Virginia Mathematics Teacher Spring 2017

Journal for Res. in Math. Education, 31 (2), 168-190 Reys, R.E. & Nohda, N. (Eds.) (1994). Computational alternatives for the 21 st century: Cross-cultural perspectives from Japan and the United States . Reston, VA: NCTM. Schoen, H.L. & Zweng, M.J. (Eds.) (1986). Estimation and mental computation. 1986 NCTM Yearbook . Reston, VA: NCTM

Jérôme Proulx Professor University de Québec proulx.jerome@uqam.ca

Recursion in Secondary Mathematics Classrooms Nicole Fratrik and Joe Garofalo

Recursive functions are advocated by both the National Council of Teachers of Mathematics (NCTM) and the Virginia Department of Education (VDOE). NCTM goes as far to state, “In grades 9 – 12, students should encounter a wide variety of situations that can be modeled recursively, such as interest-rate problems or situations involving the logistic equation for growth. The study of recursive patterns should build during the years from ninth through twelfth grade…Recursively defined functions offer students a natural way to express these relationships and to see how some functions can be defined recursively as well as explicitly” (NCTM, 2000). The VDOE Algebra II Curriculum Framework includes “Essential Understanding: Sequences can be defined explicitly and recursively” and “Essential Knowledge and Skills: Generalize patterns in a sequence using explicit and recursive formulas” (VDOE, 2009). Recursive functions can be used to explore sequences and limits, look at connections between rational and irrational numbers, model growth patterns, compute and analyze compound interest and mortgages, and learn some basic computer programming. Furthermore, tasks involving recursive functions can provide opportunities for students to link multiple representations of functions – numerical, algebraic and graphical. One way to introduce recursive thinking is through the use of continued

fractions. Continued Fractions

Evaluating continued fractions such as this one can be done in a variety of ways; hence tasks can be designed for students in different courses (see Figure 1). Pre-algebra and algebra students can use their basic knowledge of rational numbers to create a numerical sequence, with fractions of increasing complexity, look for a pattern, and then approximate the value of the continued fraction. This type of task can be done with or without technology, depending on what students are expected to do. Such an exploration can be used to introduce students to ideas about irrational numbers and limits. Students in courses beyond a first course in algebra can be asked to develop a recursive function to determine the value of the fraction and even asked to derive the fraction value algebraically. These three ways to evaluate the continued fraction in Figure 1 are shown below.

Figure 1 : An infinite continued fraction

Numerical strategy

Many students start by generating a

Virginia Mathematics Teacher vol. 43, no. 2

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