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Two Fundamental QuestionsWhat mathematics should we teach in school? 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 theor
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1. Two Fundamental Questions: A DNR Perspective Guershon Harel
University of California, San Diego
harel@math.ucsd.edu
http://www.math.ucsd.edu/~harel
2. Two Fundamental Questions
What mathematics should we teach in school?
How should we teach It?
DNR-based Instruction in Mathematics
DNR-based instruction in mathematics (DNR, for short) is a conceptual framework that provides a language and tools to formulate and address critical curricular and instructional concerns.
In this framework the mathematical integrity of the content taught and the intellectual need of the student are at the center of the instructional effort.
DNR has been developed from a long series of teaching experiments in elementary, secondary, and undergraduate mathematics courses, as well as teaching experiments in professional development courses for teachers at each of these levels. BrieflyDNR-based instruction in mathematics (DNR, for short) is a conceptual framework that provides a language and tools to formulate and address critical curricular and instructional concerns.
In this framework the mathematical integrity of the content taught and the intellectual need of the student are at the center of the instructional effort.
DNR has been developed from a long series of teaching experiments in elementary, secondary, and undergraduate mathematics courses, as well as teaching experiments in professional development courses for teachers at each of these levels. Briefly
3. 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-based instruction in mathematics (DNR, for short) is a conceptual framework that provides a language and tools to formulate and address critical curricular and instructional concerns.
In this framework the mathematical integrity of the content taught and the intellectual need of the student are at the center of the instructional effort.
DNR has been developed from a long series of teaching experiments in elementary, secondary, and undergraduate mathematics courses, as well as teaching experiments in professional development courses for teachers at each of these levels. BrieflyDNR-based instruction in mathematics (DNR, for short) is a conceptual framework that provides a language and tools to formulate and address critical curricular and instructional concerns.
In this framework the mathematical integrity of the content taught and the intellectual need of the student are at the center of the instructional effort.
DNR has been developed from a long series of teaching experiments in elementary, secondary, and undergraduate mathematics courses, as well as teaching experiments in professional development courses for teachers at each of these levels. Briefly
4. 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)
5. 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)
6.
DNR-Based Instruction in Mathematics
Instructional Principles
7.
8. 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?
9. 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?
10. Please work individually on this problem
11. What are some of the assumed ways of thinking?
What are some of the targeted ways of thinking?
12. 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?
13. 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
14. Students Responses
All students translated the farmer statement into a system equations similar to:
Attempted to construct a 4th equation, e.g.,
15. 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
16. 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)
17. 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.
18. 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.
19. 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.
20. Teachers Action 2
Construct two figures, each representing a different solution.
21. Students Response
22. Students Response
23. Teacher: you didnt try fractions
24. Teacher: you didnt try irrational numbers
25. Conjecture: The figure cannot be constructed for A=100
26. 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.
27. Students Responses
Surprise
Almost a disbelief
Formation of a second conjecture
It doesnt 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.
28. 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.
29.
30. 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.
31.
DNR-Based Instruction in Mathematics
Instructional Principles
32. 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.
33. 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.
34. 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
35. 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
The concept of intrinsic motivation was conceptualized as a result of the failure of mechanistic theories to account for human behaviors.
Mechanistic theories view the energy for what humans do in terms of physiological drives. Sex and hunger according to Freud; sex, hunger, thirst, and avoidance of pain, according to Hull. Learning according to these mechanistic theories amounts to accruement of associative bonds between stimulus and response.
The modern theories of motivation, on the other hand, are organismic, in that they view humans as active creature with interest and desire for self determination. The concept of intrinsic motivation was conceptualized as a result of the failure of mechanistic theories to account for human behaviors.
Mechanistic theories view the energy for what humans do in terms of physiological drives. Sex and hunger according to Freud; sex, hunger, thirst, and avoidance of pain, according to Hull. Learning according to these mechanistic theories amounts to accruement of associative bonds between stimulus and response.
The modern theories of motivation, on the other hand, are organismic, in that they view humans as active creature with interest and desire for self determination.
36. 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. The concept of intrinsic motivation was conceptualized as a result of the failure of mechanistic theories to account for human behaviors.
Mechanistic theories view the energy for what humans do in terms of physiological drives. Sex and hunger according to Freud; sex, hunger, thirst, and avoidance of pain, according to Hull. Learning according to these mechanistic theories amounts to accruement of associative bonds between stimulus and response.
The modern theories of motivation, on the other hand, are organismic, in that they view humans as active creature with interest and desire for self determination. The concept of intrinsic motivation was conceptualized as a result of the failure of mechanistic theories to account for human behaviors.
Mechanistic theories view the energy for what humans do in terms of physiological drives. Sex and hunger according to Freud; sex, hunger, thirst, and avoidance of pain, according to Hull. Learning according to these mechanistic theories amounts to accruement of associative bonds between stimulus and response.
The modern theories of motivation, on the other hand, are organismic, in that they view humans as active creature with interest and desire for self determination.
37.
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.
38. Categories of intellectual need
Need for certainty
Need for causality (enlightenment)
Need for computation
Need for communication-formulation- formalization
Need for structure
39. Goal: To necessitate the e-N definition of limit
Teacher: What is and why?
Students:
Teacher:
40.
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.
41. 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.
42. 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?
43. 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.
Lets try some numbers:
44. 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?
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:
45. 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?
46.
Necessitating Deductive Reasoning
47. 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.
48. 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.
49. 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: 30,693,385,322,765,657,197,397,208
50. 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: 30,693,385,322,765,657,197,397,208
Students view a counterexample as an exceptionin their view it does not affect the validity of the statement.
51.
How Do We Intellectually Necessitate the Transition from Empirical Proof Schemes to Deductive Proof Schemes?
A Dissertation Topic!
52. 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.
53. 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.
56. Logical Justification and Intellectual Need
57.
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)