Academic journal article The Mathematics Enthusiast

Mathematical Creativity: The Unexpected Links

Academic journal article The Mathematics Enthusiast

Mathematical Creativity: The Unexpected Links

Article excerpt

(ProQuest: ... denotes formulae omitted.)

1 Introduction

In 2004, Sriraman conducted a qualitative study in which he interviewed five creative mathematicians to get an insight on some characteristics of mathematical creativity. He ended his study with an inspiring conjecture that captured our interest:

"It is my conjecture that in order for mathematical creativity to manifest itself in the classroom, students should he given the opportunity to tackle nonroutine problems with complexity and structure - problems which require not only motivation and persistence but also considerable reflection. " (Sriraman, 2004)

We believe that problems demonstrating unexpected links between two or more domains in mathematics represent one type of the non-routine problems addressed by Sriraman in his conjecture. Through well designed problems of this kind, students could be trained to find unusual connections between seemingly far domains in mathematics in order to reach a solution.

This approach in solving a mathematical problem is viewed in the work of many creative mathematicians. In 1896, Hadamard (Hadamard, 1893; Hadamard, 1896) and De la Vallée Poussin (De la Vallée Poussin, 1896) established and proved (independently) the famous Prime Number Theorem using complex analysis. This theorem describes the general distribution of prime numbers among positive integers. It states: If tt(x) is the number of primes less than or equal tox, then Irai... that is, tt(x) is asymptotically equal to ... Although there is no clear connection between complex analysis and the distribution of prime numbers, the proof depends greatly on Riemann's zeta function from complex analysis.

Another example is related to the infinity of primes. Fürstenberg defined a topology on the set of integers Z and linked it with elementary properties on numbers to prove that the set of prime numbers is infinite (p:5) (Aigner & Ziegler, 2010).

Moreover, in graph theory, Erdös and Rényi introduced for the first time the probabilistic methods to prove the existence of some graphs that are usually difficult to find (Aigner & Ziegler, 2010). The usage of this method represents the unexpected link between graph theory and probability.

In addition, the friendship theorem is a real situation problem that was translated into a graph theoretical problem and then solved using both graph theory and linear algebra techniques. It states: "Suppose in a group of people we have the situation that any pair of persons has precisely one common friend. Then there is always a person who is everybody's friend. "

Also, in graph theory, Tverberg (1982) established a proof of a theorem about the decomposition of a complete graph into complete bipartite graphs. This proof makes use of a system of linear equations to show that the minimum number of stars necessary to cover a complete graph is n - 1. The following example illustrates this theorem for the complete graph K, :

The decomposition of K5 into the four stars K14,Kli,K12 and Kxl respectively:

After reviewing the above examples, one might think that such approach to solve mathematical problems can only be used by professional mathematicians or postgraduate math students since they possess deep knowledge in mathematics, yet we claim that such approach can be inserted in the educational curriculum of intermediate and secondary students. In this way students will be able to experience such type of creativity in solving mathematical problems by constructing links among distinct domains. This is clearly shown in the content of this paper where examples are given and tested to prove the credibility of this claim.

2 Literature Review

The examples mentioned above, and others, provided us with the very first flame that ignited the idea that there is a correspondence between solving mathematical problems using unexpected links and mathematical creativity.

2.1 Mathematical Creativity

For a long period of time, the dominant view was that creativity in mathematics is limited to "genius" individuals (p. …

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