**ERIC Number:**EJ1044119

**Record Type:**Journal

**Publication Date:**2014-Mar

**Pages:**5

**Abstractor:**ERIC

**Reference Count:**6

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

Using Covariation Reasoning to Support Mathematical Modeling

Jacobson, Erik

Mathematics Teacher, v107 n7 p515-519 Mar 2014

For many students, making connections between mathematical ideas and the real world is one of the most intriguing and rewarding aspects of the study of mathematics. In the Common Core State Standards for Mathematics (CCSSI 2010), mathematical modeling is highlighted as a mathematical practice standard for all grades. To engage in mathematical modeling, beginning algebra students must learn to use their understanding of arithmetic operations to make mathematical sense of problem situations and to relate this sense making to functions represented by equations, tables, and graphs. The word problems commonly used in beginning algebra courses give opportunities to practice mathematical modeling. Further, the ability to reason with quantities as well as numbers is an important capacity for students to develop. Two kinds of quantitative reasoning have a special relevance for beginning algebra students. The "correspondence" perspective deals with the question, How is one quantity related to another? A correspondence understanding of speed might be expressed as the rule that relates each value for time with a unique value for distance, such as the equation y = 25x, where x represents time and y represents distance. By contrast, the key question for "covariation" reasoning is, How does one quantity change as another quantity changes? A covariation understanding of speed would focus on how distance and time change together--that is, the distance covered increases by 25 meters as the elapsed time increases by 1 second. Both kinds of reasoning are important goals for algebra students. Correspondence is a fundamental piece of mature reasoning about functions, and covariation is critical for developing the rate-of-change concept. Presented in this article are two sessions from Ms. Holmes's classroom (the teacher's name is a pseudonym) in which seventh graders intuitively used covariation to begin to make sense of word problems. The passages show how students' covariation reasoning might surface in the classroom and illustrate some of the teaching strategies that Ms. Holmes used to support her students' reasoning. The sessions also provide a foundation for the discussion of classroom strategies, which summarizes research-based strategies for supporting students' use of covariation reasoning to build robust mathematical models.

Descriptors: Mathematics Instruction, Mathematical Models, Mathematical Logic, Teaching Methods, Relevance (Education), Algebra, Word Problems (Mathematics), Mathematical Concepts, Problem Solving, Grade 7, Secondary School Mathematics

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: orders@nctm.org; Web site: http://www.nctm.org/publications/

**Publication Type:**Journal Articles; Guides - Classroom - Teacher; Reports - Descriptive

**Education Level:**Grade 7; Junior High Schools; Middle Schools; Elementary Education; Secondary Education

**Audience:**Teachers

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A