The operationalisation of creativity is a desired outcome within any mathematical educational setting. Sriraman (2009, p. 13) emphatically states that “mathematical creativity ensures the growth of the field of mathematics as a whole”. No one can dispute technological innovations
in modern society have been owed to the inspirational creativity of scientists and professional mathematicians (Nadjafikhan, Yaftian & Bakhshalizadeh, 2012). In Scotland, creativity has a high profile in education and fits very well within the broad framework of CfE, although ironically, the vast majority of pupils would not associate the domain of mathematics with creativity.
Nevertheless, numerous pupils’ classroom experiences of mathematics entail working with practitioner-driven material and sequential tasks or being passive observers of mathematics (Boaler, 1997). Based on her ethnographic case studies of teaching approaches at two different English secondary schools, Boaler (1998, p. 59) cautions against the stereotypical limitations of using only standard mathematical methods when she warns “students developed an inert, procedural knowledge that was of limited use to them in anything other than textbook situations”. Often, the creative side of mathematics education is neglected, as instruction normally has an imitative and reproductive character since it is focussed on rudimentary activities with a dependency on routine skills, where pupils are encouraged to think in narrow domains (Haylock, 1987).
It is important to reflect on what is epitomised by mathematical creativity. Previous research has suggested that it may be confined to the employment of professional mathematicians when they formulate a problem that has not been solved before (Hadamard, 1945; Poincare, 1948). However, the conceptualisation of creative learning fluctuates due to the diversity of perspectives of creativity. Ervynck (1991) deems that mathematical creativity cannot occur in a vacuum and needs a context in which the individual moves forward through previous experiences which provide a suitable environment for creative development. Ervynck (1991) asserts that creativity plays a vital role in the full cycle of advanced mathematical thinking:
It contributes in the first stages of development of a mathematical theory when possible conjectures are found as a result of individual experiences of the mathematical connects; it also plays a part in the formulation of the final edifice of mathematics as a deductive system with clearly defined axioms and formally constructed proofs (p. 42).
Silver (1997) views creativity as an orientation or disposition towards mathematical activity that can be fostered in the general school population. He proclaims the “connection to creativity lies not so much in problem posing itself, but rather the interplay between problem posing and problem solving. It is in this interplay of formulating, attempting to solve,
reformulating, and eventually solving a problem that one sees creative activity” (p. 76). Silver discusses previous research by Getzels & Jackson (1962), Balka (1974) and Skinner (1991) amongst others which demonstrate valuable samples of problem posing. He proposes a didactical paradigm in which classroom practitioners can relate to three practical core assessment components of creativity i.e. fluency, flexibility and originality (novelty) as displayed in Figure 3.2. However, Kontorovich et al. (2011) argues that these indicators do not fully capture the essence of pupils’ creativity and suggest that aptness be included as an additional quantitative evaluation. Silver (1997) presents a task which requires showing that the product of any four consecutive integers is divisible by 24. Whist this particular illustration is more emblematic of problem solving, it can be easily adapted to provide a problem posing activity. For example, generate as many problems as you can using the terms ‘four’, ‘consecutive integers’, ‘divisible’ and ‘24’. Silver (1997, p. 79) claims through the use of an inquiry based approach, “teachers can assist students to develop greater representational and strategic fluency and flexibility and more creative approaches to their mathematical activity”.
At school level, Jenson (1973) maintains that mathematically creative pupils should be able to pose mathematical questions that extend and deepen the original problem as well as solve the problem using multiple methods. Likewise, Krutetskii (1976) portrayed creativity in the context of problem formation, invention, independence, originality and associates mathematical creativity with giftedness. In a study of 359 Cypriot pupils (aged 9-12 years) by Kattou et al. (2013), the researchers found a strong positive correlation between mathematical creativity and mathematical ability. In contrast, Skemp (1987, p. 64) argues that all learners have the ability to demonstrate mathematical creativity “since all new learning in mathematics by the method of concept-building consists of the formation by individuals of new ideas in their own minds, it is creative from their point of view”. In the same vein, Mann (2006) warns that without providing for creativity in teaching mathematics, all learners are denied the option to appreciate the beauty of mathematics.
Figure 3.2 Core indicators of creativity (Adapted from Silver, 1997) Problem Solving Mathematical Creativity Problem Posing Students explore open-ended problems, with many interpretations, solution methods or answers
Students solve (or express or justify) in
one way, then in other ways Students discuss many solution methods Students examine many solution methods or answers (expressions or justifications); then generate another that is different Students generate many problems to be solved Students share their
posed problems
Students pose problems that are
solved in many different ways. Students use “What if not?” approach to
pose problems
Students examine several posed problems; then pose a problem that
is different
Fluency
Flexibility
Novelty
Logically, in order to cultivate mathematical creativity, teachers should select contexts that offer pupils opportunities to pose their own problems. Singer & Voica (2015) found that within the context of problem posing, mathematical creativity is a special type requiring abstraction and generalization. Jay & Perkins (1997, p. 257) maintain “the act of finding and formulating a problem is a key aspect of creative thinking and creative performance in many fields, an act that is distinct from and perhaps more important than problem solving”. Another illustration of creativity is found in the work of Runco (1994, p. ix) when he expressed creativity as a multifaceted construct involving both “divergent and convergent thinking, problem finding and problem solving, self-expression, intrinsic motivation, a questioning attitude, and self-confidence”. Alternatively, Torrance (1988) proclaimed that creativity is almost limitless and occurs whenever a solver has no learned solution for an existing problem.
While a number of researchers (e.g. Cai & Cifarelli, 2005; Singer et al., 2011; Siswono, 2011) have endorsed the connection between creativity and problem posing, this perspective is not universally shared (e.g. Haylock, 1997; Leung, 1997). Yuan & Sriraman (2011, p. 25) note “there might not be consistent correlations between creativity and mathematical problem- posing abilities or at least that the correlations between creativity and mathematical problem posing abilities are complex”.
Within my professional practice, I have adapted two problem posing activities (Figure 3.3) from Christou et al. (2005b) that have stimulated the developmental growth of mathematical creativity between S1 and S2 pupils. Whilst I cannot verify the impact of such creativity on achievement, these tasks have promoted deep critical thinking and have generated many interesting and enjoyable learning experiences.
Figure 3.3 Examples of problem posing activities (Adapted from Christou et al., 2005b)
(a) Write a question to the following story so that the answer to the problem is ‘75 pounds’: Lachlann had 150 pounds. His mother gave him some more. After buying a book for 25 pounds he had 200 pounds.
(b) Write an appropriate problem for the following: (2300 + 1100) – 790 = n