ࡱ> ``bjbjss BV#   4%%%8%TD&l4~&b''''f)b)$)}~~~~~~~$h^~ /^)b)//~ ''h~R<R<R</ R' '}~R</}~R<R<!y " !~'& 0Nڼ% 8>zPA~<~0~A{>^:>h!~> !~ *dd+R<b,.-f***~~;^***~////444%444%444   PROMOTING CREATIVITY FOR ALL STUDENTS IN MATHEMATICS EDUCATION: AN OVERVIEW LINDA JENSEN SHEFFIELD Abstract: This paper gives an overview of the approximately forty papers written expressly for ICME 11 Discussion Group 9 on the topic of Promoting Creativity for All Students in Mathematics Education. These papers are loosely divided into four sections: Section 1: Goals and Definitions Section 2: The Role of the Teacher Section 3: The Use of Problems and Assessment Section 4: Technology and the Environment Several of the papers address more than one aspect of creativity. All papers are designed to instigate discussion about goals, research and exemplary practices for promoting mathematical creativity for all students. Key words: mathematical creativity, promise, talent, giftedness, and innovation; teaching and assessing creativity; problem solving and problem posing; technology and environmental influences INTRODUCTION Imagination is more important than knowledge. Albert Einstein Creativity and innovation are often cited along with problem solving, critical thinking, communication and cooperation as critical skills, knowledge and expertise that students need to be successful in work and in life in this twenty-first century world. In the United States, as cited in reports such as Rising Above the Gathering Storm, the Partnership for 21st Century Skills, and the Business Roundtable Education Innovation Initiative, Tapping Americas Potential, the need for creative, innovative individuals with a strong foundation in mathematics is overwhelming. As seen in the approximately 40 papers from 24 countries and 60 participants in Discussion Group 9, this topic is indeed of international interest and concern. In todays world, it is not enough to be proficient at computation or at memorizing rote procedures to solve routine problems. These skills are important, but even more important are the abilities to recognize and define problems, generate multiple solutions or paths toward solution, reason, justify conclusions, and communicate results. These are not simply abilities that one is born with and they do not generally develop on their own. For students to become creative mathematicians, these talents must be cultivated and nurtured. The papers for Discussion Group 9 address a variety of issues surrounding this topic, beginning in Section 1 with several definitions of mathematical creativity and a discussion of whether all students can and should be mathematically creative. In the second section, the papers address the critical role of the teacher and the goals involved in recognizing and promoting mathematical creativity. In Section 3, a variety of mathematical problems are introduced along with suggestions for using problems to develop, evaluate and assess mathematical creativity. In addition to the teacher, there are several other factors that are important in the development of mathematical creativity and some of these, including technology, are discussed in Section 4. It is hoped that the papers presented here are one step further toward a long-term professional collaboration among mathematics educators at all levels from all parts of the world in the research and dissemination of information concerning this critical topic. GOALS AND DEFINITIONS The moving power of mathematical invention is not reasoning but imagination. Augustus de Morgan One way to think of the development of mathematical creativity is to imagine pupils moving along the following continuum:  innumerates doers computers consumers problem problem creators solvers posers (Sheffield, 2003, p. 5) In this model, we can think of students as moving from being virtually innumerate, lacking basic concepts and skills to being able to do some basic mathematics and to compute proficiently. At one point in our history, schools focused on producing students who knew enough mathematics to be intelligent consumers. In the twentieth century, as technology progressed rapidly, schools in much of the world realized that proficient computation and consumer skills were not enough, and problem solving became an important goal and means of instruction in mathematics education. Many educators also realized that problem solving was insufficient, and problem posing and the creation of mathematics that was new to the student became critical, especially as we moved into the twenty-first century. The papers in this section address the question of what it means to be mathematically creative. Aralas discusses the connections of creativity to mathematical imagination, noting that mathematical imagination can enhance pupils creative work and appreciation of a diversity of mathematical structures. Meissner also includes imagination as well as intuitive and mainly unconscious components in his description of mathematical creativity. In addition, he compares and contrasts what it means to be mathematically creative and mathematically gifted. Choi and Do, Daniel, Lenart, and Meletea also discuss the connection of mathematical creativity to giftedness and Tsvetkova applies creativity to team competitions in mathematics. Other views of creativity are presented by Kadijevich noting creativity is manifested in non-conventional problem solving, Hansen-Smith defining mathematics as an art that engages young students in playing with its functions and expanding its boundaries, and Mina characterizing creativity as the ability of man to establish new relationships and change reality. These papers set the stage for a discussion of a variety