The effect of worked examples on student learning and error anticipation in algebra

The present study examines the effectiveness of incorporating worked examples with prompts for self-explanation into a middle school math textbook. Algebra 1 students (N=75) completed an equation-solving unit with reform textbooks either containing the original practice problems or in which a portion of those problems were converted into correct, incorrect, or incomplete examples. Students completed preand posttest measures of conceptual understanding, procedural problem-solving skill, and error anticipation. Results indicate the example-based textbook assignments increased students’ procedural knowledge and their ability to anticipate errors one might make when solving problems. Differences in students’ anticipation of various types of errors are also examined.

[1]  M. Frese,et al.  Organizational error management culture and its impact on performance: a two-study replication. , 2005, The Journal of applied psychology.

[2]  Sidney D'Mello,et al.  Confusion and its dynamics during device comprehension with breakdown scenarios. , 2014, Acta psychologica.

[3]  Edit Yerushalmi,et al.  Guiding students to learn from mistakes , 2006 .

[4]  M. Frese,et al.  Action errors, error management, and learning in organizations. , 2015, Annual review of psychology.

[5]  M. Tulis Error Management Behavior in Classrooms: Teachers' Responses to Student Mistakes. , 2013 .

[6]  B. Rittle-Johnson,et al.  The effectiveness of using incorrect examples to support learning about decimal magnitude. , 2012 .

[7]  Michael A. McDaniel,et al.  The generation effect: A meta-analytic review , 2007, Memory & cognition.

[8]  Paul Gorsky,et al.  The role of anomaly and of cognitive dissonance in restructuring students' concepts of force , 1994 .

[9]  J. Sweller,et al.  Structuring Effective Worked Examples , 1990 .

[10]  Chun-Yi Lee,et al.  Effects of Worked Examples Using Manipulatives on Fifth Graders' Learning Performance and Attitude toward Mathematics , 2015, J. Educ. Technol. Soc..

[11]  Cor J. M. Suhre,et al.  How indirect supportive digital help during and after solving physics problems can improve problem-solving abilities , 2009, Comput. Educ..

[12]  J. Sweller The worked example effect and human cognition , 2006 .

[13]  Edgar Erdfelder,et al.  G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences , 2007, Behavior research methods.

[14]  Stellan Ohlsson,et al.  Learning from error and the design of task environments , 1996 .

[15]  M. Frese,et al.  Action as the core of work psychology: A German approach. , 1994 .

[16]  Stellan Ohlsson,et al.  Learning from Performance Errors. , 1996 .

[17]  The negative sign and exponential expressions: Unveiling students’ persistent errors and misconceptions , 2013 .

[18]  Michael M. Mueller,et al.  Errorless Learning: Review and Practical Application for Teaching Children with Pervasive Developmental Disorders. , 2007 .

[19]  G. Peeters,et al.  Positive-Negative Asymmetry in Evaluations: The Distinction Between Affective and Informational Negativity Effects , 1990 .

[20]  J. Metcalfe Learning from Errors , 2017, Annual review of psychology.

[21]  H. Mandl,et al.  Learning from Worked-Out Examples: The Effects of Example Variability and Elicited Self-Explanations , 1998, Contemporary educational psychology.

[22]  K. Begolli,et al.  Teaching Mathematics by Comparison: Analog Visibility as a Double-Edged Sword. , 2016 .

[23]  James W. Stigler,et al.  Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education , 1993 .

[24]  R. Santagata “Are you joking or are you sleeping?” Cultural beliefs and practices in Italian and U.S. teachers’ mistake-handling strategies , 2004 .

[25]  C. Henderson,et al.  Quiz Corrections: Improving Learning by Encouraging Students to Reflect on Their Mistakes , 2009 .

[26]  Aiso Heinze,et al.  The impact of incorrect examples on learning fractions: A field experiment with 6th grade students , 2014 .

[27]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .

