1. Implementing Constructivism in Mathematics Classrooms Tomas Cometto 20 March 2008 Tomas Cometto 20 March 2008

2. Outline  Define Constructivism  Aspects of implementing constructivism in math classrooms  Math Anxiety  Role of Instructor  Development of Math knowledge  Counting  Patterns  Functions  Define Constructivism  Aspects of implementing constructivism in math classrooms  Math Anxiety  Role of Instructor  Development of Math knowledge  Counting  Patterns  Functions

3. What is Constructivism?  Theory of knowledge, assumes that “knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience” 1  Many applications: politics, philosophy, science, worldview  Specifically, we will view constructivism as a theory of teaching and learning in math classrooms  New, specific definition: constructivism dictates that students learn new math concepts though previous knowledge structures.  Theory of knowledge, assumes that “knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience” 1  Many applications: politics, philosophy, science, worldview  Specifically, we will view constructivism as a theory of teaching and learning in math classrooms  New, specific definition: constructivism dictates that students learn new math concepts though previous knowledge structures. 1 Von Glasersfeld, Ernst Radical Constructivism. Washington, D.C.: The Falmer Press, 1995

4. Math Anxiety  Common affliction in our culture  One reason - Students don’t connect “school math” to their own experiences  Through constructivism, meaning is intimately connected with experience  3 ways to alleviate math anxiety using constructivism  Connect previous knowledge (formal and informal) to new material  Introduce applications before formal computations  Bring out students’ intuitive understanding of mathematics  Common affliction in our culture  One reason - Students don’t connect “school math” to their own experiences  Through constructivism, meaning is intimately connected with experience  3 ways to alleviate math anxiety using constructivism  Connect previous knowledge (formal and informal) to new material  Introduce applications before formal computations  Bring out students’ intuitive understanding of mathematics

5. Connecting previous formal knowledge to new material  Math has a specific sequence of topics  Algebra, Geometry, Algebra 2, Pre Calculus, Calculus  Calculus 1-Calculus 2-Multivariable Calculus  “Calculus teachers lament that students find the subject difficult not because derivatives and integrals are abstruse concepts--they're just rate and accumulation--but because you can't do calculus unless algebraic operations are second nature. Most students enter the course without having learned the algebra properly and need to concentrate every drop of mental energy on that.” 2  Constructivist instructors must engage previous formal understandings, rather than just assume students mastered the previous material.  Math has a specific sequence of topics  Algebra, Geometry, Algebra 2, Pre Calculus, Calculus  Calculus 1-Calculus 2-Multivariable Calculus  “Calculus teachers lament that students find the subject difficult not because derivatives and integrals are abstruse concepts--they're just rate and accumulation--but because you can't do calculus unless algebraic operations are second nature. Most students enter the course without having learned the algebra properly and need to concentrate every drop of mental energy on that.” 2  Constructivist instructors must engage previous formal understandings, rather than just assume students mastered the previous material. 2 Steven Pinker. How the Mind Works. New York: W. W. Norton & Company 1997

6. Connecting previous informal knowledge to new material  This aspect of constructivist teaching makes students aware of the applications of math  Relating to $$$  Bob Moses’ Algebra Project  Connect previous informal knowledge (riding the subway) to new material (adding and subtracting integers)  Helps students construct their understanding of new material through previous experiences in the real world  This aspect of constructivist teaching makes students aware of the applications of math  Relating to $$$  Bob Moses’ Algebra Project  Connect previous informal knowledge (riding the subway) to new material (adding and subtracting integers)  Helps students construct their understanding of new material through previous experiences in the real world

7. Water Vase Example  Water is poured into this vase  Graph height of water vs. time  Water is poured into this vase  Graph height of water vs. time abc H T a b c

