Understanding the development of flexibility in struggling algebra students

The ability to flexibly solve problems is considered an important outcome for school mathematics and is the focus of this paper. The paper describes the impact of a three-week summer course for students who struggle with algebra. During the course, students regularly compared and contrasted worked examples of algebra problems in order to promote flexible use of solution strategies. Assessments were designed to capture both knowledge and use of multiple strategies. The students were interviewed in order to understand their rationales for choosing particular strategies as well as their attitudes toward instruction that emphasized multiple strategies. Findings suggest that students gained both knowledge of and appreciation for multiple strategies, but they did not always use alternate strategies. Familiarity, understandability, efficiency, and form of the problem were all considerations for strategy choice. Practical and theoretical implications are discussed.

[1]  A. Luchins.  Mechanization in problem solving: The effect of Einstellung. , 1942 .

[2]  Vadim Andreevich Krutet︠s︡kiĭ.  The Psychology of Mathematical Abilities in Schoolchildren , 1976 .

[3]  G. Hatano,et al.  TWO COURSES OF EXPERTISE , 1984 .

[4]  James Hiebert.  Missing Links@@@Conceptual and Procedural Knowledge: The Case of Mathematics , 1987 .

[5]  James P. Byrnes,et al.  Role of Conceptual Knowledge in Mathematical Procedural Learning , 1991 .

[6]  P. H. Miller,et al.  Strategy utilization deficiencies in children: when, where, and why. , 1994, Advances in child development and behavior.

[7]  R. Siegler,et al.  Four aspects of strategic change: contributions to children's learning of multiplication. , 1995, Journal of experimental psychology. General.

[8]  E. Goldenberg,et al.  Habits of mind: An organizing principle for mathematics curricula , 1996 .

[9]  R. Siegler.  Emerging Minds: The Process of Change in Children's Thinking , 1996 .

[10]  J. Woodward,et al.  The Effects of an Innovative Approach to Mathematics on Academically Low-Achieving Students in Inclusive Settings , 1997 .

[11]  Daniel L. Schwartz,et al.  A time for telling , 1998 .

[12]  A. Treffers,et al.  The empty number line in Dutch second grades: Realistic versus gradual program design , 1998 .

[13]  F. Paas,et al.  Cognitive Architecture and Instructional Design , 1998 .

[14]  Hung-Hsi Wu,et al.  Basic Skills versus Conceptual Understanding. A Bogus Dichotomy in Mathematics Education. , 1999 .

[15]  A. W. Blöte,et al.  Mental computation and conceptual understanding , 2000 .

[16]  A. Su,et al.  The National Council of Teachers of Mathematics , 1932, The Mathematical Gazette.

[17]  A. W. Blöte,et al.  Students' flexibility in solving two-digit addition and subtraction problems: Instruction effects. , 2001 .

[18]  B. Rittle-Johnson,et al.  Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process. , 2001 .

[19]  M. Montague,et al.  Meeting the Challenge of Mathematics Reform for Students with LD , 2002 .

[20]  Juliet A. Baxter,et al.  We Talk About It, But Do They Get It? , 2002 .

[21]  Ann Dowker,et al.  The development of arithmetic concepts and skills: Constructing adaptive expertise , 2003 .

[22]  J. Star.  Reconceptualizing procedural knowledge. , 2005 .

[23]  Jon R. Star,et al.  The development of flexibility in equation solving , 2006 .

[24]  Lieven Verschaffel,et al.  The Development of Children's Adaptive Expertise in the Number Domain 20 to 100 , 2006 .

[25]  A. Baroody,et al.  An Alternative Reconceptualization of Procedural and Conceptual Knowledge. , 2007 .

[26]  B. Rittle-Johnson,et al.  Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. , 2007 .

[27]  Lieven Verschaffel,et al.  Developing adaptive expertise: A feasible and valuable goal for (elementary) mathematics education? , 2007 .

[28]  J. Star.  Research Commentary: A Rejoinder Foregrounding Procedural Knowledge. , 2007 .

[29]  K. Holyoak,et al.  Cognitive Supports for Analogies in the Mathematics Classroom , 2007, Science.

[30]  B. Rittle-Johnson,et al.  Flexibility in Problem Solving: The Case of Equation Solving. , 2008 .

[31]  K. Newton.  An Extensive Analysis of Preservice Elementary Teachers’ Knowledge of Fractions , 2008 .

[32]  Jon R. Star,et al.  The nature and development of experts’ strategy flexibility for solving equations , 2009 .

[33]  B. Rittle-Johnson,et al.  The Importance of Prior Knowledge When Comparing Examples: Influences on Conceptual and Procedural Knowledge of Equation Solving , 2009 .

[34]  B. Rittle-Johnson,et al.  Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving , 2009 .

[35]  Citable Link,et al.  It pays to compare: An experimental study on computational estimation , 2011 .