[28]  James W. Stigler,et al.  The Learning Gap: Why our Schools are Failing and What We can Learn from Japanese and Chinese Education. , 1993 .

[29]  R. Siegler Microgenetic Studies of Self-Explanation , 2002 .

[30]  Julie L. Booth,et al.  Persistent and Pernicious Errors in Algebraic Problem Solving , 2014, J. Probl. Solving.

[31]  Julie L. Booth,et al.  Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge , 2020 .

[32]  Richard A Armstrong,et al.  When to use the Bonferroni correction , 2014, Ophthalmic & physiological optics : the journal of the British College of Ophthalmic Opticians.

[33]  J. Sweller,et al.  The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra , 1985 .

[34]  M. Frese,et al.  Effectiveness of error management training: a meta-analysis. , 2008, The Journal of applied psychology.

[35]  Julie L. Booth,et al.  Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples , 2013 .

[36]  J. Sweller,et al.  Worked example effects in individual and group work settings , 2010 .

[37]  Brian E. Townsend,et al.  THE REFLECTIVE CYCLE OF STUDENT ERROR ANALYSIS , 2006 .

[38]  Julie L. Booth,et al.  Learning Algebra by Example in Real-World Classrooms , 2015 .

[39]  Alexander Renkl,et al.  Finding and fixing errors in worked examples: Can this foster learning outcomes? ☆ , 2007 .

[40]  M. Goos,et al.  In search of practical wisdom: A conversation between researcher and teacher , 2006 .

[41]  F. Paas Training strategies for attaining transfer of problem-solving skill in statistics: A cognitive-load approach. , 1992 .

[42]  Raffaělla Borasi Capitalizing on Errors as "Springboards for Inquiry": A Teaching Experiment. , 1994 .

[43]  Arthur C. Graesser,et al.  Organizing Instruction and Study to Improve Student Learning. IES Practice Guide. NCER 2007-2004. , 2007 .

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

[45]  Richard E. Mayer,et al.  Using erroneous examples to improve mathematics learning with a web-based tutoring system , 2014, Comput. Hum. Behav..

[46]  William M. Carroll Using worked examples as an instructional support in the algebra classroom. , 1994 .

[47]  T. Perneger What's wrong with Bonferroni adjustments , 1998, BMJ.

[48]  Neil T. Heffernan,et al.  Tutored Problem Solving vs. “Pure” Worked Examples , 2009 .

[49]  Julie L. Booth,et al.  Support for Struggling Students in Algebra: Contributions of Incorrect Worked Examples. , 2016 .

[50]  Susanne M. Jaeggi,et al.  Executive function in learning mathematics by comparison: incorporating everyday classrooms into the science of learning , 2018, Thinking & reasoning.

[51]  Vera Cherepinsky,et al.  Self-Reflective Grading: Getting Students to Learn from their Mistakes , 2011 .

[52]  Hans Gruber,et al.  Learning in the workplace: new developments in the relation between working and learning , 2007 .

[53]  R. Atkinson,et al.  Transitioning From Studying Examples to Solving Problems: Effects of Self-Explanation Prompts and Fading Worked-Out Steps. , 2003 .

[54]  J. Bauer,et al.  Negative Knowledge: Understanding Professional Learning and Expertise , 2008 .

[55]  Julie L. Booth,et al.  Design-Based Research Within the Constraints of Practice: AlgebraByExample , 2015 .

[56]  R. Siegler,et al.  Differentiation and integration: guiding principles for analyzing cognitive change. , 2008, Developmental science.

[57]  Nicole M. McNeil A Change–Resistance Account of Children's Difficulties Understanding Mathematical Equivalence , 2014 .

[58]  John Dunlosky,et al.  Improving Students’ Learning With Effective Learning Techniques: Promising Directions From Cognitive and Educational Psychology , 2012 .

[59]  Jeffrey D. Karpicke,et al.  The Critical Importance of Retrieval for Learning , 2008, Science.