The Literature Review of Algebra Learning: Focusing on the Contributions to Students’ Difficulties

Author(s)
Xiong Wang

ABSTRACT

This paper reviews the research literature with respect to the contributions to the students’ difficulties in their algebra learning in order to understand the students’ difficulties in algebra learning. To start with, 29 articles selected from the database (ERIC) are categorized into a taxonomy which has been generated from the research literature, which falls into five categories including: algebra content, cognitive gap, teaching issues, learning matters, and transition knowledge. The challenges that students confront with under those categories are unpacked in the review process. In addition, the five categories adopted in this paper could serve as a framework of better understanding students’ difficulties in their algebra learning. Finally, the research gap from the literature review is discussed.

This paper reviews the research literature with respect to the contributions to the students’ difficulties in their algebra learning in order to understand the students’ difficulties in algebra learning. To start with, 29 articles selected from the database (ERIC) are categorized into a taxonomy which has been generated from the research literature, which falls into five categories including: algebra content, cognitive gap, teaching issues, learning matters, and transition knowledge. The challenges that students confront with under those categories are unpacked in the review process. In addition, the five categories adopted in this paper could serve as a framework of better understanding students’ difficulties in their algebra learning. Finally, the research gap from the literature review is discussed.

Cite this paper

Wang, X. (2015) The Literature Review of Algebra Learning: Focusing on the Contributions to Students’ Difficulties.*Creative Education*, **6**, 144-153. doi: 10.4236/ce.2015.62013.

Wang, X. (2015) The Literature Review of Algebra Learning: Focusing on the Contributions to Students’ Difficulties.

References

[1] Banerjee, R., & Subramaniam, K. (2012). Evolution of a Teaching Approach for Beginning Algebra. Educational Studies in Mathematics, 80, 351-367.

http://dx.doi.org/10.1007/s10649-011-9353-y

[2] Behr, M., Erlwanger, S., & Nichols, E. (1980). How Children View the Equals Sign. Mathematics Teaching, 92, 13-18.

[3] Booth, L. R. (1984). Algebra: Children’s Strategies and Errors. Windsor, UK: NFER-Nelson.

[4] Booth, L. R. (1988). Children’s Difficulties in Beginning Algebra. In A. F. Coxford (Ed.), The Ideas of Algebra, K-12 (1988 Yearbook, pp. 20-32). Reston, VA: National Council of Teachers of Mathematics.

[5] Cai, J., & Moyer, J. (2008). Developing Algebraic Thinking in Earlier Grades: Someinsights from International Comparative Studies. In C. Greenes, & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (70th Yearbook of the National Council of Teachers of Mathematics, pp.169-180). Reston, VA: NCTM.

[6] Cai, J., Lew, H. C, Morris, A., Mover, J. C, Ng, S. F., & Schmittau, J. (2004). The Development of Students’ Algebraic Thinking in Earlier Grades: A Cross-Cultural Comparative Perspective. Paper Presented at the Annual Meeting of the American Educational Research Association, San Diego, CA.

[7] Carry, L. R., Lewis, C., & Bernard, J. (1980). Psychology of Education Solving: An Information Processing Study. Austin: University of Texas at Austin, Department of Curriculum and Instruction.

[8] Chaiklin, S. (1989). Cognitive Studies of Algebra Problem Solving and Learning. In S. Wagner, & Kieran (Eds.), Research Issue in Learning and Teaching of Algebra (pp. 93-114). Reston, VA: National Council of Teachers of Mathemaics; Hillsdale, NJ: Lawrence Erlbaum.

[9] Davis, R. B. (1975). Cognitive Processes Involved in Solving Simple Algebraic Equations. Journal of Children’s Mathematical Behaviour, 1, 7-35.

[10] Falkner, K., Levi, L., & Carpenter, T. P. (1999). Children’s Understanding of Equality Foundation for Algebra. Teaching Children Mathematics, 6, 232-237.

[11] Filloy, E., & Rojano, T. (1989). Solving Equations: The Transition from Arithmetic to Algebra. For the Learning of Mathematics, 9, 19-25.

[12] Fischbein, E., & Barash, A. (1993). Algorithmic Models and Their Misuse in Solving Algebraic Problems. Proceedings of PME 17, 1, 162-172.

[13] Freiman, V., & Lee, L. (2004). Tracking Primary Students’ Understanding of Equal Sign. In M. Hoines, & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 415-422). Bergen: PME.

