arash rastegar department of mathematical sciences sharif university of technology

Arash Rastegar Department of Mathematical Sciences Sharif University of Technology

There are two kinds of Mathematicians Problem Solvers Contribute Local to global point of view Interesting special cases Problem Solvers Contribute Local to global point of view Interesting special cases Theoriticians Contribute Global to local point of view Various generalizations Theoriticians Contribute Global to local point of view Various generalizations

My mathematics teachers 1. School teachers: Data sources 2. Special teachers: Physicians 3. Mathematics Olympiad coaches: Coaches 4. Instructors in undergraduate level: Role models 5. Instructors in graduate school: Life companion 6. My PhD advisor: Scientists 7. My personal view: Philosophers 1. School teachers: Data sources 2. Special teachers: Physicians 3. Mathematics Olympiad coaches: Coaches 4. Instructors in undergraduate level: Role models 5. Instructors in graduate school: Life companion 6. My PhD advisor: Scientists 7. My personal view: Philosophers

Teacher as a source of information Knowledge in such a classroom is nothing but raw information and teaching and learning are nothing but transformation and communication of information. Teacher either orally explains the information to the class or writes the information on board or shows students demonstrations on computer or on data show. Some other information are made available to students by handing them books and papers. Students access to information is not always in full detail. Most of the time teacher chooses the key information and makes it known to his/her pupils. Teacher also gives some explanations on the meaning and the relations between the packages of information. This role of such a teacher could be easily played by computers. Information technology could provide students with more detailed, accurate, advanced, and accessible information.

Student versus a source of information The role of student in such a class is taking notes and trying to memorize them. Repetition helps students to keep information in mind for a longer period of time. Decisions that a teacher should make in such a classroom are very formal and can be modeled to be performed by a computer. Teacher decides about the language and tools which each piece of information shall be demonstrated to the class, and also the exercises which help students internalize the information. Teacher has to choose the key information and decide about appropriate references to introduce students for further study. Teacher has to decide about how to assess a piece of information too. We shall introduce models for different personalities of teachers in making such decisions. The role of student in such a class is taking notes and trying to memorize them. Repetition helps students to keep information in mind for a longer period of time. Decisions that a teacher should make in such a classroom are very formal and can be modeled to be performed by a computer. Teacher decides about the language and tools which each piece of information shall be demonstrated to the class, and also the exercises which help students internalize the information. Teacher has to choose the key information and decide about appropriate references to introduce students for further study. Teacher has to decide about how to assess a piece of information too. We shall introduce models for different personalities of teachers in making such decisions.

Teacher as a physician In such a classroom, teacher is trying to give special treatments to the minds of his/her students. For this kind of teacher, knowledge is only used as a means of communication in order to get information on how students think. A teacher tries to replace an unorganized mind by an ordered healthy mind. This is why he/she demonstrates some examples of truthful deductions and accurate reasonings to the class. The main occupation of the teacher in this classroom, is discovering the mental weaknesses of students and trying to bring them to a normal position in which all of their different abilities demonstrate a coherent progress during the process of mathematical education. Only a very well-trained experienced teacher can play this role in the classroom. There is some chance that some day a detailed classification of learners behavior plus a well developed theory of information engineering could partially replace the role of these teachers in the educational system.

Student versus a physician In this classroom, students try to awaken the abilities of their mind, by learning and imitating examples of deductions and reasonings presented in class. They try to cooperate with their teacher in discovering their mental weaknesses. Teacher has to have a classification of human mathematical mind so that he/she can decide each student belongs to which class and then comparing students abilities with the particular class he/she has chosen, decide about weaknesses and strengths of the student. Classification of weaknesses is an abstract procedure, because un-normal situations are exceptions; but teacher has to put similar ones in the same class, so that he/she be able to study them by their sympthoms. Making such decisions is very abstract and complicated. In this classroom, students try to awaken the abilities of their mind, by learning and imitating examples of deductions and reasonings presented in class. They try to cooperate with their teacher in discovering their mental weaknesses. Teacher has to have a classification of human mathematical mind so that he/she can decide each student belongs to which class and then comparing students abilities with the particular class he/she has chosen, decide about weaknesses and strengths of the student. Classification of weaknesses is an abstract procedure, because un-normal situations are exceptions; but teacher has to put similar ones in the same class, so that he/she be able to study them by their sympthoms. Making such decisions is very abstract and complicated.

Teacher as a coach In this classroom, teacher is trying to discover his/her students personal talents and train them in direction of perfection of these talents. The particular characteristics of the mathematical knowledge being discussed in class has an important role in this training. Teacher makes determining decisions in choosing the material to be discussed in class. Different examples demonstrated in class are aimed to train different groups of students of different talents. Teacher shall be able to teach students the particular practices and exercises which help them in perfection of their talents. Teacher supports and protects his/her students in the storm of their mental contradictions. He/she has to provide an appropriate atmosphere in classroom for treatment of students mental contradictions.

Student versus a coach The role of student in this classroom is doing mathematics. Student personally gets actively involved in enrichment of his/her knowledge upto a level in which he/she be able to overcome mental contradictions leaning on his/her own personal abilities. Students also cooperate with teacher in the process of discovery of their talents. Teacher has to introduce a classification of mathematical talents and decide about the manners in which these talents show off so that he/she can decide about talents of students and excercises which can lead students to perfection of their talents. Teacher has to decide about different levels os student access to each of these talents. The role of student in this classroom is doing mathematics. Student personally gets actively involved in enrichment of his/her knowledge upto a level in which he/she be able to overcome mental contradictions leaning on his/her own personal abilities. Students also cooperate with teacher in the process of discovery of their talents. Teacher has to introduce a classification of mathematical talents and decide about the manners in which these talents show off so that he/she can decide about talents of students and excercises which can lead students to perfection of their talents. Teacher has to decide about different levels os student access to each of these talents.

