WORKFILE MNUE

My net for Understanding EDUCATION

F Richard Singer III Current Edition: 10/2002

Website and Email:http://www.conceptualstudy.org

This edition integrates some earlier papers written while I was still actively engaged in teaching at Webster University. I have presented most the material as originally written, except for some minor correction and some changes to make the language used compatible with my current conceptual nets.

This workfile uses many of the concepts that are central to all my nets such as person, origin, events, processes, states, relationships, doing, behavior parameters, deliberate action, situations, understanding, nets, conceptual, paraceptual. However I have tried to make this workfile self contained by briefly explaining these concepts as I use them. A fuller presentation of these and related concepts can be found in the workfile “My Net for Philosophy”.

This distinction between conceptual and paraceptual statements is essential to an understanding of this workfile. Conceptual statements merely give information about the concepts being used. Such statements are correct unless they fail to use concepts as they have been presented. Paraceptual statements use concepts to propose information about some state of affairs that is not merely conceptual. When I want to stress the conceptual nature of a statement I use a capitalized form o the word “IS”. Thus the first sentence (1) below is an abbreviated way of saying sentence (2).

(1) Socialized education IS any process that a society uses to influence or direct learning.

(2) Socialized education is conceptualized in such a way that any process a society uses to
influence or direct learning is an instance of socialized education.

SECTION 0 THE PRIMARY PURPOSE OF SCHOOLING, SOCIAL OR PERSONAL

Introductory Remarks: This section introduces some concepts which I find useful in thinking about the relationship between persons and society. To illustrate them, I have chosen one arena that I find significant, the arena of schools. I act within a world which stresses the social purpose of schools. In contrast my attitude towards learning is such that I stress the personal purpose of education. This present section examines this attitude and clarifies my own concepts and strategies. I hope it might prove useful to other persons with similar concerns. Later sections show results of this attitude in practice.

I have tried to only make paraceptual claims that persons with ordinary information about schools will find highly plausible. Anyone doubting any claims in this paper, probably has not understood what I am proposing. This does not mean that most persons would approve of my strategies. Objections to them should occur even to persons with similar ideals if they doubt the utility these strategies might have for enhancing these ideals. Furthermore anyone with opposing ideals will object to my strategies if they deem them as potentially effective. I make no claims about whether my ideals are more likely to result in good than evil, and I only conjecture that my strategies are useful for enhancing my ideals. For a more detailed discussion of ideal an their role see “My Net For Doing”.

The Primary Purpose of Schooling: Schooling IS socialized education that takes place in any institution that is a school. Schools include elementary schools thru universities. The primary purpose of schooling IS the primary purpose exhibited by the social practices within these schools. Since these are concepts, they cannot imply any paraceptual or normative links between schooling and its purposes. Statements by Willoughby and Faeber make paraceptual claims about what seems to be the primary purpose of schooling in our current educational system.

Education has developed differently from other professions in our society. Probably because of its greater importance to our society, education was socialized very early in history, while other professions remain under the free enterprise system. You, as an individual, may believe that it is important that you have excellent medical or legal aid when you need it, but to society it is more important that you have an excellent education. The legal and medical professions will attempt to help you whether your future contribution to society is likely to be positive or negative, but it is the goal of the education profession to help make your contribution to society more positive.
--Stephen S. Willoughby, see [A]--

School is where you let the dying society put its trip on you..... School is a genetic mechanism for society, a kind of DNA process that continually recreates styles, skills, values, hangups; and so keeps the whole think going. The dying part of society, the society that has been, molds the emerging part more or less in its own image, and fashions the society that will be. Our schools may seem useful to make children into doctors, sociologists, engineers...to discover things. But they are poisonous as well. They exploit and enslave students; they petrify society, they make democracy unlikely. And its not what you’re taught that does the harm but how you’re taught. --Jerry Faeber, see [B]--

In spite of their differences, Willoughby and Faeber agree that the primacy of schooling is the socializing purpose of education, i.e. the social purpose of preparing students to become useful members of society from the perspective of what society considers useful. Willoughby approves of this purpose and of the way schools try to implement this purpose, and his essay goes on to suggest how the implementation could be more effective. Faeber does not approve of the socializing purpose he observes. If you read his essay you will find that he wants education to serve the purpose of transforming society, preparing persons to fit into a radically better society, rather than fitting them into society as it exists.

I also find the evidence too strong for me to seriously doubt the paraceptual claim that the primary purpose of schools is to prepare persons to become useful members of society. Major school practices are directed by this purpose, even though they may not always work to enhance it. If you go to a doctor, are you expected to show that you benefited from these services? In school you earn grades and credits, which become part of a record which you are expected to make available to others. Why? Because it is not primarily for you. While most educators favor helping individual students, they often view this as doing what is good for the student, not according to the student’s view, but according to socially held views. Schools subordinate personal goals to social goals and to society’s goals for persons. Serving persons is an important secondary purpose of schools, supplementing their primary purpose when the needs of a person happen to be compatible.

My Primary Purpose as an Educator: The primary purpose of schooling is not my primary purpose as an educator. I have no interest in preparing persons merely to fit into a social system or to serve the purposes of society. I am interested in the fact that education can serve and be guided by the personal purposes, and in particular the purposes of students. My primary purpose for education is to exemplify this aspect of education, by being a self-directed student and also by assisting any students showing an inclination to take charge of their own education. This is descriptive of my purposes. It is not a normative claim or a recommendation. At most I would recommend experiments in shifting the primary purpose of schooling in that direction. I see no essential conflict in acting within a schooling system whose primary purposes differs from my own, altho this does present me with some difficult problems.

I teach because I am fascinated by thinking, by the creations and imaginings of persons. I teach because I favor creative involvement, personal discipline, love, courage, wisdom. A person behaves as an origin in a situation to the extent that they take initiative, are active rather than passive, proactive more than reactive, focus on creation more than maintenance. My origin ideal envisions more supportive environments for persons acting as origins to deliberately enhance their own person characteristics in directions which increase their behavior options. The main precept guiding my strategy for implementing this ideal is to avoid giving social purposes precedence over the acknowledged purposes of the persons I encounter. I call this the strategy of radical originship. My primary purpose as an educator is rooted in my origin ideal and I use this strategy in support of that purpose. This purpose is to challenge persons to enhance their originship by developing a variety of characteristics, and to support this process. As an educator I have chosen to emphasize the personal rather than the social purposes of education.

While emphasizing personal purposes can involve conflict with our schooling system and its social purposes, this is not my main problem. My problem is that even though social and personal purposes are often supplementary, energy that could be devoted to personal purposes gets drained by working within a system structured to automatically channel resources towards the social purposes of education. How can I channel more energy into the personal purposes I want to make primary? Later sections focus on strategies which provide a partial answer to this question. For now I merely acknowledge it as a problem.

Conceptual Nets: A net is a related set of concepts and conceptual distinctions. A conceptual proposition merely proposes information about relation within some such net, while a paraceptual proposition uses the net to propose information about some states beyond the net. This paper uses the concepts of person and society and state of affairs from the conceptual net of descriptive psychology. This net is a systematic refinement of the net we commonly use to think about persons and how they behave in the world, so my use of this net should not depend on the reader having any explicit knowledge of that net. The concept of a state of affairs will be referred to as a state.

I have already used my concepts of the primary purpose of schooling, the social purpose of education, the personal purpose of education, my purpose for education, and my origin ideal. The names used for these concepts suggest concepts in more standard nets, and altho my net differs from these, I hope that these differences are not so great as to obscure the main thrust of my thinking.

Other concepts I use include radical behavior and conservative behavior, but these concepts are somewhat ambiguous in many ordinary nets. To prevent blatant confusion I briefly present each of these main concepts before using it. In particular I focus on the concepts of conservative and radical in a way that makes them conceptually orthogonal rather than conceptual opposites. For this purpose I conceptualize two dimensions, stability and depth. I then make some conjectures about likely relationships between conservative and radical behavior, under certain types of conditions.

Depth Dimension: The depth dimension involves the extent to which a persons behavior relates to a surface level versus a root level examination of some state. Radical behavior IS behavior whose want parameter includes a strong desire to get at the roots of some state and whose competence parameter includes distinctions involving these roots and what might affect them. Loosely speaking, to have the person characteristic of being a radical IS to have a significant history of radical behavior. The opposite pole to radical behavior could be called conventional behavior. Neither concept is intended in either a positive or negative sense. There are times when radical behavior is useful, times when conventional behavior is useful, and times when it is most useful to act somewhat conventional and somewhat radical.

Stability Dimension: The stability dimension involves the extent to which a persons action relates to changing or maintaining some state. Conservative behavior IS behavior whose want parameter includes a strong desire to maintain some state and whose competence parameter includes distinctions about actions doing this and requires some skill in implementing these actions. To have the person characteristic of being a conservative IS to have a significant history of conservative behavior. The opposite pole to conservative behavior could be called modifying behavior, and a person acting that way IS a modifier. Again, neither concept is intended in either a positive or negative sense.

Comments: In this conceptualization it is perfectly consistent for behavior to be both radical and conservative, as it would be when we think an root level examination of factors relating to some state that a person favors would be useful in maintaining this state. One can also act as a modifier without acting as a radical, but if the change is likely to affect the roots of some state then the changes that occur may come as a surprise to the modifier. Such surprise might be less likely if the modifier had engaged in some competent radical thinking behavior.

The General Role of Persons and Societies in Human Evolution The statements below form a conjecture which I find plausible enough to act upon.

¨ Novelties tend to emerge thru the thoughts and actions of persons and are then filtered thru highly conservative social mechanisms.

¨ This conservative role of society is not essentially antagonistic to radical behavior of persons or to their radical ideals and goals.

¨ Conservative and radical are often supplementary.

¨ For modifier, society can be more like a powerful game opponent than a war enemy.

¨ Such an opponent presents a challenge to experiment with new wild moves that might affect the root of some state.

These conjectures do not suggest a dichotomy between personal and social purposes, but it can be important to focus on which purpose is taken as primary. Think of two sides of a coin. Neither exists without the other. These sides are not identical and the question is which side to consider as heads and which as tails. At times personal purposes may directly conflict with social purposes, but even when they do not, effort directed towards one cluster of purposes may use resources which could be directed towards the other. Even when two clusters of purposes are supportive, which purposes are emphasized can make a considerable difference to the extent to which each is enhanced. Thus while I partially endorse some aspects of the social purpose of education, I work most actively on its personal purpose.

An Analogy: Adam Smith claimed that the general welfare of society would follow if persons were as free as possible to pursue their own economic welfare. Lenin claimed that the economic welfare of persons could only be obtained in a system that stressed central planning for the economic welfare of society, at least during the period when capitalism was being destroyed. Both claimed to be concerned with the economic welfare of persons and society and to see these as positively related. However, they differed radically in which purposes should be emphasized, and thus advocated radically different economic systems.

Some Minor Evidence For My Conjecture: I conjectured that originship tends to be enhanced thru radical experimentation in an environment with strong support systems, where persons provide the experiments and society provides the support system. This conjecture is based on non-systematic observations of social and personal states of affairs. It is also based on some general paraceptual conjecturing that has been suggested by conceptual analysis.

I do not see how a society can be strong unless it is highly conservative. The strength of society seems to reside in traditions and customs whose workings are so complex that conceptual analysis seldom provides any workable alternative. Social reformers and revolutionaries seldom see that the mechanisms which they think have evil consequences may be linked in a complex manner to other mechanism supportive of values essential to the existence of the society. This is not to say that there are no better ways to enhance social purposes than those which tradition has evolved. It is merely to acknowledge that most of the time deliberate change based on conceptual analysis and paraceptual conjecturing probably will not produce the results intended. Fortunately most attempted reforms produce almost no results. When they do, at least with any regularity, this is more likely to lead to social instability than to utopia. A strong society is like a massive gyroscope which is likely to maintain its stability and return close to its normal motion in spite of the shocks it receives. This may be frustrating to potential reformers or revolutionaries, but should be comforting to the rest of us. As a radical I find it extremely comforting.

My personal radicalism leads me to act as a modifier only in respect to certain states. In many situations I am essentially conservative. I need a strong support system in order to experiment with being a modifier. Without this support, I tend to become security oriented, and this narrows the options I imagine. I am able to devote my energy towards enhancing originship only because society is able to maintain itself without my deliberate support of its purposes. If the majority of the people in the world suddenly adopted my strategy, I would be appalled. I might even become an educational conservative, reluctantly because my talents in this regard seem to be minimal in comparison to the talents many other people exhibit.

An Analogy: Suppose the majority of people in the world decided to devote the major part of their energy to developing curriculum in mathematics. Other things, which I now depend on for my survival, would not be done. I suspect that I would need to shift my energy towards providing things I now take for granted. Perhaps I have some latent talents for these things and the rest of the world has hidden mathematical talents that I have not observed. I suspect not.

