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Part 1: A three week mini-course Chapter One: On the nature of numeration (one week, three days a week) Section 1: On the history of counting 1.1: Most first semester liberal arts mathematics classes have a good history of the counting system, including the origins of counting in India and Persia, counting in Mayan and Inca culture, Arabic numerals, Roman numerals and the adoption of Arabic numerals at the beginning of the European renaissance. This material will be covered from a textbook or as a handout (written by the author). 1.2: At the last day of the first week, learners will be asked to write an informal paper on one of the following topics a) Cultural anthropology and counting b) Numeration’s role in developing economic systems c) Comparison between counting with integers and natural language d) Axiomatization of the positive integers by Peano e) The limitations of counting numbers f) How I feel about mathematics The paper can be hand written and of any length. Part of the pedagogy is to allow the student, or require the student, to make up his/her own mind, and to make decisions. So they may need to express dissatisfaction with how the course is being taught, and they are encouraged to do this. Having made this expression, the student can look beyond the decision. Some students will take on deeper challenges, such as cultural anthropology. It is important to allow the student to earn credit towards positive grades. Section 2: The use of position of digits in base-10 2.1: The class will be guided, using the Socratic method, to identify what is necessary for a base-10 number system. The elements are a set of digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } and the positional convention is that 456 in base 10 means 4(10)2 + 5(10)2 + 6(10)0. It is vital that all students catch on that this class is governed by the Socratic method. 2.2: (10)0 = 1 is discussed, at length and some of the properties of 0 are explored. We introduce the very deep issue of how addition and multiplication work together. Something never see, or quite appreciated is sought. 2.3: The use of a symbol to represent an arbitrary base is discussed. 2.3: (a)m (a)n = (a)m+n is justified. Justifying (a)m (a)n = (a)m+n could take as few as 15 minutes, but it is important that each student comes to assert that the use of “a”, “n”, and “m” is this way is both reasonable, and useful. The class could be asked to write on a piece of paper some example or a justification. The papers would be opinions and not for a grade. 2.4: (a)0 = 1 is discussed and the use of proof is discussed. At this point we find one justification for the formula in #2.3, and this justification involves the use of the logic that no matter which natural number, “a”, (a) 0 = 1. 2.5: Students are given the problem of showing that if (a) m (a)n = (a)m+n is justified then (a)0 = 1. 2.6: The notion of formal justification is discussed. At this point there will be students who have an intuitive feels about the foundations of mathematics and logic. If this is recognized by the teacher, and consequently by the student his or her self, then it is possible to allow that student to develop this intuition a bit further. 2.7: Assignment: Be prepared to justify that (a)0 = 1 in class on paper the next class day. Section 3: Counting in base 5, and 8 3.1: Students will work in small groups and with props like a bag of beans. 3.2: Students will be asked to discover how to express the count of beans, in their group’s bean-bag, as a base-8 and a base-5 number. No instruction will be given since the discovery has to come from the groups. The concept that there is one “correct” answer is examined. This “correctness” is compared to question of cultural viewpoints where often the “correctness” is contextual. 3.3: Suppose that the count in base-5 is (p)5 and in base-8 the count is (q)8 . What is the sequence of digits that p and q stand for? Here is a chance to re-enforce fundamental mathematical knowledge. Students are encouraged to write an answer to this question, and to also let each student realize the discovery / remember process. 3.4: The use of a symbol to represent an arbitrary number is discussed again. 3.5: Assignment: Now that each group has agreed as to the count in base-5 and in base-8, is there a way to check the count directly by using the meaning of the digits? This is the key learning objective. In fact there are a number of quite different ways to validate that an answer is correct or not. This fact, that there is more than one way to solve the problem, allows the student to take charge of the learning process, to make up problems and solve them without having to appeal to the back of the book answers. Section 4: Positional convention for arbitrary number base 4.1: Learners are given the opportunity to explain to the other learners how to convert (p) 5 to (q)8 . The discovery process must be guided carefully so that each learner finds answers for him or her self and in some cases helps a classmate. 4.2: The Socratic method is presented formally and a hand out on Greek philosophy is provided to the learners so that the principle is understood. 