of questions including: What is mathematical creativity and which mathematics students can and should be creative? Is mathematical creativity a property of a person, a problem, a solution, a process or a teaching technique? Is mathematical creativity domain specific? What does it mean to think like a mathematican; is this creative by definition? How does mathematical creativity relate to general concepts of mathematics, mathematical problem solving, problem posing, research, and creativity? Can concepts be creative, rather than simply problems or solutions? What would this mean? Should mathematical creativity be something new to the world or can it be just new to the creator? Is it enough to have ideas that are novel and innovative or must creative mathematics be applied to mathematical problem solving? Can all mathematics students be creative? Is mathematical creativity dependent on mathematical talent or is it a distinct trait? Is an in-depth knowledge of mathematics a prerequisite for becoming mathematically creative? Can creativity be developed? Is it innate? Can it be taught? THE ROLE OF THE TEACHER To raise new HYPERLINK "http://creatingminds.org/quotes/questions.htm"questions, new possibilities, to regard old HYPERLINK "http://creatingminds.org/quotes/problems.htm"problems from a new angle, requires creative HYPERLINK "http://creatingminds.org/quotes/imagination.htm"imagination and marks real advance in science. HYPERLINK "http://creatingminds.org/quoters/quoters_e.htm"Albert Einstein Teachers play a critical role in the development of students mathematical creativity. In some cases, education does more to destroy creativity than to enhance it, but the promotion of creativity and innovation are increasingly important to the future of the world. The following heuristic (Sheffield, 2003, p. 15) is one that teachers might use to encourage students to think like creative, investigative mathematicians.  Using this heuristic when solving a problem, students may start at any point on the diagram and proceed in any order. One possible order might be: Relate the problem to other problems that you have solved. How is this similar to other mathematical ideas that you have seen? How is it different? Investigate the problem. Think deeply and ask questions. Evaluate your findings. Did you answer the question? Does the answer make sense? Communicate your results. How can you best let others know what you have discovered? Create new questions to explore. What else would you like to find out about this topic? Start a new investigation. Papers in this section address a variety of other techniques and strategies that teachers might use to nurture students creativity. As Foong notes, teachers conceptions of mathematical creativity have a strong influence on their teaching strategies. She found that the majority of preservice teachers saw creativity as an event linked to problem solving where more experienced teachers noted the importance of teachers actions and not just the problems presented. Gal, et al and Karsenty and Friedlander discuss their work with preservice and experienced teachers and the techniques they use to prepare them to work with mathematically gifted and talented students. Millman and Jacobbe also look at work with preservice teachers when they discuss mathematical habits of mind that need to be developed for creative problem solving including exploring mathematical ideas, formulating questions, constructing examples and problem solving approaches, generalizing concepts and reflecting on answers. Ong also discusses habits of mind noting the importance of persistence and thinking interdependently. Building on the idea of thinking interdependently, Toncheva discusses the importance of students talking to each other and the teacher about their intuitive arguments. Beswick addresses this as well stating that classrooms are complex systems where teachers should structure classrooms to maximize creativity of the class as a whole as well as of individuals. Alfonso and Martinez discuss the use of formative evaluation and learning strategies to move college students from the reproductive stage of solving routine problems through the practical level based on traditional mechanisms and theoretical understanding to understanding at the investigative level. Teoh suggests ways to create opportunities for developing creativity by helping teachers make connections that allow children to revel in the magic of mathematics. Abdounur takes teachers out of the classroom into a museum where students can creatively explore relationships among mathematics and music as they make and test conjectures, solving and creating problems as they establish and express analogies between mathematics and music. All the papers in this section raise a variety of questions related to the connections among teaching, learning and mathematical creativity including the following. What is the role of the teacher and others in recognizing and promoting mathematical creativity? What is the goal in doing this? How should we prepare teachers to foster mathematical creativity in all students? Is there a difference between the creative teacher and the productive teacher? between the creative teacher and the teacher of mathematical creativity? Must the teacher be mathematically creative to foster student creativity? Should a teacher to demonstrate his or her mathematical creativity? If so, how and when should this be done? What might teachers, students, parents, or others do to foster (or inhibit) creativity? Should mathematical creativity be made an explicit goal/critical area in mathematic education? If so, how? Will a focus on mathematical creativity distract