Middle School Students' and Mathematicians' Judgments of Mathematical Typicality

K-12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K-12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can limit inductive generalization. Middle school students largely did not differentiate between everyday and mathematical typicality – they did not view special mathematical properties as limiting generalization, and rated items similarly regardless of context. These results suggest directions for learning mathematical argumentation and represent an important step towards understanding the nature of typicality in math.

[1]  D. Kuhn.  Children and adults as intuitive scientists. , 1989, Psychological review.

[2]  P. Herbst.  Establishing a custom of proving in american school geometry: evolution of the two-column proof in the early twentieth century , 2002 .

[3]  D. Medin,et al.  Categorization and Reasoning among Tree Experts: Do All Roads Lead to Rome? , 1997, Cognitive Psychology.

[4]  C. Hoyles,et al.  A Study of Proof Conceptions in Algebra , 2000 .

[5]  Kristen N. Bieda,et al.  Middle School Students' Production of Mathematical Justifications , 2009 .

[6]  B. H. Pillow,et al.  Children's and adults' evaluation of the certainty of deductive inferences, inductive inferences, and guesses. , 2002, Child development.

[7]  Lois W. Sayrs.  Interviews : an introduction to qualitative research interviewing , 1996 .

[8]  Marjorie Rhodes,et al.  Sample diversity and premise typicality in inductive reasoning: Evidence for developmental change , 2008, Cognition.

[9]  S. Gelman,et al.  The importance of knowing a dodo is a bird: Categories and infer - ences in two - year olds , 1990 .

[10]  Caroline C. Williams,et al.  Adolescent Reasoning in Mathematical and Non-Mathematical Domains: Exploring the Paradox , 2012 .

[11]  Charles W. Kalish,et al.  Adolescent Reasoning in Mathematics: Exploring Middle School Students' Strategic Approaches in Empirical Justifications , 2011, CogSci.

[12]  Shawn L. Ward,et al.  Semantic Familiarity, Relevance, and the Development of Deductive Reasoning , 1990 .

[13]  Jaguthsing Dindyal,et al.  The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof , 2005 .

[14]  O. Zaslavsky,et al.  Is this a coincidence? The role of examples in fostering a need for proof , 2011 .

[15]  Edward E. Smith,et al.  Category-Based Induction , 1990 .

[16]  V. Sloutsky,et al.  Conceptual Development , 2014 .

[17]  J. Sandefur,et al.  Generating and using examples in the proving process , 2013 .

[18]  Dana Pomykal Franz,et al.  Common Core State Standards for Mathematics and Assessments: (mis)Alignment , 2012 .

[19]  G. Stylianides.  AN ANALYTIC FRAMEWORK OF REASONING-AND-PROVING , 2008 .

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

[21]  Interventions in Children's Deductive Reasoning with Indeterminate Problems , 1997, Contemporary educational psychology.

[22]  Kurt VanLehn,et al.  Seeing Deep Structure From the Interactions of Surface Features , 2012 .

[23]  N. Chater,et al.  Rational models of cognition , 1998 .

[24]  D. Kahneman,et al.  Conditions for intuitive expertise: a failure to disagree. , 2009, The American psychologist.

[25]  Ian Hacking,et al.  An Introduction to Probability and Inductive Logic: Contents , 2001 .

[26]  Karen J. Graham,et al.  Focus in High School Mathematics: Reasoning and Sense Making in Algebra , 2010 .

[27]  John Mason,et al.  Mathematics as a Constructive Activity: Learners Generating Examples , 2005 .

[28]  David M. Sobel,et al.  A theory of causal learning in children: causal maps and Bayes nets. , 2004, Psychological review.

[29]  D. Epstein,et al.  Experimentation and proof in mathematics , 1995 .

[30]  Kenneth R. Koedinger,et al.  Conjecturing and Argumentation in High-School Geometry Students , 2012 .

[31]  Andreas J. Stylianides.  Proof and Proving in School Mathematics , 2007 .

[32]  G. Lakoff,et al.  Women, Fire, and Dangerous Things: What Categories Reveal about the Mind , 1988 .

[33]  Edward E. Smith,et al.  The development of category-based induction. , 1992, Child development.

[34]  Gary Lupyan,et al.  The difficulties of executing simple algorithms: Why brains make mistakes computers don’t , 2013, Cognition.

[35]  J. Piaget.  Judgement and Reasoning in the Child , 1962 .

[36]  Sharon Lee Armstrong,et al.  What some concepts might not be , 1983, Cognition.

[37]  Susan Carey,et al.  The emergence of reasoning by the disjunctive syllogism in early childhood , 2016, Cognition.

[38]  T. Dreyfus.  Why Johnny Can't Prove , 1999 .

[39]  V. Reyna.  A new intuitionism: Meaning, memory, and development in Fuzzy-Trace Theory. , 2012, Judgment and decision making.

[40]  Evan Heit,et al.  Inductive reasoning: Experimental, developmental and computational approaches , 2007 .

[41]  Robert S. Stawski,et al.  Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling (2nd Edition) , 2013 .

[42]  A. Watson,et al.  The structuring of personal example spaces , 2011 .

[43]  J. Feldman.  Bias toward regular form in mental shape spaces. , 2000, Journal of experimental psychology. Human perception and performance.

[44]  Alison G. Lynch,et al.  Mathematicians’ Example-Related Activity when Exploring and Proving Conjectures , 2016 .

[45]  Keith Porteous.  What do children really believe? , 1990 .

[46]  Understanding students' similarity and typicality judgments in and out of mathematics , 2011 .

[47]  John Mason,et al.  Shedding light on and with example spaces , 2008 .

[48]  I. Hacking.  An Introduction to Probability and Inductive Logic , 2001 .

[49]  G. Murphy,et al.  The Big Book of Concepts , 2002 .

[50]  Caroline C. Williams,et al.  A Framework for Mathematicians’ Example-Related Activity When Exploring and Proving Mathematical Conjectures , 2012 .

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

[52]  Elisabeth Dévière,et al.  Analyzing linguistic data: a practical introduction to statistics using R , 2009 .

[53]  Matthew Inglis,et al.  Doctoral students’ use of examples in evaluating and proving conjectures , 2008 .

[54]  G. Harel.  Students' proof schemes revisited , 2007 .

[55]  Joan Ferrini-Mundy,et al.  Principles and Standards for School Mathematics: A Guide for Mathematicians , 2000 .

[56]  A. Strauss,et al.  The discovery of grounded theory: strategies for qualitative research aldine de gruyter , 1968 .

[57]  S. Gelman.  The development of induction within natural kind and artifact categories , 1988, Cognitive Psychology.