Creativity: An Essential Element in Your Mathematics Classroom

We have known for some years now…that most children’s mathematical journeys are in vain because they never arrive anywhere, and what is perhaps worse is that they do not even enjoy the journey. (Whitcombe, 1988)

As a classroom teacher I often wondered why my students did not share my enjoyment of mathematics. For me, mathematics was both challenging and intellectually stimulating and I wanted my students to share the pleasure I found in tackling a tough problem. I became increasingly frustrated, not only by my students lack of interest, but also the discovery that many of my fellow teachers and most parents shared the view that math was hard and something to endure, rather than to explore and enjoy. Most shared the view Whitcombe captured in the opening quote to this article.

Literature is filled with references to the beauty and the creativity that is the foundation of mathematics. The vision of an ideal mathematics classroom is one where “students confidently engage in complex mathematical tasks…draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress” (NCTM, 2000, p. 3). Yet, many students find their time filled watching as mathematical methods are demonstrated and committing to memory facts and algorithms (Pehkonen, 1997). These students often develop the conception of mathematics as a discipline where knowledge is complete and the mastery of mathematics is simply a digestive process, not a creative one (Dreyfus and Eisenberg, 1996). Allowing creativity back into our classrooms is essential to rekindle an interest in mathematics.

Ginsburg (1996) saw the essence of mathematics not as producing the correct answers, but thinking creatively. Yes, accuracy is important as the students’ responses must fit the context of the problem and be mathematically correct, but strict emphasis on accuracy discourages students from taking risks and creating their own contextual understanding of mathematics. All too often I hear both teachers and employers comment about the inability of our students to use mathematics productively, yet how often do we provide such opportunities in our classrooms?

Most of the mathematical concepts we routinely teach in our classrooms were born in controversy, often debated for long periods of time by the mathematicians of earlier eras. Yet we expect our students to memorize and accept without question the rules and algorithms that were the product of those debates. One such debate was the concept of negative numbers. In the 19th century, Busset attributed the introduction of negative numbers as the reason for failures in the teaching of mathematics in France (Boye, n.d.). It was often common practice to ignore negative answers as meaningless. Our students today still struggle with negative numbers and may revisit the same debate. Rather than distill the concept to a set of rules, and in doing so imply that mathematics is all about the application of rules, why not let our students know that their questions are the same as those that puzzled mathematicians for centuries?

Mathematical creatively can be thought of as “the ability to see new relationships between techniques and areas of application and to make associations between possibly unrelated ideas” (Tammadge as cited in Haylock, 1987, p. 60). Balka’s 1974 article in the Arithmetic Teacher, Creative Ability in Mathematics, is frequently cited in literature on mathematical creativity. In this brief, three-page article, Balka offers five attributes of creativity. These attributes, qualities we must strive to develop in all students, are the ability to:

1. Formulate mathematical hypotheses
2. Determine patterns
3. Break from established mindsets to obtain solutions in a mathematical situation
4. Sense what is missing and ask questions
5. Consider and evaluate unusual mathematical ideas, to think through the consequence from a mathematical situation (divergent)

At the 2006 NAGC national convention, I offered sample activities for each of these attributes. A copy of that presentation, The Essence of Mathematics, is available on my web site at http://www.edci. purdue.edu/elmann/. There you will also find links to some of my favorite mathematics education resources that you may find useful. Two other programs that emphasize mathematical creativity are Project M3: Mentoring Mathematical Minds (http://www.projectm3.org) and Extending Process Skills for Able Mathematicians (http://www.franassisi.rbkc.sch.uk/curriculum/ centre_for_excellence.htm). Finally, Rachel McAnallen, is a name very familiar to the veterans of the University of Connecticut’s CONFRATUE (http://www. gifted.uconn.edu/confratu.html), Purdue University’s DISCOVER! (http://www.geri. soe.purdue.edu/profdev/discover institute/ default.html) or Edufest (http://www. edufeast.org), her website (http://www. mathchannel.com/) offers a wealth of information to help you instill in your students a love of the wonders of mathematics.

Students often tell me they are no good at math. My response is, “How do you know?” to which they reply with comments about performance in school or difficulty in solving problems quickly. Over the years, these students have learned to equate mathematics with algorithms, the learning of rules, and ability to find the answer the teacher is expecting. A sixth grader’s comment that, “It doesn’t make much sense. But, we are in math class, so I guess it does here,” or a calculus student’s comment that, “In math, I do things just the opposite way from what I think it should be and it almost always works” (Linquist, as cited in Heibert et al., 1997, p. vii), are illustrative of the impact such instruction can have. If taught that there is only one right answer or only one correct method, a student’s concept of mathematics as only the application of mathematical techniques is reinforced.

The way we teach mathematics is a significant contributor to this perception. Köhler (1997) illustrates this point in a discussion with an elementary classroom teacher about a student who had arrived at the correct answer in an unexpected way. Rather than delight in the student’s creativity, the teacher responded:

While going through the classroom, that pupil asked me [the teacher] whether or not his solution was correct. I was forced to admit that it was. That is what you get when you don’t tell the pupils exactly what to do….” The teacher now reproaches himself for not having prevented this solution. He is obviously influenced by an insufficient understanding of what is mathematics, by the image of school as an institution for stuffing of brains…. (p. 88 (emphasis added))

Constant emphasis on sequential rules and algorithms may prevent the development of creativity, problem solving skills and spatial ability (Pehkonen, 1997). If we want to deepen our students’ understanding of mathematics, then we need to recognize that the mastery of rules, algorithms, rules and strategies is not the end goal of mathematics education. Our students should use these procedural tools to explore, test, revise and defend their solutions to meaningful problems.

Mathematics is meant to be performed, not just practiced. In sports, language arts, or music we practice to improve performance; not just for the sake of practice. Yet how often do our students see mathematics this way? Bogomolny’s (2000) comments capture the change we need to make in our classrooms to restore a love of mathematics and develop the mathematical potential of our students.

“Any” fruit of human endeavor shows creativity, if you think about it. The interesting question to me is this: Why is it that a student who is only playing other people’s music instinctively understands that those composers were creative, and that s/he might aspire to the same kind of creativity -- or, in English class, instinctively understands that those writers were creative, even when s/he is just reading their creations and answering quiz questions about them -- but doesn’t have the same instinctive understanding that Euclid and Newton and Pascal and Gauss and Euler were creative mathematicians? The most obvious answer has to do with the way these disciplines are taught.