I believe that most people are even more conservative than I am, at least a majority of the time and in most ways. I expect radical experimentation with originship or anything else to come from persons, but only in a limited manner. Some persons may cultivate radical ideas and find they are satisfied with most states they feel they can influence. Most of the time they will behave conservatively. Others may only have radical ideas in specialized areas, accepting more conventional ones in other realms of endeavor. Some people may behave in radical ways and have conservative ideas. Almost any mixture of radicalism and conservatism may occur in a person, yet there is a tendency for radicalism about a state to lead to a desire for change. This is because we tend to see possibilities for change when we go to the root things. On examination we can usually imagine favorable change in almost any state. Thus each person will probably at some time experiment with something which is both radical and novel, while in most respects being more conventional than radical, and more concerned with conservation than novelty.

My conjecture that radical experiments can come primarily from persons depends only on these occasional flirtations with radicalism and novelty. There is little advantage for more extensive experimentation. Most radical experiments are more likely to self-destruct than to be creative. That is why I think society is a necessary support for any origin ideal. It checks the destructive effects of my creative behavior, but it accommodates itself at least some the positive effects. Society may resist new experiments, but once they are established, it protects and conserves them as its own. Radical modification activity need not be rooted in a disdain for the conservative function of society. Rather it can be rooted in a willingness to gamble, and to hope that society can maintain itself regardless of the most reckless experiments with originship that we might have the courage to try.

Conclusion: I am aware of my deep conservatism. To become more of an origin I must cultivate my radicalism, or I will easily be overwhelmed by my conservative tendencies. The strategy I have chosen to implement my primary purpose for education is a form of radical originship, that is I try to never favor the purposes of society over the acknowledged purposes of any person with whom I am interacting. This is a personal strategy which I adopt for a variety of reasons. It is not a strategy that I would recommend to others unless they are willing to take certain risks. It is an experimental strategy and it keeps me in contact with my conjectures about how originship might be enhanced. My experiment is to place primary emphasis on the goals of persons, and to let society protect conventional values as best it can. I do not know whether such a strategy can seed fundamental changes in our social structures. Nor is that my primary concern. I am more interested in the immediate effect on myself and on the persons that I encounter. Given the conservative characteristics that most of us so often exhibit, I doubt that my experiment in would be a major threat to social continuity.

Organization: The sections of this paper this paper can be read in any order. Section 1 focuses on general themes in my educational behavior as an adult with an emphasis my purposes as an educator. The appendix augments this by indicating one of the policies I have adopted in an attempt to implement my purpose within a system which takes the social purpose of education as the primary purpose of education, while I merely take this as my secondary purpose as an educator. Again, I stress that I am using a conceptual net in a way that my conceptual claims are correct, and that any paraceptual claims I make should seem highly plausible. Furthermore, conjectures I make are intended to be questionable, so they should not be taken as claims. Section 2 focuses on the evolution of my educational ideals and purposes, primarily by tracing my development as a learner. Section 4 presents the concept of an informal learning center as a way of illustrating my educational ideals. Section 5 focuses on why and how I teach mathematics. This section is also integrated into another one of my main workfiles available on my website, namely “My Net For Mathematics”. Anyone interested in mathematics education might want to read this section in conjunction with that workfile.

SECTION 1 MY EDUCATIONAL BEHAVIOR

General Perspective: I teach so I may learn. I learn in order to teach, so I may learn again in more depth. There is an irreducible element of choice each time I reaffirm my decision to learn-teach, for I have chosen to idealize learning-teaching both for its own sake and because learning-teaching is supportive of other fundamental ideals I have chosen.

There are also a variety of factors influencing my educational behavior at any point in time, such as my physical condition, my emotional state, external pressures, my needs, etc. To examine these would be a complex task far beyond the scope of this workfile. I will leave it to you to conjecture what such factors might be operative at various times, if you find my analysis too superficial. The purpose of this section is merely to provide a sketch of what I believe are the two most important continuing roots of my educational behavior, namely part my working theories and my ideals.

I distinguish between ideals and working theories as follows. Ideals are the patterns a person would choose for shaping some state. As such they make no claims, not even claims that the states they envision is better for persons than some of the alternatives they can imagine. The concept of a theory is a broad ordinary one which allows theories to vary in the extent to which they are loosely or systematically organized. Theories help a person think about what reality might be like. A working theory, that a person holds at a given time, is a collections of one or more paraceptual conjectures which a person is willing to act on because the expected value of doing so seems greater than the expected value of acting on imagined alternatives. Thus it often makes sense to adopt a working theory that seems less plausible than some of its alternative, such as the theory involved in Pascal’s wager, or more simply the theory one might adopt in playing a lottery. A theory can only be related to behavior if certain purposes or ideals are chosen, as may be illustrated by the following oversimplified example.

Example: Suppose I hold the theory that rote drill is the most effective way to develop algorithmic skills, but that it undermines a person’s potential for obtaining intuitive insight about mathematical ideas. If my dominant ideal is to enhance a students potential for mathematical insight I will seldom suggest pure rote drill. If my dominant ideal is to help a student master certain skills then I might assign such drill.

Ideals: My purposes as an educator are rooted in ideals about persons. These ideals involve the idea of an origin. To act as an origin relative to some state IS to intervene with some deliberate purpose of altering it. If a ball is rolling down a hill toward a pond and you try to stop the ball so it will not go into the water then you are acting as an origin. On the other hand, if the ball stops merely because it ran into you then you are a chance factor rather than an origin relative to that system. It is probably sufficient to think of originship competence as the ability to live effectively within the states you are likely to encounter by developing the characteristics to imagine and create options. This is distinct from merely having social or political liberties which may be somewhat supportive of such originship, but which cannot guarantee it. Originship takes its ultimate support from the person who is acting as an origin.

My most fundamental ideal envisions the creation of environments which are vastly more supportive of persons who are expanding their originship powers to levels beyond any which currently exist. I call this my origin ideal. Altho this ideal may sound individualistic, it has significant social components. I think that originship is enhanced by developing richer personal relationships. I also think it is enhanced by trying to create a society more supportive of persons. I see social goals and person goals a supplementary, as long as personal goals are considered as primary.

Since I sense a strong tendency to sacrifice personal goals to social ones, one main precept guiding my strategy for the implementation of my origin ideal is to never give social purposes precedence over the acknowledged purposes of the persons I encounter. I call this the strategy of radical personism. My primary purpose as an educator is to challenge persons to expand their inner resources, and if they so choose to assist in this process. Thus my primary purpose as an educator is to use education directly in support of my origin ideal.

Until I was about 23 year old most of my educational behavior included more emphasis on learning than on teaching. My learning behavior can be fairly well illustrated a number of statements.

(B1) I was highly involved in my own learning, asking many questions and trying to obtain
knowledge about anything I was exposed to.

(B2) I integrated learning with play and recreation.

(B3) I drew heavily on educational institutions as a learning resource, at the same time trying to
use my wits to minimize the requirements they placed on me.

(B4) I directed my learning behavior toward my own purposes, mostly this meant the satisfaction
of my own curiosity, altho sometime in school I directed it towards other tasks.

(B5) The dominant activity in my learning behavior was figuring things out, trying to reduce
detail by seeing a pattern I could understand.

While (B1) thru (B5) are generally descriptive of my behavior over a long time period, a more detailed description would show a great deal of variance. For example, at age 9 I was fascinated by bigness and maps, so I learned relative sizes of countries like Russia, U.S., China, Brazil, and Canada. I learned that Greenland was smaller than each of them, even though on Mercator maps it looked bigger than South America. At age 17 I read Adam Smith’s “Wealth of Nations” to find out about the origins of the theory of free enterprise. Both behaviors fit under B1, but at 17 I was exposing myself and being exposed to different things than when I was 9. In general as I get older my curiosity became more directed towards the abstract and general.

During my teenage years my learning behavior expanded to a form of informal teaching behavior with my younger brothers and with fellow students, in the following sense.

(B6) I spent a considerable amount of time explaining to others how I thought about things.

At age 23 I took my first job as math teacher, and (B6) that I tried to make most pervasive. However, in practice when I taught secondary school my behavior could be described as follows:

(B7) I taught students who were assigned to me according to an institutional schedule.

(B8) I spent most of my time on material in the text I was given.

(B9) Each day I spent some time trying to make things reasonable to my students, but mostly
I tried to make sure that my students temporarily acquired specific skills, even though
I felt their understanding was minimal.

(B10) I assigned class work or home work daily and gave weekly quizzes. I devised a
grading system to reward skill, but to more highly reward understanding.

(B11) I encouraged a few students to go beyond the test to ideas I knew to be more significant.

When I moved from teaching secondary math to college math, my behavior changed considerably. Altho I still worked within an institutional schedule, I had control over when my classes were scheduled and some control over who I took as students. I made less use of texts, often developing my own materials, and using texts only for supplemental purposes. I freely modified curriculum to focus on helping students expand their power to reason. I emphasized acquisition of skill only if accompanied by understanding. I designed test questions and homework problems that could not be solved by mere mastery of skills. I tried to encourage students to delve into ideas more significant than those of the standard curriculum. I also spent a large amount of time and effort trying to help restructure educational programs and institutions. I tried to help create or imagine various structures that would make learning options more flexible and place more responsibility for decisions about learning with the individual student.

One program which I played the major role in redesigning was the master of arts in teaching mathematics. Earlier I had helped design a program in which there was a definite sequence of courses a student would take, depending on the student’s background. There were many options within these courses, but each had a definite focus and met for a block of time related to the credits offered. My first change was to break up time blocks and try to allocate time according to perceived educational needs rather than credits or course title. Students registered for different courses might meet together for certain purposes. Students registered for the same course might not all be meeting together. Amount of meeting time varied with regard to purpose rather than credits. Later, I totally dropped the procedure of registering for courses, and had students register according to a category. If they weren’t working on a degree, they merely registered for general studies. If they were working on a degree they registered for advanced or basic masters studies, merely on the basis of how long they had been in the program. There were no grades in the program, merely credit or no credit. The teacher had no responsibility for even assigning credit. Credit was automatic, unless the student withdrew or definitely selected no credit.

At that time there were a variety of theoretical ideas that had a major impact on my educational behavior. I shall label those that I held as (T1), (T2), etc. I shall label some of the beliefs of others which influenced by behavior as (OT1), (OT2), etc. These influenced me even though I did not agree with them.

(T1) Knowledge is an account of the way things are, and thus grounded in reality rather
than authority, and is accessible thru reasoning and experience.

(T2) Understanding is the key to efficient knowing, and since all normal humans are potentially
rational in the same way they can obtain the same knowledge.

(T3) Knowledge is the main source of personal power and personal virtues, and hence is
good for every human being.

(T4) Reliance on authority and rote memory diminishes the capacity to obtain significant knowledge.

(T5) Knowledge is good for society, being one source of human well being and social progress.

(T6) Altho all normal human beings have a potential to become interested in understanding,
social and educational traditions reinforce behavior incompatible with this potential.

(T7) External reward systems, (grades, praise, etc.) can be designed which will enhance anyone’s
capacity to become a better learner. Thus an accreditation systems can be used to adequately
indicate learning accomplishments. Furthermore, it would be good to use it in this manner.

(T8) Educational institutions have the potential to help most people significantly expand
their learning and their power to learn.

(T9) In practice, educational institutions don’t even begin to approach the potential they
have to enhance learning.

(OT1) There is a collection of skills which everyone needs, primarily in order to fit in and
function in society, and the standard school curriculum embodies these.

(OT2) It is good for persons to become useful members of society.

(OT3) Alto schools could improve, they do an adequate job of teaching the standard curriculum.

(OT4) Radical change in the structure of education is more likely to interfere with learning
than to enhance learning.

I think it should be fairly easy to see that most of my learning behavior meshed well with my ideals and my own theoretical ideas. My teaching behavior, on the other hand, tended to lag behind the development of both my ideals and my theoretical orientation. This behavior was more strongly influenced by habit, custom, tradition, external pressure, the theoretical ideas of others. This was especially true of my behavior as a secondary school teacher. Even as I articulated my ideals and theories I had to struggle to root my behavior in them.

My earlier behavior as a college teacher involved shifting toward such a rooting. I was aware that I had previously influenced by (OT1) thru (OT4) and was deliberate in my rejection of those. With experience some of my theoretical idea changed. In particular I came to reject (T7).

About age 40 my attitudes changed in a radical fashion. Looking back on (T1) thru (T9), I acknowledge that when I talked that way most of these sentences expressed propositions with more than a minimal amount of truth. However whatever truth I may have glimpsed at that time, is muddled by conceptual nets much less powerful than my ones. Nevertheless, on reflection, I think that I was more wrong than right about most of my theoretical beliefs. I also have changed my way of thinking about ideals. I used to think of ideals as being basically theoretical claims of a special kind, claims about what was good. I now no longer can understand how I thought this made sense. I stopped thinking of ideals as correct or incorrect, but as facts, partially created by my choices. These ideals like any facts may be good or bad. Furthermore I decided to regard my ideals as more fundamental than considerations of good and evil. Before making this decision, my most basic purpose had been to do what was good for myself and other persons. This now became a secondary purpose.

