Introduction. Islands. Afterword. The maximum number of rectangular islands -. Experiments in the classroom. Attila Máder. University of Szeged, Bolyai Institute.
Introduction Islands Afterword
The maximum number of rectangular islands Experiments in the classroom Attila M´ader University of Szeged, Bolyai Institute
CSM - The First Conference of PhD Students in Mathematics June 29 - July 2, 2010.
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Motto
”... mathematics apperas here as a close relative to the natural sciences, as a sort of ”observational science” in which observation and analogy may lead to discoveries ...” ¨ rgy Po ´ lya: Mathematical Discovery Gyo
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
Islands in 1 dimension ¨ ldes and N. M. Singhi: On instantaneous codes, Journal S. Fo of Combinatorics, Information and System Sci., 31 (2006), 317-326.
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
1 × n board
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
1 × n board
Islands - 1 × n board We say that some consecutive cells constitute an island, if the integers in them are all greater than the integers in the neighbouring cells.
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
Islands in 2 dimension ´ bor Cze ´dli: The number of rectangular islands by means of Ga distributive lattices, European Journal of Combinatorics 30 (2009), 208-215. [1]
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background Definition: Islands on a m × n board We say that some consecutive cells forming a rectangle constitute an island, if the integers in them are all greater then the integers in the neighbouring cells.
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
Theorem: Islands on a m × n board The maximum number of rectangular islands in a m × n rectangular board on square grid: mn + m + n + 1 f (m; n) = 2
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Historical background
Theorem: Islands on a m × n board The maximum number of rectangular islands in a m × n rectangular board on square grid: mn + m + n + 1 f (m; n) = 2 Proof: Islands on a m × n board Using ”higher” mathematics: lattice theoretical methods
classroom
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
What’s the problem? We would like to the teach ”science” (e.g. islands), but the rigorous ”definition - example - theorem - proof” maths does not work in the classroom.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
What’s the problem? We would like to the teach ”science” (e.g. islands), but the rigorous ”definition - example - theorem - proof” maths does not work in the classroom. Lattice theory couldn’t be used also.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
What’s the problem? We would like to the teach ”science” (e.g. islands), but the rigorous ”definition - example - theorem - proof” maths does not work in the classroom. Lattice theory couldn’t be used also. Let us use Experimental Mathematics!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview
The first step
Computers and Mathematics Notices of the American Mathematical Society (Keith Devlin)
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview Experimental Mathematics - Journal
Experimental Mathematics - 1992 ”Theory and experiment feed on each other, and the mathematical community stands to benefit from a more complete exposure to the experimental process.”
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview Experimental Mathematics - Books
Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004)
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3]
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
4
Testing and especially falsifying conjecture
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
4
Testing and especially falsifying conjecture
5
Exploring a possible result to see if it is worth a formal proof
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
4
Testing and especially falsifying conjecture
5
Exploring a possible result to see if it is worth a formal proof
6
Suggesting approaches for formal proof
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
4
Testing and especially falsifying conjecture
5
Exploring a possible result to see if it is worth a formal proof
6
Suggesting approaches for formal proof
7
Replacing lengthy hand derivations with computer based derivations
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experimental Mathematics - a short overview The methodology of Experimental Mathematics [3] 1
Gaining insight and intuition
2
Discovering new patterns and relationships
3
Using graphical displays to suggest underlying mathematical principles
4
Testing and especially falsifying conjecture
5
Exploring a possible result to see if it is worth a formal proof
6
Suggesting approaches for formal proof
7
Replacing lengthy hand derivations with computer based derivations
8
Confirming analytically derived results
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
A little bit of didactics summary, before starting
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
A little bit of didactics summary, before starting Let us use IBL/T techniques!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
In the classroom.... - What can be done in a classroom?
A little bit of didactics summary, before starting Let us use IBL/T techniques! Let us use the computer not only instead of chalk, but as a real (experimental) mathematician!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
The first step: Catching the attention
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
The first step: Catching the attention
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
The first step: Catching the attention
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
The first step: Catching the attention
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
The first step: Catching the attention
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments: Building the definition
Definitions Try not to give, but to make the definitions!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments: Building the definition
Definitions Try not to give, but to make the definitions! An island rises up out of the sea, each of its points is higher than any of the surrounding points.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments: Building the definition
Definitions Try not to give, but to make the definitions! An island rises up out of the sea, each of its points is higher than any of the surrounding points. Definitions Make the definitions in an intuitive way, with the help of heuristical thinking.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Find all islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Find all islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Find all islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Find all islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - What can be done in a classroom? The main problem We have too many islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - What can be done in a classroom? The main problem We have too many islands! Solution Let us consider only rectangular islands!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments: Building the definition
An appropriate way of teaching in this phase Searching for islands on a rectangular grid, if the height function is given, Inverse problems: Searching for a suitable height function, if a rectangle system is given, Searching for a suitable height function for a fixed number of islands on a grid, Searching for a suitable bounded height function, if a rectangle system is given.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Islands and the water level
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Proof:
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Proof: I1 is an island, so a > b.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Proof: I1 is an island, so a > b. I2 is an island, hence
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Proof: I1 is an island, so a > b. I2 is an island, hence b > a.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Using graphical displays to suggest underlying mathematical principles Our first theorem! Two different islands either are far from each other (we can swim between them) or the bigger one contains completely the cells of the smaller one in an island configuration. [5]
Proof: I1 is an island, so a > b. I2 is an island, hence b > a. Contradiction.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Discovering new patterns and relationships, supporting the learning process
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results - The second theorem
The maximum number of rectangular islands on an 1 × n grid is n.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results - The second theorem
The maximum number of rectangular islands on an 1 × n grid is n.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Discovering new patterns and relationships - We can cut!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Discovering new patterns and relationships - We can cut!
