Discovering mathematics
Motivation
Discovering Mathematics
Summary
Discovering Mathematics
Marie Demlová
Czech Technical University, Prague
Kuopio, September 18, 2004
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Contents
Motivation
Why Mathematics
Mathematical Education
Discovering Mathematics
What it is
For whom it is
Structure of a module
Examples
Summary
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
’... mathematics has two faces;... Mathematics presented in the
Euclidean way appears as a systematic, deductive science; but
mathematics in the making appears as an experimental, inductive
science. Both aspects are as old as mathematics itself.’
G. Polya
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
’... mathematics has two faces;... Mathematics presented in the
Euclidean way appears as a systematic, deductive science; but
mathematics in the making appears as an experimental, inductive
science. Both aspects are as old as mathematics itself.’
G. Polya
’A problem? If you can solve it, it is an exercise; otherwise it’s a
research topic’.
R. Bellman
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Why Mathematics
Mathematical Education
Problem solving includes:
◮
◮
◮
◮
◮
experiments
trial-and-error
simplification
analogy
abstraction
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Problem solving includes:
◮
◮
◮
◮
◮
◮
Why Mathematics
Mathematical Education
experiments
trial-and-error
simplification
analogy
abstraction
Often succesfully replaced by mathematical model.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Problem solving includes:
◮
◮
◮
◮
◮
◮
◮
Why Mathematics
Mathematical Education
experiments
trial-and-error
simplification
analogy
abstraction
Often succesfully replaced by mathematical model.
A mathematical model is a collection of
◮
◮
◮
concepts,
their attributes,
their interrelations
such that the behavior of the object is imitated.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
What is learning
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
What is learning
◮
’Learning is seen as the individual coming to new ways of
conceptualising, comprehending, seeing or understanding the
phenomenon under study; coming to see new features and
relate them to one another and to the whole, as well as to the
wider world.’
S. Booth
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
◮
separately taught maths topics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
◮
separately taught maths topics
’The result [of teaching] is that nobody can use anything
from the material even in the simplest examples ...’
R. Feynman
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
estimate of difficulty
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
◮
a textbook of a specific mathematical topic
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
◮
a textbook of a specific mathematical topic
◮
an exercise book
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
◮
Two modules finished
◮
◮
Finite Sums
Infinite Sequences
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
◮
Two modules finished
◮
◮
◮
Finite Sums
Infinite Sequences
Authors of the two modules:
Jiřı́ Gregor et al.
from Department of Mathematics,
Faculty of Electrical Engineering,
Czech Technical University, Prague
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
want to use mathematics in a creative way
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
◮
want to use mathematics in a creative way
need to know how to solve problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
◮
◮
want to use mathematics in a creative way
need to know how to solve problems
want to deepen their understanding of maths
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
◮
Theory of equations (differential and difference)
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
◮
Theory of equations (differential and difference)
◮
Discrete mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
Can’t solve a problem?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
◮
Have you solved a problem?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
◮
◮
Have you solved a problem?
Try more difficult ones.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
Suggestions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
Suggestions
Problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
Basic definitions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
Basic definitions
Some useful theorems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Hints how to use Mathematica
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Hints how to use Mathematica
Answers to problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
◮
Find the number of integer points lying on a segment
(including end points) with the given integer end points in a
plane. How would you characterize a segment containing no
integer point except its end points?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
◮
Find the number of integer points lying on a segment
(including end points) with the given integer end points in a
plane. How would you characterize a segment containing no
integer point except its end points?
◮
Optimists say that a savings account with an interest rate of
p % p.a. will double the initial investment in 70
p years.
70
Pesimists say that the prices will double r years when an r %
inflation rate can be predicted. Can you give some reasons for
either of the two opinions?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
↓3
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
↓3
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
↑ 1, ↑ 2
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
↓3
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
↓3
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
↑ 1, ↑ 2
The structure is represented by hyperlinks.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
◮
Without Discovering Maths
no real maths education!
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
◮
Without Discovering Maths
no real maths education!
◮
:-)
Marie Demlová
Discovering Mathematics
Discovering Mathematics
Summary
Discovering Mathematics
Marie Demlová
Czech Technical University, Prague
Kuopio, September 18, 2004
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Contents
Motivation
Why Mathematics
Mathematical Education
Discovering Mathematics
What it is
For whom it is
Structure of a module
Examples
Summary
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
’... mathematics has two faces;... Mathematics presented in the
Euclidean way appears as a systematic, deductive science; but
mathematics in the making appears as an experimental, inductive
science. Both aspects are as old as mathematics itself.’
G. Polya
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
’... mathematics has two faces;... Mathematics presented in the
Euclidean way appears as a systematic, deductive science; but
mathematics in the making appears as an experimental, inductive
science. Both aspects are as old as mathematics itself.’
G. Polya
’A problem? If you can solve it, it is an exercise; otherwise it’s a
research topic’.