My behavior as a teacher until then involved trying to redesign educational institutions and programs within these institutions. I currently have only a passing interest in this, but I would be happy to discuss it in detail with anyone who is interested. My way of thinking has shifted too radically. When I look back, my older way of thinking seems rather naive.

Current Concepts and Theories: I distinguish three processes as follows. Learning is an automatic and continuing process. You learn because you adapt on the basis of experience. Learning can be functional, but this is not always the case, so it can be useful to structure experiences in order to guide learning. This is what education attempts, and we can define education as the deliberate attempt to structure experiences in order to guide learning. Education is often a highly socialistic enterprise, that is structured to serve the purposes of society and to draw extensively on social mechanisms. I shall define schooling as socialistic education, in that sense. Note that large corporations are also socialistic in that sense, so I am not using socialistic as opposed to capitalistic.

I try to use the word education in a fairly precise way that will distinguish it from two other related processes which I call learning and schooling. The first distinction is probably easier to remember because it should be apparent from everyday experience that considerable learning takes place without education. The second distinction may be easy to forget because schooling currently plays such a prominent role in education.

The concept of learning I use is broad in the sense commonly used in the literature on learning theory. Learning consists of almost any established changes in an person’s perspective, attitudes, ideas, skills, information, values, or general response tendencies. As a process, learning tends to occur in conjunction with any activity and to be regulated by automatic feedback mechanism. For learning to accompany an activity, it is definitely not necessary for the person or anyone else to perceive of the activity as directed towards learning. In fact many would claim that more learning takes place when the activity is perceived as fulfilling at least some other purposes than mere learning, and for many types of learning this may be the case. For example, I often make little progress when I try to learn some new mathematical ideas, yet make considerable progress learning these ideas when I try to use them to solve some problem.

Much of what we learn does not depend on a conscious effort or decision to learn. However we can make deliberate decisions which affect learning. We can supplement natural learning processes. We can structure situations which we hope will increase the likelihood of selected learning goals. I use the word education to refer to a multitude of processes which are linked by a common thread, namely they all involve a deliberate attempt to influence or channel learning.

Education can be a highly personal enterprise, as in the case of a person who deliberately tries to learn thru personal observation. On the other hand it can be a highly socialistic enterprise, as in the case of the student in a typical classroom. Most educational processes involve both socialistic and personal components. For example, I continue to educate myself thru personal writing and problem solving, but I also use books and other resources which have been produced thru highly organized social processes. Altho I recognize that my education still draws on social resources, it is not organized around any specific cluster of social mechanisms and is largely under my personal control, so I would classify it as a primarily personal enterprise.

In conversations with students and teachers I sense an implicit identification of education with a special type of education which can be called schooling. Schooling involves students and teachers in an institutional setting. In this setting teachers are expected to assume responsibility for the direction and evaluation of the students learning. The degree to which students are also expected to be responsible for their education may vary, but in all schooling they are expected to acknowledge some authority on the part of the teachers in such matters. Furthermore, both students and teachers are expected to accept institutional authority and usually this authority is supposed to be at least somewhat rooted in broader social purposes. In practice schooling is the type of education stressed by most schools, such as elementary, secondary, colleges, universities, training centers, etc.

The fact that the primary purpose of education is currently social is probably related to the fact that schooling is the predominant form of education. Even people who are not currently involved with schooling tend to implicitly identify education with schooling and both private and public resources for education tend to be channeled thru schools. Yet schooling is not the only form of education, and if other forms begin to flourish the primary purpose of education might possibly shift. One of my strongest interests in education is in the alternative which I call informal education. This is discussed in Section 4.

By its very nature schooling tends to be a socialistic enterprise. This does not mean that it must place a predominant emphasis on social purposes, but it makes such an emphasis more likely. For my own idealistic reasons I would advocate a shift towards a greater emphasis on personal purposes. I believe that some shift in this direction can come, but only slowly, and I doubt that my primary purpose as an educator is likely to become the primary purpose of schooling the foreseeable future. The attitudes in which the primary purposes of schooling are rooted are extensive and beyond my comprehension, so any projections I make about such a shift should be highly tentative.

Current Ideals: Many radical educators favor universal human autonomy. They try to justify this on practical or ethical grounds. I neither favor nor oppose universal human autonomy. My related origin ideal is limited in scope, and applies only to persons who choose originship a primary goal. Perhaps it is neither practical nor good to expand originship as radically as I would see it expanded. Even as applied to me, I am not sure that my origin ideal is practical or good; however my choice of ideals is transrational, and thus it is not a choice I judge in terms of prudential or ethical considerations. My ideals are grounded in me, rather than in external criteria. As long as I continue to reaffirm this choice, it provides the main basis in which my primary purpose as an educator will be rooted. Other considerations are relevant only to the strategies I use in trying to implement this purpose.

The fact that one of my primary purposes as an origin is to make my ideals take precedence over ethical considerations is itself grounded in my origin ideal. Our ethical principles have evolved and are changing. One dangerous aspect of origin activity is to experiment with alternative to ethical behavior, perhaps giving birth to new ethical principle, perhaps living from a more radical perspective. The relationship between ethics and an origin ideal is considered in “My Net for Doing”.

I think it is important for me to stress that the main reason I would have education take the purposes of the persons it serves as primary is that I think this would serve my origin ideal. This comment is not intended to help me obtain much support. I do not believe many people share this ideal. However I am not interested in having other people support methods which enhance my ideals at the expense of their own values. Such support is not reliable. I prefer allies who either share my ideals or who at least find their ideals supported by the same strategies which support mine.

To those persons who value the majority of social purposes which have evolved in our society, I can only offer a few feeble reasons for supporting some shift to a greater emphasis on the purposes of persons, especially when these purposes seem to conflict with social purposes. I can only say that the conservative mechanisms of society are probably strong enough to allow the risks and that there may be social benefits in allowing persons to pursue such purposes, as long as doing so is not too dangerous. Purposes are often complex and the apparent conflict may only be due to a surface perspective. Society might find that its purposes are being served in spite of the apparent conflict. Furthermore a society which trusts the wisdom of its purposes can trust that they are good for most persons. Often the only way some persons can discover this social wisdom to experiment with deviant purposes and fail. I shall not press such reasons. They do not support the radical personalism I favor, but only a milder controlled personalism in which social purposes are still primary. This is what I would advise as the primary purpose of our socialistic educational system, altho personally I choose to work for a more radical personalism.

I end this section with some comments on personalism within a socialistic enterprise. In addition to stressing social purposes, most socialistic enterprises stress the purposes of persons who hold key positions in the enterprise, and they can even stress the purposes of everyone interacting with the enterprise. For example, the Gestapo was a socialistic enterprise, but it is easy to conjecture that its primary purpose was to serve the interests and ideals of a few top Nazi leaders. In general, social turmoil tends to shift the primary purposes of socialistic enterprises towards the purposes of persons who are in a position to exploit these resources. However such a shift towards person purposes is not likely to be a shift which I would find supportive of my origin ideal. I am interested in the kind of shift in which socialistic enterprises deliberately stress personal purposes because they see social purposes primarily as a support system for all personal purposes. I think that such a perspective might actually do more to strengthen society and enhance social purposes than a direct preoccupation with social purposes. I do not think it would be directly supportive of my ideal, but it is more likely to provide an environment in which such originship can be enhanced than any of the alternatives I can currently imagine.

Coexistence with Educational Institutions: My current behavior as an educator can be described as coexistence with educational institutions. This behavior is rooted in a radical new perspective on my fundamental ideals and a radical shift in my theories about education. The radical aspects of this behavior are often not noticed, since I they tend to bypass rather than challenge institutional structures. To serve my primary purpose as an educator, I try to make my teaching behavior supportive of those persons who show a tendency to use education for their own purposes. There are a variety of factors which can distract me from this strategy. The persons who I would assist are usually a minority among my students, and the structure I create to support them can result in less learning and more dissatisfaction for the majority of my students than other structures I could provide. The persons I would serve are often registered in courses whose titles suggest purposes that are not very important to these persons. Few of them can easily ignore this fact and make their education serve their purposes to the extent they would choose, so they often experience feelings or failure or frustration.

None of my colleagues actively share my primary purpose as an educator. They are all moved, much more than I would choose to be, by practical and ethical considerations. Even those who are oriented more towards the personal rather than the social purpose of education, tend to think in terms of what is good for most of their students, and their attitudes influence me at an unconscious level. Working within

a system tends to draw me into a concern about the repair or modification of the system, at least the part of it with which I most often interact, and it is often difficult for me to tell whether or not this even indirectly helps me serve the purposes of my students. It is only by emphasizing that my origin ideal is more important to me than ethical or practical considerations, that I can keep my primary purpose as an educator in focus and avoid being too much distracted by all these factors.

In spite of the fact that my primary purpose as an educator differs radically from what I observe as the primary purpose of education in our educational system, I continue to work within this system. Some of my involvement is due to inertia, stemming from the time when my nets were too primitive to allow me to analyze my involvement as precisely as I now can. At that time I claimed that my primary purpose as an educator was the true purpose of education, and I wanted to demonstrate this claim by creating islands within the system supportive of this true purpose. I now know that this supposed claim was too imprecise to have much cognitive content. Both my reason for favoring and my strategies with respect to the creation of islands supporting my educational purposes have shifted. I now favor their creation because of personal ideals, but since my ideals are tentative long range blueprints, I no longer feel any sense of urgency in creation. I expect the kind of islands that I favor are more likely to be created outside of our educational system rather than on its fringes where I first hoped to create them. My current deliberate choice to work within our system is because this encourages me to continue thinking, and at the same time tap certain social resources. It enables me to meet my material needs while placing fewer restrictions on my actions and giving me more free time than any other option that is currently available. Furthermore, it brings me in contact with people who are at least somewhat involved in education.

Altho I choose to work within our educational system, I do not assign my highest priority to the work, except when I feel that the work I am doing in the system is directly supportive of my most basic ideals. To allow myself more time for these basic ideals, I have chosen part time employment within the system. Such a choice is possible for me because I keep my own material needs extremely low in comparison to the standards of my culture and because my family keeps their needs somewhat below this average.

My choice to give higher priority to the work I am not paid to do than to the work I am paid to do is a fundamental choice that I feel is essential to my radicalism. My opportunities for employment arise from within the conservative institutions of my culture. Money is available for doing things which seem to be directly linked to that which has been established. It would not be realistic for me to expect much monetary support for experimentation with radical ideals. The fact that I can find some such support thru part time employment in an institution which is somewhat removed from the mainstream of our culture is a bonus which I do not intend to rely on. I will continue to structure my life in such a way that most of my choices are independent of my material needs. This is a strategy I would also recommend to other radicals. This recommendation is made primarily on long term pragmatic grounds.

The main problem I have in working within our educational system is that I can be distracted from my radical personal purposes because of a structure which automatically tends to channel efforts toward the social purposes of education. I have tried to develop a strategy for dealing with this problem which involves two major components. The first component is simply for me to continue to examine and develop my own framework for thinking about education. The second is to examine this framework in the context of the choices I make and the problems I confront while working within the system.

I develop the first component of my strategy primarily thru writing, reading what I have written, thinking about it. This is part of a more general process in which I write about and try to develop a framework for thinking about my thoughts and actions. I want to develop a way of thinking which takes me beyond the more conservative way of thinking which is part of my biological and cultural heritage, so I average at least five hours a week writing. I also feel it would be useful to broaden the extent to which I discuss my ideas with others. One of the reasons I try to take my writing beyond a rough draft form is so other people will read what I have written and initiate conversation with me about such matters.

There are several ways in which I develop the second component of my strategy. This component involves conceptual analysis because I am examining my net for education, but it also involves trial and error behavior because I am trying to solve specific problems. These specific problems relate to one main general problem. How to channel most of the energy which I devote to teaching toward helping persons who might try to expand their own personal power and originship, and how to do this in ways that are helpful to them from their own perspective. I think of this main problem in terms of several general kinds of sub-problems; namely the maintenance of my own motivation, the need for available external resources, the distasteful aspects of the structure in which I operate, the discrepancy between my behavior and the expectations of others. These sub-problems are much more interrelated than they may appear to be in the brief analysis which follows.

The main way I maintain my own motivation is by reminding myself that my seemingly other oriented purposes as an educator are in most cases really something I do for my own reasons more than something I do for my students or for the general good. The primary reason that I work with others is because this helps me develop the kind of allies I would choose in my efforts to implement my own ideals. My decision to write about ideas also helps me maintain my motivation, both because it helps me to think and because it encourages others to talk to me. Input from others about my ideas is something which I find motivating, regardless of whether the input is favorable or unfavorable.

