CONSTRUCTIVIST LEARNING RESOURCES
John Pais and F Richard Singer III edited 3/2005
mailto:richardsinger3@sbcglobal.net website: conceptualstudy.org
mailto:pais@kinetigram.com website: kinetigram.com
Overview: This paper has two equally important parts, the main text with supplements and the appendices. They are related and can be read in either order. The main text in this paper makes a distinction between various resource types that apply both to the design and use of constructivist learning resources. Our perspective on what is involved in concept construction and how it occurs will be briefly discussed. This will be related to several parameters for thinking about concepts and how these parameters relate to these resource types. The main parameter we will use is the experiential parameter which relates a concept to its proximity to ordinary experience. We also relate constructivist learning resources to the significance and achievement parameters from the Descriptive Psychology behavior description concept. The appendices illustrate three of these resource types in terms of some mathematical activities. These activities are directed towards some favorite problems of mathematical hobbyists, namely finding magic squares. These problems are easy to understand, but many related questions are not easy to answer. The activities will provide ideas for exploring them. More significant, these activities may help the learner construct an expanded concept of a mathematical function whose proximity will seem manifest rather than remote from his or her experience.
Resource Types: For easy reference, some types of constructivist learning resources are indicated below. Altho many learning resources can be classified in terms of these types, they are not intended to be exhaustive.
Ordinary Type: Activities that are likely to be encountered as we participate in various social practices involving some realm of interest and that focus on doing rather than on learning.
Group-plan Type: Plans designed to provide flexible group activities that involve some cluster of conceptual distinctions and that will use situations that are likely to enhance concept construction
Group-descriptive Type: Accounts of interactions of groups engaged in constructivist learning activities that were designed to focus some cluster of concepts and conceptual distinctions
Dialog Type: Fictional dialogues in which groups engage in constructivist learning activities that were designed to focus on some cluster of concepts and conceptual distinctions
Individualized Type: Materials that provide individualized activities and exercises designed to focus on concepts and conceptual distinctions, and that a variety of targeted learners are likely to find manifest enough for some concept construction to occur
Expository Type: Expositions that primarily present concepts and conceptual distinctions as the designer thinks they can be understood and related to other concepts by most targeted learners
Note: All caps verbs such as ‘IS’ will be used to focus on purely conceptual distinctions. You can think of ‘IS’ as short for ‘is conceptualized for the purpose of this paper in such a way that’.
Concept Construction: A conceptual net IS a collection of related concepts and conceptual distinctions that can be used to think and communicate about some realm of interest. Personal concept construction IS any modification by a person in any conceptual net used by that person. This may involve learning new concepts, but it primarily involves the modification and expansion of the conceptual distinctions and conceptual relationships previously used by the person. Personal concept construction occurs as a person learns to make conceptual distinctions and apply conceptual relationships by participating in courses of action in which these play a part.
Shaping a conceptual net for some realm of interest occurs in conjunction with the development of competence in relation to that realm. This involves understanding factual information, and the acquisition and use of such information is often more the center of our attention than is concept construction. Constructing an adequate conceptual net for almost any realm goes hand in hand with becoming adequately competent in relation to that realm. However having a conceptual net is only a small part of what it takes to become competent, and without competence, an adequate conceptual net is unlikely to emerge. With competence, concept construction may occur without deliberate effort. Thus, constructivist learning resources may place their primary focus on the acquisition of competence.
Study & Concept Construction: Study IS any activity that focuses on understanding of some state of affairs or realm of interest. (This differs from the common use of the word study, which is often applied when the desire to understand is absent.) At times, understanding a state of affairs or a realm of interest may depend heavily on concept construction. Even so, the majority of our concepts are acquired without deliberate study. They often emerge as they are needed to participate in commonly occurring social practices. This is the way we acquire most of the concepts for our ordinary realms of interest. However ordinary realms can be used to deliberately enhance concept construction.
Example: Casino is a game in which you can use a card to capture cards with that sum. Rob taught his daughter Jan to play when counting was her only number concept. He was preparing her for enjoyable activities in which she would acquire basic arithmetic concepts. 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 into a deliberately designed constructivist resource for that purpose. This was a way to easily transform an ordinary activity into an effective constructivist learning resource. Jan used this resource with her children for the same purpose. Rob's sister also used it with her children, even modifying the rules in order to enhance its numerical concept construction potential. In so doing it even became a more interesting game, which is occasionally played just for enjoyment. 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.
Fiction: The concept of a social practice includes any type of activity learned thru interaction with others. Reading a novel is a solitary social practice in which a reader may vicariously acquire concepts without deliberate study. In The Parrot’s Theorem by Denis Guedj (St. Martins Press), the characters become engaged in an amateur discussion of mathematical concepts because of other aspects of their life. Mr. Ruche is a bookseller with little mathematical perspective. He receives an extensive set of historical books on mathematics from an old friend. Ruche, along with his assistant and her children and a parrot, must learn some basic mathematical concepts in order to organize this collection of books. The need to learn concepts also emerges in an attempt to find clues in the history of mathematical concepts in order to solve the novel’s central mystery. Altho the mathematical discussions tend to overshadow the story in places, this novel could be a useful ordinary constructivist learning resource for any reader who finds the story interesting.
Achievement and Significance: Descriptive Psychology is a network of theory neutral conceptual tools. Thus it does not compete with any psychological theories. One of these tools is a parametric formulation of a behavior description concept. A behavior description IS given by an observer for a course of action X by some person P. Parameters for this concept are developed briefly in Supplement 2, which also relates some of them to concept construction. The main text only makes direct use of the achievement and the significance parameters. The achievement parameter in a faithful description may include anything that could reasonably be considered as a noteworthy outcome of X, including unintended ones. A description of X can indicate not only the most direct features of X, but also the significance that X has for P. To consider this parameter the observer asks, "In doing X what else is P intentionally doing?" This can sometimes be organized in terms of one or more chains. This means that a link in the chain entails in some fashion also doing the next link. Further links may or may not be more significant than nearer ones. They are just less directly a part of what is intentionally being done.
Example: Let X be Ray taking a walk in the woods. In doing X, Ray is getting his allotted aerobic exercise for the day. Doing this will help him keep his blood sugar under control. By keeping his blood sugar under control, Ray hopes to prevent some serious health problems from arising. This he hopes could enhance the quality of his life in later years. This significance chain for what he is doing is part of the significance parameter for describing X. In doing X, Ray is also learning the territory near his farm. This part of the significance parameter is not in that chain. His primary achievement is completing his walk. The extended achievement parameter also includes meeting his exercise quota and locating some cedar trees he can later transplant. His blood sugar level was under 120 the next morning. The broader health and life quality achievements are unknown. The achievement also had a negative component. While crossing a creek on a log, he slipped and soaked his walking shoes.