Teacher as a role model In this classroom various forms of mathematical knowledge are discussed. Teacher tries to openly demonstrate the process of thinking to the class. Students could learn how to solve problems, only if they can openly watch their teacher wrestle with hard problems. Teacher must be able to think loudly so that students could imitate his/her skills of problem solving. Teacher could serve as a good role model, only if students can watch his/her thoughts inside out. In this classroom, teacher shall do mathematics which is not familiar with, in front of the students. Teacher tries to make students involved in his/her process of problem solving. If this role could be successfully played in classroom, students have been given the best possible education for group thinking. A teacher for this class should be very experienced and knowledgeable in mathematics.

Student versus a role model The role of students in such a classroom is doing mathematics while keeping the teacher in mind as a role model. Students are able to construct their mathematical abilities personally and guide the direction of progress in their mental skills. Most of the decisions that such a teacher should make are personal decisions which he/she makes in problem solving and the class is supposed to imitate him in making personal decisions. The role of students in such a classroom is doing mathematics while keeping the teacher in mind as a role model. Students are able to construct their mathematical abilities personally and guide the direction of progress in their mental skills. Most of the decisions that such a teacher should make are personal decisions which he/she makes in problem solving and the class is supposed to imitate him in making personal decisions.

Teacher as a life companion In this classroom, teacher tries to apply mathematics to students everyday life situations. To this teacher, knowledge is what is practically useful. Teacher tries to introduce mathematics as a human endeavor. The aim of teacher is to translate problems of everyday life situations to the abstract language of mathematics and slove them using mathematical machinery and then translate the solution back to the language of everyday life. This ability should be internalized in students so that they use their mathematical skills outside school. In such a class, the human history of evolution of mathematical concepts is under focus. Also, the process of discovery in problem solving should be recorded carefully by students. Strategies of problem solving and skills of decision making in course of discovery are taught carefully in this classroom.

Student versus a life companion Students engage in groups in solving everyday life problems using mathematics. Learning is a group activity but each student has a particular interest and perspective in doing mathematics. Every student helps his/her colleagues during the learning process and takes advantage of the helps offered by other colleagues. Teacher has to decide about different skills that students should learn and about different levels of suffistication in these skills. Teacher has to decide about the skills which each of the students could learn better by having a classification of students learning strategies. Students engage in groups in solving everyday life problems using mathematics. Learning is a group activity but each student has a particular interest and perspective in doing mathematics. Every student helps his/her colleagues during the learning process and takes advantage of the helps offered by other colleagues. Teacher has to decide about different skills that students should learn and about different levels of suffistication in these skills. Teacher has to decide about the skills which each of the students could learn better by having a classification of students learning strategies.

Teacher as a scientist In this classroom, teacher unveils the mathematical order of the universe we live in. For this teacher, mathematical knowledge is based on ideas borrowed from nature. Teacher makes sure his/her students learn to observe nature, discover mathematical ideas from it and use them in development of mathematics. History of science is under focus. Students should learn how mathematical ideas taken from nature have affected human civilization. Proposing reasonable conjectures based on scientific methodology is an important skill which students should learn. The aim of teacher is not only familiarizing students with applications of mathematics to other branches of science, but also teaching them skills of discovering new applications of mathematics to other areas of science. These applications do influence the practice of scientists in other areas.

Student versus a scientist Students in such a classroom search in groups for mathematical ideas in nature. They recognize mathematics as the language of nature and try to get a better understanding of nature through this language. Student also tries to apply mathematics to other branchers of science. Teacher has to decide about how he/she can motivate students to become interested in nature and its order. Teacher has to decide about the branches of science that a student is fascinated by, so that he/she could insert mathematical ideas to teach the student how to apply mathematics in basic different branches of science. Students in such a classroom search in groups for mathematical ideas in nature. They recognize mathematics as the language of nature and try to get a better understanding of nature through this language. Student also tries to apply mathematics to other branchers of science. Teacher has to decide about how he/she can motivate students to become interested in nature and its order. Teacher has to decide about the branches of science that a student is fascinated by, so that he/she could insert mathematical ideas to teach the student how to apply mathematics in basic different branches of science.

Teacher as a philosopher In this classroom, mathematics is an abstract knowledge independent of nature which is governed by the mathematical essence of the world of creation. Mathematical ideas could be borrowed not only from nature, but also from beyond. Teacher tries to discover the mathematical order with the aim of demonstrating his/her students unity in the essence of creation. Teacher tries to find similar ideas in different parts of mathematics and use them to unveil the unity beyond these ideas. In such a class, the abstract science of mathematics is formulated independent of nature. Mathematics has many abstract layers and teacher tries to connects these layers by unifying different mathematical theories in each layer. This is a new perspective to learning mathematics in which each student does mathematics in a particular layer of abstractness.

Student versus a philosopher Student tries to structure the abstract mathematical nature of his/her mind and is interested in knowing the essence beyond the mathematical ideas. Relating different branches of mathematics and performing commutations and comparing them in different branches are tools which students use to extend the abstract structure of their mind. Learning and doing mathematics is no longer possible in groups because of the high level of abstractness of this kind of mathematical activity. Student is learning mathematics for reasons beyond mathematics. So, again the particular mathematical content of mathematics does not play any important role in such a classroom. In such a classroom, teacher has to decide about the particular level of abstractness each student could handle. Teacher also has to make sure if unification of a few different examples and creation of a more abstract generalization has actually occurred for a particular student. Teacher has to have a personal knowledge of different layers of abstractness and be able to decide each particular mathematical content belongs to which of these abstract layers. Teacher chooses a particular mathematical content which he/she is most familiar with its abstract layers. Student tries to structure the abstract mathematical nature of his/her mind and is interested in knowing the essence beyond the mathematical ideas. Relating different branches of mathematics and performing commutations and comparing them in different branches are tools which students use to extend the abstract structure of their mind. Learning and doing mathematics is no longer possible in groups because of the high level of abstractness of this kind of mathematical activity. Student is learning mathematics for reasons beyond mathematics. So, again the particular mathematical content of mathematics does not play any important role in such a classroom. In such a classroom, teacher has to decide about the particular level of abstractness each student could handle. Teacher also has to make sure if unification of a few different examples and creation of a more abstract generalization has actually occurred for a particular student. Teacher has to have a personal knowledge of different layers of abstractness and be able to decide each particular mathematical content belongs to which of these abstract layers. Teacher chooses a particular mathematical content which he/she is most familiar with its abstract layers.

