A CONCEPTUAL NET FOR EDUCATIONAL PHILOSOPHY

The 4 by 6 white rectangle is called a 1white. The dotted lines indicating internal 1 inch square cells do not appear on most of your pieces.

A piece that is composed of two such white rectangles
is called a 2white.

CONCEPTS FOR UNDERSTANDING EDUCATION

by F. Richard Singer

website and email: conceptualstudy.org

Current Edition: 9/2006

By a net, I mean a network of conceptual distinctions and conceptual relationships that can be used to think and communicate about at least one significant realm of interest. This book is a personal tool helping me to understand my attitudes and my own concepts for understanding education and to shape these into coherent net. Perhaps it might be useful to others who would like to examine concepts in their own personal net for understanding education. The net I am developing also contains a portion of the kind of public net for understanding education that I would like to see emerge. In describing my personal net I illustrate ways in which its concepts are used in my own thinking and actions. This is one reason that this book is personal rather than scholarly. However in order to focus primarily on concepts in the main text, I have placed all of the more extensive personal material in the appendices.

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. Since the main text of this book is primarily conceptual, it should be clear from context when statements are conceptual from within the net being developed. Altho this book uses many of the specialized versions of concepts that are central to all my nets, I have tried to make this book self contained by briefly explaining concepts as I use them. Hopefully, this should be sufficient for most purposes. For anyone wanting a more refined account, I included more about some concepts in Supplement 2. An even fuller presentation of these and related concepts can be found in the Concept Encyclopedia‑Dictionary and in my book A Personal Approach to Conceptual Philosophy, both of which can be downloaded from the website.

TABLE OF CONTENTS

Preface

Section 0 Basic Concepts

Section 1 Cognitive Oriented Educational Processes

Section 2 Making Decisions About Educational Processes

Section 3 The Primary Purpose of Schooling, Social or Individual

Section 4 Evaluating Educational Processes

Appendix 1 Convergent Reasoning About Grading Policies

Appendix 2 My Educational Behavior

Appendix 3 Some Steps in My Educational Liberation

Appendix 4 Informal Educational Centers

Appendix 5: Why I Teach Mathematics

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

The appendices can be read in any order. Appendix 1 focuses on a policies I 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. Appendix 2 focuses on the broader perspective that I take when I am acting as an educator. Appendix 3 focuses on the evolution of my educational ideals and purposes, primarily by tracing my development as a learner. Appendix 4 presents the concept of an informal learning center as a way of illustrating my educational ideals. Appendix 5 focuses on why and how I teach mathematics. This Appendix is also integrated into another one of my main booklets available on my website, namely My Net for Understanding Mathematics. Anyone interested in mathematics education might want to read this section in conjunction with that book.

Supplement 1: Introducing The Realm Of Colored Pieces

Supplement 2: More About Some Concepts and Terminology

Altho I have tried to keep concepts and terminology close to that used in ordinary language, some modifications are useful in order to make distinctions often ignored in our ordinary nets. For instance, the ordinary connotation of the term pupil usually involves being in school at a pre-college level, while the term student is used more broadly. I use these terms to make a different type of distinction. Hopefully this and other terminological and conceptual matters are clarified in the main text, but in some cases they are further highlighted in Supplement 2.

Preface

Many of the specialized concepts that I use are (largely faithful) version of those from Descriptive Psychology, a public net of theory neutral concepts and conceptual tools created by Peter Ossorio, and henceforth referred to as PNDP. The concepts are refinements of those widely used by people to think about persons and their behavior, and an ordinary comprehension of these concepts should be adequate for a basic understanding of their use in this book. Altho everything I say about education could be said without out recourse to PNDP, for a deeper understanding using concepts from PNDP provides a distinct advantage over their ordinary counterparts. Being more systematic, they suggest ideas that might not otherwise be considered, and they help bring these ideas into sharper focus. This has helped me begin to construct a more systematic net for thinking about education and the function of education in a culture. For more information go to the Descriptive Psychology part of my website or the website of the Society for Descriptive Psychology sdp.org.

In Place, Ossorio gives over a hundred maxims that relate to concepts that we use in thinking about persons. This is followed by extensive commentary indicating how these maxims give concise information about relationships between these and providing reminders of noteworthy feature to be observed in using theses concepts. I will quote versions of a few of these maxims that I find useful in thinking about educational concepts and issues. My versions will sometime differ slightly from those he has given, primarily in order to concisely add a perspective gleaned from his commentary. Another technique that I have adopted from PNDP is the use of parametric analysis. Altho the idea of parameters did not originate in PNDP, and many of the parameters that I use are not given there, seeing how parametric analysis has been developed in PNDP suggested a wider usage than I had previously imagined. A technique called a paradigm case formulation was also suggested to me by PNDP. This technique allows us to develop a complex concept in a fairly precise manner. I have used it for formulating the concept of a behavior description and for formulating the concept of an educational community.

The CS Group: Some of the illustrations make use of a small learning group taken from my book Understanding Fractions for Adults. Bob and Jan and Kay and Roy are studying fractions with a mentor named Jo. This group is called the CS Group because its primary focus is on conceptual study. The main fraction model consists of a set of colored pieces. The piece representing the number 1 is a
4 inch by 6 inch white rectangle. Since 2 red pieces take up the same space as this white, a red represents ½. Supplement 1 gives information about this model that should be sufficient for following the example in which it is used. The CS Group is imaginary and anything I say about them is clearly fictional. Since this group is intentionally idealized as exceptional, some of what they do may seem unlikely. However I am not using them as some kind of model. I am using them to communicate ideas to adults, primarily about fraction concepts and secondarily about the kind of learning group in which the mastery of such concepts would be likely to flourish. I also use this CS Group for similar purpose in several other books and learning resources, all of which may be downloaded without charge from my website. In this present book my purpose for this group is more to illustrate ideas about educational concepts rather than mathematical ones. Since the idealized nature of this group makes it far from typical, it does so only in a narrow manner. I have added a few illustrations that are more realistic, but these seem far from sufficient. I welcome any help in developing further illustrations or in expanding the scope of any portion of the net for understanding education that I would like to see emerge.

SECTION 0 BASIC CONCEPTS

Overview: The concepts of a person and a behavior description and behavior roles as formulated in PNDP are introduced in this section. These and other concepts from PNDP are used to formulate a part of net for thinking about education. This net will include the concepts of an educational process, various types of educational processes, the related learner and educator roles, constructivist and receptive educational activities, educational communities. In shaping a net for thinking about education, no claim is being made that this only way to formulate such a net or that it is a net already being implicitly used. However it is hoped that it will be close enough to nets being used to be at least somewhat more useful for similar purposes. I want to emphasize that it is a net that is being developed, rather than a theory, and that this net is theory neutral. To the extent that this net becomes comprehensive it could be used to present various educational theories and discuss issues on which they agree or disagree.

The Person Concept: The main feature of the person concept is given below.

A person is an individual with a history in which deliberate action is a major factor.

A Behavior Description Concept: Closely related to the person concept is the concept of a behavior description. I begin with a version of this concept that I will used to formulate some basic educational concepts. The paradigm case of a behavior description uses all of the parameters below to describe a course of action X by a person called the actor. The person giving the description is called the describer. Describer and actor are roles which may or may not be taken by the same person.

¨ Identity (I) specifies who is the actor for X.

¨ Wanting (W) indicates what the actor intends to achieve by X.

¨ Knowledge (K) has to do with facts and distinctions in relation to X on which the actor is acting.

¨ Know‑how (KH) has to do with the competencies the actor displays in relation to X.

¨ Performance (P) encompasses the processes the actor is implementing.

¨ Achievement (A) is what X accomplishes, what difference it makes.

¨ Characteristics (C) involve which of the actor’s characteristics are being expressed by doing X.

¨ Significance (S) includes what else is being done in doing X, what importance X has for the actor.

Allowable Transformations: A describer can give a behavior description for a course of action involving more than one actor. Furthermore, an actor need not be a person. For instance, an actor could be an animal or a robot. The describer can be a team working together to give a behavior description.

Deliberate Action: A describer can give a behavior description that does not use all of these parameters. A behavior description that uses at least the first five of these parameters is an intentional action description. To be a deliberate action description, an intentional action description must use a K‑parameter that indicates that the actor has distinguished at least two options. As I said earlier my observations about education could be said without out recourse to PNDP. However using the distinction between deliberate action and non‑deliberate intentional action provided an idea that I might not have considered. Attention to this distinction could help a teacher formulate a strategy. Of course, recognizing when an action is or is not deliberate does not guarantee that a useful strategy for dealing with any concerns will be recognized nor that a teachers concerns are justified. It merely gives a tool for thinking about options that might not otherwise have occurred. Like any tool, a conceptual tool does not do a job. Achievement depends on how the tool is used.

Example: Jo poses a problem. Roy immediately relies on algorithm, giving no thought to alternatives, i.e. his use of the algorithm is intentional but not deliberate. Jo considers this as part of a behavior pattern, and believes that if Roy could learn to imagine alternatives his understanding would be enhanced. She decides to ask a question where Roy might use an algorithm that might get challenged. She chooses one that she thinks Roy can correctly solve with an algorithm that Bob will find puzzling. Had she judged that most of the time Roy deliberately used algorithms over other options she would have adopted a different strategy. On another occasion, Jan vacillates between two options and neglects solving the problem at hand. Judging this tendency to vacillate as a major trait for Jan, Jo suggest that Jan might try the first thing that comes to mind and reflect on alternate ways of solving the problem later. In general she wants Jan to become comfortable with non‑deliberate action without neglecting useful deliberation.

The C‑parameter: This parameter is organized it into three categories, each of which includes several types of characteristics. See Supplement 2 for my divergence from the PNDP power types.

Dispositions: {Traits, Attitudes, Interests, Styles}

Powers: {Functional Abilities, Cognitive Competencies, Operational Values}

Derivatives: {Embodiment, Capacities, States}

Example: Both traits and attitudes involve frequency of behavior. The distinction between them is that attitudes involve a specific focus and traits do not. Roy’s tendency to use algorithms is an attitude that he has towards arithmetic questions. He does not use algorithms to answer questions about baseball. Jan’s tendency to vacillate is a trait because it occurs in a variety of unrelated situations and has nothing to do with whether or not the situation involves using fractions. If she only vacillated when solving fraction problems, it would be an attitude towards such problems rather than a trait. Kay has a tendency to try to understand concepts. Altho this is an attitude towards concepts, Kay also has a general trait of trying to understand whatever she encounters.

Major Behavior Roles: PNDP uses three major roles {actor, observer, critic} that a person P may take in relation to behavior. The actor’s job is to do something that P has one or more reasons to do, and in doing this P is spontaneous and creative, responding to and acting on whatever P finds relevant. The observer’s job is to note what an actor is doing, what is happening, what is relevant to this. If the observer describe any of this then we may call the observer a describer. The critic’s job is to appraise how things are going and how this relates to future courses of action. If things seem satisfactory in that regard then the critic takes no present action, except perhaps to think about how to do even better in the future. If anything seems unsatisfactory then the critic prescribes corrective action. To be an observer or a critic is to be so in relation to some actor who may or may not be some other person.

Educational Processes: An educational process involves activities of one or more persons who are engaged in two types of roles called educator and learner. An educational process is intentionally directed towards the development or alteration of one or more dispositions or powers called the goals of the process. Anyone attempting to direct an educational process is taking the role of an educator towards a pupil, where a pupil is anyone whose dispositions or powers are intended to be developed or altered by the process. A pupil takes the learner role a process if that pupil is actually engaged in an activity relevant to its goals. The term learner is used not only in relation to a specific process, but also in reference to someone who normally takes a learner role in the educational processes in which he or she is a pupil. Thus being an educational process is characterized primarily the C‑parameter and P-parameter of the learner or learners and by the W‑parameter of an educator or educators. Furthermore developing or altering a characteristic is an achievement, so this W‑parameter is directed towards a preferred A‑parameter for the pupil or pupils. The other parameters are also important in helping understand what educators and pupils are actually doing during some educational process, and especially in thinking about what could go right or wrong.
As conceptualized it should be clear that unless at least one pupil is actually a learner there is no educational process, regardless of the intent of any educators. Likewise altho vast amount of learning takes place without educators, no educational process occurs without an educator. For instance, members of the CS Group learned to respect Jo without anyone intentionally trying to help them develop this attitude, i.e. no educational process was involved. It should also be noted that there is no reason that a person cannot take both the educator and learner roles. In fact, in many effective educational processes the learner is also one of his or her own educators. A person who decides to cultivate a new attitude towards snakes, say to regard them with respect rather fear, takes the role of educator in deciding what to do and the role of learner in so doing. This is an educational process with only one person who plays both roles. In an interactive educational community, switching between educator and learner roles may be fairly frequent.

