One of the most important problems when learning and teaching mathematics is the formation of the ability to solve text mathematical tasks. Solving mathematical tasks is a creative process, which requires a productive activity. It is not enough to learn how to solve a set of standard tasks available in the curriculum, rather than learning an automated process which can be applied to any situation in life, which requires a solution.
Taking the example of statistics, the process of solving a problem can be related to the processes of testing a hypothesis, and thus the general steps and guidelines followed in the first case can be applied to the latter. In that regard, in order to have a closer look at the process of coming up with a solution, this paper analyzes the steps of solving a problem, which were outlined by the Hungarian mathematician George Polya, and published in his book “How to Solve It” (1956). During the analysis the steps proposed by Polya will be compared to the steps taken in hypothesis testing, as well as the possible application of Polya’s steps to other areas in life.
George Polya was born in 13 December, 1887 in Budapest, Hungary, to Jewish parents Anna Deutsch and Jakab Polya. The achievements of Polya include working in probability, analysis, number theory, geometry, combinatorics and mathematical physics while at the University of Budapest, Brown University, and Stanford University.
Polya was enrolled at the University of Budapest in 1905, where he studied language and literature before he turned to physics and mathematics. He also attended mathematics lectures by Wirtinger and Mertens at the University of Vienna, after which he returned to Budapest and earned his doctorate in mathematics. Polya collaborated with Szegö on writing a book of analysis problems, the success of which was followed by a two volume work in 1926 “Aufgaben und Lehrsätze aus der Analysis”, which was “a mathematical masterpiece that assured their reputations. “ (“George Pólya”).
Polya also published the English version of “How to Solve It” in the United States, which sold over one million copies over the years, and was translated in 17 languages. The achievements of Polya include working in probability, analysis, number theory, geometry, combinatorics and mathematical physics while at the University of Budapest, Brown University, and Stanford University.
How to Solve a Problem
The steps outlined by Polya, although comprise of four steps, emphasize on three major stages, which are known as the 3 R’s, i.e. Request-Response-Result. The steps sequence is also of major importance, thus it is not only following the principles, but also following in a particular order. The steps can be outlines as follows (“Summary from G. Polya, “How to Solve It””):
- Understand the problem (Request) – This step, despite its simplicity, emphasizes on the correct statement of the problem, which can be achieved by understanding the linguistic part of the problem, recognizing the problem’s question, and gathering enough information on the subject.
- Devising a plan (Response) – This step implies choosing the correct strategy, through finding the connection between the available data and the unknown. The step is a matter of selection, where the strategy can be chosen through finding similar solved problems, using previous results, solving related problems or others.
- Carrying out the plan – This step is basically implying the execution of the strategy outlined in the previous step. The step also implies persistency with the pan, assuming that each step performed is acknowledged as correct.
- Looking Back (Results) – The steps implies the examination of the obtained solution of the problem. The questions that could be asked at this stage are related to the ability to check the results and/or the argument, and the ability of using the results or the selected strategy in solving other problems.
Relating hypothesis testing to the issue of problem solving, it can be said that hypothesis testing is used to solve the problem of judging whether a statement about a population is true based on the data from a sample. (Utts and Heckard). Thus, taking in consideration that all problems represent questions, hypothesis testing can also be considered as a method of solving the question.
The first step in Polya’s method, i.e. understanding the problem, in the context of hypothesis testing can be related to the formulation of the hypothesis statement, i.e. the determination of the null and the alternative hypothesis. The null and the hypothesis statement are two choices of answers to the research statement. The similarity between the two steps is evident, as both rely on correct understanding of the problem. Both steps require decomposing the problem into various parts, representing the data, and the unknown.
The differences can be seen in that the formulation of the hypothesis statement, by default implies the formulation of the alternative that should be accepted once the hypothesis statement proved wrong. Additionally, a correct hypothesis statement relies on a correct formulation of the research question, and the hypothesis of the research itself, which might vary in terms of formulation.
The next step in Polya’s method is devising the plan, which can be related to the verification of necessary data conditions and the identification of test statistics. The similarities of both steps can be seen through obtaining all the necessary information and choosing a method based on which this information will be used to solve the problem.
In that regard, the information in case of hypothesis testing is represented in the data summary, test statistics which can be obtained through various methods of data collection. Additionally, both steps rely on establishing a connection to previously known data. In the case of problem solving it might be represented by previous solutions of identical or similar problems, while in hypothesis testing this might be represented by the designated level of significance.
Carrying out the plan, as a step that implies certain actions, can be related to two steps in hypothesis testing, which is finding the p value and the decision whether the value is statistically significant. The difference can be seen through the difference in logic, where the logic of hypothesis testing is similar to the “”presumed innocent until proven guilty” logic.” (Utts and Heckard). In the method of problem solving outlined by Polya, carrying out the plan the same logic can be one of many other strategies, where problem solving, from his perspective is not limited only to one strategy. On the other hand, this can be justified by the fact that, the question solved in hypothesis testing is always directed in one direction, whether the results of an experiment are caused by sampling error or chance.
The fourth step, i.e. looking back can be related to reporting “the conclusion in the context of the situation.”Both steps are similar in that the researcher might conclude based on the results, whether the solution of the problem served its purpose, where in case of a positive answer the results and the methods might be used as references for other cases.
The usage of Polya’s method in other purposes can beneficial as the steps are generalized, and merely serve as guidelines. Hypothesis testing can be used as one of the possible strategies based on which a solution can be reached. One of the areas in which Polya’s method can be applied is the area of programming. Where the steps taken in writing a program can correspond to the steps outlined by Polya as follows:
- Identifying the problem – Stating the programs purpose
- Devising a plan –writing an algorithm of the programs execution
- Conducting the plan –writing the code of the program
- Looking back – testing the program.
It can be concluded that the methods outlined by George Polya can be applicable to many areas, and in that regard the reason of its popularity is not the content of each step. The steps outlined are not informative in terms of telling what to do, rather in telling in what sequence the problem should be approached. The relation of Polya’s steps to hypothesis testing shown that despite some differences in content, the sequence of the steps similarly represent the three R’s (Request-Response-Result).
“George Pólya“. 2002. The MacTutor History of Mathematics archive. Web.
“Summary from G. Polya, “How to Solve It“”. 2009. University of California, Riverside. Web.
Utts, Jessica M., and R. F. Heckard. Statistical Ideas and Methods. Belmont, CA: Thomson-Brooks/Cole, 2006.