Concept Construction and Behavior Description Parameters: Altho much of the activity during which concepts are learned has a major linguistic component, this component seldom focuses on explaining concepts. For most activities in which concept construction occurs, the construction will be part of the achievement parameter without being part of the significance parameter. Even in most study activities, understanding of the state of affairs is central and concept construction is either absent from the significance parameter or is at least one link out on a significance chain. The main exception to this is in study episodes where the state of affairs to be understood is a network of conceptual distinctions and conceptual relationships. However an understanding of conceptual distinctions and conceptual relationships is often acquired by study activities centered on using, rather than on constructing concepts. Having concept construction as explicitly a part of the significance parameter is one way that concept construction can be more effective. For instance, a law student must use legal concepts to study cases. Altho the study centers on the cases, a deliberate focus on the understanding concepts may enhance the understanding of the cases as well. If this is significant for the student then fruitful concept construction is more likely to occur.
Example: Al learned a ‘maul’ concept by hearing Ray call a tool being used to split a log a maul. Altho the only course of action directly involved was one simple act of observation, his previous firewood activities prepared Al to automatically integrate this concept into his conceptual net for thinking about preparing firewood. The effort involved in constructing his concept of a maul was so insignificant that Al was not aware of constructing a new concept in this net. Had Ray not used a name for the tool, Al would still have constructed a new concept for this net, altho he might have asked what it was called. Even so, Al could have learned this concept and this word without awareness of doing so.
Example: Ann learned most of the concepts for the games she plays by a brief study episode followed by playing them many times. Study episodes involved an ordinary resource consisting of a sample demo game. She acquired concepts without using a deliberately designed resource such as a sheet of rules. In learning, a major part of the significance was becoming prepared to play the game. Altho playing the game involved additional concept construction, most of the time this was not an immediate purpose. It occurred without even being part of a significance parameter, altho at times she would play with the intent of enhancing an understanding of concepts. Altho Ann acquired these concepts without using a deliberately designed resource, she always liked having a sheet of rules to help bring the concepts into sharp focus. A sheet of rules also allowed her to learn games without needing to find someone who already knew how to play the game. This was how she acquired the concepts for poker, thru study involving a deliberately designed expository resource. She used an encyclopedia to find the rules.
History is a realm that Ann finds of interest. Shaping her conceptual net for history involves familiarity with a large amount of historical information and how it relates to the culture of various historical periods. Ann cannot go back and participate in their social practices. She can read authentic historical novels that are rich with cultural information. Since reading such novels is a hobby and since reading is at least a limited way of interacting with the author, the potential for her to read historical novels is an ordinary constructivist learning resource. She supplemented this by reading from history texts or by looking up historical materials in an encyclopedia. Since reading such materials is a commonly occurring activity for her, the potential to read them as an ordinary resource. The fact that the major focus was informational is merely another case in which concept construction occurs without a deliberate study of concepts. In both reading historical novels and in reading about historical information, concept construction was never her most immediate purpose, and most of the time it was not even a link in the significance chain for such activities.
Concept Parameters: Various parameters can be used to think about concepts and how they are acquired. Each parameter involves a relationship between a person and any concept that person can consider. Such relationships can change with time, but for the most part, they change slowly.
- experiential parameter: indicates how a concept relates to various types of a person’s experience, the proximity of a concept to specific features of experience that a person finds easy to identify
- explanation parameter: indicates the way a person could or would be able to explain a concept
- integration parameter: indicates the ways the concept fits within a person’s conceptual nets
- precision parameter: indicates the ways a person understands the precision of a concept
- utility: indicates both the scope and types of uses a person has for a concept.
Example: Al’s maul concept illustrates these parameters. A more detailed general description of these parameters is given in Supplement 1. In relation to the experiential parameter, Al’s ‘maul’ concept easily relates to experience he finds easy to identify. In fact it was constructed directly from a single such experience. When he later saw a monster maul, he made a slight modification in this concept, but the modified concept remained highly manifest. If Al were ask to explain what a maul is. He would tend to explain it analytically as a tool designed in a certain way to split logs. He would also be inclined to add a synthetic component to the explanation by showing a maul and demonstrating how it is used. The is typical of the way we explain our concepts. This concept was easily integrated into his conceptual net for firewood. Specifically he easily related mauls to other tools for splitting logs and to concepts for thinking about what tools to use. His maul concept is clearly in focus and he understands it precisely enough to say of almost any object whether or not it is a maul. It utility is narrow, related only to splitting logs. However for his firewood network it was a highly useful addition.
Example: Al’s concept of a billion is also highly manifest because he had been fascinated with numbers since early childhood. Unlike his concept of a maul, Al’s it was constructed thru simple but deliberate study whose most noteworthy purpose was concept construction. Apart from a brief linguistic interaction with his father, most of the construction was unaccompanied by others. His first ‘billion’ concept was purely numerical. He simply thought about a billion as a thousand millions and a million as a thousand thousands. Later his concept of a billion changed slightly as he learned to think about powers of ten. Altho Al found purely numerical concepts highly manifest, he did not have a physically manifest concept of a billion. He deliberately decided to relate this concept to something he could visualize. Thinking about billions of people or billions of miles, while numerically easy, did not help make this concept visually manifest. Since a million centimeter cube will form a cubic meter block that would easily fit on a table. A thousand such blocks could be arranged as a cube whose sides were ten meters. Since Al could visualize both this cube and a centimeter block, he now had a way to visualize a billion. What enabled him so easily to think of the sugar cube example was that he found purely numerical concepts manifest. Being manifest depends on which features of experience are in a person’s comfort zone, and most people are not totally comfortable in the realm of numbers. In fact, Al constructed this example not so much for himself, but to help make the concept manifest to others. His concept of a billion is as precise as a concept can get, and is related in a definite way to other numerical concepts such as powers of ten. To explain this concept he would tend to first define it as 109 or as a thousand millions. Unless he was acting as a mentor, he would be unlikely to add anything to this explanation, since he usually thinks of a billion in an almost purely analytic manner.
Bridging: In discussing resource type we will focus on the proximity aspect of the experiential parameter. Proximity can vary from highly manifest to extremely remote. A concept is manifest to a person to the extent that it is close to experiences that the person finds easily accessible. It is remote to a person to the extent that the person finds it removed from such experiences. There is a gap between thought that uses concepts that a person finds manifest and those that this person finds remote. In the course of ordinary living we repeatedly bridge these gaps. Concepts that once would have seemed remote become manifest. For a small boy the concept his mother is manifest, but the concept of mother is too remote to grasp. Yet the concept of mother will emerge as manifest without deliberate effort. How does this happen? He learns that his mother is also his sister’s mother. He learns that his mother has a mother, that his friends have mothers. Gradually his concept of a mother is not restricted to those he knows. The bridge from a highly manifest concept to one that once was too remote to grasp has been bridged. Looking at this bridge we see that it is built from a variety of smaller bridges, each of which has a small enough span between what is manifest to something slightly more remote. Of course how many small bridges are needed and how wide a gap they can span will vary from person to person.