Example: Jo asks a question designed to help pupils reflect on fractions sizes. Taking the learner role, Jan says that ¹/₆ is smaller than ¹/₃. Roy says that for unit fractions the one with the larger denominator is the smaller. He does so primarily to help the group better understand the concept fraction sizes, but also to solidify his own understanding. He is acting primarily as an educator, but he is also acting as a learner. A pupil making this comment merely to show off would be neither. When Kay adds her reason for understanding this, she is also both an educator and a learner, altho if she had only hoped to solidify her own understanding she would not have been an educator. During this Jo is primarily an educator in relation to the main educational processes for the group, she is also both an educator and a learner in the process of developing a deeper understanding of these learners.

Relation of Learner and Educator Roles to Major Behavior Roles: The learner role may be taken in conjunction with any of the major behavior roles. For instance, Roy adds ²/₃ and ³/₄ and obtains ⁵/₇. Roy says that he added numerators and denominators. He is puzzled because ⁵/₇ is less than l and asks Jo to tell him what went wrong. Thus he is engaged as actor in adding, observer of what he did, critic in suspecting something is wrong and prescribing the alternative of asking Jo. The educator role may also be taken in conjunction with any of the major behavior roles. In asking questions Jo is taking an actor role, altho her questions are often preceded by observation and appraisal of what her pupils are doing. In her response to Roy, Jo in the role of critic tells him to use the fraction pieces. Later in the self‑critic role she decided that in this case it might been better to ask the others if they could explain what went wrong. Another alternative that occurred to her was to ask him how he could figure this out for himself, and decided in the future she needed to pay especial attention to Roy’s trait of wanting authoritative answers. In general, Jo tries to imagine educational processes that will influence all powers and dispositions that are relevant to there having greater behavior potential.

Derivatives: An educational process could be conceptualized to include derivatives in addition to powers and disposition. For instance, when my leg was in a state that prevented me from walking easily I went to a therapist who directed me in a process designed alter this state. Likewise a person engaged in a weight lifting program could be thought of as engaged in an educational process designed to develop embodiment characteristic. I have no problem in extending the concept of an educational process to include these. However ordinary language does not refer to a person altering most derivatives as learning, and since processes involving derivatives will not be discussed in this book, I will only refer to powers and dispositions when referring to educational processes.

Change Maxims: To further develop some educational concepts, I consider slightly modified versions of three maxims given by Ossorio in his book entitled Place. For instance I use ‘develop or alters’ where he uses ‘acquire’, because this has a somewhat broader process connotation that includes acquisition. These maxims remind us that characteristics are often developed or altered without an educational process being involved. The first applies to all characteristics while the other two apply primarily to knowledge and abilities. Furthermore characteristic are usually developed or altered in combinations rather in isolation. For instance, Jo’s attitudes and knowledge in relation to her pupils
developed concurrently. What distinguishes an educational process is that it is intentionally directed towards the development of characteristics and often with a primary focus.

Maxim D1: An individual develops or alters a given individual characteristic by virtue of having the prior capacity and the relevant intervening history.

Maxim A6: A person develops or alters knowledge of the world by observation or being informed and by further thought.

Maxim D4: A person develops or alters concepts and abilities by practice and experience that involve the use of these concepts or the exercise of these abilities.

Receptive Educational Activities: Jo asks her pupils to look at the colored pieces and record which colors are being used in descending order of size. All of them do so and thru this observation easily acquire some new knowledge about this small realm of colored pieces. Instead, Jo might have merely given them this information. Had she done so, the activity of listening would have been purely receptive, i.e. they would have acquired this knowledge by merely by being told. Instead Jo had them observe, but only in order to more fully engage them as actors. Jo still expected this activity to be primarily receptive, since she expected them to merely add this information to their knowledge by a simple observation. Bob observes that 4 blue pieces to make a white, and so a blue can be used to picture ¹/₄. He is adding to his knowledge by thought, as are the others when they follow his thinking. Again, the activities involved are mostly receptive. In general, when an activity enables a learner to develop a characteristic in a merely add on manner without any need to alter or utilize any prior characteristics in a noteworthy manner it is a receptive learning activity. A receptive learning activity that is part of an educational process is a receptive educational activity.

Altho Maxim A6 is a reminder that a vast a person’s knowledge is developed thru receptive learning activities, not all observation or thought is merely receptive, and even being informed may not be merely receptive. In fact any of these activities may be more involved. Furthermore, most receptive learning activities are not educational. For instance, Jan once watched a game of hearts and immediately developed an interest in the game.

Constructivist Educational Activities: An educational activity is constructivist to the extent that it effectively utilizes prior characteristics of the learner in a noteworthy manner in order to help the learner further develop these characteristics and acquire additional characteristics. The term constructivist indicates that the learner is engaged in more or less complex a building process that is related to an existing foundation that perhaps may also have to be altered in a substantial manner. The construction may not be recognized by the learner, unless the learner is also engaged as observer or critic. After being puzzled by getting an unacceptable answer for ²/₃ and ³/₄ Roy had to alter his simple algorithmic attitude and construct a new understanding of what it means to add fractions. Had Jo merely given him a correct algorithm he might have been able to acquire a knowledge of how to add fractions in a purely receptive manner. In Roy’s case she wanted more than this. At another time she merely gave Kay the standard algorithm for extracting square roots. She knew that Kay would probably be puzzled about why this algorithm worked and would try to figure this out for herself. The fact that she did not think Kay would succeed actually was one of Jo’s reasons for doing so.

Example: Casino is a game in which you can use a card to capture cards with that sum. Meg’s father, acting as an educator, taught Meg to play when counting was her only number concept. He was preparing her for enjoyable activities in which she would acquire basic arithmetic concepts without studying. While playing, he would casually suggest ways to think about numerical combinations. Altho casino was not designed to help anyone construct number concepts, he used the activity of playing it as a deliberately designed constructivist resource for that purpose. This was a way to easily transform an ordinary activity into an effective constructivist learning resource. Meg later used this resource with her children for the same purpose. Meg’s uncle also used it with his children, even modifying the rules

in order to enhance its numerical concept construction potential. In so doing it even became a more interesting game. Learning basic number concepts involves learning numerical facts. Only a few of these are used in playing casino, and these emerge slowly, usually only after playing many games. This is not surprising, since most of the attention is on features of the game other than the need to make additive combinations. Of course, using casino as a resource for learning addition concepts depends on the learner finding the game a realm of interest.

Types of Educational Processes: Altho educational processes can be directed towards any powers or dispositions or combination of these, they often have a specific focus and this has a major impact on how the process gets implemented. For this reason I will distinguish four types of educational processes, depending on the primary focus. These are not intended as either exhaustive and an educational process may be a mixture of any of these types.

{Dispositional, Value, Functional, Cognitive}

I will briefly sketch what is meant by these type. Section 1 turns in more depth to the last, which is the one on which this book places its main attention. The first 2 are exactly what their names suggest, i.e. the first is directed towards any of the dispositions or combination of them, and the second is directed towards one or more values. A functional oriented educational process is oriented to abilities that are not primarily cognitive. For instance, in learning how to split logs I was engaged in a functional educational process. Likewise a basketball coach often takes the role of an educator in a functional oriented educational process. This is not to say that learners do not alter their cognitive powers, but that this is not the primary goal in a functional educational process. In fact in any educational process almost any characteristic may be altered. A cognitive oriented educational process is directed towards cognitive competence. Here this term is use broadly to include all types of knowledge and cognitive know‑how. Bloom’s Taxonomy of Educational Objectives for the Cognitive Domain gives an overview of what is included in cognitive competence. This taxonomy is given in the next section.

Communities: PNDP uses some or all of the seven parameters below to characterize a community (or type of community) and to differentiate it from other communities (or type of communities). Supplement 3 gives a simplified account of these parameters.

{members, statuses, concepts, locutions, choice principles, social practices, world}

Except for the last parameter, an ordinary understanding of most of these terms should be sufficient. Having a strong sense of membership in a group that share a world in a sense encompasses the others. In describing what we do about we use elements that we think of as {objects, events, processes, states of affairs} and we talk about of relationships among them. A world for a community is a large interrelated set of such elements that its members are willing to act on in a way that is relevant to being in that community. They react to this world by manifesting values and attitudes and interests that are similar in a multitude of ways. They make distinctions in a common manner. They have shared practices and choice principles. Altho crucial for community purposes, the ideas involved in these parameters are not restricted to such purposes. A person’s world may differ in various ways from the shared world of the communities that he/she belongs to.

Educational Communities: A highly functional elementary school class is more than just a group of pupils with a teacher. Its members have a sense of belonging. The class has educational processes as a central focus. Members assume various roles in this process and have various educational statuses. They have concepts and ways of talking about matters relevant to these processes. The class has practices and ways of interacting and making choices that are supportive of these processes. In this and other ways class members share an educational world. In short, they form an educational community. Not only is a highly functional elementary school class an educational community, what I said about it indicates the general idea of an educational community. However there are many variations. A sports team can be an educational community with a shared educational world, as long as developing the skills of individual players is not seen merely as an extrinsic goal subservient to winning.
Paradigm Case of an Educational Community: Not only is a highly functional elementary school class an educational community, what I said about it could be taken as an example of the general paradigm case formulation of an educational community C which has the features given below.

(0) The majority of the main activities of C are educational activities that center on educational processes, with the educators and learners being members of C.

(1) Members of C know that they belong to C and feel a sense of belonging. They also recognize all others who belong to C. They regard each other as fully members of C.

(2) Altho at times any member may be able to assume either the role of educator or learner, most will have their main status as either learner or educator. Such a status may be qualified in any number of ways. For instance an educator might have the status demanding teacher, disorganized teacher, etc. There may be other educationally relevant statuses, such as skeptic, teacher’s helper, peer coach, etc.

(3) All members of C share and understand a variety of concepts that are central to being an educational community and that relate the educational processes occurring in C. As educational processes occur, new concepts will become central to C and will be acquired by all members of C.

(4) For the concepts central to C as an educational community, locutions and ways of talking are used and understood by all members of C. Altho most of the concepts central to C and the locutions regarding them are used in some other communities, some of them may be distinctive to C or used in ways peculiar to C.

(5) The noteworthy choice principles of C include accepted ways of making choices that are related to educational processes. They also include accepted ways of making choices that are supportive of the welfare of C and the realization of the goals of educational processes. Furthermore these choice principles are understood and usually appropriately applied and seldom violated.

(6) C has well established practices based on the choice principles for ways of interacting and behaving. Some of these are directly supportive of educational processes and are core practices used by all members of C.

(7) In the ways above, and in other ways, members of C share an educational world.

Allowable Transformations: An allowable transformation indicates a way of relaxing or broadening one of these features. I want to stress that these are ways rather than exact prescriptions. Changes that are in the spirit of these ways, including less pronounced changes, are also allowable. What is definitely not allowed is changes going too far beyond those indicated. For instance, in feature (0) changing ‘the majority of’ to ‘a few’ or to ‘none’ is definitely not allowed.

In (0), change ‘the majority of’ to ‘many of’ or change ‘center on educational processes’ to ‘significantly involve educational processes’.

In (1) change ‘members’ to ‘most members’; change ‘recognize all others’ to ‘have fairly effective ways to recognize others who’; weaken the sense of belonging; omit or weaken the last sentence.

In (2) change ‘the main status of all members’ to ‘a noteworthy status for most members’; allow more or less flexibility in assuming the roles related to these statuses.

In (3) and (4) allow considerable variation in the extent to which concepts central to C and locutions in regard to them are shared and understood. This applies especially to central concepts that educational processes are expected to help learners develop and utilize in later educational processes.

In (5), change ‘accepted’ to ‘largely accepted’ or to ‘largely understood’; weaken ‘usually appropriately applied’; change ‘seldom violated’ to ‘seldom violated without sanctions’.

In (6), omit ‘well established’; change all members to most members.

In (7), change ‘the ways’ to ‘reasonable variations of the ways; omit ‘and in other ways’.

Comment: I want to stress that the concept of an educational community has been formulated in a way that allows considerable latitude in what qualifies. The educational activities can have any mix of contructivist and receptive educational activities. No criteria of the effectiveness of its educational processes are required. Any set of dispositional or power characteristics can be included or excluded from its educational goals. It can be authoritarian or permissive, traditional or progressive, etc. However a clear-cut case must unequivocally satisfy the formulation just given and for a borderline case it must at least come close. For instance, altho a typical small town is a community, at most a small portion of its main activities will involve educational activities in a significant manner. Thus it is not an educational community or even a borderline case of one. For a more relevant case of a community that is not an educational community, consider what might be called a pseudo-educational community. By this I mean a community that imitates an educational community but clearly does not meet the criteria because there is only a pretense that it is seriously involved in educational processes.