[14] Greeno, J. G. (1982). A Cognitive Learning Analysis of Algebra. The Annual Meeting of the American Educational Research Association, Boston, MA.

[15] Herscovics, N. (1989). Cognitive Obstacles Encountered in the Learning of Algebra. In S. Wagner, & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 60-86). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum.

[16] Herscovics, N., & Linchevski, L. (1994). A Cognitive Gap between Arithmetic and Algebra. Educational Studies in Mathematics, 27, 59-78.

http://dx.doi.org/10.1007/BF01284528

[17] Hoz, R., & Harel, G. (1989). The Facilitating Role of Table Form in Solving Algebra Speed Problems: Real or Imaginary? In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceeding of the 13th International Conference for the Psychology of Mathematics Education (pp. 123-130). Paris: G. R. Didactique, CNRS.

[18] Khng, K. H., & Lee, K. (2009). Inhibiting Interference from Prior Knowledge: Arithmetic Intrusions in Algebra Word Problem Solving. Learning and Individual Differences, 19, 262-268.

http://dx.doi.org/10.1016/j.lindif.2009.01.004

[19] Kieran, C. (1992). The Learning and Teaching of School Algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419). New York: Macmillan Publishing Company.

[20] Kieran, C. (2004). Algebraic Thinking in the Early Grades: What Is It? The Mathematics Educator, 8, 139-151.

[21] Kiichemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children’s Understanding of Mathematics (pp. 11-16). London: John Murray.

[22] Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

[23] Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of Differences in Children’s Understandings of Mathematical Equality: Comparative Analysis of Teacher Guides and Student Texts in China and the United States. Cognition and Instruction, 26, 195-217.

http://dx.doi.org/10.1080/07370000801980845

[24] Linchevski, L., & Herscovics, N. (1996). Crossing the Cognitive Gap between Arithmetic and Algebra: Operating on the Unknown in the Context of Equations. Educational Studies in Mathematics, 30, 39-65.

http://dx.doi.org/10.1007/BF00163752

[25] Napaphun, V. (2012). Relational Thinking: Learning Arithmetic in Order to Promote Algebraic Thinking. Journal of Science and Mathematics Education in Southeast Asia, 35, 84-101.

[26] Ng, S. F., & Lee, K. (2009). The Model Method: Singapore Children’s Tool for Representing and Solving Algebraic Word Problems. Journal for Research in Mathematics Education, 40, 282-313.

[27] Rachlin, S. L. (1989). Using Research to Design a Problem-Solving Approach for Teaching Algebra. In S. T. Ong (Ed.), Proceedings of the 4th Southeast Asian Conference on Mathematical Education (pp. 156-161). Singapore: Singapore Institute of Education.

[28] Radford, L. (2012). Early Algebraic Thinking Epistemological, Semiotic, and Developmental Issues. 12th International Congress on Mathematical Education, Seoul, South Korea.

[29] Radford, L., & Puig, L. (2007). Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking. Educational Studies in Mathematics, 66, 145-164.

http://dx.doi.org/10.1007/s10649-006-9024-6

[30] Reed, S. K. (1987). A Structure-Mapping Model for Word Problems. Journal of Experimental Psychology: Learning, Memory and Cognition, 13, 124-139.

http://dx.doi.org/10.1037/0278-7393.13.1.124

[31] Reed, S. K., Dempster, A., & Ettinger, M. (1985). The Usefulness of Analogous Solution for Solving Algebra Word Problems. Journal of Experimental Psychology: Learning, Memory and Cognition, 11, 106-125.

http://dx.doi.org/10.1037/0278-7393.11.1.106

[32] Sadovsky, P., & Sessa, C. (2005). The Adidactic Interaction with the Procedures of Peers in the Transition from Arithmetic to Algebra: A Milieu for the Emergence of New Questions. Educational Studies in Mathematics, 59, 85-112.

http://dx.doi.org/10.1007/s10649-005-5886-2

[33] Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22, 1-36.

http://dx.doi.org/10.1007/BF00302715

[34] Sfard, A., & Linchevski, L. (1993). Processes without Objects—The Case of Equations and Inequalities. The Special Issue of del Seminario Matematico de U’Universita edel Politecnico di Torino.