8. Formal Methods of Computation vs. Application  Traditional method: Formal methods then applications  Reverse would likely yield more effective learning  Students must have a familiar concept to relate new material to  Derivative - position, velocity, acceleration  Concavity - Water vase example  Perhaps alleviate math anxiety  Traditional method: Formal methods then applications  Reverse would likely yield more effective learning  Students must have a familiar concept to relate new material to  Derivative - position, velocity, acceleration  Concavity - Water vase example  Perhaps alleviate math anxiety

9. Bring out students’ intuitive understanding of math  Many students understand math to be following rules to guarantee correct answers  Math should be about solving problems!  Short math puzzles  Math talk  Many math classrooms don’t involve discussion  Makes students’ thinking visible  “an exchange of ideas and mutual control are essential for children’s development of logic.” 3  Many students understand math to be following rules to guarantee correct answers  Math should be about solving problems!  Short math puzzles  Math talk  Many math classrooms don’t involve discussion  Makes students’ thinking visible  “an exchange of ideas and mutual control are essential for children’s development of logic.” 3 3 Kamii, Constance. “Constructivism and Beginning Arithmetic (K-2).” In Teaching and Learning Mathematics in the 1990s: 1990 Yearbook. T. J. Cooney, Ed. NCTM, 1990

10. Role of Instructor LecturerFacilitator Dictates materialGuides Students to the material TellsAsks/Involved in dialogue with students Teaches from the frontSupports by walking around Gives answersProvides guidelines Learner is passiveLearner is active Focus on the materialFocus on the student

11. Development of math knowledge  “Math knowledge is constructed, at least in part, through a process of reflective abstraction, and those cognitive structures are under continual development.” 4  “Mathematics is ruthlessly cumulative, all the way back to counting to ten.” 5  Specifically, we will see how students construct their understanding of counting, patterns, and functions.  “Math knowledge is constructed, at least in part, through a process of reflective abstraction, and those cognitive structures are under continual development.” 4  “Mathematics is ruthlessly cumulative, all the way back to counting to ten.” 5  Specifically, we will see how students construct their understanding of counting, patterns, and functions.  4 Ferrini-Mundy, J. & Graham, K. “Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals.”  In Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA Notes Number 33.  Mathematical Association of America. 1994 5 Steven Pinker. How the Mind Works. New York: W. W. Norton & Company 1997

12. Counting  Jean Piaget’s theory of how children construct number concepts  Order integers and objects into 1-1 relationship  “Children become able to think about the objects as ‘eight’ only when they can impose their logico-mathematical knowledge, that is, self-created relationships, on the set.” 6  Many adults have forgotten this process, they consider counting intuitive.  Jean Piaget’s theory of how children construct number concepts  Order integers and objects into 1-1 relationship  “Children become able to think about the objects as ‘eight’ only when they can impose their logico-mathematical knowledge, that is, self-created relationships, on the set.” 6  Many adults have forgotten this process, they consider counting intuitive. 6 Kamii, Constance. “Constructivism and Beginning Arithmetic (K-2).” In Teaching and Learning Mathematics in the 1990s: 1990 Yearbook. T. J. Cooney, Ed. NCTM, 1990

13. Patterns  Robert Quinn’s lesson plan:  * * * * * *  * * * * * * * * * 4 th = ?  * * * * * * * * * * * *  * * * * * * * * * * * * * * *  “they [students] should be allowed to explore situations in which pattern recognition plays a vital role in the construction of important mathematical knowledge.” 7  “construct numerical relationships through their own natural ability to think.” 7  Different formulas for the n th group of dots:  a n = 6 + 4n  a n = 10 + (n-1)*4  a n = (n) + (n+1) + (n+2) + (n+3)  Previous + 4, or a n = a n-1 + 4, a 0 = 10  Robert Quinn’s lesson plan:  * * * * * *  * * * * * * * * * 4 th = ?  * * * * * * * * * * * *  * * * * * * * * * * * * * * *  “they [students] should be allowed to explore situations in which pattern recognition plays a vital role in the construction of important mathematical knowledge.” 7  “construct numerical relationships through their own natural ability to think.” 7  Different formulas for the n th group of dots:  a n = 6 + 4n  a n = 10 + (n-1)*4  a n = (n) + (n+1) + (n+2) + (n+3)  Previous + 4, or a n = a n-1 + 4, a 0 = 10 7 Quinn, Robert. “A Constructivist Lesson to Introduce Arithmetic Sequences with Patterns.” In Australian Mathematics Teacher, v61 n4. Adelaide: Australian Association of Mathematics Teachers (AAMT), 2005 p18-21