Dysfunctional Educational Communities: An educational community is dysfunctional to extent that it fails to meet the goals of its educational processes. For instance, Jo believes that CS would become somewhat dysfunctional if she decided to ignore color pictures and applications, and focus primarily on deducing the algorithms for fractions from the basic laws for rational numbers. She is aware of the reactions to the so called modern mathematic projects. While she know that some of the dysfunctional results were due to the fact that many teachers did not understand the concepts involved, she also thinks that a great deal of constructivist activity must take place in order to appreciate a deductive perspective on mathematics.

Borderline Cases: Altho the use of fairly loose allowable transformations gives borderline cases, using more precise criteria would be likely to draw a rather arbitrary boundary. Furthermore the existence of borderline cases adds some perspective that I find useful. One type of borderline case includes organizations where learning is a necessary part of its main mission but this is not the focus of this mission. Another type of borderline case is when an educational group does not clearly qualify as a community. However the main borderline cases that I want to consider have borderlines that are between being a dysfunctional educational community and a pseudo-educational community.

Example: Organizations where learning is a necessary part (but not the focus) of its main mission include some departmental faculties at a university. Altho faculty in their teaching may be involved in educational processes the learners will not be part of the faculty community. Thus most faculties whose mission does not take research as a major goal are not even borderline cases. On the other hand, a faculty placing a major emphasis on research may be an educational community. However this is not sufficient, research processes may involve developing powers of the researcher, but their main focus may be on developing public knowledge. Unless developing powers of the researcher is a significant secondary focus it would not be an educational process. In addition, unless faculty members are jointly involved in intentionally helping each other develop such powers, merely having each involved in their own studies is not enough for it to be considered an educational community. Other examples of borderline cases include some engineering work groups and many service organizations.

Example: I was once involved in a tutoring center whose purpose was to help adults prepare for a secondary school equivalence exam. Our activities clearly centered on cognitive educational processes and the group of tutors and pupils involved satisfied reasonable transformations of most features of the paradigm case. However this was at best a borderline case of an educational community, primarily because it was a borderline case of a community. What was questionable whether the membership feature was satisfied. Altho the group was small, the sense of belonging was extremely weak. For similar reasons the set of pupils and teachers in a large city school system may be a borderline case even if the educational process feature is satisfied. On the other hand even a large secondary school within such a system is likely to be a fairly clear case of an educational community if some reasonable transformation of that feature is satisfied. Of course, some such schools could be borderline cases, especially multicultural schools.

Note: The term well-functioning is used descriptively rather than as an approbation. Likewise the term dysfunctional is descriptive. They relate to the way the particular educational community is intended to function, whether or not the describer would endorse it purposes. Furthermore a well-functioning group may or may not be effective when it comes to achievement of its educational goals and even if its goal are achieved this may or may not be good for the learners or society. The educational community I would have hoped for in the example below would have been a teacher controlled educational community largely devoted to receptive educational activities. This contrasts sharply with what I later came to hope for and which would have been more like the CS Group which is intended to function as an interactive student centered educational community that relies heavily on constructivist educational activities. Section 3 will focus on these types of educational goals in some detail.

Example: During my first year as a teacher I was given a ninth grade general math class as an overload until they could hire a replacement for a teacher that had unexpectedly resigned. Let C denote this class, including myself as a member. Altho C may have constituted a community, I would not even consider C as a borderline case of an educational community. Instead it was a pseudo-educational community in which we made some pretense at having educational activities. During group time I presented concepts and techniques, asked questions, gave assignments, etc. During individualized time I supervised them as they supposedly worked on the assignments. A few pupils listened and responded during group time and did a minimal amount of the assignment during individualized time. My goal soon became to get by until a permanent teacher could be found. In the six weeks I was with them, at most a small number of activities involved educational processes in a noteworthy manner. Most of my effort involved trying to maintain order and most of their effort involved avoiding work and disrupting the class and entertaining each other. Had I been faced with the prospect of being in C for the year I might have tried harder to involve more of them in educational processes. Altho C probably would not have become a clear case of an educational community, perhaps it might have become a borderline case. To even achieve this much I faced two major barriers. Ninth grade general math repeats mathematical topics from previous years in order to satisfy a graduation requirement for pupils who generally dislike math and have done poorly in the past. I was a first year teacher with little potential to work with resistant pupils.

A Related Borderline Case: I now sketch a simplistic scenario for what the above class C might have become. This scenario is imaginary, largely because after over fifty years I cannot remember that much detail about my first years of teaching. However it is realistic in the sense that is based on my overall experiences during that year. I then consider the extent to which the class might have satisfied allowable transformations each of the features of the paradigm case. Given the barriers I faced, the C I can imagine is a borderline case at best, and if so a highly dysfunctional educational community.

In C I can now devote more than half of my effort to teaching rather than to giving busy work and maintaining order. I roughly classify the 38 pupils into four main types; 10 conforming learners, 13 sporadic learners, 9 passive resistors, 6 disrupters. Conforming learners spend a noteworthy part of the class time engaged to some extent in educational processes, altho seldom with a significant desire to increase there cognitive competence. Furthermore even conforming learners are easily diverted from engaging in educational activities. Sporadic learners pay little attention to material being presented, altho they at least remain quiet most of the time and they will try to answer questions if I call on them. During individualized time sporadic learners tend to dawdle. They work on the assignment mainly when I am aware of them and become noisy if I am not. The passive resistors merely sit killing time in various manners. Except on rare occasions, the disrupters are totally disengaged from any educational activities, devoting most of their energy to what there name implies. I merely tolerate what I can and eject one or more of them if they become too disruptive.

Since many of the pupils are learners for at least some of the time perhaps a majority of the main activities involve educational processes in a significant manner. This entails a major weakening of educational process feature, but perhaps it still sufficiently satisfies this feature.

Altho members know that they belong and recognize all others who belong, many have little sense of belonging. I consider the membership feature to be satisfied well enough to classify C as a loose community.

Altho about 40% of the members do not have the status of learner or educator, a majority at least engages in educational activities enough to be assigned the status as learners. Perhaps this is sufficient to satisfy feature the educational status feature.

There is a major problem in C in relation to understanding many mathematical concepts and language and applications that I am trying to teach. However all pupils understand some mathematical concepts and language; and they can talk about aspects of an assignment and their textbook, being in the ninth grade, teachers, etc. Perhaps this is enough to satisfy the educational concept and locutions features.

The basic educational choice principle for the educational processes is to pay some attention during group time and work on assignments during individualized time. These are understood by almost all members, and are seldom violated by conforming learners or deliberately violated by sporadic learners. However they are seldom applied by the passive resistors or disrupters. Another educational optional choice principle that I try to promote is for a pupil to ask questions when he/she does not understand. This principle, while seeming to be understood by all, is not used nearly as often it is called for.

The core educational practice is to engage in educational processes. This is at most well-established for 60% of C so the practices feature is at best minimally satisfied. In addition to obvious aspects of what this core practice entails, there are well understood supportive practices of common courtesy; such as being silent when others are speaking, talking quietly during individualized time, etc. These are barely well-established among conforming learners, barely established among sporadic learners, ignored or deliberately violated by the resistors.

That the members of C share a world seems fairly clear. The extent to which it is an educational world is questionable.

Educational Processes and Educational Worlds: I am reading The First Man in Rome by Colleen McCullouch. This is a highly authentic novel with a supplement containing a considerable amount of information about the Roman Republic. In consulting the supplement I am engaged in an educational process in both the roles of educator and learner, for I am trying to cultivate my knowledge and understanding of the Roman Republic. I am also engaged in an educational process when reading the novel, altho this is often of secondary significance. These educational processes take place in one of my small educational worlds. This educational world includes all the reality elements that are relevant to these educational processes. Major objects in it include the book and me. It also includes objects such as my glasses and the lamps I use when reading. It includes states of affairs such as having certain attitudes toward the book and having choice principles for reading it and using the supplement. It includes having concepts for thinking about Roman history, having prior knowledge and understanding of Roman history, etc. My educational world includes this world along with a number of others, some of which are much larger, such as my educational world for developing my conceptual competence. In general a person’s educational world includes anything that is relevant to the educational processes that person is has been engaged in or is planning to be engaged in. A person educational world may have a number of more limited worlds in which various sets of related educational processes takes place.

SECTION 1: COGNITIVE ORIENTED EDUCATIONAL PROCESSES

Overview: This section develops some additional concepts that are relevant to cognitive oriented educational processes and distinguish such a process from other types of educational process. Such concepts include student, teacher, study, cognitive competence, types of knowledge, concept mastery.

Students and Study: In a cognitive oriented educational processes a learner is called a student and the educator working most closely with the student is called a teacher. One intended activity for students is to study, altho the teacher may also encourage other learning activities. Study is any activity in which the student wants to develop cognitive competence and this want directs the performance. See Supplement 2 for comments on the terminology being used. Thus study is characterized by the want and performance parameters. The acquisition of cognitive competence may depend heavily on concept development. Even so, the majority of our concepts are acquired without deliberate study. They often emerge as they are needed to participate in commonly occurring practices. Since this is the way we acquire most of the concepts for our ordinary realms of interest, ordinary realms can be used by a teacher to intentionally help a student enhance concept development, and this may be done whether or not the student is engaged in study. This is one of the reasons that a teacher may decide to encourage learning activities other than study.

Most of the time when we talk about study we are thinking of a study episode rather than a single action. To say that you studied history last night would normally suggest that you did more than look up a historical fact in the encyclopedia. However, the concept of study includes extremely casual activity. Even watching a game of chess and asking how a pawn is allowed to move qualifies as fleeting study. What counts is the W-parameter and its relationship to the P-parameter. In particular, the A-parameter is not relevant. A person may be engaged in study without developing additional cognitive competence. In fact, much of a person’s cognitive competence is achieved without study. Study is directed to the acquisition of cognitive competence that might not just accrue by living or that might not be as efficiently acquired without study.

Conceptual & Paraceptual: The distinction between conceptual and paraceptual statements is important for understanding cognitive educational processes. Conceptual statements merely give information about the concepts being used. Paraceptual statements use concepts to propose information about some state of affairs that is not merely conceptual. That first cousins share a pair of grandparents is a relation between concepts used in our net for ordinary family relationships. This statement is conceptual, since it is independent of any state of affairs in the realm of families. Saying that Joe and Sue are first cousins uses this concept, but tells about a state of affairs that this concept helps us think about. Information of this type could be called empirical, but this term has connotations about how the information is obtained and perhaps about how it is verified. So I have coined the term paraceptual for such information. Paraceptual information my be correct or incorrect. It may have been obtained by careful observation, casual experience, intuition, hearsay, etc. Saying that it is 1pm Central Standard Time proposes paraceptual information whether or not it is correct and whether it is merely a guess or obtained by using a clock. Clearly understanding it depends on conceptual information about how we think about time.

In conceptual study, the only noteworthy information is conceptual. So conceptual study focuses on concepts and relationships between concepts. Mathematics is the most prevalent example in schools where most of the study is conceptual, altho much of what goes on is algorithm training rather than study. The study of grammar is also conceptual. Other types of study involve noteworthy information about some realm that is not purely conceptual. Paraceptual study presuppose some concepts and focuses on some realm that these concepts are intended to help access. Many study episodes are a mixture of paraceptual study and conceptual study, with the emphasis being on the paraceptual components. Furthermore the ordinary acquisition of concepts normally occurs with a paraceptual

emphasis. These way of acquiring concepts work well enough for many purposes, but there are times when this does not lead to adequate concept mastery. Since understanding concept is a basis for understanding paraceptual knowledge, this is one of the ways an educational process can go wrong. This my main reason in this book for making the {conceptual study, paraceptual study} distinction. One decision that an educator might make when the goal is concept acquisition is on what mixture of conceptual study and paraceptual study is likely to be most useful. In introducing size names, Jo focused primarily on conceptual study. Earlier she had worked with them on fraction concepts by referring to some ordinary paraceptual applications.

Bloom’s Taxonomy of Educational Objectives for the Cognitive Domain: Some of the concepts relating to Cognitive Oriented Educational Processes are adequately developed in the of six levels {Knowledge, Comprehension, Application, Analysis, Synthesis, Evaluation} indicated in Bloom’s Taxonomy of Educational Objectives for the Cognitive Domain. The account of these given below is taken from (http://faculty.washington.edu/krumme/guides/bloom.html).

Knowledge of terminology; specific facts; ways and means of dealing with specifics (conventions, trends and sequences, classifications and categories, criteria, methodology); universals and abstractions in a field (principles and generalizations, theories and structures):
Knowledge is (here) defined as the remembering of appropriate, previously learned information.

defines; describes; enumerates; identifies; labels; lists; matches; names; reads; records; reproduces; selects; states; views.