Routinely Acquired Concepts: Since most human behavior involves making conceptual distinctions and since persons have the capacity to do so as they participate in various social practices, most of our concepts for ordinary realms of interest are routinely acquired using resources that were not deliberately designed. We may not even think of ordinary activities as potential resources for concept construction. Personal concept construction occurs not only without deliberately designed resources, it also usually occurs without study. The concepts most likely to be automatically acquired are those that we find immediately manifest and easily integrated into a conceptual net for some realm of interest. They will be close to specific features of experience that we find easy to identify, well enough in focus to be used for the activities from which they emerged, and directly connected to a variety of other concepts we can use. Furthermore, we would tend to explain them more synthetically than analytically, i.e. by relating them to the activities from which they emerge rather than by reducing them to concepts taken as more basic and we may or may not have these concepts in sharp focus.
Deliberately Designed Resources: Since there are limitations involved in this ordinary type of concept acquisition, concept acquisition can be augmented by other ways of deliberately helping a person construct concepts. One purpose of study and deliberately designed resources is to bring concepts into focus and to add analytic aspects to the way they might be explained. Study can also be used without deliberately designed resources to help in concept construction. Such ordinary study is likely to be casual and to occur in brief episodes.
The central feature of any deliberately designed resource is that its main purpose relates directly to a conceptual net that the designer wants learners to acquire. One advantage of such resources is that many of the concepts needed may not easily emerge from activities in some realm currently of interest for the learner. So this more direct approach can be more efficient. This direct approach of using deliberately designed resources in relation to some realm has a major latent pitfall. Altho a realm may be of interest for the designer, it may be a realm of little or no interest to some potential learners.
Group-plan type resources focus on activities that involve oral interaction with other people. Descriptive and dialog types also focus on such interactions, but only as a spectator. They provide a vicarious way of participating with others. Unless used with a mentor, individualized and expository types only involve the learner drawing on a resource designed by another person. Focusing on these types ignores the fact that persons have the power to deliberately shape or modify the concepts and conceptual distinctions that they use without drawing on such resources. They can do so by reflecting on the concepts they use, as Al did when he decided to think about how to augment his concept of a billion by linking it to something that he would find physically manifest. They can draw on educational materials that were designed without any focus on concept construction. Even a description of an algorithm that ignores its rationale can be a resource for concept construction. For instance, some learners who encounter the standard algorithm for long division gain a perspective on the relationship between division and subtraction, altho many do not.
In learning the concepts in his firewood net, Al constructed all of his
concepts via ordinary type resources. In learning his concept of a billion, the
only resource he can recall was one of an expository type. He was told to think
of a billion as a thousand millions. His experiential background was such that
the construction of an adequate concept of a billion followed easily. It was
only much later that he realized he thought of an activity to make this concept
more manifest. This activity could be used as part of a group-plan
constuctivist learning resource on ways to think about large number concepts.
It could also be used in any other deliberately designed learning resource. For
instance it could be used in a purely expository resource in which it was
presented pretty much like I presented it above. Altho we cannot merely
transfer a concept to another person, sometimes the most efficient way to help
someone acquire a concept is a direct exposition telling how you understand the
concept. To make it a more constructivist exposition, we would explain how we
saw that a million centimeter cubes would form a cubic meter block. In an
individualized resource we would do his by asking questions.
Comparing Various Types of Deliberately Designed Resources: The rest of the main text centers directly on deliberately designed constructivist learning resources, with an emphasis on comparing and relating them. This is done in two main parts. One part sketches a resource that centers on activities involving magical squares. As you read this, you may want to examine Appendices 1 thru 3 where these activities are presented in detail. You may even want to study them before reading the rest of the main text. The other part gives a more general discussion of these resource types. Altho we start with the magical square resources because they seem manifest, these parts can be read in either order.
A Group-Plan for Using Magical Squares: The appendices along with a plan for using them, can be used to form a group-plan resource working with magical squares. Below is one such plan.
Read Appendix 1, which is an expository resource whose single purpose is to help a learner expand his or her concept of a function. Discuss the function concept, along with additional examples of functions that you can imagine.
Look at Appendix 2, which is an individualize type resource for exploring 3´ 3 magical squares. Tentatively indicate an approach in working with magical squares that you might want to try.
After taking some time to work alone or with others, present and discuss any results obtained.
Discuss the questions for consideration that are given in Session 2 of Appendix 2.
Read and discuss Appendix 3, which is a fictional dialog type resource giving an account of characters working on magical squares and relating them to what they had read about functions.
Designers’ Significance Parameter: Using a significance chain is one of our ways of organizing a perspective on what we are doing as we design a learning resource. Designing a plan and the related resources for working with magical squares was a joint course of action involving many hours during a period of several months. The significance of what we were doing evolved over this time, and thinking about this evolution had an impact on the resources we produced. There are a number of components of the significance parameter we could mention. A significance chain that had a major impact relates directly to the prospective mathematical outcomes considered for some potential resource users. We were trying to provide a resource that could result in the outcomes below.
(1) an interesting experience working on a mathematical problem
(2) a deeper appreciation of using multiple approaches to solving a problem
(3) more competence in mathematical reasoning and problem solving
(4) an expanded perspective on the function concept used in contemporary mathematics
(5) a slightly expanded concept of mathematics
(6) a foundation for an understanding of the concept of a group of permutations under composition
(7) a basis for understanding how the contemporary concept of mathematics differs from the concept commonly used by most non-mathematicians
Outcomes for Users: In a faithful description for a course of action, 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 us as we designed the resource. It expresses our hopes for the achievement users of this resource, altho it is expected that this will differ for different users. A significance parameter for a user might include components that are similar to any or none of these. However the outcomes we suggested are more likely to occur for users who explicitly find them significant. This is especially the case if (1) and (2) are significant for a user. Some of the other outcomes may then occur without any particular intent on the part of a user. Those who consider their personal significance in using these resources are likely have a richer experience. They might find it useful to look at what we found significant in designing the resource. This was influenced by our understanding of mathematics and the significance that the function concept has in contemporary mathematics. This is sketched in Supplement 3.
A General Discussion of Resource Types: We now turn to some general considerations about various resource types.
Group-Plan Type: The deliberate use of activities in some realm of interest R0 to enhance concept construction for some other realm R1 is one potent way of shaping ordinary resources into group-plan resources for constructing concepts that apply to R1. Doing so provides a deliberately designed resource that can be used to bridge from manifest concepts for R0 to more remote concepts for R1. Such a group-plan activity can often be used largely without any activity whose primary significance parameter for the learner is social interaction rather than study. Such a group plan resource acts in the same manner as an ordinary resource. In either case concept construction can be enhanced if some part of the significance parameter is study.
The common way to design group-plan resources is to develop a cluster of group activities whose main purpose relates directly to concepts for a realm that the designer wants learners to acquire. As indicated earlier, one advantage of this more direct approach is that it can be more efficient, since the concepts needed for competence in a realm may not easily emerge from activities in some other realm of interest for the learner. One way group-plan resources differs from other deliberately designed resources is that they allow concept construction to occur with immediate feedback from others. In this way, they are more like most common social practices. This has the advantages inherent in ordinary resources along with the additional advantage mentioned above. One limitation of both ordinary and group plan resources is that the concepts may not come into focus while using them. This is especially the case when the designer is trying to help learners acquire concepts that need to be precise. Also the analytic aspects of concepts are likely to be ignored, and this is especially a problem with many mathematical concepts whose analytic aspects need to be understood.