4.3: Class is dismissed early. One day will be spent on review and then there will be the first examination. The dismissal of the class early has a pedagogical value. The acquired learning disability is real, has real conceptual and perhaps even accommodating neurological support. However, the primary reinforcement is cultural and social. The reinforcement is cultural because our culture makes the assumption that only a few children have aptitude for “mathematics”. The educational system, as a collective property, forgets that arithmetic and algebra is not the “same as” mathematics. Arithmetic and algebra is, if learned properly only the very beginning of what a mathematician might regard as mathematics. Given this cultural setting it is little wonder that most freshman students have difficulty with freshman mathematics classes. The reinforcement is also social in that students reinforce the learned disability. They constantly express to each other and great dislike of anything that has to do with learning math. The primary challenge is to subvert this socialization and to allow the students to gain a collective pride in doing something that is different from the expected. End of Chapter Summary: 1) Counting plays an essential role in the development of even primitive culture 2) Positional notation has various expressions 3) Letter symbols are sometimes used to talk about number properties 4) Each member of the class has be asked to discover base-b positional notation 5) The learning method called the Socratic method has been discussed and learners are aware that the requirement for this class is the personal discovery of knowledge (without being told what the knowledge is by someone else.) Chapter Two: Number base conversions (one week) Section 1: The use of position of digits in base-b (one day) 1.1: The class identifies what is necessary for a base-b number system. The elements are a set of digits from { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, c, d, e, … } and the positional convention is that 6d8 in base-b means 6(b)2 + d(b)2 + 8(b)0. Counting is reviewed in base 15. 1.3: Numbers in arbitrary bases are converted to an different base, for example (425) 7 (q) 9 . 1.4: Learners as asked to develop a method for checking the answer. Check (425) 7 (258) 9 . use (425) 7 (4(72) +2(71) + 5(70) ) 10 = (215) 10 and (258) 9 (2(92) +5(91) + 8(90) ) 10 = (215) 10 Section 2: The discovery of what addition and multiplication means in base-b. 2.1: The addition table for base-b 2.2: The multiplication table for base-b. 2.3: Practice in addition and multiplication and bases other than 10. 2.4: Checking the multiplication in base-8 by converting the numbers to base-10, doing the arithmetic in base-10 and then converting base to base-8 to check the result. Section 3: The final test on addition and multiplication, in bases other than 10, and on using base conversions to check the answer. A short essay is also required to explain what the learner has learned in the three-week period. End of Chapter Summary: 1) A new and unexpected arithmetic skill is gained. 2) The learner is empowered with a method that allows a non-trivial check of the first result so that the learner, and not a textbook or teacher, can check to see if the answer is correct. 3) The foundational concepts of arithmetic are deeply and thoroughly experienced using the Socratic method. 4) Certain practices, such as the use of a letter to designate any number, are used in a way that unblocks the learner’s resistance to this practice. 5) The student is allowed to see, that what appears as, very difficult problems can be understood and solved by thinking about the meaning of the formalism of counting numbers, addition and multiplication. They learn about the nature of arithmetic, perhaps for the very first time. Part II: Modified curriculum for Liberal Arts mathematics Chapter Three: Set Theory Section 1: The notion of a set is developed, including the philosophical/logical history of set theory 1.1: Set membership and philosophical notions of category theory 1.1.2: Partitioning a bag of things into categories 1.1.3: Oppositional scales, “she loves me she loves me not” 1.1.4: Information science and sorting of items into ranked lists 1.2: Venn diagrams 1.2.1: Unions and intersections 1.2.2: Universal sets and complementation 1.2.3: The algebra of sets 1.2.4: Lattice of sets 1.2.5: Sequences of sets 1.3: The notion of a fuzzy set and a rough set 1.3.1: Fuzzy logic Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth -- truth values between "completely true" and "completely false". 1.3.2: The Artificial Intelligence Dream, myth or reality 1.3.2.1: The brief overview history of Artificial Neural Networks and Artificial Intelligence 1.3.2.2: The issues claimed by AI supporters 1.3.2.3: The neuro-cognitive science perspective 1.3.3: Rough sets The rough set concept (cf. Pawlak (1982)) is a new mathematical tool to reason about vagueness and uncertainty. The rough set theory bears on the assumption that in order to define a set we need initially some information (knowledge) about elements of the universe - in contrast to the classical approach where the set is uniquely defined by its elements and no additional information about elements of the set is necessary. End of Chapter Summary 1) Section 1 builds an inquiry into the notion of a set. This inquiry is intended to upset the learner’s sense that set theory is both not interesting personally and is well understood simply because the student mastered Venn diagrams at one point in school. Information science may be thought to suffer from an over simplification of the concept of category membership, for example. Most individuals will make sense of the limitations that common knowledge, or lack or knowledge, of the formal processes of categorization has on information technology. 