from other critical areas of mathematics education? Should mathematical concepts and skills be learned creatively or should they be memorized before students are encouraged to be creative? What methods of instruction might stimulate students to create new problems, solve problems uniquely, conduct research work in mathematics, etc.? What is the role of motivation in creativity? When do we look for creativity? How? Where? Is it ever too late to search for creativity? What might the benefits be of teaching creativity in mathematics, for students who will not become mathematicians? Are we developing creative students simply for the benefit of society (or of the economy)? Or is there an intrinsic value to the individual of developing the faculty of creativity? THE USE OF PROBLEMS AND ASSESSMENT Too often we give students answers to remember rather than problems to solve. Roger Lewin Rich tasks or problems are critical for encouraging mathematical creativity. Some characteristics of these problems might be: Tasks should ask questions that make students think deeply, even about simple concepts, not questions that make them guess what the teacher is thinking. Tasks should have an entry point for all students and lend themselves to challenging even the most advanced students. Tasks should enable children to build on previous knowledge and to discover previously unknown mathematical principles, concepts, and generalizations. Tasks should be connected to core standards and benchmarks in mathematics. Tasks should be rich, with a wide range of opportunities for children to explore, make mistakes, reflect, extend, and branch out into new related areas. Tasks should give children the opportunity to demonstrate abilities in a variety of ways, verbally, geometrically, graphically, algebraically, numerically, etc. Tasks should allow children to use their abilities to question, reason, communicate, solve problems, and make connections to other areas of mathematics as well as to other subject areas and real world problems. Tasks should make full, appropriate, use of technology such as calculators and computers as well as mathematical manipulatives and models. Tasks should give time for individual reflection and problems solving as well as time for group exploration and discovery. Tasks should be interesting and should actively involve the child building a variety of thinking and learning styles. Tasks should be open with more than one right answer and/or more than one path to solution. Tasks should encourage continued exploration once the original question has been answered. Students should question the answers and not just answer the questions to develop deep mathematical sense. (Sheffield, 2003, p. 6) The papers in this section present a variety of tasks and problems that might be used to develop and evaluate mathematical creativity. Kim presents problems of rabbits and chickens, Pascals Triangle, and the use of pattern blocks for introducing fractions that are familiar to educators around the world and discusses how these might be used to promote creative problem solving. This idea of digging creatively into a simple problem is extended in Holtons paper with the six circle problem and in Chans paper with the model-eliciting biggest box problem. In a similar manner, the Grupo MUSA.E1 use examples from Gaussian theory and Pythagorean theory in their discussion of the development of creativity. Bilchev presents a chain of steps for guiding students to pose problems and create a group of connected problems that lead to mathematical results that are new to the creator and even new to the world. He gives examples from his extensive work with the Rousse Mathematical Circles as well as his university work. Sedrakyan illustrates this idea of making connections and generalizing from special cases to a wide range of related problems with examples of geometric inequalities. Cuador Gil extends the concept of creativity to the geosciences fields using geostatistics for engineering students. Sinitsky looks at essential features of activities for eliciting creativity and notes that any creative activity must take the student, the problem and the teacher all into consideration. Gogovska and Malcevski note the importance of using counter-examples, Hall discusses using ill-defined problems, and Li emphasizes open-mind questions, all to develop students mathematical creativity. Swirski, Wood, Carmody and Godfrey discuss the use of creativity in the assessment of engineering students and Fong notes the importance and difficulty of incorporating more creative questions in assessments. Cibulis and Lace present problems, games and toys used to recognize and assess mathematical creativity. Many of the papers address the idea of teaching less so students might learn more, giving students more time to delve deeply and creatively into topics of interest. The papers in this section introduce a number of questions related to the use of problems in the development and assessment of mathematical creativity, including the following: How might mathematical problems be used to develop mathematical creativity? How might mathematical creativity be assessed? How do we evaluate our success in developing mathematical creativity in all students? What are examples of good investigations and problems that can be useful for promoting mathematical creativity? How might problems best be used to develop mathematical creativity? Can all mathematical problems be used to evoke a creative response? Should they be used in this way? How might assessment be used to promote rather than inhibit mathematical creativity so that all students might be creators and not simply consumers of