1. 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 Introduction 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 o Students construct their own knowledge o Problem solving o Collaborative

5. Comparing K-3 & 4-8 Instruction K-3 Children should create their own algorithms o Waiting until 3 rd 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 o Textbook-based problems rely on routine o 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) Ask the class to solve a problem and give an explanation of their method (Kamii, p. 201) o Allow students time to explore their own methods o Let students express theories (Kamii, 1993) o As a teacher do not agree or disagree with a procedure o 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) Present problems in meaningful contexts (Carroll, p. 371) o Provide manipulatives to “support children’s thinking” (Carroll, 1993) When evaluating student-invented algorithms ensure the procedures are: 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: o 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. 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. 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. 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 (2 nd Edition). Sausalito, CA: Math Solutions Publications Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn grades k-6 (2 nd 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. 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. 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. 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. 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. 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. 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. 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 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, 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: Van de Walle, J.A, Karp, K.S, & Bay-Williams, J.M (2009). Elementary and middle school mathematics: Teaching developmentally (7 th edition). Boston: Allyn & Bacon. Yim, J. (2010). Children's strategies for division by fractions in the context of the area of a rectangle. 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.