2) Section 2 develops the traditional Venn diagram curriculum, but quickly moves on to motivate the learner by showing several easily accessable concepts that are important in topology and the foundations of mathematics and science. 3) Section 3 is an optional section that is designed to give the awakening learner easy access to the two major variations of modern set theory: fuzzy set and rough sets. This introduction is motivated by an examination of the notion that a computer program can become a sentient being. Chapter Four: Arithmetic in arbitrary number bases This chapter reinforces the first two chapters can returning to the elementary number bases and providing very challenging problems while at the same time teaching the student how to learn mathematics properly. Section 1: Review of positional notational and addition/multiplication 1.1: Introduction to two models of learning behavior 1.1.2: Categorization of topics into { known, not known, not know that not know } 1.1.3: { motivated, bored, fearful } model of learning behavior 1.2: An extended curriculum on number theory in arbitrary number base is provided (as a hand out). This curriculum will be redeveloped as part of the author’s teaching effort’s this year. The following issues are noted: 1.2.1: This curriculum is designed to ground learning about an “unknown” set of topics that are accessable in steps. During the process of discovery, each learner will develop a private log on his/her experiences, frustrations and successes. 1.2.2: The nature and causes of learned disability with respect to learning the skills of mathematics. This is presented as a conjecture to the students. Once a learner’s interest and motivation has been sparked then a personal transformation can occur. 1.2.3: The exploration of base conversions is the subject of several recent patents in information theory. 1.2.4: The learning strategy happens to be ideal for an embodiment into a distance learning program. 1.3: The development of this curriculum involved the following steps (actually accomplished in a class room setting – at least partially) 1.3.1: The development of learner ability to easily add and multiple in a base other than 10. 1.3.1.1: Acquiring this skill requires an almost constant mental attention to “practicing” in the other base. This practicing is all that separates ANY student (no matter what the “natural” aptitude of the learner) and this skill. 1.3.1.2: The author’s repeated experiences offer the hope that the skill can (ALWAYS) be learned once the student’s learned inhibition to thinking about arithmetical concepts has been turned off. This turning off of this inhibition and the learner’s learning how to learn mathematics is the objective of the first two chapters 1.3.2: Students where repeatedly faced with a new problem that was at first both surprising and that no student in the class could solve when first posed. Students were often either bored or fearful. But in each case, individual students and then the class as a whole came to understand what the problem was and in most cases the student developed new skills. Those who did not were still bored. Example 1: the notion of a negative exponent is introduced by examining the rules: (a)m (a)n = (a)m+n and (a)0 = 1. The question of what (a)-1 must mean is asked. Because the novelty of the number base conversion has been high, and there has been several (perhaps as many as 10) cycles of being fearful/bored motivated/knowledgeable, there is more than one student (of average ability) who will all of a sudden start to “need” to convince the others that (a)-1 must mean 1/a. This will happen in the middle of a class if properly primed. The instructor can then ask the question (1/4)6 + (1/3) 6 = ?, and then dismiss the class. Example 2: The computer cannot represent 1/3. The computer can not represent (1/3)10 in base 10 since the computer can not develop a finite number of steps. This fact introduces the Greek paradoxes on quantification and comparison, such as Zeno’s paradox. But if one changes the base to 6 then (1/3)10 = (1/3)6 = (0.2) 6 . This leads quickly to the fundamental theorem in a new area of research in number base conversions and computer round-off error. This theme will be continued in the last chapter. Example 3: The replacement of the information (database) search/retrieval problem with a set membership problem. In this example, the instructor can set up the information (database) search/retrieval problem: Given a database column of 1,000,000 records, each holding a ASCII string with between 1 to 40 ASCII characters, and given a randomly selected ASCII string having between 1 and 40 ASCII characters; specify a process that identifies whether or not the randomly selected string is in the column. Can this be done in less that 21 fetch cycles in a serial computer? (Answer is yes.) This is NOT how traditional SQL databases do the information (database) search/retrieval problem, and as a consequence these traditional databases are not optimal. 