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MANIFEST
RESOURCES FOR MATHEMATICIANS
Mathematicians: The concept of a mathematician used here is broader than the concept of a professional mathematician. It includes anyone who has shown a significant interest in studying mathematics with a highly conceptual attitude and who has demonstrated a capacity for such a study. To be a mathematician is to embrace a way of thinking, a way involves a basic understanding and appreciation of both the power and the limitations of deductive reasoning. In this sense, we would say that an essential aspect of being liberally educated is to be able to think as a mathematician. This is not a widespread attitude. Most people who study what they call mathematics do so primarily thru drill and practice, so much so that they may think of math as a collection of techniques. Because of this, the majority of them do not even have an initial understanding of what it like to be a mathematician.
What was called the modern mathematics movement was motivated by a desire to
help the majority of students understand mathematical thinking. The difficulty
in accomplishing this was under estimated for a variety of reasons. The resources
we designed focus on only one aspect of this difficulty, namely on the problem
that mathematical reasoning focuses on concepts that most people initially find
remote. The main purpose of our manifest mathematical resources is to provide a
variety of materials for relating the remote mathematical concepts to concepts
that are more manifest. These units are written for anyone who wants to think
about mathematics from a manifest perspective. We now discuss what manifest
reasoning means. We then give an elementary example indicating its role in
helping us construct more remote concepts. You can skip this
and go to the last part where we indicate several types of mathematicians and
how we hope that they might find some our resources of use.
Manifest vs Remote Reasoning: One dictionary gives the literal meaning of the word ‘manifest’ as
1 : readily perceived by the senses and especially
by the sight
2 : easily understood or recognized by the mind
Ordinary reasoning is often manifest, being guided by situations that provide some immediate feedback. Such reasoning is directed to immediate purposes. Highly manifest reasoning includes reasoning which is likely to occur when planting a garden, repairing a broken vase, building a bookcase, playing chess, driving a car to some new destination. We also engage in more remote reasoning, i.e. in the kind of reasoning in which our attention is directed away from our present environment. Ordinary examples include analyzing the strategy used in an earlier game of chess, predicting the weather for the coming weekend, defending a proposed tax reform, or thinking about causes of the Civil War. Even more remote, would be reasoning about general strategies for chess, weather prediction theories, the general causes of war. The main advantage of manifest reasoning is that it provides direct feedback about what we are doing. This is also its main limitation. Unless we relate it to some remote reasoning, it may provide little insight beyond its immediate application.
What seems remote to one person may seem close at hand to another. Anyone familiar with the concepts involved in deductive reasoning find such reasoning remote only in the sense that it involves very little feedback from their surroundings. Anyone less familiar with these concepts might find deductive reasoning more remote. This may not make the reasoning about a particular situation hard, but for most people remoteness is one factor that increases difficulty. Perhaps remote reasoning abilities are an evolutionary extension of manifest reasoning abilities; but in any case, most people obtain a better grasp of remote reasoning when they can relate it to manifest reasoning. Mathematics uses the kind of reasoning which is about as remote from ordinary feedback as we can imagine. In order to relate mathematical reasoning to reasoning that is less remote, our resources focuses on reasoning about information which can be analyzed by both manifest and remote methods. This can be done at all levels in the study of mathematics, including the level of highly advance mathematics. At a fairly basic level, we can relate the reasoning about the use of common denominators to add fractions to common denominators manifest reasoning with colored pieces use to represent fraction. At a much more advance level, our Think-a-Do Group unit relates the fundamental decomposition theorem for abelian groups to an elegant solution of a set of puzzles that most people would first solve by trial and error.
Constructing Remote Concepts: To understand involves more than just gathering information. It also involves having a network of related concepts. Unlike information, which a person can receive directly from another, a person must acquire conceptual relationships by augmenting and transforming ideas they already understand. To expand our understanding, we need experiences that challenge us to construct new ways of thinking. As a preliminary illustration of this, consider the question below and some answers that might be given. In a traditional classroom setting this question might be given to see if the students could multiply ½ times ½ obtain the answer. While people learn to do this, it often does not fit with their concept of multiplication, and the result does not fit with their idea that a product should not be smaller than the numbers being multiplied. Instead of posing this question after they have studied fraction multiplication, this question can be posed early in their experience with fractions. The more a person can learn to think about fraction concepts before being asked to calculate, the easier it will be to understand and use fraction concepts.