[35] Sfard, A., & Linchevski, L. (1994). The Gains and the Pitfalls of Reification—The Case of Algebra. Educational Studies in Mathematics, 26, 191-228.

http://dx.doi.org/10.1007/BF01273663

[36] Wang, X. (2014). The Transition from Arithmetic to Algebra: Cognitive Gap, Prealgebraic Conceptualization, and Teacher Preparation. Edmonton: University of Alberta. (Unpublished Essay).

[37] Welder, R. M. (2012). Improving Algebra Preparation: Implications from Research on Student Misconceptions and Difficulties. School Science and Mathematics, 112, 255-264.

http://dx.doi.org/10.1111/j.1949-8594.2012.00136.x

[38] Wenger, R. (1987). Cognitive Science and Algebra Learning. In A. Schoenfeld (Ed.), Cognitive Science and Mathematics Education (pp. 217-251). Hillsdale, NJ: Lawrence Erlbaum.

[1] Banerjee, R., & Subramaniam, K. (2012). Evolution of a Teaching Approach for Beginning Algebra. Educational Studies in Mathematics, 80, 351-367.

http://dx.doi.org/10.1007/s10649-011-9353-y

[2] Behr, M., Erlwanger, S., & Nichols, E. (1980). How Children View the Equals Sign. Mathematics Teaching, 92, 13-18.

[3] Booth, L. R. (1984). Algebra: Children’s Strategies and Errors. Windsor, UK: NFER-Nelson.

[4] Booth, L. R. (1988). Children’s Difficulties in Beginning Algebra. In A. F. Coxford (Ed.), The Ideas of Algebra, K-12 (1988 Yearbook, pp. 20-32). Reston, VA: National Council of Teachers of Mathematics.

[5] Cai, J., & Moyer, J. (2008). Developing Algebraic Thinking in Earlier Grades: Someinsights from International Comparative Studies. In C. Greenes, & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (70th Yearbook of the National Council of Teachers of Mathematics, pp.169-180). Reston, VA: NCTM.

[6] Cai, J., Lew, H. C, Morris, A., Mover, J. C, Ng, S. F., & Schmittau, J. (2004). The Development of Students’ Algebraic Thinking in Earlier Grades: A Cross-Cultural Comparative Perspective. Paper Presented at the Annual Meeting of the American Educational Research Association, San Diego, CA.

[7] Carry, L. R., Lewis, C., & Bernard, J. (1980). Psychology of Education Solving: An Information Processing Study. Austin: University of Texas at Austin, Department of Curriculum and Instruction.

[8] Chaiklin, S. (1989). Cognitive Studies of Algebra Problem Solving and Learning. In S. Wagner, & Kieran (Eds.), Research Issue in Learning and Teaching of Algebra (pp. 93-114). Reston, VA: National Council of Teachers of Mathemaics; Hillsdale, NJ: Lawrence Erlbaum.

[9] Davis, R. B. (1975). Cognitive Processes Involved in Solving Simple Algebraic Equations. Journal of Children’s Mathematical Behaviour, 1, 7-35.

[10] Falkner, K., Levi, L., & Carpenter, T. P. (1999). Children’s Understanding of Equality Foundation for Algebra. Teaching Children Mathematics, 6, 232-237.

[11] Filloy, E., & Rojano, T. (1989). Solving Equations: The Transition from Arithmetic to Algebra. For the Learning of Mathematics, 9, 19-25.

[12] Fischbein, E., & Barash, A. (1993). Algorithmic Models and Their Misuse in Solving Algebraic Problems. Proceedings of PME 17, 1, 162-172.

[13] Freiman, V., & Lee, L. (2004). Tracking Primary Students’ Understanding of Equal Sign. In M. Hoines, & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 415-422). Bergen: PME.

[14] Greeno, J. G. (1982). A Cognitive Learning Analysis of Algebra. The Annual Meeting of the American Educational Research Association, Boston, MA.

[15] Herscovics, N. (1989). Cognitive Obstacles Encountered in the Learning of Algebra. In S. Wagner, & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 60-86). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum.

[16] Herscovics, N., & Linchevski, L. (1994). A Cognitive Gap between Arithmetic and Algebra. Educational Studies in Mathematics, 27, 59-78.

http://dx.doi.org/10.1007/BF01284528

[17] Hoz, R., & Harel, G. (1989). The Facilitating Role of Table Form in Solving Algebra Speed Problems: Real or Imaginary? In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceeding of the 13th International Conference for the Psychology of Mathematics Education (pp. 123-130). Paris: G. R. Didactique, CNRS.