Cut - Recursion a1 = 2, a2 = 3, an = an−2 + 3, (n > 2).
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - Solving the recursion Let us use Mathematica again:
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - Solving the recursion Let us use Mathematica again: RSolve[{a[n] == a[n-2]+3, a[1] == 2, a[2] == 3}, a[n], n]
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - Solving the recursion Let us use Mathematica again: RSolve[{a[n] == a[n-2]+3, a[1] == 2, a[2] == 3}, a[n], n]
a[n] →
1 4
The answer: 4 − (−1)n − 3(−1)2n + 6n
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - Solving the recursion Let us use Mathematica again: RSolve[{a[n] == a[n-2]+3, a[1] == 2, a[2] == 3}, a[n], n]
The answer: a[n] → 14 4 − (−1)n − 3(−1)2n + 6n FullSimplify[a[n]/.RSolve[{a[n]==a[n-2]+3,a[1]==2,a[2]==3}, Assumptions→ {n∈ Integers}] We get: an =
6n + 1 + (−1)n+1 . 4
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - m × n grid Cutting
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - m × n board Recursion an = an−2 + m + 1 (n > 2) Solving the recursion
mn + m + n − 1 an = 2
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Replacing lengthy hand derivations with computer based derivations - m × n board Recursion an = an−2 + m + 1 (n > 2) Solving the recursion
mn + m + n − 1 an = 2
Result: The third theorem We get at least an islands. We get a lower bound.
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Experiments - Discovering new patterns and relationships
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results: The structure of the island systems The structure of the island systems Islands on a given grid form a partially order set with inclusion. [5]
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results: The structure of the island systems The structure of the island systems Islands on a given grid form a partially order set with inclusion. [5]
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results: The structure of the island systems The structure of the island systems Islands on a given grid form a partially order set with inclusion. [5]
After a little bit of mathematics [2] (through experiments) we get an upper bound: mn + m + n − 1 2 Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Results: The number of rectangular islands
Our fourth theorem: The number of rectangular islands The maximal number of islands on a m × n grid is mn + m + n − 1 2
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Open problems
The number of square islands Islands with bounded hight (for partial results, see [6]) Islands and puddles What shall we do?
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
Historical background Experimental Mathematics Islands in the classroom
Open problems
The number of square islands Islands with bounded hight (for partial results, see [6]) Islands and puddles What shall we do? Let’s make experiments!
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
The future? References Thanks
Summary Use the computer to catch attention! Let the children act (play, ask, etc.)! Make experiments, use the computer as an experimental mathematician! - The use of computers can support the shift from frontal teaching to project based, self-paced or active small group learning. Use IBL/T techniques! Do not forget the ”White-Box / Black-Box Principle”! - In the ”white-box” phase of teaching the pertinent parts of the SC systems should not be used, while in the ”black-box” phase (in which the students completely master the new topic), it is essential for modern teaching of math to use these systems. (Bruno Buchberger) Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
The future? References Thanks
The future - Automated proofs
CreaComp Computer-Supported Experiments and Automated Proving in Learning and Teaching Mathematics
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
The future? References Thanks
The future - Automated proofs
CreaComp Computer-Supported Experiments and Automated Proving in Learning and Teaching Mathematics The Theorema System Theorema: A System for Computer Supported Mathematical Theorem Proving and Theory Exploration
Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
The future? References Thanks
References G. Cz´edli, The number of rectangular islands by means of distributive lattices, European Journal of Combinatorics 30 (2009), 208-215. J. Bar´ at, P. Hajnal and E. K. Horv´ ath, Elementary proof techniques for the maximum number of islands, submitted. J. M. Borwein, The Experimental Mathematician: The Pleasure of Discovery an the Role of Proof, International Journal of Computers for Mathematical Learning, 10(2), Springer, 2005, 75-108. E. K. Horv´ ath, A. M´ ader, A. Tepavˇcevi´c, Introducing Cz´edli-type islands, The College Mathematical Journal, submitted. A. M´ ader, R. Vajda, Elementary Approaches to the Teaching of the Combinatorial Problem of Rectangular Islands, International Journal of Computers for Mathematical Learning, submitted. A. M´ ader, G. Makay, The maximum number of rectangular islands, The Novi Sad Journal of Mathematics, submitted. Attila M´ ader
Experiments in the classroom
Introduction Islands Afterword
The future? References Thanks
Thank you for your attention!
Attila M´ ader
Experiments in the classroom
