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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 fx = -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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