Implementing Constructivism in Mathematics Classrooms

Implementing Constructivism in Mathematics Classrooms

Tomas Cometto

20 March 2008

Tomas Cometto

20 March 2008

OutlineOutline

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

What is Constructivism?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.

1Von Glasersfeld, Ernst Radical Constructivism. Washington, D.C.: The Falmer Press, 1995

Math AnxietyMath 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

Connecting previous formal knowledge to new materialConnecting 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

Connecting previous informal knowledge to new material

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

Water Vase ExampleWater 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

a b c

H

T

a

b

c

Formal Methods of Computation vs. Application

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

Bring out students’ intuitive understanding of math

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

Role of InstructorRole of InstructorLecturer Facilitator

Dictates material Guides Students to the material

Tells Asks/Involved in dialogue with students

Teaches from the front Supports by walking around

Gives answers Provides guidelines

Learner is passive Learner is active

Focus on the material Focus on the student

Development of math knowledgeDevelopment 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

CountingCounting

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

PatternsPatterns Robert Quinn’s lesson plan: * * * * * * * * * * * * * * * 4th = ? * * * * * * * * * * * * * * * * * * * * * * * * * * * “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 nth group of dots:

an = 6 + 4n an = 10 + (n-1)*4 an = (n) + (n+1) + (n+2) + (n+3) Previous + 4, or an = an-1 + 4, a0 = 10

Robert Quinn’s lesson plan: * * * * * * * * * * * * * * * 4th = ? * * * * * * * * * * * * * * * * * * * * * * * * * * * “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 nth group of dots:

an = 6 + 4n an = 10 + (n-1)*4 an = (n) + (n+1) + (n+2) + (n+3) Previous + 4, or an = an-1 + 4, a0 = 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

Functions before calculusFunctions 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 = 81.98

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 = 81.98

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.

Functions in CalculusFunctions 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

y

x

-4 -2 2 4x

-4

-2

2

4

y

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

yy

x

Conclusions of this studyConclusions 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

Functions after calculusFunctions 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

Closing RemarksClosing 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.

Thank youThank 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

ReferencesReferences 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

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

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

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

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