Welcome to the Developing Creativity room of MathematiCreativity!
Tom Barone and Elliot Eisner listed the following as attributes of creativity: recognizing patterns, taking risks, challenging assumptions, taking advantage of chance, seeing in new ways and making connections. All of these attributes, in a mathematics learning environment, can give students the freedom to develop a deeper understanding of the math that they are learning.
Recognizing patterns:
Mathematics is full of patterns that, once discovered and understood, make learning mathematics much simpler. Students should be encouraged to look for patterns and helped in verifying their discoveries. Once a student understands a pattern for one area of math, the student might also apply it in other areas.
Taking risks:
Too much of an emphasis on the right answer can make students wary of taking the risk to offer an answer. Math classes need to be places where it is okay to make mistakes, where the process is as important as the answer, where students are encouraged to take risks in searching out patterns and trying to make new connections between what they are learning and what they have previously learned.
Challenging assumptions:
Zero was a radical concept! Innovation takes place when people challenge what they previously thought was true. Both teachers and students bring assumptions about numbers, mathematics and learning with then when they enter the classroom, and some of those assumptions may hinder learning.
Some students believe that when they multiply two numbers, the answer will always be larger than the original numbers. Others have heard that algebra is difficult to learn and that proofs are impossible. Teachers should address students' erroneous assumptions when they come up, and encourage students to monitor their own assumptions about mathematics and learning.
Teachers may also carry some assumptions that can limit learning. Some teachers think that high school students should not ever need manipulatives and that young children can not understand concepts such as negative numbers. Other teachers may believe that students can not learn complicated mathematics by working together, but always need direct instruction from the teacher. Some teachers force students to solve problems only one way, even if another way is valid. When teachers begin to challenge their own assumptions, they may be surprised to find that student learning increases!
Taking advantage of chance:
Student engagement can ebb and flow, so teachers should take advantage of times when students are digging deeper, excited for a new challenge, making connections and wanting to know more. Sometimes the lesson plan needs to be set aside to take advantage of student excitement and enthusiasm to get into a problem or concept deeper. An animated ten minute discussion may yield more interest and engagement and more powerful learning later on!
Seeing in new ways:
Sometimes, when you get stuck while working on a problem, looking at the problem from a fresh perspective can help find a solution! Some people like to brainstorm, which means to make a list of as many methods as they can think of, and then pick one to try. Others may try the same problem with different numbers or a similar problem, to get an idea of how to start. For some problems, working backwards from the solution helps the student see the process underlying the problem. Teachers should model a variety of methods of "seeing in new ways" and be open to new methods that students devise, in their problem solving efforts.
Ms. Math (featured on the People floor of MathematiCreativity!) challenges students to see numbers and numerical operations in new ways by providing the answer to the problem and asking students to find multiple new ways to get to the answer.
Making connections:
Mathematics is a system of interrelated numbers and operations, held together by a set of rules. All too often, math is taught as a series of individual facts and operations, leaving the connectedness untouched and untaught. Teachers need to help students see the connectedness of mathematics and to understand how that can be used to make learning more robust. Students should be encouraged to look for connections throughout their mathematics education and to continue to do so as they use mathematics throughout their life.
Developing an environment that fosters creativity in mathematics
Math teachers can:
- Ask open ended questions
- Allow more than one way to solve a problem
- Encourage students to find new ways of solving a particular problem
- Help students to test out their new methods to figure out if they are valid
- Deliberately express numbers in a variety of ways
- Provide students with puzzles and puzzling situations that require out of the box thinking
- Offer different approaches to a problem and ask the students to find the connections between the methods: Numeric (using numbers, or raw data, like in a table), Analytic (using equations), Graphical (using a pictoral representation like a diagram or graph, and Descriptive (writing sentences to explain what is going on)