Comprehension: Grasping (understanding) the meaning of informational materials.

classifies; cites; converts; describes; discusses; estimates; explains; generalizes; gives examples; makes sense out of; paraphrases; restates (in own words); summarizes; traces; understands.

Application: The use of previously learned information in new and concrete situations to solve problems that have single or best answers.

acts; administers; articulates; assesses; charts; collects; computes; constructs; contributes; controls; determines; develops; discovers; establishes; extends; implements; includes; informs; instructs; operationalizes; participates; predicts; prepares; preserves; produces; projects; provides; relates; reports; shows; solves; teaches; transfers; uses; utilizes.

Analysis: The breaking down of informational materials into their component parts, examining (and trying to understand the organizational structure of) such information to develop divergent conclusions by identifying motives or causes, making inferences, and/or finding evidence to support generalizations.

breaks down; correlates; diagrams; differentiates; discriminates; distinguishes; focuses; illustrates; infers; limits; outlines; points out; prioritizes; recognizes; separates; subdivides.

Synthesis: Creatively or divergently applying prior knowledge and skills to produce a new or original whole.

adapts; anticipates; categorizes; collaborates; combines; communicates; compares; compiles; composes; contrasts; creates; designs; devises; expresses; facilitates; formulates; generates; incorporates; individualizes; initiates; integrates; intervenes; models; modifies; negotiates; plans; progresses; rearranges; reconstructs; reinforces; reorganizes; revises; structures; validates.

Evaluation: Judging the value of material based on personal values/opinions, resulting in an end product, with a given purpose, without real right or wrong answers.

appraises; compares & contrasts; concludes; criticizes; critiques; decides; defends; interprets; judges; justifies; reframes; supports.

Cognitive Competence: The term ‘cognitive competence’ is used in a broad sense. It includes not only what is given in the knowledge and comprehension levels of Bloom’s Taxonomy of Educational Objectives for the Cognitive Domain, but also what is given in the levels above these. Thus cognitive competence includes the ability to use and make judgements about factual knowledge. It includes the mastery of and the competence in using concepts. In fact one of the most basic distinctions relevant to a cognitive educational process is a distinction between conceptual and paraceptual competence. Conceptual competence involves knowing conventions, games, concepts, classification schemes, abstractions, principles, structures. It also involves skills like the ability to analyze, synthesize, evaluate. Paraceptual competence involves having information about various states. It involves the ability to recognize the need for additional information and skill in obtaining such information. It involves understanding states from both a remote and manifest perspective. Altho conceptual competence often leads to paraceptual knowledge and paraceptual competence, it can also be developed in isolation from the paraceptual. Most of my mathematical ability has developed in this manner. On the other hand, most paraceptual ability is dependent in some fashion on conceptual ability. Even immediate factual knowledge, such as knowing there are eggs in the refrigerator, would elude us if we did not know the concept of an egg.

For various reasons, I use the term ‘knowledge’ more broadly than it is used in the knowledge level of Bloom’s Taxonomy. I will distinguish various types of knowledge along with some closely related types of understanding. These are sketched below and developed in Supplement 2.

Propositional Knowledge: Information that can be expressed by statements that can be classified as true or false is the least complex form of knowledge. I call this type propositional knowledge.

Informational Knowledge: Informational knowledge includes but goes beyond propositional knowledge. It includes information that Z can act upon, even if Z cannot formulate this linguistically.

Note: It is easy to regard your informational knowledge as propositional. After all, if you try to tell someone about your informational knowledge you are likely to formulate it using propositions. One of my favorite examples of informational knowledge that is not propositional is the information I often need to reach some destination. Propositions giving road names and turns is much less useful to me than a map. Altho a map implicitly contains information that I could make propositional, it also has information that is pictorial and more basic for me. For an example of informational knowledge that is not at all propositional, consider the knowledge of a child who has not yet learned to talk.

Process Knowledge: Process knowledge is the type of knowledge that is acquired thru practice and that enables an actor to implement various processes. It includes the cognitive aspects of Z’s action capabilities, or the cognitive aspect of Z’s competence.

Realm Knowledge: Realm knowledge is gestalt knowledge of a realm. Knowing a realm means being reasonably familiar with it. It is the kind of knowledge that we have in mind when we say that a reporter really knows city hall.

Relational Understanding: Relational understanding involves understanding relationships between concepts. It is essential to having well integrated concepts. Relational understanding also involves an understanding of relationships in various states of affairs that are not purely conceptual, such as understanding the relationship between nutrition and health.

Operational Understanding: A person has operational understanding of a realm to the extent that he or she understands relevant concepts and previously learned information well enough to apply them as needed in situations related to the realm, including situations not previously encountered.

Parametric Analysis for Concept Mastery: Below I indicate 6‑parameters for thinking about concepts. The are intended primarily to indicate a relationship between a person Z and a concept C, and specifically the ways in which Z masters C. While these parameters are conceptually independent, they are often interrelated in practice. For instance, the concepts that Z finds most manifest are also likely to be included in those that Z uses in a fairly precise manner. These parameters are developed more fully in my book A personal Approach to Conceptual Philosophy.

¨ proximity: indicates how C relates to Z’s ordinary experience

¨ realm: indicates the realms or type of realms which Z can relate to C

¨ utility: indicates both the scope and types of uses Z has for C

¨ integration: indicates how C is connected to other concepts within the conceptual nets used by P

¨ explanation: refers to way Z could or would be able to explain C

¨ focus: refers to the ways Z understands the precision of C

The Proximity Parameter: This parameter indicates the proximity of a concept to ordinary experience or to specific features of experience that a person finds easy to identify. Proximity can vary from highly manifest to extremely remote. C is manifest to Z to the extent that it is close to Z’s easily accessible experiences. C is remote for Z to the extent that Z finds it removed from such experiences. Suppose Meg is a very young girl with a new baby sister Ann. The concept ‘my baby sister’ will be highly manifest for Meg. The concept of concept of a sister will initially be extremely remote, or even absent. As Meg gains experience, her sister concept will normally become manifest, but probably not as manifest as her my sister concept. Before her sister concept becomes manifest, there are likely to be other manifest concepts involving particular sisters. Meg might know that she is a sister to Ann prior to understanding the more remote sister concept. The bed Meg uses will be a highly manifest concept of my bed for Meg. The concept of a bed will be somewhat less manifest, but since the function of a bed is easy to understand in terms of ordinary experience and since other examples of bed are easy to observe, it should not be difficult for the bed concept to become highly manifest. Likewise Meg will find a multitude of direct abstractions from highly manifest concepts. This will include not only object type concepts such as bicycles and rocking chairs, but also other type concepts such as fear, holidays, hunger, etc. She will also obtain fairly manifest concepts not directly related to ordinary experience but that she can easily imagine in terms of ordinary experience, such as a great white shark or a saber‑tooth tiger. At a slightly higher level, but still manifest, might be her concept of a unicorn. On the other hand, even if the concept of an ion is explained to her it may be one that she finds extremely remote.

The Realm Parameter: This parameter indicates type of realms which Z can appropriately relate to C. It may also describe limited mastery by indicating realms to which the concept might apply but which Z would not consider. Mastery of the ace of diamonds concept would normally involve understanding that this concept applies in the realm of playing cards. Two types of realms that are broad and especially significant are the personal and the impersonal. A realm is personal to the extent that it is used to think about persons and their actions. For instance, the usual concept of frugality makes sense only in the realm of person. Meg’s my sister concept is a personal concept that applies to the realm of family relationships. Meg’s concept of a bed is likely to also be somewhat personal, since she will associate a bed in relation to personal use. To the extent that her concept of a bed relates to its physical features rather than on its use, her concept will relate to impersonal realms. In general, an impersonal realm is one in which the actions of persons do not seem central. Altho all concepts are acquired thru personal experience, some have little else to do with persons. This is especially the case with many concepts that are primarily perceptual, such as red or wet.

The Utility Parameter: This parameter indicates the scope and type of uses Z has for C. At one extreme Z may depend on C so much that C is essential for Z. At another extreme Z may use C so seldom and with no interest that C may be expendable. Of course what is expendable for one person may could be essential for another. For instance Meg’s concept a sister would be essential for her, but a hermit might find a sister concept expendable. My concept of a fireplace essential because I use one to heat my cabin. For many of my friends the fireplace concept is relevant but far from essential. The utility parameter may also indicate anything that might provide a perspective on how Z might be able to use C. I acquired my concept of a maul by splitting wood rather than as an interested observer. Furthermore I acquired it as an amateur who normally splits just enough wood for my own personal use. Such information could be included as components of the utility of my maul concept.

The Integration Parameter: This parameter indicates the place C has within Z nets. C is disconnected to the extent that it does not relate to other concepts within these nets. C is connected to the extent that it relates to other concepts within one or more of these nets. C is appropriately connected to the extent that these relationships are coherent and correspond to those used in public nets. C is adequately integrated if C is appropriately connected and these connections are those that Z would most commonly need in order to understand and communicate with others. C is well-integrated to the extent that C appropriately connected and exhausts the reasonable ways in which C can be appropriately connected. Meg’s concept of a sister began as connected only to her concept of Ann. Thus it was appropriately connected but also highly disconnected. It became slightly more connected when she thought of herself as Ann’s sister. Only when she understood the relationship between concepts in {sister, brother, male, female, same parents} did it become adequately integrated for most purposes. To become well-integrated this concept would need to be related to a vast number of concepts in a net for family relationships, such as aunt, half sister, step sister, cousin, etc.

The Explanation Parameter: This parameter indicates the ways Z could or would be able to explain C. To the extent that Z understands how to explain C by reducing it to more concepts, C is analytic for Z. To the extent that Z can explain C in other ways, perhaps by relating to a multitude of ways that it is used or by illustrating it with examples, Z has a synthetic mastery of Z. A concept can be both analytic and synthetic for a person. For instance, my concept of a sister is mostly analytic because I would usually reduce it explicitly to the concepts of female and same parent. It is also somewhat synthetic because I might first explain it to a small child first giving familiar example. Meg’s early sister concept was almost totally synthetic and she would have explained it almost exclusively by giving examples.

The Focus Parameter: C is in focus for Z to the extent that Z know when and how to use C and also understands the limitations in using C. A major component of this understanding involves knowing the extent to which a concept is precise or vague, and if Z also has the ability to articulate this knowledge then the concept is highly in focus. C is precise for Z to the extent that Z applies it consistently and coherently without having to allow for indeterminate cases. C is vague to the extent C is not precise. As with many extremely manifest concepts, Meg’s concept of her sister is in focus and precise, altho she would not have been able to articulate the fact that her use of this concept was precise. In general, for most persons their highly manifest concepts tend to be precise and in focus, but often not highly in focus since they are taken so much for granted. A concept may be somewhat vague for Z and still be highly in focus, if Z knows how it is vague. For instance, since I know how to use my concept of an educational community in many situations and understand the extent of its precision, it is highly in focus. Altho I could have chosen transformations to formulate a more precise concept of an educational community precise, I deliberately used transformation that left this concept somewhat vague. These transformations from the paradigm case provide some clear-cut additional cases of educational communities and some clear-cut cases of communities that are not educational communities. However I deliberately left borderline cases, such as the high school my son attended.
Example 1 from Supplement 1 Activity Set 1: Altho Jo had briefly worked with her pupils on some very basic fraction concepts, she introduced colored pieces without at first relating them in any way to fractions. Activity Set 1 merely introduces the concept of a color name and the concept of trading color pieces. In doing this set of activities, the group found these concepts highly manifest and related them to the impersonal routine realms of colors and sizes. Kay immediately saw how color pieces could be used to think about fractions, and so she also related these concepts to the realm of fractions. Kay assigned some mathematical utility to these concepts, but considers them expendable. The others found them highly expendable, altho Bob enjoys visualizing and found them somewhat useful for hedonic reasons. For all members, these concepts were adequately integrated in relation to their routine realms, but only Kay had them well‑integrated in relation to fraction concepts. For the others, these concepts were disconnected from fraction concepts, altho this changed as soon as they move on to later activities in this set. Most of the members would have given a synthetic explanation of these concept by showing examples like those given in Appendix 1. In explaining trading they might have given the analytic explanation of equivalent pieces as pieces of the same size. The concept of color names and trading were in focus and precise for all members of the group.

