A Question of Access

 

Paul S. Prueitt, PhD

version date: February 16, 2004

 

 

The first two chapters will supplement standard curriculums for traditional college algebra or liberal arts mathematics.  The curriculum takes three weeks (or 1/6 the semester), after which time the usual curriculum is to be covered.  The curriculum can also be used as a remediation that occurs concurrent with the first three weeks of any freshman mathematics class.  The purpose of this three-week curriculum is to learn how to learn mathematics.

 

The complete book is designed as a first course in college mathematics. 

 

The material is rigorous and can be used to introduce computer science and college algebra.  There is about a 75% over lap with this traditional freshman course for liberal arts majors.

 

The material is presented as a university level course syllabus. Pedagogic remarks and notes are made regarding a teaching philosophy that is designed to re-open access to the foundations of mathematics.

 

Chapter One: On the nature of numeration (two weeks, three days a week)

 

Section 1:  On the history of counting (one week)

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.  Several of this learner papers will be selected for distribution back to the entire class (with permission form the author).

Section 2:  The use of position of digits in base-10 (one day)

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. 

2.2: (10)0 = 1 is discussed, at length and some of the properties of 0 are explored.

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

2.4: (a)0 = 1 is discussed and the use of proof is discussed.

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.

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 (one day)

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 group.

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?

3.4: The use of a symbol to represent an arbitrary number is discussed.

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?

Section 4: Positional convention for arbitrary number base (one day)

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 help 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, with the instruction to reflect on what is being learned.

 

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 à (q) 9 .

1.4: Learners as asked to develop a method for checking the answer. 

Check (425) 7 à (258) 9 .

use (425) 7 à (4(72) +2(71) + 5(70) ) 10 = (215) 10

and

(258) 9 à (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.

 

 

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 (optional)

1.3.1: Lofti Zadeh’s fuzzy logic and the dream of computing with words

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".  It was introduced by Dr. Lotfi Zadeh of UC/Berkeley in the 1960's as a means to model the uncertainty of natural language.

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: Pawlak’s 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

1.4: Category theory (optional)

1.4.1: Issues related to partitioning of data sets

1.4.2: History related to early attempts at theory of biology

1.5:  Mill’s logic (optional)

1.5.1: Over view of the history of formal logic

1.5.2: Divergences over theories of inference

1.5.3: Overview of cognitive neuroscience view of cognition

 

End of Chapter Summary

1)     Section 4.1 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, of lack or knowledge, of the formal processes of categorization from information technology.

2)     Section 4.2 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 4.3 is an optional section that is designed to give the awakening learner easy access to the two major variations of 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. 

4)     Sections 4.4 and 4.5 is an optional section that is designed to give a view into the substantial and philosophical issues related to the application of formal reasoning to models of human cognition. 


 

Chapter Four: Arithmetic in arbitrary number bases

 

Section 1:  Review of positional notational and addition/multiplication

1.1: Introduction to two models of learning behavior

1.1.2: Nodal Forest 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: Prueitt’s conjecture is on the nature and causes of learned disability with respect to learning the skills of mathematics.  This conjecture is very much connected to the conjecture that once a learner’s interest and motivation has been sparked then a personal transformation will occur.

1.2.3: It is true that the extended curriculum has many topics that have not been explored by the professional mathematics community.  It is also true that the exploration of base conversions is the subject of several recent patents in information theory.

1.2.4: The Nodal Forest learning strategy happens to be ideal for a embodiment into a distance learning program.  A delivery paradigm for Nodal Forest based curriculum was implemented in a prototype distance learning system for the US State Department in 1999.   

End of Chapter Summary

1)     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. 

2)     The development of this curriculum involved the following steps (actually accomplished in a class room setting – at least partially)

a.      The development of learner ability to easily add and multiple in a base other than 10.

                                                                          i.      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. 

                                                                        ii.      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

b.     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 where 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 à 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.  The class will seek the answer without being required by the teacher.

Example 2: 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.

c.     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?

 

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.

 

Chapter Six:  Brief introduction to college algebra 

 

Summary:  The first elements of curriculum in 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 very difficult challenges that when solved provide a great sense of accomplishment. 

 

1)     The most elementary notions of relations and functions are developed, including permutations, functional composition and bi-jections (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. 

5)     Example: Algebraically, find the intersection between two lines while using only calculations 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.

 

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 of conic sections is developed.

 

Chapter Eight:  Discrete Mathematics

 

Summary:  The material in this last 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.

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 do college algebra

 

A Sample Teaching Session

 

Teaching/Learning Methodology

 

 

 

Our thesis is that slow and often un-inspired instruction in arithmetic causes an inhibition of most children’s’ ability to think about arithmetic.  As entering freshman, this inhibition is often reinforced by peer pressure and social philosophy.  By changing the stimulus one creates a natural cognitive arousal to the changed stimulus.  Over time, a paradox is created in which individual students may come to learn about the inhibition.  At that point, a natural curiosity about mathematics and science will return.  This re-kindling of interest then may propagate within the members of the class and within the University.

