Thinking in Terms Ways of Thinking

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Thinking in Terms Ways of Thinking Powered By Docstoc
					Two Fundamental Questions:
    A DNR Perspective

            Guershon Harel
      University of California, San Diego

             harel@math.ucsd.edu
   http://www.math.ucsd.edu/~harel
        Two Fundamental Questions

1. What mathematics should we teach in
   school?
2. How should we teach It?



  DNR-based Instruction in Mathematics
                      What is DNR?
DNR is a system of three categories of constructs:

     – Premises:
               explicit assumptions, most of which are taken from or based on
               existing theories.

     – Concepts:
               definitions oriented within the stated premises.

     – Claims:
               statements formulated in terms of the DNR concepts, entailed from
               the DNR premises, and supported by empirical studies.

           •   Instructional principles: claims about effects of teaching
                                             practices on student learning.

           The term DNR refers to three foundational instructional principles:
                          –The Duality Principle
                          –The Necessity Principle
                          –The Repeated Reasoning Principle
                                DNR Premises
Mathematics
   1. Mathematics: Knowledge of mathematics consists of all ways of understanding and
        ways of thinking that have been institutionalized throughout history.
Learning
   2. Epistemophilia: Humans—all humans—possess the capacity to develop a desire to
         be puzzled and to learn to carry out mental acts to solve the puzzles they create.
         Individual differences in this capacity, though present, do not reflect innate
         capacities that cannot be modified through adequate experience. (Aristotle)
   3. Knowing: Knowing is a developmental process that proceeds through a continual
         tension between assimilation and accommodation, directed toward a (temporary)
         equilibrium. (Piaget)
   4. Knowing-Knowledge Linkage: Any piece of knowledge humans possess is an
         outcome of their resolution of a problematic situation. (Brousseau)
   5. Context-Content Dependency: Learning is context and content dependent. (Cognitive
         Psychology)
   Teaching
Teaching
   6. Learning scientific knowledge (such as mathematics) is not spontaneous. There will
         always be a difference between what one can do under expert guidance or in
         collaboration with more capable peers and what he or she can do without
         guidance. (Vygotsky)
Ontology
   7. Subjectivity: Any observations humans claim to have made is due to what their
          mental structure attributes to their environment. (Piaget)
   8. Interdependency: Humans’ actions are induced and governed by their views of the
          world, and, conversely, their views of the world are formed by their actions.
          (Piaget)
 Conceptual Tools                      DNR Premises                               Subject Matter

Mathematics
    1. Mathematics: Knowledge of mathematics consists of all ways of understanding and
          ways of thinking that have been institutionalized throughout history.
Learning
    2. Epistemophilia: Humans—all humans—possess the capacity to solve puzzles
           AND to develop a desire to be puzzled. (Aristotle)
           Individual differences in this capacity, though present, do not reflect innate capacities that
           cannot be modified through adequate experience.
    3. Knowing: Knowing is a developmental process that proceeds through a continual tension
           between assimilation and accommodation, directed toward a (temporary) equilibrium. (Piaget)
    4. Knowing-Knowledge Linkage: Any piece of knowledge humans possess is an outcome of their
           resolution of a problematic situation. (Brousseau)
    5. Context-Content Dependency: Learning is context and content dependent. (Cognitive Psychology)
Teaching
    6. Learning scientific knowledge (such as mathematics) is not spontaneous. There will always be a
          difference between what one can do under expert guidance or in collaboration with more
          capable peers and what he or she can do without guidance. (Vygotsky)
Ontology
    7. Subjectivity: Any observations humans claim to have made is due to what their mental structure
           attributes to their environment. (Piaget)
    8. Interdependency: Humans’ actions are induced and governed by their views of the world, and,
           conversely, their views of the world are formed by their actions. (Piaget)
          Instructional Principles


      DNR-Based Instruction in
           Mathematics


Duality       Repeated Reasoning

     Necessity
           The Necessity Principle

For students to learn the mathematics we intend
to teach them, they must have a need for it,
where ‘need’ refers to intellectual need, not
social or economic need.