Example 2 from Supplement 1 Activity Set 2: Activity Set 2 introduces slash names for color pieces and relates these to sizes. From a novice perspective, a size name is just another way to name color pieces, altho the similarity between size names and fractions will be easily noted by all members of the group. A size name has been called a bridge. Bridges cross a gap. Jo used this bridge to prepare her pupils for crossing a gap between manifest color pieces and the more remote numbers named using fractions. There are subtleties involved that make the concept of a fraction harder to master than any of them realize. Specifically the distinction between a number and the name of a number is difficult to master. Altho the distinction between a name and what is named is easily acquired in ordinary matters, this distinction is almost too obvious to be in focus and thus would seem to be of little at most trivial utility. People would never feed the name of their dog instead of feeding their dog. However many people will confuse a fraction with the number it names. I suspect that this is partially because the kind of number that a fraction names is more remote than what we ordinarily name. Activity 2 begins by focusing on alternate size names for color pieces and uses an intuitive notion of same size. It then bridges to the intuitive notion of size, letting size names also name sizes. For people doing these activities with only a casual knowledge about a few common fractions, the more mathematical notion of a rational number will be imprecise and disconnected and remote. At this point I would not even expect that the realm of rational numbers would be one of their realms.

Note: These parameters can also be used as if they were attributes of a concept. A concept is expendable to the extent that it is expendable for most people and if little or nothing in their life would be different if nobody used the concept. A concept is analytic to the extent that most people who understand the concept would explain it analytically. Etc. Thus the concept of a googolplex remote, numerically applied, analytic, appropriately connected, precise, expendable. Altho the vast majority of people might find the concept of atomic fusion expendable, it has had too much of an impact on our life to be classified as expendable, altho it is also not essential. I would also classify this concept as remote, applicable to physics, analytic, appropriately connected, precise.

SECTION 2 APPLICATIONS TO EDUCATIONAL PROCCESES

Limitations of Receptive Educational Activities: Recall that an attitude involves frequency of behavior with a specific focus. Before working with Jo each of the pupils had acquired both attitude toward algorithms and certain power in relation to them. These characteristics were acquired primarily thru receptive educational activities. Bob’s attitude was one of dread. He expected to find algorithms meaningless. He expected to be confused by them, to have difficulty remembering them, to have something go wrong in using them. Jan had a positive attitude towards algorithms involving whole numbers, but did not realize that this was because they made sense to her on an intuitive level. One indication of this was that she did not at first like the standard algorithm for long division. It was not until she encountered the algorithm that allowed choice in which multiple to subtract that she developed a favorable attitude. Kay took a critical attitude towards algorithms, demanding that they made intuitive sense and valuing them as a convenience when they seemed the best way to obtain a result. Roy took a positive uncritical attitude towards algorithms, valuing them for their own sake over all other aspects of mathematics. He found them easy to use and remember.

Having talked with these pupils about mathematics before she began working with them, Jo was aware of these characteristics. In the interchange below gave her an indication of how these characteristics might affect their work with fractions. After this interchange she felt that, except for Kay, they would all benefit from modifying their characteristics regarding algorithms. She also felt that what she hoped for was unlikely to happen thru purely receptive educational activities.

Jo: You have been driving all day and have completed half of a trip you are taking. You want to take it easy tomorrow, so you plan to only go half as far as you went today. What part of this trip are you expecting to complete tomorrow?

Roy: ½·½ = ¼.

Kay: If you went ½ of the trip today, then the other ½ of the trip is still left. Since ½ of ½ is ¼, they are expecting to complete ¼ of the trip tomorrow.

Jan: I imagined the whole trip as 100 miles. Half of this is 50 miles. Half of this is 25 miles. This is one fourth of 100. It also works for trips of other sizes.

Bob: I used some white blocks to picture the trip. I replaced half of them by red blocks for half the trip. I then used half as many blues to picture the next day’s trip. This is one fourth the trip.

g g g g g g g g g g g g

Keeping the following maxim in mind Jo formulated a constructivist strategy for each of her pupils.

Maxim B7: If the situation calls for a person P to do something that P cannot do then P will do something P can do - if P does anything at all.

Engaging the Actor: One thing Maxim B7 reminds us of is to be aware when we try to structure activities that we are may fail to engage the actor in the activity we have in mind. Assigning all even numbered problems at the end of the section may be more likely to engage the actor in getting them out of the way rather than in thinking about the ideas involved or even in useful practice of the algorithms involved. Many pupils may not have the requisite characteristics to think about the ideas involved when facing a long assignment. The alternative of suggesting that the actor work on these problems for 30 minutes might have the intended result, altho it might merely result in clock watching instead. Finding a good strategy to engage the actor in working on these problems may involve helping the actor first modify characteristics so the intended activity is something he or she could do.

Constructivist Strategies: In the past when a situation called for Bob to use an algorithm Jo realized that he often did something else instead, namely since he could not really use the algorithm, he simulated using it. What she meant by this was that he was not really applying an algorithm to get a result, but merely hoping that would nobody would realize that he was merely going thru some motions that he hoped would allow him to get by. Since he was excellent at pictorial reasoning there were times when he knew the answer, and since teachers often only checked his answers, he got by more often than might have been expected. Using color pieces was ideal for working with Bob. A receptive strategy of just showing him algorithms or trying to tell him that he needed to change his attitude seemed unlikely to be useful. So Jo not only allowed him to ignore algorithms, she encouraged him to solve fraction problems via color pieces, even when the others tired of his pictorial comments. She hoped that being allowed to ignore algorithms, while at the same time seeing others use them, Bob would develop a comfortable attitude towards them, use his pictorial abilities to understand them, and even value the advantages they offered in solving problems.

Unlike her earlier experiences, when Jan first started working with fraction in a situation that called for using an algorithm, Jan changed it to one she could think about using whole numbers, as she did with Jo’s question about the trip. Because of this Jo often asked questions that could be solved almost as easily using whole numbers as by using fractions. She once asked how much meat was needed to make 12 quarter pound hamburger patties. Jan responded 48 ounces which makes 3 pounds. When Kay said 12·¹/₄ = 3·4·¹/₄ = 3, Jan thought that this made good sense. With this and other similar experiences Jo expected that Jan would construct a positive attitude towards fraction algorithms.

Altho she thought that Kay already had balanced attitude towards algorithms, Jo realized that her Kay’s understanding of them was based on an implicit rather than explicit understanding of the underlying mathematical structure. Furthermore Jo thought that Kay had the characteristics that would allow her to make her understanding more explicit. The strategy Jo used with Kay was designed to encourage Kay to construct an explicit contemporary conceptual perspective using her intuitively understanding as a basis. There is a subtly in relating mathematics to any application, and Jo thinks that Kay might be ready to appreciate this. Stated simplistically the arithmetic of fractions depends on conceptual decisions about our numerical structure and at most only indirectly on any intended model or application. What we can say about color pieces or any application is that if fractions are to apply to it as we intend then our conceptual decisions must yield this arithmetic of fractions. What Jo wants Kay to realize is that we decided to extend our numerical structure so that any positive number will have a multiplicative inverse. She will mention this to all of her pupils, but for the time being she will suggest extra activities to help Kay understand and appreciate a purely mathematical perspective on the algorithms for fractions. Using the notation /x instead of 1/x for the multiplicative inverse of x, and focusing on the conceptual law x·/x & /x·x = 1, along with other basic laws, partially account for the way Kay thinks about fraction problems. For instance, in saying that 12·¼ = 3·4·¹/₄ = 3, Kay was thinking of 12·¼ as 3·4·/4.

Jo adopts an opportunistic strategy in relation to Roy’s attitude towards algorithms. Most of the time he will be able to apply fractions to solving problems and will use algorithms that work. In fact because he has an older brother who shows him algorithms, Roy already knows how to use many fraction algorithms by rote. However not understanding why they work, he occasionally uses an incorrect version. For instance knowing the rule for multiplication of fractions, he once added numerators and denominators in when adding fractions. If a situation call for Roy to explain an algorithm he will usually describe it instead. The only way he is likely to be induced to take a critical attitude is on the rare occasion when the algorithm does not work. Still he see the dissatisfaction of others when he describes rather than explains, and seeing how the others are then able to make sense of what he did, Jo hopes will begin to construct a more balanced attitude.

Engaging the Observer and or Critic: Even more than with most activities, educational activities are apt to only engage the actor, altho not necessarily in the intended activities. At times an educator might think about the extent to which the observer or the critic is engaged. The typical question about what is this good for is more likely to mean ‘why do we have to do this’ rather than ‘in doing this what am I achieving’. It is the latter observer or critic question that an educator might welcome. One way for a student to be engaged as an observer is for the student to reflect on achievement. Seeing a greater complexity in the causes for the civil war is an achievement, whether or not the student observes this achievement. Observing it can add a deeper appreciation and even trigger the role of critic if the student then thinks about how to build on this achievement.

Behavior Description: Altho much more could be said about engaging the various behavior roles, I now turn our attention to some behavior descriptions. I relate these to an interaction sequence, sketching Jo’s behavior descriptions for her pupils. It should be noted that in these descriptions, only some of what could be said is actually stated. In giving a behavior description, completeness is neither possible nor desirable. For instance altho Jo could specify that the ability to count is part of the KH‑parameter for Jan, this is hardly noteworthy. What Jo gives depends on the purpose at hand. In fact most of the behavior descriptions she uses are implicit rather than explicit. As it will be seen she has a special reason for making these explicit. She does not mention P‑parameters, since this parameter is merely what they said and a did in the interchange below.

Jo: In a 2 mile race, Barb ran ²/₃ of the first mile and Mat ran ⁵/₈ of the first mile. Who was ahead at this point? The race ended in a tie. How much of a lead did the runner who was behind overcome?

Jan: We need to know which is larger, ²/₃ or ⁵/₈. For ²/₃use a 2yellow. Use a 5pink for ⁵/₈. I cannot tell from just looking.

Bob: One way to see that a 2yellow is larger than a 5pink is to use a different arrangement. We can even see that a 2yellow exceeds a 5pink by a 1violet.

Jan: So Barb is ahead at that time and Martin had to make up ¹/₂₄ of a mile or 220 feet. I could have traded the 5pink for 15violet and the 2yellow for a 16 violet to see which is larger.

Kay: Since a 1pink is the size of a 5violet, a 5pink trades for a 15violet. Likewise 2yellow trades for a 16violet. Writing this with fractions ⁵/₈ = ¹⁵/₂₄ and ²/₃ = ¹⁶/₂₄.

Roy: Here is the way to change ⁵/₈ and ²/₃to ¹⁵/₂₄ and ¹⁶/₂₄.

3´5

8´2

3´8

8´3

Description for Jan: The W‑parameter is to answer the question. The K‑parameter involves knowing the relation between fraction concepts and color pieces. The KH-parameter is knowing how to interpret the color pieces and to make correct trades. The A‑parameter is correctly answering the question, altho only after trading. The most noteworthy parts of the C‑parameter are those characteristics related to her attitude toward whole numbers. The S‑parameter includes what else is intentionally being done in what Jan is doing. Jo decided not to assign any value to the S‑parameter. Altho what Jan did influence what Bob did, Jo did not take this as part of Jan’s intent. As with any behavior description, she might have been mistaken in this or in some of her other decisions about parameter assignments.

Description for Bob: The W‑parameter is wanting to answer the question and explain his answer by using color pieces in a way that goes beyond what Jan observed. The K‑parameter involves knowing the relation between fraction concepts and color pieces and the KH-parameter is knowing how to easily and flexibly use an interpret the color pieces. The A‑parameter is correctly answering the question and demonstrating the visual reasoning. The most noteworthy part of the C‑parameter are those characteristics already described about Bob’s powers and disposition in relation to visual reasoning. Altho Jo suspects that nothing other than answering is of noteworthy importance to Bob, he was probably intentionally trying to help Jan see another perspective. So Jo used this for the S‑parameter. Talking later with Bob, she found that she was correct in this assessment.

Description for Kay: Altho the W‑parameter is not directed towards answering a question, since Kay is already satisfied with. What Kay want is to explain Jam’s answer and expand on it by articulating the fraction concepts involved. Jo does not bother to explicitly formulate any of the other parameters for Kay’s actions in this interchange.

Description for Roy: The W‑parameter is wanting to use an algorithm for raising fractions, to impressively demonstrate that he can use this algorithm, to show the others how to use it. The K and KH parameters involve knowing this algorithm and being able to easily perform and articulate it. The A‑parameter includes succeeding in a correct use of the algorithm and demonstrating to Jo that he can use it, altho if he knew that she has reservations about his understanding he would know that he had not impressed her in the way he desired. As he will later discover, he also had a largely negative achievement in respect to showing Bob and Jan how to use this algorithm. The C‑parameter includes those powers and dispositions relating to algorithms that have been previously mentioned. It also includes Roy’s general character trait of wanting to be impressive and his attitude towards there being a best way to do something.

