The Concept of Invariance and Change

Our world is a constant stream of different phenomena. Human attempts to understand the world and the various phenomena that it entails, often lead to a need to determine which properties remain invariant despite changes in a situation. This quest arises from the understanding that some properties and characteristics of an object can change while others remain invariant. Obviously, if one state is exactly the same as another, then no doubt exists that all aspects are exactly the same in both. However, once any changes are observed, considerations arise regarding the source, causes, and forces that lead to them.

[1]  Liora Linchevski,et al.  A cognitive gap between arithmetic and algebra , 1994 .

[2]  Jacqueline Grennon Brooks,et al.  The Courage To Be Constructivist. , 1999 .

[3]  Hung-hsi Wu The Mis-Education of Mathematics Teachers , 2011 .

[4]  D. Schifter,et al.  Classroom Stories: Examples of Elementary Students Engaged in Early Algebra , 2017 .

[5]  Michał Głażewski,et al.  "Ways of Learning to Teach. A Philosophically Inspired Analysis of Teacher Education Programs", Shlomo Back, Rotterdam/Boston/Taipei 2012 : [recenzja] / Michał Głażewski. , 2012 .

[6]  L. Vygotsky Mind in Society: The Development of Higher Psychological Processes: Harvard University Press , 1978 .

[7]  Luis Radford,et al.  Algebraic thinking from a cultural semiotic perspective , 2010 .

[8]  E. Fischbein,et al.  Intuition in Science and Mathematics: An Educational Approach , 2014 .

[9]  James G. Greeno,et al.  The development of semantic categories for addition and subtraction , 1982 .

[10]  Luis Radford,et al.  The Anthropology of Meaning , 2006 .

[11]  Liora Linchevski,et al.  Algebra with Numbers and Arithmetic with Letters: A Definition of Pre-algebra. , 1995 .

[12]  J. Mason Expressing Generality and Roots of Algebra , 1996 .

[13]  M. Blanton,et al.  Characterizing a Classroom Practice that Promotes Algebraic Reasoning. , 2005 .

[14]  Arthur W. Collins,et al.  Kant and the Critique of Pure Reason , 1999 .

[15]  Suzanne R. Harper,et al.  Promoting Appropriate Uses of Technology in Mathematics Teacher Preparation , 2000 .

[16]  Liping Ma,et al.  Knowing and Teaching Elementary Mathematics Teachers' Understanding of Fundamental Mathematics in China and the United States , 2010 .

[17]  Ruth Stavy,et al.  How Students Mis/Understand Science and Mathematics: Intuitive Rules (Ways of Knowing in Science Series) , 2000 .

[18]  S. Drake,et al.  Galileo at Work: His Scientific Biography , 1969 .

[19]  Diana F. Steele,et al.  Seventh-grade students’ representations for pictorial growth and change problems , 2008 .

[20]  Hyman Bass,et al.  Knowing Mathematics for Teaching Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? , 2005 .

[21]  Z. Usiskin Why Is Algebra Important to Learn , 1995 .

[22]  Rina Zazkis,et al.  Generalization of patterns: the tension between algebraic thinking and algebraic notation , 2002 .

[23]  J. Piaget The Child's Conception of Number , 1953 .

[24]  D. Tirosh,et al.  Evoking Cognitive Conflict to Explore Preservice Teachers' Thinking about Division. , 1990 .

[25]  B. Hofer Instructional Context in the College Mathematics Classroom: Epistemological Beliefs and Student Motivation. , 1999 .

[26]  B. Flexer PREDICTING EIGHTH-GRADE ALGEBRA ACHIEVEMENT , 1984 .

[27]  A. Morris Factors Affecting Pre-Service Teachers' Evaluations of the Validity of Students' Mathematical Arguments in Classroom Contexts , 2007 .

[28]  Richard Gunstone,et al.  The fluid/gravity correspondence , 2011, 1107.5780.

[29]  Jeanne L. Tunks,et al.  Changing practice, changing minds, from arithmetical to algebraic thinking: an application of the concerns-based adoption model (CBAM) , 2009 .

[30]  P. Freire Pedagogy of the Oppressed, 30th Anniversary Edition , 1970 .

[31]  Carl B. Boyer,et al.  A History of Mathematics. , 1993 .

[32]  Kenneth R. Koedinger,et al.  An Investigation of Teachers' Beliefs of Students' Algebra Development , 2000, Cognition and Instruction.

[33]  Lieven Verschaffel,et al.  Pre-service Teachers' Preferred Strategies for Solving Arithmetic and Algebra Word Problems , 2003 .

[34]  Edmund Husserl,et al.  Experience and judgment: Investigations in a genealogy of logic , 1973 .

[35]  R. Skemp Relational Understanding and Instrumental Understanding. , 2006 .

[36]  David W. Carraher,et al.  Early Algebra Is Not the Same as Algebra Early , 2017 .

[37]  Ralph T. Putnam,et al.  Learning to teach. , 1996 .

[38]  L. Shulman Those Who Understand: Knowledge Growth in Teaching , 1986 .