In addition to writing about my net for education, I spend a considerable amount of time developing educational materials designed to help me implement my general purposes as an educator in specific situations. I invest more effort in the design of these materials than in all of my other actions as an educator combined, with one exception. My greatest priority as an educator is to spend as much time as possible talking with anyone who I feel might become important allies in quest for radical originship. I will not elaborate on details about how the materials I write exemplify my general purposes as an educator; but I would encourage any reader who is interested to select some specific materials and discuss them with me.

The main thing I find distasteful about the structure I work within is the existence of grades and credits. I am opposed to the main values and purposes in which the practice of grades and credits is rooted. My main strategy in dealing with this aspect of the structure in which I work has been to elude it, by shifting the responsibility to other faculty when team teaching or to students when teaching alone. This strategy has often relegated grades and credits to a minor nuisance, but usually it does not work very well. In particular I have never been able to keep the existence of grades and credits from interfering with one of my main goals as an educator, the goal of helping students develop and rely on their own highly personal and complex educational standards.

Another feature of the structure in which I work which I find distasteful is the fact that being an educator is a paid profession. This creates subtle pressure to work for the institution rather than to work for personal ideals and values. I do not believe that this is bad for most people, but it does tend to make it more difficult for me to find allies who are radical personalists. I shall deal with this problem in a paper entitled “Free-Work and Eco-Work. Currently I have no working solution to the problem that I work in an educational structure which is dominated by professionals.

There was a time when I tried to solve the problem of working within a distasteful structure by changing the structure, or at least the part of this structure nearest me. I have now decided that this was not an effective way to channel my energy, because the support for this structure came not only from above but from the attitudes of the vast majority of people affected by the structure. There seems to be a complex feedback relationship in which our socialistic tendencies tends to produce socialistic structures which then tend to increase our socialistic tendencies.

My second general attempt to solve the problem of being a radical personalist in such a structure was to be aware of the structure and the fact that it was intimately related to the needs of others, but make my own choices as a radical personalist. This is a partially effective solution, but it has a major drawback. It means that my behavior often runs counter to the expectations of others. Furthermore, this often happens without their being explicitly aware of it, and it leads to a kind of nebulous confusion. This can interfere with some of my major goals, in particular it can screen me from potential allies, so I am experimenting with other solutions.

My current solution is an extension of my previous one; but it involves an additional component. Since I see personalism and socialism as supplementary, I intend to be more imaginative in trying to transcend the gap between me and others. I shall illustrate my general intentions in this regard by making some final comments on how I intend to cope with the following problem.

Problem: Most of my students expect that I will set educational standards for them and that they feel that such standards are related to grades.

The first thing I would do to cope with this problem is to try to articulate my primary goals as an educator, and convince students that I personally have no standards which I know to be applicable to any person who is a stranger to me. I spite of this fact I know that most students feel a need for standards, and that it may be better to have highly inappropriate standards than none at all. Therefore I shall specify standards which I expect for the average student in that situation might be more useful than no standards at all. Furthermore, in spite of the fact that I feel that linking such standards to grades and credits tends to prevent the evolution of powerful personal standards, I will accept the fact that most of my students may need such a link. I could elaborate on my solution to this problem in some detail in Appendix1

I have no easy solution to being a radical personalist in a world where primary purposes are so often socialistic. It may take eons before personalism becomes the head side of the coin in human affairs. Fortunately for me it is not the total impact of my personalism which concerns me. I try to create a universe which enhances radical personalism, but in spite of this outward focus, I know that it is really me that I am trying to create.

The full radical impact of this change is still to come. For now, I am still subject to habits rooted in earlier theories and ideals. However I can project a trend. I have stopped trying to have any impact of an educational institution. Instead, I am beginning to ignore them, except as a resource and a hunting ground. They still provide me a resource for my own education, but perhaps one on which I will rely less as time passes. As a hunting ground they may or may not be very useful. Perhaps elsewhere I can better find others with whom to share my quest for learning.

SECTION 2 SOME STEPS IN MY EDUCATIONAL LIBERATION

Liberty IS the absence of external constraints. I won my struggle for educational liberation when I walked out on my doctoral comprehensive exam, and then decided that working towards a Ph.D. was interfering with my education. I still struggle, but the chains that bind me are internal, perhaps forged long ago in my struggle for liberation, perhaps merely a result of human frailty. I shall not deny the value of my liberation, but there is something I value more, something positive which I call freedom. I am free to the extent that I have live options. A live option IS one which is more than a mere conceptual possibility. An option IS not live unless I personally can imagine it, and unless it is within the realm of my capacities to bring this option into existence if I so choose.

I do not use liberty and freedom as synonyms. They are related. Lack of liberty can inhibit freedom, and the struggle for and creation of liberty can enhance freedom. However there are situations in which liberty increases and freedom decreases.

Imagine a child who likes his teacher, and is not threatened by typical school situations but has been conditioned to learn mathematics only under the supervision of a teacher. Suppose his teacher makes definite regular reasonable assignments, insisting that the assignments be turned in on time with most of the work done carefully and correctly. This child has some significant live educational options. He can meet the minimal expectations and learn some mathematics. He can cooperate earnestly and eagerly and learn even more. Replace this teacher by one who gives the child no direction and lots of free time. The child’s liberty has been expanded for the external constraints have been loosened. He is at liberty to learn mathematics. However he is unlikely to understand or appreciate this, and he is even more unlikely to possess the internal resources to exercise it. Unless he is a highly unusual child, learning mathematics has ceased to be a live option. His freedom to learn mathematics has been almost eliminated.

I shall not elaborate further on the relationship between freedom and liberty in the abstract. The purpose of this paper is to trace some steps in my own educational liberation. In the process I hope to obtain greater insight into how this struggle helped me develop freedom, but also on how it lead me to neglect the development of certain resources that would have expanded my freedom to an even greater extent.

I know that writing this paper will help me both as a student and as a teacher. I hope it may help other students in their quest for educational freedom. Even more I hope it will help teachers to reflect on what they are doing. Specifically, I hope they will put themselves into the role of my teachers and ask how they might have helped me achieve greater freedom than I was able to obtain in what, except for the support of my parents, was often a lonely struggle for liberation. In this struggle no one ever helped me focus my attention on the fact that it was freedom not liberty that was really the prize. It was only later that I explicitly formulated this distinction and realized that only part of my quest had been for freedom and that I neglected significant portions of this quest by an obsession with the struggle for mere liberty.

Perhaps the first time I took an educational liberty, rather than merely accepting a liberty when granted, was when I was doing column addition in the primary grades. We were supposed to go straight down the column, but I knew it was better to skip around and take advantage of special combinations. Seeing an 8 I would look for a 2 to make 10 or a 7 and a 5 to make 20. Sometimes I would use a 3 with an 8 saying to myself 3 and 8 is 10, with a small voice reminding me to use the extra 1 somewhere. I know I felt this would horrify my teacher, especially the thought that 3 plus 8 was 10. I also knew enough to keep such thoughts to myself, for at least implicitly I knew that this secret revolt could not be challenged.

Having eliminated teacher influence over my thoughts in doing mathematics, I had clearly expanded my educational liberty. From that time on I always did mathematics my own way. Usually only the results were tested. I had some difficulties when I encountered teachers who insisted that I write the process in some specific way, but teachers who merely asked for answers were easy to cope with. I implicitly knew the answers were independent of the whims and opinions of teachers and that I could reach these answers without their interference. (This may sound too harsh, some of my teachers may have been helpful, or at least neutral in my mathematical development.)

Judging from my present perspective I also gained educational freedom in this struggle, but I also sowed the seed of a bad habit, namely to exercise my independence reactively rather than actively. A large part of my reinforcement came from the thrill of intellectual pride, the feeling I was smarter than the teacher. This is not a solid foundation for educational freedom. While this may be harmless or even useful in specific cases, in the long run it tends to be a diversion from what is really significant.

This is merely one typical early example of my struggle for educational liberty which helped me expand my freedom. However this is not merely a case of constraints challenging my inner resources and then of me lifting myself by my bootstraps. I had external support. My self confidence was born earlier in educational experience with my mother. She started teaching me numbers and addition at a very young age. She claims that at age 4 I mystified her by announcing that four eights made thirty two, even though she had taught me nothing about multiplication. She says I backed up this statement by an argument which she did not follow. I have only vague memories of this situation which I am sure are colored by hearing the story so often. As I recall I obtained the result simply by thinking 8+8 = 16 & 16 + 16 = 32. Perhaps I didn’t articulate my reasoning, or perhaps she merely wanted to believe in my talents. I cannot make sense out of the facts from which this story emerged, but I do know that this type of experience encouraged me to trust my own reasoning powers.

My father also backed me in my decision to trust my own reasoning powers. He was convinced that the only authority in mathematics was reason, and I suspected that he believed that to memorize mathematics you did not understand was an affront to your intelligence. I don’t know if he would have put it that strongly, but I felt it that strongly. In the eighth grade I learned the algorithm for extracting square roots, but I did not figure out why it worked. I prided myself on forgetting it, something I would have never done for any process I understood. Another factor in my fathers attitude which reinforced my rebellion was his skepticism about the mathematical competence of most elementary school teachers. He taught high school math before going into education administration so I had authority as well as reason on my side in my mathematical rebellions.

Thruout my elementary and secondary education my parents always supported my struggle for liberation. They gave me guidance and direction, but I never felt constraints. Furthermore, they viewed school as a legitimate but extremely fallible institution whose purpose was to assist me in obtaining an education. I even had the educational advantage (altho social disadvantage) of being in a high school where my father was the principal. From my present radical perspective I view my father as a good liberal and good conservative in educational matters. He worked hard to use and enhance the best in what I think is an extremely coercive system. I feel that I was damaged much less than most because of the extensive support I received from my parents.

Returning to specifics, the first open revolt I remember was in the seventh grade. I argued at some length about the solution to a mathematical word problem with my teacher. He finally put the dispute to the class vote, and the class decided he was correct. I did not find his authority, even when backed by the class, very convincing and went home furious. At this time I began to explicitly formulate my attitude toward the educational system.

By the time I was in 9^th or 10^th grade I had decided that most educational demands made by teachers were rather trivial, and that by merely understanding what was really involved in a subject these demands could be met with only a minimal interference with my free time. I was locked into school five hours a day so I just used this time efficiently and made it a point of honor never to take school work home, and whenever possible to use study hall for my own purposes. I took the minimum requirements for graduation, and did homework for my classes in other classes when possible.

My freshman and sophomore years were the best. We had 45 minute periods, so I only had to spend 3 hours a day in class. I arranged my schedule to have last hour study hall, and since I was an honor student I was allowed to leave school after 7^th period. Furthermore, I was interested in the ideas I was exposed to so high school allowed me much greater liberty than I had previously experienced. The situation got worse the last two years when we changed to hour periods and when the town’s attitude resulted in cancellation of the policy that allowed me to leave school early. However things never got oppressive. I managed to rebel against the system by using it to enhance my own goals, while most students rebelled by merely dragging their feet.

The most vivid memory I have illustrating this is my course in bookkeeping. I took the course merely to fill my schedule. I found the concepts trivial. I seldom did double entry checking because I knew in advance which answers should be the same. I merely obtained the ones involving the easiest computations. I finished the assigned work in class as soon as possible and then sat back and read something I considered more appropriate to my intelligence. This always disturbed my teacher, who thought I had prospects as an accountant. She would then find some extra work, which I would do with total disregard for the scruples which motivate good bookkeepers and then return to my reading. Since she also had to cope with the rest of the class, I could usually find considerable reading time in this fashion. I guess I could have refused to do the extra work. The teacher was afraid for her job and somewhat in awe of the fact I was the principal’s son, so she probably would have backed down in a direct confrontation. Furthermore, I knew my father had hired her only as a last resort. I don’t think I was really against her. I was just for me.

Note: I usually knew my father’s opinions about teachers. Perhaps some would think that his professional duty would have prevented that, but they came out of some specific situations. I never discussed his opinions about the school or teachers with anyone else. I think we both implicitly understood that gaining insight into educational institutions as an outgrowth of people was just part of my education.

There were a variety of incidents in high school that reinforced the notion that I was more competent to guide my education than my teachers were. I had a biology teacher who rejected evolution on biblical grounds.

My teacher for sophomore English was a shop teacher who just happened to be certified in English but was not very comfortable with grammar. He used to consult two of us about which answers were correct when he was unsure of himself. This is not meant as a criticism. He was open about his limitations, and the fact that he was teaching English only because no one else was available that year. I think it was the only year he taught English.

I had a history teacher who spent most of our first hour class reading the morning paper, but he assigned two term papers that got me deeply involved in American History. Except for my father, who I had for only one semester of mathematics, I had a mathematics teacher who was competent at doing math but showed no intuitive insight as far as I could tell.