Using Behavior Descriptions for Educational Activities: One purpose in formulating a behavior description for a pupil is to check on the understanding one already has of that pupil’s behavior and characteristics. Those just given merely verify what Jo had already observed. A teacher’s own A‑parameter is largely dependent her understanding of what her pupils are doing as they engage in various activities. Focusing on the A‑parameter of their activities is central to this, but attention to the other parameters is also very useful. Roy answers the question correctly, but is it only because he already has a rote know how of an algorithm? Does Bob know how to trade color pieces to solve the problem, but wants to find a shortcut instead. Did Jan want to solve the problem without trading or was her know‑how in relation to trading not yet well established? Jo’s behavior descriptions are not adequate for answering such questions. However she could have given more elaborate descriptions, labeling various parts in terms of plausibility estimates. For her current purposes this did not seem to warrant the effort that would have been involved. What is warranted depends on what Jo is trying to achieve, and to see this we need to examine her behavior description for herself.

Behavior Description for Jo: Jo also gives a behavior description for herself. The W‑parameter is to formulate a question that will be clear to all of her pupils and result in some useful mathematical thinking by them. Among other things, the K‑parameter involves knowing what color pieces to associate with this question, knowing what fraction concepts are involved, knowing how her pupils might relate to this question and their relevant characteristics. She also knows that reducing fractions seems natural but that raising them may seem artificial. The KH-parameter involved her ability to formulate questions relating to the mathematical perspective she deems important, as well as all the know‑how in relation to the mathematics involved. The C‑parameter involves the characteristic she shares with many mathematicians and her characteristics as a constructivist educator that will be sketched later. The S‑parameter for Jo used for herself goes far beyond trying to ask a question her that pupils could immediately relate to faction concepts. Jo was also trying to do some of the thing for her pupils that are indicated below.

(1) provide an interesting experience working on a mathematical question

(2) help them obtain a deeper appreciation of using multiple approaches to solving a problem

(3) provide a reason that they would see for the utility of raising fractions

(4) help them more competence in mathematical reasoning and problem solving

(5) help then obtain a slightly expanded concept of mathematics

Each of these is a want parameter for what else Jo is doing, and each could be expanded by specifying values for the other types of parameters for what else she is doing. For example in relation to the second of these the K‑parameter includes knowing that ⁵/₈ and ²/₃ are close enough in size that at least one of her pupils will be unlikely to tell the difference without trading them for violets.

The Significance Parameters: In a faithful description for a course of action X, a significance parameter only includes what else the person is intentionally doing in doing X. Outcomes may occur that are not related to the significance X has for the person. These would be indicated in an extended achievement parameter. The above significance chain applies to Jo. It expresses her hopes for the achievement of her pupils, altho she expects that this will differ for different pupils. A significance parameter for a pupil might include components that are similar to any or none of these. However the outcomes Jo has in mind are more likely to occur to pupils who explicitly find them significant. This is especially the case if (1) and (2) are significant for a pupil. Some of the other outcomes may then occur without any particular intent on the part of a pupil.

In general, students who consider the personal educational significance of their studies are likely have a richer learning experience. In first looking at the derivation of the quadratic formula my own S‑parameter included having solved all quadratic equations. In remarking to a classmate that we had now solved all quadratic equations to a classmate, I discovered that this had not occurred to her. Evidently in looking at this derivation we had different S‑parameters. In fact most of my mathematical studies involve significance parameters beyond the task at hand. As soon as I heard about Gödel’s Incompleteness Theorem I knew that my study of mathematical logic was intentionally preparing me to understand that result. In contrast doing bookkeeping assignments in secondary school had a different S‑parameter. I got them out of the way as quickly as possible so I could get back to something interesting. This may be a more typical S‑parameter in relation to school work. On multiple occasions I have observed that in doing assignment, the S‑parameter seemed to be getting it out of the way.

Jo’s Teacher Characteristics: Jo used characteristic categories to formulate and organize a detailed cluster of interrelated characteristics that she considered relevant to her behaving consistently as a contructivist teacher. When she initiated this process she only expected to make explicit what she already implicitly understood about herself. She was surprised to find many characteristics that she had not even implicitly recognized. She also found that there were some that she wanted to modify and

new ones that she wanted to cultivate. Furthermore she realized that this project would have a major impact on her strategies and tactics as a teacher. Since her complete formulation is useful primarily to her, the sketch below gives is only a small sample. I give it mainly to illustrate what such a formulation it might contain. To see the value of such a formulation, a teacher would probably find their own more useful than the one I have imagined that Jo made for herself. However looking at characteristic formulations by other teacher might be useful, and I would certainly welcome a few such formulations and add them as supplement in the next edition of this book.

Traits: I am creative and flexible and patient. These traits are exhibited in various ways in my role as a teacher. I continually look for a variety of questions and problems that she thinks will be appropriate for the present characteristics of pupils and will help them further enhance their characteristics. I wait for them to work out for themselves what they can. I expect pupils to acquire relevant characteristics in their own way and in their own time, and see my role as providing a state of affairs that is conducive to this process.

Attitudes: I care about pupils and listens carefully to them. I enjoy communicating about concepts and their applications. I enjoy their questions and comments.

Interests: I am interested in expanding my own understanding of a variety of realms and in teaching about them. I am interested in people and their behavior potential, and especially in all of their characteristics that are relevant to learning. I am interested in concepts, and in all the ways a concept can be acquired.

Styles: I am non‑authoritarian, suggesting rather than demanding. I am interactive, and I encourage both student‑student interaction and student‑teacher interaction. In group interaction, I facilitate. I prefer saying less than most of the rest of the group, inserting comments judiciously or answering questions often with a suggestion rather than a definitive answer.

Abilities: I know how to listen and discern, to formulate questions that are relevant to the students’ current characteristics. I can anticipate pitfalls and formulate their countermeasures.

Knowledge: I know relevant characteristics of my students. I know mathematics in depth and can look at concept from several points of view. I know various applications, but need to know more.

Values: I value multiple approaches to problems solving, I value finding new methods both in my own thinking and in that of students. I value student success and student creativity.

Mathematicians Characteristics: Jo has a cluster of interrelated characteristics that are shared by many mathematicians. These characteristics make them especially competent in learning mathematics. They relate to some of the concept parameters, involving the dispositions and powers sketched below.

Traits: They live comfortably in an ideal world of highly connected concepts made precise by axioms and definitions and bearing no particular relationship to anything outside of the world of mathematics.

Attitudes: They enjoy looking for general results. For instance after finding a few even perfect numbers mathematicians conjectured that all perfect number are even. Considerable effort has been applied towards proving this or finding a counterexample. A person remarked that given two examples, a mathematician generalizes. A mathematician replied that one example will do. Another mathematician remarked, who needs an example?

Interests: They find mathematical abstractions and ideas of interest in their own right, with this interest related primarily to conceptual and esthetic reasons. The fact that nobody can see why knowing if all perfect numbers are even has any practical applications does not make the search for an answer to the problem of determining if there is an odd perfect number less interesting.

Styles: Their preferred style of presentation tends to be of the theorem‑proof and definition format, with some examples thrown in. This is also their style when organizing their own mathematical understanding. Their style used in obtaining the understanding is much more intuitive.

Abilities: They can acquire and connect concepts that most people find remote merely by being exposed to a clear analytic presentation of them, and they can enhance their understanding of these concepts by constructing examples and problem which add synthetic components to these concepts.

Knowledge: They have an understanding of mathematics and a knowledge base that makes a multitude of concepts seem manifest and that allows newly acquired concepts to be easily connected to those already acquired.

Values: They value mathematical concepts and results without regard to what most persons consider as utility. They also value precise analytic reasoning.

My Personal Perspective on the C‑parameter and Learning Mathematics: These characteristic not only influence how I learn but how I act as an educator. I have a tendency to design resources that are most useful to others having similar characteristics. I must reflect on the fact that most pupils do not share these characteristics. These same characteristics can be integrated into designing constructivist resources that are more broadly useful. One thing I try to keep in mind is that my style in organizing my mathematical understanding is not the main style I use in constructing new portions of my net for mathematics. The style I use then is flexible and multifaceted. I create a multitude of examples and explore many questions. I think about intuitive connections and relationships. Essentially, I focus on what might help those with different characteristics also cultivate some of these characteristics.

Perhaps the key disposition for learning mathematics is the trait of being comfortable in the world of mathematics, and closely related is the disposition to be interested in mathematical activities as exercise in thinking. Without this, the other characteristics are unlikely to develop. One way to help a person acquire this trait is to suggest a variety of mathematical activities that they may find manifest and conceptual, but also somewhat challenging. Note that I said suggest, rather than prescribe. Choice is important, for without interesting options, comfort and positive attitudes are less likely to occur. Nor are the powers that enable the construction of mathematical concepts likely to be enhanced. While I should realize all of this without using PNDP, having the C‑parameter organized as it is gives me a tool for being more systematic about reflecting on and guiding a course of action.

The C‑parameter is also a tool for thinking about the differences in learning mathematics. Since my main style when doing mathematics is algebraic, I must be careful to recognize that potential learners may be less comfortable with this style. I used Bob and Jan in my books to make them appropriate for those with visual or numerical styles. The fact that many persons have an algorithmic style is why I included Roy. This is especially significant for me, because many persons have acquired a blind algorithmic style rather than a thoughtful one.

Example: That the sum of two fractions with the same denominator is the sum of their numerators over this denominator follows directly from the distributive law, at least if you have the mathematician like characteristics indicated above. There was a time when I would have considered this as the way to understand adding fractions this way. Keeping styles in mind, Understanding Fractions for Adults does not even mention this perspective. It leaves this to Understanding Ordinary Algebra for Adults. Instead, I indicate visual and counting perspectives. Another activity would be to suppose we decided to add fractions by adding the numerators and adding the denominators. This can be related to something manifest. If you made 7 out of 10 free throws and then made 4 more out of the next 5 then you have made 11 out of 15. Does this meet the usual intent for adding fractions? Given this way of adding fractions what would we get if we added a fraction to itself, say ¾+¾?

Example: The invert and multiply rule for dividing fractions also follows easily from basic laws, and the reason it works is easy to understand if you have the mathematician characteristics indicated above. Yet it is one of the rules about fractions that most non‑mathematicians do not understand. Knowing this how little this rule is understood, was what motivated me to design alternate ways of dividing fractions that is easy to understand for anyone with a visual style of learning.

SECTION 3 THE PRIMARY PURPOSE OF SCHOOLING, SOCIAL OR INDIVIDUAL

Introductory Remarks: Schooling is socialized education that takes place in any institution that is a school. Schools include elementary schools thru universities. Schooling is so significant in our society that when people talk about education they are usually thinking about schooling, altho on reflection they will usually acknowledge that education is broader than schooling. For many years I acted within a world which stresses the social purpose of education as exemplified in schools. In contrast my attitude towards learning is such that I stress the individual purpose of education. I often wonder about the extent to which acting in such worlds can serve my purposes as an educator. This present section examines this attitude and clarifies my own concepts and strategies. It sketches and then utilizes some concepts that I find useful in thinking about the relationship between individuals and societies. I hope it might prove useful to others with similar concerns.

I have tried to only make paraceptual claims that persons with ordinary information about schools will find highly plausible. Anyone doubting any of these 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 those 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.

The Primary Purpose of Schooling: By the primary purpose of schooling I mean the primary purpose exhibited by the practices within schools. Since this is merely a conceptualization, it cannot entail 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 Willoughby, Accountability Threat or Opportunity, Mathematics Teacher Nov. 1972.

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, hang‑ups; 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 it is not what you’re taught that does the harm but how you’re taught. ‑‑Jerry Faeber, The Student in Society, as essays in his book entitled The Student As Nigger.

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 pupils 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 members of society. Major school practices are directed by this purpose, altho 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 and which you are expected to make available to others. Why? Because it is not primarily for you. While most educators favor helping individual pupils, they often view this as doing what is good for the pupil, not according to the pupil’s view, but according to socially held views. Schools subordinate individual goals to social goals and to society’s goals for individuals. Serving individuals is an important secondary purpose of schools, supplementing their primary purpose when social and individual needs 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 primarily 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 purposes of individuals, 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 students who show an inclination to take charge of their own education. This is descriptive of my purposes, rather than a normative claim or a recommendation about the purpose of education. 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 presented me with some extremely 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 acts as an origin in a situation to the extent that he or she takes initiative, is active rather than passive, proactive more than reactive, focuses 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 diverse 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 individuals that 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 individual rather than the social purposes of education.

While emphasizing individual purposes can involve conflict with our schooling system and its social purposes, this is not my main problem. My problem is that altho social and individual purposes are often supplementary, energy that could be devoted to individual 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 individual purposes I want to make primary? The appendices focus on strategies which provide a partial answer to this question. For now I merely acknowledge it as a problem.

Some Concepts: I have already used my concepts of the primary purpose of schooling, the social purpose of education, the individual purpose of education, my purpose for education, and my origin ideal. 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 person’s behavior relates to a surface level versus a root level examination of some aspect of the world. Radical behavior is behavior whose want parameter includes a strong desire to get at the roots and whose competence includes the ability to make 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 laudatory or pejorative 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 person’s action relates to changing or maintaining some aspect of the world. Conservative behavior is behavior whose want parameter includes a strong desire to maintain and whose competence includes the ability to make distinctions about actions doing this and 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 progressive behavior, and a person with a noteworthy history of acting that way is a progressive. Again, neither concept is intended in either a laudatory or pejorative sense.

