Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking

This paper describes the importance of developing students’ reasoning about samples and sampling variability as a foundation for statistical thinking. Research on expert–novice thinking as well as statistical thinking is reviewed and compared. A case is made that statistical thinking is a type of expert thinking, and as such, research comparing novice and expert thinking can inform the research on developing statistical thinking in students. It is also posited that developing students’ informal inferential reasoning, akin to novice thinking, can help build the foundations of experts’ statistical thinking.

[1]  John Mason,et al.  LOCAL AND GLOBAL THINKING IN STATISTICAL INFERENCE , 2008 .

[2]  Lyn D. English,et al.  Handbook of International Research in Mathematics Education , 2002 .

[3]  Dani Ben-Zvi,et al.  Junior high school students' construction of global views of data and data representations , 2001 .

[4]  Ian Putt,et al.  A Framework for Characterizing Children's Statistical Thinking , 2000 .

[5]  Jere Confrey,et al.  COMPARING TWO DISTRIBUTIONS: INVESTIGATING SECONDARY TEACHERS' STATISTICAL THINKING , 2002 .

[6]  Ernesto Sanchez,et al.  TRAINING IN-SERVICE TEACHERS TO DEVELOP STATISTICAL THINKING , 2010 .

[7]  Lynn Arthur Steen,et al.  Heeding the call for change : suggestions for curricular action , 1992 .

[8]  R. Kozma,et al.  Multimedia and understanding: Expert and novice responses to different representations of chemical phenomena , 1997 .

[9]  Beth Chance,et al.  Reasoning about Sampling Distribitions , 2004 .

[10]  Jason Martin Differences between experts’ and students’ conceptual images of the mathematical structure of Taylor series convergence , 2012, Educational Studies in Mathematics.

[11]  L. Resnick Treating Mathematics as an Ill-Structured Discipline. , 1988 .

[12]  Dani Ben-Zvi,et al.  The Role of Context in Developing Reasoning about Informal Statistical Inference , 2011 .

[13]  Uri Wilensky,et al.  What is Normal Anyway? Therapy for Epistemological Anxiety , 1997 .

[14]  Herbert A. Simon,et al.  Models of Competence in Solving Physics Problems , 1980, Cogn. Sci..

[15]  John H. Maindonald,et al.  This Passionate Study: A Dialogue with Florence Nightingale , 2004 .

[16]  J. Lave Cognition in Practice: Outdoors: a social anthropology of cognition in practice , 1988 .

[17]  Carmen Batanero Bernabeu Joint ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges for Teaching and Teacher Education , 2011 .

[18]  Paul Cobb,et al.  Symbolizing and communicating in mathematics classrooms : perspectives on discourse, tools, and instructional design , 2000 .

[19]  Daniel L. Schwartz,et al.  EFFICIENCY AND INNOVATION IN TRANSFER , 2005 .

[20]  Clifford Konold,et al.  Teaching Concepts Rather Than Conventions , 2002 .

[21]  D. Perkins,et al.  Informal reasoning and education , 1991 .

[22]  Beth Chance,et al.  Components of Statistical Thinking and Implications for Instruction and Assessment , 2002 .

[23]  J. Mestre Transfer of learning from a modern multidisciplinary perspective , 2005 .

[24]  Nicholas P. Vitalari,et al.  Differences Between Novice and Expert Systems Analysts: What Do We Know and What Do We Do? , 1998, J. Manag. Inf. Syst..

[25]  Chris J. Wild The Concept of Distribution. , 2006 .

[26]  Allan J. Rossman,et al.  REASONING ABOUT INFORMAL STATISTICAL INFERENCE: ONE STATISTICIAN’S VIEW , 2008 .

[27]  Joan Garfield,et al.  Developing statistical modelers and thinkers in an introductory, tertiary-level statistics course , 2012 .

[28]  Maxine Pfannkuch,et al.  INFORMAL INFERENTIAL REASONING , 2006 .

[29]  Dani Ben-Zvi SCAFFOLDING STUDENTS' INFORMAL INFERENCE AND ARGUMENTATION , 2006 .

[30]  Dani Ben-Zvi,et al.  The Reasoning Behind Informal Statistical Inference , 2011 .

[31]  George W Cobb,et al.  The Introductory Statistics Course: A Ptolemaic Curriculum? , 2007 .

[32]  Kim I. Melton Statistical Thinking Activities: Some Simple Exercises with Powerful Lessons , 2004 .