Perspectives towards mathematics education 1. Mathematics education has an important role in development of mental abilities. 2. Mathematics education is an efficient tool in developing the culture of scientific curiosity. 3. We use mathematics in solving everyday problems. 4. Development of the science of mathematics and our understanding of nature are correlated. 5. Doing mathematics has an important role in learning and development of mathematics. 1. Mathematics education has an important role in development of mental abilities. 2. Mathematics education is an efficient tool in developing the culture of scientific curiosity. 3. We use mathematics in solving everyday problems. 4. Development of the science of mathematics and our understanding of nature are correlated. 5. Doing mathematics has an important role in learning and development of mathematics.

6. Development of tools and Technology is correlated with development of mathematics. 7. We use mathematics in studying, designing, and evaluation of systems. 8. We use mathematical modeling in solving everyday problems. 9. Group thinking and learning is more efficient than individual learning. 10. Mathematics is a web of connected ideas, concepts and skills. 6. Development of tools and Technology is correlated with development of mathematics. 7. We use mathematics in studying, designing, and evaluation of systems. 8. We use mathematical modeling in solving everyday problems. 9. Group thinking and learning is more efficient than individual learning. 10. Mathematics is a web of connected ideas, concepts and skills.

Some skills of mathematical thinking 1. Mathematical intuition 2. Making arguments 3. Making assumptions 4. Describing mathematical objects 5. Recognition of common structures 1. Mathematical intuition 2. Making arguments 3. Making assumptions 4. Describing mathematical objects 5. Recognition of common structures

6. Mathematical modeling in mathematics 7. Constructing mathematical structures 8. Performing calculations 9. Mathematical modeling of everyday problems 10. Doing mathematics in different parallel categories 6. Mathematical modeling in mathematics 7. Constructing mathematical structures 8. Performing calculations 9. Mathematical modeling of everyday problems 10. Doing mathematics in different parallel categories

Arash Rastegar Mathematics Educator Young Scholars Club

How to solve a problem! It is rarely the case that advices are helpful. The only way to make sure that students use them is to make them compare and see the benefit of taking advice. One can compare the performance of students, or make a student redo the problem solving process under teachers supervision and observe the progress.

1) Writing neat and clean Make your students write their arguments. Show them their first draft and give them reasons why they would have performed better if they had written a clean draft. Tell them that, what they write represents what goes on in their mind. A clean paper represents a clean mind. Make them compare their drafts on different problems and see for themselves that the clearer they think, the cleaner they write.

2) Writing down the summary of arguments It happened often that students have the key piece of argument on hand, but they use it in the wrong place. Not writing down the summary of these arguments makes them forget to think of them in the right time. Show them examples of their own first drafts which contain the main ideas needed to complete the solution of a problem which they failed to solve.

3) Clarifying the logical structure Make students to solve a problem under your supervision while you ask them questions like: what is the nature of what we are trying to prove? What kind of argument do you expect to imply such a result? What kind of information you have on hand? How can one relate this kind of information and such implications? Can one use local or global arguments, or perform estimations to get the result?

4) Drawing big and clean figures Hand out a nice figure when your students fail to solve a problem. They will see the benefit of drawing nice figures themselves. Ask them to draw figures which suit the arguments better. Ask them to draw figures which show what happens at limiting cases. Ask them to draw figures which show different components of the arguments needed to solve the problem. Ask them to draw an appropriate figure after solving the problem in order to explain the argument to other students.

5) Recording the process of thinking It is not the case that only the final solution is of any importance to the students. Recording the process of solving the problem could help the students to know their mathematical personality and their personality of problem solving better. Help them understand better what they were trying to do while they were thinking about the problem. Supervise their thinking and ask them constantly of what they are trying to do. Help them how to summarize and record the answer.

6) Deleting irrelevant remarks and explanations Read their first drafts and delete irrelevant remarks, and explain to them why you think a particular explanation could not be helpful in solving the problem. Make them read other students drafts and correct them under your supervision.

7) Writing down side results In the course of problem solving many side results are proven which may or may not be of use in the final argument. It is wise to write down all of these little results. Side results could be forgotten when they are the most useful. Show students drafts in which forgotten side results have failed students to solve the problems.

8) Putting down the full proof after finishing the arguments Students usually write down the proof while they are still thinking about the solution. This makes them not to think about how to present the proof in written form. This is why one should stop students after solving the problem and give them a break and then ask them to decide many ways to explain the arguments and choose the most appropriate one.

9) Notifying important steps in form of lemmas Lemmas are the best tools to simplify explanation of a complicated argument. Students usually believe that there is a unique way to divide a proof into lemmas. Show them examples of proofs with many suggestions for important lemmas and ask students to decide which form is the most appropriate.

10) Considering the mind of reader Students are usually not taught the fact that they do not write for themselves only. They usually write for others to read and understand their results. Read students solution sheet and let them know about the parts which could be unclear for a reader which is ignorant of the final solution of the problem. Ask the students to write down the proof again minding what would come to the mind of a reader. Read and correct the improved version again.