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

¨ Conservative and progressive are often supplementary.

¨ For progressive, 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 aspect of the world.

Altho these conjectures do not suggest a dichotomy between individual and social purposes, it can be important to focus on which purpose is taken as primary. At times individual 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 individual purposes.

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 were concerned with the economic welfare of persons and society and saw these as positively related. 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 progressive experimentation in an environment with strong support systems, where individuals provide the experiments and society provides the support system. This conjecture is based on non‑systematic observations. 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 wellbeing 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 who is selectively progressive I find it extremely comforting.

My personal radicalism leads me to act as a progressive only in respect to certain aspects of the world. In many situations I am essentially conservative. I need a strong support system in order to experiment with being a progressive. 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.

The Role of Schooling: ??

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 experimentation with radical originship or anything else to come from persons, but only in a limited manner. Some persons may cultivate progressive 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 progressive 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 progressive 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. 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 originship would be a major threat to social continuity.

SECTION 4 EVALUATING EDUCATIONAL PROCESSES AND WORLDS

Overview: This section uses concepts developed in earlier sections, including concepts from PNDP, to discuss considerations that could be involved in evaluating educational processes and that might remain unrecognized with less intricate conceptual tools. The discussion will focus on some imagined examples taken from memory of my own educational experience and observations. Currently (2006), I am only engaged as an educator with myself as learner, and it has been about 15 years since I worked with more than on learner at a time. Thus these will only be illustrative of some of the ideas, and they will be oversimplified and lacking in depth. Given a good in depth evaluation that some educator would find useful, I would add it to this section. Before looking at these examples I sketch some of the considerations to be considered.

Since educational processes have development goal evaluation centers on the achievement of goals. Evaluation can occur before or during or after an educational process. Prior evaluation of a process involves considering if it seems likely to achieve its goal or needs to be modified before implementation. Evaluation during implementation may involve deciding if it should be modified or even if it should be abandoned. After implementing an educational process, evaluation can be used to make decisions about future processes and whether to modify developmental goal.

The ability to achieve developmental goals and evaluate the achievement depends at least on the extent to which the goals are in focus. It also depends on knowing what factors may interfere or be supportive of these goal. This in turn depends on the present characteristics of the pupil, including not only dispositions and powers, but also derivatives. Since educational processes usually occur in a broader developmental context, evaluating a process may involve examining its goals to see if they seem to be compatible with and supportive broader the goals. In the study of fractions by the CS group, Jo decided that an educational process that involved introducing negative rational numbers would probably interfere with the goals involved in their study by focusing on considerations that might be distracting. Had one of her students raised certain questions, she might have involved them in such a process. The choice of processes depends on developmental goal and these need not be static.

Furthermore almost any aspect of a pupil’s world or worlds can be interrelated with the characteristics that to be developed by an educational process, and so the ability to determine which aspect are relevant and how they are relevant can be a major factor in evaluating a process. One obviously important world in this regard is the pupil’s educational world. If this world significantly involves educational community then the way in which that community is part of the pupil’s world is crucial to effective evaluation. However even an educational process involving one person who takes both the roles of learner and educator has an educational world. In such a case, the status of educator is self assigned rather than recognized by an educational community. However the version of that status that the person assumes is still an important aspect in evaluating the educational processes in which the person is involved. A perosn working alone and who assigns himself the status of an unqualified educator, may find that this is a major reason why the educational processes work poorly.

Example 1: This example considers a secondary school American History. The educational processes primarily involved the teacher giving an assignment and letting the students work on it while he read the morning newspaper. He also assigned a term paper and gave a few tests. His development goals for pupils were primarily cognitive, altho he was largely indifferent to how much development occurred.

a cognitive educational community, i.e. the most notable goals involved developing cognitive competence. Except on testdays the teacher class taught by a basketball

Example 2: This example considers a third grade class in the year 1942, in which there was one teacher Ann and about 30 pupils. This was a cognitive educational community, i.e. the most notable goals involved developing cognitive competence. Most of these would be at the knowledge level of Bloom’s Taxonomy or would involve competence in routine cognitive skills. Example include being able to spell a multitude of words, knowing states and their capital, knowing a variety of geographical fact, being able to accurately add columns of multi digit numbers knowing the multiplication tables. As with any such class some effort was directed towards developing other characteristics that would be supportive of the cognitive goals. For instance, pupils were expected to develop a respect for classroom order, and this was partially towards

Example 3: This example involves what I believe are some accurate memories of the first seminar I taught on Galois Theory. Several features stand out. This was a cognitive educational community, i.e. the primary goals were cognitive. However I had a non-cognitive goal that was not only directed to developing other characteristics that would be supportive of the cognitive goals, but was also as important to me as the cognitive ones. Specifically I wanted the pupils to become their own primary educations, not only for the seminar but for the rest of their lives.

Evaluation by the learner.

APPENDIX 1 ADOPTING A GRADING POLICY

Grading Policies: Altho I am opposed to the use of grades, when acting as an educator in a school I found it necessary to adopt some type of grading policy. A grading policy involves standards for various grades and someone to evaluate which standards have been satisfied. I consider only two policies, the one that works best for me and one like those that are widely used. Clearly variations of these and some intermediate types of policies could be considered. The policy that seems to work best for me is one in which a student is responsible for the choice and application of these standards. The main purpose for such a grading policy is to encourage students to take the major responsibility for their own education. This is contrasted with a teacher dominated grading policies. My reasons for adopting a student responsibility policy over a teacher dominated policy are then indicated.

A Student Responsibility Policy: One version of such a policy allows the student to choose any grading standards and evaluation methods that the teacher will allow. This could include effort and or achievement. The term ‘goal’ refers to any appropriate list of achievement goals that is relevant to this realm of study. Recommended competency lists for formulating goal may be suggested by the teacher. Students may use or modify these to obtain personal competencies goals, or they may design an alternate list of similar breadth or depth. These may be revised by the student at any time. For students wanting guidance the following default option can be used or modified. Effort can most easily be related to time spent. The term ‘hours’ refers to any time spent on any work relevant to this realm of study, including time in class. Regular weekly reports, with both the time for the week and cumulative time reported, is sufficient evidence of time spent. The student’s personal estimate as to competence in relation to the personal competency list is sufficient evidence of competence, however the evaluation should be consistent with additional relevant evidence, such as class participation, suggested assignments, other written work, etc. Without the weekly reports or detailed estimate of competence, the responsibility will default to the teacher who will simply make a reasonable conservative estimate of effort or competence on the basis of whatever information is easily available.

A: 120 hours for A or excellent competence in relation to the students goals

B: 105 hours or 90 hours plus functional competence for most of the students goals

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

D or I: less than 60 hours (A grade of F will not be used)

Comments: The above is only one of many possible policies that encourage students to set their own standards, and the extent to which it is a student responsibility grading policy will depend on how flexible the teacher is in deciding what standards to allow. One of my students seriously proposed that her standard for a grade of 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 policies that encourage students to set their own standards would differ from the option I used 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 taking a major responsibility for their own education, these are more likely to be a ‘nobody as responsible’ grading policies. The extreme policy of telling students to just assign their own grade is likely to be such a policy.

Teacher Dominated 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. 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 the one. 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%.

This may be augmented by specifications like the following. Any late homework assignment will have its grade reduced by 20%. The 2 lowest homework grades will not be used in computing the homework grade for the course. No makeup quizzes will be given, but 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.

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

In hardship cases, such as prolonged illness, the teacher may decide that there are factors which warrant exceptions, 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: In order to make considerations easier to present, I shall pretend that SR (Student Responsibility) and TD (Teacher Dominated) are the only options. 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. 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 SR over TD, I have chosen the following for analysis.

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

Background: This claim 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 I give for this claim 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: to minimize time I must spend on tasks extraneous to my goals as educator

P2: to maintain my reputation within university well enough to act effectively

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

P4: to maintain a classroom atmosphere that is supportive to as many students as possible, including those who do not choose to take responsibility for their own education

P5: to act mainly as a resource for students who choose to take responsibility for their own education.

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

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

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

S53: SR make it explicit that a teacher wants students to take responsibility for their own learning.

S54: A number of origin 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 SR. These statements provide support for my claim that TD tends to adversely affect P4, and that SR is compatible with a strategy for accomplishing P4.

S41: At least half of the students are more comfortable with SR than TD, and a significant minority finds SR much more motivating.

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

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, TD offers no noteworthy advantage over SR. This applies only to my decision to adopt SR, and is not intended to be taken as support for a claim that the use of TD by others does not offer such an advantage over the adoption of SR. 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 does not 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, TD offers no noteworthy advantage over SR. Again this applies only to my decision to adopt SR.

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.

APPENDIX 2 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 book. I 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 appendix 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.

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 is one that a person holds at a given time. It is a collection 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 simplistic 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 originship. 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 prevent it going 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. 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 it was (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 homework 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 math. Earlier I had helped design a program in which there was a definite sequence of courses, 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 graduate 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 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 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) Altho 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, etc. 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 current 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 my self 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 trans‑rational, 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.

That one of my 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 originship is to experiment with alternative to ethical behavior, perhaps giving birth to new ethical principle, perhaps living from a more radical perspective. relationship between ethics and an originship is considered in A Personal Net for Conceptual Philosophy.

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 extreme personal focus I favor, but only a milder controlled personal focus 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 personal focus.

I end this section with some comments on having a personal focus 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 personal purposes is not likely to be a shift which I would find supportive of originship. 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 that the kinds 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 educational ideals. 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 my progressive educational 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 that 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 progressive educators. 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 personal purposes for education 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 net for thinking about education. The second is to examine this 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 understanding 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 understanding 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 take an extreme personal focus. I 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 working within a distasteful structure was to be aware of the structure and the fact that it was intimately related to the needs of others, while rooting my own choices in my ideal of taking an extreme focus on the personal purposes of education. 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 a personal focus and social focus 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 to be given standards, and that it may be better to give highly inappropriate standards than to give none at all. Therefore I shall specify standards that I expect might be more useful than no standards at all for the average student in that situation. Furthermore, in spite of the fact that I feel that linking such standards to accreditation 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 Appendix 1.

I have no easy solution to having a personal focus in a world where primary purposes are so often socialistic. It may take eons before a personal focus becomes the head side of the coin in human affairs. Fortunately for me it is not the total impact of my personal focus which concerns me. I try to create a universe which enhances radical originship, 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.

APPENDIX 5: WHY I TEACH MATHEMATICS

Teaching Math as Conceptual Study: 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 conceptual nets ever created. It is powerful both because it is a tool and because it has become an extensive art form. 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 Mathematics in Western Culture gives an excellent introduction to such matters. What interests me most is the way in which a person can 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 studies indicated there was little evidence that taking courses in algebra or geometry increased the student’s 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 that is meaningful to students. Read Carl Bereiter’s 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. Altho most mathematical work draws on an understanding of numbers, 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 many substructures of the ring of complex numbers form 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 utilize some mathematical thinking that involves no use of numbers, because this illustrates how mathematical thinking does not depend on numerical content.

Rings: The concept of a commutative ring is one of the most basic concepts of contemporary mathematics. A commutative ring is a type of mathematical structures that satisfy some of the same algebraic laws those satisfied by ordinary number systems. A commutative ring has 2 binary operation called addition and multiplication, since these operations satisfy many of the same laws of ordinary addition and multiplication. In particular, both addition and multiplication are associative and commutative. A ring has identity elements for each of these operations. It has a unary operation giving additive inverses. Furthermore multiplication distributes over addition.

Potential Topics: There are a number of topics that 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. Of these, I think that either the study of finite cyclic rings or of boolean algebra has the greatest advantages for an initial perspective. In any study of mathematics, I also recommend a number of books and articles on the history and nature of mathematics. The Newman volumes on the world of mathematics contain some excellent materials that could be used to supplement either a beginning or follow up study of mathematics as the rigorous exploration of conceptual nets.

Finite Cyclic Rings: For any natural number n, the integers modulo n is a commutative ring whose set is {0,1,¼,n‑1}. These are the finite cyclic rings, i.e. the elements can all be obtained by adding 1 a finite number of times. I illustrate this for n = 6, and show an application of this structure to the solution of a simple puzzle. Below is the addition table for this ring. The multiplication table can be obtained from the addition table using the ring laws. For example, since 2 = 1+1, by the distributive law and the identity law, we must have 2·x = x+x; which can be seen as follows 2·x = (1+1)·x = 1·x+1·x = x+x.

