Two Fundamental Questions: A DNR Perspective

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Two Fundamental Questions A DNR Perspective

• Guershon Harel
• University of California, San Diego
• harel_at_math.ucsd.edu
• http//www.math.ucsd.edu/harel

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• Two Fundamental Questions
• What mathematics should we teach in school?
• How should we teach It?
• DNR-based Instruction in Mathematics

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• 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

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• 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 Humansall humanspossess 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)

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• 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 Humansall humanspossess 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)

Subject Matter
Conceptual Tools
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Instructional Principles

• DNR-Based Instruction in Mathematics

Duality
Repeated Reasoning
Necessity
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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. D N R
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• 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?

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• 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
• One day the farmers 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 fathers statement?

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• Please work individually on this problem

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• What are some of the assumed ways of thinking?
• What are some of the targeted ways of thinking?

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• 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?

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• The Lesson Actual Classroom Episodes
• Students responses
• Explanations in terms of
• students current knowledge
• the way students have been taught
• nature of learning
• Teachers actions
• Explanations in terms of
• the teachers conceptual framework DNR-based
  instruction in mathematics

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• Students Responses
• All students translated the farmer statement into
  a system equations similar to
• Attempted to construct a 4th equation, e.g.,

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• Students Responses
• Attempted to construct a 4th equation, e.g.,
• 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

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• 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 Humansall humanspossess 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)

Subject Matter
Conceptual Tools
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• Teachers 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.

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• Explanations of the teachers action in terms of
  DNR
• Teaching actions in DNR are determined largely by
  students current knowledge.
• Teachers 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.

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• Teachers 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?
• DNRs Approach
• Present the student with a new task that puts
  them in a disequilibrium.

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• Teachers Action 2
• Construct two figures, each representing a
  different solution.

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• Students Response

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140
5
15
14015?1500
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• Students Response

10
70
10
30
7030?1500
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• Teacher you didnt try fractions

1002/3150
7002/31050
2/3
2
10502?1500
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• Teacher you didnt try irrational numbers

100v2
700v2
v2
300(100v2)3v2
(3v2) (700v2) ?1500
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• Conjecture The figure cannot be constructed for
  A100

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• Teachers action-students response 3
• Prompted by the teachers, some students offered
  to use a variable t.
• 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.

100t
700t
t
300(100t)3t
3t(700t)2100?1500
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• Students Responses
• Surprise
• Almost a disbelief
• Formation of a second conjecture
• It doesnt work with A100 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.

100t
700t
t
300(100t)3t
3t(700t)2100?1500
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• Teachers 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.

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At
(A600)t
t
(A200)(At) (A200)t/A
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• Outcomes
• Students were broughtagainto a conceptual stage
  where the use of variable was necessitated
  intellectually.
• Second conjecture was settled for the students
  (i.e., proved)againby algebraic means.

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Instructional Principles

• DNR-Based Instruction in Mathematics

Duality
Repeated Reasoning
Necessity
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• Teachers action 5
• Reflective discussion
• Why did our first approach to solving the problem
  fail?
• The need to attend to the figures form
• versus
• Objective 2
• To advance the way of thinking
• In representing a problem algebraically, all of
  the problem constraints must be represented.

D
A
C
B
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• Summary
• What are the ways of thinking targeted by this
  Lesson?
• Ways of Thinking
• In representing a problem algebraically, all of
  the problem constraints must be represented.
• 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.
• 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.
• Algebraic means are a powerful tool to remove
  doubtsthat is, to prove or refute conjectures.
• Proof schemes (how one removes doubts about
  mathematical assertions) constitute the third
  category of ways of thinking.

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• Concepts learned/reinforced through intellectual
  need to solve a problemreach an equilibrium
• Operations with fractions and irrational numbers
• Algebraic manipulations
• The concepts of variable and parameter

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• 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

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• Intrinsic Motivation
• Need for autonomy
• The need for freedom to follow ones inner
  interest rather than being control by extrinsic
  rewards.
• Need for competence
• The need for having an effectfor being
  effective in ones interactions with the
  environment.
• Need for relatedness
• The need for a secure relational base with
  others.

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• 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.

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• Categories of intellectual need
• Need for certainty
• Need for causality (enlightenment)
• Need for computation
• Need for communication-formulation- formalizatio
  n
• Need for structure

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• Goal To necessitate the e-N definition of limit
• Teacher What is and why?
• Students
• Teacher

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• 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.

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• Need for Computation
• Towns A and B are 300 miles apart. At 1200 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 200 PM.
• They will meet 160 miles from A.

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• Towns A and B are 300 miles apart. At 1200 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?

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• Towns A and B are 300 100 miles apart. At 1200
  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.
• Lets try some numbers

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• Towns A and B are 118 miles apart. At 1200 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.
• Lets try some numbers

Necessitating the concept of variable
Repeated Reasoning
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• Local Necessity versus Global Necessity
• First Course in Linear Algebra
• Focus Linear systemsscalar and differential
• The problem (general investigation)
• Given a linear system
• How do we solve it?
• Can we solve it efficiently?
• Is there an algorithm to solving such systems?
• How do we determine whether a given system has a
  solution?
• If a system is solvable, how many solutions does
  it have?
• If the system has infinitely many solutions, how
  do we list them?

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• Necessitating Deductive Reasoning

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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.

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• Students view a counterexample as an exceptionin
  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.

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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 is an integer is
  false for . The first value for which
  the statement is true is 30,693,385,322,765,657,1
  97,397,208

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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 is an integer is
  false for . The first value for which
  the statement is true is 30,693,385,322,765,657,1
  97,397,208
• Students view a counterexample as an exceptionin
  their view it does not affect the validity of the
  statement.

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• How Do We Intellectually Necessitate the
  Transition from Empirical Proof Schemes to
  Deductive Proof Schemes?
• A Dissertation Topic!

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• 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.

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• From a chapter on eigen theory
• In this section we consider the problem of
  factoring an nn 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.

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Necessitating the Concept of Diagonalization
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• Logical Justification and Intellectual Need

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• Grassmanns theory of extension (1844, 1862)
• Grassmanns idea of a sound foundation for
  mathematics designed to support both the method
  of discovery and the method of proof.
• Grassmanns 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)