Decisions to be made What makes good problem solvers successful is to make right decisions at the right time. It is not the case that students always race against time to solve problems. Sometimes they have to race against depth of vision of other people who thought about the same problems or against their own mathematical understanding and abilities regarding a specific problem. Anyhow, decision making in the process of problem solving keeps students in control of the flow of their mathematical thoughts. This makes them recognize their own mathematical personality is the course of decision making.

11) Where to start Make your student suggest many ways to start thinking about a problem. The goal is to understand the problem better. Working out illuminating examples, understanding assumptions, considering implications of the statement of the problem are examples of good starting points. Ask students to decide among several starting points and give reasonable indication why their choices are the most efficient.

12) Listing different strategies to attack the problem There are many strategies which could be chosen by students in the course of problem solving. Drawing a picture, trial and error, organizing data regarding particular cases, reducing to a few simpler problems, solving similar simpler problems, using algebra to translate the problem to symbolic language, are all examples of strategies to attack a problem. Students should decide how to attack the problem after they gain some understanding of the problem.

13) Mathematical modeling in different frameworks In many cases, there are different possibilities to mathematically formulate a problem. One could think of a problem geometrically or suggest different ways for algebraic c0-ordinatisation of the problem, or extract the combinatorial essence of a mathematical situation or translate it to analytical language. It is crucial to choose the best formulation. In some problems different parts are best to be done in different formulations.

14) Using symbols or avoiding symbols Ask students for introducing different mathematical symbols in order to solve the problem algebraically and make them decide which is better and if it is appropriate to use symbols or not. They must decide if it is appropriate to introduce new symbols or not.

15) Deciding what not to think about When racing with time, thinking about a problem from particular perspectives could be confusing. There could be mathematical languages which are unsuitable for particular problems, because they do not reveal the infra structure or because they disregard some of the important information. Students should be taught to foresee difficulties which may arise in particular approaches towards the problem.

16) Organizing the process of coming to a solution After a while when student worked out some details about the problem, it is time to formulate a solution plan. A solution plan is a strategy for attacking a problem and overcoming the obstacles and finally solving the problem. Make students to write down their solution plans for a particular problem which has many solutions and then compare their solution plans.

17) How to put down the proof Make different students put down a particular proof and then compare their drafts. This makes them decide on how to put down a particular proof. They may decide that some lemmas should have easier proofs or decide where to put a lemma or the proof of the lemma or where to put the finishing argument in order to make the poof more readable. Ask them to write down such decisions and give them guidance.

Habits to find Good habits are a result of supervised practice. Students should be provided with long term supervision and guidance in order to secure good habits of problem solving. One can teach how to become a good problem solver, but it takes time. Teachers should personally gain these habits and show off in their performance in front of the class. It is suggested that from time to time, the teacher should try to solve problems proposed by students by the board so that students have role models in problem solving.

18) Tasting the problem For students to make comparison between relevant problems, they should find an understanding of the logic of problem, estimate the level of difficulty, and locate the center of complicacy. This is an intuitive process. The more students are experienced in problem solving, the better they are in tasting a problem.

19) Gaining personal view towards the problem The process of understanding a problem is not independent of the mathematical personality of the students. Each student has personal ways to get in touch with the core of problem. Make the students translate problems into the language of personal preference and rewrite it in this new language.

20) Talking to oneself Supervise students by asking them questions which could guide them through the process of solving problems. Then make them ask these questions themselves and make students conscious of the decisions they make in the process of thinking. They should tell themselves what to do and what not to do.

21) Considering all the cases Solving a problem by case studies is a particular ability which is most effective in computational proofs. It happens often for students who are not experienced that they forget to consider some simple limiting or extreme cases which are naturally eliminated in the course of computations.

22) Checking special cases Recognizing special cases which are easy enough to simplify the computations and are complicated enough to show the core of the problem, is an intuitive art which is gained by experience. Ask the students to suggest special cases of many problems and solve these particular cases and see if this can help solution of the problem in general.

23) Performing a few steps mentally Students have different abilities in mental computations. Some of them are able to perform and relate quite a few steps mentally. This helps them to foresee the path each decision could lead to. This ability could be improved by supervised practice. Teachers should propose problems for students which are appropriate for their level of understanding to be solved mentally without the use of paper and pencil.

24) Thinking simple Nature always chooses the simplest way. This could be imitated as a rule for efficient problem solving. There are many experienced problem solvers who are unable to think simple. All the complicated strategies and computations show off in their mind when thinking about a problem and influence how they react to the problem. This is why sometimes students of young age perform better in solving some problems.

Personality of good problem solvers Good problem solvers have certain common characteristics which governs their success in this art. It is not possible to teach a particular person to have a certain personality. Some students naturally have these characteristics and some have hidden capacities which are developed by experience. The role of teachers is to face students with these personalities and motivate them to work hard for gaining them.

25) Patience It happens often that a student has all the ideas needed to solve the problem but does not have enough patience to perform the computations or to relate different ideas to each other. When this happens students tries to solve the problem by forcing it into predetermined methods instead of trying to consider as an intellectual challenge. This impatience not only taken students mind away from the solution but also makes student nervous and angry. This is why many students dislike exams.

26) Divergent thinking New ideas and scientific breakthroughs only come to the mind of researchers with divergent thinking. It has happened often that a student in a high class competition has proposed a much simpler solution to a problem that a group of experts has been working on it for hours. Divergent thinking is the main ingredient of a creative mind.

27) Criticizing conjectures Many students jump into conclusions and waste time on trying to follow the path suggested by these conclusions. The most efficient students in problem solving are those whom are self critical. They try to find different independent reasons for following a solution plan. One can criticize conjectures by considering special cases, or by philosophical considerations.

28) Looking for equivalent formulations Every student according to his or her mathematical personality chooses a particular language to think about the problem. Students are usually imprisoned in their choice of the language which is a very limiting characteristic for a problem solver. Good problem solvers are fluent in translating the set up to different languages and they do this on several occasions in the process of problem solving to think of the problem in a language most appropriate for computations.