Descriptive-Plan Type: A
descriptive-plan resource is normally an account of a group using a group-plan
resource. Such accounts may be in written or other formats. Altho these
accounts may be intended as a resource for educational ideas, they can also be
a resource for the construction of the concepts the group was working with. The
Dialogue Type: This idea
occurred to Richard Singer while he and Andrea Rothbart were developing
materials for teaching concepts and problem solving in elementary number
theory. The book she developed from these materials, The Theory of
Remainders, is a combination dialog and individualized type resource that
includes a fictional account of a couple who discuss mathematical ideas in the
course of ordinary activities. This is because one of the characters is an
amateur mathematician and the other is interested in puzzles. This book has
been used for years at
Unlike the book mentioned above, a dialog resource can be an account of fictional characters engaged in mathematical activities that have no specified relationship to anything else they are doing. These could be characterized as dialog resources that give a fictional account of using a group-plan resource. The development of such resources can be extremely time consuming and difficult in comparison to the development of other types. To directly design a dialog resource, the designer needs to understand various useful perspectives on the concepts and ways in which these concepts can be misunderstood. The designer also needs to think of activities and discussions that might occur and which will bring these perspectives into focus and clarify common misunderstandings. One option is to design an individualized resource for the concepts and then use it as part of a group-plan resource, try it with some groups, and then formulate a dialog resource based partially on these experiences.
Like group-plan resources, interaction plays a major role in dialogue resources. As with group-plan, dialogue has the advantage of efficiency, since it relates directly to a conceptual net that the designer wants learners to acquire. It also has the latent pitfalls of all deliberately designed resources. Unlike a group-plan resource, the interaction involved in using a dialogue resource is vicarious. Altho there is a danger that the interaction will seem artificial and contrived, as in a bad movie or novel, vicarious involvement can be both useful and satisfying. Well-designed fiction takes us into a world we might not otherwise encounter, and it does so in a way that simulates ordinary experience.
Altho the designer structures the interaction in both group-plan and dialogue resources, the vicarious interaction in dialogue resources allows the designer more control. One disadvantage is that it is less like ordinary and may easily miss important perspectives that might emerge from actual interaction. Another disadvantage is that the learner may only act vicariously and not personally engage in the activities being discussed by the fictitious characters.
One advantage of vicarious interaction is that it avoids many interchanges that might be irrelevant or even counterproductive. Thus dialogue resources resemble individualized and expository resources in being able to bring concepts into focus and further the understanding of their analytic aspects, altho both dialogue and individualized do so less directly than expository. To the extent dialogue is designed so the learner is likely to do most of the activities done by the fictitious characters, it can have all the advantages of individualized, along with a greater ability to simulate interaction. One way to accomplish this is to first have the learner use a group-plan or individualized resource that involves doing these activities. The dialogue resource can then be used to enhance the perspective.
Individualized and Expository Resources: Potential resources of these types abound in combination with resources designed for other purposes such as providing information or developing skills. They include parts of most textbooks. They include portions of lectures and explanations given in traditional classrooms. Altho these provide a resource for concept construction with some learners, their designers were often insensitive to this possibility. Some expository resources are not merely parts of materials designed for some other purpose. A rulebook for a game like football is an ordinary example, altho this is not the way most fans acquire their football concepts. The main text of this paper is another example. It is hard to find brief mathematical expository resources that are not embedded in materials designed for other purposes. Appendix 1 gives such a single purpose expository resource. It was deliberately designed to help learners acquire the function concept used in contemporary mathematics. It is constructivist in the sense that it was designed to help the learner bridge to a concept that many find remote from concepts that they are likely to find manifest. It differs from the individualized resource because it does not suggest a flexible set of activities for bridging. The individualized resource in Appendix 2 is also designed with a primary purpose of helping the learner develop an expanded function concept. However it also serves a number of other broader concept construction purposes.
One obvious feature of these resources is that communication is only from designer to learner. Thus the feedback from interaction and the motivation inherent in the desire to interact with others are both missing. Some interaction can be simulated in an individualized resource by having activities that make a variety of strategy suggestions and ask questions followed by possible answers. This is attempted in Appendix 2, with what success is yet to be determined. Partially this depends on whether magical squares can become a realm of interest for the user.
Altho individualized and expository resources are designed so that a learner can use them independently, both can be used as part of or to augment a group-plan resource. Group activities could involve a discussion of some brief expository resource or some brief portion of such a resource. They could include all or some of the activities from an individualized resource along with discussion to form a group-plan resource. This was indicated in our discussion of the resource that we designed for use with 3´ 3 magical squares.
Altho expository resources can be used in conjunction with group activities, they are important primarily because the ability to read and understand these resources provides a wide access to concepts, especially in more specialized realms of interest. Advance texts tend to differ radically from most lower level texts. Altho most post calculus mathematical books are not written from a constructivist perspective, they are actually extended expository resources that focus mainly on concepts as commonly understood by the community of mathematicians. It is harder to find expository resources for lower level mathematics that focus mainly on concepts.
A special problem with expository resources is that most of our concepts for ordinary realms of interest are constructed in the context of doing something that uses them. For most purposes, this is even the way most of us prefer to acquire them. Even people who are fascinated with focusing directly on conceptual nets may still prefer an initial stage where concept acquisition occurs in the process of using them. Even when their primary purpose is concept construction, they may want a heavy dose of concept application, and individualized resources are more likely to supply this need. As mentioned earlier most people prefer to learn a game without the tedium of reading the rules, but serious players want a clear account of the rules to later bring them into sharp focus. For this reason many expository resources should be brief enough to use in a single session or should be structured as components that can be briefly consulted.
One major use of a brief expository resource is to help bring a concept into focus and enhance the ability to explain it analytically. This is especially important for concepts that need to be precise, as is the case with many mathematical concepts. This does not mean that understanding a mathematical concept does not also involve being able explain it synthetically. The ability to state an analytic definition does not automatically mean that the concept has been adequately constructed. A person who can define the concept of an even integer but does not see that 0 is even, does not understand the concepts involved. Because the synthetic aspects of understanding a concept are so important, expository resources are most helpful to a learner who can easily construct this aspect of a concept without external resources. Even then other types of resources can be extremely helpful. One option is to develop an individualized resource accompanied by easy to find brief expository resources. These can be embedded, but separating them into an appendix or a supplement can also be useful.
Purposes For Resources: By an intended user for a constructivist resource we mean a person whose understanding of the concepts involved is less sophisticated than that of the designer. Altho an obvious purpose for a constructivist resource is for intended users, it can serve other important purposes. Suppose a designer finds that a resource does not resonate with intended users. Altho disappointing, the design experience can still be valuable.