14. Functions before calculus  “When Ted got home from his waiter job,he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much did Ted make per hour?” 8  “Starting with some number, if I multiplied it by 6 and then added 66, I get 81.9. What number did I start with?” 8  Solve for x: x*6 + 66 = 81.9 8  66% correctly answered the story problem, 62% correctly answered the word problem, and only 43% correctly answered the equation. 8  “When Ted got home from his waiter job,he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much did Ted make per hour?” 8  “Starting with some number, if I multiplied it by 6 and then added 66, I get 81.9. What number did I start with?” 8  Solve for x: x*6 + 66 = 81.9 8  66% correctly answered the story problem, 62% correctly answered the word problem, and only 43% correctly answered the equation. 8 8. National Research Council of the National Academies. How students learn: Mathematics in the classroom. Washington, DC: The National Academies Press, 2005.

15. Functions in Calculus  Require a much deeper understanding of functions than in previous courses.  Many beginning calculus students have a “primitive understanding of functions.” 9  Functions “trigger” a search for an algebraic formula defining the function. 9  Require a much deeper understanding of functions than in previous courses.  Many beginning calculus students have a “primitive understanding of functions.” 9  Functions “trigger” a search for an algebraic formula defining the function. 9 9 Ferrini-Mundy, J. & Graham, K. “Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals.” In Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA Notes Number 33. Mathematical Association of America. 1994

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18. Conclusions of this study  “Calculus students will actively formulate their own theories, build their own connections, and readily construct meaning for problem situations. These processes seem to be influenced strongly by previous experience and knowledge. There are powerful tendencies to call upon familiar examples and frequently-used patterns” 10  “Startling inconsistencies exist between performance, particularly on procedural items, and conceptual understanding. Traditional means of assessment in calculus are almost certain to mask the nature of student understanding” 10  “Calculus students will actively formulate their own theories, build their own connections, and readily construct meaning for problem situations. These processes seem to be influenced strongly by previous experience and knowledge. There are powerful tendencies to call upon familiar examples and frequently-used patterns” 10  “Startling inconsistencies exist between performance, particularly on procedural items, and conceptual understanding. Traditional means of assessment in calculus are almost certain to mask the nature of student understanding” 10 10 Ferrini-Mundy, J. & Graham, K. “Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals.” In Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA Notes Number 33. Mathematical Association of America. 1994

19. Functions after calculus  Discrete math:  Relation between two sets  Classify functions  Equivalence relation (reflexive, symmetric, transitive)  Even/odd functions  Multivariable calculus: Functions with more than one variable  Functions develop through every year of math instruction  Discrete math:  Relation between two sets  Classify functions  Equivalence relation (reflexive, symmetric, transitive)  Even/odd functions  Multivariable calculus: Functions with more than one variable  Functions develop through every year of math instruction

20. Closing Remarks  Traditional model: ideal student is passive, quiet and hard-working  Constructivism is a progressive educational theory, students should bring more to the classroom than hard work.  Traditional model: ideal student is passive, quiet and hard-working  Constructivism is a progressive educational theory, students should bring more to the classroom than hard work.

