Arash Rastegar Department of Mathematical Sciences Sharif
University of Technology
Slide 2
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
Slide 3
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
Slide 4
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.
Slide 5
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.
Slide 6
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.
Slide 7
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.
Slide 8
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.
Slide 9
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.
Slide 10
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.
Slide 11
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.
Slide 12
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.
Slide 13
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.
Slide 14
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.
Slide 15
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.
Slide 16
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.
Slide 17
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.
Slide 18
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.
Slide 19
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.
Slide 20
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
Slide 21
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
Slide 22
Arash Rastegar Mathematics Educator Young Scholars Club
Slide 23
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.
Slide 24
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.
Slide 25
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.
Slide 26
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?
Slide 27
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.
Slide 28
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.
Slide 29
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.
Slide 30
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.
Slide 31
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.
Slide 32
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.
Slide 33
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.
Slide 34
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.
Slide 35
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.
Slide 36
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.
Slide 37
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.
Slide 38
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.
Slide 39
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.
Slide 40
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.
Slide 41
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.
Slide 42
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.
Slide 43
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.
Slide 44
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.
Slide 45
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.
Slide 46
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.
Slide 47
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.
Slide 48
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.
Slide 49
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.
Slide 50
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.
Slide 51
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.
Slide 52
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.
Slide 53
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.
Slide 54
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.
Slide 55
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.
Slide 56
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.
Slide 57
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.
Slide 58
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.
Slide 59
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
Slide 60
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.
Slide 61
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.
Slide 62
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.
Slide 63
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.
Slide 64
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.
Slide 65
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.
Slide 66
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.
Slide 67
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.
Slide 68
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.
Slide 69
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.
Slide 70
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.
Slide 71
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.
Slide 72
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.
Slide 73
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.
Slide 74
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.
Slide 75
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.
Slide 76
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
Slide 77
Arash Rastegar Philosophy of Science Group Sharif University of
Technology
Slide 78
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
Slide 79
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
Slide 80
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
Slide 81
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
Slide 82
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
Slide 83
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
Slide 84
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
Slide 85
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
Slide 86
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
Slide 87
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
Slide 88
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
Slide 89
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
Slide 90
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
Slide 91
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
Slide 92
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
Slide 93
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
Slide 94
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
Slide 95
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
Slide 96
Rene Descartes Human layers of existence Mind: immortal Body
Mind: immortal Body
Slide 97
Thomas Hobbes Layers of existence Human soul is material
Slide 98
Brauch Spinoza Layers of existence Divine essence Body Divine
essence Body
Slide 99
Gottfried Leibniz Layers of existence Everything is made of
monad
Slide 100
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
Slide 101
Emanuel Kant Layers of existence
Slide 102
Friedrich Hegel Layers of existence
Slide 103
Charles Darwin Layers of existence
Slide 104
Karl Marx Layers of existence Human is a social entity
Slide 105
Friedrich Nietzsche Layers of existence We can not understand
the essence
Slide 106
Henri Bergson Layers of existence Wisdom and vision
Slide 107
Martin Heidegger Layers of existence Existence through
time
Slide 108
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
Slide 109
Arash Rastegar Educator Ministry of Science, Research and
Technology
Slide 110
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
Slide 111
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.
Slide 112
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.
Slide 113
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.
Slide 114
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.
Slide 115
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.
Slide 116
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.
Slide 117
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.