	+	0	7	6	5	4	3
Picture these numbers arranged as in a clock. To obtain a+b, start at a and move b positions clockwise For example, 2+3 is 5, but 4+3 is 1. We call this operation addition because it satisfies the same basic laws that addition satisfies for the integers. The table 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

Comment: The use of numerical names for the elements in a cyclic ring is a conceptual convenience. Among other advantages, it makes it easier to focus on both the similarities and the difference between these rings and the ring of integers. However the elements in these rings are not numbers in the usual sense of being applicable to quantities in the usual sense. Note that 2·3 = 0, something that we would not expect to be able to apply to quantities. However we also use numbers for positional considerations, and adding in a cyclic ring can be related to positional considerations as in the puzzle below.

The Tea Party Puzzle: Ms Army, Ms Banjo, Ms Clive, Ms Dumont, Ms Elk, Ms Fish 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 Fish sat between the realistic and the slim one. Ms Elk 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 0 as 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 +. That slim is left of unreliable gives s = u+1. That Banjo is opposite unreliable gives b = u+3. The translation of between is subtler. Check some instances to see that a is between d and b implies d+b = a+a. The tea party information gives the following equations.

b = u+3, p = c+3, q+u = c+c, s = a+3, s = u+1, p = s+1, r+s = f+f, j = e+1, j = t+3

Since we chose s = 0, these equation give u = 5, p = 1, r = f+f, 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 = 4 and r = 2, f = 1, j = 1. This means that Jan is Ms Fish.

Differences Between the Integers and the Integers Modulo 6: As remarked above 2·3 = 0, so the integers modulo 6 has a pair of non‑zero elements whose product is 0. This prevents this ring from satisfying a basic cancellation law satisfied by ordinary numbers. To see this note that 3 and 0 are different elements of this ring and that if we multiply both of them by 2 we obtain the same answer. For another example of the failure of cancellation, 3·1 = 3·5 but 1 ¹ 5. Altho the numerical names used did not include ^‑1, the element 1 does have an additive inverse, namely 5. So we can also name 5 as ^‑1. It is easy to show that 5·5 = 1. So this gives an example a ring in which ^‑1·^‑1 = 1, but in which ^‑1 is not thought of as a negative number.

Reference: For a beginning study of finite cyclic rings, I recommend The Theory of Remainders by Andrea Rothbart. This book focuses on the study in a way that challenges the students to adopt a contemporary perspective and to do some significant thinking from this perspective. However it accomplishes this without requiring the student to have much mathematical background. Finite cyclic rings are presented as algebraic structures in their own right, rather than from the older congruence perspective. These rings 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 the multiplicative cancellation law fails in some of these rings, the presence of zero products without zero factors can be contrasted with their absence in more familiar structure. This is related to finding roots of equations in both types of structures. That these structures are not ordered rings provides 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 finite rings 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 these rings. This illustrates the morphism concept and use of it 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 nets. 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 a students perspective on mathematics so they would see mathematics as a specialized kind of conceptual study.

Boolean Group: A commutative group is a structure with one binary operation sometimes denoted as +. This operation is associative and commutative. There is also an identity element for this operations and a unary operation giving inverses. A commutative group where each element is its own additive inverse is called a boolean group. Altho I have used numerical names for the elements, they do not represent quantities or even position in any usual manner. When first invented boolean structures had no practical applications. They now have many applications.	+	0 1 2 3 4 5 6 7
	0	0 1 2 3 4 5 6 7
	1	1 0 3 2 5 4 7 6
	2	2 3 0 1 6 7 4 5
	3	3 2 1 0 7 6 5 4
	4	4 5 6 7 0 1 2 3
	5	5 4 7 6 1 0 3 1
	6	6 7 4 5 2 3 0 2
	7	7 6 5 4 3 2 1 0

Nim: Conceptual study of boolean structures does not depend on the existence of any application, altho such applications can motivate the conceptual study. A variety of applications of boolean concepts can be found in the boolean algebra section of my website. Here I give a trivial application to Nim. Nim is a game for 2 players who alternate turns. It uses 3 rows of items. One version begins with rows of sizes of 3 and 5 and 7 items. On a turn a player selects one of the 3 rows and removes 1 or more items from that row. The player to take the last item from the last non‑empty row is the winner.

With any position in a Nim game assign an element of the ring by taking the numbers in each row and adding them using this table. For example, the ring element for initial position (3,5,7) is 3+5+7 = 2 and for the position (3,5,6) it is 0. To win at Nim always leave a position whose ring element is 0. Why this works, how it was discovered a more efficient naming scheme can be found in the resource entitled Nim Groups. My purpose above was merely to illustrate an algebraic structure in which addition does not have much to do with the quantitative or positional ideas usually associated with addition.

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 can be used to show that the middle number must be 4.

Adding the second column and the second row and the two diagonals gives 48.

The middle number gets used 4 times while each other number is used exactly once.

This gives 36 plus 3 times the middle number, so the middle number is 4.

Since the only ways to use 0 in a sum to make 12 is with 4 and 8 or with 7 and 5, we can also deduce that 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. Using symmetry, there are exactly 7 more magic squares, which can all be obtained by rotating and flipping the one above.

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.

^‑4

^‑3

^‑2

^‑3

^‑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 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 introduced to a double magic diamond. The challenge is to fill the circle with the numbers from 0 thru 13 in such a way that each connected set of three squares has the same sum. I used my understanding of algebra to discover such an array and some interesting facts about it. I have a partially developed resource on the exploration of magic arrays.

Attribute Logic: My book A Manifest Approach to Mathematical Logic used the set of 8 attribute items pictured below. Items vary by: size, color, shape.

l u l u l u l u

Suppose that one of these items was hidden in a box and we had the following clues.

Clue 1. If it is small then it is a diamond.

Clue 2. If it is circle then it is red.

Clue 3. If it is blue then it is large.

Clue 4. If it is large then it is a circle.

Clue 5. If it is a diamond then it is blue.

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. Clue 2 eliminates the 2 blue circles, one of which was also eliminated by clue 1.Continuing, we find that the only remaining item is the large red circle.

Clue 1 eliminates the small circles.	l	u	l	u	u	u
Clue 2 eliminates the blue circles.		u	l	u	u	u
Clue 3 eliminates the small blues.		u	l	u		u
Clue 4 eliminates large diamonds.			l			u
Clue 5 eliminates red diamonds.			l

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 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. It also shows the conceptual effect of each clue and indicated that the order in which they are used will not affect the answer.		lbc	lrc	lbd	lrd	sbc	src	sbd	srd
	Clue 1					X	X
	Clue 2	X				X
	Clue 3					X		X
	Clue 4			X	X
	Clue 5				X				X

Manifest vs Remote Reasoning: The solutions above both used highly manifest reasoning. A somewhat more remote analysis can be used to focus on extracting information without immediately think about the items involved. Using clues 1 and 5, we can infer that if it is small then it is blue. Using this with clue 3 we see that the item cannot be small. Since there are only two sizes the item must be large, and thus by clues 2 and 4 it must also be a circle and red.

Like any mathematical reasoning abstraction 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 more remote reasoning just used has that feature. For simple puzzle this is at most a minor advantage. However developing alternate methods for simple puzzles makes it possible to imagine various options for extending the ideas involved to more complex ones.

Ideographic Language: As remarked earlier, many people think of symbols when they think of mathematics. This is an important part of mathematics, but primarily because remote analysis would be extremely tedious without using this type of language. Use of ideograms also helps the novice to avoid extraneous connotation from ordinary language. However ideographic language is a mathematical tool rather than the core of mathematics. This tool allows for precise compact representations that focus on the aspects of the information that are relevant to our reasoning. The deduction below for our puzzle illustrates the use of ideographic language. Altho such language is not the essence of mathematics, it can often extend the power of our thinking. ‘&’ 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

(6) S Þ B (1)(5)

(7) ØS (6)(3)

(8) L (7)

(9) C (8)(4)

(10) R (9)(2)

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

There are also more complicated puzzles about attribute items involving relations and variables. In a situation involving 2 different items i and j, suppose we knew had the following information:

(1) i is large. (2) j is a diamond. (3) They differ in color. (4) They have the same size and shape.

It is apparent that they are the large red diamond and the large blue diamond. However there are a variety of such puzzles where the solution is not so obvious.

Manifest Analysis Manifest analysis is tedious. One way is to make a table for all possible ordered pairs of items, placing an X on the diagonal since the items are distinct. The first clue eliminates half of these pairs as indicated. The second eliminates half of the remaining ones. The third leaves only 6 pairs. The last clue then leaves only (lbd,lrd) and (lrd,lbd).		lbc	lbd	lrc	lrd	sbc	sbd	src	srd
	lbc	X	3	2	4	2	3	2	4
	lbd	2	X	2		2	3	2	4
	lrc	2	4	X	3	2	4	2	3
	lrd	2		2	X	2	4	2	3
	sbc	1	1	1	1	X	1	1	1
	sbd	1	1	1	1	1	X	1	1
	src	1	1	1	1	1	1	X	1
	srd	1	1	1	1	1	1	1	X

More remote reasoning could be given as follows. It is essentially the same as the manifest reasoning, but it focuses only on the items to be found and their attributes, ignoring those that do not satisfy the clues.

(1) iL (2) jD (3) ØiCLj (4) iSZj & iSHj clues

(5) iL&jL (1)(4)

(6) iD&jD (2)(4)

(7) iB&jR Ú iR&jB (3)

(6) (i = lbd & j = lrd) Ú (i = lrd & j = lbd) (5)(6)(7)

An Attribute Game: This game uses attribute tokens along with 6 labels: {S, L, R, B, C, D}. The game is played on a board laid out with 2 intersecting circles that 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.

An Attribute Game: This game uses attribute tokens along with 6 labels: {S, L, R, B, C, D}. The game is played on a board laid out with 2 intersecting circles that are used as a Venn diagram. 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. For example, if X is red and Y is small the partition would be as indicated. The game rules are not relevant to the comments I am about to make. For anyone interested these rules are given at bottom of this page.

Imagine two labels X and Y turned upside down. Suppose we know that src goes inside X and outside Y. We can deduce that lbd belongs outside X and inside Y as follows.

lbd differs from src in all 3 attributes.
Since src is inside X, lbd cannot be inside X.
Since src is outside Y lbd cannot be outside Y.
Thus lbd is outside X and inside Y.

Altho the reasoning makes no appeal to numbers, mathematicians would agree that it was mathematical. The 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.)

Note: I have used this game in a variety of workshops and few persons have observed that given the placement of a piece the placement of a piece that differs from it in each attribute is determined. Furthermore, an optimal strategy is rarely discovered prior to my asking questions about strategies.

Attribute Game Rules: The game involves teams A and B, with 2 to 4 people on each. To start, Team A selects 2 labels of different attribute types and places them face down beside X and Y. Team B chooses any token. Team A must put this token in the correct set. Team B tries to place the remaining tokens in the 4 indicated sets and to identify the labels. On each trial, Team A either verifies the choice or removes the token if it is in the wrong set and gives it back to Team B. Team A receives one point each time they return a token. If they return a token incorrectly or allow it to be placed incorrectly, they forfeit the game. Team B may try a rejected token elsewhere or try a different token. Once Team 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 The teams then reverse their roles and play is repeated. If there is no forfeit, the team with the most points is the winner. For a more advanced game use 3 circles and a larger set of attribute items and labels.

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: 0ú¾¾¾¾¾¾¾¾¾¾) 1

There is a standard way involving the idea of a limit to conceptualize infinite bit strings as names for such numbers. One day, having nothing better to do, I invented a 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 try 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 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 that Fred can reach in a finite number of jumps. What makes Fred supernatural is that he can make an infinite number of jumps in an hour, if he needs to. In order to code what Fred does, we use a one to indicate when Fred jumps and zero when Fred rests.

Example: Fred wants to get to the 9/16 ft position. Fred jumps in the first 1/2 hr, landing at the 1/2 ft position. Fred does not jump in the next 1/4 hr, since this would put Fred at the 3/4 ft position, which is past the 9/16 ft position. 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 ft position. 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 100100000¼, which can serve as a name for the 9/16. In a similar manner the bit string 11100000¼ indicates that Fred went for the 7/8 ft position.

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 rests 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 rests 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 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 materials 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 that 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 incorrect paraceptual opinion, 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 a 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 information. 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 mathematics. 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 conceptual study 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 conceptual study also makes it easier to understand why we decided to let 0! = 1 and a⁰ = 1, as well as a variety of mathematical ideas that students often resist because of their naive attitude towards mathematics.

I discussed the difference between 19^th and 20^th century mathematics with several mathematicians. They agreed in principle with the conceptual versus paraceptual distinction that I 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 being used 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 conceptual study 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 that could be used by any psychological theory, much as mathematics provides a net that can be used by any theory in physics. In developing this net, Ossorio uses conceptual methods that 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 that 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.