While I developed a pride in my own ability and a certain amount of independence, I did not develop any capacity for systematic independent study in high school. The only exception to this was that I did study American History systematically, but only because I knew that Washington University had a policy giving credit by proficiency in this area. My only other attempt at systematic independent study in high school was college algebra. I couldn’t take this as a high school course since there were not enough students to offer it. I also could not get college credit for it. My self discipline only lasted a few weeks.

The only teacher I encountered in high school who stimulated my thinking and who really encouraged me as a student was my father. However lead me to realize that the main portion of my education would come outside of schooling. As a mathematics teacher he inspired me to understand the standard materials taught rather than take them on faith. I needed more than this. I needed the challenge to go beyond them. My father also had a wide range of interests outside of mathematics, and while he encouraged me to read widely, he made no definite suggestions. He encouraged me to think about political and social questions, but in terms of an already fixed philosophy which I do not believe it occurred to him to seriously question. I was extremely interested in religious and philosophical questions, but neither he nor any other teacher encouraged me in this directions, so I merely did incidental reading. Likewise, no teacher ever challenged me to examine science critically, and I obtained a view of science that should have gone out of date with the advent of quantum theory. The fact that my view of science was as modern as the 19^th century was more due to my own insight than the help of my teachers. In English classes I only obtained a nodding acquaintance with good writing. What I needed was a teacher with wide knowledge and an appreciation of literature and with enough insight to suggest reading that matched my interests and enough flexibility to suggest rather than assign.

In brief, I was in a high school in which I had no constraints beyond my powers to cope with, and so I developed the confidence to pursue my own limited educational goals within the confines of an educational institution. I did not learn to go beyond the resources of the institution in a systematic fashion, nor to draw on the resources available in a creative way. I also did not learn to even formulate my own educational goals. I think with external support in this direction I could have.

A crucial step towards liberation was taken early in college. I conceptualized and acted on the distinction between education and accreditation. I thought I needed college for both, but I also realized that education was a life long process and that college was only a start. I saw accreditation as something desirable and something to be obtained as rapidly as possible. I obtained my A.B. in 2½ years and planned to work straight thru a Ph.D. Fortunately, I had to leave graduate school for financial reasons, so I never received the final accreditation which I would now regret for ideological reasons.

Even though I decided to obtain my degree rapidly I decided not to let this interfere with my education. I made it a point of honor not to study more than fifteen minutes for a test, to take at most a few pages of notes in a course, and to never memorize anything. I also decided never to do required course work except during work hours (8 to 5, Monday thru Friday), and to make sure that I averaged less than half an hour outside class for every hour in class on required course work. I formulated these principles during my first year of college and seldom compromised them. The idea was to leave me free to pursue ideas, mostly arising from my course work, but not required by my teachers.

At first I had to struggle with the factor of grades. My father encouraged me to make good grades in high school, not because he thought they were educationally significant, but because they would help me obtain a scholarship. I started college in the summer, making two A’s in the first summer term. The second summer term I took only a philosophy course, and I made a B. This was a shock. I had a better grasp of the material than any students in the course, but I had never learned to write. I think this helped me learn that grades were a poor measuring device. I never felt that my A’s indicated much, but I believed I would have been really inadequate if I had received a lower grade. However, this B did not hurt my self image and I decided it would be okay to make a few B’s.

The greatest step in my liberation from grades came in my second year. My most cherished grade is the F I received in advanced calculus. Prior to this course I had received only A’s in my high school and college math courses. An F in math was at first a shock to my confidence, especially coming from a teacher who stated that the purpose of the course was to separate mathematicians from non-mathematicians. However I refused to accept his judgement, and took advanced calculus from a different teacher the next semester.

The stance I finally evolved toward grades in college was to ignore them if possible and learn according to my own criteria of what was valuable. I tried to take the attitude that if I did this good, grades would follow and if not, so much the worse for the grading system. In fact, while I cherish my F and my C’s, I was always hurt by a B, so I did not become emotionally acclimated to my principles in the matter of grades, at least not until much later.

Looking back on college, I realize that my struggle for educational liberation increased my freedom. However, I was far from liberated. I still felt the need to be accredited, and when accreditation conflicted with education, I had to struggle to choose education. Also, I was still dependent on teacher approval. It was important to me to be academically talented and I wanted recognition. Therefore, I still put myself into a position where external constraints could affect my education. I went to graduate school.

At that time I would have had difficulty pursuing my own education systematically outside of graduate school. I could have studied by reading but I did not have the discipline to study systematically. Also, I did not have the vision necessary to study mathematics and science on my own. My perspective was too limited. That might have changed however, even without graduate school. I received no inspiration from teachers in science or math while in college. My view of science was still 19^th century. My view of math was 20^th century, but hardly deep. However three of my teachers in philosophy really inspired me and helped me obtain a basis from which I could proceed on my own. In particular, my road to becoming a mathematician took a route thru symbolic logic. The area was revealed to me by philosophy. This happened in my last semester of college and was the whole story of my first year of graduate school. During that year I learned how to create mathematics. I also learned you can do original research that is very good and not get a masters degree because your advisor doesn’t support you. He wouldn’t approve my thesis and he didn’t help me get my fellowship extended. I don’t want to speculate on why this happened. I took half my work in logic and logic research with my advisor, was told by him that my results were significant enough to publish, and received A’s on all this work.

Looking at my research I knew that I had become a mathematician, and that I could at least do routine research competently. However I also knew that I did not have the temperament to devote myself to such research and that I had no reason to think I could make an significant research contributions. At this point, I left graduate school. It was six years before I was to return except for an occasional course.

During those six years I learned one thing about accreditation that helped liberate me. I had been out of graduate school for three years and still had no masters degree. I was teaching junior high school at the time and during these three years I had totally abandoned systematic study, except in one area. I continued to study mathematical logic because I knew that I wanted a deeper understanding Godel’s incompleteness theorem. Since a masters degree would give me a salary raise I decided to go ask for one. I went to my former philosophy teachers at Washington University with a thesis I had written at the University of Illinois and with a plan which would allow them to immediately grant me a masters degree. They agreed to the plan with one minor provision, that I take a 3 hours independent study, but that only postponed the degree one semester and it allowed me to work again with one of my favorite teachers. He knew nothing about my thesis topic, but was able to learn what I had done. I didn’t add to the results, but with his help I polished the presentation. The fact that I was able to obtain a degree at one institution for a thesis written at another encouraged me to feel less constrained by the need for certification. Also during the next three years I learned to do systematic study on my own.

When I returned to graduate school I was competent as a mathematician, and I had a greater understanding of the graduate school system that the other students. I took courses in areas I had already studied on my own. I don’t think I took a single course in which all the ideas were new to me. This helped me solidify my knowledge in a pleasant way and still receive credits. I decided after one year that I didn’t like the qualifying exam system in math, didn’t want to obtain proficiency in French and German, an didn’t have an interest in writing a thesis in an area which was likely to get approved. I decided to try for an Ed.D. instead of a Ph.D. There was no language requirement and the exam system in education looked easier.

The education department agreed to accept all my previous course work in mathematics and philosophy and to let me study in education without having to take very many courses. They also agreed that I could take a modified version of their comprehensive which would include a section on mathematics. I studied educational psychology for two years, and while I was passively interested in their research never found a thesis topic of interest. I walked out on the first part of the comprehensive exam because the questions seemed dull and I did not want to spend several hours writing on them. None of this bothered my advisor. Everybody seemed willing to help me receive a doctorate, that is everybody but me. Then it occurred to me, very simply, that I did not want the degree. At that point if somebody would have said to me to study anything I wanted as long as I liked and come in and tell them when I thought I deserved a degree, I would have accepted the offer and taken the degree. However I was not willing to do one thing that did not fit in with my educational goals in order to receive the degree. The degree still seemed desirable, but not if it would interfere with my education. Only my advisor understood this decision, but had I been totally alone I would have made it anyway. That was the next to last step in my educational liberation.

If I knew how, and it wasn’t too much trouble, I would give back my other degrees. In 1991 I realized that I would not take a doctors degree even if it was offered free. To discover this I took a final step towards obtaining this degree. I went with a proposal to Washington University which would have given an interdisciplinary Ph.D. I asked that my previous academic and professional work be recognized as replacing all the standard requirements except a thesis, that I could as my thesis committee a professor from mathematics, a professor from philosophy, a professor from. education, that my thesis was to consist of “My Net For Philosophy” along with “My Net for Understanding” and “My Net for Doing”, and that these could remain as creative rather without documentation or references. I made this proposal because I wanted to establish the fact that an individual could obtain a high level of accreditation with a reasonable non-standard plan. I was also hoping that my committee would give me feedback that would help me in my goal of improving these workfile. The proposal was accepted, and the members of my committee were very encouraging. However they were all very busy, and it was difficult to get then together to discuss my work. Since I was not enrolled in the university, I was hesitant to press for much of their time. However the main reason that I did not complete these files at that time was because, I was still in the middle of my third collapse of will I had made my proposal during an interlude in which I mistakenly thought that my will had emerged. When I recovered, two years later, one of my advisors had retired and was no longer in the area. Most important I finally overcame any duality about accreditation, realizing that my ideals about accreditation were incompatible with receiving any degrees, even if they were awarded on terms that I found educationally reasonable. I do not know to give back the degrees I have been awarded, but at least I do not have what has been called the terminal degree.

I am past 16, so compulsory education no longer applies. I have no desire for accreditation. Any desire I have for academic acclaim is too minimal to move me in any fashion. I can think of no external constraints that directly affect my education. Yet while I feel my educational liberation is complete, my quest for educational freedom bas barely begun, but at least the focus has become clearer. I am no longer blinded by the desire for liberty, but I am still crippled by false pride and lack of self discipline. I am also overwhelmed by the fact that my surface desire for knowledge runs far ahead of what appears that I can ever accomplish. I have not learned to be at ease with the stubborn fact that for every answer received in my search for knowledge, many new questions always emerge.

The greatest regret I have about my educational struggle is that I became oriented to answers rather than to questions. Perhaps someday I can overcome this limitation.

SECTION 3 INFORMAL EDUCATIONAL CENTERS

Purpose The main purpose of this section is to present the concept of an informal educational center. Some currently existing ones include: museums, libraries, zoos, planetariums. These and various other types of institutions collect and organize resources which they use to provide educational opportunities for the general public. Unlike schools which stress formal education, these institutions stress what I shall call informal education. I call such institutions informal educational centers.

The types of center mentioned illustrate the concept of an informal educational center, but they do not exhaust this concept. In order to more fully develop the concept I will speculate on some new directions in this regard to the expansion of informal education. These new directions could be taken by already existing informal educational centers, either by extension of their current function or thru the creation of special new divisions. Institutions which now stress formal education might also find reasons to incorporate more divisions which would stress informal education. However it might also be useful to establish some new independent informal educational center whose primary function would be the creation, distribution, and utilization of new resources and ideas for informal education.

The distinction between informal and formal education IS a matter of emphasis. The terms formal and informal, like hot and cold, have meaning in relation to each other and to some current perspective. Rather than try to characterize the concept of informal education, I describe some of the common features of currently established informal educational centers.

Generally speaking, such centers are structured so you can just come and use available resources. Some special resources may only be available at scheduled times, but many resources are available any time the center is open. The user takes the initiative in how and when and in what depth to utilize the resources provided. You do not need to point to any past educational achievements to show you are prepared to benefit from the resources provided. No one checks to see if you have met certain standards of achievement when using these resources. You may stay for a long time or only a few minutes. You can come with a specific purpose or just to browse. You may be superficially involved, or you may be engaged in systematic study. The staff available to help you feels no personal responsibility toward you except in terms of specific requests on your part.

Most currently existing centers tend to be specialized, both with respect to the methods they utilize in communication and the areas in which they tend to focus. Museums tend to stress exhibits. They also tend to be specialized in their areas of focus; that is we have art museums, museums of natural history, museums of science and industry, etc. Educational television and libraries tend to rely on specialized means of communication, but they offer a much broader range of resources. Libraries are limited primarily by the fact that they still rely heavily on written materials. However this is changing. Libraries are also limited because they organize and utilize already existing resources, but they play almost no role in the creation of these resources.

There are various reasons why currently established centers do not reach out in new directions. Financial considerations are relevant, but tradition and habit are often more basic. Most centers could take small steps in new directions if they so chose, but it is hard to progress in new directions. This involves the creation of resources, and usually there is no tradition which encourages the creation of the types of resources which are most needed, and there is no established pattern for their distribution and utilization.

New Direction in Informal Education: To facilitate the expansion of informal education, it might be useful to create some new centers which were not too directly tied to past traditions and patterns. Such centers would need to be concerned both with the creation of resources and the creation of ideas for using these resources with the general public in new ways. Such a center could be small, having a single new thrust. It could be a large organization taking a comprehensive approach which would integrate a multiplicity of new and old resources for the purpose of informal education. I think it would be useful to have centers of both types. In proposing new directions, I shall ignore questions about organizational structure and merely sketch some new directions that seem interesting to me.

