D E F I N I T I O N:A n a l g o r i t h m i s a s t e p - b y - s t e p p r o c e s s t h a t g u a r a n t e e s t h e c o r r e c t s o l u t i o n t o a g i v e n p r o b l e m , p r o v i d e d t h e s t e p s a r e e x e c u t e d c o r r e c t l y ( B a r n e t t , 1 9 9 8 a s c i t e d i n M o r r o w , 1 9 9 8 , p . 6 9 ) .

E x p l o r i n g t h e d i ff e r e n c e b e t w e e nTr a d i t i o n a l & I n v e n t i v e A l g o r i t h m si n t h e c l a s s r o o m

Introduction

Presentation by James Becker & Jennifer Sluke

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)

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)

Critical Aspects

Traditional Algorithms

Can be memorized Step-by-step procedure Several steps that repeat Order critical

• Inventive Algorithms

o Students construct their own knowledgeo Problem solvingo Collaborative

Comparing K-3 & 4-8 Instruction

K-3 Children should create their own algorithms

o 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

o Textbook-based problems rely on routineo Tasks focus on low-level thinking skills (Chamberlin, 2010)

Suggestions for Instruction

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 methodso Let students express theories (Kamii, 1993)

o As a teacher do not agree or disagree with a procedureo 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)

o Provide manipulatives to “support children’s thinking” (Carroll, 1993)

When evaluating student-invented algorithms ensure the procedures are: Efficient Mathematically valid Generalizable (Salinas, 2009)

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 followingEveryday Math curriculum

(textbook & journal)

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)

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 (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.

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.