knowledge? What are the effects of standardized or standards-based assessment on mathematical creativity? How do we know when creativity has happened? How do we put this phenomenon to use? What criteria might be used to recognize, encourage, and assess creativity? How do we recognize a creative act? Can we teach our students to recognize one? Is it the surprising solution? The brief solution? The solution that uses advanced notions? Or the simplest notions? How do we evaluate whether we have been successful at developing mathematical creativity? TECHNOLOGY AND THE ENVIRONMENT Science and technology revolutionize our lives, but HYPERLINK "http://creatingminds.org/quotes/memory.htm"memory, tradition and myth frame our response. HYPERLINK "http://creatingminds.org/quoters/quoters_s.htm"Arthur Schlesinger The environment plays a critical role in the need for as well as the nurturing of mathematical creativity. No longer do individuals work in isolation on the most critical problems. Technology gives students the opportunity to quickly access information, peers and mentors around the world as well as actively investigate problems that would be nearly impossible without the technology. Successful problem solving in the twenty-first century will require that students are able to work efficiently and creatively with ill-defined problems, large amounts of information, calculators, computers, and others around the globe. The articles in this section look at a variety of ways that technology and the environment affect students mathematical creativity. Velikova describes a successful creative teacher-training program for preservice secondary teachers of Mathematics and Informatics where preservice teachers create a variety of multimedia applications, essays, and presentations to promote not only their own mathematical creativity, but also that of the secondary students whom they will teach. Watanabe compares secondary students problem solutions using paper and pencil to practice solving problems where the teachers have both the questions and the answers to creatively solving the same problems using graphing calculators or computers. One goal is to increase student interest in mathematics and to reduce their dislike for it. Bonka and Andzans also acknowledge the effect of interest and note several tools for fostering mathematical creativity ranging from positive emotions and textbooks to correspondence courses, summer camps, and contests presented in newspapers and on the Internet. Johny also describes the importance of improving attitudes and self-concepts while reducing mathematical anxiety. Using dynamic geometry software to develop mathematical creativity is discussed in articles by Flores Samaniego; Wurnig; and Kakihana, Fukuda, and Watanabe, while the use of computer algebra systems to stimulate mathematical creativity is investigated in articles by Windsteiger, Wurnig, and Siller. The articles in this section raise a number of questions about ways in which the environment affects students mathematical creativity including the following: How do technology, other resources, and the environment affect the mathematical creativity of the student? Does the use of technology promote or inhibit students mathematical creativity? How might technology and other resources, including those outside the school setting, best be used? What environment best nurtures creativity? How does this environment differ from student to student, from grade to grade, from level to level, from subject area to subject area? How do these environments (or even the goals of these environments) vary in different cultures? Does creativity mean the same thing in different social contexts? How is the creative individual viewed in different cultures? How do political contexts affect our work? How can we adapt to the constraints set up by our political situation? How can we influence the making of political decisions that affect us? NEXT STEPS The best way to predict the future is to create it. Alan Kay Discussions during this conference may raise more questions than it answers. What have we learned and where do we go from here? Can we summarize our findings? Are there major suggestions for future research? Should we explore a permanent professional society? Should it include both mathematical creativity and mathematical giftedness, talent, or promise? What about mathematical challenge or competitions? Other questions or suggestions? REFERENCE Sheffield, L. J. (2003). Extending the challenge in Mathematics: Developing mathematical promise in K - 8 pupils. Thousand Oaks, CA: Corwin Press. ABOUT THE AUTHOR Linda Jensen Sheffield, Ph. D. Regents Professor Emeritus Northern Kentucky University Highland Heights, KY 41099 Phone: 859-466-7983 E-mail:  HYPERLINK "mailto:Sheffield@nku.edu" Sheffield@nku.edu Websites:  HYPERLINK "http://www.nku.edu/~sheffield" www.nku.edu/~sheffield and  HYPERLINK "http://www.projectm3.org" www.projectm3.org  PAGE 328 PAGE 328 PAGE 328 PAGE 328 PAGE 328    Linda Jensen Sheffield Promoting Creativity for All Students in Mathematics Education: An Overview PAGE 370 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 5 PAGE 371 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 369 ICME 11, Mexico, 2008 +,OPghp @ X Z >?RS껯ꗋ}rfrZrfr}h3XhMCJH*aJh3XhM6CJaJh3XhMCJaJhP~hM56CJaJhP~hM5CJaJhhM5CJaJh'VhM6CJaJhhM6CJaJhhM56CJaJh'VhM5CJ aJ hM5CJ$aJ$h'VhM5CJ$aJ$h'VhM5CJaJhM5CJaJ,OPghg  $8d\$]^8a$gdM$d\$]^a$gdMd\$gdM $d\$a$gdM^`` ?RSqq3$d$d%d&d'd1$7$8$H$NOPQ\$a$gdM $d\$a$gdM$d\$]a$gdM $]a$gdMd\$gdM-$d$d%d&d'dNOPQ\$a$gdM S>x=> z $ & F x]^`a$gdM'$ & F x]^`a$gdM$d\$]a$gdM$d\$]a$gdM $d\$a$gdM $d\$a$gdMSx<>z ! !!! 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