Designing a resource can enhance the understanding of these concepts for unintended users, such as the designer or others who already have a sophisticated understanding of these concepts. The very act of designing a resource is likely to influence the integration and explanation parameters for the concepts involved. It may also enhance their precision and even make them highly manifest for the designer. Given a level of understanding comparable to or greater than the designers, looking at a resource can have similar effects.
For the community of constructivist educators, the continuing enhancement of their own concepts might be the most important factor in their ability to help learners construct their own concepts. Not only does this enable the educator to design better resources of each type, it also models the attitude that understanding concepts is valued. This is essential for the educator who wants to establish an environment in which the educator is viewed as a co-learner.
Even a resource that did not work with intended users or was not even tested can be a resource for designing further resources. No resource needs to be regarded as immutable. Having a multitude of resources, organized in relation to conceptual clusters and available in a readily modifiable form, could be a resource for developing resources of various types. Having a wealth of other types would be especially useful for developing and supplementing group-plan resources.
Note by Richard: Most of the resources I developed for mathematical concepts begin as an expository or individualized resource. Understanding Fractions for Adults was first written as individualized resources and then transformed into its current dialogue format supplemented with expository appendices. The only resources I have directly designed as a dialogue resource are those chapters that I have written in Understanding Ordinary Algebra for Adults. Of the nine chapters planned, only a three have been developed. None has been tested with intended users. One of the most extensive resources that I designed is a collection of resources for learning logic. Altho these are an individualized resource with expository components, some have been revised to dialog form as chapter in A Manifest Approaches to Mathematical Logic. I have also designed an extensive resource for boolean algebra. It is intended for secondary mathematics teachers, both as a resource they can use for personal concept construction and for ideas they can use to add perspective to concepts when they teach ordinary algebra.
There are three links in the main significance chain I would normally use to describe my course of action when designing constructivist learning resources. The first link is to enhance the understanding of these concepts for myself. The second is to provide a resource for others who already have a sophisticated understanding of these concepts. It is not until the third link that I am also designing a resource for intended users. Enhancing my own understanding is more manifest, and thinking about the perspective of colleagues is easier than thinking about the perspective of intended users. Recall that later links may or may are not be more significant than nearer ones. They are just less directly a part of what is intentionally being done. Once I have used a resource as it was being developed with a number of learners, the third link usually becomes the most significant.
All of the resources that I have
designed are on my website. I welcome comments. These resources can be freely
copied and used, as long as they are not used for commercial purposes.
For direct links use Ordinary Algebra or Mathematical Logic or Boolean Algebra
The supplements which follow are
revised versions of papers written by Richard
SUPPLEMENT 1: PARAMETERS
FOR CONCEPTS
Notation: P is a variable ranging over the set of persons. C is a variable ranging over the set of concepts that P can use, at least in some manner.
Parametric Analysis: Altho the idea of parameters did not originate in Descriptive Psychology, seeing how parametric analysis has been developed there suggested a wider usage than previously imagined. Their use of parametric analysis motivated the formulation of the parameters sketched in the main text. The most current version of these parameters is developed in more detail in Chapter 1 of A Net for Understanding by Richard Singer. While these parameters are conceptually independent, they are often interrelated in practice. For instance, the concepts that P finds experientially manifest are also likely to be included in those that P uses in a fairly precise manner.
The Proximity Parameter: This main component parameter indicates the proximity of a concept to ordinary experience or to specific features of experience that a person finds easy to identify. A concept IS manifest to P to the extent that it is close to P’s easily accessible experiences. It IS remote to the extent that P finds it removed from such experiences. Altho this parameter is one dimensional in an ordering sense, it is not intended to be subject to numerical analysis. Suppose Jo is a very young girl with a new baby sister Ann. The concept ‘my baby sister’ will be highly manifest for Jo. The concept of concept of a sister will at first be extremely remote, or even totally absent. As Jo gains experience, the ‘sister’ concept will normally become manifest, but probably not as manifest as the concept of my sister. Before the ‘sister’ concept becomes manifest, there are likely to be other manifest concepts involving particular sisters. For instance Jo might know that she is a sister to Ann prior to understanding the more remote ‘sister concept’. The bed Jo uses will be a highly manifest concept of ‘my bed’ for Jo. 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.
The Realm Parameter: This parameter indicates the realm or type of realm to which P applies the concept. Two opposite broad type are called personal and impersonal. A personal concept IS one used to think about persons and their actions. For instance the concept of frugality makes sense only when thinking about a person. An impersonal concept IS one whose use has been influenced primarily by experience 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 concepts that are primarily perceptual, such as red or wet. Some concepts that are primarily functional, such as the concept of a lever, are intermediate. This concept is personal because we think about a person would use a lever. It is impersonal because we classify something as a lever basically because of its potential to produce an effect on a state that is largely impersonal. Furthermore we can recognize something as acting as a lever even when no person is involved.
One of the most manifest personal concepts I can currently imagine is my concept of how I felt as I was waiting to take several shot for a summer camp many years ago. This suggests slightly less manifest personal concepts such as fears. Altho the concept ‘my fears’ seems fairly manifest, the concept ‘fear’ seems more remote, as I am clearly unable to directly experience the fears of other persons. I also have a number of such remote personal concepts. Most of these seem to be synthetic. Examples include my concepts of loving, wanting, feeling, suffering, enjoying.
The Explanation Parameter: This parameter refers to way a person could or would be able to explain a concept. A concept IS analytic for a person to the extent that the person understands how to explain it by reducing it to more basic concepts. It IS synthetic to the extent that a person understands how it can be explained in some other way, perhaps by relating to a multitude of ways that it is used. A concept can be both analytic and synthetic for a person. For instance, my ‘sister’ concept is almost totally analytic because I would normally explicitly reduce it 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. The concept of a chair is partially analytic for me because I would partially explain it as furniture used for sitting. However this does not totally explain my concept because it does not eliminate other types of furniture used for sitting. With effort, I could probably totally reduce a ‘chair’ concept to more basic concepts, but I would be more likely to give a partial analysis with some counterexamples.
This parameter also indicates the extent to which a person P can talk about a concept C. To the extent that P uses C but cannot talk about it, C IS implicit. C IS explicit to the extent that P has terminology for C and can discuss how and when C is used and can be used. Language allows us to become aware of concepts, making implicit concepts more explicit.
The Integration Parameter: This parameter indicates the place the concept has within a person’s conceptual nets. A concept IS disconnected to the extent that it does not relate to other concepts within these nets. It IS connected to the extent that it relates to a multitude of other concepts within these nets. It IS appropriately connected to the extent that its relationships are coherent and correspond to those used in public conceptual nets. It IS adequately integrated if it is appropriately connected and if these connections are those that the person would most commonly need in order to understand and communicate with others. It IS well integrated to the extent that it is appropriately connected and exhausts the reasonable ways in which it can be appropriately connected. Jo’s concept of my 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 sister and brother and male and female and same parents did it become adequately enough 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. It should be noted that being adequately integrated depends on the worlds in which a person participates. A person having no contact with the world of bridge can have an adequately integrated ‘finesse’ concept without knowing anything about how this concept relates to playing bridge. This would be an inadequately connected concept for a person talking to a bridge player.