                  DNR
             DNR in the Classroom
• What does it mean to think of mathematics
  teaching and learning in terms of both ways of
  understanding and ways of thinking?
• How does DNR is used to advance desirable
  ways of understanding and ways thinking with
  students?
• When should one start targeting particular
  ways of thinking with students?
Rectangular Land Problem
A farmer owns a rectangular piece of land. The land is
divided into four rectangular pieces, known as Region A,
Region B, Region C, and Region D, as in the figure:
                            A        C


                            B        D




One day the farmer’s daughter, Nancy, asked him, what
is the area of our land? The father replied:
    I will only tell you that the area of Region B is 200 larger than
    the area of Region A; the area of Region C is 400 larger than
    the area of Region B; and the area of Region D is 800 larger
    than area of Region C.
What answer to her question will Nancy derive from her
father’s statement?
Please work individually on this
            problem
What are some of the assumed
 ways of thinking?
What are some of the targeted ways
 of thinking?
Assumed Way of Thinking:
   The problem-solving approach of representing a
   given problem algebraically and applying
   known procedures (such as procedures to
   solve systems of linear equations) to obtain a
   solution to the problem.
       Algebraic representation approach

Objective 1:
   Reinforce this way of thinking.

Problem solving approaches constitute one
category of ways of thinking.
   How do we help students acquire the algebraic
   representation approach way of thinking?
The Lesson: Actual Classroom Episodes

• Students’ responses
   – Explanations in terms of
      • students current knowledge
      • the way students have been taught
      • nature of learning
• Teacher’s actions
   – Explanations in terms of
      • the teacher’s conceptual framework:
        DNR-based instruction in mathematics
Students’ Responses
All students translated the farmer statement
into a system equations similar to:
                 B  A  200
                
                C  B  400
                 D  C  800
                

Attempted to construct a 4th equation, e.g.,

B  C  D  ( A  200)  ( B  400)  (C  800)
Students’ Responses
Attempted to construct a 4th equation, e.g.,
A  C  D  ( A  200)  ( B  400)  (C  800)
Explanation of students’ behavior:
    Most students’ conception of a system of equations
    includes the constraints:
        • the number of unknowns must equal the number of
          equations.
        • there is always one solution.
These are largely didactical obstacles: obstacles
caused by how students are taught.
                didactical obstacles
                       versus
             epistemological obstacles
Conceptual Tools               DNR Premises                           Subject Matter

Mathematics
   1. Mathematics: Knowledge of mathematics consists of all ways of understanding and
        ways of thinking that have been institutionalized throughout history.
Learning
   2. Epistemophilia: Humans—all humans—possess the capacity to solve puzzles
         AND to develop a desire to be puzzled. (Aristotle)
         Individual differences in this capacity, though present, do not reflect innate
         capacities that cannot be modified through adequate experience.
   3. Knowing: Knowing is a developmental process that proceeds through a continual
         tension between assimilation and accommodation, directed toward a (temporary)
         equilibrium. (Piaget)
   4. Knowing-Knowledge Linkage: Any piece of knowledge humans possess is an
         outcome of their resolution of a problematic situation. (Brousseau)
   5. Context-Content Dependency: Learning is context and content dependent. (Cognitive
         Psychology)
Teaching
   6. Learning scientific knowledge (such as mathematics) is not spontaneous. There will
         always be a difference between what one can do under expert guidance or in
         collaboration with more capable peers and what he or she can do without
         guidance. (Vygotsky)
Ontology
   7. Subjectivity: Any observations humans claim to have made is due to what their
          mental structure attributes to their environment. (Piaget)
   8. Interdependency: Humans’ actions are induced and governed by their views of the
          world, and, conversely, their views of the world are formed by their actions.
          (Piaget)
Teacher’s action 1: classroom discussion of
the students’ proposed solution
Outcome:
   Agreement: there are infinitely many solutions:
    every choice of A gives a value for the total area.