R. Bellman
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Why Mathematics
Mathematical Education
Problem solving includes:
◮
◮
◮
◮
◮
experiments
trial-and-error
simplification
analogy
abstraction
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Problem solving includes:
◮
◮
◮
◮
◮
◮
Why Mathematics
Mathematical Education
experiments
trial-and-error
simplification
analogy
abstraction
Often succesfully replaced by mathematical model.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
◮
Problem solving includes:
◮
◮
◮
◮
◮
◮
◮
Why Mathematics
Mathematical Education
experiments
trial-and-error
simplification
analogy
abstraction
Often succesfully replaced by mathematical model.
A mathematical model is a collection of
◮
◮
◮
concepts,
their attributes,
their interrelations
such that the behavior of the object is imitated.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
What is learning
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Mathematical Education
Maths education involves
◮
knowledge of concepts
i.e. notions, definitions, theorems, etc.
◮
ability to use the concepts to solve problems
What is learning
◮
’Learning is seen as the individual coming to new ways of
conceptualising, comprehending, seeing or understanding the
phenomenon under study; coming to see new features and
relate them to one another and to the whole, as well as to the
wider world.’
S. Booth
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
◮
separately taught maths topics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Why Mathematics
Mathematical Education
Troubles of math education:
◮
decline of mathematical abilities and skills
of secondary students
◮
surface learning approach to learning maths
◮
assessments more of skills than of understanding
◮
lack of undestanding
◮
separately taught maths topics
’The result [of teaching] is that nobody can use anything
from the material even in the simplest examples ...’
R. Feynman
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
estimate of difficulty
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
◮
a textbook of a specific mathematical topic
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
What is Discovering Mathematics
◮
Collection of modules
◮
Each module is a network of interrelated problems
A problem is equipped with
◮
◮
◮
◮
◮
◮
estimate of difficulty
lists of predecessors and successors
theoretical background
suggested plan of solution (if needed)
solution
What is not Discovering Mathematics
◮
a textbook of a specific mathematical topic
◮
an exercise book
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
◮
Two modules finished
◮
◮
Finite Sums
Infinite Sequences
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Present state
◮
Two modules finished
◮
◮
◮
Finite Sums
Infinite Sequences
Authors of the two modules:
Jiřı́ Gregor et al.
from Department of Mathematics,
Faculty of Electrical Engineering,
Czech Technical University, Prague
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
want to use mathematics in a creative way
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
◮
want to use mathematics in a creative way
need to know how to solve problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
For whom it is
Discovering Mathematics is for
◮
teachers of mathematics for non-mathematical specialists
◮
interested students willing to understand
all those who does not have deep maths knowledge and
◮
◮
◮
◮
want to use mathematics in a creative way
need to know how to solve problems
want to deepen their understanding of maths
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
◮
Theory of equations (differential and difference)
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Requirements
A user should have basic knowledge of
◮
Calculus
◮
Elementary and analytic geometry
◮
Linear algebra
◮
Theory of equations (differential and difference)
◮
Discrete mathematics
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
Can’t solve a problem?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
◮
Have you solved a problem?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Role of precedesors and successors
◮
Preceding problems
◮
◮
◮
Can’t solve a problem?
Try it’s predecessors.
Succeeding problems
◮
◮
Have you solved a problem?
Try more difficult ones.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
Suggestions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
Suggestions
Problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
Basic definitions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
Basic definitions
Some useful theorems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Hints how to use Mathematica
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Technically each module consists of
◮ Part I - Examples
◮
◮
◮
Suggestions
Problems
Part II - Supportive items
◮
◮
◮
◮
◮
◮
Basic definitions
Some useful theorems
Plans of solutions
Further references
Hints how to use Mathematica
Answers to problems
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
◮
Find the number of integer points lying on a segment
(including end points) with the given integer end points in a
plane. How would you characterize a segment containing no
integer point except its end points?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Examples from module Finite Sums
◮
How many elements belong to the set
T [n] = {(i, k); i ≥ 0, k ≥ 0, i + k < n}
provided that i, k and the fixed number n are integers.
◮
Find the number of integer points lying on a segment
(including end points) with the given integer end points in a
plane. How would you characterize a segment containing no
integer point except its end points?
◮
Optimists say that a savings account with an interest rate of
p % p.a. will double the initial investment in 70
p years.
70
Pesimists say that the prices will double r years when an r %
inflation rate can be predicted. Can you give some reasons for
either of the two opinions?
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
↓3
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
↓3
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
↑ 1, ↑ 2
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
What it is
For whom it is
Structure of a module
Examples
Example of the structure of successors and predecessors
1. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y2 ), (x2 , y1 ).
↓3
2. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the rectangular triangle with vertices
(x1 , y1 ), (x1 , y2 ), (x2 , y1 ).
↓3
3. Let xi , yi , i = 1, 2 be integers. Find the number of integer
points located inside the trapezium with vertices
(x1 , 0), (x1 , y1 ), (x2 , y2 ), (x2 , 0).
↑ 1, ↑ 2
The structure is represented by hyperlinks.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
◮
Without Discovering Maths
no real maths education!
Marie Demlová
Discovering Mathematics
Motivation
Discovering Mathematics
Summary
Summary
◮
A problem is not used to illustrate a theory.
◮
Instead, several theories are used to solve a problem
◮
Without Discovering Maths
no real maths education!
◮
:-)
Marie Demlová
Discovering Mathematics