This parameter also indicates how a concept supports other concepts in a person P’s conceptual net for a realm of interest. In a conceptual net some concepts may be so basic that they provide a substrate supporting each other and all the other concepts in the net, altho often only implicitly. These concepts permeate P’s thinking about a realm, giving coherence to the way in which P thinks about the realm. Since they lie submerged and implicitly support a significant part of the meaning of other concepts used, I refer to them as subconcepts for the net. For P’s ordinary conceptual net, the subconcepts consist of intimately intertwined conceptual strands that permeate P’s thinking and mediate all of P’s experience. No sharp boundary is intended between subconcepts and less basic ones. Subconcepts in P’s ordinary net will be subconcepts for all of P’s conceptual nets, but most such nets will also have other subconcepts. For instance, my ordinary net has a subnet consisting of the concepts used to think about family relationships. The concepts of parent and male and female are subconcepts in this subnet, but not in my ordinary net. There they are supported by other concepts. This parameter also involves support relationships between concepts. My aunt and uncle concepts are supported by my brother and sister concepts, and these are supported by my parent and gender concepts.
The Precision Parameter: This parameter refers to the ways P understands the precision of C. C IS precise for P to the extent that P applies it consistently and coherently without having to allow for indeterminate cases. C IS vague to the extent C is not precise. C IS in focus to the extent that P knows exactly when and how the to use C and also understands the limitations in using C. A major component of this understanding involves the ability to tell to what extent a concept is precise or vague.
As with most of my extremely manifest concepts, the concept of the chair on which I am currently sitting is precise and easy to bring into focus. Because of borderline case possibilities, my concept of a chair is less precise, but still precise enough for most purposes. Part of the precision parameter involves knowing this and knowing what to do if I encounter borderline cases. My concept of a modern art is extremely vague. I would only know that something was modern art if I was told by a reliable critic that it was modern art and I also have only a vague concept of a reliable critic. Vague concepts serve important purposes, especially when recognized as vague and used with caution.
The Utility Parameter: This parameter indicates the scope and type of uses P has for C. At one extreme P may depend on C so much that C is essential for P, such as P’s concept of P. At another extreme P may use C so seldom that C may be expendable. Of course what is expendable for one person may could be essential for another. For instance a concept of a supernova is expendable for me, but there are astronomers who would find it essential. This parameter may also indicate anything that might provide a perspective on how P might be able to use C. Al acquired his concept of a maul because he was interested in splitting wood. He acquired it as a wood splitter rather than as an interested observer. Furthermore he acquired it as an amateur who normally splits just enough wood for his own personal use. Such information could be included as components of the utility.
Mathematical Nets: Suppose P has an adequate conceptual net for some mathematical realm. All of the concepts will be impersonal and only a few will be subconcepts. There will be a precise support structure for concepts, with a multitude of connections. Many concepts will be subject to analytic explanation. P will also be able to give synthetic explanations. Altho a large number of the concepts will be remote from ordinary experience, they will seem manifest because mathematical thinking is an easily accessible experience for P. Some of the mathematical concepts will also be directly connected to ordinary concepts. In a net for ordinary algebra, consider the concept of multiplication as extended to the rational numbers. The fact that this concept is remote from P’s ordinary experience will not make it seem remote to P. It will be connected to the basic algebraic laws and supported by these laws. It will be seen as the only extension of multiplication that preserves the laws for natural numbers, and also allows for inverses. This concept will also be connected to applications. These ideas are developed in the chapter on the basic algebra for integers in Understanding Ordinary Algebra for Adults.
SUPPLEMENT 2 BEHAVIOR
DESCRIPTIONS & CONCEPT CONSTRUCTION
Descriptive Psychology: Peter Ossorio created Descriptive Psychology as a network of theory neutral conceptual tools. Since Descriptive Psychology is used in similar ways by a notable number of people and is designed for use by a wider public, I refer to it as PNDP, the Public Net for Descriptive Psychology. Persons, Behavior, and the World by Mary Shideler gives a comprehensive introduction to these tools (see the Society for Descriptive Psychology website sdp.org). Some of these tools are also described in the Descriptive Psychology part of my website. Below I indicate the PNDP behavior description concept that I use. By factual knowledge, I mean the type of knowledge that can be expressed using propositions and which is acquired by observation and thinking. Instead of KF, PNDP calls this the K parameter. This difference is primarily linguistic.
A Behavior Description Concept: 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 observer. The observer and actor can be the same person.
Identity (I) specifies who is the actor for X.
Wanting (W) indicates what the actor intends to achieve by X.
Knowledge of Facts (KF) has to do with facts the actor uses in relation to X.
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: An observer 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 KF parameter that indicates that the actor has distinguished at least two options.
An observer 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 observer can be a team working together to give a behavior description.
The C Parameter: PNDP calls this the PC-parameter for person characteristics and organizes it into three categories, each of which includes several types of characteristics. I will discuss some of them later as I use them.
Dispositions: {Traits, Attitudes, Interests, Styles}
Powers: {Abilities, Knowledge, Values}
Derivatives: {Embodiment, Capacities, States}
Course of Action: The concept of a course of action includes not only a sequence of actions but also what we would usually think of as a single action, such as turning on a light. In fact, the distinction between what we think of as single action and a sequence is a matter of perspective. For instance, I may think about cutting down a tree as a single action or as a sequence of actions. The sequence might include obtaining the tools, taking them to the tree, looking for obstructions, notching the tree, sawing the tree, testing to see if it was ready to fall, etc.
Behavior: The concept of a behavior description is not a conceptualization of the concept of behavior. The concept of behavior is presupposed. The behavior description concept merely provides a tool for bringing various aspects of behavior into focus. Which parameters to use and the detail to which they are developed will depend on the observer’s purposes in giving the description. Unlike theories, conceptual tools can be judged almost exclusively by their utility. Thus there is no need to use a tool for the purposes it was designed to serve and no essential reason not to modify a tool if this serves your purpose. Because of my interest in concept construction and various epistemic concepts, I use a modified version of the power category with Understandings instead of Knowledge. I include various types of knowledge and comprehensions as components of understanding. Propositional knowledge IS the kind of knowledge that can be expressed by propositions. I also use process knowledge and realm knowledge. Process knowledge IS the type of knowledge that is acquired thru practice and that enables an actor to implement various processes. It is the cognitive aspect of competence. Realm knowledge IS the knowledge of a realm that is obtained by living within that realm. It includes propositional and process knowledge as components but is much broader. It cannot be reduced to a set of component parts. It is the kind of knowledge that we have in mind when we say that a reporter really knows city hall. It is the basis needed for powerful understanding of concepts for a realm. Relational comprehension INVOLVES an understanding of relationships between concept, and is essential to having well integrated concepts. Unlike propositional knowledge, which can often be expressed using simple propositions, relational comprehension can vary from minimal to extremely sophisticated. Relational comprehension also INVOLVES an understanding of relationships in various states of affairs that are not purely conceptual. For instance, understanding aspects of the relationship between nutrition and health. All of these concepts are developed in A Net for Conceptual Philosophy and in A Net for Understanding. These can be downloaded as word files from my website.