[33]  Joan Garfield,et al.  The Role of Technology in Improving Student Learning of Statistics , 2007 .

[34]  Dani Ben-Zvi,et al.  REASONING ABOUT VARIABILITY IN COMPARING DISTRIBUTIONS 4 , 2004 .

[35]  Developing Statistical Thinking in a Secondary School: A Collaborative Curriculum Development , 2009 .

[36]  Christian Rüede How secondary level teachers and students impose personal structure on fractional expressions and equations—an expert-novice study , 2013 .

[37]  Randall E. Groth An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value , 2005 .

[38]  Todd Swanson,et al.  Development and Assessment of a Preliminary Randomization-Based Introductory Statistics Curriculum , 2011 .

[39]  Rolf Biehler,et al.  TOWARDS REQUIREMENTS FOR MORE ADEQUATE SOFTWARE TOOLS THAT SUPPORT BOTH: LEARNING AND DOING STATISTICS , 2007 .

[40]  Anna Reid,et al.  Students' Conceptions of Statistics: A Phenomenographic Study , 2002 .

[41]  P. Nesher,et al.  Shifts in reasoning , 1986 .

[42]  J. Greeno,et al.  Thinking Practices in Mathematics and Science Learning , 1998 .

[43]  Joan Garfield,et al.  The challenge of developing statistical literacy, reasoning and thinking , 2004 .

[44]  Arthur Bakker,et al.  Lessons from Inferentialism for Statistics Education , 2011 .

[45]  Dani Ben-Zvi,et al.  Students’ emergent articulations of uncertainty while making informal statistical inferences , 2012 .

[46]  Richard Noss,et al.  Designing for Mathematical Abstraction , 2010, Int. J. Comput. Math. Learn..

[47]  Dani Ben-Zvi,et al.  Technology for enhancing statistical reasoning at the school level , 2012 .

[48]  C. Wild,et al.  Statistical Thinking in Empirical Enquiry , 1999 .

[49]  A. Tversky,et al.  BELIEF IN THE LAW OF SMALL NUMBERS , 1971, Pediatrics.

[50]  L. R. Novick Analogical transfer, problem similarity, and expertise. , 1988, Journal of experimental psychology. Learning, memory, and cognition.

[51]  Joyce J. Elam,et al.  A Comparative Study of How Experts and Novices Use a Decision Aid to Solve Problems in Complex Knowledge Domains , 1992, Inf. Syst. Res..

[52]  Andee Rubin,et al.  A FRAMEWORK FOR THINKING ABOUT INFORMAL STATISTICAL INFERENCE , 2009, STATISTICS EDUCATION RESEARCH JOURNAL.

[53]  Arthur Bakker,et al.  REASONING ABOUT SHAPE AS A PATTERN IN VARIABILITY , 2004, STATISTICS EDUCATION RESEARCH JOURNAL.

[54]  Jane Watson,et al.  Developing Reasoning about Samples , 2004 .

[55]  Kevin F. Collis,et al.  Evaluating the Quality of Learning: The SOLO Taxonomy , 1977 .

[56]  Joan Garfield,et al.  Developing Students’ Statistical Reasoning: Connecting Research and Teaching Practice , 2008 .

[57]  John W. Tukey,et al.  Exploratory Data Analysis. , 1979 .

[58]  A. Schoenfeld Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint) , 2009 .

[59]  Edward A. Silver,et al.  The teaching and assessing of mathematical problem solving , 1988 .

[60]  John R. Anderson,et al.  How people learn to skip steps. , 1996 .

[61]  Ann L. Brown,et al.  How people learn: Brain, mind, experience, and school. , 1999 .

[62]  J. Roschelle,et al.  Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition , 1994 .

[63]  Dani Ben-Zvi,et al.  Developing Teachers’ Statistical Thinking , 2011 .

[64]  Paul J. Feltovich,et al.  Categorization and Representation of Physics Problems by Experts and Novices , 1981, Cogn. Sci..

[65]  Jeremy Kilpatrick,et al.  Third International Handbook on Mathematics Education , 2003 .

[66]  Patrick W Thompson,et al.  Conceptions of sample and their relationship to statistical inference , 2002 .

[67]  Robert G. Gould,et al.  VARIABILITY: ONE STATISTICIAN'S VIEW , 2004, STATISTICS EDUCATION RESEARCH JOURNAL.

[68]  Paul Cobb,et al.  Symbolizing, Modeling, and Instructional Design , 2000 .