Critique of an article describing creativity
Submitted by Sylvia Zinser for SPED451
1. Synopsis
This article by Richard Kalman with the title ’Revisiting the Sum of Odd Natural Numbers’ describes how the author encourages creativity in a math class. The problem is a very simple one: add the first twenty odd numbers. It is given to the students in a graphical way: they are supposed to visualize rows of stars, each row two stars longer than the previous, and add the number of stars. The students described are in fourth or fifth grade.
The author describes the various ways his students used to solve a problem, the wrong solutions as well as the right ones. Four correct solutions, found by the students, are described. These solutions are graphical (grouping the stars in different ways) and arithmetical (adding numbers in various orders). The author helps his students to find a fifth solution which relates the star pattern to square numbers.
2. Strengths and weaknesses of the article
This is a practitioner’s article; the author writes what he did with his students, how it worked and at which results the children arrived. The purpose of the article is not clearly stated, which would be a weakness of it if the author had not pointed out the purpose of his lesson: He writes that he used this lesson to encourage creativity and to teach the children that the standard way to calculate a problem is not always the only way, and that there might be less tedious but more creative ways. Richard Kalman also wants to teach the students to try out a new solution, “even if they do not know where it is headed” (Kalman, 2003, p 61). The journal in which this article is published (Teaching Mathematics in the Middle School) is aimed at teachers interested in new sources and strategies of teaching. Teachers who read this article are to be motivated to try similar problems with their class.
The success of the whole lesson is thoroughly documented. This tedious description of the evolution of the whole lesson is a strength of this article: The author gives a good and detailed description of the thinking process of his students, including the errors they made at first. This can help to understand the different thinking procedures children might apply. He also mentions the positive reactions of his pupils to the whole process, how well they cooperated, and how they explained their results to the other kids. In this context the author’s goal to encourage teachers to try this approach towards math instruction probably will be met.
Students learn better by applying their math skills to problems they are interested in. The success of the concept of problem based learning confirms this: If children have a reason to apply different techniques they will do it. If the skills are not sufficient, the children show higher motivation to work on them like one of the boys who “spent more time considering how to add correctly” (Kalman, 2003, p. 59). This emphasis on learning by doing is another strenth of the article: Teachers are encouraged to use problem based instruction as a learning tool.
The children learn creativity in math, especially because the teacher did not stop them after one creative solution. He encouraged fluency which is an essential component of divergent thinking (Torrance, as described on page 218 in Davis, 1999). Every solution was valued the same. Even the last solution which the teacher helped his students to find was not valued higher. The students will probably show high motivation to come up with many ideas, if every single idea is listed and valued. Achieving a variety of solutions, using totally different ways of thinking might teach the children, for the next similar problem, to consider these various approaches. Richard Kalman also pronounces another important component of creativity: risk taking, i.e. to be ready to take the risk that the solution runs into a dead end. The kids were encouraged to play with the stars, rearranging them etc. for finding the solution. The explicit description of those different components of creativity or divergent thinking like fluency, playfulness and risk taking which the children used solving the problem is a strength of the article, because it causes awareness of other teachers towards creativity: A teacher who reads this article might for example develop a higher tolerance against alternative solutions for other math problems. He or she also might be inclined to let students play with one problem for a longer time.
The article gives only one example of a possible task for the children, but the references show a few sources for similar applicable math problems. The reference list is very short. More resources for similar problems could help teachers to choose the topics fitting to their class’s needs. This lack of resources is a weakness of the article, because teachers might use the given sources and not find time to find more similar problems for their classes. The author seems to be aware of this problem, because he asks the readers at the end of the article for submission of similar projects.
Yet, alltogether the article is a good encouragement for teachers to try out similar approaches to creativity in math.
References
[DavisDavis1999] Davis, G. A. 1999. Creativity is Forever. USA: Kendall/Hunt Publishing Company.
[KalmanKalman2003] Kalman, R. 2003. Revisiting the Sum of Odd Natural Numbers. Mathematics Teaching in the Middle School, 9(1), 58-61.
