Discovering mathematics

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Discovering Mathematics

Marie Demlov´a

Czech Technical University, Prague

Kuopio, September 18, 2004

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Motivation

Structure of a module Examples

Summary

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’... 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

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’... 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

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◮ experiments

◮ trial-and-error

◮ simplification

◮ analogy

◮ abstraction

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Problem solving includes:

◮ Often succesfully replaced by mathematical model.

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◮ 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.

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Maths education involves

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◮ knowledge of concepts

i.e. notions, definitions, theorems, etc.

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◮ ability to use the concepts to solve problems

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What is learning

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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

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Troubles of math education:

◮ decline of mathematical abilities and skills of secondary students

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◮ surface learning approach to learning maths

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◮ assessments more of skills than of understanding

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◮ lack of undestanding

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◮ separately taught maths topics

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◮ separately taught maths topics

’The result [of teaching] is that nobody can use anything from the material even in the simplest examples ...’

R. Feynman

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Collection of modules

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◮ Collection of modules

◮ Each module is a network of interrelated problems

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◮ Each module is a network of interrelated problems ◮ A problem is equipped with

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◮ estimate of difficulty

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◮ lists of predecessors and successors

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◮ lists of predecessors and successors

◮ theoretical background

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◮ lists of predecessors and successors ◮ theoretical background

◮ suggested plan of solution (if needed)

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◮ suggested plan of solution (if needed)

◮ solution

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◮ suggested plan of solution (if needed) ◮ solution

Whatis notDiscovering Mathematics

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◮ a textbook of a specific mathematical topic

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◮ a textbook of a specific mathematical topic

◮ an exercise book

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◮ Two modules finished

◮ Finite Sums

◮ Infinite Sequences

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◮ Two modules finished

◮ Finite Sums ◮ Infinite Sequences

◮ Authors of the two modules: Jiˇr´ı Gregor et al.

from Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Prague

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Discovering Mathematics is for

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◮ teachers of mathematics for non-mathematical specialists

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◮ interested students willing to understand

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◮ all those who does not have deep maths knowledge and

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◮ want to use mathematics in a creative way

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◮ want to use mathematics in a creative way

◮ need to know how to solve problems

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◮ want to use mathematics in a creative way ◮ need to know how to solve problems

◮ want to deepen their understanding of maths

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A user should have basic knowledge of

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◮ Calculus

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◮ Calculus

◮ Elementary and analytic geometry

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◮ Calculus

◮ Linear algebra

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◮ Calculus

◮ Linear algebra

◮ Theory of equations (differential and difference)

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◮ Calculus

◮ Linear algebra

◮ Theory of equations (differential and difference)

◮ Discrete mathematics

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◮ Preceding problems

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◮ Can’t solve a problem?

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◮ Can’t solve a problem?

◮ Try it’s predecessors.

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◮ Can’t solve a problem? ◮ Try it’s predecessors.

◮ Succeeding problems

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◮ Have you solved a problem?

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◮ Have you solved a problem?

◮ Try more difficult ones.

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Technically each module consists of

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Technically each module consists of ◮ Part I - Examples

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◮ Suggestions

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◮ Suggestions

◮ Problems

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◮ Suggestions ◮ Problems

◮ Part II - Supportive items

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◮ Basic definitions

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◮ Basic definitions

◮ Some useful theorems

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◮ Basic definitions ◮ Some useful theorems

◮ Plans of solutions

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◮ Basic definitions ◮ Some useful theorems ◮ Plans of solutions

◮ Further references

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◮ Basic definitions ◮ Some useful theorems ◮ Plans of solutions ◮ Further references

◮ Hints how to use Mathematica

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◮ Basic definitions ◮ Some useful theorems ◮ Plans of solutions ◮ Further references

◮ Hints how to use Mathematica

◮ Answers to problems

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◮ 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 numbern are integers.

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◮ How many elements belong to the set

T[n] ={(i,k);i ≥0,k ≥0,i+k <n}

◮ 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?

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◮ How many elements belong to the set

T[n] ={(i,k);i ≥0,k ≥0,i+k <n}

◮ 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.

Pesimists say that the prices will double 70_r years when an r% inflation rate can be predicted. Can you give some reasons for either of the two opinions?

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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).

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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).

points located inside the rectangular triangle with vertices (x1,y1), (x1,y2), (x2,y1).

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(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).

points located inside the trapezium with vertices (x1,0), (x1,y1), (x2,y2), (x2,0).

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(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

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(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.

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◮ A problem is not used to illustrate a theory.

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◮ Instead, several theories are used to solve a problem

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◮ Without Discovering Maths no real maths education!

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◮ Without Discovering Maths no real maths education!

◮ :-)