                     B  A  200
                    
                    C  B  400
                     D  C  800
                    
    Total Area  A  B  C  D 
    A  ( A  200)  ( B  400)  (C  800) 
    4 A  2200
– Explanations of the teacher’s action in
  terms of DNR
   • Teaching actions in DNR are determined
     largely by students’ current knowledge.
      Teacher’s actions depends on the solutions
      proposed, consensus or dispute among
      students about a particular solution or idea,
      etc.
      Agreements and disputes must be explicitly
      institutionalized in the classroom.
   • Public Debate versus Pseudo-Public
     Debate.
Teacher’s action 1: classroom discussion of
the students’ proposed solution
Outcome:
   Agreement: there are infinitely many solutions:
    every choice of A gives a value for the total area.
             Total Area = 4A + 2200

What can and should the teacher do with this
outcome?

DNR’s Approach:
Present the student with a new task that puts
them in a disequilibrium.
Teacher’s Action 2
Construct two figures, each representing a
different solution.
                  B  A  200
                 
                 C  B  400
                  D  C  800
                 
    Total Area  A  B  C  D 
    A  ( A  200)  ( B  400)  (C  800) 
    4 A  2200
Students’ Response
                  20               140

            100                  700
  5


            300                  1500
 15




                                140×15≠1500


       B  A  200
      
      C  B  400
       D  C  800
      
      Total Area  4 A  2200
Students’ Response
                  10               70

            100                  700
 10


            300                  1500
 30




                                70×30≠1500


       B  A  200
      
      C  B  400
       D  C  800
      
      Total Area  4 A  2200
Teacher: you didn’t try fractions
              100÷2/3=150           700÷2/3=1050

             100                  700
 2/3


             300                  1500
 2




                                 1050×2≠1500


        B  A  200
       
       C  B  400
        D  C  800
       
       Total Area  4 A  2200
Teacher: you didn’t try irrational numbers
               100÷√2               700÷√2

                  100                  700
       √2


                  300                  1500
300÷(100÷√2)
=3×√2



                                (3×√2) × (700÷√2) ≠1500


       B  A  200
      
      C  B  400
       D  C  800
      
      Total Area  4 A  2200
Conjecture: The figure cannot be constructed
            for A=100
Teacher’s action-students’ response 3:
      Prompted by the teachers, some students offered to use a variable t.


                 100÷t                           700÷t

         t          100                             700



                    300                             1500
300÷(100÷t)=3t




                                             3t×(700÷t)=2100≠1500

• Students were brought to a conceptual stage where the use
  of variable was necessitated intellectually
      –it was a natural extension of their current activity.
• Conjecture was settled for the students (i.e., proved) by
  algebraic means.
Students’ Responses
                  100÷t                              700÷t

                      100                               700
       t


                       300                              1500
300÷(100÷t)=3t




                                                 3t×(700÷t)=2100≠1500
• Surprise
      – Almost a disbelief
• Formation of a second conjecture
      – It doesn’t work with A=100; perhaps it would work with a different
        value of A.
• Trial-and-error approach repeated
      – Students try different values for A, but ran into the same conflict:
           none of the values chosen led to a constructible figure.
Teacher’s action 4:
  Discussion: How to resolve the question of
  whether the figure is constructible for any values
  of A.




  The class, led by the teacher, repeats the same
  activity with the variable A.
                 A÷t                    (A+600)÷t

                       A                   A+600
       t


                       A+200               A+1400
(A+200)÷(A÷t)=
(A+200)t/A


                               ( A  200)t A  600
                                                    A  1400
                                    A         t
                               A2  800 A  12000  A2  1400 A
                               12000  600 A
                               A  200
Outcomes:
•Students were brought—again—to a
 conceptual stage where the use of variable
 was necessitated intellectually.
•Second conjecture was settled for the
 students (i.e., proved)—again—by algebraic
 means.
          Instructional Principles


      DNR-Based Instruction in
           Mathematics


Duality       Repeated Reasoning

     Necessity
Teacher’s action 5:
   Reflective discussion:
     • Why did our first approach to solving the problem fail?
     • The need to attend to the figure’s form:
                               A          C