Side Comment: As an alternative type of behavior description, I once used a U parameter for Understanding, comprehend and have {propositional knowledge, process knowledge, realm knowledge, other} as a set of sub-parameters. On reflection, I realized a potential redundancy in having a U parameter, and in even having the KF and KH parameters. Because of the power category, all could be accounted for when describing the C parameter. While conceptually possible, doing so would make it easy to miss the prominent roles that specific factual knowledge and know-how play in many behavior descriptions. The broader aspects of understanding or of realm knowledge seldom play such a prominent role. When prominent, such as in a course of action centered on concept construction, it may be just as convenient to focus on them by using the C parameter, altho using a U parameter could help focus on interrelated aspects of understanding.
A Simple Illustration: It is Jill’s turn in a game of Gin Rummy. The face up card would improve her hand, but the card on top of the deck might be even more useful. For an illustrative purpose, we give a description in which the parameters are simple. Each could be expanded if we had reasons to give a more elaborate description. The I-parameter in our description is Jill. The W-parameter is her desire to improve her hand. By saying that Jill knows that she can take either card, our description is a deliberate action description. The KF-parameter also includes knowing what is in her current hand and which cards would improve this hand. The KH-parameter includes knowing how to count the points she would be caught with if her opponent goes down or gins. She takes the card from the deck, the P-parameter. Since the card Jill draws is useless, the A-parameter is the negative achievement of failing to enhance the hand to the extent the face up card would have done. One value of the S-parameter is that Jill is trying to win the game. One noteworthy instance of the C-parameter is Jill’s risk taking attitude while playing games. To see why this might be used as part of the behavior description, consider a person with the same value for the other parameters, but who usually avoids risks. For such a person taking the card that is face down would involve a different value of the C-parameter. Perhaps that person was in a state of frustration that overcame the aversion to risk taking.
Another Illustration: This illustration focuses on the course of action X of me cutting down a dead tree one afternoon. I am also the observer giving the behavior description. Altho this course of action is more involved than the act of drawing a card (and may thus add some additional perspective), it is not crucial for using behavior descriptions in relation to developing constructivist learning resources. I first focus on partially indicating the primary values that I used for the parameters, i.e. those not associated with what else is being done as indicated in the significance parameter.
I-parameter: me
W-parameter: wanting the tree down
KF-parameter: knowing that I could safely delay taking this tree down, knowing that it is dead and thus there could be a danger from falling limbs, knowing the relation between notching the tree and where it is likely to fall, knowing that there was an obstacle blocking the fall, etc.
KH-parameter: knowing how to use tools, knowing where to cut with the saw once the tree was notched with the ax, etc.
P-parameter: getting my tools at 2pm, cutting a notch with the ax and then using the saw on the opposite side to cut slightly above the bottom of the notch.
A-parameter: the tree coming down where I wanted it to fall
C-parameter: my dislike of chain saws, the value I place on staying physically active, my habit of engaging in physical activity in the afternoon, etc.
S-parameter: enhancing resources ® making firewood easier to obtain ® potentially saving money on his fuel bill ® potentially having more security, getting exercise, preventing a hazard
The P-parameter could be expanded as indicated earlier.
The first two items in C-parameter add an important perspective on what I did. This attitude and this value account for the fact that I used an ax and a crosscut saw and the timing of X.
Above, I indicated one significance chain that can be described as follows. In doing X, I am obtaining a usable firewood resource. Having this resource will make it easier to obtain more firewood. One reason for wanting this is that it could later save money on my fuel bill. Saving money would provide some extra security. This significance chain is part of the S-parameter for describing X. In doing X, I am also getting exercise and removing a potential hazard to the road. These parts of the S-parameter are not in that chain. Each value given in the significance parameter indicates what else I was doing in doing X. Thus we can think of the values of this parameter as intentional actions and use one or more of the parameters to bring them into focus. These values are included in the extended parameters for X. Values for the extended identity and performance parameters are the same as the primary ones for X.
Consider action X1 of trying to prevent a road hazard. The values for W1 and A1 should be apparent. KF1 includes knowing if this tree falls it is likely to fall across the road. C1 includes the value I place on the safety and convenience of others.
Consider action X2 of trying to make firewood easier to obtain. A1 includes achieving this, altho not to the degree I expected because I have not yet processed it and it is beginning to rot. KF2 includes a knowing that once a tree is down it can be converted into firewood in small stages. C2 includes my more positive attitude toward processing firewood than to taking down dead trees.
Consider action X3 of trying to obtain more security. A1 includes the small increase in security due to having a fallen dead tree, but since I did not process it, this increase was not as large as expected. KF3 includes a comprehension of the relationship between financial resources and security. C3 includes my trait of frugality.
Behavior Description & Developing Constructivist Learning Resources: I now focus on why I find the behavior description concept useful for thinking about developing constructivist learning resources. I illustrate this by describing some aspects of my course of action in writing this present paper, which I regard as a mixed type multipurpose constructivist learning resource. I want to again stress that the behavior description concept is a tool, much as Bloom’s Taxonomy of Educational Objectives is a tool. I was once asked how the behavior description concept explains why persons do what they do. The answer I gave was simple. It doesn’t, because it is a conceptual tool rather than a theory. If the observer wants an explanation, then the observer must decide what counts as an explanation for the purpose at hand. The observer can then use a behavior description to help construct such an explanation. As an analogy, if I were asked how my ax and saw cut down a tree, I would say they don’t, and I do not expect them to. Cutting the tree down is what I do, and these tools help me implement this. The way I use a behavior description concept also depends on the purposes at hand. It is a useful tool for various purposes primarily because it has enough parameters to help me focus my attention more clearly than I could do with a less systematic tool. This also helps me focus on and organize relevant observations. However, as with any tool, results depend on the skill with which it is used. Even a magic wand could be used badly.
Developing This Present Paper: Altho designing a plan and the related resources for developing the concepts in this paper was a joint course of action involving many hours during a period of several months, I describe this only in terms of myself as the actor. During this course of action, the primary values of the want parameter was to develop some conceptual distinctions that would help me think about my own work in developing constructivist learning resources. Many of the component acts were guided by a more immediate want, usually to play with and polish the ideas most directly at hand.
The significance of what I was doing evolved over time, and thinking about this evolution had an impact on the resources that I produced and my understanding of the concepts I was developing. Thus I will organize my behavior description around this parameter. My course of action involved three main strands or sub-courses of action.
The mathematical sub-course involved writing the appendices, and most of my behavior description will focus on this strand. However, the strand in designing this paper that was both most immediate and most significant had nothing to do with mathematics. It centered on clarification of and expansion of my own perspective on constructivist learning and constructivist learning resources. This was the main reason that I wrote this paper. Even the mathematical strand was motivated by this purpose, altho that sub-course of action took on a life of its own. The third strand involved relating the conceptual tools from PNDP and my own conceptual net for understanding to the development of constructivist learning resources. This sub-course of action involved writing the supplements. Altho I spent the least amount of time on this strand, it was almost as significant as the second strand and was even integrated somewhat into that strand.