29) Fluency in working with ideas and concepts Good problem solvers are able to solve the problem in the language of concepts without any need for reference to particular symbols or mathematical formulations. This is a very abstract ability. Also translation of an idea to the language of mathematical symbols and performing computations using these algebraic symbols is an ability of good problem solvers.

30) Looking for simpler models One shall constantly check if there is a possibility to use symbolizations which makes the problem look simpler than it is. This tendency towards using the simplest models could be recognized in many students occasionally, but only good problem solvers have developed this personality in a way that it is constantly in work.

Intuition This is a mysterious behavior which has roots in external inspirations. Intuition does not work like personality. One should concentrate deep enough on a problem to become ready for receiving unconscious inspirations. There have been scientists who believed that thinking is nothing but getting prepared for inspirations. Only a healthy and patient mind can put itself in such a position and only by working hard. Intuitive thinking and logical thing are complements of each other and as skill they are independent. One can not have a good intuition and at the same time, be weak in arguments.

31) Geometric imagination By geometric imagination we mean pictorial thinking which is against verbal or logical thinking. Good problem solvers always make pictures in their mind, even when doing discrete mathematics or algebra. The importance of pictorial intuition in mathematics is due to the geometric nature of the physical world. This does not mean that good problem solvers are not god logical thinkers.

32) Recognizing simple from difficult A good problem solver after tasting a problem, estimates the level of difficulty of the problem, and then chooses solution plans accordingly. Doing particular cases of the problem introduces good indications about the level of difficulty of the problem. Only intuition is able to judge about simplicity of a problem before solving it. Similar problems could also give help for judging accurately

33) Decomposition and reduction to simpler problems Reduction a problem to simpler problems usually can be done in several ways. Usually, it is completely unclear how one should decompose a problem to smaller ones. This is because some problems do not easily reveal their essence. This is when we need intuition to help us. Sometimes you have to translate the problem to a new language in order to make the reduction simpler.

34) Jumps of the mind It is not at all the case that the process of solving a problem can always be entirely explained and understood. There are many instances that intuition makes the mind of the problem solver jump to new perspectives which can not be easily explained by the available data. Divergent thinkers are usually better jumpers. Convergent minds always do the trivial things.

35) Estimating how much progress has been made Until the time the problem is solved, it is impossible to make sure how much progress is made. Intuition if supported by an accurate solution plan can reveal the level of progress. There are differences in opinions about recognizing parts of the solution to be more important than the other parts. Therefore this kind of estimation only provides personal information.

36) Finding the trivial propositions quickly This quality is usually recognized as being smart or a fast thinker. But it has to do with intuition and fluency in arguments and embodying them in the form of propositions. The more experienced the problem solver is, the better are the guesses about what trivial propositions could be. One should think about the nature of such propositions.

37) Formulating good conjectures Good problem solvers are often good in formalizing conjectures. A good conjecture should be tried against challenging cases and also formulated in a way that it introduces a better perspective. Intuition is the only tool our mind has possessed to introduce conjectures which are potential propositions. A good guesser chooses the best conjecture from a list of potential conjectures.

38) Being creative and directed in constructions Introducing new constructions needs intuition. Because, one needs to foresee what one is trying to construct. By creativity in construction we mean combing different elements borrowed from different construction to introduce a new construction with new abilities. By being direct we mean going straight to the action and experiencing through the constructions.

39) Understanding an idea independent of the context Students are always more comfortable with the language of mathematical symbols, rather than with the network of ideas. Only very abstract minds translate the problem to the language of concepts, and only some mathematicians are able to think through a problem only working with concepts related to the problem avoiding symbols. This is not a very concrete kind of understanding. It is rather intuitive.

40) Imagination comes before arguments and computations Many people believe that proofs give insight. But it is never the case. Always insight comes before the arguments. This is what makes the mind of a problem solver an intuitive mind before being a logical mind. Giving a proof for a proposition is like finding an address which explains how to get to the facts mentioned in the proof. Therefore, it is by no means unique. One shall get rid of it as soon as one is familiar with the surrounding area around the facts mentioned in the proposition. A problem solver who is good in arguments is good in imagination.

Stages of problem solving by atlas of concepts 1-Student translates the problem to the language of concepts and their relations. 2-Student compares the concept map to atlas of concepts and finds possible relations between these concepts which are relevant to the problem. 3-Student uses the old concepts or creates new concepts to form the expected relationships between already recognized concepts. 4-Student recognizes new relationships between concepts in the concept map. 5-Student goes through the steps 2 to 4 over and over until the concept map is extended enough to solve the problem. 6-Student summarizes the concept map to the subset which covers the solution. 7-Student translates the solution from the language of concepts and their relations to the language of the problem. 1-Student translates the problem to the language of concepts and their relations. 2-Student compares the concept map to atlas of concepts and finds possible relations between these concepts which are relevant to the problem. 3-Student uses the old concepts or creates new concepts to form the expected relationships between already recognized concepts. 4-Student recognizes new relationships between concepts in the concept map. 5-Student goes through the steps 2 to 4 over and over until the concept map is extended enough to solve the problem. 6-Student summarizes the concept map to the subset which covers the solution. 7-Student translates the solution from the language of concepts and their relations to the language of the problem.

Translation to the language of concepts and their relations. Level 1-Student recognizes the concepts relevant to the problem. Level 2-Student recognizes the relations between the concepts relevant to the problem. Level 3-Student is able to translate the assumptions of the problem to the language of concepts. Level 4-Student is able to translate the problem completely to the language of concepts. Level 1-Student recognizes the concepts relevant to the problem. Level 2-Student recognizes the relations between the concepts relevant to the problem. Level 3-Student is able to translate the assumptions of the problem to the language of concepts. Level 4-Student is able to translate the problem completely to the language of concepts.

