculum is often referred to as Investigations or simply TERC. Patterned after the NCTM standards for mathematics, it is among the most widely used of the new reform mathematics curricula. As opposed to referring to textbooks and having teachers impose methods for solving arithmetic problems, the TERC program uses a constructivist approach that encourages students to develop their own understanding of mathematics. The curriculum underwent a major revision in 2005–2007.

Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.

Connected Mathematics is a comprehensive mathematics program intended for U.S. students in grades 6–8. The curriculum design, text materials for students, and supporting resources for teachers were created and have been progressively refined by the Connected Mathematics Project (CMP) at Michigan State University with advice and contributions from many mathematics teachers, curriculum developers, mathematicians, and mathematics education researchers.

Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics (NCTM). The NCTM document Curriculum and Evaluation Standards for School Mathematics (CESSM) set forth a vision for K–12 mathematics education in the United States and Canada. The CESSM recommendations were adopted by many local- and federal-level education agencies during the 1990s. In 2000, the NCTM revised its CESSM with the publication of Principles and Standards for School Mathematics (PSSM). Like those in the first publication, the updated recommendations became the basis for many states' mathematics standards, and the method in textbooks developed by many federally-funded projects. The CESSM de-emphasised manual arithmetic in favor of students developing their own conceptual thinking and problem solving. The PSSM presents a more balanced view, but still has the same emphases.

<span class="mw-page-title-main">Diagrammatic reasoning</span>

Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.

<span class="mw-page-title-main">Virtual manipulatives for mathematics</span>

Virtual manipulatives for mathematics are digital representations of concrete mathematics manipulatives. These virtual manipulatives are based on the physical manipulatives used in classrooms. They are generally used to introduce mathematical concepts using visuals and are used for teaching students new topics that may be more difficult to explain in other ways.

Modern elementary mathematics is the theory and practice of teaching elementary mathematics according to contemporary research and thinking about learning. This can include pedagogical ideas, mathematics education research frameworks, and curricular material.

<span class="mw-page-title-main">Embodied design</span>

Embodied design grows from the idea of embodied cognition: that the actions of the body can play a role in the development of thought and ideas. Embodied design brings mathematics to life; studying the effects of the body on the mind, researchers learn how to design objects and activities for learning. Embodiment is an aspect of pattern recognition in all fields of human endeavor.

Bootstrap is based at Brown University (USA), and builds on the research and development done there. Bootstrap curriculum consists of 4 research-based curricular computer science modules for grades 6-12. The 4 modules are Bootstrap:Algebra, Bootstrap:Reactive, Bootstrap:Data Science, and Bootstrap:Physics. Bootstrap materials reinforce core concepts from mainstream subjects like Math, Physics and more, enabling non-CS teachers to adopt the introductory materials while delivering rigorous and engaging computing content drawn from Computer Science classes at universities like Brown, WPI, and Northeastern.

References

  1. 1 2 Goldin, Gerald A. (2014), "Mathematical Representations", in Lerman, Stephen (ed.), Encyclopedia of Mathematics Education, Springer Netherlands, pp. 409–413, doi:10.1007/978-94-007-4978-8_103, ISBN   978-94-007-4978-8
  2. 1 2 S. Ainsworth, P. Bibby, and D. Wood, "Information technology and multiple representations: New opportunities – new problems," Journal of Information Technology for Teacher Education 6, no. 1 (1997)
  3. 1 2 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, ERIC   ED413184
  4. 1 2 3 "Archived copy". Archived from the original on 2011-07-15. Retrieved 2010-07-19.{{cite web}}: CS1 maint: archived copy as title (link)
  5. "Sharing: Multiple Representations of Systems". 25 April 2010.
  6. "- YouTube". YouTube .
  7. "Student work. Multiple representations" (PDF). losmedanos.edu.[ dead link ]
  8. Hegarty, M., and Kozhevnikov, M. (1999). Types of Visual-Spatial Representations and Mathematical Problem Solving. Journal of Educational Psychology v91, no 4 p.684 – 689.
  9. J. Schultz and M. Waters, "Why Representations?" Mathematics Teacher 93, no. 6 (2000): 448–53
  10. "Grades 6 - 8: Representation". Archived from the original on 2000-10-28.
  11. "Bar-04 | CTL - Collaborative for Teaching and Learning". 11 March 2019.
  12. "ETA/Cuisenaire - Manipulatives, science materials, and teacher resources for grades K-12". Archived from the original on 2000-08-15. Retrieved 2010-07-19.
  13. "雷竞技ray86_雷竞技app官方版苹果_雷竞技app官网网址入口".
  14. "Math Textbooks in the Future".
  15. "How Can Instructional Technology Make Teaching and Learning More Effective in the Schools? | ioste2008.com". Archived from the original on 2010-07-25. Retrieved 2010-07-19.
  16. "Carnegie Learning - Secondary Curricula". Archived from the original on 2010-07-02. Retrieved 2010-07-19.
  17. "Info". Archived from the original on 2010-03-09.
  18. M. Hohenwarter and J. Preiner, "Dynamic mathematics with GeoGebra," Journal of Online Mathematics and its Applications 7 (2007)
  19. "Interactivate: Activities".