One cluster of ideas is related to the creation and use of brief unpublished written materials. I begin with this cluster because it involves some ideas which can be implemented on a small scale and with a modest cost. A major reason for these materials to be brief and unpublished is to maximize flexibility. The kinds of materials I have in mind are already being produced sporadically, but established informal educational centers do not utilize such materials. The examples illustrating these ideas all relate to my favorite conceptual hobby of mathematics. While this may make them seem somewhat specialized, I am sure you can supply ideas for brief written materials relating to other areas of interest.

Example A: Problem Solving Papers: Such a paper differs from the usual published educational materials in several ways. First, it takes a more limited focus. Its aim is to provide the background for the understanding and appreciation of a single important concept, or perhaps a small set of related concepts. It uses problem solving primarily in the initial aspects of learning rather than as a drill exercise to reinforce a concept after it has been explained and illustrated. I have used various formats for such papers. One format, is to begin with, a concise statement of the major purpose of the paper and then pose one or more problems which will be the major focus of the paper. I try to find problems whose statement and partial solution presuppose as little conceptual background as possible, and certainly do not presuppose any knowledge of the concepts which motivated me to write the paper. I also try to use problems which admit to varying degrees of partial solution. The reader is told to use any strategy he can devise. He is also provided a special section on strategies and a section devoted to careful presentations of solutions and partial solutions to problems. It is not until the last section that I would focus on an explication of the concepts which motivated writing the paper. I might also include one or more appendices for special purposes.

Example B: Perspective Papers Such a paper presupposes that the reader has already partially mastered a specific set of skills or concepts. The purpose of such a paper is to help the reader find relationships between these skills and concepts, as well as their significance in some broader perspective.

Example C: Skill Development Units: Such written material can combine brief programmed instructions with background activity sheets and special drills. The programmed materials would differ from the usual programmed text in that it would have a limited and easily recognized purpose.

The second cluster of ideas I want to mention, is merely an extension of the first to include additional communications media. Written material can be combined with audiovisual. They can also be integrated into activity oriented exhibits. One idea which I find interesting, is the combined use of physical and written materials in a unit which the Madison Project calls an Educational Shoebox. This idea involves writing some activity cards which can be used with some physical materials to provide a small self-contained educational resources, all of which will fit in a shoebox. The Madison Project tried to popularize this idea with schools, however I think the idea has a greater potential as a resource for informal education.

The most powerful new communications media, in my opinion, is the computer. A number of projects have been working on ideas for the educational use of computers. Mostly they have been thinking in terms of schools, however I think the computer has a greater potential as a medium for informal education. In particular it could be used simultaneously as a device for access to problem solving papers and a device to assist in problem solving. It could also be integrated with skill development activities, for educational games, and various types of simulation. This is only a glimpse of the potential ideas for writing computer programs for educational purposes. Currently there is no established patterns for use of such programs. I wish every library had some computer terminals from which you could call a wide variety of educational programs, as well as some good means to help the user find programs appropriate to his own personal needs and interests.

A third cluster of ideas relates to the utilization of people as a resource for informal education. Often I have wished I could discuss something or obtain help from someone who had knowledge or skill that I did not have. Why do we not have a wider variety of people available in informal centers? When I go to a physics library there are librarians who will give me some limited help, but no physicists for me to draw on. I can think of various schemes to make human resources more broadly available. My favorite way to be available would be to have some work space at an informal learning center and then to be there at specified times. When no one wanted to draw on me, then I could work on developing informal learning resources. I suspect many other people would enjoy being a resource in a similar fashion, either as a full or part-time job or as a volunteer activity. Another way to utilize volunteer resources would be to have available at the center, a catalog of human resources. Such a catalog could give a sketch of background, special competence and interest, and how and when these people were willing to be available.

A fourth cluster of ideas, relates to special programs where the emphasis is to provide something of interest to a whole group at a scheduled time. Many ideas for special programs could merely be featuring of opportunities the center was making generally available on an individual basis. For example, a comprehensive informal educational center might have a wide variety of workshops or lectures featuring people from their catalog of human resources, book discussion groups, films or film series, group simulation games, etc. It could be a center where something was always happening. You could just drop in or you could dial their computer and question it about the special programs available.

I conclude with a suggestion that perhaps the time is now ripe for the expansion of resources for informal education. Formal education is intertwined with society’s assumed needs for economic production. Informal education could be a major key to living in a complex society in which the need for the production of material goods was no longer a major problem and where economic considerations might cease to play such a dominate role in our lives.

I suspect that the internet has the potential to be one of the most useful tools for informal education. What is needed is someway to easily find the resources the available via the net and that are relevant to the learner’s purpose. The back of a novel I am reading contains a map showing Poland at its greatest extent. Yesterday I decided to use the internet to see if I could find more historical maps of Poland. It was easy to find a multitude of sites with historical maps, but I could not figure out how to find the maps I wanted or even to tell if such maps were available.

I also have a broad interest in what it was like to live in various historical periods, and I find that reading an authentic well written historical novel is an excellent resource for this purpose. Consulting a guide to historical fiction has been helpful for some of the periods I wanted to understand, but it did not give me enough novels set in Anglo-Saxon England. By accident I discovered that there was an excellent historical novel that provided me the vicarious experience of being emerged in England during the time of Alfred the Great. This novel can be read on the website http://www.octavia.net/ , and this site also provides a large number short essays and excellent references about this historical period. It does not, nor would I expect it to, provide me with references to novels by other authors that are set in this time period.

What would make the internet a more useful tool for informal education would be a central website devoted to that purpose. Persons with specialized educational sites could ask to have links to this website. One feature this central website could be an informal educational menu. At the top level I would be able to select literature or history and obtain a submenu from which I could select historical fiction or historical maps and from there select some county or historical period. If I selected novels whose setting was Anglo Saxon England, I would not expect most of these novels to be available for reading on the web, but it would be nice if excerpts could be there. I would also hope that it would also be easy to find other informational resources relating to this period, including email addresses of persons willing to share their expertise.

SECTION 4 WHY I TEACH MATHEMATICS

Teaching Math as CS My reasons for teaching mathematics are rooted in my origin ideal. The way I teach is rooted in this ideal and related ideals. I teach mathematics because I believe it is one of the most powerful net created by persons. It is powerful both because it is a tool and because it has become an extensive art form. Mostly mathematics is powerful because it has expanded beyond being a subject matter and has evolved into a way of thinking. I will not discuss the broad social impact of mathematics, since Morris Kline’s book “Mathematics in Western Culture” gives an excellent introduction to such matters. What interests me most is the way in which a person draw on mathematics to enhance originship. My own experience in learning and living with mathematics suggests that there is a great potential for truth in the old fashioned claim that the study of mathematics is one of the best ways to expand your thinking powers and thinking processes. Before expanding on this I would like to comment on an objection to this claim.

Early in studies indicated there was little evidence that taking courses algebra or geometry increased the students ability to think. Some people drew the conclusion that studying mathematics did not increase thinking ability. I am inclined to draw the conclusion that merely studying the results of mathematics does not increase thinking ability. In school most of the work concentrates on the results. Even teachers who know that the results without mastery of process are of minimal utility often fail to stress process in a way which is meaningful to students. Read Carl Bereiter, “Does Math Have to be so Awful?” for an elaboration of this point. This explains why most people who aren’t proficient in mathematics identify mathematics with a subject matter. In the name of mathematics, they have only been exposed to a collection of facts about numbers and routines for manipulating numbers and symbols. This is not even an adequate characterization of the results of mathematics much less the process. At present, most mathematical work draws on an understanding of numbers, but only a small portion of it is directly concerned with numbers. This is not to say that results about numbers are not an important part of the curriculum. It is only to say that these results are only a minor part of mathematics.

My main goal as I teach mathematics is to help students expand their reasoning powers. Other results of this thinking are merely an additional benefit. I work primarily with a numerical subject matter when teaching the basic aspects of mathematical thinking because numbers form one of the simplest and thus most accessible subject matters to which mathematical thinking can be applied. Furthermore the results about numbers, if adequately understood can be used in applying mathematical thinking to a variety of other subject matters. However even for novices I would teach some mathematical thinking which involves no use of numbers, because this illustrates how mathematical thinking does not depend on numerical content. Later I present two illustrations of this type, as well as one in which mathematical thinking does involve numbers.

Potential Topics There are a number of topics which could be used to introduce mathematics as conceptual study, rather than as a study of paraceptual claims. I have developed many such materials and am in the process of developing more. These include modular structure, boolean algebra, gaussian integers, matrix algebra, quaternions, functions algebras, symmetry groups, hyper-real numbers, elementary mathematical logic. Of these, I think the study of modular structures has the greatest advantages, although the study of boolean algebra and selected subjects from some of these other areas could make and excellent follow up or supplement. Furthermore a study of a number of books and articles on the history and nature of mathematics also come to mind. For example, the Newman volumes on the world of mathematics, “What is Mathematics” by Courant and Robbins, the Halmos article on mathematics as a creative art, all contain some excellent material which could be used to supplement either a beginning or follow up study of mathematics as the rigorous exploration of conceptual nets.

Modular Structures The integers modulo 6 structure uses the set Z₆ = {0,1,2,3,4,5} and an operation Å. Picture these numbers arranged as in a clock. To obtain a Å b, start at a and move b positions clockwise.

	Å	0	1	2	3	4	5
For example, 2 Å 3 is 5, but 4Å3 is 1. We call this operation addition because it satisfies the same basic laws that addition satisfies for the integers. The table below gives the complete addition table for the ring of integers mod 6.	0	0	1	2	3	4	5
	1	1	2	3	4	5	0
	2	2	3	4	5	0	1
	3	3	4	5	0	1	2
	4	4	5	0	1	2	3
	5	5	0	1	2	3	4

The Tea Party Puzzle Ms Army, Ms Banjo, Ms Clive, Ms Dumont, Ms Ekwall, Ms Fisk had a tea party. They were seated at a circular table. One of these women was pretty, one realistic, one slim, one talkative, one unreliable, one quiet. Ms Banjo sat opposite the unreliable one. The pretty one sat opposite Ms Clive, who sat between the quiet one and the unreliable one. The slim one sat opposite Ms Army, next to the pretty one, to the left of the unreliable one. Ms Fisk sat between the realistic and the slim one. Ms Ekwall sat to the right of Jan who was opposite the talkative one. Who is Jan?

Solution Use {0,1,2,3,4,5} to label the position around the table, with 0as the position of the slim one. Let j be Jan’s position. Also let the first letter of each last name and each characteristic be their positions. Clues involving ‘left of’ or ‘opposite’ can easily be translated in terms of Å. That slim is left of unreliable gives s = uÅ1. That Banjo is opposite unreliable gives b = uÅ3. The translation of ‘x is between y and h’ is more subtle. Check some instances to see that this implies y+h = x+x. The tea party information gives the following equations.

b = uÅ3, p = cÅ3, qÅu = cÅc, s = aÅ3, s = uÅ1, p = sÅ1, rÅs = fÅf, j = eÅ1, j = tÅ3

Since we chose s = 0, these equation give u = 5, p = 1, r = fÅf, a = 3, b = 2, c = 4, q = 3. Since t is different from each of {s,u,p,q}, t = 2 or t = 4. If t = 2 then j = 5 and e = 4 = c. Thus t ¹ 2. This leaves t = 4, and thus r = 2, f = 1, j = 1. Thus Jan is Ms Fisk.

For a beginning study of modular structures, I recommend Andrea Rothbart’s book “The Theory of Remainders”. This book focuses on the study of modular structures in a way that challenges the students to adopt a 20^th century perspective and to do some significant thinking from this perspective. However it accomplishes this without requiring the student to have much mathematical background. Modular structures are presented as algebraic structures in their own right, rather than from the older perspective of congruences. Since these structures are rings, they have the properties needed to use and reinforce many standard ideas of ordinary algebra. The fact that these ideas are inherent in the net for rings, rather than somehow being specialized to subsystems of the complex numbers is well illustrated. Since some of modular structures are not cancellation rings, the presence of zero divisors can be contrasted with their absence in more familiar systems, and this can be related to finding roots of equations in both types of structures. The fact that these structures are not ordered rings, gives a plausible reason to examine the concept of negative numbers from the broader perspective of additive inverses. That some of these structures are fields illustrates the algebraic nature of multiplicative inverses. In addition to comparing and contrasting modular to more familiar structures, this book uses the remainder function as a morphism to solve problems about integers by mapping them to simpler problems in modular structures. This illustrates the concept and use of a morphism at a very elementary level.

The above comments only indicate a few of the specific ways that this book focuses on a contemporary approach to mathematics in an elementary way, and in a manner that is appropriate for an introduction to mathematics as the study of math net. In particular it deals with a unified math net in an elementary, but challenging manner. Supporting computations take imagination but are not tedious. They compare and contrast easily with ideas from ordinary algebra. This book on the theory of remainders can provide a significant step in seeing mathematics as a conceptual net and in understanding the difference between conceptual and paraceptual claims. This could serve as an excellent first step in expanding peoples perspective on mathematics so they would see mathematics as a specialized kind of conceptual study.