 

Counting in base-6;  Organize into groups and acquire a bag of beans or a bag of coins.  Figure out what is the base-6 number that accounts for the bag’s contents. 

 

The set of digits for base 6 is { 0, 1, 2, 3, 4, 5, }

 

Positional notation:  

 

(54)6 = 5*(61) + 4*(60)

 

Zero-th power:   xn  *  xm  =  xn + m  ,  implies that   x0  =  1  no matter what x is, except if x = 0.   

 

This exception is an actual paradoxical fact of mathematics that has deep consequences that some students, even students who are very poorly prepared for college, will see if pointed out carefully.  This issue is often regarded as an obscure philosophical issue.  However, the resolution to this most interesting issue may carry more than one student far into the very foundations of mathematics. Why should a student be expected to just accept this “fact” since the fact has been “troublesome” to mathematicians?  School teachers should be knowledgeable about this fact and should be able to introduce students to a relevant part of the history of mathematics. This has often not been the case in school mathematics classes.  But, in turn, the mathematics education process has not produced the quantity of qualified teachers that is required in the school systems, particularly in rural settings.

 

The legitimization of students’ natural concerns, such as with the meaning of the zero-th power of any number, is vital. The school teacher must build the confidence of each student. 

 

In the curriculum being developed, we will see other opportunities to demonstrate this confidence building.  In each case, we have an opportunity to turn something that has been typically perplexing into something that opens access to the rich history and substance of arithmetic.

 

Review of positional notation for decimals: 

 

(243.21)6 = 2*(62) + 4*(61) + 3*(60) + 2*(6-1) + 1*(6-2)

 

where (6-2) = 1/(6-2)  is by definition and is consistent to the rule:

 

x-n  *  xn  =  x0  . 

 

The principle of consistency and the logic of consistency and completeness are being introduced here and can be picked back up later.  It is NOT assumed that NO student enrolled in a developmental mathematics class will become a dedicated mathematics major. 

 

Addition in base-6:  The table for addition is to be developed.

 

 

Addition Table, base-6

 

 

Multiplication in base-6: The table for multiplication is to be developed.

 

Exercise:      Find a , b, c

 

(97) 10 = ( ? )6   =  a*(62) + b*(61) + c*(60)

 

Hint:

 

63 = 216

62 = 36

61 = 6

60 = 1

 

Exercise:      (43) 6 + ( 45 )6   =   ( ? )6  

 

Exercise:       (416) 8 = ( ? )6  

 

Exercise:     How can we check this result using a non-trivial calculation?

 

Division in base 7: 

 

                        (426) 7 / (5)7   = ( ?  ) 7

 

                   Confirm that “long division” works in base 7.

 

Return to the question of fractions.

 

Exercise:     In base-7 what is 1/3 + 1/4?

 

Exercise:     In base-6 what is 1/3 * 1/4?

 

Now one can ask a question that has some deep confusion in the minds of most individuals who learned to add in grade one and was told that multiplication is to be learned later.  Multiplication is learned after addition “because” multiplication is more difficult that addition.  This is what many children “learn” in grade one.  Of course this is true in a sense if one is talking only about whole numbers. 

 

Exercise:    Compare the difficulty of doing fractional arithmetic and fractional multiplication.  Which is more difficult and why?

 

Polling instruments have shown, in research that the author has conducted in class, that this fundamental confusion actually is responsible for a certain percentage, perhaps as high as 25%, of students giving up on arithmetic when fractions are introduced. For some of the students a fact that they learned in grade one, regarding which is more difficult, is now wrong.  There is a subtle claim here that in some cases the natural emotional bonding to the first grade teacher is transferred to a belief that whatever was said about multiplication being more different than addition must apply also to fractions.  The teacher, on the other hand, is simply misunderstood by the young child.  The context that he/she had was correct, and the consequence not anticipated by the teaching profession or literature.

 

This seems a small matter until one sees the data.  The author has to admit that the analysis of data is not peer reviewed as yet. Making this data ready for peer review is part of what our current work in educational philosophy is designed to achieve. 

 

Liberating adults from this specific confusion is often accompanied by a change in self-image (see the work of Albert Bandura on Social Learning Theory and self image) with respect to the perceived ability to learn and be comfortable with arithmetic.  This is one step in a series that can be made in a complete liberation of the adult student’s image of achievement in this regard. 

 

The Challenge Problem

 

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 on number base conversions and computer round-off error. 

 

Theorem:  Any ratio of integers can be represented with a finite number of digits arranged in positional notation. 

 

Definitions:  If  bn is in base n, we can write it in with the “decimal” immediately to the right of the first digit.  For example we might write 32,400 as 3.24 * 10^ 4.  In base ten this is called scientific notation. 

 

 

Conjecture:  Suppose an and bm are in this “scientific notation”.   Then there exists a number cz  in base z such that z is minimal. 

 

( b * m^x  ) * ( a * n^y )  =   c * q^z

 

Implication:  Round off error due to indeterminacy can be controlled in a von Neumann computer.