                               B          D




     versus
                                                              D
                      A

                                                 C
                                   B




   Objective 2:
     • To advance the way of thinking:
              In representing a problem algebraically, all of the problem
              constraints must be represented.
                               Summary
What are the ways of thinking targeted by this Lesson?
Ways of Thinking:
 1. In representing a problem algebraically, all of the problem
    constraints must be represented.
 2. Algebraic representation way of thinking:
    Reinforcing the problem-solving approach of representing a given problem
    algebraically and applying known procedures (such as procedures to solve
    systems of linear equations) to obtain a solution to the problem.
      Problem solving approaches constitute one category of ways of thinking.
 3. Mathematics involves trial and error and proposing and refining
    conjectures until one arrives at a correct result.
      Beliefs about mathematics constitute the second category of ways of
      thinking.
 4. Algebraic means are a powerful tool to remove doubts—that is, to
    prove or refute conjectures.
      Proof schemes (how one removes doubts about mathematical
      assertions) constitute the third category of ways of thinking.
  Concepts learned/reinforced through
  intellectual need: to solve a problem—
  reach an equilibrium



• Operations with fractions and irrational
  numbers
• Algebraic manipulations
• The concepts of variable and parameter
     Intellectual Need versus Intrinsic Motivation

Intrinsically motivated activities are defined as those that
individuals find interesting and would do in the absence of
operationally separable consequences.

Intrinsic motivation is conceptualized in terms of three innate
psychological needs:

    Need for autonomy


    Need for competence


    Need for relatedness
            Intrinsic Motivation


Need for autonomy:
    The need for freedom to follow one’s inner
    interest rather than being control by extrinsic
    rewards.
Need for competence:
    The need for having an effect—for being
    effective in one’s interactions with the
    environment.
Need for relatedness:
    The need for a secure relational base with
    others.
                  Intellectual Need

Intellectual need refers to the perturbational stage
in the process of justifying how and why a
particular piece of knowledge came into being.
   It concern the genesis of knowledge, the perceived a
   priori reasons for the emergence of knowledge.
Categories of intellectual need
    (a) Need for certainty
    (b) Need for causality (enlightenment)
    (c) Need for computation
    (d) Need for communication-formulation-
                 formalization
    (e) Need for structure
Goal: To necessitate the e-N definition of limit
                          1
Teacher: What is     lim
                     n  n
                                 and why?
Students:
        1                                      1
   lim  0 because the larger n gets the closer is to zero.
   n  n                                      n
Teacher:                       4




                               2




                                    5       10



                   fx = -1
                               -2




                               -4


         1                                       1
    lim  1 because the larger n gets the closer is to -1.
    n  n                                       n
Algebraic approach way of thinking:
   Reinforcing the problem-solving approach of representing a given
   problem algebraically and applying known procedures (such as
   procedures to solve systems of linear equations) to obtain a
   solution to the problem.
              Need for Computation
Towns A and B are 300 miles apart. At 12:00
PM, a car leaves A toward B, and a truck leaves
B toward A. The car drives at 80 m/h and the
truck at 70 m/h. When and where will they meet?
Students’ reasoning:
After 1 hour, the car drives 80 miles and truck 70
miles.
Together they drive 150 miles.
In 2 hours they will together drive 300 miles.
Therefore,
They will meet at 2:00 PM.
They will meet 160 miles from A.
Towns A and B are 300 miles apart. At 12:00
 PM, a car leaves A toward B, and a truck
 leaves B toward A. The car drives at 80 m/h
 and the truck at 70 m/h. When and where will
 they meet?
Towns A and B are 300 100 miles apart. At
  12:00 PM, a car leaves A toward B, and a truck
  leaves B toward A. The car drives at 80 m/h
  and the truck at 70 m/h. When and where will
  they meet?
Students’ reasoning:
It will take them less than one hour to meet.
It will take them more than 30 minutes to meet.
They will meet closer to B than to A.
Let’s try some numbers:
                   50           50      ?
                        80          70  100
                   60           60

