Arash Rastegar
Department of Mathematical Sciences
Sharif University of Technology
There are two kinds of
Mathematicians
 Problem Solvers
 Theoriticians
Contribute
Contribute
Local to global
point of view
Global to local
point of view
Interesting
special cases
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
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. Student’s 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.
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 learner’s 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 student’s 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.
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.
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.
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.
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.
Perspectives towards
mathematics education
 1. Mathematics education has an important role in
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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
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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
 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 teacher’s 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 student’s 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 student’s 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
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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.
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.
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.
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.
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.
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.
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.
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
…
Arash Rastegar
Philosophy of Science Group
Sharif University of Technology
Confucius
Layers of existence
Human is alone
 Tao Essence: truth
 Common
 Spirit: Knowledge
 Immortal
 Heart: Faith
 Inward Communication with
 Soul: Educate
spirits
 Outward communication
 Outward communication
 Body: Action
Buda
Layers of existence
Everything is a human
 Krishna
 Truth
 Brahman
 Annihilation-transpersonal
 Atman
 Social spirit
 Self
 Social self
 Body
 personal
Zartosht
Layers of existence
Human parallel to universe









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
Pythagoras
Layers of existence
Human is divine
 Spirit
 Immortal
 Soul
 Feelings
 Body
 Material: Atomism-Monad
Human
 Spirit
 Soul
 Body
God
 Universal spirit
 Soul: Mathematical thoughts
 Body: Creation
Plato
Layers of existence
Human-Universe isomorphic
 Wisdom: immortal
 Global Wisdom
 Spirit or Soul
 Spirit or Soul: universal
 Body
 Body
 Human communication:
 Communication with human:
in all layers
in all layers
Aristotle
Human layers of existence
Universal layers of existence
 Soul
 God
Human soul
Animal soul
Vegetal soul
 Spirit
 Body
 Initial Wisdom
 Nature: realm of
communication
Kendi
Human layers of existence
Universal layers of existence
 Wisdom
 Soul or spirit
 Soul or spirit:
Realm of communication
Wisdom-Mind-Feeling
 Body
 Nature
Farabi
Human layers of existence
Universal layers of existence
 Wisdom
11 layers of existence
 Soul or spirit
 Soul or spirit:
Realm of communication
Classification of forces
 Body
 Nature
Avicenna
Human layers of existence
Universal layers of existence
 Wisdom
 Wisdom
11 layers of existence
 Soul or spirit:
Human soul
Classification of powers
Animal soul
Classification of powers
Vegetal soul
Classification of powers
 Body
11 layers of existence
 Universal Soul or spirit
 Nature
Ibn-Heitham
Human layers of existence
Universal layers of existence
 Soul
 Nature:
Human soul
Animal soul
Vegetal soul
 Spirit
 Body
Human is part of nature
Independent of Human
Ghazzali
Human layers of existence
Universal layers of existence
 Essence, wisdom, spirit,
 Wisdom
heart, soul
 Body
 Soul or spirit
 Nature
‫ جبروت‬
‫ ملکوت‬
‫ ملک‬
Human is image of God
Ibn-Roshd
Human layers of existence
Universal layers of existence
 Soul
 God
Human soul
Animal soul
Vegetal soul
 Spirit
 Body
 Initial Wisdom
 Nature: realm of
communication
Sohrevardi
Human layers of existence
Universal layers of existence
 Light
 Light
many layers of existence
 Soul or spirit
light or darkness
Avicenna’s classification
many layers of existence
 Universal Soul or spirit
also light
Towards light
Towards darkness
 Body
 Darkness
Ibn-Arabi
Human layers of existence
Universal layers of existence
 Light
 Light
many layers of existence
Human is a mirror of God’s
names
many layers of existence
Human is the spirit of
universe
Nasireddin Toosi
Human layers of existence
Universal layers of existence
 Wisdom
 Wisdom
 Soul or spirit:
 Soul
Essence or personality
Human soul
Classification of powers
Animal soul
Classification of powers
Vegetal soul
Classification of powers
 Body
 Nature
Molla-Sadra
Human layers of existence
Universal layers of existence
 many layers of existence
 many layers of existence
 Wisdom
 Wisdom
 Soul
 Soul
 Body
 Body
Saint Agustine
Human layers of existence
Universal layers of existence
 Essence, wisdom, spirit,
 Wisdom
heart, soul: Global concepts
 Body :Feeling
 Soul or spirit
Human could incarnate
through history
 Nature
Saint Thomas Aquinas
Human layers of existence
Universal layers of existence
 Soul
 God
Human soul
Animal soul
Vegetal soul
 Spirit
 Body
Humanity is incarnation of
Christ in history
 Initial Wisdom
 Nature
Rene Descartes
Human layers of existence
 Mind: immortal
 Body
Thomas Hobbes
Layers of existence
 Human soul is material
Brauch Spinoza
Layers of existence
 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
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
 Essence
 Light
 Light
 Wisdom
 Wisdom
 Spirit
 Spirit
 Heart
 Heart
 Soul
 Soul
 Body
 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
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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
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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
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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
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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
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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
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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
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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
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
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.