Magic Squares.	In a magic square the sum of each row, each column, and each diagonal is the same. For example, in the 3 by 3 magic square these sums are all 12	5	0	7
		6	4	2
		1	8	3

Suppose you were searching for a 3 by 3 magic square, but had not seen the above example. Your search would be easier if you could predict that the sum was 12 before trying to arrange the numbers into a magic square. This can be deduced as follows.

Adding all the numbers gives 3 times the row sum.

Since each number 0 thru 8 was used once, the result must be 36.

But if 3 times the row sum is 36, then each row sum is 12.

While the above analysis uses the arithmetic fact 0+1+2+3+4+5+6+7+8 = 36, most of the solution involves reasoning rather than computation. Additional reasoning, given below if you are interested, can be used to show that the middle number must be 4. There are only two ways to use 0 in a sum to make 12, namely with 4 and 8 or with 7 and 5. Thus 0 cannot be in a corner. By symmetry 0 could go in any non-corner, so place 0 in the middle of the first row. To make 12, 5 and 7 must be in the same row as 0. By symmetry either 5 or 7 could go in the top left corner. Using 5 leaves exactly one way to complete the square. Further reasoning shows that there are exactly 7 more magic squares, which by symmetry can all be obtained by rotating and flipping the one above.

Showing That The middle number is 4 Add the second column and the second row and the two diagonals. Since we know that each of these sums is 12 the total will be 48. Furthermore the middle number gets used 4 times while each other number is used exactly once. This gives 36 plus 3 times the middle number. Thus 3 times the middle number must be 12, and so the middle number is 4.

Remarks We can also obtain a 3 by 3 magic square whose middle number is 0 and whose sums are all 0. Just subtract 4 from each number in the magic square above.	1	-4	-3
	2	0	-2
	-3	4	-1

Magic squares come in various sizes. For a 4 by 4 magic square using the numbers 0 thru 15, we can show that the sums are 30. Just reason as with the 3 by 3 case. Observe that the 4 rows sums total to 120 because it is the sum of 0+1+...+14+15. Further analysis is more challenging, because even if you find one such magic square there are a multitude of others that cannot be obtained from it by rotating and flipping.

Magic Squares are merely one type of magic array. An array called a magic hexagon is one that I found challenging. In high school I was introduce to an array called a double magic diamond, and I used my understanding of algebra to discover such an array and some interesting facts about it. There is a partially developed unit which relates the solving of magic arrays to certain aspects of PNCM.

Attribute Logic I have designed a set of units for learning a math net for mathematical logic by using an simple math net for a set of 8 attribute items. Items vary by size, color, shape. The sizes are large and small. The colors are blue and red. The shapes are circle and diamond. Suppose that an item was hidden in a box and we had the following clues, which I have stated in both ordinary and ideographic language.

Clue 1. If it is small then it is a diamond. S Þ D

Clue 2. If it is circle then it is red. C Þ R

Clue 3. If it is blue then it is large. B Þ L

Clue 4. If it is large then it is a circle. L Þ C

Clue 5. If it is a diamond then it is blue. D Þ B

We could determine the item by manipulating tokens representing these items, using each clue to eliminate items incompatible with that clue. For instance clue 1 eliminates the 2 small circles. This strategy can be used without actually having tokens. Arrange item names and clues as below, placing an X whenever a clue eliminates an item. Note that lrc is the only item left		lbc	lrc	lbd	lrd	sbc	src	sbd	srd
	S Þ D					X	X
	C Þ R	X				X
	B Þ L					X		X
	L Þ C			X	X
	D Þ B				X				X

This solution is like using tokens except that it more symbolic than physical. This may seem like a minor difference, but it indicates that what we are doing is primarily conceptual. Also like any mathematical reasoning it encourages you to ignore irrelevant information such as tokens are made of wood. Of course even in solving the puzzle by manipulation you ignore such irrelevant facts. A powerful mathematical method allows you to deal with a number of individual items at the same time. The method just used does not have that feature. For simple situations involving a small number of items this is a minor limitation. However if we develop alternate methods for simple puzzles it is often possible to extend the ideas involved to more complex ones. Remote reasoning for solving such puzzles can be illustrated by the deduction below; ‘&’ denotes ‘and’, ‘Ø’ denotes ‘not’, ‘Þ‘ denotes ‘then’ .

(1) S Þ D (2) C Þ R (3) B Þ L (4) L Þ C (5) D Þ B clues

¾¾¾¾¾¾¾¾¾

(6) D Þ L by (5) and (3)

(7) ØS by (6) and (1)

(8) L by (7)

(9) C by (8) and (4)

(10) R by (9) and (2)

(11) L&R&C by (8),(9),(10)

This deduction illustrates the use of ideographic language, which is not the essence of mathematics, but which can often extend the power of a process. It also is applicable to more complicated situations involving relations and variables. In a situation involving 2 item, suppose we knew that these items are the same size. This could be given concisely as ‘xSZy’. The unit on relational logic uses a variety of such relations and has the user define them in terms of more primitive ones. xSZy means (xL&yL)or(xS&yS).

An Attribute Game This game uses attribute tokens along with 6 labels: S, L, R, B, C, D. The game can be played on a board like the one indicated below, but it is usually played on a board laid out with 2 intersecting circle which are used as a Venn diagram. The game rules are not relevant to the comments I am about to make, however for anyone interested these rules are given at bottom of this page.

X = Y =

XÇY = { , } ØXÇØY = { , }

XÇØY = { , } ØXÇY = { , }

This board is intended to partition the attribute tokens into 4 subsets each having 2 elements, depending on the values chosen for X and Y. The symbols Ø and Ç indicate complement and intersection.

For example, if X is red and Y is small the partition should be imagined as follows:

X = R Y = S XÇY = {src, srd} ØXÇØY = {lbc, lbd} XÇØY = {lrc, lrd} ØXÇY = {sbc, sbd}

Imagine that in a game we know that src goes in XÇØY. We can deduce that lbd belongs in ØXÇY as follows.

lbd differs from src in all 3 attributes.
Since src is in X, lbd cannot be in X.
Since src is in ØY, lbd must be in Y.
Thus lbd is in ØXÇY.

The reasoning makes no appeal to numbers yet any mathematician would agree that it was mathematical. His criteria might involve such observations as:

A. We reason about a definite math net. (No fair using a label of dog).

B. The reasoning is general, since similar reasoning could be applied to any token and its opposite,
regardless of where it was placed.

C. The reasoning is conceptual and cannot be contradicted by paraceptual facts. (If I turn the card over
and find lbd doesn’t go where I claimed then I am not really wrong because my opponent violated
the rules.)

Attribute Game Rules The game involves teams A and B, with 2 to 4 people on each. To start, A selects 2 labels of different attribute types and places them face down beside X and Y. B chooses any token. A must put this token in the correct set. The object is for B to identify the labels giving A as few points as possible in the process. B must place the remaining tokens in the 4 indicated sets. On each trial, A either verifies the choice or removes the token if it is in the wrong set and gives it back to B. A collects one point each time they return a token, however if they return a token incorrectly or allow it to be placed incorrectly, they forfeit the game. B may try a rejected token elsewhere or try a different token. Once B has placed all the tokens they must either correctly identify the labels or forfeit the game. If they are able to do this then A and B reverse roles and play is repeated. If there is no forfeit the team with the most points is the winner.

An Imaginary Perspective on Bit String Names for Real Numbers Let C denote the set of real numbers starting at 0 and continuing up to but not including 1. C is called a continuum because it can be represented as a continuous set of points, packed in such a way that no additional points can be inserted, at least from the most widely used net for mathematical analysis.

continuous set C: ú¾¾¾¾¾¾¾¾¾¾)

0 1

There is a standard way in involving the idea of a limits to conceptualize infinite bit strings as names for such numbers. One day, having nothing better to do, I invented an supernatural jumping frog as a way of imagining such names. Having no place to use this invention, I decided to write it up as a curiosity. I make no claim that it will help anyone better understand such a naming scheme

Fred the Frog Fred’s main activity is to start at 0 and trying to get to some point x in C in a period of 1 hr, where C is one foot long. Fred can only rest or make a single forward jump in a given time period. Furthermore in each period there is exactly one possible jump Fred can make. The first period is 1/2 hr. Fred can either rest or make a 1/2 ft jump in this period. The second period is 1/4 hr. Fred can either rest or make a 1/4 ft jump in this period. The third is 1/8 hr. Fred can either rest or make a 1/8 ft jump in this period. In general each time period is half as long as the preceding one and the length of the jump half as far. We start with an example of a point Fred an reach in a finite number of jumps. However makes Fred supernatural is that he can make an infinite number of jumps in an hour, if he need to. In order to code what Fred does, we use a one to indicate when Fred jumps and zero when Fred rests.

Example Suppose Fred wants to get to the 9/16 ft point. Fred jumps in the first 1/2 hrs, landing at the 1/2 ft point. Fred does not jump in the next 1/4 hr, since this would put Fred at the 3/4 ft point which is past the 9/16 ft point. Likewise Fred does not jump in the next 1/8 hr, but does jump in the following 1/16 hr, landing neatly on 9/16. Having achieved this goal, Fred does rests in the next 1/32 hr, in the next 1/64 hr, in the next 1/128 hr, etc. This gives the bit string 100100000..., which can serve as a name for the 9/16 ft 1/64.

Example In a similar manner 11100000... indicates that Fred went for the 7/8 ft point..

Example Suppose Fred wants to reach the 1/3 ft point. Let us begin to code what Fred must do.

Since 1/2 > 1/3 the first bit is 0. However the 1/3 > 1/4, so the next bit is 1: 01?…

Since 1/4 = 3/12 and 1/3 =4/12, Fred is now 1/12 short of 1/3.
Thus he rest in the 1/8 period and jumps in the 1/16 period: 0101?….

Since 1/4+1/16 = 5/16 =15/48 and 1/3 = 16/48, Fred is now 1/48 short of 1/3,
so he rest in the 1/32 period and jumps in the 1/64 period: 010101?….

Since 1/4+1/16 +1/64 = 21/64 =63/192 and 1/3 = 64/192, Fred is short 1/192,
so he rest in the 1/128 period and jumps in the 1/256 period: 01010101?….

Extending the arithmetic, the claim that the alternating bit string 0101010... names 1/3, should seem plausible. (This can be checked by using the formula for the sum of an infinite geometric series).

Can you convince yourself that the other alternating bit string 1010101... names 2/3?

Example Suppose Fred wants to reach the 7/12 ft point. Since 7/12 is 1/3 +1/4, Fred can modify what he would do to reach the 1/3 ft point by also jumping in the 1/2 hr period rather than in the 1/4 hr period. This gives 100101010101… as the code for 7/12.

Comment This scheme also names points that cannot be named as fractions. One such string is 1001000010000001..., which has ones in positions 1, 4, 9, 16, 25, etc. The proof that this number is irrational follows from the theorem that a number is rational if and only if at some position there is a finite a finite bit string that repeats itself forever from that point on.. In the case of 9/16 the bit string 0 starts at position 5 and goes on forever. In the case of 1/3 the bit string 01 starts immediately and repeats forever. In the case of 1/12, 01 begins at position 3 and continues forever.

Concluding Remarks One thing these material illustrate is how conceptual reasoning yields information not explicitly given about the initial situation, and it does this without appeal to paraceptual information. To accomplish this the reasoning must be precise, and to be precise it cannot deal with too many things at the same time. In each case this was accomplished by limiting the range under consideration to states which are much simpler than the complex situations of ordinary experience. To think mathematically, we create remote idealizations, rather than consider manifest states. We do this even in the simplest cases. For example, ‘2 plus 3 is 5’ is a remote proposition compared to the ‘2 dogs plus 3 dogs is 5 dogs’. The latter refers more directly to a manifest state. The former abstracts a quantitative aspect from many similar states involving many different kinds of items.

In a sense all deductive reasoning is about some remote state, for it avoids many of the complexities of any manifest state. However in most non-mathematical type reasoning our interest and intent is still directed towards paraceptual information about a state. When we reason mathematically we turn within, to extremely remote math nets, to idealized creations of the imagination. This is why such reasoning cannot be contradicted by paraceptual information. It makes no claim of direct applicability to such matters. If on the basis of mathematical reasoning we hazard some paraceptual opinion which is wrong, we do not conclude that our mathematical reasoning is wrong. We merely conclude that it does not apply to that state. Having reasoned that the token hidden in the box is the large red circle. Suppose we look in the box we find a medium size green triangle. This does not contradict our reasoning. We were not reasoning about an paraceptual state, but a conceptual one. In the realm we imagined, in the imaginary box of this realm, there is a large red circle.