21. Thank you  Dr. Adelina Alegria  Dr. Ron Solorzano  Professor McDonald  Professor Buckmire  All the math professors who helped me construct my understanding of mathematics  Dr. Adelina Alegria  Dr. Ron Solorzano  Professor McDonald  Professor Buckmire  All the math professors who helped me construct my understanding of mathematics

22. References  1. Ferrini-Mundy, J. & Graham, K. “Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals.” In Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA Notes Number 33. Mathematical Association of America. 1994  2. Hanley, Susan. “On Constructivism”. 1994. Maryland Collaborative for Teacher Preparation. http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt  3. Kamii, Constance. “Constructivism and Beginning Arithmetic (K-2).” In Teaching and Learning Mathematics in the 1990s: 1990 Yearbook. T. J. Cooney, Ed. NCTM, 1990  4. Mathematical Association of America. The Concept of Function: Aspects of Epistemology and Pedagogy. First printing by the Mathematical Association of America, 1992  5. Matthews, Michael R. “Constructivism in Science and Mathematics Education”. 1997. University of New South Wales http://wwwcsi.unian.it/educa/inglese/matthews.html http://wwwcsi.unian.it/educa/inglese/matthews.html  6. Moses, Bob Radical Equations. Boston: Beacon Books Press, 2001  7. National Council of Teachers of Mathematics. Curriculum and evaluation standards for school mathematics. Reston: The National Council of Teachers of Mathematics, Inc, 1989  8. National Research Council of the National Academies. How students learn: Mathematics in the classroom. Washington, DC: The National Academies Press, 2005.  9. Noll, James Wm; Elklind, David; Carson, Jamin. Clashing Views on Educational Issues. Fifteenth edition. New York: McGraw Hill, 2008  10. Quinn, Robert. “A Constructivist Lesson to Introduce Arithmetic Sequences with Patterns.” In Australian Mathematics Teacher, v61 n4. Adelaide: Australian Association of Mathematics Teachers (AAMT), 2005 p18-21  11. Von Glasersfeld, Ernst Radical Constructivism. Washington, D.C.: The Falmer Press, 1995  12. Zaslavsky, Claudia. Fear of Math: How to Get Over It and Get On with Your Life. New Brunswick, New Jersey: Rutgers University Press, 1994.  13. Steven Pinker. How the Mind Works. New York: W. W. Norton & Company 1997  1. Ferrini-Mundy, J. & Graham, K. “Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals.” In Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. MAA Notes Number 33. Mathematical Association of America. 1994  2. Hanley, Susan. “On Constructivism”. 1994. Maryland Collaborative for Teacher Preparation. http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/Essays/Constructivism.txt  3. Kamii, Constance. “Constructivism and Beginning Arithmetic (K-2).” In Teaching and Learning Mathematics in the 1990s: 1990 Yearbook. T. J. Cooney, Ed. NCTM, 1990  4. Mathematical Association of America. The Concept of Function: Aspects of Epistemology and Pedagogy. First printing by the Mathematical Association of America, 1992  5. Matthews, Michael R. “Constructivism in Science and Mathematics Education”. 1997. University of New South Wales http://wwwcsi.unian.it/educa/inglese/matthews.html http://wwwcsi.unian.it/educa/inglese/matthews.html  6. Moses, Bob Radical Equations. Boston: Beacon Books Press, 2001  7. National Council of Teachers of Mathematics. Curriculum and evaluation standards for school mathematics. Reston: The National Council of Teachers of Mathematics, Inc, 1989  8. National Research Council of the National Academies. How students learn: Mathematics in the classroom. Washington, DC: The National Academies Press, 2005.  9. Noll, James Wm; Elklind, David; Carson, Jamin. Clashing Views on Educational Issues. Fifteenth edition. New York: McGraw Hill, 2008  10. Quinn, Robert. “A Constructivist Lesson to Introduce Arithmetic Sequences with Patterns.” In Australian Mathematics Teacher, v61 n4. Adelaide: Australian Association of Mathematics Teachers (AAMT), 2005 p18-21  11. Von Glasersfeld, Ernst Radical Constructivism. Washington, D.C.: The Falmer Press, 1995  12. Zaslavsky, Claudia. Fear of Math: How to Get Over It and Get On with Your Life. New Brunswick, New Jersey: Rutgers University Press, 1994.  13. Steven Pinker. How the Mind Works. New York: W. W. Norton & Company 1997