[18] Khng, K. H., & Lee, K. (2009). Inhibiting Interference from Prior Knowledge: Arithmetic Intrusions in Algebra Word Problem Solving. Learning and Individual Differences, 19, 262-268.

http://dx.doi.org/10.1016/j.lindif.2009.01.004

[19] Kieran, C. (1992). The Learning and Teaching of School Algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419). New York: Macmillan Publishing Company.

[20] Kieran, C. (2004). Algebraic Thinking in the Early Grades: What Is It? The Mathematics Educator, 8, 139-151.

[21] Kiichemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children’s Understanding of Mathematics (pp. 11-16). London: John Murray.

[22] Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

[23] Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of Differences in Children’s Understandings of Mathematical Equality: Comparative Analysis of Teacher Guides and Student Texts in China and the United States. Cognition and Instruction, 26, 195-217.

http://dx.doi.org/10.1080/07370000801980845

[24] Linchevski, L., & Herscovics, N. (1996). Crossing the Cognitive Gap between Arithmetic and Algebra: Operating on the Unknown in the Context of Equations. Educational Studies in Mathematics, 30, 39-65.

http://dx.doi.org/10.1007/BF00163752

[25] Napaphun, V. (2012). Relational Thinking: Learning Arithmetic in Order to Promote Algebraic Thinking. Journal of Science and Mathematics Education in Southeast Asia, 35, 84-101.

[26] Ng, S. F., & Lee, K. (2009). The Model Method: Singapore Children’s Tool for Representing and Solving Algebraic Word Problems. Journal for Research in Mathematics Education, 40, 282-313.

[27] Rachlin, S. L. (1989). Using Research to Design a Problem-Solving Approach for Teaching Algebra. In S. T. Ong (Ed.), Proceedings of the 4th Southeast Asian Conference on Mathematical Education (pp. 156-161). Singapore: Singapore Institute of Education.

[28] Radford, L. (2012). Early Algebraic Thinking Epistemological, Semiotic, and Developmental Issues. 12th International Congress on Mathematical Education, Seoul, South Korea.

[29] Radford, L., & Puig, L. (2007). Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking. Educational Studies in Mathematics, 66, 145-164.

http://dx.doi.org/10.1007/s10649-006-9024-6

[30] Reed, S. K. (1987). A Structure-Mapping Model for Word Problems. Journal of Experimental Psychology: Learning, Memory and Cognition, 13, 124-139.

http://dx.doi.org/10.1037/0278-7393.13.1.124

[31] Reed, S. K., Dempster, A., & Ettinger, M. (1985). The Usefulness of Analogous Solution for Solving Algebra Word Problems. Journal of Experimental Psychology: Learning, Memory and Cognition, 11, 106-125.

http://dx.doi.org/10.1037/0278-7393.11.1.106

[32] Sadovsky, P., & Sessa, C. (2005). The Adidactic Interaction with the Procedures of Peers in the Transition from Arithmetic to Algebra: A Milieu for the Emergence of New Questions. Educational Studies in Mathematics, 59, 85-112.

http://dx.doi.org/10.1007/s10649-005-5886-2

[33] Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22, 1-36.

http://dx.doi.org/10.1007/BF00302715

[34] Sfard, A., & Linchevski, L. (1993). Processes without Objects—The Case of Equations and Inequalities. The Special Issue of del Seminario Matematico de U’Universita edel Politecnico di Torino.

[35] Sfard, A., & Linchevski, L. (1994). The Gains and the Pitfalls of Reification—The Case of Algebra. Educational Studies in Mathematics, 26, 191-228.

http://dx.doi.org/10.1007/BF01273663

[36] Wang, X. (2014). The Transition from Arithmetic to Algebra: Cognitive Gap, Prealgebraic Conceptualization, and Teacher Preparation. Edmonton: University of Alberta. (Unpublished Essay).

[37] Welder, R. M. (2012). Improving Algebra Preparation: Implications from Research on Student Misconceptions and Difficulties. School Science and Mathematics, 112, 255-264.

http://dx.doi.org/10.1111/j.1949-8594.2012.00136.x

[38] Wenger, R. (1987). Cognitive Science and Algebra Learning. In A. Schoenfeld (Ed.), Cognitive Science and Mathematics Education (pp. 217-251). Hillsdale, NJ: Lawrence Erlbaum.