Development of Algebraic Thinking at the Middle School Level

Principal Investigator: 
Project Overview
Background & Purpose: 

The purpose of the study is to provide a developmental account of generalization involving patterning activity among a cohort of middle school children over the course of three years.

Setting: 

Urban middle school in Northern California.

Research Design: 

The research design for this project is longitudinal, cross sectional, and comparative, and is designed to generate descriptive evidence using case study, design research, ethnography, observation, phenomenological research, and design experiments under Realistic Mathematics Education framework. This project collects original data using assessments of learning/achievement tests, personal observation, videography, and survey research, including paper and pencil self-completion questionnaires, and structured interviewer-administered questionnaires. Instruments used include mostly open ended assessments. Qualitative methods will be used for analysis, constant comparative method. Originally-collected data will be made available for use by others through workshops and a website.

Findings: 

Progressive mathematization and formalization of pattern generalization among middle school children; Use of multiplicative thinking in assisting students to capture the structure of patterns in an efficient manner.

Publications & Presentations: 

Rivera, F. (2011). Neural Correlates of Gender, Culture, and Race and Implications to Embodied Thinking in Mathematics. Advances in Mathematics Education Volume 3. Netherland: Springer.

Rivera, F. (2011). Explaining Differences in Second Grade Students' Patterning Competence Using Parallel Distributed Processing. Research Report. PME 35 in Ankara, Turkey.

Rivera, F. (2010). Toward a Visually-Oriented School Mathematics Curriculum: Research, Theory, Practice, and Issues (Mathematics Education Library Volume 49). Dordrecht, Netherlands: Springer.

Rivera, F. (2010). Visual Templates in Pattern Generalization Activity. Educational Studies in Mathematics, 73(3), 297-328.

Rivera, F. (2010). There is More to Mathematics than Symbols. Mathematics Teaching, 218, 42-47.

Rivera, F. & Becker, J. (2010). Formation of pattern generalization among middle school students: Results from a Three-Year Study. To appear in J. Cai & E. Knuth (eds.), Early algebra: Advances in Mathematics Education. Netherlands: Springer.

Rivera, F. & Becker, J. (2009). Algebraic Reasoning Through Patterns. Mathematics Teaching in the Middle School, 15(4), 212-221.

Rivera, F. (2008). On the pitfalls of abduction: Complicities and complexities in patterning activity. For the Learning of Mathematics, 28(1), 17-25.

Rivera, F. & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM, 40(1), 65-82.

Rivera, F. & Becker, J. R. (2007). Abductive-Inductive (Generalization) Strategies of Preservice Elementary Majors on Patterns in Algebra. Journal of Mathematical Behavior, 26(2), 140-155.

Rivera, F. (2007). Visualizing as a Mathematical Way of Knowing: Understanding Figural Generalization. Mathematics Teacher, 101(1), 69-75.

Rivera, F. (2007). Accounting for Students’ Schemes in the Development of a Graphical Process for Solving Polynomial Inequalities in Instrumented Activity. Educational Studies in Mathematics, 65(3), 281-307.

Rivera, F. (2006). Changing the Face of Arithmetic: Teaching Children Algebra. Teaching Children Mathematics, 12(6), 306-311.

Rivera, F. & Becker, J. R. (2005). Figural and Numerical Modes of Generalizing in Algebra. Mathematics Teaching in the Middle School, 11(4), 198-203.

Rivera, F. (2006). Structures of elementary algebraic thinking: Evidence from sixth grade students and implications for classroom practice. Intersection, 7(2), 2-13.

Rivera, F. (2010). Second Grade Students' Preinstructional Competence in Patterning Activity. PME Brazil.

Rivera, F. & Becker, J. (2009a). Visual Templates in Pattern Generalization. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, 473-480). Thessaloniki, Greece: PME. 

Becker, J. & Rivera, F. (2009b). Justification Schemes of Middle School Students in the Context of Generalization of Linear Patterns. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, 9-16). Thessaloniki, Greece: PME.

Rivera, F. & Becker, J. R. (2008a). Sociocultural Intimations on the Development of Generalization Among Middle School Learners: Results from a Three-Year Study. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 4, pp. 193-200). Morelia, Mexico: Cinvestav-UMSNH.

Becker, J. R. & Rivera, F. (2008b). Nature and Content of Generalization of 7th- and 8th-Graders on a Task That Involves Free Construction of Patterns. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 4, pp. 201-208). Morelia, Mexico: Cinvestav-UMSNH.

Becker, J. R. & Rivera, F. (2007a). Factors Affecting Seventh Graders’ Cognitive Perceptions of Patterns Involving Constructive and Deconstructive Generalizations. In J-H Woo, H-C Lew, K-S Park, & D-Y Seo (Eds.), Proceedings of the 31th Conference of the International Group for the Psychology of Mathematics Education (Vol.4, pp. 129-136). Seoul, Korea: The Korea Society of Educational Studies in Mathematics.

Becker, J. R. & Rivera, F. (2007b). Seventh Graders’ Generalization Strategies Involving Decreasing Linear Patterns: Cognitive Complexities of Transfer. Proceedings of the 29th Conference of the North American Chapter of the Psychology of Mathematics Education. Lake Tahoe, Nevada.

Rivera, F. & Becker, J. R. (2007c). Abduction in Pattern Generalization. In J-H Woo, H-C Lew, K-S Park, & D-Y Seo (Eds.), Proceedings of the 31th Conference of the International Group for the Psychology of Mathematics Education (Vol.4, pp. 97-104). Seoul, Korea: The Korea Society of Educational Studies in Mathematics.

Becker, J. R. & Rivera, F. (2006a). Establishing and Justifying Algebraic Generalization at the Sixth Grade Level. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol.4, pp. 465 -472). Prague, Czechovslovakia: Charles University.

Becker, J. R. & Rivera, F. (2006b). Sixth Graders' Figural and Numerical Strategies for Generalizing Patterns in Algebra. In S. Alatorre, J. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol.2, pp. 95-101). Mérida, México: Universidad Pedagógica Nacional.

Rivera, F. & Becker, J. R. (2006c) Sixth Graders’ Ability to Generalize Patterns in Algebra: Issues and Insights. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol.1, p. 320). Prague, Czechovslovakia: Charles University.
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Rivera, F. & Becker, J. R. (2006d). Accounting for Sixth Graders’ Generalization Strategies in Algebra. In S. Alatorre, J. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol.2, pp. 155-157). Mérida, México: Universidad Pedagógica Nacional.

Becker, J. R. & Rivera, F. (2005). Generalization Schemes in Algebra of Beginning High School Students. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 4-121 – 4-128). Melbourne, Australia: University of Melbourne.

Other Products: 

PD materials, curriculum materials on patterning and algebraic thinking

Scholarly Book on visual epistemology in school mathematics learning