Strangely enough the power of mathematics is enhanced by the fact that it does not try to analyze paraceptual states. Important actual situations may be too complicated to analyze directly, however they can often be approached by using a variety of simplified remote models. Furthermore the same remote models can often be partially applied to a variety of situations. This is why mathematics is not a subject matter, but rather a conceptual net that can be applied to a variety of subject matters.

The history of our attempt to cope with our world is interwoven with the development of mathematics. But mathematical thinking can also help a person P increase P’s own powers. Whether this happens may depend on P’s awareness of the nature of PNCM. P must be aware of the process as well as the results. Furthermore P must be aware of the limitations of mathematical reasoning. Even good mathematicians often forget that their reasoning applies to models rather than actual states. This can lead to extremely unintelligent behavior. To yield power, mathematical reasoning must be seen in perspective. It must be contrasted with and augmented by other means of thinking, feeling, perceiving. In this fashion it can be one of the key resources for intelligent behavior, and thus a component of personal power.

There are several reasons to broadly expand the awareness that mathematics now focuses on the invention and rigorous study of math net. It provides a perspective that removes certain barriers we encounter when we teach traditional materials. It provides a basis for obtaining a historical perspective on the evolution of our ideas about the nature and limitations of our mathematical knowledge. It helps us speculate on how this trend in the evolution of mathematics may spill over to our whole way of thinking about human knowledge. Let us consider each of these reasons.

A perspective that mathematics is CS helps remove biases that interfere with learning mathematical ideas. For example, students often learn and fixate on the idea that multiplication must be repeated addition. Since multiplication is something we conceptualize, we can extend this concept beyond the natural numbers in any way that proves convenient. Basically we want multiplication to satisfy certain laws that will allow us to employ our familiar algebra. If introduced carefully this should make the algebra of negative and complex numbers easier to understand. Treating mathematics as CS also makes it easier to understand why 0! = 1 and a⁰ = 1, as well as a variety of mathematical ideas that students often resist because of their naive attitude towards mathematics.

In discussing the difference between 19^th and 20^th century mathematics with several mathematicians they all agreed in principle with the conceptual versus paraceptual distinction mentioned earlier. Consider what could be the broader impact of the paradigm shift in our attitude toward mathematics. In most areas of study conceptual and paraceptual concerns have been so intertwined, that little effort has been made to isolate the conceptual net for study strictly in its own right. Because of this, the distinction between concept and theory has often been hard to sort out. This both limits the sophistication of the net and makes it difficult to study alternative nets. The paradigm shift in mathematics gives evidence of the advantages of pure conceptual study. There is some evidence that this may be spilling over to other areas of study. The examples most closely related to mathematics are portions of computer science, such at automata theory. In other areas CS is both using and going beyond the conceptual methods developed in mathematics.

Much of mathematics evolved in connection with the evolution of physics, and provides a large portion of the net for physics. Of course mathematics has also been used in many areas other than physics. The net for physics also contains a number of concepts not currently classified as mathematical, partially because they are intertwined with the paraceptual claims of some physical theory. Recently this has been changing, as more concepts once considered as physical are being treated from a purely mathematical perspective. Excellent examples of this include “A Deductive Theory of Space and Time” by Saul Basri, as well as purely geometric treatments of parts of relativity and algebraic treatments of quantum mechanics.

Over the last 30 years Peter Ossorio has been developing a conceptual net called descriptive psychology. This net is purely conceptual, and does not constitute a psychology theory. Instead it provides a net which could be used by any psychological theory, much as mathematics provides a net which can be used by any theory in physics. In developing this net, Ossorio uses conceptual methods like those used in mathematics, but which significantly broaden the kinds of conceptual nets that can be somewhat rigorously developed. For example, the material organized by Larry Wright is an excellent example of a conceptual net for informal logic, and one which could be enhanced by some of Ossorio’s conceptual methods. Much of the work in contemporary philosophy might also benefit from a separation of conceptual from paraceptual concerns, and from the use of Ossorio’s methods.

Appendix 1 CONVERGENT REASONING ABOUT GRADING POLICIES

A Student as Origin Grading Policy: A grading policy involves standards for various grades and someone to evaluate which standards have been satisfied. A student as origin policy IS one in which a student controls the choice and application of these standards. The main purpose for such a grading policy is to encourage students to take a major responsibility for their own learning. One possible version of student as origin grading policy is described below. Other types of grading policies are also discussed. Convergent reasoning about such policies are then illustrated. Grading policies are directed towards the student as a person, so the word ‘you’ refers to any student to whom the policy might apply.

Policy SAO: You may choose any grading standards and evaluation methods that your teacher will allow. Students who do not choose their own standards methods will be graded via the following default standards and methods.

Default Criteria for Grading The term hours refers to any time you spend on any work relevant to this realm of study, including time in class. Some teacher endorsed competency lists are available. You may use or modify these to obtain your personal competencies goals, or you may design an alternate list of similar breadth or depth.

Credit Only: 90 hours or functional competence in relation to an appropriate list of goals.

A: excellent competence for a most of your goals

B: 90 hours plus functional competence for most of your goals

C: 75 hours or functional competence for a few of your goals

D: 60 hours

I: less than 60 hours

Evaluation Methods: Regular weekly reports, with both time for the week and cumulative time reported, is sufficient evidence of time spent. Your personal estimate as to your competence in relation to each item on your personal competency list is sufficient evidence of competence, however your evaluation should be consistent with additional relevant evidence, such as class participation, suggested assignments, other written work, etc. Without the weekly reports or your detailed estimate of competence, the teacher will simply make a reasonable conservative estimate of effort or competence on the basis of whatever information is easily available.

Constructing Your Personal Grading Standards: One way to construct personal standards is to modify the default standards for those grades that are relevant to you. If your primary goal is an A or B, and you want to focus on effort rather competence, you might change 90 hours to 105 hours for B and 120 hours for A and delete functional competence. If the primary goal you want to focus on is depth quality rather than broad mastery, you might delete the time standards and change the competence standards for an A to the production of one product of excellent quality, and a B if it is only of good quality.

Other Student as Origin Grading Policies: SAO is only one of many possible student as origin grading policies, and the extent to which it is a student origin policy depends on how flexible the teacher is in deciding what standards to allow. One of my students seriously proposed that her standards for an A was to attend every class, since this was a feat she found extremely difficult to accomplish. After some discussion she withdrew this proposal, but I do not think that she could have convinced me to allow this standard. Most other student as origin grading policies would differ from SAO primarily in their specification of a default option and in how much they provide support for a student in the identification of relevant standards. Unless they do this to some degree, or the students are already skilled in acting as origins in learning situation, they are more likely to be a nobody as origin grading policies. The extreme policy of telling students to just assign their own grade is likely to be such a policy.

Student Origin with Teacher Control Grading Policies: A policy may offer a student various options for earning a grade, with the teacher both defining the options and determining when the standards have been met. A cumulative point system can be used to implement such a policy. In such a system points are awarded for various achievements. For example, if points are awarded by testing for certain competencies, the testing could be repeated until the student gave up or reach the level specified.

Teacher as Dominate Grading Policies: A teacher dominated grading policy IS one in which the teacher specifies the grading standards and also decides to what extent the student meets these standards. SAO stresses the student as origin. The one I am about to describe is primarily teacher dominated. A cumulative point system tends to be somewhere in between. Teacher dominated grading policies vary in their regard for student concerns, such as the desire for a clear and reasonable set of expectations. Fairly traditional types of such teacher dominated grading policies are quantitative weighting policies, such as TAD below.

Policy TAD: The student’s grade is determined by a weighted average of the following components, which will be graded on a numerical basis.

Homework 10%, Class participation 10%, Weekly Quizzes 10%, Mid-Term Exam 20%, Final Exam 30%, Term Paper 20%.

Any late homework assignment will have its grade reduced by 20%, however the 2 lowest homework grades will not be used in computing the homework grade for the course. No makeup quizzes will be given, however only the grades on the top 5 quizzes will be use to compute the grade for quizzes. The standards used to grade the term paper are included with the list of topic options. A term paper received after the deadline will have its grade reduced by a certain amount according to the guidelines given. Makeup exams will be given only if the teacher determines that the student had a legitimate reason for missing the exam. The letter grade for the course will be determined on the basis of the scale below. Students taking the course under the CR vs NC option must indicate this by the second week of class. A grade of CR is equivalent C or better.

A: 90 to 100, B: 80 t0 89, C: 70 to 79, D: 60 t0 69 NC: under 60

In hardship cases, such as prolonged illness, the teacher may decide that there are factors which warrant exceptions to the above system, however this will only be done if the students demonstrate in some other acceptable manner the appropriate level of mastery of the course materials.

Adopting a Policy: Altho I have adopted SAO over a number of rival policies, I shall pretend that OGR1 has only one rival, namely TAD. I have chosen this limited analysis in order to make the considerations involved easier to present. Similar analysis would apply if I were to choose between any other reasonable policies of these two types. In a more complete analysis I would also eliminate a variety of rivals such as some intermediate type of cumulative point system.

A Claim: To adopt a policy is a decision, rather than a claim, and hence neither true nor false. Of the various claims that might influence my decision to adopt SAO over TAD, I have chosen the following for analysis.

C: SAO is a much better policy for me to adopt than TAD.

To be a claim, a statement must propose clear information. As formulated above, it is hard for me to keep the information I am proposing in focus. C is intended as a short way to propose the following information.

There are a number of my general purposes that could be affected by a decision to adopt a policy for grading. TAD tends to adversely affect some of the ones that are most important to me, while SAO is compatible with a strategy for accomplishing these purposes. For the other relevant purposes TAD offers no significant advantage over SAO.

Background: Claim C relates to purposes that can be affected by a grading policy. Altho the number of such purposes is vast, many are unlikely to be affected in any significant manner. The support for C, which I give is based on the presupposition that it is sufficient for me to consider the effects on the purposes listed below. Those that are more important to me have higher numbers.

P1: minimize time I must spend on tasks extraneous to my goals as educator

P2: maintain reputation within university sufficient to act effectively

P3: not unduly undermine reputation of the university in a way that might threaten
its ability to meet my educational objectives

P4: maintain a classroom atmosphere that is supportive to as many students as possible,
including those who do not choose originship ideal

P5: act primarily as a resource for students choosing originship ideals

Support: To support conclusion C, consider each of these purposes separately, starting with P5, since it is the one which is most important to me. The following provides support for my claim that TAD tends to adversely affect P5, and that SAO is compatible with a strategy for accomplishing P5.

S51: TAD encourages students spend time worrying about what will be on tests.

S52: TAD encourages students to place a higher priority on teacher expectations
than on what they expect of themselves.

S53: SAO make it explicit that the teacher prefers students to act as origins
in regard to their own learning.

S54: A number of origins oriented students have convinced me that they learn better with me
than with most of their traditional teachers, often citing grading as a significant factor.

By referring to students in the statements below, I mean students I have worked with in courses where I used a policy like SAO. These statements provide support for form my claim that TAD tends to adversely affect P4, and that SAO is compatible with a strategy for accomplishing P4.

S41: At least half of the students are more comfortable with SAO than TAD,
and a significant minority find SAO much more motivating.

S42: With assistance on my part, most students find that the can be as comfortable
and as well motivated with SAO as with TAD.

S43: Most students become much more aware of goals accomplished by focusing
on competencies rather than on grades.

I now provide support for my claim that for P3, TAD offers no significant advantage over SAO. This applies only to my decision to adopt SAO, and is not intended to be taken as support for a claim that the use of TAD by others does not offer such an advantage over the adoption of SAO. Such a claim is not part of claim C. The numbering indicates which purpose is most relevant.

S31: Altho I am open about the grading policy I adopt,
almost nobody other than my students has bothered to learn about it.

S32: Most grades given at my university are A’s or B’s,
so my policy doesn’t significantly inflate grades relative to other policies.

S33: For me to use a grading system which suggests that a one dimensional quantitative scale
is a useful way to evaluate anything as complex as learning is not something that
I find intellectually defensible

I now provide support for my claim that for P1 and P2, TAD offers no significant advantage over SAO. Again this applies only to my decision to adopt SAO.

S21: My role in the university is a minor part of my life.

S22: My reputation at the university has been established and is maintained thru a wide variety
of my actions, and a carefully articulated policy, no matter how radical is unlikely,
to affect it in ways that will make me less effective.

S11: Record keeping is something I tend to neglect.

S12: To use a grading policy which makes no sense to me is likely to involve me
in a time consuming discussion over matters that are unimportant to me.

REFERENCES

[A] Willoughby, Stephen. Accountability Threat or Opportunity “Mathematics Teacher” Nov. 1972.

[B] Faeber, Jerry. The Student In Society, from a collection of essays in his book entitled The Student As Nigger.

[C] DeCharms, Richard. Personal Causation

[D] Shidler, Mary. Persons Behavior and the World