Prospective Middle School Mathematics Teachers’ Covariational Reasoning for Interpreting Dynamic Events During Peer Interactions

This study investigated the covariational reasoning abilities of prospective middle school mathematics teachers in a task about dynamic functional events involving two simultaneously changing quantities in an individual process and also in a peer interaction process. The focus was the ways in which prospective teachers’ covariational reasoning abilities re-emerge in the peer interaction process in excess of their covariational reasoning. The data sources were taken from the individual written responses of prospective teachers, transcripts of individual comments, and transcripts of conversations in pairs. The data were analyzed for prospective teachers in terms of the cognitive and interactive aspects of individual behavior and also interaction. The findings revealed that prospective teachers at different levels working in pairs benefited from the process in terms of developing an awareness of their own individual and also a pair’s understanding of covarying quantities. Furthermore, the prospective teachers had opportunities to develop their knowledge on the connection between variables, rate of change, and slope. The prospective teachers’ work in pairs provided salient explanations for their reasoning about the task superior to their individual responses.

[1]  D. Ball.  Prospective Elementary and Secondary Teachers' Understanding of Division. , 1990 .

[2]  Marilyn P. Carlson,et al.  Applying Covariational Reasoning While Modeling Dynamic Events: A Framework and a Study. , 2002 .

[3]  N. Webb,et al.  Constructive Activity and Learning in Collaborative Small Groups. , 1995 .

[4]  Ruhama Even.  Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept. , 1993 .

[5]  Charis Psaltis,et al.  Conservation and Conversation Types: Forms of Recognition and Cognitive Development. , 2007 .

[6]  Carolyn Kieran.  The learning and teaching of school algebra. , 1992 .

[7]  James D. Klein,et al.  Computer-mediated instruction: a comparison of online and face-to-face collaboration , 2008 .

[8]  Trudie Aberdeen.  Yin, R. K. (2009). Case study research: Design and methods (4th Ed.). Thousand Oaks, CA: Sage. , 2013, The Canadian Journal of Action Research.

[9]  Jere Confrey,et al.  Exponential functions, rates of change, and the multiplicative unit , 1994 .

[10]  K. Keene.  A Characterization of Dynamic Reasoning: Reasoning with Time as Parameter. , 2007 .

[11]  A. C. Kruger.  Peer collaboration: conflict, cooperation, or both? , 1993 .

[12]  Richard De Lisi,et al.  Implications of Piagetian theory for peer learning. , 1999 .

[13]  D. Zeidler,et al.  Tangled up in views: Beliefs in the nature of science and responses to socioscientific dilemmas , 2002 .

[14]  Patrick W Thompson,et al.  Students, Functions, and the Undergraduate Curriculum † , 1994 .

[15]  Heather Lynn Johnson,et al.  Reasoning about variation in the intensity of change in covarying quantities involved in rate of change , 2012 .

[16]  N. Webb.  Task-Related Verbal Interaction and Mathematics Learning in Small Groups. , 1991 .

[17]  Marilyn P. Carlson,et al.  The Precalculus Concept Assessment: A Tool for Assessing Students’ Reasoning Abilities and Understandings , 2010 .

[18]  Mary Budd Rowe,et al.  Graphing Skills among High School Physics Students , 1993 .

[19]  P. W. Thompson,et al.  RE-THINKING COVARIATION FROM A QUANTITATIVE PERSPECTIVE: SIMULTANEOUS CONTINUOUS VARIATION , 1998 .

[20]  Noreen M. Webb,et al.  Developing productive group interaction in middle school mathematics , 1999 .

[21]  J. Houssart,et al.  Conducting task‐based interviews with pairs of children: consensus, conflict, knowledge construction and turn taking , 2011 .

[22]  R. Yin.  Case Study Research: Design and Methods , 1984 .

[23]  Malcom Swan.  The Language of Functions and Graphs An Examination Module for Secondary Schools , 1982 .

[24]  M. Goos,et al.  Socially mediated metacognition: creating collaborative zones of proximal development in small group problem solving , 2002 .

[25]  J. Creswell.  Qualitative inquiry and research design: Choosing among five approaches, 2nd ed. , 2007 .

[26]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[27]  Draga Vidakovic,et al.  Understanding the limit concept: Beginning with a coordinated process scheme , 1996 .

[28]  Richard Lesh,et al.  Integrating a Models and Modeling Perspective With Existing Research and Practice , 2003 .

[29]  R. Leikin.  The wholes that are greater than the sum of their parts: employing cooperative learning in mathematics teachers’ education , 2004 .

[30]  Open Ended Realistic Division Problems, Generalisation and Early Algebra. , 2003 .

[31]  J. Confrey,et al.  Splitting, covariation, and their role in the development of exponential functions , 1995 .

[32]  David Moshman Molly Geil,et al.  Collaborative Reasoning: Evidence for Collective Rationality , 1998 .

[33]  Aysel Sen Zeytun,et al.  Mathematics Teachers' Covariational Reasoning Levels and Predictions about Students' Covariational Reasoning Abilities * , 2010 .

[34]  Marie E. Karlsson.  Emotional identification with teacher identities in student teachers’ narrative interaction , 2013 .

[35]  Alan H. Schoenfeld,et al.  Research in Collegiate Mathematics Education. I , 1994 .

[36]  Resolving cognitive conflict with peers: Is there a difference between two and four? , 2007 .

[37]  Christian D. Schunn,et al.  The effects of skill diversity on commenting and revisions , 2013 .

[38]  Shlomo Vinner.  The Pseudo-Conceptual and the Pseudo-Analytical Thought Processes in Mathematics Learning , 1997 .

[39]  B. Schwarz,et al.  Constructing and Consolidating of Algebraic Knowledge within Dyadic Processes: A Case Study , 2006 .

[40]  Ed Dubinsky,et al.  The Concept of Function: Aspects of Epistemology and Pedagogy [MAA Notes, Volume 25] , 1992 .

[41]  Angela M. O'Donnell,et al.  Cognitive Perspectives on Peer Learning , 1999 .

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

[43]  O. Zaslavsky,et al.  Cooperative problem solving in combinatorics: the inter-relations between control processes and successful solutions , 2003 .

[44]  R. Yin,et al.  Case Study Research: Design and Methods (4th ed. , 2009 .