An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Abstract Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.

[1]  Jerry P. Becker,et al.  Elementary School Practices , 1996 .

[2]  Anna Craft.  Creativity Across the Primary Curriculum: Framing and Developing Practice , 2000 .

[3]  Robert M. Groves,et al.  Actors and Questions in Telephone and Personal Interview Surveys , 1979 .

[4]  M. Patton.  Qualitative research and evaluation methods , 1980 .

[5]  DaiYoung Kwon,et al.  Using puzzles: problem-solving and abstraction , 2007, SIGITE '07.

[6]  D. Johanning.  Supporting the development of algebraic thinking in middle school: a closer look at students’ informal strategies , 2004 .

[7]  Mary Grace Kantowski.  PROCESSES INVOLVED IN MATHEMATICAL PROBLEM SOLVING , 1977 .

[8]  Patricia S. Wilson,et al.  Research Ideas for the Classroom: High School Mathematics. , 1993 .

[9]  M. Pajares,et al.  Teachers’ Beliefs and Educational Research: Cleaning Up a Messy Construct , 1992 .

[10]  Roger W. Shuy,et al.  26 In-Person Versus Telephone Interviewing , 2001 .

[11]  James A. Holstein,et al.  Handbook of Interview Research: Context and Method , 2001 .

[12]  Thomas J. Cooney.  A Beginning Teacher's View of Problem Solving. , 1985 .

[13]  Kenneth R. Koedinger,et al.  Teachers' and Researchers' Beliefs about the Development of Algebraic Reasoning. , 2000 .

[14]  Anthony Ralston,et al.  Knowing and teaching elementary mathematics , 1999 .

[15]  A. Thompson.  The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice , 1984 .

[16]  Mark J. Driscoll,et al.  Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10 , 1999 .

[17]  Shigeru Shimada,et al.  The Open-Ended Approach: A New Proposal for Teaching Mathematics , 1997 .

[18]  Peter Kloosterman,et al.  Measuring Beliefs about Mathematical Problem Solving. , 1992 .

[19]  Mathematical Investigation: Description, Rationale, and Example. , 1988 .

[20]  G. Pólya,et al.  How to Solve It. A New Aspect of Mathematical Method. , 1945 .

[21]  L. Lesser,et al.  Mathematical Power: Exploring Critical Pedagogy in Mathematics and Statistics. , 2007 .

[22]  Jesse Goodman.  Reflection and teacher education: A case study and theoretical analysis , 1984 .

[23]  Ann V. McGillicuddy-De Lisi,et al.  Gender differences in advanced mathematical problem solving. , 2000, Journal of experimental child psychology.

[24]  Merle L. Borrowman,et al.  Teacher education in America : a documentary history , 1966 .

[25]  J. Dewey.  How we think : a restatement of the relation of reflective thinking to the educative process , 1934 .

[26]  Kaye Stacey,et al.  Learning the Algebraic Method of Solving Problems , 1999 .

[27]  L. Shulman.  Knowledge and Teaching: Foundations of the New Reform , 1987 .

[28]  Jeremy Kilpatrick,et al.  International handbook of mathematics education , 1997 .

[29]  Juliet M. Corbin,et al.  Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory , 2008 .

[30]  James A. Holstein,et al.  Handbook of Interview Research , 2001 .

[31]  Stephen Krulik,et al.  Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12: A Resource for the Mathematics Teacher , 2008 .

[32]  Jinfa Cai,et al.  The evolution of mathematical explorations in open-ended problem-solving situations , 2005 .

[33]  Robert Karplus.  Proceedings of the fourth International Conference for the Psychology of Mathematics Education , 1980 .

[34]  E. Silver.  Metacognitive and Epistemological Issues in Mathematical Understanding , 2013 .

[35]  Capturing the structure of japanese mathematics lessons: some findings of the international comparative studies , 2009 .

[36]  J. Kilpatrick.  The mathematics teacher and curriculum change , 2009, PNA. Revista de Investigación en Didáctica de la Matemática.

[37]  Jeremy Kilpatrick,et al.  Adding It Up: Helping Children Learn Mathematics , 2013 .

[38]  M. G. Jones,et al.  An Investigation of African American Students' Mathematical Problem Solving. , 1998 .

[39]  Alan H. Schoenfeld,et al.  Mathematical Problem Solving , 1985 .

[40]  J. Lannin.  Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities , 2005 .

[41]  A. Strauss,et al.  Grounded theory , 2017 .

[42]  John G. Hedberg,et al.  Teachers' pedagogies and their impact on students' mathematical problem solving , 2005 .

[43]  Grayson H. Wheatley,et al.  The role of reflection in mathematics learning , 1992 .

[44]  Edward A. Silver,et al.  Teaching and Learning Mathematical Problem Solving : Multiple Research Perspectives , 1985 .

[45]  T. P. Carpenter,et al.  Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics , 1996 .

[46]  A. Strauss,et al.  Basics of Qualitative Research , 1992 .

[47]  R. Allen,et al.  Reinventing critical pedagogy , 2006 .

[48]  D. Mewborn.  Reflective Thinking among Preservice Elementary Mathematics Teachers. , 1999 .

[49]  D. Johanning.  Is There Something to be Gained from Guessing? Middle School Students' Use of Systematic Guess and Check , 2007 .