Question: 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?
Kay If you went ½ of the trip today then you must go the other ½ of the trip tomorrow. Since ½ of ½ is ¼, they are expecting to complete ¼ of the 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 yellow 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 of the trip.
g g g g g g g g g g g g
g g g g g g g g g g g g
g g g g g g g g g g g g
Solutions like these are more likely to occur if we are encouraged to think in terms that we find manifest. Each of these students understand the concepts of a half and a fourth. Kay already finds fraction concepts manifest. She seems very comfortable with these concepts and how they are related. To relate this question to the concept of multiplication, she will need at most a small extension of her fraction concepts. Both Jan and Bob seem to find fraction concepts somewhat remote. Jan finds whole number concepts manifest. She can use reasoning involving whole numbers to help her make concepts she initially finds remote become manifest. To make something manifest, Bob finds visualization extremely helpful. Hopefully as he does this less visual concepts will also become manifest to him.
Potential User: There are several types of mathematicians who we hope might find these resources useful.
Amateur mathematicians: The fractions book and the units on ordinary algebra are designed specifically for two types of amateur mathematicians. They are primarily for adults who feel dissatisfied with their mathematical education and would like their liberal education to include the ability to think as a mathematician. We hope they would be especially helpful to anyone wanting to mentor younger persons in their study of mathematics. Secondarily, they are for younger persons who have not yet begin to study those topic and who have an adult mentor to help them in their studies. In addition there are enrichment topics for secondary level mathematics. These are also designed both for younger persons and their mentors.
Undergraduate mathematics majors: Using a few specific examples, the concept of a finite group of permutation can make this concept manifest to most undergraduate mathematics majors. The unit we developed on permutation groups is intended to make this concept even more manifest. It goes far beyond what most professional mathematicians would ever encounter. The group of permutations of {0,1,2,3}has 30 subgroups which fall into 11 conjugate classes. This unit provides a path the student can use to calculate these conjugate classes and exhibit a member of each. It then goes on to provide a similar path in relation to the19 conjugates classes of subgroups of S5. In the process the student will obtain considerable experience and practice in calculating with cycle notation. The purpose of all of this is to make permutation groups highly manifest to the student, primarily so the more general concepts of group theory will seem less remote. For example altho the Sylow theorems are not used in finding conjugate subgroups, when the student does encounter these theorems the concept of conjugate subgroups should not seem remote.
If several units of this type were used in a typical course in abstract
algebra it would probably be impossible to cover the usually set of topics.
What use would such units have then have for undergraduate mathematics majors?
A conservative answer to this question would be to offer an elective seminar in
which different students would explore different manifest units and then
present various parts to the class.
Graduate students in mathematics: Graduate students in mathematics often focus primarily on mastering definitions and theorems and proofs. Some example are used in the process, and some students spend time developing and exploring their own examples. Time constraints may prevent developing a manifest basis for much of what is covered, and students succeed in passing qualifying exams without such a basis. Once they get to thesis research, concepts that seem remote to those working in other areas become manifest. The thesis is written in general terms, but discovery usually involves considerable work with specifics. Persons cannot do quality research unless there own specialty is something they have made manifest. Why then would graduate students use the kind of manifest resources we are designing? One reason is that this could add perspective on areas of outside of their specialty. This could be especially important because they are likely to teach materials that they have only examined from a perspective that their students would find remote. Of course the pressure of graduate work may make it seem better to postpone such considerations until later. Given the time need to organize effective materials this reservation is easy to understand. However it takes much less time to work thru manifest materials that are designed by someone else, and working thru some such materials might even save time in mastering the more remote materials required for the qualifying exams.
Professional Mathematicians: A person does not become a professional mathematicians without their own specialties seeming manifest, and unless they are teaching the main reason they may have no professional reasons to look at the kind of resources we are designing. We would be pleased if some of them would just find parts of what we developed interesting and would share their reactions. For those whose primary professional responsibility is teaching students whose mathematical sophistication is in its initial stages, we hope that our units could be helpful. We would welcome any of them in our endeavor to create additional manifest resources. We would also welcome help with those currently being developed, and this includes all of the ones on our websites because all of them could use additional work. They especially need work to be augmented in ways that will make them more manifest to more students.
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