Comparing the concept map to atlas of concepts Level 1-Student is able to find each of the concepts in his/her atlas of concepts. Level 2-Student is able to find a concept map in his/her atlas of concepts which has the same concepts as the problem. Level 3-Student compares the relations holding between the concepts in atlas and the relations between the concepts relevant to the problem. Level 4-Student chooses a concept map in his atlas best fitting to the concepts and their relations in the problem. Level 1-Student is able to find each of the concepts in his/her atlas of concepts. Level 2-Student is able to find a concept map in his/her atlas of concepts which has the same concepts as the problem. Level 3-Student compares the relations holding between the concepts in atlas and the relations between the concepts relevant to the problem. Level 4-Student chooses a concept map in his atlas best fitting to the concepts and their relations in the problem.

Using the old concepts or creating new concepts Level 1-Student recognizes concepts which are related in the atlas but not in the setting of the problem. Level 2- Student is able to relate these concepts using other concepts in the atlas compatible to the setting of the problem. Level 3-Student is able to relate these concepts by creating new concepts. Level 4-Student is able to relate these concepts by creating new concepts which are relevant to the setting of the problem. Level 1-Student recognizes concepts which are related in the atlas but not in the setting of the problem. Level 2- Student is able to relate these concepts using other concepts in the atlas compatible to the setting of the problem. Level 3-Student is able to relate these concepts by creating new concepts. Level 4-Student is able to relate these concepts by creating new concepts which are relevant to the setting of the problem.

Recognizing new relationships between concepts Level 1-Student finds relations between concepts using the concepts in the atlas. Level 2-Student relates a new concept to the old concepts. Level 3-Student relates the new concepts to each other. Level 4-Student relates the concepts in direction of the solution of the problem. Level 1-Student finds relations between concepts using the concepts in the atlas. Level 2-Student relates a new concept to the old concepts. Level 3-Student relates the new concepts to each other. Level 4-Student relates the concepts in direction of the solution of the problem.

5-Going through the steps 2 to 4 over and over Level 1-Student is able to extend the concept map of the problem. Level 2-Student extends the concept map in direction of the solution of the problem. Level 3-Student makes a decision whether he/she has to go through steps 2 to 4 again. Level 4-Students recognizes when the extended concept map is enough to solve the problem. Level 1-Student is able to extend the concept map of the problem. Level 2-Student extends the concept map in direction of the solution of the problem. Level 3-Student makes a decision whether he/she has to go through steps 2 to 4 again. Level 4-Students recognizes when the extended concept map is enough to solve the problem.

Summarizing the concept map so that it still covers the solution. Level 1-Student recognizes the portion of the map needed for solution of the problem. Level 2-Student is able to rewrite the concept map as a series of maps in which new concepts are relating the old concepts to form the old relations between concepts. Level 3-Student is able to summarize the new maps in the atlas. Level 4-Student is able to summarize the new maps in the atlas in direction of simplifying the solution. Level 1-Student recognizes the portion of the map needed for solution of the problem. Level 2-Student is able to rewrite the concept map as a series of maps in which new concepts are relating the old concepts to form the old relations between concepts. Level 3-Student is able to summarize the new maps in the atlas. Level 4-Student is able to summarize the new maps in the atlas in direction of simplifying the solution.

Translating from the language of concepts to that of the problem. Level 1-Student is able to translate the new concepts to the language of the problem. Level 2-Student is able to translate the new relations between concepts to the language of the problem. Level 3-Student is able to translate the solution to the language of the problem. Level 4-Student is able to translate the process of solving the problem to the language of the problem. Level 1-Student is able to translate the new concepts to the language of the problem. Level 2-Student is able to translate the new relations between concepts to the language of the problem. Level 3-Student is able to translate the solution to the language of the problem. Level 4-Student is able to translate the process of solving the problem to the language of the problem.

Problem Solving Perspectives Human problem solving Mathematical problem solving Learning by problem solving Teaching by problem solving Assessment of problem solving Problem solving in groups Problem solving strategies Gifted problem solvers Human problem solving Mathematical problem solving Learning by problem solving Teaching by problem solving Assessment of problem solving Problem solving in groups Problem solving strategies Gifted problem solvers

Arash Rastegar Philosophy of Science Group Sharif University of Technology

Confucius Layers of existence Human is alone Tao Essence: truth Spirit: Knowledge Heart: Faith Soul: Educate Body: Action Tao Essence: truth Spirit: Knowledge Heart: Faith Soul: Educate Body: Action Common Immortal Inward Communication with spirits Outward communication Common Immortal Inward Communication with spirits Outward communication

Buda Layers of existence Everything is a human Krishna Brahman Atman Self Body Krishna Brahman Atman Self Body Truth Annihilation-transpersonal Social spirit Social self personal Truth Annihilation-transpersonal Social spirit Social self personal

Zartosht Layers of existence Human parallel to universe Ahouramazda Divine wisdom Social soul Personal soul Body Human Layers Wisdom Soul Body Ahouramazda Divine wisdom Social soul Personal soul Body Human Layers Wisdom Soul Body Eternal First creation Soul of creation Perfect Human: Zoroastria Universe Layers: By Superposition Wisdom Soul Body Eternal First creation Soul of creation Perfect Human: Zoroastria Universe Layers: By Superposition Wisdom Soul Body

Pythagoras Layers of existence Human is divine Spirit Soul Body Human Spirit Soul Body Spirit Soul Body Human Spirit Soul Body Immortal Feelings Material: Atomism-Monad God Universal spirit Soul: Mathematical thoughts Body: Creation Immortal Feelings Material: Atomism-Monad God Universal spirit Soul: Mathematical thoughts Body: Creation

Plato Layers of existence Human-Universe isomorphic Wisdom: immortal Spirit or Soul Body Human communication: in all layers Wisdom: immortal Spirit or Soul Body Human communication: in all layers Global Wisdom Spirit or Soul: universal Body Communication with human: in all layers Global Wisdom Spirit or Soul: universal Body Communication with human: in all layers

