ࡱ> 796 @|<bjbj >Z؝؝=2>86<r4OC^:JJJBBBBBBB$_ERGB ~~ BJJ4 C%%% JJB% B%%>@J _q]j#0r?2BC0OC?\H$H0@@H@8` %lzBB j%X FOSTERING CREATIVITY IN PRESERVICE TEACHERS THROUGH MATHEMATICAL HABITS OF THE MIND RICHARD S. MILLMAN AND TIM JACOBBE Abstract. This article describes and gives some examples of the concept of mathematical habit of the mind and its connection to exemplification and Polya problem solving principles. It recommends MHM as a path to creativity, and then focuses on the use of MHM for pre-service elementary and middle school teachers. Key Words: Mathematical Habit of the Mind, Exemplification, Polya principles INTRODUCTION Creating mathematical habits of the mind (MHM) is an approach to promoting mathematical creativity in children through the development of a culture which utilizes sound mathematical questioning in present and future teachers. MHM will be described briefly and methods will be sketched and approaches will be presented to help preservice teachers (PSTs) develop MHM that will be passed on to their future students. This article concludes with a narrative taken from a book by Fields Medal winner Terence Tao which gives insight into the MHM of a superb mathematician. While we believe that MHM is appropriate at all levels, in order to narrow the scope of this article, the focus of this paper will be introducing the idea of a MHM during the preparation of PSTs. OVERVIEW OF THE CONCEPT OF A MATHEMATICAL HABIT OF THE MIND In order to help students develop MHM, teachers should work with their students to: 1) explore mathematical ideas; 2) formulate questions; 3) construct examples (Watson & Mason, 2005); 3) identify problem solving approaches that are useful for large classes of problems; 4) ask themselves whether there is something more (a generalization) in the mathematics on which they are working; and 5) reflect on their answer to see whether they have made an error (Jacobbe, 2007). These five traits can be called MHM which is a term that is featured prominently in an important report on the mathematical education of future teachers (CBMS, 2001). Furthermore, the process standards set forth by NCTM (2000) in the Principles and Standards for School Mathematics include problem solving, communication, reasoning and proof, making connections and representations all of which can be described as MHM. MHM as described above are more general than problem solving which frequently starts with a specific problem but does not generally go on to encourage generalizations or class of solutions. Although he does not use the phrase directly, all of Polyas (2007, 1973) foundational writing on problem solving are based on MHM. Polyas problem solving process consists of four steps: understanding the problem, devising a plan, carrying out the plan, and looking back (Polya, 1973, xvi-xvii). It is the looking back step that is very much the heart of MHM as described above. By looking back at the completed solution, by reconsidering and re-examing the result and the path that led to it, they [students] could consolidate their knowledge and develop their ability to solve problems (Polya, 1973, pp. 14-15). Jacobbe (2007) used the looking back step as a way to help structure students thinking in order to overcome translation difficulties. In his study, PSTs were required to outline their thinking at each step of Polyas process with particular attention to the looking back step. Utilizing this method and helping PSTs develop MHM increased their performance on translation tasks from 21 to 86% (Jacobbe, 2007). Can MHM be introduced at different levels to help foster creativity in all students? The infusion of MHM can provide a way to make mathematical creativity an explicit goal of mathematics education. It will be important to carefully study the development of the strategy of MHM to see if there is scholarly evidence to support a place for that concept as a platform for promoting student creativity. A review of the literature has revealed no studies on this subject except for those which are revolve around exemplification (Watson & Mason, 2005). There have also been discussion pieces centered at the high school level (Cuocco, 1996, 2001). There is a close connection between the notion of MHM and the idea of exemplification in mathematics. In their work on exemplification, Watson & Mason (2005) carefully explain the background and research base of this approach. One strategy suggested by Watson and Mason consists of having learners construct their own examples as a way to encourages creativity with PSTs. MHM as outlined in this paper does not require a deep mathematical background, only an inquiring mind which will focus on thinking in a mathematically creative way. Focusing on the creation of MHM is a way to structure, infuse, and talk about some aspects of mathematical creativity. EXAMPLES OF MHM AT THE ELEMENTARY AND MIDDLE SCHOOL LEVELS This section presents problems that can be used to promote mathematical creativity among PSTs and foster their creation of MHM that may be passed on to their future students. In a content course on problem solving for PSTs (future teachers of children ages 5 to 14) teachers, a task which introduced MHM was given. After proving that the sum of even numbers is even, the sum of odd numbers is even, etc., the class was asked to show that the product of even numbers is even. At the end of that in-class group exercise, the instructor asked the students to think back on what happened and whether there was anything more going on. After more conversations in their groups, and a series of hints from the instructor to some of the groups, the future teachers realized that the product was actually divisible by four, not just two and that, as one of them said, the result is free meaning that the same proof worked for the stronger result. We have given similar Is there more? questions, such as ones concerning the square of an odd number, in the content math course required of all future elementary school teachers. Two other examples, which come from a textbook designed for future elementary school teachers, are provided. While these are phrased as questions, not examples, the analogies between this type of exercise and the approach of Watson & Mason are clear. The first asks the PSTs to recognize what it means for a statement to always be true, to see that algebra can be used for this purpose, and to change a statement so that it