Introduction

1. Introduction DEFINITION: An algorithm is a step-by-step process that guarantees the correct solution to a given problem, provided the steps are executed correctly (Barnett, 1998 as cited in Morrow, 1998, p.69). Exploring the difference between Traditional & Inventive Algorithms in the classroom Presentation by James Becker & Jennifer Sluke

2. Research • Traditional algorithms prevent children from developing number sense (Kamii, p. 202) • Studies fail to determine the effectiveness of individual programs • Restricted number of studies for any particular curriculum • Uneven quality of the studies (NRC, 2002) • 40% of U.S. ten-year-olds didn’t understand subtraction using the standard “borrowing” algorithm (The University of Chicago, n.d.) • Special educators should use concrete strategies when helping students learn a real-world approach (Sayeski & Paulsen, 2010)

3. Pros & Cons PROS: • Algorithms are efficient • Produce accurate results • Process can be repeated with similar problems • Students will use algorithms • Sometimes “invented procedures” are widely used algorithms • Other times invented procedures are not generalizable (Chapin, O’Connor & Anderson, 2009) CONS: • Traditional algorithms are digit-oriented rather than developing number sense • They often read right-to-left • They are rigid – can only be done “one right way” (Van de Walle, Karp & Bay-Williams, 2009)

4. Critical Aspects • Traditional Algorithms • Can be memorized • Step-by-step procedure • Several steps that repeat • Order critical • Inventive Algorithms • Students construct their own knowledge • Problem solving • Collaborative

5. Comparing K-3 & 4-8 Instruction K-3 • Children should create their own algorithms • Waiting until 3rd grade allows students to do their own problem solving (Kamii, 1993) • The understanding children gain from invented strategies will make it easier for you to teach the traditional algorithms (Chamberlin, 2010) 4-8 • The curricula has an overreliance on routine procedures • Textbook-based problems rely on routine • Tasks focus on low-level thinking skills (Chamberlin, 2010)

6. Suggestions for Instruction • Ask the class to solve a problem and give an explanation of their method (Kamii, p. 201) • Allow students time to explore their own methods • Let students express theories (Kamii, 1993) • As a teacher do not agree or disagree with a procedure • Strategies can be done alone, in small groups and then as a large group (Carroll, p. 371) • Present problems in meaningful contexts (Carroll, p. 371) • Provide manipulatives to “support children’s thinking” (Carroll, 1993) • When evaluating student-invented algorithms ensure the procedures are: • Efficient • Mathematically valid • Generalizable (Salinas, 2009)

7. K-8 Textbooks • Highline School District: • No active textbook used with students K-4/5 • Teacher text have fluctuated between constructivism approach and traditional algorithms (Addison-Wesley Mathematics to Building on Numbers You Know –TERC) • Presently using CMP2 (Connected Math) • In process now to determine future math curriculum and text • Seattle School District: • Strict adherence to textbooks in elementary and following Everyday Math curriculum (textbook & journal)

8. Conclusions • Knowing algorithms increases students’ mathematical power (NCTM, 1989) • It is essential to understand algorithms rather than just applying them in a rote fashion (Chapin, O’Connor & Anderson, 2009) • Reflecting on other students’ invented procedures encourages the belief that mathematics is creative and sensible (The University of Chicago, n.d.) •  Teachers are encouraged to allow students to explore and create algorithms before the traditional algorithms are introduced (Salinas, 2009)

9. References: Basturk, S. (2010). First-year secondary school mathematics students’ conception of mathematical proofs and proving. Educational Studies, 36 (3), 283-298. Carrol, W., & Porter, D. (1993). Invented strategies can develop meaningful mathematical procedures. Teaching Children Mathematics, 3 (7), 370-374. Chamberlin, S. (2010). Mathematical problems that optimize learning for academically advanced students in grades k-6. Journal of Advanced Academics, 22 (1), 52-76. Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn grades k-6 (2nd Edition). Sausalito, CA: Math Solutions Publications Curcio, F. R., & Schwartz, S. L. (1998). There Are No Algorithms for Teaching Algorithms. Teaching Children Mathematics, 5 (1), 26-30. Kamii, C., Lewis, B., & Livingston, S. J. (1993). Primary arithmetic: Children inventing their own procedures. The Arithmetic Teacher, 41 (4), 200-203. Mokros, J., Russell, S., & Economopoulos, K. (1995). Beyond arithmetic: Changing mathematics in the elementary classroom. Cambridge: Pearson Education. Philipp, R. (1996). Multicultural mathematics and alternative algorithms. Teaching Children Mathematics, 3 (3), 128-33. Randolph, T. D., & Sherman, H. J. (2001). Alternative algorithms: Increasing options, reducing errors. Teaching Children Mathematics, 7 (8), 480-484. Salinas, T. M. L. (January 01, 2009). Beyond the right answer: exploring how preservice elementary teachers evaluate student-generated algorithms. Mathematics Educator, 19 (1), 27-34. Sayeski, K., & Paulsen, K. (2010). Mathematics reform curricula and special education: Identifying intersections and implications for practice. Intervention in School and Clinic,46 (1), 13-21. The National Council of Teachers of Mathematics. (1998). The teaching and learning of algorithms in school mathematics (1998 Yearbook ed.). (L. Morrow, & M. Kenney, Eds.) Reston, VA: NCTM. The University of Chicago. (n.d.). Algorithms in everyday mathematics. Retrieved April 1, 2010, from University of Chicago School Mathematics Project: http://everydaymath.uchicago.edu/about/research/ Van de Walle, J.A, Karp, K.S, & Bay-Williams, J.M (2009). Elementary and middle school mathematics: Teaching developmentally (7th edition). Boston: Allyn & Bacon. Yim, J. (2010). Children's strategies for division by fractions in the context of the area of a rectangle. Educational Studies in Mathematics,73 (2), 105-120.