Aristotle Human layers of existence Universal layers of existence Soul Human soul Animal soul Vegetal soul Spirit Body Soul Human soul Animal soul Vegetal soul Spirit Body God Initial Wisdom Nature: realm of communication God Initial Wisdom Nature: realm of communication

Kendi Human layers of existence Universal layers of existence Soul or spirit: Realm of communication Wisdom-Mind-Feeling Body Soul or spirit: Realm of communication Wisdom-Mind-Feeling Body Wisdom Soul or spirit Nature Wisdom Soul or spirit Nature

Farabi Human layers of existence Universal layers of existence Soul or spirit: Realm of communication Classification of forces Body Soul or spirit: Realm of communication Classification of forces Body Wisdom 11 layers of existence Soul or spirit Nature Wisdom 11 layers of existence Soul or spirit Nature

Avicenna Human layers of existence Universal layers of existence Wisdom 11 layers of existence Soul or spirit: Human soul Classification of powers Animal soul Classification of powers Vegetal soul Classification of powers Body Wisdom 11 layers of existence Soul or spirit: Human soul Classification of powers Animal soul Classification of powers Vegetal soul Classification of powers Body Wisdom 11 layers of existence Universal Soul or spirit Nature Wisdom 11 layers of existence Universal Soul or spirit Nature

Ibn-Heitham Human layers of existence Universal layers of existence Soul Human soul Animal soul Vegetal soul Spirit Body Human is part of nature Soul Human soul Animal soul Vegetal soul Spirit Body Human is part of nature Nature: Independent of Human Nature: Independent of Human

Ghazzali Human layers of existence Universal layers of existence Essence, wisdom, spirit, heart, soul Body Human is image of God Essence, wisdom, spirit, heart, soul Body Human is image of God Wisdom Soul or spirit Nature Wisdom Soul or spirit Nature

Ibn-Roshd Human layers of existence Universal layers of existence Soul Human soul Animal soul Vegetal soul Spirit Body Soul Human soul Animal soul Vegetal soul Spirit Body God Initial Wisdom Nature: realm of communication God Initial Wisdom Nature: realm of communication

Sohrevardi Human layers of existence Universal layers of existence Light many layers of existence Soul or spirit light or darkness Avicennas classification Towards light Body Light many layers of existence Soul or spirit light or darkness Avicennas classification Towards light Body Light many layers of existence Universal Soul or spirit also light Towards darkness Darkness Light many layers of existence Universal Soul or spirit also light Towards darkness Darkness

Ibn-Arabi Human layers of existence Universal layers of existence Light many layers of existence Human is a mirror of Gods names Light many layers of existence Human is a mirror of Gods names Light many layers of existence Human is the spirit of universe Light many layers of existence Human is the spirit of universe

Nasireddin Toosi Human layers of existence Universal layers of existence Wisdom Soul or spirit: Essence or personality Human soul Classification of powers Animal soul Classification of powers Vegetal soul Classification of powers Body Wisdom Soul or spirit: Essence or personality Human soul Classification of powers Animal soul Classification of powers Vegetal soul Classification of powers Body Wisdom Soul Nature Wisdom Soul Nature

Molla-Sadra Human layers of existence Universal layers of existence many layers of existence Wisdom Soul Body many layers of existence Wisdom Soul Body many layers of existence Wisdom Soul Body many layers of existence Wisdom Soul Body

Saint Agustine Human layers of existence Universal layers of existence Essence, wisdom, spirit, heart, soul: Global concepts Body :Feeling Human could incarnate through history Essence, wisdom, spirit, heart, soul: Global concepts Body :Feeling Human could incarnate through history Wisdom Soul or spirit Nature Wisdom Soul or spirit Nature

Saint Thomas Aquinas Human layers of existence Universal layers of existence Soul Human soul Animal soul Vegetal soul Spirit Body Humanity is incarnation of Christ in history Soul Human soul Animal soul Vegetal soul Spirit Body Humanity is incarnation of Christ in history God Initial Wisdom Nature God Initial Wisdom Nature

Rene Descartes Human layers of existence Mind: immortal Body Mind: immortal Body

Thomas Hobbes Layers of existence Human soul is material

Brauch Spinoza Layers of existence Divine essence Body Divine essence Body

Gottfried Leibniz Layers of existence Everything is made of monad

John Locke Layers of existence We should not study humans We should study what he/she understands We should not study humans We should study what he/she understands

Emanuel Kant Layers of existence

Friedrich Hegel Layers of existence

Charles Darwin Layers of existence

Karl Marx Layers of existence Human is a social entity

Friedrich Nietzsche Layers of existence We can not understand the essence

Henri Bergson Layers of existence Wisdom and vision

Martin Heidegger Layers of existence Existence through time

A Suggested Hierarchy Human layers of existence Universe layers of existence Essence Light Wisdom Spirit Heart Soul Body Essence Light Wisdom Spirit Heart Soul Body Essence Light Wisdom Spirit Heart Soul Body Essence Light Wisdom Spirit Heart Soul Body

Arash Rastegar Educator Ministry of Science, Research and Technology

Different Languages in action 1- The language of formulas and symbols. 2- The language of mathematical concepts and their relations. 3- The language of growth and deformation of mathematical concepts. 4- The language of deformation space which is the ambient space for concepts growth and deformation. 5- The language of the logical system which is relevant to the problem. 6- The language of connection which exists between our mind and logical system of mathematics. 7- The language of creation of mathematical systems 1- The language of formulas and symbols. 2- The language of mathematical concepts and their relations. 3- The language of growth and deformation of mathematical concepts. 4- The language of deformation space which is the ambient space for concepts growth and deformation. 5- The language of the logical system which is relevant to the problem. 6- The language of connection which exists between our mind and logical system of mathematics. 7- The language of creation of mathematical systems

Formulas and symbols Level 0- Student is not able to work with formulas and symbols, and is not familiar with the rules governing them. Level 1- Student has translated the problem to the language of formulas and symbols, but is not able to compute or make an argument. Level 2- Student has solved the problem accurately after translating it to mathematical symbols. Level 3- Student is able to translate the solution to the language of the problem. Level 4- Students has considered all the concepts behind every step of calculations and arguments. Level 0- Student is not able to work with formulas and symbols, and is not familiar with the rules governing them. Level 1- Student has translated the problem to the language of formulas and symbols, but is not able to compute or make an argument. Level 2- Student has solved the problem accurately after translating it to mathematical symbols. Level 3- Student is able to translate the solution to the language of the problem. Level 4- Students has considered all the concepts behind every step of calculations and arguments.