1.3.3: The first elements of foundations theory and number theory can be done in an arbitrary base. Moreover, there happens to be number theory about number base conversions which is surprising and that can teach an “awakened learner” about foundational thinking and the nature of mathematics. Example: Given that a number expressed in base n is prime, and the number is converted to a different base m; is the new expression prime? End of Chapter Summary The original development of this curriculum was in order to test that hypothesis that a well-posed challenge, using completely novel curriculum, will shut off an inhibition of motivation. Chapter Five: Word problems Summary: Culturally relevant work problems are developed 1) Word problems continue to carry forward the notion that there are many unique and unexplored real world problems that the learner can both understand and become comfortable with. 2) The Chapter will take a good two weeks to cover and involves motivation from economics and accounting curriculums. In both cases, the motivation will involve some reading of and discussion of economic theory and the methods used in accounting. Chapter Six: Brief introduction to college algebra Summary: The first elements of curriculum in traditional college algebra can be done in arbitrary bases. This Chapter is designed to use the novelty of non-base-10 to bring the student “up against” the steps that are often over looked when this material is covered. The Chapter presents several very difficult challenges that, when solved, provide a great sense of accomplishment. Each of these challenges can be addressed somewhat independently. A Nodal Forest listing of topics { known, not known, not known that not known } 1) The elementary notions of relations and functions are developed, including permutations, functional composition and bi-jections (e.g., the one to one functions). 2) Example: One can introduce the notion of an x-axis and y-axis as a geometrical/topological constraint on a set of points and then assign a base to the expression of these points. 3) Example: The entire process of finding the slope of a line that contains two points can be done in base 7. The point is two-fold. Any student can be guided to be able to do this within one semester. Doing this is a huge accomplishment by the student, partially because anyone who has not taken the curriculum will NOT be able to demonstrate a superior skill at any of a large number of curious problems. 4) The slope plus one point formula for finding the equation of a line is covered in a base other than base-10. This requires a deep appreciation of the arithmetic including arithmetic on fractions. 5) Example: Algebraically, find the intersection between two lines while using only calculations performed outside of base-10. Chapter Seven: The polynomial equations in base 10. Summary: The quadratic equation is derived and the intersection between two quadratic equations is determined. This chapter is considered to be the core of business mathematics as well as one of the foundational prerequisites for the study of the calculus. 1) Motivation will come from word problems taken from economics (as found in traditional business mathematics text books) 2) The full definition of a polynomial is given. The addition and multiplication of polynomial forms are developed. 3) We restrict ourselves to first and second order polynomial equations. 4) Intersections between two lines, a line and a quadratic, and two quadratics are computed. 5) Discussion of complex numbers is developed. 6) The geometry and equations of the conic sections is developed. Chapter Eight: Discrete Mathematics Summary: The material in this chapter is designed to introduce the learner to computer science. 1) Finite state machines and transition state tables 2) Properties of relations, equivalence and partitions 3) Category theory, the fundamental notions 4) The Integers and the well-ordering principle 5) Principle of Mathematical Induction Chapter Nine: Number Theory Summary: The beginning of classical number theory (on the positive integers) is presented. 1) Sequences and series 2) Use of Induction 3) Prime numbers and composite numbers 4) The division algorithm 5) Greatest common divisor and least common multiplier 6) The Euclidean algorithm 7) The Fundamental Theorem of Arithmetic Chapter Ten: Number base conversions Summary: This Chapter will bring the curriculum full circle with the first two chapters. 1) The learner will look closely at the procedure of long division, but in a base other than base-10. 2) Each of the topics in Chapter Eight will be redone with examples that require the learner to compute completely outside of base-10. 