Closely related to my desire to clarify and expand my own perspective was my desire to communicate this to others. This relates to one crucial value of the C-parameter, namely that I have a much stronger interest in writing to enhance my personal understanding than in writing in order to communicate. One of my prevalent dispositions when writing is to get carried away with my own interests. Thus, I must pay more attention to communication than I would normally be inclined to do. However as a prelude to communication I must enhance my own understanding, and the primary performance parameter for this course of action centered on organizing and developing my own personal perspective. The primary value for the achievement parameter was a high level of success in that regard. I cannot judge the extent to which the secondary communication achievement will be realized, altho it has been realized to some minor extent.
The Mathematical
- an interesting experience working on a mathematical problem
- a deeper appreciation of using multiple approaches to solving a problem
- more competence in mathematical reasoning and problem solving
- an expanded perspective on the function concept used in contemporary mathematics
- a slightly expanded concept of mathematics
- a foundation for an understanding of the concept of a group of permutations under composition
- a basis for understanding how the contemporary concept of mathematics differs from the concept commonly used by most non-mathematicians.
Focusing on the wants associated with these, I tried to make a resource whose core would be suitable for any potential user that I could imagine and rich enough to be useful to readers with a more sophisticated perspective. For instance the resource in Appendix 1, was designed not only to make the function concept more manifest to users who find the contemporary function concept remote. However it is also intended to help those for whom this concept is manifest broaden their concept by adding an additional manifest perspective. In designing Appendix 1, I deliberately tried to suggest concepts that most potential users would find highly manifest.
There is a cluster of interrelated characteristics, typical of many mathematicians. These relate directly 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 it was conjectured that all perfect number are even. A person remarked that given two examples, a mathematician generalizes. Another replied that one example will do. A third replied, who needs an example?
Interests: They find mathematical abstractions and ideas of interest in their own right, as long as they are esthetically pleasing.
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 understanding of mathematics. The 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.
Understandings: 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 without regard to what most persons consider as utility. They value precise analytic reasoning.
Altho these characteristics are instrumental in learning mathematics, unless recognized they tend to interfere with designing constructivist learning resources. This is primarily because mathematicians tend to design resources that are most useful to others having similar 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 conceptual net for mathematics. The style I use in constructing is flexible and multifaceted. I create a multitude of examples and explore many questions. I think about intuitive connections and relationships. I often wish that I had more resources that would assist me in these endeavors, but often the main one I can locate are axiomatic presentations.
Essentially, what I try to do when I design mathematical learning resources is to focus on what might help those with different characteristics also cultivate some of these characteristics. Perhaps the key disposition 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 potential users of
constructivist learning resources. 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 Appendix 3 to make this
resource appropriate for those with visual or numerical styles. The fact that
many persons have an algorithmic style is why I included
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 characteristics indicated above. Yet it is of the rules about fractions that most non-mathematicians do not understand. Knowing this, was what motivated me to design alternate ways of dividing fractions that is easy to understand for anyone with a visual style of learning.
SUPPLEMENT 3
MATHEMATICAL SIGNIFICANCE
Personal Significance: Altho a multitude of factors will influence the personal significance involved in the design of any constructivist learning resource, a major factor will be the designers understanding of the mathematical significance of concepts and the value placed on helping intended users appreciate this. Specific to the design of the resources involving magical squares was our understanding the significance that the function concept has in contemporary mathematics. This includes the relationship that concept has to the concept of a group of permutations and the role this played in transforming mathematics during the 19th and 20th century. Our design was also influenced by the value we place on helping expand the awareness of the nature of contemporary mathematics to more people.
The Function Concept: The contemporary concept of a function did not emerge in mathematics until the 20th century. Before then, the concept of a function was limited primarily to numerical functions that can be represented by formulas. Even now, this is the concept that many people use. The more remote contemporary function concept is often only presented via an abstract definition, perhaps with some examples. The magic square resources were designed to provide an experience in using coding and permutations as problem solving tools. One hope is that using them will help make the function concept seem more manifest.
The Concept of a Group: A group is a structure having an associative binary operation with an identity element and an inverse operation. Addition is an associative binary operation with 0 as the identity element, and each integer has an additive inverse. A similar remark applies to multiplication. Thus the concept of a group can be abstracted from what addition and multiplication for certain numerical structures have in common. Had familiar numerical groups been the main motivation for the group concept, this concept would have included a commutative law. Instead, the group concept emerged from the operation of composing permutations, and composition is not a commutative operation. A permutation is a one-to-one function from a set onto itself. Permutations can be composed to provide simple examples of non-commutative groups. Altho the concept of a group is beyond the scope of the resources in this paper, manifest work involving permutations can provide a basis for a later acquisition of this concept. We have designed a resource for exploring magic squares of larger sizes that explicitly uses the concept of a group. We also have designed resources for solving manifest puzzles that involve using permutations and that focus directly on the concept of a group. Historically the concept of a group emerged early in the 19th century. It was first used to show to study formulas for solving algebraic equations via formulas that involves only radicals and the 4 basic operations of algebra. Abel and Galois showed the general 5th degree equation had no such solution.
The Concept of Mathematics: Sciences are commonly characterized by identification with some special subject matter. Biology studies organisms, economics the production and distribution of goods and services, physics matter and energy. Such characterizations are in dictionaries and textbooks. Experts in these sciences tend to find such statements satisfactory as an initial and simplistic way to identify their sciences. It is common practice to identify mathematics by a subject matter, say quantitative and spatial relationship. I have yet to find a mathematician who would accept this. Furthermore, they find such statements misleading rather than merely simplistic. How do mathematicians characterize mathematics? Why is their perspective so divergent from the common one, and does this divergence matter? Contemporary mathematics is primarily a method of thinking, rather than a specific subject matter. To appreciate the attitude adopted by contemporary mathematicians, one needs mathematical experiences unlike those usually encountered. The concept of a group is one of the most basic concepts of contemporary algebra. An appreciation of this concept could provide a basis understanding of the contemporary concept of mathematics.
Magic Squares and Group Orbits This link takes you to a paper by John Pais and Richard Singer which is published in the online journal Visual Mathematics. This paper is an essay in visual mathematics. It is a guided discovery in which the learner constructs his/her own math concepts, first intuitively, exploring, constructing, and counting 4 by 4 visual magic squares, and second analytically, using numerical representations and group orbits to classify and generate these visual magic squares. It shows how the resources in the appendixes can be extended to concepts with greater mathematical significance.
Finding 3by3 Magical Squares This link takes you to the materials John Pais and Richard Singer used for 1½ hour workshop at 1he 2004 annual ACT conference. It contains worksheet for 2 by 2 magical squares which are similar to those suggested in Appendix 2, but which use the tokens that can be made using the MS Word file Token Template.