is always true (i.e. make a conjecture). The area of number theory (at a level for elementary PSTs) is a way of providing conjectures and involving MHM (see Numbers 1, 2, and 3 in initial list of five traits of MHM). The first example is a slightly more general version of two problems presented in a textbook for a mathematics content course for PSTs (Long, DeTemple, & Millman, 2008, 243). Consider the following statements (where n, b, and c are integers): If n = bc and p is an integer which divides n, show that p divides either b or c. If the primes p and q both divide the integer m, show that pq divides m. To PSTs, Statements A and B sound like they should be true. Students, working in small groups, find almost immediately that A is false when they try a few values of n and p (Strangely, they generally do not use  EMBED Equation.DSMT4  to give a counterexample). The next step in showing a MHM is to ask if something like Statement A is true. By asking the small groups to see what is common in their examples for cases in which the statement actually is true, they discover that p needs to be a prime number. On the other hand, for B, it is only with copious hints that they realize that B is false. They have rarely come to the conclusion that p  EMBED Equation.DSMT4  q gives both a necessary and sufficient condition for Statement B to be valid. We also emphasize that, as in A and B, the use of the Fundamental Theorem of Algebra gives a number of ways of generating interesting and challenging problems for more creative students. MHM doesnt have to come only from problems or find the right generalization. The following discussion provides a technique which is neither an example nor an exercise. The MHM comes from asking PSTs to think about the unexpected relationships between two different concepts. In this case, the ideas underneath the prose are that we can derive a formula by relating two quantities which seem unrelated and that areas or volumes can be approximated by adding very small areas or volumes. (Calculus is not generally required for a teaching certificate in elementary school education in the U.S.) The formula for the surface area of a [right, circular] cone can be derived from the surface area formula for a pyramid. To see how, imagine that the cone is closely approximated by a right regular pyramid. An example is shown with a dodecagon (12-gon) as its base. (Long, DeTemple, & Millman, 2008, 819) EXAMPLES OF MHM AT THE HIGH SCHOOL LEVEL We shall discuss only one of those from Tao (2006), but many more examples may be found there. What is fascinating about the book is Taos writing which includes the way in which he thinks through and approaches the problems. The book gives insights on the mathematical thought of a Fields Medal winner when dealing with problems that he was able to do as a teenager. We give a brief description of his solution to a problem presented in his book (Tao, 2006, 45). Suppose that a1, a2, , an are integers. Prove that any polynomial of the form  EMBED Equation.DSMT4  cannot be factorized into two non-trivial polynomials each of which has integer coefficients. He first remarks that, since f is a square plus one, it must be positive. He then notes that there are linear terms and asks Can we use these (x ai)s to our advantage (Tao, 2006, 45)? If f(x) factors into p(x)q(x) then p(ai)q(ai) must be one. The next observation is crucial: What does this mean? Very little, unless one remembers that p and q have integer coefficients and that the ai are integers too (Tao, 2006, 45). From this last step, the polynomials, p and q, are determined by their value at n points since they are polynomials of degree at most n and the conclusion that p or q must be a constant follows. What is remarkable is the prose of the book as it shows how his mind operates. END REMARKS This article advocates the inclusion of the notion of MHM in the courses for PSTs as a way to stimulate mathematical creativity in their students. If PSTs are not exposed to sophisticated examples, problems, and discussions to push their thinking beyond the memorization of algorithms, they may be satisfied with only knowing how to do mathematics without realizing why we do mathematics. Mathematics is an art that takes years to master; however creating MHM in PSTs will help them create such ways of thinking in all students across grade levels. The field of mathematics is dependent upon the creativity of young students that will lead to future discoveries. The first author has used MHM in the ALGEBRA CUBED project (NSF grant 0538465) at the University of Kentucky. REFERENCES Conference Board of the Mathematical Sciences (2001). The Mathematical Education of Teachers. American Mathematical Society and Mathematical Association of America: Providence, RI & Washington, D.C. Cuoco, A. (2001). Mathematics for teaching. Notices of the American Mathematical Society, vol. 100, 1169-1172. Cuoco, A., Goldenberg, E. P., Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15, 375-402. Jacobbe, T. (2007). Connecting research to teaching: Using Polya to overcome translation difficulties, Mathematics Teacher, 101, 390-393. Long, C., DeTemple, D., Millman, R. (2008). Mathematical Reasoning for Elementary Teachers, Addison-Wesley, Fifth Edition. National Council of Teachers of Mathematics (2000). Principles and standards or school mathematics. Reston, VA: Author. Polya, G. (2007). Mathematics as a subject for learning plausible reasoning. Mathematics Teacher, vol. 100, 36-39 special issue (originally published in German in Gymnasium Helveticum, 1956) Polya, G. (1973). How to Solve It, Princeton University Press: Princeton, NJ. Tao, T. (2006). SolvingMathematical Problems, a Personal Perspective, Oxford University Press, Oxford, U.K. Watson, A., Mason, J. (2005). Mathematics as a Constructive Activity: Learners Generating Examples, Lawrence Erlbaum Associates Publishers. ABOUT THE AUTHORS Richard Millman, Ph.D Dept. of Mathematics University of Kentucky Lexington, KY 40506-0027 USA  HYPERLINK "mailto:millman@ms.uky.edu" millman@ms.uky.edu 1-859-257-6821 Tim Jacobbe, Ph.D. Dept. of Curriculum and Instruction University of Kentucky Lexington, KY 40506-0027 USA  HYPERLINK "mailto:jacobbe@coe.uky.edu" jacobbe@coe.uky.edu 1-859-257-1643     Fostering Creativity in Preservice Teachers through Mathematical Habits of the Mind Richard S. 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