Mathematical concepts and their relations Level 0- Student has considered the concepts and the relations behind calculations and arguments and the relations between symbols. Level 1- The process of problem solving is affected by the relations discovered between the relevant concepts. Level 2- Student has chosen key concepts and has solved the problem by considering their relation to other concepts relevant to the problem. Level 3- Student has considered the relations between concepts, during the process of problem solving and after the problem is solved. Level 4- Student has solved the problem and understands the problem and solution in terms of concepts and their relations and independent of formulas and symbols. Level 0- Student has considered the concepts and the relations behind calculations and arguments and the relations between symbols. Level 1- The process of problem solving is affected by the relations discovered between the relevant concepts. Level 2- Student has chosen key concepts and has solved the problem by considering their relation to other concepts relevant to the problem. Level 3- Student has considered the relations between concepts, during the process of problem solving and after the problem is solved. Level 4- Student has solved the problem and understands the problem and solution in terms of concepts and their relations and independent of formulas and symbols.

Growth and deformation of mathematical concepts Level 0- After solving the problem some of the concepts are grown and understood better. Level 1- Student has considered the relations between the extended concepts and other concepts relevant to the problem. Level 2- Student has considered the new relations through the process of problem solving, after the problem is solved. Level 3- After solution of the problem, some concepts are extended so much that they have absorbed other concepts as a special case. Level 4- Student has compared the relations between concepts after and before the solution. Level 0- After solving the problem some of the concepts are grown and understood better. Level 1- Student has considered the relations between the extended concepts and other concepts relevant to the problem. Level 2- Student has considered the new relations through the process of problem solving, after the problem is solved. Level 3- After solution of the problem, some concepts are extended so much that they have absorbed other concepts as a special case. Level 4- Student has compared the relations between concepts after and before the solution.

Deformation space which is the ambient space for concepts Level 0- Student has interfered the process of growth and deformation of concepts. Level 1- Student has affected the growth and deformation of concepts in direction of solving the problem. Level 2- Student understands the limitations in growth and deformation of concepts. Level 3- Student has taken advantage of the relations of concepts in growth and deforming particular concepts. Level 4- Student successfully guides growth of concepts in a way that it absorbs a pre-destined concept as a special case. Level 0- Student has interfered the process of growth and deformation of concepts. Level 1- Student has affected the growth and deformation of concepts in direction of solving the problem. Level 2- Student understands the limitations in growth and deformation of concepts. Level 3- Student has taken advantage of the relations of concepts in growth and deforming particular concepts. Level 4- Student successfully guides growth of concepts in a way that it absorbs a pre-destined concept as a special case.

Logical system which is relevant to the problem Level 0- Students recognizes a system isomorphic but different from the mathematical system of the problem. Level 1- Student is able to translate the data from this system to the system of the problem. Level 2- Student uses the translation from one logical system to the other in solution of the problem. Level 3- Student accurately translates the problem to an isomorphism logical system. Level 4- Student understands the logical system of the problem independent of concepts and their relations. Level 0- Students recognizes a system isomorphic but different from the mathematical system of the problem. Level 1- Student is able to translate the data from this system to the system of the problem. Level 2- Student uses the translation from one logical system to the other in solution of the problem. Level 3- Student accurately translates the problem to an isomorphism logical system. Level 4- Student understands the logical system of the problem independent of concepts and their relations.

Connection between our mind and the logical system Level 0- Student is able to reconstruct a logical system in his/her mind which is similar to the logical system of the problem. Level 1- Student directly understands the logical system in his/her mind in a way that is useful in solution of the problem. Level 2- Student has taken role in reconstruction of the logical system in his/her mind. Level 3- Student has directed the process of reconstruction of the logical system in his/her mind in a way that is useful to the solution of the problem. Level4- The reconstruction of the logical system by student leads to the solution. Level 0- Student is able to reconstruct a logical system in his/her mind which is similar to the logical system of the problem. Level 1- Student directly understands the logical system in his/her mind in a way that is useful in solution of the problem. Level 2- Student has taken role in reconstruction of the logical system in his/her mind. Level 3- Student has directed the process of reconstruction of the logical system in his/her mind in a way that is useful to the solution of the problem. Level4- The reconstruction of the logical system by student leads to the solution.

Creation of logical systems Level 0- Student has expanded the logical system of the problem. Level 1- Student has connected the logical system of the problem to other mathematical system in order to expand it. Level 2- Expansion of the system is lead to generalization of the problem. Level 3- Student has compared the expanded system with other mathematical systems and related the generalization of the problem to similar problems in other systems. Level 4- Student has created a new mathematical system to solve the problem. Level 0- Student has expanded the logical system of the problem. Level 1- Student has connected the logical system of the problem to other mathematical system in order to expand it. Level 2- Expansion of the system is lead to generalization of the problem. Level 3- Student has compared the expanded system with other mathematical systems and related the generalization of the problem to similar problems in other systems. Level 4- Student has created a new mathematical system to solve the problem.

arash rastegar department of mathematical sciences sharif university of technology

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