                    40          40      ?
                         80         70  100 (Yes !)
                    60          60
Towns A and B are 118 miles apart. At 12:00
 PM, a car leaves A toward B, and a truck
 leaves B toward A. The car drives at 80 m/h
 and the truck at 70 m/h. When and where will
 they meet?
                    Necessitating the concept of variable
Students’ reasoning:
It will take them less than one hour to meet;
It will take them more than 30 minutes to meet;
They will meet closer to B than to A.
                                           ?
Let’s try some numbers:      50
                                80 
                                     50
                                        70  118
                             60          60

                                                ?
                             40          40
                                  80         70  118
                             60          60

                                                ?
                             30          30
                                  80         70  118
                             60          60
Local Necessity versus Global Necessity

First Course in Linear Algebra
• Focus: Linear systems—scalar and differential
• The problem (general investigation):
   Given a linear system
  1. How do we solve it?
  2. Can we solve it efficiently?
     – Is there an algorithm to solving such systems?
  3. How do we determine whether a given system has a
     solution?
  4. If a system is solvable, how many solutions does it have?
  5. If the system has infinitely many solutions, how do we list
     them?
Necessitating Deductive Reasoning
  Students and Teachers’ Conceptions of
         Proof: Selected Results
• Students and teachers justify mathematical
  assertions by examples

• Often students’ and teachers’ inductive
  verifications consist of one or two example,
  rather than a multitude of examples.

• Students’ and teachers’ conviction in the
  truth of an assertion is particularly strong
  when they observe a pattern.
•   Students view a counterexample as an
    exception—in their view it does not
    affect the validity of the statement.

•   Confusion between empirical proofs and
    proofs by exhaustion.

•   Confusion between the admissibility of
    proof by counterexample with the
    inadmissibility of proof by example.
Teaching Actions with Limited Effect
  – Raising skepticism as to whether the
    assertion is true beyond the cases
    evaluated.
  – Showing the limitations inherent in the use
    of examples through situations such as:

    The conjecture “ 1141y  1 is an integer” is false for 1  y  10 .
                             2                                      25




     The first value for which the statement is true is:
     30,693,385,322,765,657,197,397,208
Why showing the limitations inherent in
 the use of examples is not effective?
• Students do not seem to be impressed
  by situations such as:
    • The conjecture “ 1141y  1 is an integer” is false for
                             2
                                                                          .
                                                               1  y  1025

      The first value for which the statement is true is:
      30,693,385,322,765,657,197,397,208


• Students view a counterexample as an
  exception—in their view it does not affect
  the validity of the statement.
 How Do We Intellectually Necessitate the
Transition from Empirical Proof Schemes to
         Deductive Proof Schemes?



          A Dissertation Topic!
Linear Algebra Textbooks
     “So far we have defined a mathematical system
       called a real vector space and noted some of its
       properties ....
     [In what follows], we show that each vector space
       V studied here has a set composed of a finite
       number of vectors that completely describe V. It
       should be noted that, in general, there is more
       than one such set describing V. We now turn to
       a formulation of these ideas.”
Following this, the text defines the concepts:
     Linear independence
     Span
     Basis
And rigorously prove all related theorems.
From a chapter on eigen theory
  “In this section we consider the problem of factoring
    an n×n matrix A into a product of the form XDX-1,
    where D is diagonal. We will give necessary and
    sufficient condition for the existence of such a
    factorization and look at a number of examples. We
    begin by showing that eigenvectors belonging to
    distinct eigenvalues are linearly independent.”
Necessitating the Concept of
     Diagonalization
Logical Justification and Intellectual Need
Grassmann’s theory of extension (1844, 1862)

Grassmann’s idea of a sound foundation for mathematics
designed to support both the method of discovery and the
method of proof.

Grassmann’s [approach] appears to be much more than a
device to aid the reader; he appears to regard the pedagogical
involvement as an essential part of the justification of
mathematics as science.

(Lewis, 2004, p. 17: History and Philosophy of Logic, 25, 15-
36)

				
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