3) The purpose of this chapter is to bring the learner to a full appreciation of the depth of knowledge of arithmetic that is really required to master college algebra. Appendix Three: One-Hour Demonstration of Prueitt’s Teaching/Learning Methodology September 6, 2002 Teaching Objective (in one hour): Engage the audience in the question: “In base-6 what is 1/3 + 1/4?” Anticipated Audience: Professors of mathematics, mathematics education and university administrators. There are five “agendas”. First: We ask the students to engage in the most elementary tasks found in basic arithmetic. Each student will appreciate the sophistication that is fundamental to arithmetic when arithmetic is fully comprehended. The switch to bases other than base-10 is used to expose the students to an extreme form of novelty while at the same time a series of problems that at first appear to be unlearnable. The first example is the question In base-6 what is 1/3 + 1/4? Second: We expose the student to what appears to be a series of very challenging problems. But the challenge of these problems is due simply to the students not be familiar with arithmetic in various bases. So a model of learning is taught to the students. This model is then used over and again to gain confidence in the new learning theory. It is in fact necessary to teach not only the mathematics slightly differently than in traditional courses, but to also teach the learning theory at least just enough to give the individual student the ability to see that the problem, of poor arithmetic skills, does not lie entirely with the student. D/S Model of learning: Things-to-learn are difficult or simple, depending only on one’s experience. In case of counting and doing arithmetic in bases other than 10, experience can be developed by ANYONE over a period of between two or three weeks. The path to acquiring this experience is at first a private experience with a few students. If circumstances are proper, this experience becomes a re-enforced social activity. The author has observed that the novelty of the learning experience provides positive social attention to those who learn first. This positive re- enforcement can spread to those who pick up on the experience. The ease of the first learning tasks, counting, then addition and then multiplication represented significant personal achievements that can be shared between the class members. But these early tasks are followed by other tasks to lead directly into higher mathematics such as computer science, topology, number theory, and category theory. Note on expectations: Positive social attention and increased interest in the new curriculum was observed (in previous teaching experiences at Hampton University, St Paul’s University and several community colleges (1989- 1994)). Given a period of two to three years, it is perhaps possible to make measurable differences in the University wide outcomes from mathematics training. We also expect to make contributions to statewide mathematics education programs and to the scholarly literature in cognitive science and mathematics education. Third: We ask that the student come to understand one’s own history regarding experience in mathematics classes. We have used writing across the disciplines (in vogue in the early 1990s) to ask that students write about personal feelings towards mathematics (and science) during the first week of class. At mid-term and then again at the end of the semester the student is asked to develop a private log of how they are feeling towards mathematics. Fourth: We ask that the students realize that not every one will see the light at the same time and in the same way. So the Teaching/Learning Methodology puts some rules down in regards to grades. Rule 1: Any test may be retaken given the student write an essay as to why the test was failed or done poorly. Rule 2: Students may reject the notion of learning and then come to a private understanding of the importance of breaking out of the inhibition of his/her interest in arithmetic, science, economics, etc. When that moment occurs, everything should be negotiable. Rule 3: Any activity that aids students in awaking interest in arithmetic, science, economics, etc should be supported. Fifth: The last agenda is related to eternalizing the process of learning. Most students “learn” that math is to be memorized and that theory is unknowable. Ask the freshman class and they will tell you this. Use a polling instrument and one will find that, on average, on college campus that 60% - 90% of incoming freshman will say that mathematics theory should not be talked about outside of class and that good grades in math is merely a question of rote memorization. But, we hold that the only way to make arithmetic difficult is to deny the student a clear understand of how arithmetic works. The author’s proposed developmental mathematics textbook creates learning tasks that: 1) Can be visualized as a student is walking to a friend’s house or to the store. 2) Can be rehearsed serendipitously. 3) Have a specific way to check the answer to problems that are made up randomly by the student. This last agenda has a neuro-cognitive basis to suggest that if this agenda become effective within a student body that the students will positively change behaviors in mathematics, science, economics, etc classes. We are looking for this type of result.